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IND U19 vs AUS U19 Live Semi Final: यश ढल ने जड़ा शतक, रशीद की 94 रन की पारी से कंगारुओं को मिला 291 रन का लक्ष्य भारत और ऑस्ट्रेलिया के बीच खेले जा रहे अंडर19 विश्व कप 2022 के सेमीफाइनल मुकाबले में टीम इंडिया के लिए कप्तान यश ढल ने शानदार बल्लेबाजी की. उन्होंने 110 रन की शतकीय पारी खेलकर टीम को मुश्किल से निकाला. बल्लेबाज शेख रशीद ने भी 94 रन की अहम पारी खेली. हालांकि वो अपने शतक से महज छह रन से चूक गए. अफगानिस्तान पर डीएलएस नियम की मदद से महज 15 रन से करीबी जीत दर्ज करने के बाद इंग्लैंड की टीम पहले ही खिताबी मुकाबले में अपनी जगह बना चुकी है. इस वक्त अंडर19 विश्व कप 2022 का दूसरा सेमीफाइनल मैच खेला जा रहा है. आज जीतने वाली टीम को सीधे फाइनल में प्रवेश मिलेगा. भारत ने आज के मैच में टॉस जीतकर पहले बल्लेबाजी का फैसला किया. टीम की शुरुआत खास अच्छी नहीं रही. 16 रन पर पहला विकेट गंवाने के बाद 37 रन तक पहुंचेपहुंचते भारत के दो बल्लेबाज प्वेलियन पहुंच चुके थे. सलामी बल्लेबाज अंगक्रिश रघुवंशी ने छह रन बनाए. दूसरे सलामी बल्लेबाज हरनूर सिंह के बल्ले से 16 रन आए. ढलरशीद के बीच 204 रन की साझेदारी सस्ते में दो विकेट गंवाने के बाद कप्तान यश ढल ने तीसरे नंबर के खिलाफ शेख रशीद के साथ मिलकर मोर्चा संभाला. दोनों के बीच 204 रन की अहम साझेदारी बनी. यश ने 10 चौकों और एक छक्के की मदद से 110 गेंदों पर 110 रन बनाए. वहीं रशीद के बल्ले से 108 गेंदों पर 94 रन बनाए. उन्होंने आठ चौके और एक छक्का लगाया. दिनेश बाना का अंतिम ओवर में जलवा 49 ओवरों तक एक गेंद पर चौका लगाकर खेल रहे दिनेश बाना का जलवा आखिरी ओवर में देखने को मिला. इस ओवर में तीन छक्के और दो चौके आए और 27 रन बने. दिनेश ने चार गेंदों पर 20 रन और निशांत सिंधू ने 10 गेंदों पर 12 रन की पारी खेली. 49 ओवरों में 2635 के ओसत स्कोर से 50 ओवरों के बाद 2905 तक पहुंचाने का श्रेय पूरी तरह से दिनेश बाना को ही जाता है. | hindi |
പ്രവാസികള്ക്ക് ആശ്വസിക്കാം, സൗദിയില് നിന്ന് രണ്ട് ഡോസ് വാക്സിനെടുത്തവര്ക്ക് മടങ്ങാം റിയാദ്: സൗദി അറേബ്യയില് നിന്ന് രണ്ടു ഡോസ് കൊവിഡ് വാക്സിനെടുത്തവര്ക്ക് തിരികെ വരാമെന്ന് റിയാദിലെ ഇന്ത്യന് എംബസി സ്ഥിരീകരിച്ചു. രണ്ടു ഡോസ് വാക്സിനും സൗദിയില് നിന്ന് സ്വീകരിച്ചവര്ക്ക് 14 ദിവസം മറ്റൊരു രാജ്യത്ത് കഴിയേണ്ടതില്ലെന്നാണ് സൗദി വിദേശകാര്യമന്ത്രാലയം അറിയിച്ചിരിക്കുന്നത്. എല്ലാ എംബസികള്ക്കും കോണ്സുലേറ്റുകള്ക്കും അയച്ച സര്ക്കുലറില് ഇക്കാര്യം അധികൃതര് വ്യക്തമാക്കിയിട്ടുണ്ട്. ഇന്ത്യയടക്കം പ്രവേശന നിരോധനമുള്ള രാജ്യങ്ങളിലേക്ക് റീ എന്ട്രിയില് പോയ സൗദി ഇഖാമ ഉള്ളവര്ക്ക് മടങ്ങിയെത്താനുള്ള അവസരമാണ് പുതിയ തീരുമാനത്തിലൂടെ കൈവന്നിരിക്കുന്നത്. എന്നാല് ഈ അനുമതി എന്നു മുതല് പ്രാബല്യത്തില് വരുമെന്ന കാര്യം വ്യക്തമല്ല. സൗദിയില് നിന്ന് പുറപ്പെടുന്നതിന് മുമ്ബ് തന്നെ രണ്ട് ഡോസ് വാക്സിന് സ്വീകരിച്ച് ഇമ്യൂണ് ആയിരിക്കണമെന്നതാണ് പ്രധാന നിബന്ധന. ഇതിനൊപ്പം മറ്റ് കൊവിഡ് പ്രോട്ടോക്കോളുകളും പാലിക്കേണ്ടിവരും. എന്നാല് നാട്ടില് നിന്നും വാക്സിന് എടുത്തവര്ക്കും ഒരു ഡോസ് വാക്സിനെടുത്ത ശേഷം നാട്ടിലേക്ക് വന്നവര്ക്കും ഇപ്പോഴത്തെ നിലയില് മടങ്ങാനാവില്ല. ഇവര്ക്ക് അനുമതി എപ്പോള് ലഭിക്കുമെന്ന കാര്യത്തില് വ്യക്തതയില്ല. | malyali |
{\bm e}gin{document}
\title{Error Analysis of ZFP Compression for
Floating-Point Data hanks{Submitted to the editors 25 January
2018. unding{This work was performed under the auspices
of the U.S. Department of Energy by Lawrence Livermore National
Laboratory under Contract DE-AC52-07NA27344 and was supported by the
LLNL-LDRD Program under Project No. 17-SI-004, LLNL-JRNL-744818-DRAFT.}
{\bm e}gin{abstract}
Compression of floating-point data will play an important role in
high-performance computing as data bandwidth and storage become dominant
costs. Lossy compression of floating-point data is powerful, but
theoretical results are needed to bound its errors when used to store
look-up tables, simulation results, or even the solution state during
the computation. {\bm l}ack{In this paper, we analyze the round-off error
introduced by ZFP, a
lossy compression algorithm.} The stopping criteria for ZFP depends on the compression mode specified
by the user; either fixed rate, fixed accuracy, or fixed precision
\cite{zfp-doc}. While most of our discussion is focused on the fixed precision mode
of ZFP, we establish a bound on the error introduced by all three compression
modes. In order to tightly
capture the error, we first introduce a vector space that allows us to
work with binary
representations of components. Under this vector space, we define
operators that implement each step of the ZFP compression and
decompression to establish a bound on the error caused by
ZFP.
To conclude, numerical tests are provided
to demonstrate the accuracy of the established bounds.
\end{abstract}
{\bm e}gin{keywords}
Lossy compression, floating-point representation, error bounds
\end{keywords}
{\bm e}gin{AMS}
65G30, 65G50, 68P30
\end{AMS}
\section {Introduction}
\label{sec:introduction}
For several reasons, the trade-offs to obtain high performance in computing
have shifted. Traditionally, the emphasis on algorithmic complexity in numerical
computation has focused on operation counts, which was justifiable when
processor clock rates were increasing and memory was cheap and
plentiful. With the end of Dennard
scaling~\cite{752522}, clock speeds have
frozen (or even reduced), so more capability, in terms of
FLOPs, is now being obtained by adding more processing units~\cite{BorkarChien2011}.
Simultaneously, the ubiquity of hand-held devices and the power requirements of
extreme-scale supercomputers is encouraging a shift to lower-power
processors and co-processors.
Unfortunately, advances in memory and memory bandwidth are not
increasing apace with the advances in processors. Thus, the memory per core
and the bandwidth per core are decreasing as the number of processing
units increases~\cite{Xcutting2010,XDMAV2011}. The on-node cost of data motion (cache and
main memory accesses), both in time and
power, is increasingly the limiting factor in many
calculations~\cite{Xcutting2010,Williams:2009:RIV:1498765.1498785}.
Since off-node and I/O data motion have
historically been orders of magnitude slower than on-node data motion,
the movement of data anywhere on a computer system must now be seriously
considered as the leading-order cost.
An obvious approach to address this challenge would be to consider data
compression techniques. Indeed, lossless data compression is routinely used in
network communications. However, for floating-point data typical of the
scientific calculations done on high performance computers, standard
lossless compression techniques such as Lempel--Ziv~\cite{1055714,1055934},
DEFLATE~\cite{deflate,katz91}, Lempel–-Ziv–-Welch~\cite{1659158}, fpzip~\cite{fpzip} and other variants,
which reproduce the original data with
no degradation, struggle to produce significant compression
rates~\cite{Ratanaworabhan:2006:FLC:1126009.1126035,fpzip}.
Lossy compression algorithms for floating-point data, e.g.,
SZ~\cite{sz} and ZFP~\cite{zfp}, allow an inexact approximation of
the original data to be reconstructed from the compressed data. Lossy
data compression typically produces a much higher rate of data reduction
than lossless compression at the cost of introducing additional
approximation error into the data.
Lossy floating-point compression may be a useful tool in reducing data
motion costs, particularly if there is a schema that allows for
progressive (cf. global) decompression of data on demand. Certainly,
for storage (e.g., tabular data) and I/O operations (data and restart
files), there may be much to gain by using lossy compression provided
that the data retain sufficient accuracy for the intended purposes. We
propose that, in addition, solution state data in a simulation could be
stored in a compressed state and be decompressed, operated on, and
recompressed in a lossy way inline during each time step or iteration of a numerical
algorithm. Numerical simulation is fundamentally about approximation,
and the solution state already contains truncation, iteration, and other
roundoff errors. However, the repeated application of compression and
decompression does generate an additional error, and it must be shown
that these lossy compression errors can be bounded to prove that such a
process is stable.
As a first step towards this goal, we consider the ZFP lossy compression
algorithm and develop an approach to analyze and bound the error resulting from lossy
compression and decompression. {\bm l}ack{While recent works have provided empirical studies of ZFP and other lossy compression algorithms on real-world
data sets \mbox{\cite{gmd-9-4381-2016, Laney:2013:AED:2503210.2503283, Lindstrom2017}},
this paper establishes the first closed form expression for bounds on the error introduced by ZFP.
It is expected that our} approach can be generalized to other algorithms involving the manipulation of components represented using bits. ZFP, which operates on blocks of $4^d$ values, can encode and truncate data using one of
three modes: fixed rate, fixed accuracy, or fixed precision. The
fixed rate mode compresses a block to a fixed number of bits, the
fixed precision compresses to a variable number of bits while retaining
a fixed number of bit planes, and fixed accuracy mode compresses a block
with relation to the tolerated maximum error. The goal of this paper is
to provide an error analysis for the fixed precision mode, as it
is the simplest to represent algebraically. However, as a result of the analysis of
the fixed precision mode we are able to develop bounds on the error
introduced by the fixed accuracy and fixed rate compression modes.
The remainder of this paper is structured as follows. In the next
section, we describe the ZFP compression algorithm.
In~\secref{sec:notation}, we introduce the notation, definitions, and
lemmas that we will use in~\secref{sec:analysis} to prove bounds on
the error introduced at each stage of the ZFP compression algorithm.
In~\secref{sec:bounds}, we derive the error bounds for the fixed precision mode for the composite
compression and decompression action, as well as an error bound for both fixed accuracy and fixed rate modes. Finally, we demonstrate
the validity of these results numerically in~\secref{sec:results}.
\section{ZFP Data Compression}
\label{sec:zfp}
We first provide a brief overview of the current ZFP compression algorithm. Further details of the implementation of ZFP can be found in \cite{zfp} with modifications described in the software documentation \cite{zfp-doc}. {\bm l}ack{For clarity purposes, a small example of ZFP is provided in Appendix \ref{sec:appendixa}. }
{\bm e}gin{itemize}
\item[\it Step 1:]The $d$-dimensional array is partitioned into arrays of dimension $4^d$, called \emph{blocks}. A 2-d example is depicted in Figure \ref{fig:cut4dblock}. If the $d$-dimensional array cannot be partitioned exactly into blocks, then the boundary of the $d$-dimensional array is padded until an exact partition is possible.
{\bm e}gin{figure}
\centering
\includegraphics[width=.45\textwidth]{figures/step1.pdf}
\caption{Deconstruction of a $10 \times 10$ 2-dimensional array into independent $4\times4$ blocks. If the data is not divisible by 4 the data at the boundaries is padded (shown in orange).}
\label{fig:cut4dblock}
\end{figure}
{\bm e}gin{figure}
\centering
\includegraphics[width=.8\textwidth]{figures/step3.pdf}
\caption{Floating-point bit representation in single precision converted to a block-floating-point representation and its corresponding signed integers. Note that, depending on the relative disparity of the 16 numbers, some truncation may occur for the numbers of the smallest magnitude. }
\label{fig:bfprepresentation}
\end{figure}
\item[\it Step 2:] The floating-point values in each block are converted
to a block-floating-point representation using a common exponent for
each block \cite{MitrablockRoundingError} and integers in two's complement
format. The block-floating point representation is then shifted and
rounded to $4^d$ signed integers as seen in Figure
\ref{fig:bfprepresentation}.
\item[\it Step 3:]The integers are decorrelated using a custom, high-speed, near orthogonal transform that is similar to the discrete cosine transform. {\bm l}ack{The idea is that continuous fields tend to exhibit autocorrelation which can be viewed as redundant information. The decorrelating transform removes these redundancies via a change in basis resulting in a ``sparser" representation with smaller magnitude coefficients. The many leading zeros in the small coefficients offer an opportunity for compression (see Section \ref{Step3Sec} for details.)}
\item[\it Step 4:] {\bm l}ack{Coefficient magnitude tends to correlate (inversely) with sequency. A 2-d example of total sequency can be seen in Figure \ref{fig:totalsequency}. Sequency ordering is done to place the coefficients roughly in order of decreasing magnitude, which tends to group ones together and zeros together in each bit plane. This facilitates compression as often small coefficients tend to share leading zeros.}
{\bm e}gin{figure}
\centering
\includegraphics[width=.5\textwidth]{figures/step4.pdf}
\caption{Total sequency ordering for a 2-dimensional array, which groups the diagonal elements together. }
\label{fig:totalsequency}
\end{figure}
\item[\it Step 5:] {\bm l}ack{The sign bit is typically the left most bit in any traditional binary representation, which does not provide any useful information until the leading one-bit is encountered, {\bm l}ack{i.e., the transition from 0 to 1 (for positive values) or from 1 to 0 (for negative values)}. However, the first nonzero bit encountered in a negabinary representation immediately informs of the sign and magnitude. For example, if the leftmost one-bit in negabinary is at position $e$, then the magnitude of the number is in $2^e\cdot[1/3, 4/3]$. If $e$ is even, then the number is positive; otherwise it is negative. Thus, the two's complement signed integers (the standard integer representation) are converted to their negabinary representation \cite{Knuth}. The negabinary representation also ensures that the error caused by the remaining steps is mostly centered around zero with a slight bias depending on the index of truncated bit-plane. }
\item[\it Step 6:] The bits that represent the list of $4^d$ integers
are transposed so that they are ordered by bit plane, from most to
least significant bit, instead of by coefficient.
\item[\it Step 7:] Each bit plane is compressed losslessly using
embedded coding, which exploits the property that the transform
coefficients tend to have many leading zeros. {\bm l}ack{The idea is to encode groups of zero-bits together using a single bit to indicate that the whole group consists of zeros.} As this step is
lossless, the encoding details are omitted.
\item[\it Step 8:]The embedded coder emits one bit at a time until
stopping {\bm l}ack{criterion} are satisfied. The exact stopping criteria is dependent on the mode of ZFP compression: either fixed rate, fixed precision, or
fixed accuracy.
\end{itemize}
Most of our discussion and error analysis in Section~\ref{sec:analysis}
will focus on Steps 2, 3, and 8 as these steps are likely to introduce
additional round-off error during compression and, in some cases, during
decompression. Our approach for bounding the round off error introduced
by ZFP compression and decomporession is to define an operator for each
step of the algorithm, compose these operators to form a ZFP compression
and a ZFP decompression operator, compute the value returned after
compressing and decompressing an arbitrary input, and finally compare
this value to the value of the original input. The next section will
focus on introducing notation and definitions that will be useful in
defining mathematical operators for each step of the ZFP compression
algorithm.
\renewcommand\arraystretch{1.2}
{\bm e}gin{table}[htp!]
\caption{Notation Table}
{\bm e}gin{center}
{\bm e}gin{adjustbox}{width=1\textwidth}
{\bm e}gin{tabular}{|c l c|}
\hline
Symbol & Description & Location \\
\hline
\hline
$d$ & dimension of the input data & \S \ref{Step1Sec} \\
$k$ & the number of IEEE mantissa bits, including the leading one-bit& \S \ref{Step2Sec}\\
$q$ & the number of consecutive bits used to represent an element in the block-floating point transform & \S \ref{Step2Sec} \\
${\bm e}ta$ & number of bit planes kept in Step 8 & \S \ref{Step8Sec} \\
\hline
$\mathcal{I}$ & active bit set & \S \ref{sec:notation} \\
$\mathcal{B}^n$ & infinite binary vector space & \S \ref{sec:bitvectorSpace} \\
$\mathcal{B}^n_k$ & subset of $\mathcal{B}^n$ with finite active bit set & \S \ref{sec:bitvectorSpace} \\
$\mathcal{N}^n$ & infinite negabinary vector space & \S \ref{sec:bitvectorSpace}\\
$\mathcal{N}_k^n$ & subset of $\mathcal{N}^n$ with finite active bit set & \S \ref{sec:bitvectorSpace}\\
$\bf{0}_\mathcal{B}$,$\bf{0}_\mathcal{N}$ & additive identity in $\mathcal{B}^n$ and $\mathcal{N}^n$, respectively & \S \ref{sec:bitvectorSpace}\\
$\bf{1}_\mathcal{B} $, $\bf{1}_\mathcal{N}$ & multiplicative identity in $\mathcal{B}^n$ and $\mathcal{N}^n$, respectively & \S \ref{sec:bitvectorSpace}\\
$\| \cdot \|_{\mathcal{B}, p}$, $\| \cdot \|_{\mathcal{N}, p}$ & $p$-norm with respect to $\mathcal{B}^n$ and $\mathcal{N}^n$, respectively & \S \ref{NormsSubsection} \\
\hline
$f_\mathcal{B}$, $f_\mathcal{B}^{-1}$, $f_\mathcal{N}$, $f_\mathcal{N}^{-1}$ & bijective maps from $\mathcal{B} \rightarrow \mathbb{R}$, $\mathbb{R} \rightarrow \mathcal{B}$, $\mathcal{N} \rightarrow \mathbb{R}$ and $\mathbb{R} \rightarrow \mathcal{N}$, respectively & Eqn. (\ref{eqn:f2})\\
{\bm l}ack{$F_\mathcal{B}$,$F_\mathcal{B}^{-1}$, $F_{\mathcal{N}}$,$F_\mathcal{N}^{-1}$} & {\bm l}ack{bijective maps from $\mathcal{B}^n \rightarrow \mathbb{R}^n$, $\mathbb{R}^n \rightarrow \mathcal{B}^n$, $\mathcal{N}^n \rightarrow \mathbb{R}^n$ and $\mathbb{R}^n \rightarrow \mathcal{N}^n$, respectively} & \S \ref{sec:bitvectorSpace}\\
$\oplus $ &bit vector addition & Lma. \ref{FieldLemma}\\
$\odot$ &bit vector multiplication & Lma. \ref{FieldLemma}\\
\hline
$s_l$, $S_l$ & shift operator on $\mathcal{B}$ and $\mathcal{B}^n$, respectively & \S \ref{ShiftSubsection} \\
$t_{\mathcal{S}_k},T_{\mathcal{S}_k} $ & truncation operator with respect to the set $\mathcal{S}_k$ & \ref{DefCOperators} \\
{$r $} & rounding operator for two's complement representation & \S \ref{Step3Sec} \\
$e_{min}$, $e_{max}$ & min and max exponent of the floating-point representation of the block & Def. \ref{def:emaxemin}\\
$\epsilon_m$ & constant $\epsilon_m := 2^{1-m}$, with $m \in \mathbb{N}$ & \S \ref{sec:bitvectorSpace} \\
\hline
$L$, $L_d$ & one and $d$-dimension forward decorrelating linear transform & Eqn. \ref{eqn:T} \\
$\tilde{L}$, $\tilde{L}_d$ & floating-point arithmetic approximation of $L$ and $L_d$ & \S \ref{Step3Sec} \\
$L_d^{-1} $, $\tilde{L}^{-1} _d$ & $d$-dimension backward decorrelating linear transform and the floating-point arithmetic approximation of $L^{-1}$& \S \ref{Step3Sec} \\
\hline
$C_k$, $\tilde{C_k}$ & lossless/lossy operator for Step $k$ of ZFP compression & \S \ref{sec:analysis} \\
$D_k$, $\tilde{D_k}$ & lossless/lossy operator for Step $k$ of ZFP decompression & \S \ref{sec:analysis} \\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\label{table:notation}
\end{table}
\section{Preliminary Notations, Definitions, and Lemmas}
\label{sec:notation}
\label{IntroSec:seqspace}
{\bm l}ack{As noted in Section \ref{sec:zfp},} many of the steps of ZFP are defined by direct
manipulation of the bits used to represent each component of the
input. While we could attempt to define operators that imitate the steps
of ZFP over the real vector space, it would be more straightforward to
work with the bitwise representation that is manipulated at each step of
the ZFP compression algorithm. Hence, in order to define operators for
each step of ZFP as actions on the bitwise representation of each
component, we construct vector spaces under which the components
correspond to binary or negabinary representations of the real
numbers. Accordingly, let $\mathbb{B} = \{ 0, 1 \}$ and define
{\bm e}gin{align}
\mathcal{C} := \left\{ \{ c_i \}_{i = -\infty}^{\infty} : c_i \in \mathbb{B} \ \text{for all} \ i \in \mathbb{Z} \right\} \label{eqn:infinitebitvector}.
\end{align}
For {\bm l}ack{$c \in \mathcal{C}$,} we define the \emph{active bit set of
$c$} by $\mathcal{I} (c) := \{ i \in \mathbb{Z} : c_i = 1 \}.$
Additionally, we define the following operators on $\mathcal{C}$ that
will be used as building blocks for defining each step of ZFP.
{\bm e}gin{definition}\label{DefCOperators}
Let $\mathcal{S} \subseteq \mathbb{Z}$. The \emph{truncation operator},
$t_{\mathcal{S}} : \mathcal{C} \to \mathcal{C}$, is defined by
{\bm e}gin{align*}
t_{\mathcal{S}} (c)_i = \left\{ {\bm e}gin{array}{ccc}
c_i &: & i \in \mathcal{S} \\
0 &: & i \not\in \mathcal{S} \\
\end{array} \right., \ \ \ \text{for all} \ c \in \mathcal{C} \ \text{and all} \ i \in \mathbb{Z}.
\end{align*}
Let $\ell \in \mathbb{Z}$. The \emph{shift operator}, $s_{\ell} :
\mathcal{C} \to \mathcal{C}$, is defined by
{\bm e}gin{align*}
s_{\ell} (c)_i = c_{i + \ell}, \ \ \ \text{for all} \ c \in \mathcal{C} \ \text{and all} \ i \in \mathbb{Z}.
\end{align*}
\end{definition}
\noindent From these definitions, it follows that $t_{\mathcal{S}}$ is a
nonlinear operator and {\bm l}ack{$s_{\ell}$ is a linear operator.} {\bm l}ack{These operators
can be extended to operators on $\mathcal{C}^n$
by defining
$T_{\mathcal{S}} : \mathcal{C}^n \to \mathcal{C}^n$ and $S_{\ell} : \mathcal{C}^n \to \mathcal{C}^n$ by
{\bm e}gin{align*}
T_{\mathcal{S}} (\bm{c}) = {\bm e}gin{bmatrix}
\ t_{\mathcal{S}} ({{\bm c}}_1) \ \\
\ \vdots \ \\
\ t_{\mathcal{S}} ({{\bm c}}_n) \ \
\end{bmatrix} \ \ \ \ \ \ \text{and} \ \ \ \ \ \ S_{\ell} ({\bm c}) = {\bm e}gin{bmatrix}
\ s_{\ell} ({{\bm c}}_1) \ \\
\ \vdots \ \\
\ s_{\ell} ({{\bm c}}_n) \ \
\end{bmatrix}\text{, for all } {\bm c}\in \mathcal{C}^n.
\end{align*}}
\noindent{\bm l}ack{For clarity, note that the components of $\bm{c}$ are each binary sequences as ${\bm c}_i \in \mathcal{C}$, for $1 \leq i \leq n$.} Additionally, note that $S_{\ell}$ is invertible with $S_{\ell}^{-1} : \mathcal{C}^n \to \mathcal{C}^n$ given by $S_{\ell}^{-1} = S_{-\ell}$.
\subsection{Defining Signed Binary and Negabinary Bit-Vector Spaces}
\label{sec:bitvectorSpace}
Let $x \in \mathbb{R}$ be given. Then there exist $c, d \in \mathcal{C}$ and $p \in \mathbb{B}$ such that $x$ can be represented in signed binary and negabinary as
{\bm e}gin{align}
{\text{Signed Binary: }} x = (-1)^p \sum_{i = - \infty}^{\infty} c_i 2^i \ \ \ \text{ and} \ \ \ {\text{ Negabinary: }} x = \sum_{i = - \infty}^{\infty} d_i (-2)^i. \label{CrepX}
\end{align}
As such, there exist subsets $\mathcal{A}$ and $\mathcal{N}$ of
$\mathcal{C}$ such that, for each $x \in \mathbb{R}$, there exist unique
elements $c \in \mathcal{A}$, $p \in \mathbb{B}$, and $d \in
\mathcal{N}$ such that $x$ can be represented in the binary and
negabinary form in (\ref{CrepX}) using $c$, $p$, and $d$,
respectively. In particular, we choose $\mathcal{A}$ and $\mathcal{N}$
such that $\mathcal{I} (c)$ and $\mathcal{I} (d)$ are finite whenever
possible. {\bm l}ack{This choice is made so that elements can be represented using finitely many nonzero bits.} Now define
$0_{\mathcal{C}}, 1_{\mathcal{C}} \in \mathcal{C}$ to be the elements
satisfying $\mathcal{I} \left( 0_{\mathcal{C}} \right) = \emptyset$ and
$\mathcal{I} \left( 1_{\mathcal{C}} \right) = \{ 0 \}$. It follows from
our choice of $\mathcal{A}$ and $\mathcal{N}$ that $\left\{
0_{\mathcal{C}}, 1_{\mathcal{C}} \right\} \subset \mathcal{A} \cap
\mathcal{N}$.
Defining $\mathcal{B} := \{ (p, a) \in \mathbb{B} \times \mathcal{A} :
(p, a) \neq ( 1, 0_{\mathcal{C}} ) \}$, we have that, for each $x \in
\mathbb{R}$, there exists a unique $b = (p, a) \in \mathcal{B}$ such that
$x = (-1)^p \sum_{i = - \infty}^{\infty} a_i 2^i$. Additionally, it is
clear from our choice of $\mathcal{N}$ that, for each $x \in \mathbb{R}$,
there exists a unique $d \in \mathcal{N}$ such that $x = \sum_{i = -
\infty}^{\infty} d_i (-2)^i$. We now define $f_\mathcal{B} : \mathcal{B} \to
\mathbb{R}$ by
{\bm e}gin{equation}
\label{eqn:f2}
f_\mathcal{B} (b) = (-1)^{{p}} \sum_{i = -\infty}^{\infty} a_i 2^i, \ \ \ \ \ \text{for all} \ b = ({p}, a) \in \mathcal{B},
\end{equation}
and $f_\mathcal{N} : \mathcal{N} \to \mathbb{R}$ by
{\bm e}gin{equation}
\label{eqn:fN}
f_\mathcal{N} (d) = \sum_{i = -\infty}^{\infty} d_i (-2)^i, \ \ \ \ \ \text{for all} \ d \in \mathcal{N}.
\end{equation}
{\bm l}ack{By our choice of $\mathcal{B}$ and $\mathcal{N}$, $f_\mathcal{B}$ and $f_\mathcal{N}$ are bijections and with inverses denoted by $f_\mathcal{B}^{-1} : \mathbb{R} \to \mathcal{B}$ and $f_\mathcal{N}^{-1} : \mathbb{R} \to \mathcal{N}$, respectively. We now define binary operators $\oplus_{\mathcal{B}} : \mathcal{B} \times \mathcal{B} \to \mathcal{B}$ and $\odot_{\mathcal{B}} : \mathcal{B} \times \mathcal{B} \to \mathcal{B}$ by}
{\bm e}gin{align}
\alpha \oplus_{\mathcal{B}} {\bm e}ta = f_\mathcal{B}^{-1} \left( f_\mathcal{B}( \alpha ) + f_\mathcal{B}({\bm e}ta) \right) \ \ \ \ \ \ \text{and} \ \ \ \ \ \ \alpha \odot_{\mathcal{B}} {\bm e}ta = f_\mathcal{B}^{-1} \left( f_\mathcal{B}(\alpha) \cdot f_\mathcal{B}({\bm e}ta) \right) \label{oplustimesdef}
\end{align}
for all $\alpha, {\bm e}ta \in \mathcal{B}$, where $+$ and $\cdot$ represent standard addition and multiplication in $\mathbb{R}$. Similarly, we can define $\oplus_{\mathcal{N}} : \mathcal{N} \times \mathcal{N} \to \mathcal{N}$ and $\odot_{\mathcal{N}} : \mathcal{N} \times \mathcal{N} \to \mathcal{N}$ by replacing all $\mathcal{B}$ with $\mathcal{N}$ in (\ref{oplustimesdef}). With these definitions in place, we now have the following result.
{\bm e}gin{lemma}\label{FieldLemma}
{\bm l}ack{$(\mathcal{B}, \oplus_\mathcal{B}, \odot_\mathcal{B})$ and $(\mathcal{N}, \oplus_\mathcal{N}, \odot_\mathcal{N})$ are fields with additive and multiplicative identities $0_{\mathcal{B}} := (0,0_{\mathcal{C}})$ and $1_{\mathcal{B}} := (0, 1_{\mathcal{C}})$ and $0_{\mathcal{N}} := 0_{\mathcal{C}}$ and $1_{\mathcal{N}} := 1_{\mathcal{C}}$, respectively.}
\end{lemma}
For the remainder of the discussion, the sign bit will be omitted from elements of $\mathcal{B}$ by letting $a$ represent the element $(0,a) \in \mathcal{B}$ and $- a$ represent $(1,a) \in \mathcal{B}$. Additionally, to simplify the notation in the following sections, we will write $+$ instead of $\oplus_{\mathcal{B}}$ or $\oplus_{\mathcal{N}}$ where the operation should be clear from the context in which it is used.
Note that $f_\mathcal{B}$ and $f_\mathcal{N}$ can be generalized to vector-valued functions by defining $F_\mathcal{B} : \mathcal{B}^n \to \mathbb{R}^n$ and $F_\mathcal{N} : \mathcal{N}^n \to \mathbb{R}^n$ as {\bm l}ack{ $F_\mathcal{B}(\bm{a}) =[\ f_\mathcal{B}({{\bm a}}_1), \cdots, f_\mathcal{B}({{\bm a}}_n)]^t$ and $F_\mathcal{N}(\bm{a}) =[\ f_\mathcal{N}({{\bm d}}_1), \cdots, f_\mathcal{N}({{\bm d}}_n)]^t$, where ${\bm a} \in \mathcal{B}^n$ and ${\bm d} \in \mathcal{N}^n$, respectively.}
By definition, $F_\mathcal{B}$ and $F_\mathcal{N}$ are invertible with inverses $F_\mathcal{B}^{-1}$ and $F_\mathcal{N}^{-1}$ defined by applying $f_\mathcal{B}^{-1}$ and $f_\mathcal{N}^{-1}$ componentwise, respectively. We will let $\bm{0}_{\mathcal{B}}$ and $\bm{0}_{\mathcal{N}}$ denote the the additive identity in $\mathcal{B}^n$ and $\mathcal{N}^n$, respectively, and $I_{\mathcal{B}} : \mathcal{B}^n \to \mathcal{B}^n$ and $I_{\mathcal{N}} : \mathcal{N}^n \to \mathcal{N}^n$ denote the identity map on $\mathcal{B}^n$ and $\mathcal{N}^n$, respectively.
{\bm l}ack{To imitate floating-point representations}
we define the following subsets of $\mathcal{B}$ and $\mathcal{N}$. Given $k \in \mathbb{N}$,
{\bm e}gin{align*}
\mathcal{B}_{k} = \left\{ (p,a) \in \mathcal{B} : \mathcal{I} (a) \subseteq \{ i, i + 1, \ldots, {i + k -1} \} \ \text{for some} \ i \in \mathbb{Z} \right\}
\end{align*}
and
{\bm e}gin{align*}
\mathcal{N}_{k} = \left\{ d \in \mathcal{N} : \mathcal{I} (d) \subseteq \{ i, i + 1, \ldots, i + k -1 \} \ \text{for some} \ i \in \mathbb{Z} \right\}.
\end{align*}
Here, $k$ represents the maximum number of nonzero bits allotted for each representation. For example, in $\mathcal{B}_k$, $k$ represents the number of bits allotted for the {\bm l}ack{mantissa and $i$ indicates} the exponent in IEEE. It should be noted that $\mathcal{B}_{k}$ and $\mathcal{N}_k$ are subsets but \emph{not} subspaces of $\mathcal{B}$ and $\mathcal{N}$, respectively, as they do not satisfy the property of closure under $\oplus$ and $\odot$. As such, the analysis will take place in $\mathcal{B}$, $\mathcal{N}$, or $\mathbb{R}$ with the use of the truncation operator, $T_\mathcal{S}$, to imitate working with finite precision elements.
\subsection{Meaningful Norms on $\mathcal{B}^n$ and $\mathcal{N}^n$}\label{NormsSubsection}
From Lemma \ref{FieldLemma}, it follows that $\mathcal{B}^n$ is a vector space under $+$. {\bm l}ack{Let $\| \cdot \|_p$ be the standard $p$-norm on $\mathbb{R}^n$. Accordingly, we define $\| \cdot \|_{\mathcal{B}, p} : \mathcal{B}^n \to [0, \infty)$ by
{\bm e}gin{align}
\| \bm{a} \|_{\mathcal{B}, p} = \left\{ {\bm e}gin{array}{ccc}
{\bm i}g( \sum_{i = 1}^{n} \left| f_\mathcal{B}(\bm{a}_i) \right|^{p} {\bm i}g)^{1/p} &: & 1 \leq p < \infty \\
\max_{1 \leq i \leq n} \left| f_\mathcal{B}(\bm{a}_i) \right| &: & p = \infty \\
\end{array} \right. \DIFaddend \label{BpNorm}
\end{align}}
The following result is an immediate consequence from the definition of $\| \cdot \|_{\mathcal{B}, p}$.
{\bm e}gin{lemma}
\label{BNormLemma}
For all $\bm{a} \in \mathcal{B}^n$, $\bm{x} \in \mathbb{R}^n$, and $1 \leq p \leq \infty$, $\| \cdot \|_{\mathcal{B}, p}$ is a norm satisfying
{\bm e}gin{align*}
\| F_\mathcal{B} (\bm{a}) \|_p = \| \bm{a} \|_{\mathcal{B}, p} \ \ \ \text{and} \ \ \ \| \bm{x} \|_p = \| F_\mathcal{B}^{-1} (\bm{x}) \|_{\mathcal{B}, p}.
\end{align*}
\end{lemma}
\noindent For any $1 \leq p \leq \infty$, we now have that $\mathcal{B}^n$ is a normed vector space with norm $\| \cdot \|_{\mathcal{B}, p}$. Additionally, we can define a norm for operators defined on $\mathcal{B}^n$.
{\bm l}ack{{\bm e}gin{definition}
\label{BopNorm}
Let $m,n \in \mathbb{N}$ and $1 \leq p \leq \infty$. The induced p-norm on $\Phi : \mathcal{B}^n \to \mathcal{B}^m$ is given by
{\bm e}gin{align*}
\| \Phi \|_{\mathcal{B}, p} = \sup \left\{ \frac{\| \Phi (\bm{a}) \|_{\mathcal{B}, p}}{\| \bm{a} \|_{\mathcal{B}, p}} : \bm{a} \in \mathcal{B}^n, \bm{a} \neq \bm{0}_{\mathcal{B}} \right\}.
\end{align*}
\end{definition}}
\noindent {\bm l}ack{As an immediate consequence of this definition, $\| \Phi (\bm{a}) \|_{\mathcal{B}, p} \leq \| \Phi \|_{\mathcal{B}, p} \| \bm{a} \|_{\mathcal{B}, p}$ holds for all $\bm{a} \in \mathcal{B}^n \setminus {\bm 0}_\mathcal{B}$. In a similar manner, given $1 \leq p \leq \infty$ we can define the norm $\| \cdot \|_{\mathcal{N}, p}$ over $\mathcal{N}$ satisfying a result analogous to Lemma \ref{BNormLemma} by replacing every $\mathcal{B}$ by $\mathcal{N}$ in (\ref{BpNorm}) and Definition \ref{BopNorm}.}
\subsection{Two's Complement}
\label{TwoCompSec}
From the end of Step 2 through the beginning of Step 5, the ZFP implementation stores the integer components of each block using a normalized fixed-point, two's complement binary integer representation. For $z \in \mathbb{Z}$, the two's complement representation is of the form
{\bm e}gin{align*}
\text{Two's Complement:} \ \ \ \ z = - t_{N-1} 2^{N-1} + \sum_{i = 0}^{N-2} t_i 2^i,
\end{align*}
for some $N \in \mathbb{N}$ and some $t \in \mathcal{C}$. {\bm l}ack{Typically, the
value of $N$ is chosen beforehand to be the number of bits allotted for
storing each integer. Unfortunately, this aspect of the two's complement
representation does not lend itself to a construction of a vector
space, unlike $\mathcal{B}^n$ and $\mathcal{N}^n$ constructed in
Section~\ref{sec:bitvectorSpace}. Thus, instead of working explicitly in
two's complement, we will take care when defining operators for Steps 2
through 5 in $\mathcal{B}^n$ to ensure they mimic the behavior of the
two's complement representation used in ZFP. }
\subsection{Truncation Operator on $\mathcal{B}^n$ and $\mathcal{N}^n$}\label{ProjectionSubsection}
In this section, we consider some properties of the truncation operator, $T_{\mathcal{S}}{(\cdot)} $, over the normed vector spaces $\mathcal{B}^n$ and $\mathcal{N}^n$. The usefulness of these definitions and results will be evident during the analysis of the ZFP compression algorithm in Section \ref{sec:bounds}.
{\bm e}gin{definition}
\label{def:emaxemin}
Let $\bm{x} \in \mathbb{R}^n$. The \emph{maximum exponent of $\bm{x}$
with respect to $\mathcal{B}$} is
{\bm e}gin{align}
e_{max, \mathcal{B}}(\bm{x}) = \max_{1 \leq i \leq n} \max_{j} \left\{ j
\in \mathcal{I} \left( f_{\mathcal{B}}^{-1} \left(\bm{x}_i \right)
\right) \right\} \label{eqn:emax},
\end{align}
and \emph{minimum exponent of $\bm{x}$ with respect to $\mathcal{B}$} is
{\bm e}gin{align}
e_{min, \mathcal{B}}(\bm{x}) = \min_{1 \leq i \leq n} \min_{j} \left\{ j \in \mathcal{I} \left( f_{\mathcal{B}}^{-1} \left(\bm{x}_i \right) \right) \right\}, \label{eqn:emin}
\end{align}
provided that the minimum exists. Similarly, we define the \emph{maximum} and \emph{minimum exponent of $\bm{x}$ with respect to $\mathcal{N}$} by replacing $\mathcal{B}$ with $\mathcal{N}$ in (\ref{eqn:emax}) and (\ref{eqn:emin}).
\end{definition}
{\bm l}ack{When it is clear from context which space, $\mathcal{B}$ or $\mathcal{N}$, the vector $\bm{x}$ is in, we will simply write $e_{max}$ or $e_{min}$. The next result provides a relation between elements of $\mathcal{B}$ and $\mathcal{N}$ and $e_{max}$.}
{\bm l}ack{{\bm e}gin{lemma}
\label{ProjectionLemma1}
For $\bm{a} \in \mathcal{B}^n$, $\bm{d} \in \mathcal{N}^n$: $(i)$ $\| \bm{a} \|_{\mathcal{B}, \infty} \geq 2^{e_{max, \mathcal{B}}(F_{\mathcal{B}}(\bm{a}))} \ \ \ (ii)$ $\| \bm{d} \|_{\mathcal{N}, \infty} \geq \frac{1}{3} 2^{e_{max, \mathcal{N}}(F_{\mathcal{N}}(\bm{d}))}$.
\end{lemma}}
{\bm e}gin{proof}
$(i)$ follows immediately from the definition of $e_{max, \mathcal{B}}(\bm{x})$. Next, for $z \in \mathbb{Z}$ even we define $\mathcal{J}(z) = \{ 2j \in
\mathbb{Z} : 2j \leq z\}$ and for $z \in \mathbb{Z}$ odd we define
$\mathcal{J}(z) = \{ 2j+1 \in \mathbb{Z}:2j+1 \leq z\}$. Letting $e = e_{max, \mathcal{N}}(\bm{x})$, $(ii)$ follows from observing
{\bm e}gin{align*}
\| \bm{d} \|_{\mathcal{N}, \infty} \geq \left| (-2)^e - \sum_{i \in
\mathcal{J}(e-1)} (-2)^{i} \right| = 2^e - \frac{2}{3} 2^e =
\frac{1}{3} 2^{e_{max, \mathcal{N}}(F_{\mathcal{N}}(\bm{d}))}.
\end{align*}
\end{proof}
We now provide a result establishing the relationship between $\bm{a}$
and $T_{\mathcal{S}} (\bm{a})$ given $\bm{a} \in \mathcal{B}^n$ or
$\bm{a} \in \mathcal{N}^n$. First, note that for certain choices of $m
\in \mathbb{Z}$ the constant term $2^{1 - m}$ will regularly occur in
error bounds established throughout this paper. Hence, we will let
$\epsilon_m := 2^{1-m}$ for any $m \in \mathbb{Z}$. For example, machine
epsilon \cite{higham2002accuracy} is defined as $\epsilon_k = 2^{1-k}$
for precision $k$.
{\bm l}ack{{\bm e}gin{lemma}
\label{ProjectionLemma2}
Suppose $p, q \in \mathbb{Z}$ and let $\mathcal{S} = \{ i \in \mathbb{Z} : i > p - q \}$. If $\bm{a} \in \mathcal{B}^n$ and $\bm{d} \in \mathcal{N}^n$ then
{\bm e}gin{enumerate}
\item[$(i)$] $\bm{a} = T_{\mathcal{S}} (\bm{a}) + \Delta \bm{a}$, for some $\Delta \bm{a} \in \mathcal{B}^n$ satisfying $\| \Delta \bm{a} \|_{\mathcal{B}, \infty} \leq \epsilon_q 2^{p}$.
\item[$(ii)$] $\bm{d} =T_{\mathcal{S}} (\bm{d}) + \Delta \bm{d}$, for some $\Delta \bm{d} \in \mathcal{N}^n$ satisfying $\| \Delta \bm{d} \|_{\mathcal{N}, \infty} \leq \frac{2}{3} \epsilon_q 2^{p}$.
\end{enumerate}
\end{lemma}}
{\bm e}gin{proof}
Let $1 \leq m \leq n$. Observing
{\bm e}gin{align*}
\left| f_{\mathcal{B}} \left( \Delta \bm{a}_m \right) \right| = \left| f_{\mathcal{B}} \left( \bm{a}_m - T _{\mathcal{S}} (\bm{a})_m \right) \right| = \sum_{i \in \mathcal{I} \left( \bm{a}_m \ominus T_{\mathcal{S}} (\bm{a})_m \right)} 2^i \leq \sum_{i = -\infty}^{p - q} 2^i = 2^{{p} - q + 1}
\end{align*}
concludes the proof of $(i)$. Next, for $z \in \mathbb{Z}$ even we define $\mathcal{J}(z) = \{ 2j \in
\mathbb{Z} : 2j \leq z\}$ and for $z \in \mathbb{Z}$ odd we define
$\mathcal{J}(z) = \{ 2j+1 \in \mathbb{Z}:2j+1 \leq z\}$. Then $(ii)$ follows by observing that
{\bm e}gin{align*}
\left| f_{\mathcal{N}} \left( \Delta \bm{d}_m \right) \right| = \left|
f_{\mathcal{N}} \left( \bm{d}_m - T_{\mathcal{S}} (\bm{d})_m \right)
\right| = \left| \sum_{i \in \mathcal{I} \left( \bm{a}_m \ominus
T_{\mathcal{S}} (\bm{a})_m \right)} (-2)^i \right| \leq \left| \sum_{i
\in \mathcal{J}(p-q)} (-2)^i \right| = \frac{2}{3} 2^{p -q + 1}.
\end{align*}
\end{proof}
Hence, if $\mathcal{S} = \{ i \in \mathbb{Z} : i > {e_{\max}(F_{\mathcal{B}}(\bm{a}))} - q \}$ then the additional round-off error incurred by the truncation
operator is dependent on $q$ and the magnitude of the input. Using
Lemma~\ref{ProjectionLemma2}, we now observe that the component-wise
relative error introduced by the truncation operator is bounded by
{\bm e}gin{align}
\max_{m, f_\mathcal{B}(\bm{a}_m) \neq 0} f_\mathcal{B} \left( \left | \frac{\bm{a}_m - T_\mathcal{S}(\bm{a}_m)}{\bm{a}_m} \right | \right) & \leq \frac{ \left \|\Delta {\bm a}\right \|_{\mathcal{B},\infty} }{\min_{m, f_\mathcal{B}(\bm{a}_m) \neq 0} f_\mathcal{B}(| \bm{a}_m |)} = \epsilon_q {\bm l}ack{2^{e_{max}(F_{\mathcal{B}}(\bm{a}))-e_{min}(F_{\mathcal{B}}(\bm{a}))}}, \label{eqn:componentwiseerror}
\end{align}
for ${\bm a} \in \mathcal{B}^n $ with $F_\mathcal{B}({\bm a}) \neq {\bm 0}$. So the component-wise
error relative to the input is dependent on $q$ and the exponent range, i.e.,
$\rho =
e_{max}(F_{\mathcal{B}}(\bm{a}))-e_{min}(F_{\mathcal{B}}(\bm{a}))$. In
\cite{zfp}, it was noted that in many real-world examples, the exponent
range was reasonable ($\rho \leq 8$). Thus, depending on $\rho$, $q$
could be chosen to ensure the component-wise error relative to the input
remains smaller than machine epsilon.
\iffalse
{\bm e}gin{lemma}
\label{ProjectionLemma3}
Let ${\bm a} \in \{ {\bm b} \in \mathcal{N} ^n: i\geq 0 \quad \forall i \in \mathcal{I}({\bm b})\}$ and ${\bm e}ta \in \mathbb{N}_0$ be given. Suppose that $\mathcal{T} = \{ i \in \mathbb{N} : i \geq i_{max} - {\bm e}ta \}$, where $i_{max} = \max(\mathcal{I}({\bm a}))$ . Then
{\bm e}gin{align*}
\bm{a} = T_{\mathcal{T}} (\bm{a}) + \Delta \bm{a},
\end{align*}
for some $\Delta \bm{a} \in \mathcal{N}^n$ satisfying $\| \Delta \bm{a} \|_{\mathcal{N}, \infty} \leq 2^{i_{max} - {\bm e}ta}$ and $\| \Delta \bm{a} \|_{\mathcal{N}, \infty} \leq 2^{-{\bm e}ta+1} \| \bm{a} \|_{\mathcal{N}, \infty}$.
\end{lemma}
{\bm e}gin{proof}
Suppose $1 \leq i \leq n$. Observing that
{\bm e}gin{align*}
\left | f_{\mathcal{N}} \left( \Delta \bm{a}_i \right) \right |&= \left| f_{\mathcal{N}} \left( \bm{a}_i \ominus T_{\mathcal{T}} (\bm{a})_i \right) \right | \\
=& \left | \sum_{j=0}^{i_{max} - {\bm e}ta-1} a_i(-2)^j \right |\\
&\leq \sum_{j=0}^{i_{max} - {\bm e}ta-1} 2^j \\
&\leq 2^{i_{max} - {\bm e}ta}-1 \\
&\leq 2^{i_{max} - {\bm e}ta}. \\
\end{align*}
concludes the proof of the first claim.
Notice, $\|{\bm a}\|_{\mathcal{N},\infty} \geq 2^{i_{max}-1}$, then we can conclude the second claim
\[ \|\Delta \bm{a} \|_{\mathcal{N}, \infty} \leq 2^{-{\bm e}ta +1}\|{\bm a}\|_{\mathcal{N},\infty}.\]
\end{proof}
\fi
\subsection{Shift Operator on $\mathcal{B}^n$}\label{ShiftSubsection}
{\bm l}ack{We now wish to determine what can be said about the norm of the shift operator defined on $\mathcal{B}^n$.} For $a \in \mathcal{B}$ and $\ell \in \mathbb{Z}$, observe that
{\bm e}gin{align*}
\left| f_\mathcal{B}(s_{\ell} (a)) \right| = \left| \sum_{i \in \mathcal{I}(a)} 2^{i - \ell} \right| = 2^{-\ell} \left| \sum_{i \in \mathcal{I}(a)} 2^{i} \right| = 2^{-\ell} \left| f_\mathcal{B}(a) \right|.
\end{align*}
{\bm l}ack{This observation, together with the definition of $\| \cdot \|_{\mathcal{B}, p}$, yields the following result.}
{\bm e}gin{lemma}
\label{ShiftLemma}
Suppose $\ell \in \mathbb{Z}$ and $1\leq p \leq \infty$. Then $\| S_{\ell} \|_{\mathcal{B}, p} = 2^{-\ell}$ and $\| S_{\ell}^{-1} \|_{\mathcal{B}, p} = 2^{\ell}$.
\end{lemma}
\iffalse
{\bm e}gin{proof}
Suppose that $\bm{a} \in \mathcal{B}^n$ with $\bm{a} \neq \bm{0}_{\mathcal{B}}$. Then for $1\leq p <\infty$ we have
{\bm e}gin{align*}
\| S_l (\bm{a}) \|_{\mathcal{B}, p} = \left( \sum_{i = 1}^{n} \left| f_\mathcal{B} \left(s_l(\bm{a}_i) \right) \right|^p \right)^{1/p} = \left( \sum_{i = 1}^{n} \left| 2^{-l} f_\mathcal{B} \left( \bm{a}_i \right) \right|^p \right)^{1/p} = 2^{-l} \| \bm{a} \|_{\mathcal{B}, p}
\end{align*}
and
{\bm e}gin{align*}
\| S_l (\bm{a}) \|_{\mathcal{B}, \infty} = \max_{1 \leq i \leq n} \| s_l(\bm{a}_i) \|_{\mathcal{B}} = \max_{1 \leq i \leq n} \left\{ 2^{-l} \| \bm{a}_i \|_{\mathcal{B}} \right\} = 2^{-l} \max_{1 \leq i \leq n} \| \bm{a}_i \|_{\mathcal{B}} = 2^{-l} \| \bm{a} \|_{\mathcal{B}, \infty}.
\end{align*}
Hence, $\displaystyle \frac{\| S_l (\bm{a}) \|_{\mathcal{B}, p}}{\| \bm{a} \|_{\mathcal{B}, p}} = 2^{-l}$, for all $1\leq p <\infty$. Since $\bm{a} \in \mathcal{B}^n$ was taken to be arbitrary, we conclude that $\| S_l \|_{\mathcal{B}, p}=2^{-l}.$
As $S_l^{-1} =S_{-l}$, $\| S_l^{-1} \|_{\mathcal{B}, p}=2^{l}$ immediately follows.
\end{proof}
\fi
{\bm l}ack{To summarize, we have constructed the normed vector spaces $\mathcal{B}^n$
and $\mathcal{N}^n$ and bijective maps between $\mathcal{B}^n$ and $\mathbb{R}^n$ and
$\mathcal{N}^n$ and $\mathbb{R}^n$. We can represent floating-point or fixed-point
representations by applying the truncation operator
$T_\mathcal{S}{(\cdot)} $ on elements of $\mathcal{B}^n$ or $\mathcal{N}^n$, and
multiplication by powers of two in $\mathbb{R}^n$ is equivalent to
applying shift operator, $S_{\ell}$, to elements in $\mathcal{B}^n$.
We now have the tools to define operators for each step of the ZFP compression
algorithm, as described is Section~\ref{sec:zfp}. }
\section{Error Analysis of Individual Steps of the ZFP Compression Algorithm}
\label{sec:analysis}
The goals of this section are to define operators for each step of the ZFP
compression algorithm and to determine the error resulting from each
step. For each step of ZFP, we define an operator that carries out
the implemented version of ZFP compression and decompression as well as
a lossless version. The lossless version of each operator will be useful
in determining the error introduced at each step of the
algorithm. Decompression for steps corresponding to invertible
compression steps are merely the inverse operators of the compression
step. Since ZFP compression is lossy in nature, some of the steps
implemented in the compression phase are not invertible. For such steps,
the corresponding decompression step is defined by an injective map
that restores only the information that has not been lost to the correct
format for the next step of decompression. For the sake of brevity, any
step of the algorithm that does not affect the error analysis will not
be considered in much depth.
\subsection{Partition $d$-dimensional array into blocks of $4^d$ values}
\label{Step1Sec}
In Step 1, the $d$-dimensional array is partitioned into blocks of size $4^d$. Since Steps 2 through 8 are then applied to each $4^d$ block individually, it is not necessary to consider Step 1 in the error analysis. Accordingly, we do not define any operators for this step.
\subsection{Block-Floating-Point Transform}
\label{Step2Sec}
Suppose that ${\bm x} \in \mathbb{R}^{4^d}$ such that $F_\mathcal{B}^{-1}({\bm x}) \in
\mathcal{B}_k^{4^d}$ for some precision $k$ (i.e., every element in ${\bm x}$ can be
represented with at most $k$-consecutive bits). For frequently used
IEEE floating-point types, $k \in \{24, 53\}$. This assumption on
$\bm{x}$ implies that we are working
with a floating point representation of a real number. To perform Step
2, we first convert each component in ${\bm x}$ to its corresponding
representation in $\mathcal{\mathcal{B}}$. Each element is then shifted to the
left by a deterministic number of bits and truncated. As a by-product of
type-casting to an integer, in the implementation of ZFP, each value is
rounded down to zero. Applying the shift operator followed by the
truncation operator, as outlined above, results in the same outcome.
{\bm l}ack{The operator in Step 2 is dependent on the fixed set $\mathcal{S} := \{
i \in \mathbb{Z} : i \geq 0 \}$ and the value $q \in
\mathbb{N}$, where $q$ denotes the maximum number of nonzero consecutive
bits (precision) that can be used for the representation of each component of the
input.} ZFP requires each value to have one bit as a safe-guard
against overflow, which occurs when the calculation produces a result
that exceeds the capacity of the finite bit representation. {\bm l}ack{In the
current ZFP implementation, if the input values are IEEE single or double precision, $q \in \{30, 62\}$ in $\mathcal{B}$, since one
bit is used to represent the sign bit and another to represent the
overflow guard bit.
}Step 2 is defined by
the map $\tilde{C}_2:\mathbb{R}^{4^d} \rightarrow \mathcal{B}^{4^d}$ where
{\bm e}gin{align*}*
\tilde{C}_2 (\bm{x}) := T_{\mathcal{S}} S_{\ell} F_{\mathcal{B}}^{-1}(\bm{x}),
\text{ for all } \bm{x} \in \mathbb{R}^{4^d},
\end{align*}
where \red{$\ell = e_{max}(F_\mathcal{B}^{-1}({\bm x}))- q+1$.} We define the lossless
operator, $C_2$, by removing all noninvertible maps from
$\tilde{C}_2$. Hence,
{\bm e}gin{align*}
C_2({\bm x} ) := S_{\ell}F_{\mathcal{B}}^{-1} ({\bm x}).
\end{align*}
{The decompression operator for Step 2 converts the block-floating point back to its original floating-point representation that is representable in $\mathcal{B}_k$ for $k \in \mathbb{N}$. In IEEE, the $q\in \{30,62\}$ consecutive bits must be converted back to $k \in \{24,53\}$ with its respective exponent information. This conversion can be seen as a typical floating-point round off error. The lossy decompression operator for Step 2 is then defined by undoing the shift performed in $\tilde{C}_2$ and converting each component back to a floating point representation. Hence, $\tilde{D}_2: {\mathcal{B}}^{4^d}\rightarrow \mathbb{R}^{4^d}$ is defined by
{\bm e}gin{align*}
{\bm l}ack{\tilde{D}_2({\bm a}) := F_{\mathcal{B}} S_{-\ell} fl_k ({\bm a})}, \ \text{for all} \ \bm{a} \in
\mathcal{{\mathcal{B}}}^{4^d},
\end{align*}
where {\bm l}ack{$fl_k ({\bm a})_i = t_{\mathcal{R}_{ik}} ({\bm a}_i)$ with $\mathcal{R}_{ik} = \{ j \in \mathbb{Z}: j > e_{max, \mathcal{B}} (\bm{a}_i) - k \}$, for all $1 \leq i \leq 4^d$. Note that the $fl_k$ operator converts each component of $\bm{a}$ to a floating point representation with $k$ mantissa bits in a bit vector format.} The lossless decompression operator is then defined as $D_2: {\mathcal{B}}^{4^d}\rightarrow \mathbb{R}^{4^d}$ with
$D_2({\bm a}) := F_{\mathcal{B}} S_{-\ell}({\bm a})$, for all $\bm{a} \in
\mathcal{{\mathcal{B}}}^{4^d}$. We conclude our discussion of this step by
presenting a result that will be useful during the error analysis in
Section~\ref{sec:bounds}.
{\bm e}gin{prop}Suppose $\bm{x} \in \mathbb{R}^{4^d}$ and $\bm{a} \in \mathcal{B}^{4^d}$, such that \red{$e_{max,\mathcal{B}}(F_\mathcal{B} ({\bm a})) \geq q-1.$}
\label{Step2Prop}
{\bm e}gin{itemize}
\item[(i)] Then $\| \tilde{C}_2 {\bm x} - C_2
{\bm x} \|_{\infty} \leq 2^{-\ell} \epsilon_{q} \| \bm{x} \|_{\infty}$.
\item[(ii)] Then $\| \tilde{D}_2 {\bm a} - D_2
{\bm a} \|_{\infty} \leq 2^{\ell} \epsilon_{k} \| \bm{a} \|_{\infty}$.
\end{itemize}
\end{prop}
{\bm e}gin{proof} For $(i)$, observe that
{\bm e}gin{align}
\| \tilde{C}_2 {\bm x} - C_2 {\bm x} \|_{\mathcal{B}, \infty} = \|
T_{\mathcal{S}} S_{\ell}F_\mathcal{B}^{-1}(\bm{x}) - S_{\ell} F_\mathcal{B}^{-1}(\bm{x})
\|_{\mathcal{B}, \infty}&\leq \epsilon_q \| S_\ell F_\mathcal{B}^{-1}(\bm{x})
\|_{\mathcal{B}, \infty} \label{4.1eq0},
\\
&\leq 2^{-\ell} \epsilon_q \| \bm{x} \|_{ \infty}, \label{4.1eq1}
\end{align}
where the inequality in (\ref{4.1eq0}) follows from Lemma
\ref{ProjectionLemma2} (i) and Lemma
\ref{ProjectionLemma1} (i). From Lemma~\ref{ShiftLemma}, we have that
$\|S_{\ell} \|_{\mathcal{B}, \infty} = 2^{-\ell}$, and from
Lemma~\ref{BNormLemma}, we have
$\| F_\mathcal{B}^{-1}(\bm{x}) \|_{\mathcal{B},\infty}= \| \bm{x}
\|_{\infty}$. These together yield the inequality in (\ref{4.1eq1}).
Next, $(ii)$ follows from an argument similar to the proof of $(i)$ by observing that
{\bm l}ack{$\displaystyle \| \tilde{D}_2 {\bm a} - D_2 {\bm a} \|_{ \infty} = \|
F_{\mathcal{B}} S_{-\ell} fl_k ({\bm a}) - F_{\mathcal{B}} S_{-\ell}({\bm a})
\|_{\infty} \leq 2^\ell \|fl_k ({\bm a}) - {\bm a}
\|_{\mathcal{B}, \infty} \leq 2^{\ell} \epsilon_k \| \bm{a} \|_{\mathcal{B},\infty}$.}
\end{proof}
}
\subsection{Decorrelating Linear Transform}
\label{Step3Sec}
In Step 3, the output from Step 2 is acted on by a linear
transformation, $L$. \red{$L$ is a near-orthogonal transform that is similar to the discrete cosine transform, both of which possess the energy compaction property \cite{Rao1990}, i.e., most of the signal energy is confined to the first, lowest-frequency transform coefficients.} In $d$-dimensions, the transform operator can be
applied to each dimension separately, and the operator can be
represented as a Kronecker product. For $A \in \mathbb{R}^{n_1,m_1}$ and
$B \in \mathbb{R}^{n_2,m_2}$, the Kronecker product is defined as
{\bm e}gin{align*}
A\otimes B ={\bm e}gin{bmatrix}a_{1,1}B &a_{1,2} B & \cdots a_{1,m_1}B \\ \vdots & \ddots& \vdots \\
a_{n_1,1}B &a_{n_1,2} B & \cdots a_{n_1,m_1}B \end{bmatrix}.
\end{align*}
Then, the total forward transform operator used in of ZFP is defined as
{\bm e}gin{align*}
L_d = \underbrace{L\otimes L \otimes \cdots \otimes L}_\text{$(d - 1)$-products},
\end{align*}
where $L \in \mathbb{R}^{4\times 4}$ is defined by
{\bm e}gin{align}
L = \frac{1}{16} {\bm e}gin{bmatrix}
{\bm e}gin{array}{rrrr}
4 & 4 & 4 & 4 \\
5 & 1 & -1 & -5 \\
-4 & 4 & 4 & -4 \\
-2 & 6 & -6 & 2
\end{array}
\end{bmatrix} \quad \text{and} \quad L^{-1}= \frac{1}{4} {\bm e}gin{bmatrix}
{\bm e}gin{array}{rrrr}
4 & 6 & -4 & -1 \\
4 & 2 & 4 & 5 \\
4 & -2 & 4 & -5 \\
4 & -6 & -4 & 1
\end{array}
\end{bmatrix}.
\end{align}
Note that $\| L\|_{\infty} = 1$ and $\| L^{-1} \|_{\infty} = 15/4$.
First, we define the lossless compression operator for Step 3 by $C_3:
\mathcal{B}^{4^d} \rightarrow \mathcal{B}^{4^d}$, where
{\bm e}gin{align}\label{c3}
C_3({\bm a}) = F_\mathcal{B}^{-1}L_dF_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d}.
\end{align}
In order to define the lossy operator used in the implementation, it is
necessary to account for the finite bit constraint on a machine. Based
on Step 2 of ZFP compression, the components provided as the input for
Step 3 represent integers. Hence, for some $q \in \mathbb{N}$, it
follows that the input for Step 3 is an element of
$\mathcal{B}_{q}^{4^d}$. Here, $q$ represents the number of bits
available for storing each component. As $\mathcal{B}_q$ is not closed
under addition and multiplication, given $a, b \in \mathcal{B}_q$,
addition or multiplication of $a$ and $b$ may not result in an element
of $\mathcal{B}_q$ and must be rounded. This circumstance is referred to as round-off; error that occurs when the calculation produces a result that
exceeds the
capacity of the finite bit representation. Since the transformation
could result in round-off, the operator used in the
implementation of the algorithm will be defined as
{\bm e}gin{align}\label{tildec3}
\tilde{C}_3 = F_\mathcal{B}^{-1} \tilde{L}_dF_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d},\end{align}
where $\tilde{L}_d$ is an operator such that $\tilde{L}_d F_\mathcal{B}({\bm a}) \in \mathcal{B}^{{\bm l}ack{4^d}}_q$, for all $\bm{a} \in \mathcal{B}^{4^d}$.
As the linear transform operator, $L$, is invertible, the lossless
decompression operator $D_3: \mathcal{B}^{4^d} \rightarrow \mathcal{B}^{4^d}$ is defined
as
\[ D_3({\bm a}) = F_\mathcal{B}^{-1}L_d^{-1}F_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d}. \]
Again, since the operation $L_d^{-1}$ may result in round-off, the operator used in the implementation is defined as
\[ \tilde{D}_3({\bm a}) = F_\mathcal{B}^{-1}\tilde{L}_d^{-1}F_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d}, \]
where $\tilde{L}_d^{-1}$ is an approximation of $L_d^{-1}$.
From \cite{higham2002accuracy} (Equation (3.12)), the forward error bound of the floating-point representation of a matrix-vector product, $L_d{\bm x} \in \mathbb{R}^{4^d}$, is
{\bm e}gin{equation}
\left\| L_d{\bm x} - \tilde{L}_d {\bm x} \right \|_p \leq \gamma \left \|L_d \right \|_p \left \|{\bm x} \right\|_p, \label{eqn:T}
\end{equation}
where $\gamma = 4^d \epsilon_m/ (1 - 4^d \epsilon_m)$ and $\epsilon_m =
2^{1-m}$ represents machine epsilon with precision
$m$. From~\cite{tensor}, we have that $\|L_d\|_p \leq\|L\|_p^d$, for
$1 \leq p \leq \infty$. Hence, (\ref{eqn:T}) yields
{\bm e}gin{equation}
\left\| L_d{\bm x} - \tilde{L}_d {\bm x} \right \|_p \leq \gamma \left \|L \right \|^d_p \left \|{\bm x} \right\|_p. \label{eqn:Td}
\end{equation}
Note that $\tilde{L}_d^{-1}$ satisfies a forward error bound analogous
to (\ref{eqn:Td}). The forward error bound represented in (\ref{eqn:Td})
is the worst possible error that can occur for an arbitrary linear
transform. As ZFP uses particular transformations, we aim to establish bounds specific to the
transformations $L_d$ and $L_d^{-1}$. Accordingly, we note that the
action of $L$ and $L^{-1}$ can be written in a very efficient C
implementation. The action of $L$ and $L^{-1}$ on ${\bm a} = [ {\bm a}_1,
{\bm a}_2, {\bm a}_3, {\bm a}_4 ]^T \in \mathcal{B}^4$ under this implementation is
outlined in Table \ref{table:actionT}.
{\bm e}gin{table}[!h]
{\bm e}gin{adjustbox}{width=\textwidth}
{\bm e}gin{tabular}{| L L L || L L L | }
\hline
\multicolumn{3}{| c ||}{$L$} & \multicolumn{3}{ c |}{$L^{-1}$} \\ \hline
{\bm a}_1 \leftarrow {\bm a}_1+ {\bm a}_4 & {\bm a}_1 \leftarrow s_1({\bm a}_1) &{\bm a}_4 \leftarrow {\bm a}_4 - {\bm a}_1 &{\bm a}_2 \leftarrow {\bm a}_2 + s_1({\bm a}_4)&{\bm a}_4 \leftarrow {\bm a}_4- s_1({\bm a}_2) & \\
{\bm a}_3 \leftarrow {\bm a}_3+{\bm a}_2 & {\bm a}_3 \leftarrow s_1({\bm a}_3)&{\bm a}_2 \leftarrow {\bm a}_2- {\bm a}_3 &{\bm a}_2 \leftarrow {\bm a}_2+ {\bm a}_4 &{\bm a}_4 \leftarrow s_{-1}({\bm a}_4) &{\bm a}_4 \leftarrow {\bm a}_4- {\bm a}_2 \\
{\bm a}_1 \leftarrow {\bm a}_1+{\bm a}_3 & {\bm a}_1 \leftarrow s_1({\bm a}_1)&{\bm a}_3 \leftarrow {\bm a}_3- {\bm a}_1 &{\bm a}_3 \leftarrow {\bm a}_3 + {\bm a}_1&{\bm a}_1 \leftarrow s_{-1}({\bm a}_1) &{\bm a}_1 \leftarrow {\bm a}_1- {\bm a}_3 \\
{\bm a}_4 \leftarrow {\bm a}_4+{\bm a}_2 & {\bm a}_4 \leftarrow s_1({\bm a}_4) &{\bm a}_2 \leftarrow {\bm a}_2 - {\bm a}_4 &{\bm a}_2 \leftarrow {\bm a}_2 + {\bm a}_3 &{\bm a}_3 \leftarrow s_{-1}({\bm a}_3) &{\bm a}_3 \leftarrow {\bm a}_3- {\bm a}_2 \\
{\bm a}_4 \leftarrow {\bm a}_4 + s_1({\bm a}_2) & {\bm a}_2 \leftarrow {\bm a}_2 - s_1({\bm a}_4) & &{\bm a}_4 \leftarrow {\bm a}_4 + {\bm a}_1 &{\bm a}_1 \leftarrow s_{-1}({\bm a}_1) & {\bm a}_1\leftarrow {\bm a}_1 - {\bm a}_4 \\ \hline
\end{tabular}
\end{adjustbox}
\caption{\label{table:actionT} Bit arithmetic steps for ZFP's forward (left) and backward (right) transform (read from left to right).
}
\end{table}
This implementation is straightforward and efficient as it only requires bit addition/subtraction and division/multiplication by two.
In ZFP, the bit vectors are padded so that any overflow that may occur
is represented (i.e., for each component in the block one extra bit is
allotted to ensure that, if a calculation results is a value greater
than what can be represented in $q$ bits, then the value is not
approximated). Thus, for the following analysis, it suffices to
calculate the error due to round-off. Additionally, as the components of
the input for Step 3 represent signed integers, round-off can only occur
during division by two (i.e., one bit shift to the right using
$s_1(\cdot)$).
As noted in Section~\ref{TwoCompSec}, care must be taken in Step 3,
since the implementation of ZFP uses a two's complement representation
of each integer. For our error analysis, the main concern is that
rounding to an integer in two's complement after a right bit shift
always results in rounding towards negative infinity. However, under the
representation defined in $\mathcal{B}$, this same sequence of
operations results in rounding towards zero. So, in order to mimic the
implementation, we define the operator $r : \mathcal{B} \to \mathcal{B}$
by
{\bm e}gin{align*}
r (a) := {\bm e}gin{cases}
t_\mathcal{S} s_{1}(a) & : f_{\mathcal{B}} (a) \geq 0, \\
t_\mathcal{S} s_{1} \left( a - 1_\mathcal{B} \right) & : f_{\mathcal{B}} (a) < 0,
\end{cases}
\end{align*}
for all $a \in \mathcal{B}$. Since $s_{1}(\cdot)$ performs a single
right bit shift and $t_\mathcal{S} (\cdot)$, where $\mathcal{S} = \{ i \in \mathbb{Z} :
i\geq 0\}$, rounds the value towards zero, $r (\cdot)$ will always round
the right bit shift toward negative infinity. The following lemma
considers the error of $r (\cdot)$ when compared to $s_1 (\cdot)$.
{\bm e}gin{lemma}\label{lemma:boundr}
Suppose $\mathcal{S} = \{ i \in \mathbb{Z} : i \geq 0\}$ and $p \in \mathbb{Z}$.
If $a = f_\mathcal{B}^{-1} (p)$, then $\left\| r (a) - s_1(a)
\right\|_{\mathcal{B}, \infty} \leq \frac{1}{2}.$
\end{lemma}
{\bm e}gin{proof}
As $p \in \mathbb{Z}$ and $a = f_\mathcal{B}^{-1} (p)$, we have that
$\mathcal{I} (a) \subseteq \{ i \in \mathbb{Z} : i \geq 0 \}$. Now
suppose $p \geq 0$. Then
{\bm e}gin{align*}
\mathcal{I} ( r (a) ) = \mathcal{I} ( t_{\mathcal{S}} s_1(a) ) = \mathcal{I} ( s_1(a) ) \setminus \{ -1 \}.
\end{align*}
Hence, $\left\| r (a) - s_1 (a) \right\|_{\mathcal{B}, \infty} \leq \frac{1}{2}$.
On the other hand, suppose $p < 0$. If $p$ is even, then $p = 2k$ for
some $k \in \mathbb{Z}$, and $f_{\mathcal{B}} ( r (a) ) =
f_{\mathcal{B}} ( t_{\mathcal{S}} s_1 (a - 1_{\mathcal{B}}) ) = k.$ So
{\bm e}gin{align*}
\left\| r (a) - s_1 (a) \right\|_{\mathcal{B}, \infty} = \left| f_{\mathcal{B}} ( r (a) ) - f_{\mathcal{B}} ( s_1 (a) ) \right| = \left| k - \frac{p}{2} \right| = | k - k | = 0.
\end{align*}
If $p$ is odd, then $p = 2k - 1$ for some $k \in \mathbb{Z}$, and
$f_{\mathcal{B}} ( r (a) ) = f_{\mathcal{B}} ( t_{\mathcal{S}} s_1 (a -
1_{\mathcal{B}}) ) = k - 1$. Hence, $\left\| r (a) - s_1 (a)
\right\|_{\mathcal{B}, \infty} = \left| f_{\mathcal{B}} ( r (a) ) -
f_{\mathcal{B}} ( s_1 (a) ) \right| = \left| k - 1 - \frac{p}{2} \right|
= \frac{1}{2}$.
\end{proof}
Thus, by replacing $s_1(\cdot)$ by $r (\cdot)$ in
Table~\ref{table:actionT}, we obtain the analogous lossy operators,
denoted $\tilde{L}$ and $\tilde{L}^{-1}$, outlined in
Table~\ref{table:actionTlossy}.
{\bm e}gin{table}[!h]
{\bm e}gin{adjustbox}{width=\textwidth}
{\bm e}gin{tabular}{| L L L || L L L | }
\hline
\multicolumn{3}{| c ||}{$\tilde{L}$} & \multicolumn{3}{ c |}{$\tilde{L}^{-1}$} \\ \hline
{\bm a}_1 \leftarrow {\bm a}_1+ {\bm a}_4& {\bm a}_1 \leftarrow r({\bm a}_1) &{\bm a}_4 \leftarrow {\bm a}_4 - {\bm a}_1 &{\bm a}_2 \leftarrow {\bm a}_2 + r({\bm a}_4)&{\bm a}_4 \leftarrow {\bm a}_4- r({\bm a}_2) & \\
{\bm a}_3 \leftarrow {\bm a}_3+{\bm a}_2 & {\bm a}_3 \leftarrow r({\bm a}_3)&{\bm a}_2 \leftarrow {\bm a}_2- {\bm a}_3 &{\bm a}_2 \leftarrow {\bm a}_2+ {\bm a}_4 &{\bm a}_4 \leftarrow s_{-1} ({\bm a}_4) &{\bm a}_4 \leftarrow {\bm a}_4- {\bm a}_2 \\
{\bm a}_1 \leftarrow {\bm a}_1+{\bm a}_3 & {\bm a}_1 \leftarrow r({\bm a}_1)&{\bm a}_3 \leftarrow {\bm a}_3- {\bm a}_1 &{\bm a}_3 \leftarrow {\bm a}_3 + {\bm a}_1&{\bm a}_1 \leftarrow s_{-1}({\bm a}_1) &{\bm a}_1 \leftarrow {\bm a}_1- {\bm a}_3 \\
{\bm a}_4 \leftarrow {\bm a}_4+{\bm a}_2 & {\bm a}_4 \leftarrow r({\bm a}_4) &{\bm a}_2 \leftarrow {\bm a}_2 - {\bm a}_4 &{\bm a}_2 \leftarrow {\bm a}_2 + {\bm a}_3 &{\bm a}_3 \leftarrow s_{-1} ({\bm a}_3) &{\bm a}_3 \leftarrow {\bm a}_3- {\bm a}_2 \\
{\bm a}_4 \leftarrow {\bm a}_4 + r({\bm a}_2) & {\bm a}_2 \leftarrow {\bm a}_2 - r({\bm a}_4) & &{\bm a}_4 \leftarrow {\bm a}_4 + {\bm a}_1 &{\bm a}_1 \leftarrow s_{-1} ({\bm a}_1) & {\bm a}_1\leftarrow {\bm a}_1 - {\bm a}_4 \\ \hline
\end{tabular}
\end{adjustbox}
\caption{\label{table:actionTlossy} Bit arithmetic steps for the lossy implementation of ZFP's forward (left) and backward (right) transform.
}
\end{table}
\red{Now that all the required notation and tools have been discussed, the following lemma establishes a forward error bound for $\tilde{L}$. }
{\bm e}gin{lemma}
\label{LemmaL1DBound}
Suppose ${\bm x} \in \mathbb{Z}^4$ such that \red{$e_{max}({\bm x}) \geq q-1$} and ${\bm x} \neq {\bm 0}$. Given the bit arithmetic implementation in Table \ref{table:actionT} and Table \ref{table:actionTlossy} for ZFP's forward linear transforms, we have
\red{{\bm e}gin{align*}
\|L{\bm x} - \tilde{L}{\bm x}\|_\infty \leq \frac{7}{4} \epsilon_q \|{\bm x}\|_\infty.
\end{align*}}
\end{lemma}
{\bm e}gin{proof}
\red{First the action of ${L}$ and $\tilde{L}$ will be formed as a composite operator of each step, as depicted in Table \ref{table:actionT} and \ref{table:actionTlossy}, respectively. Then a bound on the error between the action of $L$ and $\tilde{L}$ will be constructed using the Lemma \ref{lemma:boundr} and the triangle inequality.} Define ${\bm a} = [ {\bm a}_1, {\bm a}_2, {\bm a}_3, {\bm a}_4 ]^T = F_\mathcal{B}^{-1}({\bm x}) \in
\mathcal{B}^4$ to be the representation of ${\bm x}$ in $\mathcal{B}^4$. From
Table~\ref{table:actionT}, the lossless operator for the first two steps
can be written as
\[ {\bm a}_1 \leftarrow {\bm a}_1+ {\bm a}_4 \quad \Rightarrow \quad L_{\mathcal{B},1} ( {\bm a} ) = [ {\bm a}_1 + {\bm a}_4, {\bm a}_2, {\bm a}_3, {\bm a}_4 ]^T ,\]
and
\[ {\bm a}_1 \leftarrow s_1({\bm a}_1), \quad \Rightarrow \quad L_{\mathcal{B},2} ( {\bm a} ) = [s_1({\bm a}_1), {\bm a}_2, {\bm a}_3,{\bm a}_4 ]^T , \]
where we write $L_{\mathcal{B},i}$ to represent the action at the $i$th step
of the operator $L$ on an element of $\mathcal{B}^4$ from Table \ref{table:actionT}. The composite
operator for the first two steps can now be expressed as
\[L_{\mathcal{B},2} \circ L_{\mathcal{B},1} ( {\bm a} ) = [ s_1 ({\bm a}_1 + {\bm a}_4), {\bm a}_2, {\bm a}_3, {\bm a}_4 ]^t. \]
\red{Let} $L_{\mathcal{B}}$ denote the action of $L$ in the vector space
$\mathcal{B}^4$. Continuing in the same manner as above, we have
\red{{\bm e}gin{align*}
L_\mathcal{B} {\bm a} &= L_{\mathcal{B},14} \circ \cdots \circ L_{\mathcal{B},1} ({\bm a}), \\
&= {\bm e}gin{bmatrix} s_{1} (s_{1} ({\bm a}_4 + {\bm a}_1) + s_{1} ({\bm a}_2 + {\bm a}_3)) \\
{\bm z} - s_1({\bm z}+{\bm w}) - s_1(s_1( {\bm z}+{\bm w})+ s_1({\bm z} - s_1({\bm z}+{\bm w}))) \\
s_1 ({\bm a}_2 + {\bm a}_3) - s_1 ( s_1 ( {\bm a}_4 + {\bm a}_1 ) + s_1 ({\bm a}_2 + {\bm a}_3)) \\
s_1( {\bm z}+{\bm w})+ s_1({\bm w}- s_1({\bm z}+{\bm w}))
\end{bmatrix},
\end{align*}
where ${\bm z} = {\bm a}_4 - s_1({\bm a}_4+{\bm a}_1)$ and ${\bm w} = {\bm a}_2-s_1({\bm a}_2+{\bm a}_3)$. } Next, from Table \ref{table:actionTlossy}, we obtain the analogous lossy operator, denoted $\tilde{L}_{\mathcal{B}}$. By replacing $s_1 (\cdot)$ by $r (\cdot)$ we obtain $\tilde{L}_{\mathcal{B}} {\bm a}$. Now
{\bm e}gin{align}
\|L{\bm x} - \tilde{L}{\bm x}\|_\infty = \| L_{\mathcal{B}} {\bm a} - \tilde{L}_{\mathcal{B}} {\bm a} \|_{\mathcal{B}, \infty} = \max_{1 \leq i \leq 4} | f_\mathcal{B}((L_{\mathcal{B}} {\bm a})_i) - f_\mathcal{B}((\tilde{L}_{\mathcal{B}} {\bm a})_i) |. \label{LopMax}
\end{align}
In particular, we found that the maximum in
(\ref{LopMax}) is attained for $i = 2$. Using Lemma~\ref{lemma:boundr}
and using $s_{1}(c) = f_\mathcal{B}^{-1}( f_{\mathcal{B}} (c)/2) = \frac{1}{2}c$ for all
$c \in \mathcal{B}$, we derive the following bound. Letting $\bm{y} = L_{\mathcal{B}}
{\bm a}$ and
$\tilde{\bm{y}} = \tilde{L}_{\mathcal{B}} {\bm a}$, we find that
\red{{\bm e}gin{align}
\| \bm{y}_2 - \tilde{\bm{y}}_2 \|_{\mathcal{B},1} &= \left\| \frac{1}{4}\left({\bm a}_2 +{\bm a}_3 -{\bm a}_1 -{\bm a}_4\right) - \left[{\bm z} - s_1({\bm z}+{\bm w}) - s_1(s_1( {\bm z}+{\bm w})+ s_1({\bm z} - s_1({\bm z}+{\bm w}))) \right] \right\|_{\mathcal{B},1}, \nonumber \\
&\leq \frac{1}{2} + \left\| \frac{1}{4}\left({\bm a}_2 +{\bm a}_3 -{\bm a}_1 -{\bm a}_4\right) - \left[{\bm z} - \frac{3}{2}s_1({\bm z}+{\bm w}) -\frac{1}{2} s_1({\bm z} - s_1({\bm z}+{\bm w})) \right] \right\|_{\mathcal{B},1}, \label{LErrBoundEq1}\\
&\leq \frac{1}{2} + \frac{1}{4} +\left\| \frac{1}{4}\left({\bm a}_2 +{\bm a}_3 -{\bm a}_1 -{\bm a}_4\right) - \left[\frac{3}{4}{\bm z} - \frac{5}{4} s_1({\bm z}+{\bm w}) \right] \right\|_{\mathcal{B},1},\nonumber \\
&\leq \frac{1}{2} + \frac{1}{4} + \frac{5}{8} + \left\| \frac{1}{4}\left({\bm a}_2 +{\bm a}_3 -{\bm a}_1 -{\bm a}_4\right) - \left[\frac{1}{8}{\bm z} - \frac{5}{8}{\bm w} \right] \right\|_{\mathcal{B},1}, \nonumber\\
&\leq \frac{1}{2} + \frac{1}{4} + \frac{5}{8} +\frac{1}{16} + \frac{5}{16} = \frac{28}{16}\nonumber
\end{align}}
where (\ref{LErrBoundEq1}) follows from the triangle inequality and Lemma \ref{lemma:boundr}. Similarly, \red{$\| \bm{y}_i - \tilde{\bm{y}}_i \|_{\mathcal{B},1} \leq \frac{28}{16}$}, for all $1 \leq i \leq 4$. Since \red{$\|{\bm x} \|_\infty \geq 2^{q-1}$}, it now follows that \red{
\[ \|L{\bm x} - \tilde{L}{\bm x} \|_\infty \leq \frac{28}{16} 2^{-q+1} \|{\bm x}\|_\infty = \frac{7}{4} \epsilon_q \|{\bm x}\|_\infty. \] }
\end{proof}
\red{The following result extends the 1-$d$ error caused by the lossy forward transform operator established in Lemma \ref{LemmaL1DBound} to $d$ dimensions. }
{\bm e}gin{lemma}
\label{lemma:boundT}
Suppose ${\bm x} \in \mathbb{Z}^{4^d}$ such that $e_{max, \mathcal{B}}({\bm x}) = q$. Then
{\bm e}gin{align*}
\|L_d{\bm x} - \tilde{L}_d {\bm x} \|_\infty & \leq k_L\epsilon_q \|{\bm x}\|_\infty,
\end{align*}
\red{where $k_L = \frac{7}{4} \left( 2^d - 1 \right)$. }
\end{lemma}
{\bm e}gin{proof}
Let $\Delta L$ represent a perturbation of the action of $L$ such that $ \tilde{L} = L+\Delta L$. From Lemma \ref{LemmaL1DBound}, we have \red{$\|\Delta L {\bm y} \|_\infty \leq \frac{7}{4} \epsilon_q \|{\bm y}\|_\infty$}, for all \red{${\bm y} \in \{{\bm z} \in \mathbb{Z}^4 : e_{max}({\bm z})\geq q-1\}$.} Hence,
\red{{\bm e}gin{align*}
\|(I_{4^{d}} \otimes \Delta L) {\bm x} \|_\infty = \| (\Delta L \otimes {I_{4^{d}}}) {\bm x} \|_\infty \leq \frac{7}{4} \epsilon_q \| {\bm x} \|_\infty.
\end{align*}} Using the inequalities $\|\Delta L\|_\infty \leq 1$ and $\| L\|_\infty =1$, we have
\red{{\bm e}gin{align*}
\|L_d{\bm x} - \tilde{L}_d {\bm x} \|_\infty & = \| \left({L}\otimes \cdots\otimes {L}\right) {\bm x} - \left((L+\Delta L) \otimes \cdots\otimes (L+\Delta L)\right) {\bm x} \|_\infty ,\\
& \leq \left( \sum_{i=1}^d \left ({\bm e}gin{matrix}d \\ i \end{matrix} \right )\|L\|^{d-i}_\infty \|\Delta L\|^{i-1}_{\infty} \right) \frac{7}{4} \epsilon_q \|{\bm x}\|_\infty, \\
& = \frac{7}{4} \left( \sum_{i=1}^d \left ({\bm e}gin{matrix}d \\ i \end{matrix} \right ) \right ) \epsilon_q \|{\bm x}\|_\infty, \\
& = \frac{7}{4} \left( 2^d - 1 \right) \epsilon_q \|{\bm x}\|_\infty.
\end{align*}}
\end{proof}
\red{At this point, it remains to consider ZFP's backward linear transform with respect to Table~\ref{table:actionT} and Table \ref{table:actionTlossy}. For this particular implementation of ZFP, if Steps 3 through 8 of the compression algorithm are applied before the backwards transform, no additional error occurs\footnote{\red{The first two steps of the backwards transform operator, depicted in Table \ref{table:actionTlossy}, may result in round-off. However, the additional error that may occur depends on the user-defined parameters that define the action of Step 8. If Step 8 is performed losslessly, i.e., no bit planes are discarded, then each step of the backward transform, in bit arithmetic, undoes the associated step of the forward transform.
If at least $2d$ bit planes are discarded at Step 8 (see Section \ref{Step8Sec} for details), then the first two steps of the backwards transform will not introduce additional error.
If between $1$ and $2d - 1$ bit planes are discarded, additional error may occur in the decompression step. However, since ZFP will result in a low compression ratio if only between $1$ and $2d-1$ bit planes are discarded, the remainder of the paper will assume at least $2d$ bit planes are discarded.
See Appendix \ref{sec:appendixb} for details.}}.}
The decompression operator for the particular implementation of
ZFP is defined as the corresponding lossless operator
\[ \tilde{D}_3({\bm a}) = {D}_3({\bm a})= F_\mathcal{B}^{-1}{L}_d^{-1}F_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d}. \]
\subsection{Reorder coefficients by total sequency}
\label{Step4Sec}
The fourth step performs a deterministic permutation on the components of the input. As such, it is an invertible operation. We define $C_4: \mathcal{B}^{4^d} \to \mathcal{B}^{4^d}$ to be the map that takes the components of a block in row-major order and permutes them so that the resulting block is in total sequency order \cite{zfp-doc}. The decompression operator performs the inverse permutation such that $D_4 C_4 (\bm{a}) = \bm{a}$, for all $\bm{a} \in \mathcal{B}^{4^d}$. We summarize the key details for these operators below.
{\bm e}gin{prop}\label{Step4Prop}
Suppose $\bm{a} \in \mathcal{B}^{4^d}$. Then $\| C_4 (\bm{a}) \|_{\mathcal{B}, p} = \| \bm{a} \|_{\mathcal{B}, p} =\| D_4 (\bm{a}) \|_{\mathcal{B}, p}$, for all $1 \leq p \leq \infty$.
\end{prop}
\subsection{Convert signed two's complement to negabinary}
\label{Step5Sec}
At Step 5 of the algorithm, each component is converted from its two's complement representation to a negabinary representation. As we are representing values using a signed binary representation instead of a two's complement representation for our analysis, we will need to convert each signed binary representation to a negabinary representation. Using the operators defined in Section \ref{sec:notation}, we define the operator $C_5 : \mathcal{B}^{4^d} \to \mathcal{N}^{4^d}$ by
{\bm e}gin{align*}
C_5 (\bm{a}) := F_{\mathcal{N}}^{-1} F_\mathcal{B} ({\bm a}), \ \ \text{for all} \ \bm{a} \in \mathcal{B}^{4^d}.
\end{align*}
{A valid concern for this step is that the range of representable integers for an $N$-bit two's complement representation is not the same as the range of representable integers for an $N$-bit negabinary representation, for any integer $N \geq 2$. To account for this difference, ZFP uses an $(N-1)$-bit two's complement representation with one bit left unused, called a guard bit. In Step 3, the guard bit was required for the decorrelating transform but is unnecessary for the remaining steps. Thus, when the two's complement representation is converted to a negabinary representation in ZFP the guard bit is freed and used instead for an $N$-bit negabinary representation to ensure that the integer can be represented. Additionally, since the magnitude of each component is not increased in the following steps, the components can be converted back to two's complement without introducing any error due to round off. Hence, Step 5 is lossless. Lastly, it follows that the decompression operator is defined as $D_5 := F_{\mathcal{B}}^{-1}F_\mathcal{N}$. The following result summarizes the key result from this step used in the analysis in~\secref{sec:bounds}.}
{\bm e}gin{prop}\label{Step5Prop}
Suppose $\bm{a} \in \mathcal{B}_{k}^n$. Then $\| C_5 (\bm{a}) \|_{\mathcal{N}, p} = \| \bm{a} \|_{\mathcal{B}, p}$ for all $1 \leq p \leq \infty$.
\end{prop}
\subsection{Boolean matrix transposition}
\label{Step6Sec}
Next, the bit vectors are reordered by their bit index instead of their associated binary representation.
Under the bit vector representation, this corresponds to transposing the entire block. Since this operation is lossless and does not result in altering the representation of any element in the block, we do not define an operator here. For simplicity, we will work under the assumption that the transposition did not take place.
\subsection{Embedded block coding}
\label{Step7Sec}
In Step 7, each bit plane of $4^d$ bits is individually coded with a variable-length code that is one to one and reversible (see \cite{zfp-doc} for details). For purposes of the analysis, since Step 7 is lossless, we chose not to consider the encoding in Step 7 since the error analysis can be considered in any format. Hence, for the purposes of
simplifying the analysis, we take $C_7 = D_7 = I_{\mathcal{N}}$.
\subsection{Finite-precision: Bit stream truncation}
\label{Step8Sec}
{\bm l}ack{Step 8 is dependent on one parameter, denoted ${\bm e}ta \geq
0$, and an index set dependent on ${\bm e}ta$ and the input, denoted as $\mathcal{P}$.} Here, ${\bm e}ta$ represents the number of most significant bit planes to keep
during Step 8 and any discarded bit plane is replaced with all-zero bits. {\bm l}ack{Note that
the value of ${\bm e}ta$ corresponds to the parameter \texttt{zfp\_stream.maxprec} in ZFP and can be set to any positive integer by the user in the fixed precision
mode of ZFP.} The operator for Step 8 is given by $\tilde{C}_8:
\mathcal{N}^{4^d} \rightarrow \mathcal{N}^{4^d}$ and defined as
{\bm e}gin{align*}
\tilde{C}_8(\bm{d}) := T_{\mathcal{P}} (\bm{d}), \text{ for all } \bm{d} \in \mathcal{N}^{4^d},
\end{align*}
where {\bm l}ack{$\mathcal{P} = \{ i \in \mathbb{Z} : i > q + 1 - {\bm e}ta \}$,} $q \in \mathbb{N}$ {\bm l}ack{is the value} from Step 2, and $T_\mathcal{P}$ is the truncation operator with respect to set $\mathcal{P}$. The lossless compression and decompression operators are then
defined by $C_8 := I_\mathcal{N}$ and $D_8 := I_\mathcal{N}$, respectively. {\bm l}ack{We conclude this step with a proposition that immediately follows from Lemma~\ref{ProjectionLemma2}.}
{\bm e}gin{prop}
\label{Step8Prop}
Suppose $\bm{a} \in \mathcal{B}^{4^d}$ such that $F_{\mathcal{B}} (\bm{a}) \in \mathbb{Z}^{4^d}$ and \red{$e_{max, \mathcal{B}} (F_{\mathcal{B}} (\bm{a})) \geq q-1$.} Then $\| \tilde{C}_8 C_5 \bm{a} - C_8 C_5 \bm{a} \|_{\mathcal{N}, \infty} \leq \frac{8}{3} \epsilon_{\bm e}ta \| \bm{a} \|_{\mathcal{B}, \infty}$.
\end{prop}
{\bm e}gin{proof}
Let $\bm{d} = C_5 \bm{a}$. From Lemma \ref{ProjectionLemma2} ($ii$) we have that
{\bm l}ack{{\bm e}gin{align}
\| \tilde{C}_8 \bm{d} - C_8 \bm{d} \|_{\mathcal{N}, \infty} &= \| T_{\mathcal{P}} \bm{d} - \bm{d} \|_{\mathcal{N}, \infty} \leq \frac{2}{3} \epsilon_{{\bm e}ta} 2^{q + 1}. \label{Step8PropEQ0}
\end{align}}
From the assumption \red{$e_{max, \mathcal{B}} (F_{\mathcal{B}} (\bm{a})) \geq q-1$} it now follows that \red{$\|\bm{a}\|_\infty \geq 2^{q-1}$} and
{\bm e}gin{align*}
\| \tilde{C}_8 C_5 \bm{a} - C_8 C_5 \bm{a} \|_{\mathcal{N}, \infty} &\leq \frac{8}{3 }\epsilon_{{\bm e}ta} \| \bm{a} \|_{\infty}.
\end{align*}
\end{proof}
To conclude this section, it should be noted that the inputs at Step 5 of ZFP satisfy the hypotheses of Proposition \ref{Step8Prop} as each component is encoded as an integer up to precision $q$.
\subsection{Defining the ZFP Compression Operator}
\label{ZFPCompOperator}
To conclude this section, we define the ZFP fixed precision compression
and decompression operators by composing the operators defined for each
step of the algorithm. In order to simplify the definition of each
operator, we omit $C_7$, $D_7$, $C_8$, and $D_8$ from the composition, as
they were defined to be the identity operator $I_{\mathcal{N}}$.
{{\bm e}gin{definition}
The lossy fixed precision compression operator, $\tilde{C}: \mathbb{R}^{4^d} \to \mathcal{N}^{4^d}$, is defined by
{\bm e}gin{align*}
\tilde{C} (\bm{x}) = \left( \tilde{C}_{8} \circ {C}_{5} \circ {C}_{4} \circ \tilde{C}_{3} \circ \tilde{C}_{2} \right) (\bm{x}), \ \ \ \text{for all} \ \bm{x} \in \mathbb{R}^{4^d},
\end{align*}
where $\circ$ denotes the usual composition of operators. The lossless fixed precision compression operator, $C: \mathbb{R}^{4^d} \to \mathcal{N}^{4^d}$, is defined by
{\bm e}gin{align*}
C(\bm{x}) = \left(C_{5} \circ C_{4} \circ C_{3} \circ C_{2} \right) (\bm{x}), \ \ \ \text{for all} \ \bm{x} \in \mathbb{R}^{4^d}.
\end{align*}
Lastly, the lossy fixed precision decompression operator, $\tilde{D}: \mathcal{N}^{4^d} \to \mathbb{R}^{4^d}$, is defined by
{\bm e}gin{align*}
\tilde{D}(\bm{d}) = \left( \tilde{D}_{2} \circ {D}_{3} \circ D_{4} \circ D_{5} \right) (\bm{d}), \ \ \ \text{for all} \ \bm{d} \in \mathcal{N}^{4^d},
\end{align*}
and the the lossless fixed precision decompression operator ${D} : \mathcal{N}^{4^d} \to \mathbb{R}^{4^d}$ is defined by
{\bm e}gin{align*}
{D}(\bm{d}) = \left( D_{2} \circ {D}_{3} \circ D_{4} \circ D_{5} \right) (\bm{d}), \ \ \ \text{for all} \ \bm{d} \in \mathcal{N}^{4^d}.
\end{align*}
\end{definition}}
\section{Error Bounds for ZFP Compression and Decompression}
\label{sec:bounds}
Now that the ZFP fixed precision compression and decompression operators
have been defined, we can establish a bound on the forward error for an
arbitrary input that is compressed and decompressed. We begin by
analyzing the error introduced during compression. Recall that ${\bm e}ta$
is the fixed precision parameter, i.e., ${\bm e}ta$ bits for each of the ZFP
transform coefficients will be kept during compression.
{\bm e}gin{lemma}\label{lemma:difftildeCandC} Assume ${\bm x} \in \mathbb{R}^{4^d}$ with $\bm{x} \neq \bm{0}$ such that $F^{-1}_{\mathcal{B}} ({\bm x}) \in \mathcal{B}_k^{4^d}$, for some precision $k$. Let {${\bm e}ta \geq 0$} be the fixed precision parameter. Then
{\bm e}gin{align*}
\|\tilde{C} {\bm x} - C {\bm x}\|_{\mathcal{N}, \infty} \leq 2^{-\ell} \left( \frac{8}{3} \epsilon_{\bm e}ta+ \epsilon_q \left(1+ \frac{8}{3} \epsilon_{\bm e}ta \right) \left(k_L(1+\epsilon_q)+1 \right )\right) \|{\bm x}\|_\infty,
\end{align*}
where $q \in \mathbb{N}$ is the precision for the block-floating point representation in Step 2, \red{$\ell = e_{max,\mathcal{B}}({\bm x}) - q +1$}, and \red{$k_L= \frac{7}{4} (2^d-1)$.}
\end{lemma}
{\bm e}gin{proof} Define $c(\bm{x}) := \|\tilde{C}{\bm x} - C {\bm x}\|_{\mathcal{N},\infty}$. First, we find that
{\bm e}gin{align*}
c (\bm{x}) &= \| \tilde{C}_8 C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C {\bm x} \|_{\mathcal{N},\infty}, \\
&= \| \tilde{C}_8 C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} +C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x}- C {\bm x} \|_{\mathcal{N},\infty}, \\
&\leq \| \tilde{C}_8 C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} \|_{\mathcal{N}, \infty} + \| C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty}.
\end{align*}
By the definition of $C_4$ and $C_5$, we have $\| C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty} = \| \tilde{C}_3 \tilde{C}_2 {\bm x} - C_3 C_2 {\bm x} \|_{\mathcal{B}, \infty}$. Additionally, $\| \tilde{C}_8 C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} - C_5 C_4 \tilde{C}_3 \tilde{C}_2 {\bm x} \|_{\mathcal{N}, \infty} \leq 8 \epsilon_{\bm e}ta \|\tilde{C}_3\tilde{C}_2 {\bm x} \|_{\mathcal{B}, \infty}$ follows by applying Proposition \ref{Step8Prop} and the definition of $C_4$. Hence,
{\bm e}gin{align*}
c (\bm{x}) &\leq 8 \epsilon_{\bm e}ta \| \tilde{C}_3 \tilde{C}_2 {\bm x} \|_{\mathcal{B}, \infty} + \| \tilde{C}_3 \tilde{C}_2 {\bm x} - C_3 C_2 {\bm x} \|_{\mathcal{B}, \infty} ,\\
& \leq \left( 1 + \frac{8}{3} \epsilon_{\bm e}ta \right) \| \tilde{C}_3 \tilde{C}_2 {\bm x} - C_3 \tilde{C}_2 \bm{x} \|_{\mathcal{B}, \infty} + \frac{8}{3} \epsilon_{\bm e}ta \| {C}_3 \tilde{C}_2 {\bm x} \|_{\mathcal{B}, \infty} + \|{C}_3\tilde{C}_2 {\bm x} -C_3 C_2 {\bm x} \|_{\mathcal{B}, \infty}, \\
& \leq \left( 1 + \frac{8}{3} \epsilon_{\bm e}ta \right) \| \tilde{C}_3 \tilde{C}_2 {\bm x} - C_3 \tilde{C}_2 \bm{x} \|_{\mathcal{B}, \infty} + \frac{8}{3} \epsilon_{\bm e}ta \| \tilde{C}_2 {\bm x} \|_{\mathcal{B}, \infty} + \| \tilde{C}_2 {\bm x} - C_2 {\bm x} \|_{\mathcal{B}, \infty},
\end{align*}
where the final inequality follows from the linearity of $C_3$ and $\|
C_3 \|_{\mathcal{B}, \infty} \leq 1$. By the definition of
$\tilde{C}_2$, we have that \red{$e_{max, \mathcal{B}} (\tilde{C_2}{\bm x}) \geq
q - 1$}. Hence, Lemma \ref{lemma:boundT} yields that $\|
\tilde{C}_3\tilde{C}_2 {\bm x} -C_3\tilde{C}_2{\bm x}\|_{\mathcal{B},\infty} \leq k_L
\epsilon_q \| \tilde{C}_2 \bm{x} \|_{\mathcal{B}, \infty}$. Lastly, using
Proposition \ref{Step2Prop}, we have that $\| \tilde{C}_2 {\bm x} - C_2 {\bm x}
\|_{\mathcal{B}, \infty} \leq 2^{-\ell} \epsilon_q \|{\bm x}\|_\infty$,
which yields the inequality $\| \tilde{C}_2 {\bm x}\|_{\mathcal{B}, \infty}
\leq 2^{-\ell} (1+\epsilon_q) \|{\bm x}\|_\infty$.
Combining these observations provides the desired result.
\end{proof}
The following result provides {bound on the error} resulting from compressing then decompressing a $4^d$ block using ZFP.
{\bm e}gin{theorem}\label{thm:diffDCandDC}
Assume ${\bm x} \in \mathbb{R}^{4^d}$ with $\bm{x} \neq \bm{0}$ such that $F_{\mathcal{B}} ({\bm x}) \in \mathcal{B}_k^{4^d}$, for some precision $k$. \red{Let $0\leq {\bm e}ta \leq q- 2d+2$ be the fixed precision parameter.}\footnote{\red{In other words, it is assumed that at least $2d$ least significant bit planes are discarded in Step 8. If less than $2d$ bit planes are discarded, i.e., $q- 2d +2 <{\bm e}ta <q+2 $, error will occur in Step 3 from round-off that may occur by the decompression operator, which is not taken into account in Theorem \ref{thm:diffDCandDC}. Theorem \ref{thm:diffDCandDCappendix} is the generalization of Theorem \ref{thm:diffDCandDC} for the assumption $q- 2d +2 <{\bm e}ta <q+2 $. See Appendix \ref{sec:appendixb} for details.}} Then
{\bm e}gin{align}
\| \tilde{D}\tilde{C} {\bm x} - {\bm x} \|_\infty &\leq K_{\bm e}ta \|{\bm x}\|_\infty
\end{align}
where $q \in \mathbb{N}$ is the precision for the block-floating point representation in Step 2,
{\bm e}gin{align}
K_{\bm e}ta := \left( \frac{15}{4} \right)^d \left( (1+\epsilon_k)\left( \frac{8}{3} \epsilon_{\bm e}ta+ \epsilon_q \left(1+ \frac{8}{3} \epsilon_{\bm e}ta \right) \left(k_L(1+\epsilon_q)+1 \right )\right)+\epsilon_k \right),
\end{align}
\red{and $k_L= \frac{7}{4} (2^d-1)$.}
\end{theorem}
{\bm e}gin{proof}
Observe that
{\bm e}gin{align}
\|\tilde{D} \tilde{C}{\bm x} - D C {\bm x} \|_\infty &= \| \tilde{D}_2{D}_3 D_4 D_5 \tilde{C} {\bm x} - D_2 D_3 D_4 D_5 C {\bm x} \|_\infty, \nonumber \\
&\leq \| \tilde{D}_2{D}_3 D_4 D_5 \tilde{C} {\bm x} -{D}_2{D}_3 D_4 D_5 \tilde{C} {\bm x}\|_\infty + \|{D}_2{D}_3 D_4 D_5 \tilde{C} {\bm x} - D_2 D_3 D_4 D_5 C {\bm x} \|_\infty, \nonumber \\
&\leq 2^\ell\epsilon_k\|{D}_3 D_4 D_5 \tilde{C} {\bm x}\|_{\mathcal{B},\infty}+\| D_2{D}_3 D_4 D_5\| \| \tilde{C} {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty}, \label{5.2eq1} \\
&\leq 2^{\ell} \left ( \frac{15}{4} \right)^d \left(\epsilon_k\| \tilde{C} {\bm x} \|_{{\bm l}ack{\mathcal{N}}, \infty} + \| \tilde{C} {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty}\right), \label{5.2eq2} \\
&\leq 2^{\ell} \left ( \frac{15}{4} \right)^d \left((1+\epsilon_k) \| \tilde{C} {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty} + {\bm l}ack{\epsilon_k}\|{C} {\bm x} \|_{\mathcal{N}, \infty} \right), \nonumber \\
&\leq 2^{\ell} \left ( \frac{15}{4} \right)^d \left((1+\epsilon_k) \| \tilde{C} {\bm x} - C {\bm x} \|_{\mathcal{N}, \infty} + {\bm l}ack{2^{-\ell}\epsilon_k}\|{\bm x} \|_{\infty} \right), \label{5.2eq3}
\end{align}
where (\ref{5.2eq1}) follows from Proposition~\ref{Step2Prop}($ii$) and (\ref{5.2eq2}) follows from the linearity of $D_2$, $D_3$, $D_4$, and $D_5$ and \red{$\ell = e_{max,\mathcal{B}} ({\bm x}) - q + 1$}.
Applying Lemma~\ref{lemma:difftildeCandC} in (\ref{5.2eq3}) yields the desired result.
\end{proof}
\DIFdelbegin
\DIFdelend \DIFaddbegin
\iffalse
\DIFadd{Since the constant $K_{{\bm e}ta}$ appears in the bound and it is dependent on $k$, $q$, $d$, and ${\bm e}ta$, we provide a brief discussion on how $K_{{\bm e}ta}$ is affected by changes in these values. First, since $q$ and $k$ depend on the precision of the data provided to ZFP and $d$ is the dimension of the input data it is important to note that three of the values in $K_{{\bm e}ta}$ are dependent solely on the input data. The remaining value, ${\bm e}ta$, can be set to any positive integer when using the fixed precision mode of ZFP, as noted in Section \ref{Step8Sec}. Figure \ref{fig:KBetaPlot} helps illustrate how $K_{{\bm e}ta}$ varies with respect to the dimension of the data, $d$, and ${\bm e}ta$. The lines on the surface in Figure \ref{fig:KBetaPlot} represent where the value of $K_{{\bm e}ta}$ exceeds $10^{-6}$ and $1$. As suspected from the formula for $K_{{\bm e}ta}$, we observe that a larger value of $d$ has a greater effect on the value of $K_{{\bm e}ta}$ for small values of ${\bm e}ta$.
}
{\bm e}gin{figure}[h!]
\centering
\includegraphics[width=\linewidth]{figures/KBPlot.png}
\caption{\DIFaddFL{Visualization of $K_{{\bm e}ta}$ for ${\bm e}ta \in [1, 32]$ and dimension $d \in [2, 5]$}}
\label{fig:KBetaPlot}
\end{figure}
\fi
{\bm l}ack{Since the constant $K_{{\bm e}ta}$ appears in the bound, which is dependent on $k$, $q$, $d$, and ${\bm e}ta$, we provide a brief discussion on $K_{{\bm e}ta}$ in Appendix \ref{sec:appendixc}.} Note that Theorem \ref{thm:diffDCandDC} yields the following bound on the maximum of the component-wise relative error:
{\bm e}gin{align}
\max_{i, {\bm x}_i \neq 0} \left | \frac{(\tilde{D} \tilde{C} {\bm x})_i -{\bm x}_i}{ {\bm x}_i} \right| & \leq \frac{1}{\min_{i, {\bm x}_i \neq 0}|{\bm x}_i| }\left \| \tilde{D} \tilde{C}{\bm x} -{\bm x}\right \|_\infty \leq K_{\bm e}ta {\bm l}ack{2^{e_{max,\mathcal{B}}({\bm x}) - e_{min,\mathcal{B}}({\bm x})}}.
\end{align}
{So far, the discussion and error analysis has focused on the fixed precision mode. However, as mentioned during the introduction, ZFP also has a fixed accuracy and fixed rate mode. While we will not spend much time providing details for the fixed accuracy and fixed rate modes, it should be noted that the error bound in Theorem \ref{thm:diffDCandDC} allows us to develop error bounds for the fixed accuracy and fixed rate modes.}
{In the fixed accuracy mode, the transform coefficients in each $4^d$ block are encoded up to a minimum bit plane number. The index of the minimum bit plane will be dependent on the largest absolute magnitude and the constant $K_{\bm e}ta$ found in Theorem~\ref{thm:diffDCandDC}. The following theorem is an extension of Theorem~\ref{thm:diffDCandDC} for fixed accuracy mode of ZFP.
{\bm e}gin{theorem}\label{thm:diffDCandDCFixedAccuracy}
Assume ${\bm x} \in \mathbb{R}^{n}$ with $\bm{x} \neq \bm{0}$ such that $F_{\mathcal{B}} ({\bm x}) \in \mathcal{B}_k^{n}$, for some precision $k$ and let $\hat{{\bm x}}$ represent the compressed and decompressed values from using the fixed accuracy mode of ZFP. To guarantee $b\in \mathbb{N}$ bits of accuracy, i.e., $\| \hat{{\bm x}} - {\bm x} \|_\infty \leq 2^{-b}$, ${\bm e}ta$ must satisfy:
{\bm e}gin{equation} \label{eqn:fixedaccuracy} {\bm e}ta \geq \log_2\left(\frac{ \frac{16}{3}(1+ c)} {\frac{\left(\left( \frac{4}{15} \right)^d 2^{-b-e_{max}} - \epsilon_k\right)}{(1+\epsilon_k)}- c} \right), \end{equation}
where $c = \epsilon_q \left(k_L(1+\epsilon_q)+1\right)$ and $e_{max}:= e_{max}({\bm x})$.
\end{theorem}
{\bm e}gin{proof}
Let ${\bm x}^i$ denote the $i$-th $4^d$ block of the $d$-dimensional data ${\bm x} \in \mathbb{R}^{n}$. Then we have that $e_{max}:= e_{max}({\bm x}) = \max_i e_{max}({\bm x}^i)$. From the hypothesis it follows that $K_{{\bm e}ta}\leq 2^{-b-e_{max}}$. From Theorem ~\ref{thm:diffDCandDC} and the fact that $\|{\bm x}^i\|_\infty\geq 2^{e_{max}}$ for all $i$, we conclude that
{\bm e}gin{align*}
\| \hat{{\bm x}} - {\bm x} \|_\infty &= \max_i \| \tilde{D} \tilde{C}{\bm x}^i - {\bm x}^i \|_\infty \leq \max_i K_{{\bm e}ta} \|{\bm x}^i\|_\infty \leq 2^{-b}.
\end{align*}
\end{proof}}
\noindent If we assume $k = q = \infty$, i.e., infinite precision, then Equation (\ref{eqn:fixedaccuracy}) simplifies to
{\bm e}gin{equation} {\bm e}ta \geq \log_2\left(\left( \frac{15}{4} \right)^d \frac{16}{3}2^{b+e_{max}} \right). \end{equation}
{Similarly, an upper bound for the fixed rate mode of ZFP can be obtained using Theorem ~\ref{thm:diffDCandDC}. For the purposes of understanding the following result, it suffices to know that for the fixed rate mode of ZFP the user provides a maximum rate, denoted $r$, or number of bits per value to be stored.
{\bm e}gin{theorem}\label{thm:diffDCandDCFixedRate}
Assume ${\bm x} \in \mathbb{R}^{4^d}$ with $\bm{x} \neq \bm{0}$ such that $F_{\mathcal{B}} ({\bm x}) \in \mathcal{B}_k^{4^d}$, for some precision $k$. Let $b_e$ be the number of bits to encode the exponent and let $\hat{{\bm x}}$ represent the compressed and decompressed values in the fixed rate mode with rate, $r\in \mathbb{N}$. For some ${\bm e}ta \in \mathbb{N}$, if
\[ \|\hat{{\bm x}} - {\bm x} \|_\infty \leq K_{\bm e}ta \|x\|, \]
then $r \geq\frac{4^d{\bm e}ta +b_e}{4^d} +1$.
\end{theorem}
{\bm e}gin{proof}
In the worst case scenario, the first bit plane is all-ones, which would imply $4^d - 1$ positive group tests (see \cite{zfp-doc} for details) and thus, $4^d-1$ bits to encode the the group tests. Each bit plane would then take $4^d$ bits to encode. Note that, ZFP uses one bit to indicate if the block is all zeros. Thus, if given rate $r$, there is a total of $4^d r$ bits that can be used to encode the block, $b_e$ of those bits must be used to encode the block floating-point exponent in Step 2, one bit is used for the leading all-zeros bit and $4^d-1$ bits for the group testing, leaving $(4^dr - b_e-1-(4^d-1))$ bits to encode the bit planes. Implying
{\bm e}gin{equation}{\bm e}ta \leq \frac{(4^d(r-1) - b_e)}{4^d} \Rightarrow r \geq \frac{4^d{\bm e}ta +b_e}{4^d} +1.\end{equation}
\end{proof}}
Now that we have established bounds on the error introduced by ZFP compression and decompression, we consider several numerical experiments in order to observe the tightness of these bounds.
\section{Numerical Experiments}
\label{sec:results}
In the following numerical tests, we consider two types of error, which
we will refer to as \emph{block relative error} and \emph{componentwise
relative error}, and their respective bounds:
{\bm e}gin{itemize}
\item[]{Block Relative Error:} $\displaystyle \frac{\|\tilde{D} \tilde{C} {\bm x} -{\bm x}\|_\infty}{\|{\bm x}\|_\infty} \leq K_{{\bm e}ta}$,
\item[]{Componentwise Relative Error:} $\displaystyle \max_{i, {\bm x}_i \neq 0} \frac{| \tilde{D} \tilde{C} {\bm x}_i - {\bm x}_i |} {|{\bm x}_i|} \leq K_{{\bm e}ta} {\bm l}ack{2^{e_{max}-e_{min}}}$.
\end{itemize}
As observed in Section~\ref{sec:analysis}, Steps 2, 3, and 8 of ZFP are
the sources of round-off error. So, in constructing our numerical tests,
we considered what conditions will vary the round-off error at these
steps. Information is only lost in Step 2, if, when the block is
converted from floating-point to block-floating-point, the exponent
range of components in the block is above some threshold, as can be seen
in equation (\ref{eqn:componentwiseerror}).
At Step 3, a linear transform is applied to the block, and information
is lost whenever round-off occurs. At Step 8, if the number of
compressed bit-planes does not coincide with the negabinary precision,
we will again lose information. Since varying the exponent range of the
input is an easy parameter to control, the value $e_{\max} - e_{\min}$
is used as a parameter in many of the numerical tests. Additionally, we
chose to vary the number of bit planes kept in Step 8, denoted ${\bm e}ta$,
in the following numerical experiments.
The first numerical experiment is designed to test how well the bound
established in Section~\ref{sec:bounds} captures the round-off error
introduced by ZFP as the range of exponents and the number of bit-planes
varies within a single block with dimension $d$. While the first test
works on data generated to demonstrate the worst-case behavior, the
second experiment shows the behavior for a data set taken from an actual
physical simulation.
\subsection{Generated $4^d$ Block}
\label{sec:generatedBlock}
In the first numerical test, a 4 by 4 block was formed with absolute
values ranging from $2^{e_{min}}$ to $2^{e_{max}}$. The exponent
$e_{min}$ remains stationary while $e_{max}$ varies, depending on the
chosen exponent range. The interval $[e_{min}, e_{max}]$ was divided
into 16 evenly spaced subintervals. Each value of the block was randomly
selected from a uniform distribution in the range
$[2^{e_{min}+(h-1)\frac{e_{max}-e_{min}}{16}},
2^{e_{min}+h\frac{e_{max}-e_{min}}{16}}]$ with {sub}interval
$h\in \{1,\dots,16 \}$ and uniform randomly assigned sign. The block was
then randomly permuted, using the C$++$ standard library function
{\it{random\_shuffle}}, to remove any bias in the total sequency order
and then compressed and decompressed with precision, ${\bm e}ta$. This
specific construction of data is designed to {mimic} the worst
possible input for ZFP for a chosen exponent range. For a given exponent
range, $\rho = e_{max}-e_{min} $, we expect the componentwise relative
error and block relative error to increase as the precision
decreases. However, {as the bound is only dependent on ${\bm e}ta$,}
the block relative error bound will remain constant as the exponent
range varies. The componentwise relative error should increase as the
exponent range increases as the representable numbers in a
block-floating-point representation are dependent on the largest
exponent of the block and the value of $q$.
For
Figures~\ref{fig:componenterror_precision_4by4}-\ref{fig:normerror_exp_4by4},
the data is represented and compressed in single-precision (32-bit IEEE
standard) with $e_{min} = 0$, while $e_{max}$ varies with respect to
the required exponent range. Similar results {can be produced} for
any value of $e_{min}$. Figure~\ref{fig:componenterror_precision_4by4}
shows how the {componentwise relative error} (top) and block relative
error (bottom) vary with respect to the fixed precision parameter,
${\bm e}ta$, for a fixed exponent range. For a single ${\bm e}ta$, {one}
decompressed. The blue band represents the sampled maximum and minimum
of the true componentwise relative error or block relative error, i.e.,
{\bm e}gin{align*}
\max_{i, {\bm x}_i \neq 0} \frac{|\tilde{D}\tilde{C}{\bm x}_i -{\bm x}_i|} {|{\bm x}_i|} \quad {\text{and}} \quad \frac{\|\tilde{D}\tilde{C}{\bm x} -{\bm x}\|_\infty}{\|{\bm x}\|_\infty},
\end{align*}
respectively, of all 1 million runs. The red line depicts the
theoretical bound and the dashed green line represents the asymptotic
behavior of the bound, i.e., the smallest predictive value of the
theoretical bound. For $ e_{max} - e_{min} = 0$, meaning that the
magnitude of the absolute values of the 4 by 4 block are similar, the
componentwise relative error increases as ${\bm e}ta$ decreases. As the
exponent range increases, {fewer bits will be
used to represent the smaller values in each block during Step 2 of
ZFP, which will result in a larger relative error}. As anticipated,
{in Figure \ref{fig:componenterror_precision_4by4}} the entire plot
{shifts} toward the upper right corner as the exponent range
increases, indicating that the componentwise relative error increases
with respect to the range of compressed values. With respect to the
block relative error, the block relative error remains the same for all
${\bm e}ta$ as the exponent range varies.
Similar trends can also be seen in Figure~\ref{fig:normerror_exp_4by4},
where the exponent range varies for a single ${\bm e}ta$. In
Figure~\ref{fig:normerror_exp_4by4}, for each ${\bm e}ta$, the componentwise
relative error (top) increases as the exponent range increases while the
block relative error remains constant, as expected. As ${\bm e}ta$ increases,
both the componentwise relative error and the block relative error plots
are shifted upwards, indicating an increase in error. For ${\bm e}ta = 32$, there is a gap between the bound and the observed error, which corresponds to the gap in the far right of the plots in Figure~\ref{fig:componenterror_precision_4by4}, i.e., the theoretical error bound is limited to precision of the IEEE representation. It can be concluded, in
Figures~\ref{fig:componenterror_precision_4by4} and
\ref{fig:normerror_exp_4by4}, that the theoretical bound (red) completely
bounds the maximum sampled error (blue).
Next, we repeated this same experiment using a different machine
precision. Note that Figures~\ref{fig:2d_double_1} and
\ref{fig:2d_double_4} represent the same two-dimensional test outlined
above, but the values in the block are represented using double-precision
(64-bit IEEE standard). The same relationships can be concluded for the
double precision case.
Finally, since the dimensionality of the block plays an important part
in ZFP, Figures~\ref{fig:1d_double_1} and \ref{fig:1d_double_4} and
Figures~\ref{fig:3d_double_1} and \ref{fig:3d_double_4} represent
results in double precision for one-dimensional and three-dimensional
blocks, respectively. Again, similar relationships can be seen as those
outlined in the two-dimensional, single-precision experiment.
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_single0.pdf}
\label{fig:componenterror_precision_4by4parta}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_single7.pdf}
\label{fig:componenterror_precision_4by4partb}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_single14.pdf}
\label{fig:componenterror_precision_4by4partc}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_single0.pdf}
\label{fig:normerror_precision_4by4parta}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_single7.pdf}
\label{fig:normerror_precision_4by4partb}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_single14.pdf}
\label{fig:normerror_precision_4by4partc}
\end{subfigure}
\caption{2-d Example with single precision: componentwise relative error (top) and block relative error (bottom) with respect to {the precision parameter (${\bm e}ta$) for $ e_{max}-e_{min} \in \{ 0,7,14\} $. The blue band represents the sampled maximum and minimum error, the red line depicts the theoretical bound, and the dashed green line represents the asymptotic behavior of the theoretical bound. } }
\label{fig:componenterror_precision_4by4}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_single12.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_single22.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_single32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_single12.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_single22.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_single32.pdf}
\end{subfigure}
\caption{2-d Example with single precision: componentwise relative error (top) and block relative error (bottom) with respect to the difference in exponents ($e_{max}-e_{min}$) for ${\bm e}ta \in \{12,22,32\}${. The blue band represents the sampled maximum and minimum error and the red line depicts the theoretical bound. }}
\label{fig:normerror_exp_4by4}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_2d_double14.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_2d_double14.pdf}
\end{subfigure}
\caption{2-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the precision parameter (${\bm e}ta$) for $ e_{max}-e_{min} \in \{0,7,14\}${. The blue band represents the sampled maximum and minimum error, the red line depicts the theoretical bound, and the dashed green line represents the asymptotic behavior of the theoretical bound. }}
\label{fig:2d_double_1}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_2d_double64.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_2d_double64.pdf}
\end{subfigure}
\caption{2-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the difference in exponents ($e_{max}-e_{min}$) for ${\bm e}ta \in \{32,48,64\}${. The blue band represents the sampled maximum and minimum error and the red line depicts the theoretical bound.}}
\label{fig:2d_double_4}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_1d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_1d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_1d_double14.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_1d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_1d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_1d_double14.pdf}
\end{subfigure}
\caption{1-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the precision parameter (${\bm e}ta$) for $ e_{max}-e_{min} \in\{ 0,7,14\}${. The blue band represents the sampled maximum and minimum error, the red line depicts the theoretical bound, and the dashed green line represents the asymptotic behavior of the theoretical bound. }}
\label{fig:1d_double_1}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_1d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_1d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_1d_double64.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_1d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_1d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_1d_double64.pdf}
\end{subfigure}
\caption{1-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the difference in exponents ($e_{max}-e_{min}$) for ${\bm e}ta \in \{32,48,64\}${. The blue band represents the sampled maximum and minimum error and the red line depicts the theoretical bound. }}
\label{fig:1d_double_4}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_3d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_3d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_rel_3d_double14.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_3d_double0.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_3d_double7.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/expadd_norm_3d_double14.pdf}
\end{subfigure}
\caption{3-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the precision parameter (${\bm e}ta$) for $ e_{max}-e_{min} \in \{0,7,14\}${. The blue band represents the sampled maximum and minimum of the error, the red line depicts the theoretical bound, and the dashed green line represents the asymptotic behavior of the theoretical bound. }}
\label{fig:3d_double_1}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_3d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_3d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_rel_3d_double64.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_3d_double32.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_3d_double48.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/precision_norm_3d_double64.pdf}
\end{subfigure}
\caption{3-d Example with double precision: componentwise relative error (top) and block relative error (bottom) with respect to the difference in exponents ($e_{max}-e_{min}$) for ${\bm e}ta = \{32,48,64\} ${. The blue band represents the sampled maximum and minimum of the error and the red line depicts the theoretical bound. }}
\label{fig:3d_double_4}
\end{figure}
\subsection{Real-World Example}
\label{ex:RealWorld}
For this example, we compress data from a real-world three dimensional
viscosity and density field from
a Rayleigh-Taylor instability simulation produced by Miranda
\cite{viscosity}. For the viscosity field, the average exponent range over all blocks is
approximately 7.32. This data set is a highly variable example, as the viscosity
values are signed and have a high dynamic range. The density field has a much smaller dynamic range than the viscosity field. That means the density field is a more compressible data set for ZFP. For both fields, the same value of
${\bm e}ta$ was used across all blocks during
compression to simplify the visualization of the results. In Figure
\ref{fig:viscosity} and \ref{fig:density}, the block relative error is plotted after the data
has been compressed and decompressed (blue) as a function of
${\bm e}ta$. The theoretical bound is plotted in red. Again, we conclude
that the theoretical bound completely bounds the true error for both examples. As for the compression ratio, since the density field has a smaller dynamic range, there is a substantial increase in the compression ratio for every bit plane removed, especially compared to the viscosity field. It can be concluded, that for some error tolerance, ZFP compresses at a higher ratio for data that is ``smooth," i.e., the exponent range for each block is small.
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/viscocity.png}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/viscocity_error_norm.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/viscocity_error_normratio.pdf}
\end{subfigure}
\caption{3-d viscosity field example with double precision. On the left is a 3-d rendering of the viscosity field. In the middle is the maximum block relative error as a function of the precision parameter ${\bm e}ta$. The blue line is the error from ZFP compression and decompression with fixed ${\bm e}ta$ and the red line depicts the theoretical bound. On the right is the compression ratio as a function of ${\bm e}ta$. }
\label{fig:viscosity}
\end{figure}
{\bm e}gin{figure}
\centering
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/miranda-density.png}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/density_error_norm.pdf}
\end{subfigure}
{\bm e}gin{subfigure}[b]{0.32\textwidth}
\includegraphics[width=\textwidth]{figures/density_error_normratio.pdf}
\end{subfigure}
\caption{3-d density field example with double precision. On the left is a 3-d rendering of the density field. In the middle is the maximum block relative error as a function of the precision parameter ${\bm e}ta$. The blue line is the error from ZFP compression and decompression with fixed ${\bm e}ta$ and the red line depicts the theoretical bound. On the right is the compression ratio as a function of ${\bm e}ta$. }
\label{fig:density}
\end{figure}
\section{Conclusion}
In this paper, we addressed the error introduced in the use of lossy
compression of floating-point data. An important contribution of this paper is the formulation of the problem in
a way that simplifies analysis.
The vector space, $\mathcal{B}^n$, introduced in
Section~\ref{sec:bitvectorSpace}, proved to be useful in developing
operators that accurately represent each step of the ZFP compression
algorithm. Section~\ref{sec:bounds} presented the error analysis of the
current implementation of the fixed precision mode of ZFP and the
numerical tests presented in Section~\ref{sec:results} provided a
demonstration that the theoretical bounds established in this paper
capture the error introduced by ZFP. The techniques presented in
Section~\ref{IntroSec:seqspace} and methodology from Sections~\ref{sec:analysis}
and \ref{sec:bounds} could be applied to any compression algorithm or
numerical method involving direct manipulation of bits.
In the majority of mesh-based PDE simulations, it is reasonable to
assume that most of the blocks
provided as an input to the ZFP compression algorithm will be ``smooth"
in the sense that the exponent range, $e_{max}-e_{min}$, will be small
and there will be some natural ordering correlation between the values
within the block. As Theorem \ref{thm:diffDCandDC} represents the worst
possible error achieved, the tests presented in Section
\ref{sec:generatedBlock} were constructed to provide an exposition of
the worst case scenario (i.e., not ``smooth'') of the error introduced
by ZFP compression and decompression. Even so, the error bound
established in Theorem \ref{thm:diffDCandDC} accurately, and narrowly,
bounded the error in each example.
We limited our detailed analysis to the fixed precision implementation of ZFP,
which is one of three possible compression modes implemented by ZFP. Using the bound in Theorem~\ref{thm:diffDCandDC}, we were able to provide a similar bound for the fixed accuracy and fixed rate modes. Further research is needed to determine a tighter error bound for the fixed
rate mode for real-world data as most real-world problems will produce ``smooth" values and the bound in Theorem~~\ref{thm:diffDCandDCFixedRate} is for the worst case scenario.
Additionally, given the trends in computing
hardware, methods are needed to reduce the memory capacity and bandwidth
demands in simulation codes. {\bm l}ack{One common technique is to use mixed precision algorithms, which typically require changing the underlying algorithms to achieve the same purpose. However, one technique with promise, particularly for grid-based PDE methods, is the use of lossy floating-point compression. In \cite{zfp-doc}, the C++ compressed array primitives handle the complexity of decompression, caching, and compression transparently. By using ZFP instead, we can achieve bandwidth reduction without changing the underlying structure of the algorithm. }
An extension of our work would include an error analysis of the propagation of the errors of storing the solution
state in compressed format for an iterative method, i.e., repeatedly decompressing and
recompressing the solution data at each time step or iteration of the
numerical algorithm.
Lastly, it should be noted that this paper has
analyzed the error of converting an IEEE representation to ZFP. However,
ZFP can be seen as a number representation itself, just like
IEEE. Another interesting direction for future work would be to consider
the behavior of round-off error of floating-point arithmetic conducted directly on
the ZFP format.
\appendix
\section{ZFP Toy Example}
\label{sec:appendixa}
Here we include a toy example of ZFP compression and decompression as implemeted by the operators defined in Section~\ref{sec:analysis}. As the embedded coding implemented by ZFP in Step 7 is nontrivial and lossless we exclude it from the following example. More details on Step 7 can be found in the \emph{Algorithm} section of \cite{zfp-doc}. As our analysis is focused on the fixed precision mode of ZFP and it is the simplest mode to illustrate, we only present the output for fixed precision mode at Step 8. Note that we will write $x^{(i)}$ to denote the output from step $i$ of ZFP.
\noindent \textit{Compression}: We first outline the steps for compression on $x = [ 5632, \ 3072, \ 400, \ 68 ]^T \in \mathbb{R}^4$. \\
\noindent \textit{Step 1}: As $d = 1$ and the vector is already in $\mathbb{R}^4$ there is no partitioning to be done. \\
\noindent \textit{Step 2}: For simplicity, we will use $k = 13$ and $q = 9$. First, as outlined in Section \ref{Step2Sec}, the components are converted to a bit representation in $\mathcal{B}$. Next, we apply the shift operator $S_{\ell}$ with $\ell = e_{\max} (x) - q + 1= 12 - 9 + 1 = 4$. This operation amounts to shifting each bit four positions to the right. Finally, we apply the truncation operator $T_{\mathcal{S}}$, where $\mathcal{S} = \{ i \in \mathbb{Z} : i \geq 0 \}$. Hence, any bits after the decimal are dropped which will result in the loss of some information in this example. This procedure is illustrated below in (A.1). Note that the representations in (A.1) include the guard bit so they will have $q + 1 = 10$ bits in their representation after the truncation phase. Additionally, there is one bit allotted for the sign bit which will not be represented below.
{\bm e}gin{align}
{\bm e}gin{array}{c<{\hspace{-1.3mm}} c<{\hspace{-0.5mm}} c<{\hspace{-0.5mm}} c<{\hspace{-0.5mm}} c<{\hspace{-0.5mm}} c<{\hspace{-0.5mm}} c}
{\bm e}gin{bmatrix}
5632 \\
3072 \\
400 \\
68 \end{bmatrix} &\longrightarrow &{\bm e}gin{bmatrix}
01011000000000 \\
00110000000000 \\
00000110010000 \\
00000001000100 \end{bmatrix} &\longrightarrow &{\bm e}gin{bmatrix}
0101100000.0000 \\
0011000000.0000 \\
0000011001.0000 \\
0000000100.0100 \end{bmatrix} &\longrightarrow &{\bm e}gin{bmatrix}
0101100000 \\
0011000000 \\
0000011001 \\
0000000100 \end{bmatrix}
\\
\text{Decimal} & & \text{Signed Binary} & & \text{Bit Shift} & & x^{(2)}
\end{array} \tag{A.1}
\end{align}
In addition to the bits used to store $x^{(2)}$, note that ZFP also encodes the value $e_{\max} (x) = 12$. \\
\noindent \textit{Step 3}: Using Table~\ref{table:actionTlossy} from Section \ref{Step3Sec}, with ${\bm a}_1 = 0101100000$, ${\bm a}_2 = 0011000000$, ${\bm a}_3 = 0000011001$, and ${\bm a}_4 = 0000000100$, we can compute $x^{(3)}$. This process yields the vector
{\bm e}gin{align*}
x^{(3)} = {\bm e}gin{bmatrix*}[r]
0010001111 \\
0001111000 \\
-0000100011 \\
0000010011
\end{bmatrix*}
\end{align*}
and is outlined step by step below (steps work from left to right starting with upper left entry):
{\bm e}gin{center}
{\bm e}gin{adjustbox}{width=0.9\textwidth}
{\bm e}gin{tabular}{ | l<{\hspace{-3mm}} l<{\hspace{-2mm}} r<{\hspace{2mm}} l<{\hspace{-3mm}} l<{\hspace{-2mm}} r<{\hspace{2mm}} l<{\hspace{-3mm}} l<{\hspace{-2mm}} r | }
\hline
\multicolumn{9}{| c |}{\vspace*{-4mm}} \\
\multicolumn{9}{| c |}{$\tilde{L}$} \\ \hline \hline
\vspace*{-4mm} & & & & & & & & \\
${\bm a}_1 \leftarrow {\bm a}_1 + {\bm a}_4 $ &$=$ &$0101100100$, & ${\bm a}_1 \leftarrow r({\bm a}_1) $ &$=$ &$0010110010$, & ${\bm a}_4 \leftarrow {\bm a}_4 - {\bm a}_1 $ &$=$ &$-0010101110$, \\ \hline
${\bm a}_3 \leftarrow {\bm a}_3 + {\bm a}_2 $ &$=$ &$0011011001$, & ${\bm a}_3 \leftarrow r({\bm a}_3) $ &$=$ &$0001101100$, & ${\bm a}_2 \leftarrow {\bm a}_2 - {\bm a}_3 $ &$=$ &$0001010100$, \\ \hline
${\bm a}_1 \leftarrow {\bm a}_1 + {\bm a}_3 $ &$=$ &$0100011110$, & ${\bm a}_1 \leftarrow r({\bm a}_1) $ &$=$ &$0010001111$, & ${\bm a}_3 \leftarrow {\bm a}_3 - {\bm a}_1 $ &$=$ &$-0000100011$, \\ \hline
${\bm a}_4 \leftarrow {\bm a}_4 + {\bm a}_2 $ &$=$ &$-0001011010$, & ${\bm a}_4 \leftarrow r({\bm a}_4) $ &$=$ &$-0000101101$, & ${\bm a}_2 \leftarrow {\bm a}_2 - {\bm a}_4 $ &$=$ &$0010000001$, \\ \hline
${\bm a}_4 \leftarrow {\bm a}_4 + r({\bm a}_2) $ &$=$ &$0000010011$, & ${\bm a}_2 \leftarrow {\bm a}_2 - r({\bm a}_4) $ &$=$ &$0001111000$. & & & \\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
\noindent \textit{Step 4}: Note that the components of a vector in $\mathbb{R}^4$ are already in total sequency order. \\
\noindent \textit{Step 5}: The components are converted to a negabinary representation. Note that $q+2 = 11$ bits are available for the negabinary representation as it does not require a dedicated sign bit. Hence,
{\bm e}gin{align*}
{\bm e}gin{array}{c<{\hspace{-2mm}} c<{\hspace{-4mm}} c}
x^{(5)} = &{\bm e}gin{bmatrix}
00110010011 \\
00110001000 \\
00000101101 \\
00000010111
\end{bmatrix} &.
\\
&\text{Negabinary} &
\end{array}
\end{align*}
\noindent \textit{Step 6}: The transposition is performed so that the first row corresponds to the most significant bit while the last row corresponds to the least significant bit. Hence $x^{(6)} = (x^{(5)})^T$.
\iffalse
{\bm e}gin{align*}
x^{(6)} = {\bm e}gin{bmatrix}
0000 \\
0000 \\
1100 \\
1100 \\
0000 \\
0010 \\
1001 \\
0110 \\
0011 \\
1001 \\
1011
\end{bmatrix}
\end{align*}
\fi
Note that the terminology \emph{bit plane} used in the discussion of ZFP can be realized as the rows of bits in $x^{(6)}$ or the columns of bits of $x^{(5)}$. For example, the third bit plane of $x^{(6)}$ is $1100$ and the sixth bit plane of $x^{(6)}$ is $0010$. \\
\noindent \textit{Step 8}: Recall that we have decided to exclude Step 7 from this example for simplicity. In Step 8, the user provides a fixed number of bit planes, ${\bm e}ta$, to keep from $x^{(6)}$ starting with the most significant bit plane. The remaining bit planes are then ordered in a sequence as the rows of $x^{(6)}$. Supposing the number of bit planes to keep is ${\bm e}ta = 7$, the output would be $x^{(8)} = 0000 \ 0000 \ 1100 \ 1100 \ 0000 \ 0010 \ 1001$.
\iffalse
{\bm e}gin{align*}
{\bm e}gin{array}{rcl}
x^{(8)} = \hspace*{-2.3mm} & 0000 \ 0000 \ 1100 \ 1100 \ 0000 \ 0010 \ 1011 & \hspace*{-3mm} \text{.}
\\
& \text{Bit Sequence} &
\end{array}
\end{align*}
\fi
\iffalse
{\bm e}gin{align*}
{\bm e}gin{array}{rcl}
x^{(8)} = \hspace*{-4mm} &{\bm e}gin{bmatrix}
0011001 \\
0011000 \\
0000010 \\
0000001
\end{bmatrix} & \hspace*{-5mm} \text{.}
\\
& \text{Bit Sequence} &
\end{array}
\end{align*}
\fi
It should be noted that $x^{(8)}$ is close to the compressed bit stream format of ZFP but is not in the exact compressed format since we did not perform Step 7. Note that if Step 7 had been performed then the final compressed bit sequence would have been $0 \ 0 \ 11110 \ 110 \ 000 \ 00110 \ 1011$, requiring 6 fewer bits. This completes the steps for compression. \\
\noindent \textit{Decompression}: We now highlight the steps for decompression. To decompress, we first convert the bit sequence back into a vector format and transpose. We then place zeros to the end of each row until we have the same number of bits before we dropped bit planes (in this case four zeros per row). This vector will be very similar to $x^{(5)}$, however, due to the removal of bit planes in Step 8 some information was lost that cannot be restored. Next, we convert each negabinary representation to a signed binary representation. These steps are illustrated in (A.2).
{\bm e}gin{align}
{\bm e}gin{array}{ccccc}
{\bm e}gin{bmatrix}
0011001 \\
0011000 \\
0000010 \\
0000001
\end{bmatrix} &\longrightarrow &{\bm e}gin{bmatrix}
00110010000 \\
00110000000 \\
00000100000 \\
00000010000
\end{bmatrix} &\longrightarrow &{\bm e}gin{bmatrix*}[r]
0010010000 \\
0010000000 \\
-0000100000 \\
0000010000
\end{bmatrix*}
\\
& & \text{Negabinary} & & \text{Signed Binary}
\end{array}
\tag{A.2}
\end{align}
We now use the final vector from (A.2) as the input for the routine outlined in Table~\ref{table:actionTlossy} in Section \ref{Step3Sec}.
{\bm e}gin{center}
{\bm e}gin{adjustbox}{width=0.9\textwidth}
{\bm e}gin{tabular}{ | l<{\hspace{-3mm}} l<{\hspace{-2mm}} r<{\hspace{2mm}} l<{\hspace{-3mm}} l<{\hspace{-2mm}} r<{\hspace{2mm}} l<{\hspace{-3mm}} l<{\hspace{-2mm}} r | }
\hline
\multicolumn{9}{| c |}{\vspace*{-4mm}} \\
\multicolumn{9}{| c |}{$\tilde{L}^{-1}$} \\ \hline \hline
\vspace*{-4mm} & & & & & & & & \\
${\bm a}_2 \leftarrow {\bm a}_2 + r({\bm a}_4) $ &$=$ &$0010001000$, & ${\bm a}_4 \leftarrow {\bm a}_4 - r({\bm a}_2) $ &$=$ &$-0000110100$, & & & \\ \hline
${\bm a}_2 \leftarrow {\bm a}_2 + {\bm a}_4 $ &$=$ &$0001010100$, & ${\bm a}_4 \leftarrow s_{-1}({\bm a}_4) $ &$=$ &$-0001101000$, & ${\bm a}_4 \leftarrow {\bm a}_4 - {\bm a}_2 $ &$=$ &$-0010111100$, \\ \hline
${\bm a}_3 \leftarrow {\bm a}_3 + {\bm a}_1 $ &$=$ &$0001110000$, & ${\bm a}_1 \leftarrow s_{-1}({\bm a}_1) $ &$=$ &$0100100000$, & ${\bm a}_1 \leftarrow {\bm a}_1 - {\bm a}_3 $ &$=$ &$0010110000$, \\ \hline
${\bm a}_2 \leftarrow {\bm a}_2 + {\bm a}_3 $ &$=$ &$0011000100$, & ${\bm a}_3 \leftarrow s_{-1}({\bm a}_3) $ &$=$ &$0011100000$, & ${\bm a}_3 \leftarrow {\bm a}_3 - {\bm a}_2 $ &$=$ &$0000011100$, \\ \hline
${\bm a}_4 \leftarrow {\bm a}_4 + {\bm a}_1 $ &$=$ &$-0000001100$, & ${\bm a}_1 \leftarrow s_{-1}({\bm a}_1) $ &$=$ &$0101100000$, & ${\bm a}_1 \leftarrow {\bm a}_1 - {\bm a}_4 $ &$=$ &$0101101100$. \\
\hline
\end{tabular}
\end{adjustbox}
\end{center}
Lastly, we perform a bit shift of four bits to the left to undo the shift performed during Step 2 and convert to decimal to yield the decompressed vector. This procedure is illustrated below in (A.3).
{\bm e}gin{align}
{\bm e}gin{array}{ccccc}
{\bm e}gin{bmatrix*}[r]
0101101100.0000 \\
0011000100.0000 \\
0000011100.0000 \\
-0000001100.0000
\end{bmatrix*} &\longrightarrow &{\bm e}gin{bmatrix*}[r]
01011011000000 \\
00110001000000 \\
00000111000000 \\
-00000011000000
\end{bmatrix*} &\longrightarrow &{\bm e}gin{bmatrix*}[r]
5824 \\
3136 \\
448 \\
-192
\end{bmatrix*}
\\
\text{Signed Binary} & & \text{Bit Shift} & & \text{Decimal}
\end{array}
\tag{A.3}
\end{align}
We conclude by comparing the error with the bound in Theorem~\ref{thm:diffDCandDC}. Since $d = 1$, $k = 13$, $q = 9$, and ${\bm e}ta = 7$ for this example, we have that $K_{{\bm e}ta} \approx 0.19831$. Since $K_{{\bm e}ta} \| x \|_\infty \leq (0.19832) (5632) \leq 1117$ and $\| \tilde{D}\tilde{C} x - x \|_\infty = 260$ we observe that the bound established in Theorem~\ref{thm:diffDCandDC} holds.
\section{Round-off Error of the Lossy Decorrelating Backwards Linear Transform}
\label{sec:appendixb}
There are three cases that must be considered: $(i)$ ${\bm e}ta = q+2,$ $(ii)$ $q- 2d +2<{\bm e}ta < q+2$, and $(iii)$ ${\bm e}ta \leq q- 2d+2$. We will investigate each separately in the following sections.
{\bm e}gin{itemize}
\item[$(i)$]
When ${\bm e}ta=q+2$, no information is lost at Step 8. Thus, from Table~\ref{table:actionTlossy} we observe that the last two steps for the forward transform are exactly reversed by the first two steps of the backwards transform,
$${\bm e}gin{array}{l<{\hspace{-4mm}} r l<{\hspace{-2mm}} l<{\hspace{-2mm}} l}
\tilde{L} &\text{Step 13}: & {\bm a}_4 \leftarrow {\bm a}_4 + r({\bm a}_2) & \Rightarrow {\bm a}_4 = {\bm a}_4 + r({\bm a}_2),&\\
\tilde{L} &\text{Step 14}: & {\bm a}_2 \leftarrow {\bm a}_2 - r({\bm a}_4) & \Rightarrow {\bm a}_2 ={\bm a}_2-r({\bm a}_4 + r({\bm a}_2)), &\\
\tilde{L}^{-1} &\text{Step 1}: & {\bm a}_2 \leftarrow {\bm a}_2 + r({\bm a}_4) & \Rightarrow {\bm a}_2 ={\bm a}_2-r({\bm a}_4 + r({\bm a}_2)) +r({\bm a}_4 + r({\bm a}_2) )&= {\bm a}_2,\\
\tilde{L}^{-1} &\text{Step 2}: & {\bm a}_4 \leftarrow {\bm a}_4- r({\bm a}_2) & \Rightarrow {\bm a}_4 = ({\bm a}_4 + r({\bm a}_2)) - r({\bm a}_2) &= {\bm a}_4. \\
\end{array}$$
Thus, no error occurs by applying $\tilde{L}_d^{-1}$.
\item[$(ii)$]
If $q- 2d +2<{\bm e}ta < q+2 $, then using similar techniques as for the forward linear transform operator, a bound can be found for the lossy backwards linear transform operator.
{\bm e}gin{lemma}
\label{LemmaLingBound}
Suppose ${\bm x} \in \mathbb{Z}^4$ such that $e_{max}({\bm x}) = q - 1$, ${\bm x} \neq {\bm 0}$ and $q- 2d +2<{\bm e}ta < q+2 $. Given the bit arithmetic implementation in Table \ref{table:actionT} and Table \ref{table:actionTlossy} for ZFP's backwards linear transforms, we have
{\bm e}gin{align*}
\|L^{-1} {\bm x} - \tilde{L}^{-1} {\bm x}\|_\infty \leq \frac{5}{2} \epsilon_q \|{\bm x}\|_\infty \quad and \quad \|L^{-1}_d{\bm x} - \tilde{L}^{-1}_d {\bm x} \|_\infty & \leq k_{L^{-1}}\epsilon_q \|{\bm x}\|_\infty,
\end{align*}
where $k_{L^{-1}} = \frac{5}{2} \left( 2^d - 1 \right)$.
\end{lemma}
{\bm e}gin{proof}
Use outline of proof from Lemma \ref{LemmaL1DBound} and \ref{lemma:boundT}.
\end{proof}
\item[$(iii)$]
If ${\bm e}ta \leq q - 2d+2$, then the rightmost $2d$ least significant bits of ${\bm a}_i$ are zero, for all $i$, resulting in the following equivalences for the first two steps from Table~\ref{table:actionTlossy}: $r({\bm a}_4) = s_{-1}({\bm a}_4)$, $ r({\bm a}_2) = s_{-1}({\bm a}_2)$, and $r({\bm a}_4 + r({\bm a}_2)) = s_{-1}({\bm a}_4 + s_{-1}({\bm a}_2)) $. Thus, the lossy backwards transform operator is exactly the lossless version, resulting in no additional error.
\end{itemize}
If $q- 2d +2<{\bm e}ta < q+2 $, only a modest reduction of the data will be achieved over using ${\bm e}ta = q- 2d+2$. Thus, for the analysis of this paper, we chose to assume ${\bm e}ta \leq q - 2d+2$ so that
\[ \tilde{D}_3({\bm a}) = {D}_3({\bm a})= F_\mathcal{B}^{-1}{L}_d^{-1}F_\mathcal{B}({\bm a}), \text{ for all } \bm{a} \in \mathcal{B}^{4^d}. \]
Note, Theorem \ref{thm:diffDCandDC} can be modified to accommodate the error that occurs from the lossy backwards decorrelating operator.
{\bm e}gin{theorem}\label{thm:diffDCandDCappendix}
Assume ${\bm x} \in \mathbb{R}^{4^d}$ with $\bm{x} \neq \bm{0}$ such that $F_{\mathcal{B}} ({\bm x}) \in \mathcal{B}_k^{4^d}$, for some precision $k$. Let $q- 2d +2<{\bm e}ta < q+2 $ be the fixed precision parameter. Then
{\bm e}gin{align}
\| \tilde{D}\tilde{C} {\bm x} - {\bm x} \|_\infty &\leq B_{\bm e}ta \|{\bm x}\|_\infty
\end{align}
where $q \in \mathbb{N}$ is the precision for the block-floating point representation in Step 2,
{\bm e}gin{align}
B_{\bm e}ta :=k_{L^{-1}} \epsilon_q(1+\epsilon_{k})\left( \frac{8}{3} \epsilon_{\bm e}ta+ \epsilon_q \left(1+ \frac{8}{3} \epsilon_{\bm e}ta \right) \left(k_L(1+\epsilon_q)+1 \right )\right) +K_{\bm e}ta,
\end{align}
$k_L= \frac{7}{4} (2^d-1)$, and $k_{L^{-1}} = \frac{5}{2} \left( 2^d - 1 \right)$.
\end{theorem}{\bm e}gin{proof}
Use outline of proof from Theorem \ref{thm:diffDCandDC} along with Lemma \ref{LemmaLingBound}.
\end{proof}
\section{Discussion of error bound constant $K_{{\bm e}ta}$}
\label{sec:appendixc}
First, note that $K_{{\bm e}ta}$ is a function of $q$, $k$, $d$, and ${\bm e}ta$. Since $k$ depends on the precision of the data provided to ZFP and $d$ is the dimension of the input data it is important to note that two of the variables used in computing $K_{{\bm e}ta}$ are dependent on the input data and cannot be changed by the user in ZFP. The value of $q$ depends on the precision of data and is set to a value larger than $k$. For example, if the input values are IEEE single or double precision, $q \in \{30, 62\}$, since one bit is used to represent the sign bit and another to represent the overflow guard bit, as discussed in Section \ref{Step2Sec}. The remaining variable, ${\bm e}ta$, can be set to any positive integer when using the fixed precision mode of ZFP, as noted in Section \ref{Step8Sec}. Figure~\ref{fig:KBetaPlot} helps illustrate how $K_{{\bm e}ta}$ varies with respect to ${\bm e}ta$ and the dimensionality of the data, $d$. The lines on the contour plot in Figure~\ref{fig:KBetaPlot} represent the log base 10 value of $K_{{\bm e}ta}$, i.e. $\log_{10} (K_{\bm e}ta)$. As suspected from the formula for $K_{{\bm e}ta}$, we observe that a larger value of $d$ has a greater effect on the value of $K_{{\bm e}ta}$ for small values of ${\bm e}ta$.
{\bm e}gin{figure}[h!]
\centering
\includegraphics[width=.45\linewidth]{figures/contour_kbeta.pdf}
\caption{Contour plot of $\log_{10} (K_{\bm e}ta)$ for ${\bm e}ta \in [1, 64]$ and dimension $d \in [1, 5]$ with $k = 53$ and $q = 62$.}
\label{fig:KBetaPlot}
\end{figure}
This work was funded by LLNL Laboratory Directed Research and
Development as Project 17-SI-004: \emph{Variable Precision Computing}
and was performed under the auspices of the U.S. Department of Energy by
Lawrence Livermore National Laboratory under Contract
DE-AC52-07NA27344.
{\bm i}bliographystyle{siamplain}
{\bm i}bliography{compression}
\end{document} | math |
আবারও করোনার বৃদ্ধির কারণে স্কুলের পাঠ বন্ধের জল্পনার মধ্যেই সরকারের সিদ্ধান্ত জানালেন শিক্ষামন্ত্রী টিডিএন বাংলা ডেস্ক: কিছু দিনের জন্য করোনার প্রকোপ কিছুটা কমলেও আবারও করোনার প্রভাব বৃদ্ধি পাচ্ছে করোনার বৃদ্ধির কারণে স্কুলের পাঠ আবারও কি বন্ধ রাখা হবে এমন জল্পনা শুরু হয়ে ছিল কিন্তু সেই জল্পনার মধ্যেই সরকারের সিদ্ধান্ত জানিয়ে দিলেন শিক্ষামন্ত্রী ব্রাত্য বসু এদিন এই সংক্রান্ত প্রশ্নের উত্তরে তিনি জানান এখনই স্কুলের পাঠ বন্ধের কোনো চিন্তা করছে না শিক্ষাদপ্তর স্কুলে ছাত্রছাত্রীদের জন্য আলাদা ভাবে কোনো বিধিনিষেধ বা নির্দেশিকাও দেওয়ার পরিকল্পনা নেই তাদের স্বাস্থ্য দফতর কোনও নির্দেশিকা দিলে তবেই এ নিয়ে ভাবনা চিন্তা করা হবে বলেও তিনি জানিয়েছেন এদিন শিক্ষামন্ত্রী ব্রাত্য বসু বলেন, স্কুলে আলাদা ভাবে কোনও বিধিনিষেধ নিয়ে স্বাস্থ্য দফতর থেকে এখনও কোনও নির্দেশিকা পাইনি আমরা স্কুলশিক্ষা দফতর নিজেরা কোনও সিদ্ধান্ত নিতে পারে না স্বাস্থ্য দফতর যদি কোনও নির্দেশিকা দেয়, সেই অনুযায়ী ব্যবস্থা নেওয়া হবে প্রসঙ্গত, করোনার কারণে প্রায় দুবছর স্কুলকলেজের পঠন পাঠন বন্ধ রাখা হয়ে ছিল যার ফলে ছাত্রছাত্রীদের পড়াশোনার ক্ষেত্রে ব্যাপক ক্ষতি হয়েছে যা আর পূরণ করা সম্ভব নয় এমতাবস্থায় আবারও যেন স্কুলকলেজের পঠন পাঠন বন্ধ রাখা না হয় সেই আবেদন জানিয়েছেন শিক্ষার্থী ও অবিভাবকরা | bengali |
నిహారిక భర్త అర్ధరాత్రి న్యూసెన్స్.. అపార్ట్మెంట్ వాసుల ఫిర్యాదు హైదరాబాద్: అర్ధరాత్రి ఏమైందో ఏమో కానీ మెగా డాటర్ నిహారిక అపార్ట్మెంటులో రచ్చ రచ్చ అయింది. నిహారిక భర్త న్యూసెన్స్ చేస్తున్నాడంటూ ఒక్కసారిగా కలకలం రేగింది. అపార్ట్ మెంట్ వాసులకు, నిహారిక భర్త చైతన్య జొన్నలగడ్డకు మధ్య తీవ్ర స్థాయిలో గొడవ జరిగింది. దీంతో అపార్టుమెంటు వాసులంతా చైతన్యపై బంజారాహిల్స్ పోలీస్ స్టేషన్లో ఫిర్యాదు చేశారు. అపార్ట్ మెంట్ వాసులపై నిహారిక భర్త సైతం ఫిర్యాదు చేశారు. పోలీసులు పరస్పర ఫిర్యాదులను స్వీకరించి విచారణ నిర్వహిస్తున్నారు. | telegu |
રિલીઝના કલાકો પહેલા કાશ્મીર ફાઈલ્સમાં કાપકૂપનો આદેશ કરતી કોર્ટ : છતાં શહેરમાં પ્રદર્શિત શહીદ પતિના રોલ મામલે પત્નીની રજૂઆત પછી જમ્મુ કોર્ટે કાપકૂપ કરવા જણાવ્યું કાશ્મીરી પંડીતોના નર સંહાર પર બનેલી વિવેક અગ્નિહોત્રીની ફિલ્મ ધ કાશ્મીર ફાઈલ્સમાં શહિદ સ્કવોડ્રન લીડર રવિ ખન્નાના પત્ની શાલિની ખન્નાની અરજીના આધારે જમ્મુ જિલ્લા અદાલતે શહીદ રવિ ખન્નાની ભૂમિકાને સ્પર્શતા દ્રશ્યો ન બતાવવા આદેશ આપ્યો છે. આ ફિલ્મ આજે અન્ય શહેરોની સાથે રાજકોટમાં પણ રિલીઝ થઈ છે. બોલિવૂડ હંગામાના અહેવાલ મુજબ, શાલિનીએ દાવો કર્યો છે કે ફિલ્મમાં તેના પતિના રોલને ખોટી રીતે દર્શાવવાનો પ્રયાસ કરવામાં આવ્યો છે. શાલિનીએ તેને 4 માર્ચે ફિલ્મના સ્પેશિયલ સ્ક્રીનિંગમાં જોયો હતો. રિપોર્ટ અનુસાર, શાલિનીનું કહેવું છે કે, તેણે આ મુદ્દે ફિલ્મના મેકર્સ સાથે વાતચીત કરી હતી. જો કે, ફિલ્મ નિર્માતાઓએ તેમના વાંધાને અવગણીને તેમની વાત સાંભળી ન હતી. જ્યારે શાલિનીને નિર્માતાઓ તરફથી કોઈ જવાબ ન મળ્યો, ત્યારે તેણે આ મુદ્દાને ઉકેલવા માટે કોર્ટનો સહારો લીધો. | gujurati |
اسلام باد عید کے فوری بعد عوام کو عیدی ملنے کا امکان اوگرا نے پیٹرولیم مصنوعات کی قیمتوں میں کمی کی سمری وزارت پیٹرولیم کو بھجوادی اوگرا نے پیٹرولیم مصنوعات کی قیمتوں میں رد بدل کی سمری وزارت پیٹرولیم کو ارسال کردی جس میں پیٹرول کی قیمت میں روپے 30 پیسے فی لیٹر جبکہ ڈیزل کی قیمت میں روپے 70 پیسے فی لیٹر کمی کی سفارش کی گئی ہے اوگرا سمری کے مطابق مٹی کے تیل کی قیمت میں 11 روپے فی لیٹر اور لائٹ ڈیزل کی قیمت روپے فی لیٹر بڑھانے کی تجویز دی گئی ہے پیٹرولیم مصنوعات کی قیمتوں میں رد بدل کا فیصلہ کل کیا جائے گا جبکہ اس کا اطلاق یکم جولائی سے ہوگا سماء | urdu |
বে আইনি অর্থলগ্নি সংস্থার কর্তী সহ ১৪ জন গ্রেপ্তার কলকাতা তে নিউস ঘন্টায় ঘন্টায় ওয়েবডেস্ক : তিন বছরে টাকা দ্বিগুন করে দেওয়ার প্রতিশ্রুতি সঙ্গে দামি উপহার দেওয়ার প্রলোভন দেখিয়ে কোটি কোটি টাকা হাতানোর অভিযোগে লাল বাজারের গোয়েন্দা শাখা মূল অভিযুক্ত লিজা মুখোপাধ্যায় সহ ১৪ জন কে গ্রেপ্তার করেছে নাশিক থেকে ৬ জন কে এবং কলকাতা থেকে ৮ জন কে গ্রেপ্তার করা হয় পুলিশের ধারণা প্রতারিত অর্থের পরিমান ১০০০ কোটি টাকা ,অভিযোগ দায়ের করা হয়েছে বেনিয়া পুকুর ,এন্টালি এবং কোরেয়া থানা তে প্রতারক ঝাঁ চক চকে পার্লার ও এনজিও সংস্থা খুলে এই প্রতারণা করছিলো | bengali |
\begin{document}
\title{Heisenberg-limited Rabi spectroscopy}
\begin{abstract}
The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. In these Heisenberg-limited protocols the phase between two states in an entangled superposition evolves N-fold faster than in the uncorrelated case, where N for example can be the number of entangled atoms in a Greenberger-Horne-Zeilinger (GHZ) state. Here we propose and demonstrate the use of correlated spin-Hamiltonians for the realization of Heisenberg-limited Rabi-type spectroscopy. Rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator (LO), we probe the evolution of an, initially separable, two-atom register under an Ising spin-Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as compared with uncorrelated rotation. We implement this Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of $^{88}$Sr$^+$ using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal sub-spaces of the full Hilbert space, yielding Heisenberg-limited spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The potential improvement of clock stability due to the use of entangled states depends on the details of the method used and the dominating decoherence mechanism. The use of correlated spin-rotations can therefore potentially lead to new paths for clock stability improvement.
\end{abstract}
Different quantum technologies rely on entangled states as their primary resource. In quantum metrology it was shown that entangled states can be used to reduce the uncertainty in the spectroscopy of two-level systems (pseudo-spins). The use of spin-squeezed states was shown to reduce the spectroscopy uncertainty in atomic ensembles to below the Standard Quantum Limit (SQL) \cite{Wineland1994, Kasevich1997, Oberthaler2010, Polzik2010}. In the extreme case of fully entangled spins, it was shown that using Ramsey-like spectroscopy of an N-atom GHZ state, the phase of this state with respect to a LO evolves N-fold faster, leading to Heisenberg limited estimation of the transition frequency \cite{Bollinger1996, Wineland2004, Blatt2011}.
In Ramsey spectroscopy the phase of a superposition in free-evolution is compared to a LO. An alternative to Ramsey spectroscopy is Rabi-type spectroscopy, in which the evolution of a state under a time-dependent Hamiltonian is investigated. Under the rotating-wave approximation the Rabi Hamiltonian, $H=\hbar\left[\Omega\sigma_{y} + \delta\sigma_{z}\right]$, generates spin rotations. Here, $\sigma_{i}$ are the Pauli spin operators, $\Omega$ is the Rabi frequency and $\delta$ is the detuning of the Rabi Hamiltonian from the atomic transition frequency. The initial state can be thought of as a superposition of the Rabi-Hamiltonian dressed-states \cite{CCT1992}. The $\delta=0$ point at which the Rabi Hamiltonian frequency is on-resonance is determined by the point at which spin rotation is maximal. The width of the Rabi spectrum is determined by the gap between the two dressed-states; namely the Rabi frequency $\Omega$.
One can therefore ask whether it is possible to generate Heisenberg-limited Rabi spectroscopy by acting on multi-spin registers with time-dependent Hamiltonians. The need for entanglement in Ramsey Heisenberg-limited spectroscopy suggests that the necessary time-dependent Hamiltonians are many-body interacting Hamiltonians. The simulation of many-body quantum spin-Hamiltonians is a field of growing experimental interest. The adiabatic evolution of ground-states as well as the dynamics of spin-defects under quenching were studied using these synthesized Hamiltonians \cite{Tobias2008, Monroe2011, Greiner2011, Blatt2014}. The application of correlated many-body Hamiltonians typically results in entanglement.
Heisenberg limited Ramsey spectroscopy investigates the free-evolution of superpositions in entangled subspaces. By the same token, Heisenberg limited Rabi spectroscopy can be engineered by investigating rotations of states in these entangled subspaces by many-body spin Hamiltonians \cite{OzeriQEC2013}. Similarly to single spin Rabi spectroscopy, the resonance frequency will be determined by the maximal rotation angle and the width of the resonance will be given by the gap between the two eigenstates of the spin-interaction Hamiltonian in this subspace.
In this work we show that an Ising spin-interaction Hamiltonian with a transverse field generates rotations in two orthogonal subspaces of a two-spin Hilbert space. In the spin symmetric subspace, spanned by $\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ and $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$ this Hamiltonian results in Heisenberg limited Rabi spectroscopy of the average spin transition frequency whereas in the anti-symmetric subspace spanned by $\left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ and $\left|\left\vert \uparrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$ Heisenberg limited Rabi spectroscopy of the spins frequency-difference from the mean transition frequency is performed. We implement this protocol using the optical clock electric-quadrupole transition in a two $^{88}$Sr$^+$ ion-crystal and show that the resulting correlated spin rotation spectra are indeed twice as narrow as compared with single ion Rabi spectra.
Heisenberg limited spectroscopy was shown to have limited value in the improvement of spectroscopic precision after long averaging times. The reason is that, as the sensitivity to the resonance frequency increases, so does the sensitivity to noise and dephasing rates increase \cite{Huelga1997}. In several theoretical investigations it was shown that an improvement of measurement precision or clock stability is possible, however it depends on the exact details of the noise and the spectroscopic method used \cite{Guta2012,Schmidt2017}. Hence, the development of new Heisenberg-limited spectroscopy techniques has the potential of introducing further clock stability improvement under different conditions.
We investigate a system of two interacting spins, under the influence of an Ising two-spin Hamiltonian with a transverse field,
\begin{equation}
\begin{split}
H=\hbar\left[\Omega\sigma_{y}\otimes\sigma_{y}+\delta_{1}\left(\sigma_{z}\otimes I+I\otimes\sigma_{z}\right)+\delta_{2}\left(\sigma_{z}\otimes I-I\otimes\sigma_{z}\right)\right].
\end{split}\label{Ising}
\end{equation}
Here the $\delta_{1}\left(\sigma_{z}\otimes I+I\otimes\sigma_{z}\right)$ term represent magnetic field along the z axis common to both spins and the term $\delta_{2}\left(\sigma_{z}\otimes I-I\otimes\sigma_{z}\right)$ represents the difference between the fields on each spin. The $ \Omega\sigma_{y}\otimes\sigma_{y}$ term is an Ising-type interaction which creates a correlated rotation of the two spins.\\
The Hamiltonian in Eq. \ref{Ising} commutes with $\sigma_{z}\otimes\sigma_{z}$ and therefore conserves state parity and does not mix between the even $\left\{ \left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle ,\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle \right\} $ and odd $\left\{ \left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle ,\left|\left\vert \uparrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle \right\}$ parity subspaces. In addition, the even and odd subspaces are degenerate under the operation of $\sigma_{z}\otimes I-I\otimes\sigma_{z}$ and $\sigma_{z}\otimes I+I\otimes\sigma_{z}$ respectively. As a result, superpositions of the states $\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle ,\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$ ($\left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle ,\left|\left\vert \uparrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$) are invariant to changes in $\delta_{2}$ ($\delta_{1}$).
The two subspaces above can be thought of as two super-spin-half metrological subspaces. As an example in the even subspace the two basis states of a super-spin-half system with $\left|\tilde{\left\vert \uparrow\right\ranglearrow}\right\rangle :=\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle ,\left|\tilde{\left\vert \downarrow\right\ranglearrow}\right\rangle :=\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle $. Ising spin coupling acts as a $y$ rotation in this subspace, $\tilde{\sigma}_{y}=\sigma_{y}\otimes\sigma_{y}$ and a $z$ rotation is generated by $\tilde{\sigma}_{z}=\delta_{1}\left(\sigma_{z}\otimes I+I\otimes\sigma_{z}\right)$. Rotations around $z$ in the anti-symmetric subspace are generated by $\delta_{2}\left(\sigma_{z}\otimes I-I\otimes\sigma_{z}\right)$. The Ising Hamiltonian in Eq. \ref{Ising} therefore performs Rabi spectroscopy in the two spin subspaces with a Rabi frequency $\Omega$ and a detuning $2\delta_1$ or $2\delta_2$ respectively. The factor of two in the detuning results in a two-fold narrowing of the Rabi resonance, leading to Heisenberg limited determination of the resonance frequency under spin projection noise. Notice that a general two-spin state is a direct sum of states in these two subspaces. A measurement will therefore lead to a single bit of spectroscopic information, thus increasing the standard deviation due to projection noise.
\begin{figure}
\caption{\textbf{Schematic layout of our experiment.}
\end{figure}
In our experiment, the pseudo-spin states are the two optical-clock transition levels, $5S_{j=\frac{1}{2},m_{j}=-\frac{1}{2}}$ and $4D_{j=\frac{5}{2},m_{j}=-\frac{3}{2}}$, in trapped $^{88}Sr^{+}$ ions. Our ions are trapped in a linear Paul trap and laser-cooled to the ground state of motion \cite{WinelandComprehensive1998,NitzanApplied2012,Nitzan2015} in the axial direction. We drive the optical clock transition using a 674 $\mathrm{nm}$ narrow linewidth ($<50$ Hz) laser. We address the two ion crystal with a single large-waist beam which implements both global rotations as well as the transverse Ising Hamiltonian. Alternatively we individually address a single ion of our choice using a tightly focused laser beam. The state of our ion is detected using state-selective fluorescence detection. An illustration of our experimental setup is shown in Fig. 1. More details about it can be found in \cite{NitzanApplied2012,Nitzan2015,TomThesis2016}. \\
The Ising Hamiltonian in Eq. \ref{Ising} is realized using a M\o lmer-S\o rensen- (MS) interaction \cite{MS_Gate2000}. We denote the clock transition carrier frequency of ion 1 and ion 2 as $\omega_{0}^1$ and $\omega_{0}^2$ respectively. The ions are illuminated with a bichromatic 674 $\mathrm{nm}$ laser beam at frequencies
\begin{equation}
\omega_{\pm}=\omega_{0}\pm\nu\pm\varepsilon-\delta \label{MSfreq}
\end{equation}
where $\omega_{0}=\frac{\omega_{0}^1+\omega_{0}^2}{2}$ is the average clock transition carrier frequency, $\nu$ is the axial trap frequency, $\varepsilon$ is a symmetric detuning, and $\delta$ is an asymmetric detuning from the sideband transitions (see Fig. 2c,d). We also define the center laser frequency as $\omega_{L}=\frac{\omega_{+}+\omega_{-}}{2}=\omega_{0}-\delta$. Here, we work in the regime $\varepsilon\gg\eta\tilde{\Omega}$, where $\eta$ is the Lamb-Dicke parameter of the trap axial center-of-mass mode and $\tilde{\Omega}$ is the clock-transition carrier Rabi frequency. In this regime, the coupling to motion through the red and blue sidebands can be adiabatically eliminated. In this case, two-photon coupling yields collective spin rotations and the Hamiltonian is well approximated as an Ising $\sigma_{y}\otimes\sigma_{y}$ interaction, with a $z$ transverse field due to $\delta_{1}=\delta$. If $\omega_{0}^1\ne\omega_{0}^2$, then dynamics is governed by the Hamiltonian in eq. \ref{Ising}, where $\delta_{2}$ represents the difference in detuning between ions and $\Omega=\frac{\eta^{2}\tilde{\Omega}^{2}}{\varepsilon}$ is the two-spin coupling. using the notation above, $\delta_1=\omega_{L}-\omega_{0}$ and $\delta_2=\frac{\omega_{0}^1-\omega_{0}^2}{2}$. Figure 2e,f shows a diagramatic illustration of the different detunings in this regime.\\
\begin{figure}
\caption{\textbf{Coupling in the two metrological subspaces using M\o lmer-S\o rensen interaction.}
\end{figure}
We begin by performing correlated Rabi nutation in the two subspaces using resonant Ising interaction. Here, we initialized our system in $\left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle $ or $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle $ and turned-on our MS interaction setting $\delta_{1} \simeq \delta_{2} \simeq 0$. Correlated Rabi nutation curves in the two subspaces are shown in Fig. 2a,b. As seen, coupling to states outside the subspace is minimized by the choice of large $\varepsilon$. We observe a complete correlated spin-flip at a $\pi$-time of $\tau_{\pi}=2\frac{2\pi\varepsilon}{\eta^{2}\Omega^{2}}$, which is about $1300\ \mu sec$ in this experiment.
Next we performed a wide Rabi spectroscopy scan by scanning $\delta_{1}$ from $-\varepsilon$ to $\varepsilon$; i.e. nearly to the motional sideband; by scanning the MS laser center frequency, $\omega_L$. A measurement of the populations of all four spin states vs. $\delta_{1}$ is shown in Fig. 3a-d. Here we set $\delta_{2}\simeq 0$ and the pulse time to $\tau_{\pi}$. As seen, when the system is initialized in the even subspace correlated spin rotation does not occur unless $\delta_{1} \simeq 0$. Around this resonant value, marked by a grey background, a sharp $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle\rightarrow\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ transition is observed. This correlated spin-flip resonance is enlarged in the inset of Figures 3c,d. On the other hand, when the system is initialized in the odd subspace, correlated spin-flip occurs at any value of $\delta_{1}$. This is due to the fact that the $\{\left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle\,\left|\left\vert \uparrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle\}$ subspace is insensitive to $\delta_{1}$. This odd subspace has been used several times before as a decoherence-free subspace due to this resilience to common phase noise. In both subspaces, as $\delta_{1}$ approaches $\varepsilon$, single-photon sideband transitions occur resulting in rapid population oscillations. \\
\begin{figure}
\caption{\textbf{Broad scan of $\delta_{1}
\end{figure}
We next turned to a combined scan of both $\delta_{1}$ and $\delta_{2}$. This scan was created by light-shifting the resonance frequency of only one of the ions by using an off-resonance single-addressing beam (see Fig. 2). With a detuning $\delta_{ls}/2\pi \simeq 3.5 \mathrm{MHz}$ and a Rabi-frequency which varied between $\Omega_{ls}/2\pi \simeq 0 - 40 \mathrm{kHz}$ we scanned the light-shift between $\Delta f_{ls}/2\pi\approx\frac{\Omega_{ls}^{2}}{2\pi\delta_{ls}}\simeq 0 - 400 \mathrm{Hz}$.
\begin{figure}
\caption{\textbf{Sensitivity of correlated rotations to $\delta_{1}
\end{figure}
By definition, $\delta_{2}=\pm\frac{1}{2}\Delta_{ls}/2\pi$. The sign is determined by the specific ion being light-shifted. The magnitude of $\Delta f_{ls}$, and therefore $\delta_{2}$, was scanned by varying the intensity of the individual addressing laser. For every value of $\Delta f_{ls}$ a full scan of $\omega_L$ was carried out, by changing the parameter $\delta$ in eq. \ref{MSfreq}. Figure 4 shows the population of all spin states for such a scan, when initializing in $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$ (a-d) and in $\left|\left\vert \downarrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ (e-h). As seen, in the odd subspace, a change to $\Delta f_{ls}$ causes a resonant response every time $\delta_2=0$, whereas a change of $\omega_L$ does not change the position of this resonance. In the even subspace a scan of $\omega_L$ yields a resonant response every time $\omega_L-\omega_{0}=0$. The change of $\Delta f_{ls}$ shifts the position of this resonance symmetrically with respect to the sign of $\Delta f_{ls}$, leading to the curved shape is figures 4a and 4d. This symmetry proves that in the symmetric subspace it is only the contribution of $\Delta f_{ls}$ to $\delta_1$ which changes the resonance position (see supplementary material). Note that since symmetric phase noise is more common in our experiment than differential phase noise between the two ions, the resonance in the symmetric sub-space is much noisier than in the anti-symmetric subspace.
\begin{figure}
\caption{Heisenberg-limited spectroscopy. \textbf{(a)}
\end{figure}
Finally, to determine whether our spectroscopy is Heisenberg-limited we performed a narrower scan of $\delta_{1}$ and $\delta_{2}$, each in its corresponding subspace, and compared the resulting spectrum to that of standard Rabi spectroscopy. Here $\delta_{1}$ scan in the even subspace was performed by scanning $\omega_L$, and $\delta_{2}$ scan in the odd subspace was achieved by scanning $\Delta f_{ls}$. Uncorrelated Rabi spectroscopy was performed by a regular single-ion Rabi spectrocopy scan.
The spectral shape of the excited-state population in two-level Rabi spectroscopy is \cite{AllenEberly1975},
\begin{equation}
\begin{split}
P(\left\vert \uparrow\right\ranglearrow)=A\frac{\sin^{2}\left(\frac{\Omega\tau}{2}\sqrt{1+\left(\frac{\alpha\delta}{\Omega}\right)^{2}}\right)}{1+\left(\frac{\alpha\delta}{\Omega}\right)^{2}}, \end{split}\label{spectrum}
\end{equation}
where $\Omega$ is the Rabi frequency and the contrast parameter $A$ accounts for experiment imperfections such as dephasing or noise in the Rabi coupling. $\alpha$ is the narrowing factor which is $1$ for a uncorrelated Rabi spectroscopy and $2$ for perfect two-qubit correlated Heisenberg limited Rabi spectroscopy.
The results of the different scans are shown in Fig. 5. For the $\delta_{1}$ scan in the even subspace we obtained $\alpha=1.92\pm0.02$ for a correlated rotation and $\alpha=1.01\pm0.01$ for the single ion case using a maximum likelihood fit to Eq.\ref{spectrum}. In the odd subspace, a $\delta_{2}$ scan yielded $\alpha=1.78\pm0.03$ for the correlated case and $f=0.88\pm0.02$ for single ion spectroscopy. These results show that using correlated rotation we are indeed well below the standard quantum limit and close to the Heisenberg limit of frequency estimation.\\
In this work, only a two-ion crystal was used for a proof of principle experiment. However, the features demonstrated here are general and will apply for a larger number of spins as well. The required generalized Hamiltonian that will generate correlated $N$-spin rotations will be given by,
\begin{equation}
\begin{split}
\hbar\left[\Omega\left(\sigma_{x}\right)^{\otimes N}+\sum_{i=1}^{N}\delta_{i}I\otimes...\otimes\sigma_{z}^{i}\otimes...\otimes I\right].
\end{split}\label{Ising}
\end{equation}
Using this Hamiltonian, an N-fold narrower Rabi spectrum can be measured around the average resonance frequency. The simulation of the above $N$-body correlated Hamiltonian was proposed in \cite{Porras2009,Müller2011}. In principle, a universal quantum simulator can be used to implement multi-ion Heisenberg-limited Rabi spectroscopy on any number of spins.\\
To conclude, in this work we presented and demonstrated a two-ions Heisenberg-limited Rabi spectroscopy. We initialized the ions in a separable state, and by operating with an entangling operator we obtained a spectrum narrower by a factor of $\simeq2$ with respect to conventional single ion Rabi spectroscopy. We observed that under the influence of an Ising Hamiltonian the two-ion system splits into two orthogonal subspaces that can be used as different probes for the difference and the average of the ions' optical resonance frequency, each of them with Heisenberg-limited uncertainty. We believe that the experiment presented here can be scaled up to more than a two-ion crystal, and may be useful as a spectroscopic tool for optical frequency measurements, as in optical atomic clocks.
This work was supported by the Crown Photonics Center, ICore-Israeli excellence center
circle of light, the Israeli Science Foundation, the Israeli Ministry of Science Technology
and Space, the Minerva Stiftung and the European Research Council (consolidator grant 616919-Ionology).
\pagebreak
\begin{center}
\textbf{\large Supplemental Materials}
\end{center}
\section{Section 1: light-shifting the resonance frequency of one ion }
In order to scan the frequency difference between our ions, we operated the single addressing beam on the desired ion, with frequency $3.5 \mathrm{MHz}$ off-resonance with the ion transition. This frequency shift was done with an Acousto-Optic Deflector (AOD). The light intensity was varied by changing the AOD's RF source power, such that the light-shift achieved was between 0 and about $\simeq750 \mathrm{Hz}$. In order to calibrate the common frequency shift and the difference shift between the ions resulting from the light-shift, we performed uncorrelated Rabi spectroscopy on both ions using the global beam while the light-shifting beam was on. The results are shown in Figure 6.
\begin{figure}
\caption{\textbf{Measurement of light shift.}
\end{figure}
From these measurements we were able to calibrate the mean frequency of the ions and the difference between their frequency, as a function of the light-shift beam power. The calibration results are presented in Fig. 7.
\begin{figure}
\caption{\textbf{Calibration of common frequency shift and frequency difference between the two ions.}
\end{figure}
The values taken for the vertical axis of figure 4 are the values taken from the right plot in figure 7 divided by 2 (see figure 1f).\\
As a sanity check, we verified that indeed the mean-frequency shift measured from the Rabi spectroscopy results agrees with the shift of the correlated $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle\rightarrow\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ transition, that is apparent in figure 4a and 4d. The comparison is plotted in Fig 8.
\begin{figure}
\caption{\textbf{Comparison between resonance shift of correlated Rabi spectroscopy on $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle\rightarrow\left|\left\vert \uparrow\right\ranglearrow\left\vert \uparrow\right\ranglearrow\right\rangle$ and mean frequency light shift measured on two uncorrelated ions.}
\end{figure}
\section{Section 2: Frequency difference uncorrelated Rabi spectroscopy}
In figure 5b we present a comparison between a correlated and uncorrelated frequency difference Rabi spectroscopy. The correlated part is described in the paper (see figure 1f). For this comparison we constructed a way to measure the frequency difference using two ions in an uncorrelated manner. We wanted to simulate a pure difference between the ions, without shifting of the frequency mean. Therefore, for each light-shift value we calibrated the mean frequency between the ions by performing a standard Rabi spectroscopy, fitting the spectrum and finding the mean frequency - $f_{mean}$. The experiment protocol was as follows: The ions were first initialized in the state $\left|\left\vert \downarrow\right\ranglearrow\left\vert \downarrow\right\ranglearrow\right\rangle$. Then a light-shifting beam was operated on one of the ions. Simultaneously, we operate a $\pi$ pulse with the global beam tuned to the $f_{mean}$. Lastly, a measurement of whether each ion is bright or dark was taken.\\
The measurement method is illustrated in Fig. 9.
\begin{figure}
\caption{\textbf{Measuring ion frequency difference.}
\end{figure}
\end{document} | math |
ಹೆದ್ದಾರಿ ಅತಿಕ್ರಮಣಕ್ಕಿಲ್ಲಿ ರಹದಾರಿ ನೀಡಿದವರ್ಯಾರು : ಹೆದ್ದಾರಿ ಸವಾರಿ ಆಯೋಮಯ.. Team Udayavani, Jul 3, 2022, 8:33 PM IST ಹುಣಸೂರು : ಹೆದಾರಿಯನ್ನೇ ಆಕ್ರಮಿಸಿಕೊಂಡು, ಹೋಟೆಲ್, ಅಂಗಡಿಗಳು, ಎಂ. ಸ್ಯಾಂಡ್ ದಾಸ್ತಾನು ಮಾಡಿದ್ದರೂ ಪ್ರಶ್ನಿಸಬೇಕಾದ ಲೋಕೋಪಯೋಗಿ ಇಲಾಖೆಯಾಗಲಿ, ನಗರದಲ್ಲಿ ಉತ್ತಮ ಪರಿಸರ ಕಾಪಾಡ ಬೇಕಿರುವ ನಗರಸಭೆಯಾಗಲಿ ಪ್ರಶ್ನಿಸದಿರುವುದು ಅನುಮಾನಕ್ಕೆಡೆ ಮಾಡಿದೆ. ಮೈಸೂರುಬಂಟ್ವಾಳ ರಾಷ್ಟ್ರೀಯ ಹೆದ್ದಾರಿ275 ರ ಎಪಿಎಂಸಿ ಯಾರ್ಡ್ ಮುಂಬಾಗದಿಂದ ಹಿಡಿದು ಆರ್.ಟಿ.ಓ.ಕಚೇರಿವರೆಗಿನ ಸುಮಾರು 3 ಕಿ.ಮೀ ರಸ್ತೆ ಅಕ್ಕಪಕ್ಕದಲ್ಲಿ ಅಂಗಡಿಗಳು ನಿರ್ಮಿಸಿಕೊಂಡು ಯಾರದೇ ಭಯವೂ ಇಲ್ಲದೆ ವಹಿವಾಟು ನಡೆಸುತ್ತಿದ್ದಾರೆ. ಹೆದ್ದಾರಿ ಸ್ಥಿತಿ ಆಯೋಮಯ : ದೇವರಾಜ ಅರಸು ಭವನದ ಬಳಿ, ತೋಟಗಾರಿಕೆ ಇಲಾಖೆನ್ಯಾಯಾಲಯದ ಎದುರು, ಬಸ್ ನಿಲ್ದಾಣದ ಬೈಪಾಸ್ ರಸ್ತೆ, ಕಲ್ಕುಣಿಕೆ ವೃತ್ತದಲ್ಲಿ ಅಂಗಡಿಗಳು ತಲೆ ಎತ್ತಿದ್ದರೆ. ಅರಸು ಭವನದ ಬಳಿ ಯಾರದೇ ಭಯವೂ ಇಲ್ಲದೆ ಹೆದ್ದಾರಿಯಲ್ಲೇ ಎಂ. ಸ್ಯಾಂಡ್ ಸುರಿದು ಎಗ್ಗಿಲ್ಲದೆ ವಹಿವಾಟು ನಡೆಸುತ್ತಿದ್ದಾರೆ. ಬೀದಿ ದೀಪವೂ ಇಲ್ಲದೆ ರಸ್ತೆ ಯಾವುದು, ಫುಟ್ ಬಾತ್ ಯಾವುದು ಎಂಬುದೇ ತಿಳಿಯದಾಗಿದೆ. ಹೆದ್ದಾರಿ ಬದಿಯ ಎಂ. ಸ್ಯಾಂಡ್ ದಾಸ್ತಾನಿನಿಂದಾಗಿ ಗಾಳಿ ಬೀಸಿದರಂತೂ ವಾಹನ ಸವಾರರು, ಕಾಲೇಜು ವಿದ್ಯಾರ್ಥಿಗಳು ಹಾಗೂ ನಡೆದು ತೆರಳುವ ಸಾರ್ವಜನಿಕರ ಕಣ್ಣಿಗೆ ರಾಚುತ್ತಿದ್ದು, ಹೇಳುವವರು ಕೇಳುವವರೂ ಇಲ್ಲದಂತಾಗಿದ್ದು. ಹೆದ್ದಾರಿಯಲ್ಲಿ ಸಂಚರಿಸುವ ವಾಹನ ಸವಾರರ ಸ್ಥಿತಿಯಂತೂ ಆಯೋಮಯವಾಗಿದೆ. ಜನ ನಡೆದಾಡುವುದೇ ದುಸ್ಥರ: ಇನ್ನು ನಗರದ ಪ್ರಮುಖ ರಸ್ತೆಗಳಾದ ಹೊಸ ಬಸ್ ನಿಲ್ದಾಣದ ಮುಂಬಾಗದ ರಸ್ತೆ, ಜೆ.ಎಲ್.ಬಿ.ರಸ್ತೆ, ಎಸ್.ಜೆ.ರಸ್ತೆಗಳಲ್ಲಂತೂ ವಾಹನಗಳಿರಲಿ, ಜನರು ಓಡಾಡಲು ಹರಸಾಹಸ ಪಡಬೇಕಿದೆ. ಈ ರಸ್ತೆಗಳ ಅಂಗಡಿಗಳವರು ರಸ್ತೆಗೆ ಸ್ಟೇರ್ ಕೇಸ್, ಮೆಟ್ಟಿಲುಗಳನ್ನು ಹಾಕಿಕೊಂಡಿದ್ದರಿಂದ ನಡೆದಾಡುವುದೇ ದುಸ್ಥರವಾಗಿದೆ. ಇದನ್ನೂ ಓದಿ : ಕಲಬುರಗಿ: ಕಾರು ಟ್ಯಾಂಕರ್ ಅಪಘಾತ: ವಿದ್ಯಾರ್ಥಿಗಳಿಬ್ಬರ ದುರ್ಮರಣ ಕಣ್ಣಿದ್ದೂ ಕುರುಡರಾದ ಅಧಿಕಾರಿಗಳು: ಹಿಂದೆ ಈ ರಸ್ತೆ ಲೋಕೋಪಯೋಗಿ ಇಲಾಖೆಗೆ ಸೇರಿದ್ದಾಗಿತ್ತು. ವರ್ಷದಿಂದ ರಾಷ್ಟ್ರೀಯ ಹೆದ್ದಾರಿಯಾಗಿ ಮೇಲ್ದರ್ಜೆಗೇರಿದ್ದು. ಇತ್ತ ಲೋಕೋಪಯೋಗಿಯವರೂ ಕೇಳುತ್ತಿಲ್ಲ, ಮತ್ತೊಂದೆಡೆ ಹೆದ್ದಾರಿ ಪ್ರಾಧಿಕಾರವೂ ಇತ್ತ ತಿರುಗಿ ನೋಡಿಲ್ಲ, ಇನ್ನು ನಗರಸಭೆ ವ್ಯಾಪ್ತಿಯ ಹೆದ್ದಾರಿಯನ್ನು ಅತಿಕ್ರಮಣ ಮಾಡಿಕೊಂಡು ಅಂಗಡಿ ನಿರ್ಮಿಸಿಕೊಂಡು ವ್ಯಾಪಾರ ನಡೆಸುತ್ತಿರುವ ಬಗ್ಗೆ ಹಾಗೂ ಎಂ. ಸ್ಯಾಂಡ್ ಸುರಿದು ವಹಿವಾಟು ನಡೆಸುತ್ತಿರುವ ಬಗ್ಗೆ ಪೊಲೀಸರುನಗರಸಭೆ ಅಧಿಕಾರಿಗಳು ಕಣ್ಣಿದ್ದೂ ಕುರುಡರಂತಾಗಿದ್ದು, ಇಲ್ಲಿ ಹೇಳುವವರುಕೇಳುವವರು ಇಲ್ಲದಂತಾಗಿದೆ. ಯಾರಿಗೆ ದೂರು ನೀಡಬೇಕೆಂಬ ಜಿಜ್ಞಾಸೆ ಈ ರಸ್ತೆಯನ್ನು ಅವಲಂಬಿಸಿರುವ ವಾಹನ ಸವಾರರು, ಸಾರ್ವಜನಿಕರನ್ನು ಕಾಡುತ್ತಿದೆ. ಒತ್ತುವರಿ ತೆರವಾಗಲಿ: ನಗರದ ಪ್ರಮುಖ ರಸ್ತೆಗಳು ಒತ್ತುವರಿಯಾಗಿವೆ, ಒತ್ತುವರಿ ತೆರವಿರಲಿ ಕನಿಷ್ಟ ಸುಗಮ ಸಂಚಾರಕ್ಕೂ ನಗರದೊಳಗೂ, ಹೊರಗೂ ಅವಕಾಶವೇ ಇಲ್ಲದಂತಾಗಿದೆ. ಆಡಳಿತವಂತೂ ಹಳ್ಳ ಹಿಡಿದಿದೆ. ಬೈಪಾಸ್ ರಸ್ತೆಯನ್ನು ಅತಿಕ್ರಮಣ ಮಾಡಿಕೊಂಡು ವ್ಯಾಪಾರ ನಡೆಸುತ್ತಿರುವವರನ್ನು ತೆರವುಗೊಳಿಸಿ ಸುಗಮ ಸಂಚಾರಕ್ಕೆ ಅವಕಾಶ ಕಲ್ಪಿಸಬೇಕು, ಎಂ.ಸ್ಯಾಂಡ್ ಮಾರಾಟಕ್ಕೆ ಎಪಿಎಂಸಿ ಆವರಣದೊಳಗೆ ಪ್ರತ್ರೇಕ ವ್ಯವಸ್ಥೆ ಕಲ್ಪಿಸಬೇಕು. ಪುರುಷೋತ್ತಮ್, ಕರವೆ ತಾಲೂಕು ಅಧ್ಯಕ್ಷ, ಹುಣಸೂರು. ಈಗಾಗಲೆ ಬೈಪಾಸ್ ರಸ್ತೆಯಲ್ಲಿ ಅಂಗಡಿ ನಿರ್ಮಿಸಿಕೊಂಡಿರುವವರಿಗೆ ಎಚ್ಚರಿಕೆ ನೀಡಲಾಗಿದ್ದು, ತೆರವುಗೊಳಿಸದಿದ್ದಲ್ಲಿ ಶೀಘ್ರ ಕಾರ್ಯಾಚರಣೆ ನಡೆಸಿ ತೆರವುಗೊಳಿಸಿ, ಸುಗಮ ಸಂಚಾರಕ್ಕೆ ಅವಕಾಶ ಕಲ್ಪಿಸಲಾಗುವುದು. ರವಿಕುಮಾರ್,ಪೌರಾಯುಕ್ತ.ನಗರಸಭೆ. ಸಂಪತ್ ಕುಮಾರ್, ಹುಣಸೂರು. | kannad |
[' \n \n ["ਆਓ ਇੱਕ ਲੰਬੀ ਅਤੇ ਹੌਲੀ ਕਹਾਣੀ ਲਿਖੀਏ। \\n \\n ਪਿਛੋਕਡ਼ ਇਹ ਹੈ ਕਿ ਕਹਾਣੀ ਕ੍ਰਾਸਓਕ ਨਾਮ ਦੇ ਇੱਕ ਛੋਟੇ ਜਿਹੇ ਕਸਬੇ ਵਿੱਚ ਹੈ। ਮੁੱਖ ਪਾਤਰ ਨਾਮ _ 1 ਅਤੇ ਉਸ ਦਾ ਪਰਿਵਾਰ।ਇਸ ਸ਼ਹਿਰ ਵਿੱਚ ਚਲੇ ਜਾਓ। ਨਾਮ 1 43 ਸਾਲਾ ਹਾਈ ਸਕੂਲ ਅਧਿਆਪਕ ਹੈ। ਉਸ ਦਾ ਇੱਕ ਮੁੰਡਾ ਅਤੇ ਇੱਕ ਲਡ਼ਕੀ ਹੈ। ਉਹ ਇੱਕ ਸੁੰਦਰ ਔਰਤ ਹੈ ਅਤੇ ਉਹ ਉਸ ਵੱਲ ਆਕਰਸ਼ਿਤ ਹੋਣਾ ਪਸੰਦ ਕਰਦੀ ਹੈ।ਵਿਦਿਆਰਥੀ ਅਤੇ ਮਰਦ। ਉਹ ਮਰਦਾਂ ਨੂੰ ਤੰਗ ਕਰਨ ਲਈ ਖੁੱਲ੍ਹੇ ਕੱਪਡ਼ੇ ਪਹਿਨਣਾ ਪਸੰਦ ਕਰਦੀ ਹੈ। ਅਤੇ ਕਈ ਵਾਰ ਉਹ ਉਨ੍ਹਾਂ ਨਾਲ ਫਲਰਟ ਕਰਨਾ ਪਸੰਦ ਕਰਦੀ ਹੈ ਇਹ ਵੇਖਣ ਲਈ ਕਿ ਉਹ ਕਿਵੇਂ ਪ੍ਰਤੀਕ੍ਰਿਆ ਕਰਦੇ ਹਨ। ਸਮੇਂ-ਸਮੇਂ \'ਤੇਸਮਾਂ ਮਰਦ ਵਧੇਰੇ ਚਾਹੁੰਦੇ ਹਨ ਅਤੇ ਉਹ ਉਸ ਉੱਤੇ ਇੱਕ ਕਦਮ ਚੁੱਕਦੇ ਹਨ ਪਰ ਉਹ ਆਮ ਤੌਰ ਉੱਤੇ ਉਨ੍ਹਾਂ ਨੂੰ ਕਿਸੇ ਤਰ੍ਹਾਂ ਰੋਕਦੀ ਹੈ। ਅਤੇ ਹਾਂ ਮੈਂ ਕਿਹਾ ਕਿ ਆਮ ਤੌਰ ਉੱਤੇ ਇੱਕ ਜਾਂ ਦੋ ਵਾਰ ਉਹ ਅਜਿਹਾ ਕਰਨ ਵਾਲੀ ਹੁੰਦੀ ਹੈ।ਬਲਾਤਕਾਰ ਜਾਂ ਬਦਸਲੂਕੀ ਹੁੰਦੀ ਹੈ। ਕੁੱਝ ਆਦਮੀ ਅਸਲ ਵਿੱਚ ਨਹੀਂ ਜਾਣਦੇ ਕਿ ਕਿੱਥੇ ਰੁਕਣਾ ਹੈ। ਪਰ ਇਹ ਉਹ ਹਿੱਸਾ ਹੈ ਜੋ ਨਾਮ _ 1 ਨੂੰ ਇਸ ਨੂੰ ਚਾਲੂ ਕਰਦਾ ਹੈ। ਉਹ ਰੋਮਾਂਚਕ ਚੀਜ਼ਾਂ ਕਰਨਾ ਪਸੰਦ ਕਰਦੀ ਹੈ।ਲਾਲਚ ਬਾਰੇ, ਭਾਵੇਂ ਉਹ ਜਾਣਦੀ ਹੈ ਕਿ ਉਹ ਉਨ੍ਹਾਂ ਨਾਲ ਸਮਝੌਤਾ ਨਹੀਂ ਕਰ ਸਕਦੀ। ਪਰ, ਇਸ ਨਵੇਂ ਸ਼ਹਿਰ ਵਿੱਚ, ਉਸ ਨੂੰ ਇਸ ਤਰ੍ਹਾਂ ਦੀਆਂ ਸਥਿਤੀਆਂ ਦਾ ਸਾਹਮਣਾ ਕਰਨਾ ਪਵੇਗਾ ਅਤੇ ਉਹ ਕਰੇਗੀਫੈਸਲਾ ਕਰੋ ਕਿ ਕੀ ਉਹ ਇੱਕ ਚੰਗੀ ਪਤਨੀ ਅਤੇ ਇੱਕ ਮਾਂ ਜਾਂ ਇੱਕ ਕਾਮੁਕ ਸ਼ਰਾਰਤੀ ਪਤਨੀ ਬਣੇਗੀ? \\n\\n ਕਹਾਣੀ ਕਿੱਥੇ ਸ਼ੁਰੂ ਹੁੰਦੀ ਹੈ, ਇਹ ਨਾਮ _ 1 ਦਾ ਸ਼ਹਿਰ ਵਿੱਚ ਪਹਿਲਾ ਦਿਨ ਹੈ ਉਹ ਹਨਆਪਣੇ ਸਟਾਫ ਨੂੰ ਘਰ ਭੇਜਣਾ। ਇੱਥੇ ਕਰਮਚਾਰੀ ਹਨ ਜੋ ਉਹਨਾਂ ਦੀ ਮਦਦ ਕਰਦੇ ਹਨ। ਅਤੇ ਉਹ ਇਸ ਨੂੰ ਲੁਕਾਉਣ ਦੀ ਕੋਸ਼ਿਸ਼ ਕਰ ਰਹੇ ਹਨ ਪਰ ਉਹ ਨਾਮ _ 1 ਤੋਂ ਆਪਣੀਆਂ ਅੱਖਾਂ ਨਹੀਂ ਲੈ ਸਕਦੇ। ', "ਆਓ ਇੱਕ ਲੰਬੀ ਅਤੇ ਹੌਲੀ ਕਹਾਣੀ ਲਿਖੀਏ। ਪਿਛੋਕਡ਼ ਇਹ ਹੈ ਕਿ ਕਹਾਣੀ ਕ੍ਰਾਸਓਕ ਨਾਮ ਦੇ ਇੱਕ ਛੋਟੇ ਜਿਹੇ ਕਸਬੇ ਵਿੱਚ ਹੈ। ਮੁੱਖ ਪਾਤਰ ਨਾਮ 1 ਅਤੇ ਉਸ ਦਾ ਪਰਿਵਾਰ ਇੱਥੇ ਆ ਜਾਂਦਾ ਹੈ।ਇਹ ਸ਼ਹਿਰ। ਨਾਮ 1 43 ਸਾਲਾ ਹਾਈ ਸਕੂਲ ਅਧਿਆਪਕ ਹੈ। ਉਸ ਦਾ ਇੱਕ ਮੁੰਡਾ ਅਤੇ ਇੱਕ ਲਡ਼ਕੀ ਹੈ। ਉਹ ਇੱਕ ਸੁੰਦਰ ਔਰਤ ਹੈ ਅਤੇ ਉਹ ਉਸ ਵੱਲ ਆਕਰਸ਼ਿਤ ਹੋਣਾ ਪਸੰਦ ਕਰਦੀ ਹੈ।ਵਿਦਿਆਰਥੀ ਅਤੇ ਮਰਦ। ਉਹ ਮਰਦਾਂ ਨੂੰ ਤੰਗ ਕਰਨ ਲਈ ਖੁੱਲ੍ਹੇ ਕੱਪਡ਼ੇ ਪਹਿਨਣਾ ਪਸੰਦ ਕਰਦੀ ਹੈ। ਅਤੇ ਕਈ ਵਾਰ ਉਹ ਉਨ੍ਹਾਂ ਨਾਲ ਫਲਰਟ ਕਰਨਾ ਪਸੰਦ ਕਰਦੀ ਹੈ ਇਹ ਵੇਖਣ ਲਈ ਕਿ ਉਹ ਕਿਵੇਂ ਪ੍ਰਤੀਕ੍ਰਿਆ ਕਰਦੇ ਹਨ। ਸਮੇਂ-ਸਮੇਂ 'ਤੇਸਮਾਂ ਮਰਦ ਵਧੇਰੇ ਚਾਹੁੰਦੇ ਹਨ ਅਤੇ ਉਹ ਉਸ ਉੱਤੇ ਇੱਕ ਕਦਮ ਚੁੱਕਦੇ ਹਨ ਪਰ ਉਹ ਆਮ ਤੌਰ ਉੱਤੇ ਉਨ੍ਹਾਂ ਨੂੰ ਕਿਸੇ ਤਰ੍ਹਾਂ ਰੋਕਦੀ ਹੈ। ਅਤੇ ਹਾਂ ਮੈਂ ਕਿਹਾ ਕਿ ਆਮ ਤੌਰ ਉੱਤੇ ਇੱਕ ਜਾਂ ਦੋ ਵਾਰ ਉਹ ਅਜਿਹਾ ਕਰਨ ਵਾਲੀ ਹੁੰਦੀ ਹੈ।ਬਲਾਤਕਾਰ ਜਾਂ ਬਦਸਲੂਕੀ ਹੁੰਦੀ ਹੈ। ਕੁੱਝ ਆਦਮੀ ਅਸਲ ਵਿੱਚ ਨਹੀਂ ਜਾਣਦੇ ਕਿ ਕਿੱਥੇ ਰੁਕਣਾ ਹੈ। ਪਰ ਇਹ ਉਹ ਹਿੱਸਾ ਹੈ ਜੋ ਨਾਮ _ 1 ਨੂੰ ਇਸ ਨੂੰ ਚਾਲੂ ਕਰਦਾ ਹੈ। ਉਹ ਰੋਮਾਂਚਕ ਚੀਜ਼ਾਂ ਕਰਨਾ ਪਸੰਦ ਕਰਦੀ ਹੈ।ਲਾਲਚ ਬਾਰੇ, ਭਾਵੇਂ ਉਹ ਜਾਣਦੀ ਹੈ ਕਿ ਉਹ ਉਨ੍ਹਾਂ ਨਾਲ ਸਮਝੌਤਾ ਨਹੀਂ ਕਰ ਸਕਦੀ। ਪਰ, ਇਸ ਨਵੇਂ ਸ਼ਹਿਰ ਵਿੱਚ, ਉਸ ਨੂੰ ਇਸ ਤਰ੍ਹਾਂ ਦੀਆਂ ਸਥਿਤੀਆਂ ਦਾ ਸਾਹਮਣਾ ਕਰਨਾ ਪਵੇਗਾ ਅਤੇ ਉਹ ਕਰੇਗੀਫੈਸਲਾ ਕਰੋ ਕਿ ਕੀ ਉਹ ਇੱਕ ਚੰਗੀ ਪਤਨੀ ਅਤੇ ਇੱਕ ਮਾਂ ਜਾਂ ਇੱਕ ਕਾਮੁਕ ਸ਼ਰਾਰਤੀ ਪਤਨੀ ਬਣੇਗੀ? ਕਹਾਣੀ ਕਿੱਥੇ ਸ਼ੁਰੂ ਹੁੰਦੀ ਹੈ, ਇਹ ਨਾਮ _ 1 ਦਾ ਸ਼ਹਿਰ ਵਿੱਚ ਪਹਿਲਾ ਦਿਨ ਹੈ ਜਦੋਂ ਉਹ ਜਾ ਰਹੇ ਹਨਉਹਨਾਂ ਦਾ ਸਟਾਫ ਘਰ ਨੂੰ ਜਾਂਦਾ ਹੈ। ਇੱਥੇ ਕਰਮਚਾਰੀ ਹਨ ਜੋ ਉਹਨਾਂ ਦੀ ਮਦਦ ਕਰਦੇ ਹਨ। ਅਤੇ ਉਹ ਇਸ ਨੂੰ ਲੁਕਾਉਣ ਦੀ ਕੋਸ਼ਿਸ਼ ਕਰ ਰਹੇ ਹਨ ਪਰ ਉਹ ਨਾਮ _ 1 ਤੋਂ ਆਪਣੀਆਂ ਅੱਖਾਂ ਨਹੀਂ ਲੈ ਸਕਦੇ। \n "] | punjabi |
Beautiful large format tiles make a stunning backsplash for kitchens and bathrooms. Our Adella White Satin-finished ceramic tiles have a pure white, almost icy tone that complements a wide range of natural stone countertops, floor tiles and mosaic backsplash tiles in our inventory. Used as a kitchen backsplash, it will provide an easy to clean surface, with few grout lines. | english |
from collections import namedtuple
from django_iban.utils import clean_iban
IBANPartSpecification = namedtuple('IBANPartSpecification', ["length", "data_type"])
class IBANSpecification(object):
MASK_DATATYPE_MAP = {
'a':'a',
'n':'9',
'c':'w',
}
REGEX_DATATYPE_MAP = {
'a': '[A-Z]' ,
'n': '[0-9]' ,
'c': '[A-Z0-9]',
}
def __init__(self, country_name, country_code, bank_format, account_format):
self.country_name = country_name
self.country_code = country_code
self.bank_specification = self.decode_format(bank_format )
self.account_specification = self.decode_format(account_format)
@property
def bank_field_length(self):
return sum((_.length for _ in self.bank_specification))
@property
def account_field_length(self):
return sum((_.length for _ in self.account_specification))
def field_mask(self, specification):
return " ".join([
self.MASK_DATATYPE_MAP[part.data_type] * part.length
for part in specification if part.length > 0])
def validation_regex(self, specification):
return "".join([
"%s{%s}" % (self.REGEX_DATATYPE_MAP[part.data_type], part.length)
for part in specification if part.length > 0])
@property
def bank_regex(self):
return self.validation_regex(self.bank_specification)
@property
def account_regex(self):
return self.validation_regex(self.account_specification)
@property
def iban_regex(self):
return "[A-Z]{2}[0-9]{2}" + self.bank_regex + self.account_regex
@property
def bank_field_mask(self):
return self.field_mask(self.bank_specification)
@property
def account_field_mask(self):
return self.field_mask(self.account_specification)
@property
def total_length(self):
return 4 + self.bank_field_length + self.account_field_length
def decode_format(self, data_format):
return [
IBANPartSpecification(
length = int(part[:-1]) if part[-1] in ("n", "a") else int(part),
data_type = part[-1] if part[-1] in ("n", "a") else "c")
for part in filter(bool, data_format.split())]
@staticmethod
def checksum(value):
value = clean_iban(value)
value = value[4:] + value[:2] + '00'
value_digits = ''
for x in value:
if '0' <= x <= '9':
value_digits += x
elif 'A' <= x <= 'Z':
value_digits += str(ord(x) - 55)
else:
raise Exception('{} is not a valid character for IBAN.'.format(x))
return '%02d' % (98 - int(value_digits) % 97)
IBAN_SPECIFICATION_CONFIG = {
"AD": IBANSpecification("Andorra" , "AD", "0 4n 4n", "0 12 0 "),
"AL": IBANSpecification("Albania" , "AL", "0 8n 0 ", "0 16 0 "),
"AT": IBANSpecification("Austria" , "AT", "0 5n 0 ", "0 11n 0 "),
"BA": IBANSpecification("Bosnia and Herzegovina", "BA", "0 3n 3n", "0 8n 2n"),
"BE": IBANSpecification("Belgium" , "BE", "0 3n 0 ", "0 7n 2n"),
"BG": IBANSpecification("Bulgaria" , "BG", "0 4a 4n", "2n 8 0 "),
"CH": IBANSpecification("Switzerland" , "CH", "0 5n 0 ", "0 12 0 "),
"CY": IBANSpecification("Cyprus" , "CY", "0 3n 5n", "0 16 0 "),
"CZ": IBANSpecification("Czech Republic" , "CZ", "0 4n 0 ", "0 16n 0 "),
"DE": IBANSpecification("Germany" , "DE", "0 8n 0 ", "0 10n 0 "),
"DK": IBANSpecification("Denmark" , "DK", "0 4n 0 ", "0 9n 1n"),
"EE": IBANSpecification("Estonia" , "EE", "0 2n 0 ", "2n 11n 1n"),
"ES": IBANSpecification("Spain" , "ES", "0 4n 4n", "2n 10n 0 "),
"FI": IBANSpecification("Finland" , "FI", "0 6n 0 ", "0 7n 1n"),
"FO": IBANSpecification("Faroe Islands" , "FO", "0 4n 0 ", "0 9n 1n"),
"FR": IBANSpecification("France" , "FR", "0 5n 5n", "0 11 2n"),
"GB": IBANSpecification("United Kingdom" , "GB", "0 4a 6n", "0 8n 0 "),
"GE": IBANSpecification("Georgia" , "GE", "0 2a 0 ", "0 16n 0 "),
"GI": IBANSpecification("Gibraltar" , "GI", "0 4a 0 ", "0 15 0 "),
"GL": IBANSpecification("Greenland" , "GL", "0 4n 0 ", "0 9n 1n"),
"GR": IBANSpecification("Greece" , "GR", "0 3n 4n", "0 16 0 "),
"HR": IBANSpecification("Croatia" , "HR", "0 7n 0 ", "0 10n 0 "),
"HU": IBANSpecification("Hungary" , "HU", "0 3n 4n", "1n 15n 1n"),
"IE": IBANSpecification("Ireland" , "IE", "0 4a 6n", "0 8n 0 "),
"IL": IBANSpecification("Israel" , "IL", "0 3n 3n", "0 13n 0 "),
"IS": IBANSpecification("Iceland" , "IS", "0 4n 0 ", "2n 16n 0 "),
"IT": IBANSpecification("Italy" , "IT", "1a 5n 5n", "0 12 0 "),
"KW": IBANSpecification("Kuwait" , "KW", "0 4a 0 ", "0 22 0 "),
"KZ": IBANSpecification("Kazakhstan" , "KZ", "0 3n 0 ", "0 13 0 "),
"LB": IBANSpecification("Lebanon" , "LB", "0 4n 0 ", "0 20 0 "),
"LI": IBANSpecification("Liechtenstein" , "LI", "0 5n 0 ", "0 12 0 "),
"LT": IBANSpecification("Lithuania" , "LT", "0 5n 0 ", "0 11n 0 "),
"LU": IBANSpecification("Luxembourg" , "LU", "0 3n 0 ", "0 13 0 "),
"LV": IBANSpecification("Latvia" , "LV", "0 4a 0 ", "0 13 0 "),
"MC": IBANSpecification("Monaco" , "MC", "0 5n 5n", "0 11 2n"),
"ME": IBANSpecification("Montenegro" , "ME", "0 3n 0 ", "0 13n 2n"),
"MK": IBANSpecification("Macedonia" , "MK", "0 3n 0 ", "0 10 2n"),
"MR": IBANSpecification("Mauritania" , "MR", "0 5n 5n", "0 11n 2n"),
"MT": IBANSpecification("Malta" , "MT", "0 4a 5n", "0 18 0 "),
"MU": IBANSpecification("Mauritius" , "MU", "0 4a 4n", "0 15n 3a"),
"NL": IBANSpecification("Netherlands" , "NL", "0 4a 0 ", "0 10n 0 "),
"NO": IBANSpecification("Norway" , "NO", "0 4n 0 ", "0 6n 1n"),
"PL": IBANSpecification("Poland" , "PL", "0 8n 0 ", "0 16n 0 "),
"PT": IBANSpecification("Portugal" , "PT", "0 4n 4n", "0 11n 2n"),
"RO": IBANSpecification("Romania" , "RO", "0 4a 0 ", "0 16 0 "),
"RS": IBANSpecification("Serbia" , "RS", "0 3n 0 ", "0 13n 2n"),
"SA": IBANSpecification("Saudi Arabia" , "SA", "0 2n 0 ", "0 18 0 "),
"SE": IBANSpecification("Sweden" , "SE", "0 3n 0 ", "0 16n 1n"),
"SI": IBANSpecification("Slovenia" , "SI", "0 5n 0 ", "0 8n 2n"),
"SK": IBANSpecification("Slovak Republic" , "SK", "0 4n 0 ", "0 16n 0 "),
"SM": IBANSpecification("San Marino" , "SM", "1a 5n 5n", "0 12 0 "),
"TN": IBANSpecification("Tunisia" , "TN", "0 2n 3n", "0 13n 2n"),
"TR": IBANSpecification("Turkey" , "TR", "0 5n 0 ", "1 16 0 ")}
IBAN_GROUPING = 4
IBAN_MAX_LENGTH = 34
IBAN_MIN_LENGTH = min([_.total_length for _ in IBAN_SPECIFICATION_CONFIG.values()])
| code |
വിവാദമായ സിറോ മലബാര് സഭ ഭൂമിയിടപാട് കേസില് കര്ദിനാള് ജോര്ജ് ആലഞ്ചേരി വിചാരണ നേരിടണമെന്ന് ഹൈക്കോടതി ആറു ഹര്ജികളും തള്ളി കൊച്ചി: വിവാദമായ സിറോ മലബാര് സഭ ഭൂമിയിടപാട് കേസില് കര്ദിനാള് മാര് ജോര്ജ് ആലഞ്ചേരി വിചാരണ നേരിടണമെന്ന് ഹൈക്കോടതി . ആലഞ്ചേരി നല്കിയ ആറു ഹര്ജികളും കോടതി തള്ളി. വിചാരണ നേരിടണമെന്ന കീഴ്ക്കോടതി വിധി ഹൈക്കോടതി ശരിവെച്ചു. ഭൂമി ഇടപാടു കേസില് വിചാരണ നേരിടണമെന്ന എറണാകുളം ജില്ലാ സെഷന്സ് കോടതി ഉത്തരവ് റദ്ദാക്കണമെന്ന് ആവശ്യപ്പെട്ടാണ് കര്ദിനാള് ആലഞ്ചേരി ഹൈക്കോടതിയെ സമീപിച്ചത്. കര്ദിനാള് മാര് ജോര്ജ്ജ് ആലഞ്ചേരി,അതിരൂപത മുന് ഫിനാന്സ് ഓഫീസര് ഫാദര് ജോഷി പുതുവ, ഭൂമി വാങ്ങിയ സാജു വര്ഗീസ് എന്നിവര് കേസില് വിചാരണ നേരിടണമെന്നായിരുന്നു കീഴ്കോടതി ഉത്തരവ്. തൃക്കാക്കര മജിസ്ട്രേറ്റ് കോടതി ഉത്തരവ് റദ്ദാക്കണമെന്ന് കാണിച്ച് കര്ദിനാള് ആലഞ്ചേരി നേരത്തെ ജില്ലാ സെഷന്സ് കോടതിയെ സമീപിച്ചിരുന്നു. എന്നാല് സെഷന്സ് കോടതിയും ഹര്ജി തള്ളുകയും മജിസ്ട്രേറ്റ് കോടതി വിധി ശരിവെക്കുകയുമായിരുന്നു. എറണാകുളം അങ്കമാലി അതിരൂപതയുടെ ഉടമസ്ഥതയിലുള്ള കാക്കനടുള്ള 60 സെന്റ് ഭൂമി വില്പ്പന നടത്തിയതിലൂടെ സഭയ്ക്ക് ലക്ഷങ്ങളുടെ നഷ്ടം ഉണ്ടായെന്നാണ് കേസ്. ഭൂമി വില്പ്പന സഭയുടെ വിവിധ സമിതികളില് ആലോചിക്കാതെയാണെന്നാണ് ആരോപണം ഉയര്ന്നത്. ഭൂമി ഇടപാടില് കര്ദിനാള് ആലഞ്ചേരിക്ക് വീഴ്ച സംഭവിച്ചുവെന്ന് സഭ നടത്തിയ അന്വേഷണത്തില് കണ്ടെത്തിയിരുന്നു. | malyali |
সারাক্ষণ সব জান্তা আচরণ, দেখনদারিতে ওস্তাদ এই চার রাশির ছেলে মেয়েরা প্রতিটি মানুষের জীবন জ্যোতিষের ভূমিকা বিস্তর ব্যক্তির ভবিষ্যত জানতে অনেকেই শাস্ত্রের ওপর ভরসা করেন তেমনই কার চারিত্রিক বৈশিষ্ট্য কেমন তা জানা সম্ভব শাস্ত্র মতে বৈদিশ শাস্ত্রে রয়েছে ১২ রাশির হদিশ মেষ থেকে মীন এই সকল রাশির অধিকর্তা গ্রহ ভিন্ন আর মানুষের জীবনে গ্রহের ভূমিকা বিস্তর সে কারণে সকলে স্বভাব, চরিত্র, মানসিকতা সর্বক্ষেত্রে সকলের সঙ্গে সকলের রয়েছে তফাত সে কারণে কেউ ধূর্ত তো কেউ বোকা কেউ দয়ালু তো কে স্বার্থপর আজ রইল চার রাশির কথা এর সব সময় সব জান্তা আচরণ করে থাকেন এদের এই আচরণের জন্য অনেকের কাছে এর অসহ্য হয়ে ওঠেন দেখে নিন আপনার পরিচিত কে আছে তালিকায় কর্কট রাশি রাশি চক্রের চতুর্থ রাশি হল কর্কট এদের অধিকর্তা গ্রহ হল চন্দ্র এরা নিজেদের বুদ্ধিমান মনে করেন সব সময় আর সে কারণে সব সময় বাড়াবাড়ি করে ফেলেন তবে, অনেক সময় এরা নিজেদের আবেগ নিয়ন্ত্রণ করতে বুদ্ধিমান ও স্মার্ট আচরণের প্রকাশ করেন ধনু রাশি রাশি চক্রের নবম রাশির হল ধনু এই রাশির অধিকর্তা গ্রহ বৃহস্পতি এদের প্রায়শই অপরিচিত পরিস্থিতির সামনা করতে হয় এরা সব সময় জ্ঞানী ব্যক্তির মতো আচরণ করতে পছন্দ করেন এরা নিজেদের সংস্কৃতির সঙ্গে পরিচিত একজন জ্ঞানী ব্যক্তি হিসেবে জাহির করতে পছন্দ করে থাকেনমিথুন রাশি রাশি চক্রের তৃতীয় রাশি হল মিথুন এই রাশির অধিকর্তা গ্রহ বুধ এরা উদ্যমী বালক, চঞ্চলমতি গ্রহ হয়ে থাকে এই রাশির ছেলে মেয়েরা সব ব্যাপারে নিজের মনভাব ব্যক্ত করতে চান সব সময় জ্ঞানী ব্যক্তির মতো আচরণ করতে এরা পছন্দ করে থাকেন এই রাশির ছেলে মেয়েরা নিজেদের বুদ্ধিমান হিসেবে জাহির করে থাকেন সিংহ রাশি রাশি চক্রের পঞ্চম রাশি হল সিংহ রাশি এই রাশির অধিকর্তা গ্রহ হল রবি এরা জেদি, প্রতিজ্ঞ, দয়াবান ও গম্ভীর স্বভাবের হয়ে থাকেন এরা সোশ্যাল মিডিয়ায় সব সময় ভুয়ো ও অতিরিক্ত প্রদর্শন করে থাকেন এই রাশির ছেলে মেয়েরা নিজেদের স্মার্ট হিসেবে জাহির করতে চান তবে, এরা নিজের জীবন থেকে কী চান তা বুঝতে পারেন না একেবারেই তবে এরা দেখনদারিতে করতে খুবই পছন্দ করেন আরও পড়ুন মিথুন রাশিতে শুক্রের গোচর, এই ৪ রাশিকে খুব সমস্যায় ফেলবে আরও পড়ুন স্বামীস্ত্রী মধ্যে বিবাদ হতে পারে এই তারিখের জাতকজাতিকাদের, দেখে নিন সংখ্যাতত্ত্বের গণনা আরও পড়ুন গুরু পূর্ণিমার পূণ্য তিথিতে এইভাবে পুজো করুন, জেনে নিন কী কী উপকার মিলবে | bengali |
ગાજર કરતાં તેમના પાન છે વધુ ગણકારી ગાજર અને તેના પણ ખાવાના છે અનેક ફાયદા ગાજરના પાન લીવર, હાડકાં કરે છે મજબૂત વિટામિન K1, A, પોટેશિયમ અને કેલ્શિયમથી ભરપૂર છે Their leaves are more compact than carrots તમે શિયાળામાં ગાજર ખાઓ છો, પરંતુ શું તમે જાણો છો કે તેના પાંદડામાં પણ એવા ઘણા ગુણ હોય છે જેનાથી તમને ફાયદો થાય છે. તેના પાનનો સૂપ પીવો તમારા માટે ખૂબ જ ફાયદાકારક રહેશે. તેનું નિયમિત સેવન કરવાથી વજન ઘટાડવામાં મદદ મળશે. આ સાથે, તે આંખોની રોશની વધારવા, કેન્સરનું જોખમ ઘટાડવા અને મૌખિક સ્વાસ્થ્યને વધુ સારી રીતે જાળવવામાં પણ અસરકારક છે. ગાજર અને તેના પાંદડામાં કેલ્શિયમ હોય છે, જે એલડીએલ એટલે કે ખરાબ કોલેસ્ટ્રોલને ઘટાડવામાં મદદ કરી શકે છે. તેમાં ફાઈબરનું પ્રમાણ વધુ હોય છે, જે ધમનીઓ અને રક્ત વાહિનીઓની દિવાલો પર જમા થયેલ કોલેસ્ટ્રોલને સાફ કરી શકે છે. Their leaves are more compact than carrots ગાજરની જેમ, તેના પાંદડામાં પણ લ્યુટીન, લાઇકોપીન અને વિટામિન એ વધુ માત્રામાં હોય છે. જો તમારી દૃષ્ટિ નબળી પડી રહી છે, તો તમારા માટે તેને આહારમાં સામેલ કરવું ફાયદાકારક રહેશે. લ્યુટીન અને લાઈકોપીન બંને આંખોની રોશની વધારે છે. તેમાં જોવા મળતા કેરોટીનોઈડ પ્રોસ્ટેટ અને કોલોન કેન્સરનું જોખમ ઘટાડે છે. કેરોટીનોઈડ એ એન્ટીઓક્સીડેન્ટ છે જે સ્તન કેન્સરનું જોખમ પણ ઘટાડે છે. ગાજરમાં જોવા મળતા એન્ટીઓક્સીડેન્ટ એન્થોકયાનિન કેન્સરને રોકવામાં પણ મદદ કરી શકે છે. ગાજરના પાનનું સેવન ઓરલ હેલ્થ માટે પણ સારું છે. તે દાંત અને પેઢાંને મજબૂત રાખે છે. તેનું નિયમિત સેવન કરવાથી ફાયદો થશે. ગાજરના પાનમાં રહેલા ગુણો લીવરમાં જોવા મળતી ચરબી અને પિત્તની માત્રાને ઘટાડે છે. આ લીવરની કામગીરીમાં સુધારો કરશે. તેમાં ફાઈબર સારી માત્રામાં હોય છે, જે પેટને સાફ કરે છે અને કબજિયાતની સમસ્યાને પણ દૂર કરે છે. તે વિટામિન K1, A, પોટેશિયમ અને કેલ્શિયમથી ભરપૂર છે. આ પોષક તત્વો હાડકાને તૂટતા અટકાવે છે. આ ઓસ્ટીયોપોરોસિસનું જોખમ ઘટાડે છે. ગાજરના પાનમાં હાજર ફાઇબર શરીરમાં બ્લડ સુગર લેવલને ઘટાડવામાં મદદ કરે છે. તે ઇન્સ્યુલિન પ્રતિકાર અને સંવેદનશીલતાને પ્રોત્સાહન આપે છે. ગાજરમાં હાજર આલ્ફા અને બીટા કેરોટીન બ્લડ સુગર લેવલને નિયંત્રિત કરવામાં મદદ કરે છે. The post ગાજર કરતાં તેમના પાન છે વધુ ગણકારી appeared first on Mukhya Samachar News. | gujurati |
অধিনায়কত্ব কারও জন্মগত অধিকার নয়, বিরাট প্রসঙ্গে বললেন গম্ভীর নজরবন্দি ব্যুরোঃ টিম ইন্ডিয়ার তিন ফরম্যাটের নেতৃত্ব থেকে সরে যাওয়ার পর এখন বিরাট কোহলি Virat Kohli মুক্ত আর কাঁধ থেকে এমন চাপ সরে যাওয়ার জন্যই পুরনো মেজাজে ফের একবার কোহলিকে দেখা যাবে এমনটাই মনে করেন তাঁর প্রাক্তন সতীর্থ গৌতম গম্ভীর Gautam Gambhir আরও পড়ুনঃ কে হবেন পরবর্তী টেস্ট অধিনায়ক? কি ভাবছে সৌরভের বোর্ড? একই সঙ্গে কোহলির প্রতি তাঁর পরামর্শ, অধিনায়কত্ব কারও জন্মগত অধিকার নয় গম্ভীর বলেন, মাত্র চার মাসের ব্যবধানে সব ফরম্যাটের নেতৃত্ব চলে যাওয়া বিরাটের কাছে বড় ধাক্কা কিন্তু ও বুদ্ধিমান ছেলে বিরাট আন্দাজ করেছিল যে ওর সঙ্গে এমন কিছু ঘটতে পারে তাই এমন পরিস্থিতির জন্য প্রস্তুত ছিল তাছাড়া অধিনায়কত্ব কারও জন্মগত অধিকার নয় তবে যাই হোক এখন ও চাপমুক্ত তাই পুরনো মেজাজে ব্যাট করে ফের ঝুড়িঝুড়ি রান করতে সমস্যা হবে বলে মনে হয় না আর সেটা হলে ভারতীয় দল ও বিরাটেরই লাভ হবে অধিনায়কত্ব কারও জন্মগত অধিকার নয়, বিরাট প্রসঙ্গে বললেন গম্ভীর অন্য দিকে টেস্ট অধিনায়কের পদ থেকে বিরাট কোহলি সরে দাঁড়ানোর পরে ওয়াঘার ওপার থেকেও শুভেচ্ছাবার্তা আসতে শুরু করেছে মহম্মদ আমির থেকে আজ়হার মাহমুদ, নাসিম শাহ থেকে আহমেদ শাহজ়াদ, সকলেই বিরাটকে তাঁর নতুন জীবনের জন্য শুভেচ্ছা জানিয়েছেন শুধু পাকিস্তান থেকেই নয়, আগামী ক্রিকেট জীবন নিয়ে শুভেচ্ছা জানিয়েছেন কিংবদন্তি শেন ওয়ার্ন Shane Warneথেকে এবি ডিভিলিয়ার্সও AB de Villiers | bengali |
CERVICAL SPINE, TWO VIEWS.
## INDICATION:
Status post C7 corpectomy.
## FINDINGS:
Comparison to . Postoperative changes status post C7
corpectomy and C6-T1 anterior fusion. Fibular strut graft has been placed
across the corpectomy site with no definite evidence of osseous fusion with C6
or T1. No hardware-related complication is seen. Degenerative disc disease
is noted at C4-C5 and C5-C6 with anterior osteophyte formation. Clips are
seen in the soft tissues. Lung apices are clear.
## IMPRESSION:
Status post C7 corpectomy and C6-T1 anterior fusion with no change in
alignment or evidence of hardware-related complication. Multilevel
degenerative changes, including intervertebral disc space narrowing at C4-C5
and C5-C6.
| medical |
6 ವರ್ಷದ ಬಾಲಕಿಯ ರೇಪ್, ಗುಪ್ತಾಂಗಕ್ಕೆ ಗಂಭೀರ ಗಾಯ: ಜೀವನ್ಮರಣ ಸ್ಥಿತಿಯಲ್ಲಿ ಕಂದ! ಲಕ್ನೋಆ.08: ಆರು ವರ್ಷದ ಬಾಲಕಿನ್ನು ಕಿಡ್ನಾಪ್ ಮಾಡಿದ ವ್ಯಕ್ತಿಯೊಬ್ಬ ಆಖೆಯನ್ನು ಅತ್ಯಾಚಾರಗೈದು, ಹೊಲವೊಂದರಲ್ಲಿ ಸಾಯಲು ಬಿಟ್ಟ ಪರಾರಿಯಾಗಿರುವ ಘಟನೆ ಉತ್ತರ ಪ್ರದೇಶದಲ್ಲಿ ನಡೆದಿದೆ. ಗಂಭೀರ ಸ್ಥಿತಿಯಲ್ಲಿದ್ದ ಬಾಲಕಿಯನ್ನು ಗ್ರಾಮಸ್ಥರು ಪತ್ತೆ ಹಚ್ಚಿದ್ದು, ಸದ್ಯ ಆಕೆ ಆಸ್ಪತ್ರೆಯಲ್ಲಿ ಜೀವನ್ಮರಣ ಸ್ಥಿತಿಯಲ್ಲಿ ಹೋರಾಡುತ್ತಿದ್ದಾಳೆ. ಇನ್ಸ್ಟಾಗ್ರಾಂನಲ್ಲಿ ಬ್ಲಾಕ್ ಮಾಡಿದ ಯುವತಿಯನ್ನು ರೇಪ್ ಮಾಡಿದ! ಆಗಸ್ಟ್ 6 ರಂದು ಹಾಪುರ್ನ ತನ್ನ ಮನೆಯಂಗಳದಲ್ಲಿ ಆಟವಾಡಿಕೊಂಡಿದ್ದಳು. ಈ ವೇಳೆ ಬೈಕ್ನಲ್ಲಿ ಬಂದ ವ್ಯಕ್ತಿಯೊಬ್ಬ ಆಕೆಯನ್ನು ಅಪಹರಿಸಿದ್ದಾನೆ. ಬಳಿಕ ಆಕೆಯನ್ನು ಅತ್ಯಾಚಾರಗೈದ ರಕ್ಕಸ, ಪುಟ್ಟ ಬಾಲಕಿಯ ಗುಪ್ತಾಂಗಕ್ಕೆ ಗಾಯವುಂಟು ಮಾಡಿದ್ದಾನೆ. ಇಷ್ಟಾದ ಬಳಿಕ ಬಾಲಕಿಯನ್ನು ಹಳ್ಳಿಯ ಬಳಿ ಇರುವ ಹೊಲಕ್ಕೆಸೆದು ಪರಾರಿಯಾಗಿದ್ದಾನೆ. ಇನ್ನು ಇತ್ತ ಆಟವಾಡುತ್ತಿದ್ದ ಮಗಳು ಬಹಳ ಹೊತ್ತಾದರೂ ಕಾಣದಾಗ ಆತಂಕಕ್ಕೀಡಾದ ಪೋಷಕರು ಗ್ರಾಮಸ್ಥರೊಂದಿಗೆ ಆಕೆಯ ಹುಡುಕಾಟ ಆರಂಭಿಸಿದ್ದಾರೆ. ಹೀಗಿದ್ದರೂ ಪ್ರಯೋಜನವಾಗದಾಗ ಪೊಲೀಸರಿಗೆ ದೂರು ನೀಡಿದ್ದಾರೆ. ಕಾರ್ಯ ಪ್ರವೃತ್ತರಾದ ಪೊಲೀಸರು ತಂಡವೊಂದನ್ನು ರಚಿಸಿ ಬಾಲಕಿಗಾಗಿ ಹುಡುಕಾಟ ಆರಂಭಿಸಿದ್ದಾರೆ. ಬೆಳಗ್ಗೆ 6 ಗಂಟೆವರೆರೆ ಉಡುಕಾಡಿದರೂ ಬಾಲಕಿ ಪತ್ತೆಯಾಗಲಿಲ್ಲ. ಹೀಗಿರುವಾಗ ಬಾಲಕಿ ಪ್ರಜ್ಞಾಹೀನ ಸ್ಥಿತಿಯಲ್ಲಿ ಹೊಲದಲ್ಲಿ ಪತ್ತೆಯಾಗಿರುವುದಾಗಿ ಕುಟುಂಬ ಸದಸ್ಯರಿಗೆ ಮಾಹಿತಿ ಲಭಿಸುತ್ತದೆ. ನಟಿ ಖುಷ್ಬೂಗೆ ಅತ್ಯಾಚಾರದ ಬೆದರಿಕೆ ಹಾಕಿದ ಅಪರಿಚಿತ ಪಾಠ ಕಲಿಸೋಕೆ ಮಾಡಿದ ಪ್ಲಾನ್ ಇದು? ಕೊನೆಗೂ ನಾಪತ್ತೆಯಾದ 12 ಗಂಟೆಯ ಬಳಿಕ ಆಗಸ್ಟ್ 7 ರಂದು ಬಾಲಕಿ ಪ್ರಜ್ಞಾಹೀನ ಹಾಗೂ ಗಂಭೀರ ಸ್ಥಿತಿಯಲ್ಲಿ ಪತ್ತೆಯಾಗಿದ್ದಾಳೆ. ಬಾಲಕಿ ಪತ್ತೆಯಾಘುತ್ತಿದ್ದಂತೆಯೇ ಗ್ರಾಸ್ಥರೆಲ್ಲಾ ಒಟ್ಟು ಸೇರಿ ಹೊಲದಲ್ಲೆಲ್ಲಾ ಆರೋಪಿಗಾಗಿ ಹುಡುಕಾಟದಿದ್ದು, ಗುಪ್ತಾಂಗಕ್ಕೆಎ ತೀವ್ರ ಹಾನಿಯಾಗಿದೆ ಎಂಬುವುದನ್ನು ಖಚಿತಪಡಿಸಿದ್ದಾರೆ. ಇನ್ನು ಪೊಲೀಸರು ಹಾಗೂ ಕುಟುಂಬ ಸದಸ್ಯರು ಸೇರಿ ಗಮಭೀರ ಸ್ಥಿತಿಯಲ್ಲಿದ್ದ ಬಾಲಕಿಯನ್ನು ಹಾಪುರ್ನ ಆಸ್ಪತ್ರೆಗೆ ಕರೆದೊಯ್ದಿದ್ದಾರೆ. ಇಲ್ಲಿ ಬಾಲಕಿಯನ್ನು ಪರಿಶೀಲಿಸಿದ ವೈದ್ಯರು ಆಪರೇಷನ್ ಅಗತ್ಯವಿದೆ ಎಂದು ಮೀರತ್ನ ಮೆಡಿಕಲ್ ಕಾಲೇಜಿಗೆ ಕರೆದೊಯ್ಯುವಂತೆ ಸೂಚಿಸುತ್ತಾರೆ. ಇಲ್ಲಿ ಬಾಲಕಿಯನ್ನು ತಪಾಸಣೆ ನಡೆಸಿದ ವೈದ್ಯರು ಆಕೆಯ ಮೇಲೆ ಅತ್ಯಾಚಾರ ನಡೆದಿರುವುದನ್ನು ಖಚಿತಪಡಿಸಿದ್ದಾರೆ. ಇನ್ನು ಪೊಲೀಸರು ಎಂಟು ತಂಡಗಳನ್ನು ರಚಿಸಿ ಆರೋಪಿಗಾಗಿ ಹುಡುಕಾಟ ಆರಂಭಿಸಿದ್ದು, ಎಲ್ಲಾ ರೀತಿಯ ತನಿಖೆ ಆರಂಭಿಸಲಾಗಿದೆ. ಅಲ್ಲದೇ ಅತೀ ಶೀಘ್ರದಲ್ಲಿ ಕಾಮುಕನನ್ನು ಬಂಧಿಸುವುದಾಗಿ ಪೊಲೀಸರು ತಿಳಿಸಿದ್ದಾರೆ. | kannad |
GUJARAT CORONA UPDATE: 35 નવા કેસ, 12 દર્દી સાજા થયા, એક પણ મોત નહી ગાંધીનગર : કોરોનાના આંકડા ગુજરાતમાં ઉતાર ચઢાવ જોવા મળી રહ્યો છે. નિષ્ણાંતો દ્વારા વ્યક્ત કરાયેલી ભીતિ કે જુનજુલાઇમાં કોરોનાની વધારે એક લહેર આવી શકે છે સાચી ઠરશે કે શું તેવા સવાલો થઇ રહ્યા છે. રાજ્યમાં આજે કોરોનાના નવા 35 કેસ નોંધાયા છે. બીજી તરફ 12 દર્દીઓ સાજા પણ થયા છે. અત્યાર સુધીમાં કુલ 12,13,502 નાગરિકો કોરોનાને મ્હાત આપી ચુક્યાં છે. જેના પગલે કોરોનાનો રિકવરી રેટ પણ સુધરીને 99.09 ટકાએ પહોંચી ચુક્યો છે. જો કો રસીકરણના મોરચે પણ સરકાર મજબુતીથી લડી રહી છે. રાજ્યમાં રસીના આજે કુલ 33,994 ડોઝ અપાયા હતા. ગુજરાતમાં ઘેર ઘેર હથિયારો પહોંચાડવાનો ટાર્ગેટ હતો? ATS દ્વારા મસમોટા ષડયંત્રનો પર્દાફાશ જો એક્ટિવ કેસની વાત કરીએ તો રાજ્યમાં હાલ કુલ 211 કેસ એક્ટિવ છે. જે પૈકી 02 નાગરિકો વેન્ટિલેટર પર છે. 209 નાગરિકો સ્ટેબલ છે. આ ઉપરાંત અત્યાર સુધીમાં 12,13,502 નાગરિકો ડીસ્ચાર્જ થઇ ચુક્યાં છે. જો કે કોરોનાને કારણે અત્યાર સુધીમાં કુલ 10,944 નાગરિકોનાં મોત પણ થઇ ચુક્યાં છે. જો કે રાહતના સમાચાર કહી શકાય કે, આજે કોરોનાને કારણે એક પણ નાગરિકનું મોત નથી નિપજ્યું. નવા નોંધાયેલા કેસની વાત કરીએ તો અમદાવાદ કોર્પોરેશનમાં 23, વડોદરા કોર્પોરેશન 7, ખેડામાં 2, ગાંધીનગર, સુરત કોર્પોરેશન અને વડોદરામાં 11 કેસ નોંધાયા હતા. કેરી બાદ હવે ચીકુના વેપારીએ પણ રાતે પાણીએ રોવાનો વારો આવ્યો, મફતના ભાવે પણ કોઇ લેવા તૈયાર નથી બીજી તરફ રસીકરણના મોરચે પણ સરકાર મજબુતીથી લડી રહી છે. રાજ્યમાં આજે 18 વર્ષથી વધારેની ઉંમરના 733 ને પ્રથમ તથા 14762 ને રસીનો બીજો ડોઝ અપાયો હતો. 1517 વર્ષના કિશોરો પૈકી 122 ને પ્રથમ અને 1860 ને રસીનો બીજો ડોઝ અપાયો હતો. આ ઉપરાંત 5826 ને પ્રિકોર્શન ડોઝ અપાયો હતો. 1214 વર્ષના તરૂણો પૈકી 1527 ને રસીનો પ્રથમ અને 9164 ને રસીનો બીજો ડોઝ અપાયો હતો. આ પ્રકારે કુલ 33,994 કુલ રસીના ડોઝ આપવામાં આવ્યા હતા. અત્યાર સુધીમાં કુલ 10,83,52,935 રસીના ડોઝ અપાઇ ચુક્યાં છે. | gujurati |
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நாளைய 23022021 நாள் எப்படி இருக்கும்..? இதோ உங்கள் ராசிக்கு.!! நாளைய பஞ்சாங்கம் 23022021, மாசி 11, செவ்வாய்க்கிழமை, ஏகாதசி திதி மாலை 06.05 வரை பின்பு வளர்பிறை துவாதசி. திருவாதிரை நட்சத்திரம் பகல் 12.31 வரை பின்பு புனர்பூசம். மரணயோகம் பகல் 12.31 வரை பின்பு சித்தயோகம். நேத்திரம் 2. ஜீவன் 0. ஏகாதசி விரதம். பெருமாள் வழிபாடு நல்லது. இராகு காலம் மதியம் 03.0004.30, எம கண்டம் காலை 09.0010.30, குளிகன் மதியம் 12.001.30, சுப ஹோரைகள் காலை 8.009.00, மதியம் 12.0001.00, மாலை 04.3005.00, இரவு 07.0008.00, 10.0012.00. நாளைய ராசிப்பலன் 23.02.2021 மேஷம் உங்களின் ராசிக்கு நீங்கள் எந்த ஒரு கடினமான காரியத்தையும் எளிதில் செய்து முடித்து வெற்றி பெறுவீர்கள். சிலர் புதிய வாகனம் வாங்கி மகிழ்வார்கள். உறவினர்கள் வழியாக மனம் மகிழும் செய்திகள் வந்து சேரும். திருமண சுபமுயற்சிகளில் அனுகூலப் பலன்கள் உண்டாகும். பழைய கடன்கள் வசூலாகும். ரிஷபம் உங்களின் ராசிக்கு உத்தியோகஸ்தர்களுக்கு வேலையில் ஆர்வம் குறைந்து காணப்படும். வியாபாரத்தில் எதிர்பாராத செலவுகள் ஏற்படலாம். உறவினர்களின் உதவியால் பணப்பிரச்சினை குறையும். எடுக்கும் முயற்சிகளுக்கு குடும்பத்தினரின் ஆதரவு இருக்கும். வெளியூர் பயணங்களால் அனுகூலம் கிட்டும். மிதுனம் உங்களின் ராசிக்கு உறவினர்களிடம் இருந்த கருத்து வேறுபாடுகள் மறைந்து ஒற்றுமை கூடும். பிள்ளைகள் பொறுப்புடன் நடந்து கொள்வார்கள். பூர்வீக சொத்துக்களால் அனுகூலமான பலன்கள் ஏற்படும். வேலையில் உடனிருப்பவர்கள் ஆதரவாக செயல்படுவார்கள். வியாபாரத்தில் சிறப்பான லாபம் கிட்டும். கடகம் உங்களின் ராசிக்கு குடும்பத்தில் எதிர்பாராத செலவுகள் செய்ய நேரிடும். பெற்றோரிடம் சிறு சிறு கருத்து வேறுபாடுகள் தோன்றும். உத்தியோகத்தில் இருப்பவர்களுக்கு புதிய நபர் அறிமுகம் கிட்டும். அரசு வழியாக எதிர்பார்த்த உதவிகள் கிடைக்கும். உடன்பிறந்தவர்கள் ஆதரவாக இருப்பார்கள். சிம்மம் உங்களின் ராசிக்கு உங்கள் உடல் ஆரோக்கியம் மிக சிறப்பாக இருக்கும். குடும்பத்தில் ஒற்றுமையான சூழ்நிலை உருவாகும். உத்தியோகத்தில் சிலருக்கு உயர் பதவிகள் வந்து சேரும். தொழிலில் புதிய ஒப்பந்தங்கள் கைகூடும். எடுக்கும் முயற்சிகளில் அனுகூலப்பலன் கிட்டும். புதிய பொருட் சேர்க்கை உண்டாகும். கன்னி உங்களின் ராசிக்கு குடும்பத்தில் சுபசெய்திகள் கிடைக்கப்பெற்று மனமகிழ்ச்சி அடைவீர்கள். பிள்ளைகளின் படிப்பில் நல்ல முன்னேற்றம் ஏற்படும். உத்தியோகத்தில் இருந்த பிரச்சினைகள் குறையும். தொழில் சம்பந்தமாக வெளியூர் பயணம் செல்ல நேரிடும். புதிய பொருட்களை வாங்கும் யோகம் உண்டாகும். துலாம் உங்களின் ராசிக்கு நீங்கள் சற்று சோர்வுடனும் சுறுசுறுப்பின்றியும் காணப்படுவீர்கள். நண்பர்கள் மூலம் எதிர்பார்த்த உதவிகள் ஏமாற்றத்தை அளிக்கும். உடன் பிறந்தவர்களால் அனுகூலம் கிட்டும். உத்தியோக ரீதியாக வெளியூர் பயணம் செல்ல நேரிடும். வியாபாரத்தில் கூட்டாளிகளின் ஒத்துழைப்பால் லாபம் கிட்டும். விருச்சிகம் உங்களின் ராசிக்கு நீங்கள் செய்யும் செயல்களில் தாமதப்பலனே ஏற்படும். மருத்துவ செலவுகள் செய்ய நேரிடும். உங்கள் ராசிக்கு சந்திராஷ்டமம் இருப்பதால் மற்றவர்களின் வீண் பேச்சுக்கு ஆளாவீர்கள். அறிமுகம் இல்லாதவர்களிடம் அதிகம் பேசாமல் இருப்பது நல்லது. எந்த ஒரு விஷயத்திலும் கவனம் தேவை. தனுசு உங்களின் ராசிக்கு உங்கள் திறமைகளை வெளிபடுத்தும் நாளாக இருக்கும். குடும்பத்தில் கணவன் மனைவி இடையே மனஸ்தாபங்கள் நீங்கி சந்தோஷம் அதிகரிக்கும். தொழிலில் இருந்த போட்டிகள் விலகும். உத்தியோகத்தில் உயர் அதிகாரிகளின் பாராட்டுதல்கள் கிடைக்கும். செலவுகள் கட்டுக்குள் இருக்கும். மகரம் உங்களின் ராசிக்கு வியாபாரத்தில் நண்பர்கள் மூலம் அனுகூலமான பலன் கிடைக்கும். பிள்ளைகளுடன் இருந்த மனஸ்தாபம் நீங்கும். புதிய பொருட்கள் வீடு வந்து சேரும். உத்தியோகஸ்தர்களுக்கு வேலையில் எதிர்பார்த்த சலுகைகள் கிட்டும். வெளியூர் பயணங்களில் புதிய நபர் அறிமுகம் கிடைக்கும். கும்பம் உங்களின் ராசிக்கு குடும்பத்தில் உள்ளவர்களால் வீண் பிரச்சினைகள் ஏற்படும். உடல்நிலையில் சிறு உபாதைகள் வந்து நீங்கும். தொழிலில் புதிய யுக்திகளை பயன்படுத்தி முன்னேற்றம் அடைவீர்கள். நண்பர்களின் ஒத்துழைப்பால் பொருளாதார பிரச்சினைகள் சற்று குறையும். எதிலும் பொறுமையாக இருப்பது நல்லது. மீனம் உங்களின் ராசிக்கு உங்களுக்கு பணவரவு சுமாராக இருக்கும். குடும்பத்தில் அமைதியற்ற சூழ்நிலை நிலவும். உறவினர்கள் உதவியால் இதுவரை இருந்த பிரச்சினைகள் சற்று குறையும். வியாபாரத்தில் கூட்டாளிகளின் ஒத்துழைப்பு சிறப்பாக இருக்கும். உத்தியோகஸ்தர்களுக்கு பணிச்சுமை குறையும். | tamil |
I visited Tate Modern at the weekend for one of their shows in the magnificent Turbine Hall. It was something to do with immigration and Notting Hill and the carnival and modern London life, maybe.
I captured a little video – example below.
All wonderfully surreal, but there was a moment of intense experience that exhilarated and moved me.
As the masked horde banged their shields and moved around the auditorium, there were also herding the crowd at times, and I found myself hemmed in by the horde on one side and the crowd on the other. I was waiting to be moved back, I felt pressurised, I felt in the way, and then then several people in the front line with me starting reacting in different ways.
One group started dancing to the rhythm of the banging; another linked arms and pretended to front up against the masked horde. One old man (in orange) by me started to stare down a single shield carrier and moved very slowly away, such that that part of the horde was left behind. The individual he chose did not return his gaze, she looked away and I could sense her disconcertedness even behind her mask.
We had moved from a party and performance to a (friendly) confrontation, all in the blink of an eye. I saw there and then how quickly tension escalates at protests, how policing can antagonise unwittingly.
Of course, a gentle art experience at Tate has nothing in common with the struggles of people we see around the world to feel heard – from Ferguson, USA to Tahir Square, Egypt.
But it reminded me how easy it is to divide and polarise; how them and us becomes the norm; how community fractures. In my own work here, there is a red thread of humanity. We need to be on the same side.
That connectedness is available to everyone, every day, but it takes an unyielding commitment to “engender an approach, enable the measurement of distances and walking toward,” as John Pilger said, and as I used in my blog header.
It was a real and insistent reminder, a drum beat call that there is work to be done. | english |
Ind vs SA: ಕೇಪ್ಟೌನ್ ಟೆಸ್ಟ್ನಲ್ಲಿ ಮುಗ್ಗರಿಸಿದ ಭಾರತ, ಟೆಸ್ಟ್ ಸರಣಿ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ಪಾಲು..! ಕೇಪ್ಟೌನ್ಜ.14 ಕಳಪೆ ಬ್ಯಾಟಿಂಗ್ ಹಾಗೂ ಕ್ಷೇತ್ರರಕ್ಷಣೆಗೆ ಭಾರತ ಕ್ರಿಕೆಟ್ ತಂಡ Indian Cricket Team ಕೇಪ್ಟೌನ್ ಟೆಸ್ಟ್ Cape Town Test ಪಂದ್ಯದಲ್ಲಿ ಬೆಲೆ ತೆತ್ತಿದೆ. ಭಾರತ ಹಾಗೂ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ India vs South Africa ತಂಡಗಳ ನಡುವಿನ ನಿರ್ಣಾಯಕ ಟೆಸ್ಟ್ ಪಂದ್ಯದಲ್ಲಿ ಆತಿಥೇಯ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡವು South Africa Cricket Team 7 ವಿಕೆಟ್ಗಳ ಅಂತರದ ಭರ್ಜರಿ ಗೆಲುವು ದಾಖಲಿಸಿದೆ. ಇದರೊಂದಿಗೆ ಮೂರು ಪಂದ್ಯಗಳ ಫ್ರೀಡಂ ಟ್ರೋಫಿ ಟೆಸ್ಟ್ ಸರಣಿಯು ಡೀನ್ ಎಲ್ಹರ್ Dean Elgar ನೇತೃತ್ವದ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡದ ಪಾಲಾಗಿದೆ. ಈ ಸೋಲಿನೊಂದಿಗೆ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ನೆಲದಲ್ಲಿ ಚೊಚ್ಚಲ ಟೆಸ್ಟ್ ಸರಣಿ ಗೆಲ್ಲುವ ಟೀಂ ಇಂಡಿಯಾ ಕನಸು ಮತ್ತೊಮ್ಮೆ ಭಗ್ನವಾಗಿದೆ. ಇಲ್ಲಿನ ನ್ಯೂಲ್ಯಾಂಡ್ಸ್ ಮೈದಾನದಲ್ಲಿ ಭಾರತ ನೀಡಿದ್ದ 212 ರನ್ಗಳ ಗುರಿ ಬೆನ್ನತ್ತಿದ್ದ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡವು ಮೂರನೇ ದಿನದಾಟದಂತ್ಯದ ವೇಳೆಗೆ 2 ವಿಕೆಟ್ ಕಳೆದುಕೊಂಡು 101 ರನ್ ಬಾರಿಸಿತ್ತು. ನಾಲ್ಕನೇ ದಿನದಲ್ಲಿ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡವು 111 ರನ್ಗಳ ಅಗತ್ಯವಿತ್ತು. ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡವು ಮೂರನೇ ವಿಕೆಟ್ಗೆ ರಾಸ್ಸಿ ವ್ಯಾನ್ ಡರ್ ಡುಸೇನ್ ಹಾಗೂ ಕೀಗನ್ ಪೀಟರ್ಸನ್ ಜೋಡಿ 54 ರನ್ಗಳ ಅಮೂಲ್ಯ ಜತೆಯಾಟವಾಡುವ ಮೂಲಕ ತಂಡಕ್ಕೆ ಆಸರೆಯಾದರು. ನಾಲ್ಕನೇ ದಿನದಾಟದ ಆರಂಭದಲ್ಲೇ ಜಸ್ಪ್ರೀತ್ ಬುಮ್ರಾ Jasprit Bumrah ಬೌಲಿಂಗ್ನಲ್ಲಿ ಕೀಗನ್ ಪೀಟರ್ಸನ್ ಬ್ಯಾಟ್ ಸವರಿದ ಚೆಂಡು ಸ್ಪಿಪ್ನಲ್ಲಿದ್ದ ಚೇತೇಶ್ವರ್ ಪೂಜಾರಗೆ ಕೈಗೆ ಸೇರಿತಾದರೂ, ಸುಲಭ ಕ್ಯಾಚನ್ನು ಕೈಚೆಲ್ಲುವ ಮೂಲಕ ನಿರಾಸೆ ಮೂಡಿಸಿದರು. ಆಗ ಪೀಟರ್ಸನ್ 59 ರನ್ ಗಳಿಸಿದ್ದರು. ಬಳಿಕ ಸಿಕ್ಕ ಜೀವಾದಾನವನ್ನು ಸರಿಯಾಗಿಯೇ ಬಳಸಿಕೊಂಡ ಕೀಗನ್ ಪೀಟರ್ಸನ್ 82 ರನ್ ಗಳಿಸಿ ಶಾರ್ದೂಲ್ ಠಾಕೂರ್ ಬೌಲಿಂಗ್ನಲ್ಲಿ ಕ್ಲೀನ್ ಬೌಲ್ಡ್ ಆಗುವ ಮೂಲಕ ಪೆವಿಲಿಯನ್ ಸೇರಿದರು. ತಂಡವನ್ನು ಗೆಲುವಿನ ದಡ ಸೇರಿಸಿದ ಬುವುಮಾಡುಸೇನ್: ಕೀಗನ್ ಪೀಟರ್ಸನ್ ವಿಕೆಟ್ ಪತನದ ಬಳಿಕ ನಾಲ್ಕನೇ ವಿಕೆಟ್ಗೆ ಜತೆಯಾದ ರಾಸ್ಸಿ ವ್ಯಾನ್ ಡರ್ ಡುಸೇನ್ ಹಾಗೂ ತೆಂಬ ಬವುಮಾ ಜೋಡಿ ಮತ್ತೊಮ್ಮೆ ಜವಾಬ್ದಾರಿಯುತ ಬ್ಯಾಟಿಂಗ್ ನಡೆಸುವ ಮೂಲಕ ತಂಡವನ್ನು ಸುಲಭವಾಗಿ ಗೆಲುವಿನ ದಡ ಸೇರಿಸುವಲ್ಲಿ ಯಶಸ್ವಿಯಾದರು. ಬವುಮಾ ಹಾಗೂ ಡುಸೇನ್ ಜೋಡಿ ನಾಲ್ಕನೇ ವಿಕೆಟ್ಗೆ ಮುರಿಯದ 57 ರನ್ಗಳ ಜತೆಯಾಟವಾಡಿ ಹರಿಣಗಳ ಪಡೆಗೆ ಗೆಲುವು ತಂದುಕೊಟ್ಟಿತು. ವ್ಯಾನ್ ಡರ್ ಡುಸೇನ್ 95 ಎಸೆತಗಳನ್ನು ಎದುರಿಸಿ ಅಜೇಯ 41 ರನ್ ಬಾರಿಸಿದರೆ, ತೆಂಬ ಬವುಮಾ 32 ರನ್ ಗಳಿಸಿ ಮಿಂಚಿದರು. DRS Controversy: ಡಿಆರ್ಎಸ್ ಬಗ್ಗೆ ವಿರಾಟ್ ಕೊಹ್ಲಿ ಕೆಂಡಾಮಂಡಲ..! ಹರಿಣಗಳ ನಾಡಲ್ಲಿ ಟೆಸ್ಟ್ ಸರಣಿ ಗೆಲ್ಲುವ ಭಾರತದ ಕನಸು ಭಗ್ನ: ಭಾರತ ತಂಡವು ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ಪ್ರವಾಸದಲ್ಲಿ ಇದುವರೆಗೂ ಟೆಸ್ಟ್ ಸರಣಿಯನ್ನು ಗೆಲ್ಲಲು ಸಾಧ್ಯವಾಗಿಲ್ಲ. ಸೆಂಚೂರಿಯನ್ನಲ್ಲಿ ನಡೆದ ಮೊದಲ ಟೆಸ್ಟ್ ಪಂದ್ಯದಲ್ಲಿ ವಿರಾಟ್ ಕೊಹ್ಲಿ ನೇತೃತ್ವದ ಟೀಂ ಇಂಡಿಯಾ ಭರ್ಜರಿ ಗೆಲುವು ಸಾಧಿಸುವ ಮೂಲಕ ಸರಣಿಯಲ್ಲಿ ಶುಭಾರಂಭ ಮಾಡಿತ್ತು. ಈ ಮೂಲಕ ಹರಿಣಗಳ ನಾಡಿನಲ್ಲಿ ಟೆಸ್ಟ್ ಸರಣಿ ಗೆಲ್ಲುವ ಆಸೆ ಮೂಡಿತ್ತು. ಆದರೆ ಜೋಹಾನ್ಸ್ಬರ್ಗ್ ಟೆಸ್ಟ್ ಹಾಗೂ ಕೇಪ್ಟೌನ್ ಟೆಸ್ಟ್ ಪಂದ್ಯದಲ್ಲಿ ದಕ್ಷಿಣ ಆಫ್ರಿಕಾ ತಂಡವು 7 ವಿಕೆಟ್ಗಳ ಅಂತರದ ಜಯ ಸಾಧಿಸುವ ಮೂಲಕ ಸರಣಿಯನ್ನು ಕೈವಶ ಮಾಡಿಕೊಳ್ಳುವಲ್ಲಿ ಯಶಸ್ವಿಯಾಗಿದೆ. | kannad |
ڈبلن ڈیلی پاکستان لائن مکس مارشل رٹایم ایم اے فائٹر کانر میک گریگر نے ریٹائرمنٹ کا اعلان کر دیا ہے اور کھیل سے منسلک اپنے ساتھیوں کیلئے نیک خواہشات کا اظہار کیا ہے تفصیلات کے مطابق کانر میک گریگر نے سماجی رابطوں کی ویب سائٹ ٹوئٹر پر جاری پیغام میں ریٹائرمنٹ کا اعلان کیا اور لکھا ہیلو دوستو میں نے مکس مارشل رٹ سے فوری ریٹائرمنٹ لینے کا فیصلہ کیا ہے میں اپنے تمام پرانے ساتھیوں کو اس کھیل میں گے بڑھنے کیلئے نیک خواہشات کا اظہار کرتا ہوں میں اب اس مہم میں اپنے سابق پارٹنرز کیساتھ شامل ہو رہا ہوں جو پہلے ہی ریٹائر ہو چکے ہیں کانر میک گریگر نے ریٹائرمنٹ کا اعلان اپنے خری انٹرویو کے چند گھنٹے بعد ہی کیا ہے جس میں انہوں نے اپنی اگلی لڑائی کے بارے میں گاہ کیا تھا انہوں نے کہا تھا کہ میری اگلی فائٹ سے متعلق بات چیت چل رہی ہے جو شائد جولائی میں ہو دیکھیں گے کہ کیا ہوتا ہے بہت سی سیاست چل رہی ہے لڑائی کا کھیل پاگل پن ہے لیکن جیسا کہ میں اپنے مداحوں کو پہلے بھی کہہ چکا ہوں کہ میں تیار ہوں یہاںیہ امر بھی قابل ذکر ہے کہ میک گریگر نے پہلی مرتبہ ریٹائرمنٹ کا اعلان نہیں کیا بلکہ وہ اس سے قبل 2016ءمیں بھی ایسا ہی ایک اعلان کرتے ہوئے ان کا کہنا تھا کہ میں نے نوجوانی میں ہی ریٹائر ہونے کا فیصلہ کر لیا ہے تاہم وہ دوبارہ اس کھیل میں واپس گئے کانر میک گریگر کی خری فائٹ مسلمان روسی فائٹر خبیب کیساتھ ہوئی تھی جس میں انہیں تاریخی شکست کا سامنا کرنا پڑا اور بعد ازاں ایرینا میں اس وقت کچھ نا بھولنے والے مناظر بھی دیکھنے کو ملے جب خبیب جنگلہ پھلانگ کر باہر کود پڑے اور میک گریگر کی ٹیم میں شامل افراد کی جانب سے گالیاں نکالنے پر انہیں پیٹ ڈالا | urdu |
राजधानी शिमला और कांगड़ा में महसूस किए गए भूकंप के झटके शिमला हिमाचल प्रदेश की राजधानी शिमला सहित कांगड़ा में वीरवार को भूकंप के झटके लोगों ने महसूस किए। शिमला में दोपहर 2.55 मिनट पर भूंकप के झटके महसूस किए गए। रिक्टर पैमाने पर भूकंप की तीव्रता 2.5 मापी गई। वहीं, शाम 6.04 मिनट पर कांगड़ा जिले में भूकंप के झटके महसूस किया गया। कांगड़ा में भूंकप की तीव्रता रिक्ट स्केल पर 2.7 मापी गई। हालांकि दोनों ही जगाहों से अभी तक भूकंप से किसी तरह के नुकसान की कोई खबर नहीं है। मौसम विभाग के निदेशक सुरेंद्र पॉल ने कहा कि शिमला और कांगड़ा में गुरुवार को भूकंप के हल्के झटके महसूस किए गए हैं। उन्होंने कहा कि भूकंप ज्यादा तीव्रता वाली नहीं था और कहीं से कोई नुक्सान की सूचना नहीं है। इससे पहले भी प्रदेश के किन्नौर चम्बा में भूकंप के झटके महसूस किए गए हैं। | hindi |
62 ದೇವಳಕ್ಕೆ ವ್ಯವಸ್ಥಾಪನಾ ಸಮಿತಿ ರಚನೆ ಮಂಗಳೂರು, ನ.10: ದ.ಕ. ಜಿಲ್ಲೆಯ ಪ್ರವರ್ಗ ಬಿ ಮತ್ತು ಸಿ ಗೆ ಸೇರಿದ ಒಟ್ಟು 62 ದೇವಸ್ಥಾನದೈವಸ್ಥಾನಗಳಿಗೆ ಮೂರು ವರ್ಷಗಳ ಅವಧಿಗೆ ವ್ಯವಸ್ಥಾಪನಾ ಸಮಿತಿ ರಚಿಸಲಾಗಿದೆ. ಮಂಗಳೂರು ತಾಲೂಕಿನ ಪ್ರವರ್ಗ ಬಿ ಮತ್ತು ಸಿ ಗೆ ಸೇರಿದ ಐದು ದೇವಸ್ಥಾನ, ಬಂಟ್ವಾಳ ತಾಲೂಕಿನ ಪ್ರವರ್ಗ ಸಿಗೆ ಸೇರಿದ 9 ದೇವಸ್ಥಾನ, ಬೆಳ್ತಂಗಡಿ ತಾಲೂಕು ಪ್ರವರ್ಗ ಬಿ ಮತ್ತು ಸಿ ಗೆ ಸೇರಿದ 18 ದೇವಸ್ಥಾನ, ಪುತ್ತೂರು ತಾಲೂಕಿನ 20 ದೇವಸ್ಥಾನ, ಸುಳ್ಯ ತಾಲೂಕಿನ 10 ದೇವಸ್ಥಾನಗಳೂ ಸೇರಿದಂತೆ 62 ದೇವಸ್ಥಾನದೈವಸ್ಥಾನಗಳಿಗೆ ವ್ಯವಸ್ಥಾಪನಾ ಸಮಿತಿ ರಚಿಸಿ ಆದೇಶಿಸಲಾಗಿದೆ ಎಂದು ಜಿಲ್ಲಾ ಧಾರ್ಮಿಕ ಪರಿಷತ್ನ ಸಹಾಯಕ ಆಯುಕ್ತರು ಪ್ರಕಟನೆಯಲ್ಲಿ ತಿಳಿಸಿದ್ದಾರೆ. | kannad |
ਹੈਲੋ'] ", 'ਹੈਲੋ। \n '] | punjabi |
نیویارک اردو پوائنٹ اخبارتازہ ترین اے پی پی 17 جنوری2017ء بالی اور ہالی وڈ اداکارہ پرینکا چوپڑا نے کہا ہے کہ وہ روبصحت ہیںان کو چند روز قبل کوانٹیکو کے سیٹ پر شوٹنگ کے دوران چوٹ لگ گئی تھی جس پر انھیں ہسپتال بھی جانا پڑا پرینکا نے اپنے ٹوئیٹ پر اپنے پرستاروں سے کہا ہے کہ وہ ٹھیک ہیں اور ان کا زخم بھر رہا ہے اور یہ کہ ان کو معالجین نے مرہم پٹی کے بعد ہسپتال سے فارغ کر دیا تھااداکارہ نے اپنے پرستاروں کا شکریہ ادا کرتے ہوئے کہا کہ ان سب کا شکریہ جنہوں نے ان سے خیریت دریافت کی اور ان سے نیک تمنائوں کا اظہار کیا | urdu |
கிரிப்டோ கரன்சியின் முதல் ஊழல்வாதி: தங்கமணி மீது செந்தில் பாலாஜி பாய்ச்சல் இந்திய வரலாற்றிலேயே ஊழல் பணத்தை கிரிப்டோ கரன்சியில் முதலீடு செய்த முதல் அரசியல்வாதி, தங்கமணி, என, தமிழக மின் துறை அமைச்சர் செந்தில் பாலாஜி கூறினார்.டில்லியில் , மத்திய மின்சாரம், மரபுசாரா எரிசக்தி துறை அமைச்சர் ஆர்.கே.சிங்கை, தமிழக மின் துறை அமைச்சர் செந்தில் பாலாஜி சந்தித்துப் பேசினார்.பின் அவர் கூறியதாவது:உள்கட்டமைப்புகளை பலப்படுத்திய பின், தமிழகத்தில் மாதாந்திர மின் கணக்கீடு நடைமுறை அமல்படுத்தப்படும். வரும் நகர்ப்புற உள்ளாட்சித் தேர்தலை மனதில் வைத்து, மின் கட்டணம் உயரப் போவதாக எதிர்க்கட்சிகள் கூறுகின்றன. இதில் உண்மை இல்லை.மாயமான நிலக்கரிலஞ்ச ஒழிப்பு போலீசார் சோதனைக்கான காரணம் குறித்து, முன்னாள் அமைச்சர் தங்கமணி ஒன்றைக் கூறுகிறார். அவரது கட்சியின் ஒருங்கிணைப்பாளரும், இணை ஒருங்கிணைப்பாளரும், வேறு ஒரு விஷயத்தை கூறுகின்றனர். இதில் எது உண்மை என்பது குறித்து, முதலில் அவர்களே ஒரு முடிவுக்கு வரட்டும். பின், என் மீது குற்றம் சொல்லட்டும். இந்தியாவிலேயே ஊழல் பணத்தை கிரிப்டோ கரன்சி எனப்படும் மெய்நிகர் நாணயத்தில் முதலீடு செய்த முதல் அரசியல்வாதி தங்கமணி தான்.வடசென்னை மற்றும் துாத்துக்குடி அனல் மின் நிலையங்களில் காணாமல் போன நிலக்கரி, கண்ணுக்கு தெரியாத கிரிப்டோ கரன்சி இரண்டுக்கும் அவர் முதலில் விளக்கம் சொல்லட்டும்.இவ்வாறு அவர் கூறினார்.10 ஆயிரம் டன் நிலக்கரிமத்திய அமைச்சரிடம் செந்தில் பாலாஜி அளித்துள்ள மனுவில் கூறியிருப்பதாவது:தமிழகத்திற்கு ஆண்டுக்கு 2.37 கோடி டன் நிலக்கரி மத்திய அரசால் அனுப்பப்பட வேண்டிய நிலையில், 1.71 கோடி டன் மட்டுமே வழங்கப்படுகிறது. தினமும் 10 ஆயிரம் டன் நிலக்கரி ஒதுக்கீடு செய்ய வேண்டும்.ஒடிசா மாநிலம், சந்திரபிலா சுரங்கத்தில் இருந்து நிலக்கரி ஒதுக்கீடு பெற, 2016 மார்ச் 30ல் ஒப்பந்தம் ஏற்படுத்தப்பட்டது. ஆனால், 66 மாதங்கள் முடிந்தும், நிலக்கரி எடுக்கப்படவில்லை. வனத் துறை ஒப்புதல் பெற வேண்டியுள்ளது. இதற்கான ஒப்பந்த காலத்தை நீட்டித்து தர வேண்டும்.மத்திய அரசின் நிதி நிறுவனங்களிடம் பெறப்படும் கடன்களுக்கு வட்டி விகிதங்கள் 9.50 சதவீதம் முதல் 12.65 சதவீதமாக உள்ளன.அனைத்து வகை கடன்களுக்கும் 8.50 சதவீதமாக நிர்ணயம் செய்ய வேண்டும்.நிலுவையில் உள்ள 38 கோடி ரூபாயை, டெடா எனப்படும் எரிசக்தி மேம்பாட்டு முகமைக்கு உடனே வழங்க வேண்டும். மின் கொள்முதலுக்கு உத்தரவாதமாக வங்கி உறுதி கடிதம் வழங்க வேண்டும் என்ற உத்தரவை மறுபரிசீலனை செய்ய வேண்டும்.இவ்வாறு அதில் கூறப்பட்டுள்ளது. இந்திய வரலாற்றிலேயே ஊழல் பணத்தை கிரிப்டோ கரன்சியில் முதலீடு செய்த முதல் அரசியல்வாதி, தங்கமணி, என, தமிழக மின் துறை அமைச்சர் செந்தில் பாலாஜி கூறினார்.டில்லியில் , மத்திய சமரசத்துக்கு இடமளிக்காமல்... அதிகாரத்துக்கு அடிபணியாமல்... நேர்மையான முறையில் துணிச்சலான செய்திகளை மக்களிடம் கொண்டு சேர்க்கும் இணையத்தள செய்தி ஊடகங்களுக்கு, விளம்பர வருவாயே உயிர்நாடி. அதுவே, நீங்கள் விரும்பி வாநேசிக்கும் தினமலர், இணையதளத்துக்கும்... ஆகவே அன்பிற்கினிய வாசகர்களே,ஆட்பிளாக்கர் உபயோகிப்பதை தவிர்த்து, துணிச்சலான ஊடகத்தின் நேர்மைக்கு தோள் கொடுங்கள். உங்கள் பார்வைக்கு இடையூறாக வரக்கூடிய விளம்பரத்தை மட்டும், ஸ்கிரீன் ஷாட் எடுத்து எங்களுக்கு அனுப்புங்கள். உங்களின் சிரமத்துக்கு தீர்வு காணுகிறோம். நன்றி. தினமலர் For technical contact : webmasterdinamalar.in இங்கு வெளியாகும் விளம்பரங்கள், வாசகர்களுக்கு பயனளிக்கும் என்பதாலேயே சேர்க்கப்படுகின்றன. Ad blocker போடுவதன் மூலம், பயனுள்ள பல தகவல்களை நீங்கள் தவறவிடவும் வாய்ப்புண்டு. Ad blocker ஐ தவிருங்கள். Youll usually find this icon in the upper righthand corner of your screen. You may have more than one ad blocker installed. You may have to select a menu option or click a button. | tamil |
# Change Log
## [v0.3.2](https://github.com/serulian/compiler/tree/v0.3.2) (2018-09-24)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.3.1...v0.3.2)
**Implemented enhancements:**
- Change Grok to keep the current parameter in `code` form, and instead change other parameters to be bold [\#312](https://github.com/serulian/compiler/pull/312) ([josephschorr](https://github.com/josephschorr))
- Fix formatting of long attributes in SML tags [\#310](https://github.com/serulian/compiler/pull/310) ([josephschorr](https://github.com/josephschorr))
- SourceTracker performance and code improvements [\#307](https://github.com/serulian/compiler/pull/307) ([josephschorr](https://github.com/josephschorr))
- Add a configurable timeout for Grok handle construction [\#306](https://github.com/serulian/compiler/pull/306) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Fix range lookup in Grok of externally imported types [\#311](https://github.com/serulian/compiler/pull/311) ([josephschorr](https://github.com/josephschorr))
- Fix bug where formatter was removing explicitly declared types on lambda parameters [\#309](https://github.com/serulian/compiler/pull/309) ([josephschorr](https://github.com/josephschorr))
- Fix intersection between `null` types and other types [\#308](https://github.com/serulian/compiler/pull/308) ([josephschorr](https://github.com/josephschorr))
- Fix bug where formatter would drop `const` in favor of `var` [\#305](https://github.com/serulian/compiler/pull/305) ([josephschorr](https://github.com/josephschorr))
- Fix translation of promises [\#304](https://github.com/serulian/compiler/pull/304) ([josephschorr](https://github.com/josephschorr))
- Fix various panics when accessing graphs under Grok [\#303](https://github.com/serulian/compiler/pull/303) ([josephschorr](https://github.com/josephschorr))
- Ensure that a return statement is never allowed under a generator [\#302](https://github.com/serulian/compiler/pull/302) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Some small internal fixes [\#301](https://github.com/serulian/compiler/pull/301) ([josephschorr](https://github.com/josephschorr))
## [v0.3.1](https://github.com/serulian/compiler/tree/v0.3.1) (2018-07-15)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.3.0...v0.3.1)
**Implemented enhancements:**
- Have more descriptive parser errors when caused by a lexer error [\#296](https://github.com/serulian/compiler/pull/296) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Fix panic in Grok when user is typing nominal or agent definitions [\#299](https://github.com/serulian/compiler/pull/299) ([josephschorr](https://github.com/josephschorr))
- Fix handling of nominal root operator to always return the \*root\* data type [\#298](https://github.com/serulian/compiler/pull/298) ([josephschorr](https://github.com/josephschorr))
- Fix generation bug when assigning to function variables [\#297](https://github.com/serulian/compiler/pull/297) ([josephschorr](https://github.com/josephschorr))
- Ensure that null comparison operators are wrapped under access expressions [\#295](https://github.com/serulian/compiler/pull/295) ([josephschorr](https://github.com/josephschorr))
- Fix generator bug with Streamable's under loop expressions [\#294](https://github.com/serulian/compiler/pull/294) ([josephschorr](https://github.com/josephschorr))
- Fix bug in Grok cancelation that made it appear a valid handle was being returned [\#293](https://github.com/serulian/compiler/pull/293) ([josephschorr](https://github.com/josephschorr))
- Fix small bugs that lead to various goroutines never terminating [\#291](https://github.com/serulian/compiler/pull/291) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Some small improvements [\#300](https://github.com/serulian/compiler/pull/300) ([josephschorr](https://github.com/josephschorr))
- Disable support for integrations until the Golang plugin system is fixed [\#292](https://github.com/serulian/compiler/pull/292) ([josephschorr](https://github.com/josephschorr))
## [v0.3.0](https://github.com/serulian/compiler/tree/v0.3.0) (2018-06-16)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.2.0...v0.3.0)
**Implemented enhancements:**
- Add support for cancelation [\#290](https://github.com/serulian/compiler/pull/290) ([josephschorr](https://github.com/josephschorr))
- Disallow repeated fields in structural new expressions [\#287](https://github.com/serulian/compiler/pull/287) ([josephschorr](https://github.com/josephschorr))
- Fix type completion in Grok [\#282](https://github.com/serulian/compiler/pull/282) ([josephschorr](https://github.com/josephschorr))
- Allow lambda parameters in inline lambdas to have explicitly declared types [\#277](https://github.com/serulian/compiler/pull/277) ([josephschorr](https://github.com/josephschorr))
- Add support for constant fields at the module level [\#276](https://github.com/serulian/compiler/pull/276) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Various fixes to the formatter [\#288](https://github.com/serulian/compiler/pull/288) ([josephschorr](https://github.com/josephschorr))
- Fix scope checks for lambda return types and add a generator test [\#286](https://github.com/serulian/compiler/pull/286) ([josephschorr](https://github.com/josephschorr))
- Anonymous function and SML fixes [\#284](https://github.com/serulian/compiler/pull/284) ([josephschorr](https://github.com/josephschorr))
- Fix bug in generating flow expressions that marked them as async when they weren't [\#283](https://github.com/serulian/compiler/pull/283) ([josephschorr](https://github.com/josephschorr))
- Fix panic in SRG if given a type path with more than 2 segments [\#275](https://github.com/serulian/compiler/pull/275) ([josephschorr](https://github.com/josephschorr))
- Add extra checks on the `build` command to prevent building non-Serulian source [\#274](https://github.com/serulian/compiler/pull/274) ([josephschorr](https://github.com/josephschorr))
**Closed issues:**
- Anonymous function expression fails to compile [\#264](https://github.com/serulian/compiler/issues/264)
**Merged pull requests:**
- Enable support for casting streams [\#285](https://github.com/serulian/compiler/pull/285) ([josephschorr](https://github.com/josephschorr))
- Fix toolkit for updates to downstream libraries [\#281](https://github.com/serulian/compiler/pull/281) ([josephschorr](https://github.com/josephschorr))
- Major refactoring and addition of integration tooling [\#280](https://github.com/serulian/compiler/pull/280) ([josephschorr](https://github.com/josephschorr))
- Output differences in source mapping tests using diff and fix golden file [\#279](https://github.com/serulian/compiler/pull/279) ([josephschorr](https://github.com/josephschorr))
- Code cleanup in prep for better integration support [\#278](https://github.com/serulian/compiler/pull/278) ([josephschorr](https://github.com/josephschorr))
## [v0.2.0](https://github.com/serulian/compiler/tree/v0.2.0) (2017-12-09)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.1.0...v0.2.0)
**Implemented enhancements:**
- Add support for upgrading the syntax of source code via the formatter [\#272](https://github.com/serulian/compiler/pull/272) ([josephschorr](https://github.com/josephschorr))
- Add new toolkit command: package rev [\#271](https://github.com/serulian/compiler/pull/271) ([josephschorr](https://github.com/josephschorr))
- Add a new toolkit command: package diff [\#270](https://github.com/serulian/compiler/pull/270) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Fix bug in SML scope check which allows functions with extra non-nullable parameters to be used [\#269](https://github.com/serulian/compiler/pull/269) ([josephschorr](https://github.com/josephschorr))
- Fix fmt statements [\#266](https://github.com/serulian/compiler/pull/266) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Update Serulian's syntax for the declared and return type of type and module members [\#273](https://github.com/serulian/compiler/pull/273) ([josephschorr](https://github.com/josephschorr))
- Improvements to the diff engine [\#268](https://github.com/serulian/compiler/pull/268) ([josephschorr](https://github.com/josephschorr))
- Add library for computing the difference between the contents of two type graphs [\#267](https://github.com/serulian/compiler/pull/267) ([josephschorr](https://github.com/josephschorr))
- More performant ImmutableMap [\#265](https://github.com/serulian/compiler/pull/265) ([josephschorr](https://github.com/josephschorr))
## [v0.1.0](https://github.com/serulian/compiler/tree/v0.1.0) (2017-11-02)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.0.2...v0.1.0)
**Implemented enhancements:**
- Add a local name cache to significantly speed up scoping, and add some other small optimizations [\#263](https://github.com/serulian/compiler/pull/263) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Fix error handling of invalid typegraph construction [\#261](https://github.com/serulian/compiler/pull/261) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Small FilteredQuery optimizations [\#262](https://github.com/serulian/compiler/pull/262) ([josephschorr](https://github.com/josephschorr))
## [v0.0.2](https://github.com/serulian/compiler/tree/v0.0.2) (2017-10-28)
[Full Changelog](https://github.com/serulian/compiler/compare/v0.0.1...v0.0.2)
**Fixed bugs:**
- Fix panics when using Grok due to concurrent reads and writes [\#260](https://github.com/serulian/compiler/pull/260) ([josephschorr](https://github.com/josephschorr))
- Various small fixes for discovered bugs [\#257](https://github.com/serulian/compiler/pull/257) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Additional testing of completions in Grok [\#259](https://github.com/serulian/compiler/pull/259) ([josephschorr](https://github.com/josephschorr))
- Add validation of basic types during typegraph construction [\#258](https://github.com/serulian/compiler/pull/258) ([josephschorr](https://github.com/josephschorr))
## [v0.0.1](https://github.com/serulian/compiler/tree/v0.0.1) (2017-10-26)
**Implemented enhancements:**
- Only promise of necessary [\#156](https://github.com/serulian/compiler/pull/156) ([josephschorr](https://github.com/josephschorr))
**Fixed bugs:**
- Support pluses in VCS paths for tags [\#255](https://github.com/serulian/compiler/pull/255) ([josephschorr](https://github.com/josephschorr))
- Fix module naming and pathing for paths containing dots [\#254](https://github.com/serulian/compiler/pull/254) ([josephschorr](https://github.com/josephschorr))
- Bug fixes for Grok [\#251](https://github.com/serulian/compiler/pull/251) ([josephschorr](https://github.com/josephschorr))
- Fix formatting issue on SML attributes without values [\#250](https://github.com/serulian/compiler/pull/250) ([josephschorr](https://github.com/josephschorr))
- Fix bug in generic type caching [\#249](https://github.com/serulian/compiler/pull/249) ([josephschorr](https://github.com/josephschorr))
- Small SML parsing fixes [\#244](https://github.com/serulian/compiler/pull/244) ([josephschorr](https://github.com/josephschorr))
- Parse and format fixes [\#242](https://github.com/serulian/compiler/pull/242) ([josephschorr](https://github.com/josephschorr))
- Exit code handling in the test runner was incorrect [\#241](https://github.com/serulian/compiler/pull/241) ([josephschorr](https://github.com/josephschorr))
- Fix a race condition in the packageloader [\#239](https://github.com/serulian/compiler/pull/239) ([josephschorr](https://github.com/josephschorr))
- Integer division optimization bug [\#238](https://github.com/serulian/compiler/pull/238) ([josephschorr](https://github.com/josephschorr))
- Fix version handling around corelib [\#233](https://github.com/serulian/compiler/pull/233) ([josephschorr](https://github.com/josephschorr))
- Libraries that are imported as aliases need to be imported with their kinds intact [\#224](https://github.com/serulian/compiler/pull/224) ([josephschorr](https://github.com/josephschorr))
- Fix bug in ES5 path normalization for absolute paths [\#223](https://github.com/serulian/compiler/pull/223) ([josephschorr](https://github.com/josephschorr))
- Various bug fixes around the resolve statement [\#218](https://github.com/serulian/compiler/pull/218) ([josephschorr](https://github.com/josephschorr))
- Fix panics due to invalid corelib path and entrypoint handling [\#217](https://github.com/serulian/compiler/pull/217) ([josephschorr](https://github.com/josephschorr))
- Fix formatting and generation of inline called lambda expressions [\#215](https://github.com/serulian/compiler/pull/215) ([josephschorr](https://github.com/josephschorr))
- Fix handling of non-promising dynamic accesses [\#214](https://github.com/serulian/compiler/pull/214) ([josephschorr](https://github.com/josephschorr))
- Really fix nativenew and add a test so we don't break it again [\#213](https://github.com/serulian/compiler/pull/213) ([josephschorr](https://github.com/josephschorr))
- Fixes, cleanup and additional warnings for working with VCS [\#208](https://github.com/serulian/compiler/pull/208) ([josephschorr](https://github.com/josephschorr))
- Don't offer operators in completion [\#192](https://github.com/serulian/compiler/pull/192) ([josephschorr](https://github.com/josephschorr))
- Fix parser to allow `any` values in mapping literals [\#170](https://github.com/serulian/compiler/pull/170) ([josephschorr](https://github.com/josephschorr))
- Fix parsing of multiple annotations on WebIDL declarations [\#160](https://github.com/serulian/compiler/pull/160) ([josephschorr](https://github.com/josephschorr))
- Fix lookup of members directly under modules in type graph [\#159](https://github.com/serulian/compiler/pull/159) ([josephschorr](https://github.com/josephschorr))
- Fix handling of root ops over nullable expressions [\#158](https://github.com/serulian/compiler/pull/158) ([josephschorr](https://github.com/josephschorr))
- Properly check the expressions being returned [\#150](https://github.com/serulian/compiler/pull/150) ([josephschorr](https://github.com/josephschorr))
- Fix generation of nullable access operator [\#149](https://github.com/serulian/compiler/pull/149) ([josephschorr](https://github.com/josephschorr))
- Fix construction of read-write props in interfaces [\#147](https://github.com/serulian/compiler/pull/147) ([josephschorr](https://github.com/josephschorr))
- AssignmentType should never be overridden by context [\#146](https://github.com/serulian/compiler/pull/146) ([josephschorr](https://github.com/josephschorr))
**Merged pull requests:**
- Improvements to the release script, including updating homebrew [\#256](https://github.com/serulian/compiler/pull/256) ([josephschorr](https://github.com/josephschorr))
- Add a cache for lookup of types under modules [\#253](https://github.com/serulian/compiler/pull/253) ([josephschorr](https://github.com/josephschorr))
- Scoping and grok optimizations [\#252](https://github.com/serulian/compiler/pull/252) ([josephschorr](https://github.com/josephschorr))
- Add a scopegraph test for casting to agents [\#248](https://github.com/serulian/compiler/pull/248) ([josephschorr](https://github.com/josephschorr))
- Add better formatting of nested SML loops [\#247](https://github.com/serulian/compiler/pull/247) ([josephschorr](https://github.com/josephschorr))
- Add support for mixing streams and individual children under SML tags [\#246](https://github.com/serulian/compiler/pull/246) ([josephschorr](https://github.com/josephschorr))
- Add support for inline loops in SML [\#245](https://github.com/serulian/compiler/pull/245) ([josephschorr](https://github.com/josephschorr))
- Add a loop-that-yields test to scopegraph tests [\#243](https://github.com/serulian/compiler/pull/243) ([josephschorr](https://github.com/josephschorr))
- Some small tester improvements [\#240](https://github.com/serulian/compiler/pull/240) ([josephschorr](https://github.com/josephschorr))
- Improve the test running experience [\#237](https://github.com/serulian/compiler/pull/237) ([josephschorr](https://github.com/josephschorr))
- Remove unnecessary channel in packageloader [\#236](https://github.com/serulian/compiler/pull/236) ([josephschorr](https://github.com/josephschorr))
- Simplify graph modifier now that Cayley has a dedicated AddQuad method [\#235](https://github.com/serulian/compiler/pull/235) ([josephschorr](https://github.com/josephschorr))
- Change packageloader channels to be buffered [\#234](https://github.com/serulian/compiler/pull/234) ([josephschorr](https://github.com/josephschorr))
- Test runner improvements [\#232](https://github.com/serulian/compiler/pull/232) ([josephschorr](https://github.com/josephschorr))
- Add support for an exclusive range operator [\#231](https://github.com/serulian/compiler/pull/231) ([josephschorr](https://github.com/josephschorr))
- Add better comment onto the release script [\#230](https://github.com/serulian/compiler/pull/230) ([josephschorr](https://github.com/josephschorr))
- Add a "basic" release script for releasing new versions of Serulian [\#229](https://github.com/serulian/compiler/pull/229) ([josephschorr](https://github.com/josephschorr))
- Add Makefile for creating releases of the toolkit [\#228](https://github.com/serulian/compiler/pull/228) ([josephschorr](https://github.com/josephschorr))
- Add in-code support for versions [\#227](https://github.com/serulian/compiler/pull/227) ([josephschorr](https://github.com/josephschorr))
- Make the injected script in develop mode as async [\#226](https://github.com/serulian/compiler/pull/226) ([josephschorr](https://github.com/josephschorr))
- Add better sorting of aliased imports in formatter [\#225](https://github.com/serulian/compiler/pull/225) ([josephschorr](https://github.com/josephschorr))
- Add import alias support [\#222](https://github.com/serulian/compiler/pull/222) ([josephschorr](https://github.com/josephschorr))
- Add support for nested SML attributes [\#221](https://github.com/serulian/compiler/pull/221) ([josephschorr](https://github.com/josephschorr))
- Assign statement auto-unboxing support [\#220](https://github.com/serulian/compiler/pull/220) ([josephschorr](https://github.com/josephschorr))
- Change lexer and parser to treat negative numbers correctly [\#219](https://github.com/serulian/compiler/pull/219) ([josephschorr](https://github.com/josephschorr))
- Change ES5 state machine generation to use a continue where possible [\#216](https://github.com/serulian/compiler/pull/216) ([josephschorr](https://github.com/josephschorr))
- Remove Go 1.6 support, as we now use the plugin lib [\#212](https://github.com/serulian/compiler/pull/212) ([josephschorr](https://github.com/josephschorr))
- Various compilation fixes and some small improvements [\#211](https://github.com/serulian/compiler/pull/211) ([josephschorr](https://github.com/josephschorr))
- Formal support for language integration plugins [\#210](https://github.com/serulian/compiler/pull/210) ([josephschorr](https://github.com/josephschorr))
- Start work on allowing for formal integration of other languages and services into Serulian [\#209](https://github.com/serulian/compiler/pull/209) ([josephschorr](https://github.com/josephschorr))
- Add profile flag to the toolkit [\#207](https://github.com/serulian/compiler/pull/207) ([josephschorr](https://github.com/josephschorr))
- Fix range over composed type reference in Grok [\#206](https://github.com/serulian/compiler/pull/206) ([josephschorr](https://github.com/josephschorr))
- Small Grok improvements [\#205](https://github.com/serulian/compiler/pull/205) ([josephschorr](https://github.com/josephschorr))
- Fix missing change for Grok [\#204](https://github.com/serulian/compiler/pull/204) ([josephschorr](https://github.com/josephschorr))
- Prevent PrincipalType from panicing [\#203](https://github.com/serulian/compiler/pull/203) ([josephschorr](https://github.com/josephschorr))
- Source may not be present on imports when the graph is partial [\#202](https://github.com/serulian/compiler/pull/202) ([josephschorr](https://github.com/josephschorr))
- Fix bug in optimizing binary expression operators [\#201](https://github.com/serulian/compiler/pull/201) ([josephschorr](https://github.com/josephschorr))
- Make names optional on SRGNamedScope, as they may not be available on partial graphs [\#200](https://github.com/serulian/compiler/pull/200) ([josephschorr](https://github.com/josephschorr))
- Move parameter formatting into Grok [\#199](https://github.com/serulian/compiler/pull/199) ([josephschorr](https://github.com/josephschorr))
- Remove param documentation formatting [\#198](https://github.com/serulian/compiler/pull/198) ([josephschorr](https://github.com/josephschorr))
- Fix broken tests [\#197](https://github.com/serulian/compiler/pull/197) ([josephschorr](https://github.com/josephschorr))
- Add ContainsSource to Grok Handle [\#196](https://github.com/serulian/compiler/pull/196) ([josephschorr](https://github.com/josephschorr))
- Allow the PathLoader to override the packages cache path [\#195](https://github.com/serulian/compiler/pull/195) ([josephschorr](https://github.com/josephschorr))
- Some small fixes to source and grok [\#194](https://github.com/serulian/compiler/pull/194) ([josephschorr](https://github.com/josephschorr))
- Docstring improvements in Grok [\#193](https://github.com/serulian/compiler/pull/193) ([josephschorr](https://github.com/josephschorr))
- Lock VCS write operations globally by path [\#191](https://github.com/serulian/compiler/pull/191) ([josephschorr](https://github.com/josephschorr))
- Code context and actions support in Grok [\#190](https://github.com/serulian/compiler/pull/190) ([josephschorr](https://github.com/josephschorr))
- Add call signature lookup support to Grok [\#189](https://github.com/serulian/compiler/pull/189) ([josephschorr](https://github.com/josephschorr))
- Parameter-level documentation and better code summary support [\#188](https://github.com/serulian/compiler/pull/188) ([josephschorr](https://github.com/josephschorr))
- Expression parsing improvements [\#187](https://github.com/serulian/compiler/pull/187) ([josephschorr](https://github.com/josephschorr))
- Completion improvements [\#186](https://github.com/serulian/compiler/pull/186) ([josephschorr](https://github.com/josephschorr))
- Hardening for tooling [\#185](https://github.com/serulian/compiler/pull/185) ([josephschorr](https://github.com/josephschorr))
- Change compiler to use SourceRange's instead of SourceLocation's [\#184](https://github.com/serulian/compiler/pull/184) ([josephschorr](https://github.com/josephschorr))
- More fixes and improvements for IDE tooling [\#183](https://github.com/serulian/compiler/pull/183) ([josephschorr](https://github.com/josephschorr))
- Change packageloader to support an Entrypoint type [\#182](https://github.com/serulian/compiler/pull/182) ([josephschorr](https://github.com/josephschorr))
- Add support for multiple locations in the various graphs and in Grok [\#181](https://github.com/serulian/compiler/pull/181) ([josephschorr](https://github.com/josephschorr))
- Further range improvements in Grok [\#180](https://github.com/serulian/compiler/pull/180) ([josephschorr](https://github.com/josephschorr))
- Additional changes for IDE tooling [\#179](https://github.com/serulian/compiler/pull/179) ([josephschorr](https://github.com/josephschorr))
- Change Grok symbol ordering to use a score metric [\#178](https://github.com/serulian/compiler/pull/178) ([josephschorr](https://github.com/josephschorr))
- Add additional support to Grok [\#177](https://github.com/serulian/compiler/pull/177) ([josephschorr](https://github.com/josephschorr))
- Provide a library for building easy IDE and other tooling [\#176](https://github.com/serulian/compiler/pull/176) ([josephschorr](https://github.com/josephschorr))
- Change list and map literals to be immutable Slice and Mapping literals [\#175](https://github.com/serulian/compiler/pull/175) ([josephschorr](https://github.com/josephschorr))
- Add support for structural function mapping expressions [\#174](https://github.com/serulian/compiler/pull/174) ([josephschorr](https://github.com/josephschorr))
- Statically verify agent construction location [\#173](https://github.com/serulian/compiler/pull/173) ([josephschorr](https://github.com/josephschorr))
- Validate principals of agents after composition occurs [\#172](https://github.com/serulian/compiler/pull/172) ([josephschorr](https://github.com/josephschorr))
- Add the concept of agents to the type system [\#171](https://github.com/serulian/compiler/pull/171) ([josephschorr](https://github.com/josephschorr))
- Fix casting of nulls to any and add some additional tests [\#169](https://github.com/serulian/compiler/pull/169) ([josephschorr](https://github.com/josephschorr))
- Remove support for class composition [\#168](https://github.com/serulian/compiler/pull/168) ([josephschorr](https://github.com/josephschorr))
- Proper handling of native errors [\#167](https://github.com/serulian/compiler/pull/167) ([josephschorr](https://github.com/josephschorr))
- Better handling of conflicts between WebIDL files [\#166](https://github.com/serulian/compiler/pull/166) ([josephschorr](https://github.com/josephschorr))
- Fix panic on systems with dashes in their path [\#165](https://github.com/serulian/compiler/pull/165) ([josephschorr](https://github.com/josephschorr))
- Small fixes for resolve and structs [\#164](https://github.com/serulian/compiler/pull/164) ([josephschorr](https://github.com/josephschorr))
- Support semantic versioning in import commands [\#163](https://github.com/serulian/compiler/pull/163) ([josephschorr](https://github.com/josephschorr))
- Further formatter improvements [\#162](https://github.com/serulian/compiler/pull/162) ([josephschorr](https://github.com/josephschorr))
- Improvements and additional tests for the source formatter [\#161](https://github.com/serulian/compiler/pull/161) ([josephschorr](https://github.com/josephschorr))
- Fix handling of parsing errors in WebIDL [\#157](https://github.com/serulian/compiler/pull/157) ([josephschorr](https://github.com/josephschorr))
- Basic optimization of generated ES5 code [\#155](https://github.com/serulian/compiler/pull/155) ([josephschorr](https://github.com/josephschorr))
- Property optimizations [\#154](https://github.com/serulian/compiler/pull/154) ([josephschorr](https://github.com/josephschorr))
- Depromise vars for a small performance boost [\#153](https://github.com/serulian/compiler/pull/153) ([josephschorr](https://github.com/josephschorr))
- Dependent variable initialization fix [\#152](https://github.com/serulian/compiler/pull/152) ([josephschorr](https://github.com/josephschorr))
- Change lookup of containing nodes to use the local scope context [\#151](https://github.com/serulian/compiler/pull/151) ([josephschorr](https://github.com/josephschorr))
- Better define the rules around when types can be cast and cannot [\#148](https://github.com/serulian/compiler/pull/148) ([josephschorr](https://github.com/josephschorr))
- Struct default values [\#145](https://github.com/serulian/compiler/pull/145) ([josephschorr](https://github.com/josephschorr))
- Add support for using classes as props in SML expressions [\#144](https://github.com/serulian/compiler/pull/144) ([josephschorr](https://github.com/josephschorr))
- Small bug fixes [\#143](https://github.com/serulian/compiler/pull/143) ([josephschorr](https://github.com/josephschorr))
- Structural any type reference suppot [\#142](https://github.com/serulian/compiler/pull/142) ([josephschorr](https://github.com/josephschorr))
- Reduce the noise of errors by skipping checks on invalid scoped names [\#141](https://github.com/serulian/compiler/pull/141) ([josephschorr](https://github.com/josephschorr))
- Fix generics bug when referencing a generic scope under a generic static scope [\#140](https://github.com/serulian/compiler/pull/140) ([josephschorr](https://github.com/josephschorr))
- Fix formatting of nullable op nesting [\#139](https://github.com/serulian/compiler/pull/139) ([josephschorr](https://github.com/josephschorr))
- Add an ES5 integration test with a single call state [\#138](https://github.com/serulian/compiler/pull/138) ([josephschorr](https://github.com/josephschorr))
- Use strict, bug fixes and code gen optimizations [\#137](https://github.com/serulian/compiler/pull/137) ([josephschorr](https://github.com/josephschorr))
- Remove superfluous warning [\#136](https://github.com/serulian/compiler/pull/136) ([josephschorr](https://github.com/josephschorr))
- Bug fixes [\#135](https://github.com/serulian/compiler/pull/135) ([josephschorr](https://github.com/josephschorr))
- Various small fixes [\#134](https://github.com/serulian/compiler/pull/134) ([josephschorr](https://github.com/josephschorr))
- Fix handling of paths in Docker image [\#133](https://github.com/serulian/compiler/pull/133) ([josephschorr](https://github.com/josephschorr))
- Remove .dockerignore and add badge [\#132](https://github.com/serulian/compiler/pull/132) ([josephschorr](https://github.com/josephschorr))
- Add Dockerfile [\#131](https://github.com/serulian/compiler/pull/131) ([josephschorr](https://github.com/josephschorr))
- Additional testing and implementation for formatting tool [\#130](https://github.com/serulian/compiler/pull/130) ([josephschorr](https://github.com/josephschorr))
- Fix whitespace parsing of text under SML [\#129](https://github.com/serulian/compiler/pull/129) ([josephschorr](https://github.com/josephschorr))
- Change access rules for unexported names [\#128](https://github.com/serulian/compiler/pull/128) ([josephschorr](https://github.com/josephschorr))
- Fix source mapping and add a simple test [\#127](https://github.com/serulian/compiler/pull/127) ([josephschorr](https://github.com/josephschorr))
- Add missing formatter methods [\#126](https://github.com/serulian/compiler/pull/126) ([josephschorr](https://github.com/josephschorr))
- Fix generation of static op calls under generic types [\#125](https://github.com/serulian/compiler/pull/125) ([josephschorr](https://github.com/josephschorr))
- Add a cache for the containing implemented query in the SRG [\#124](https://github.com/serulian/compiler/pull/124) ([josephschorr](https://github.com/josephschorr))
- Some small optimizations [\#123](https://github.com/serulian/compiler/pull/123) ([josephschorr](https://github.com/josephschorr))
- Add check to disallow nullable types in match statements [\#122](https://github.com/serulian/compiler/pull/122) ([josephschorr](https://github.com/josephschorr))
- Update .gitignore [\#121](https://github.com/serulian/compiler/pull/121) ([josephschorr](https://github.com/josephschorr))
- Nullable if and not operator support [\#120](https://github.com/serulian/compiler/pull/120) ([josephschorr](https://github.com/josephschorr))
- Match type statement [\#119](https://github.com/serulian/compiler/pull/119) ([josephschorr](https://github.com/josephschorr))
- Change `match` statements to `switch` statements [\#118](https://github.com/serulian/compiler/pull/118) ([josephschorr](https://github.com/josephschorr))
- Add prefix-parsing for nullable and streams in type references [\#117](https://github.com/serulian/compiler/pull/117) ([josephschorr](https://github.com/josephschorr))
- Fix for Cayley changes [\#116](https://github.com/serulian/compiler/pull/116) ([josephschorr](https://github.com/josephschorr))
- Simplify generation of structs [\#115](https://github.com/serulian/compiler/pull/115) ([josephschorr](https://github.com/josephschorr))
- Struct clone [\#114](https://github.com/serulian/compiler/pull/114) ([josephschorr](https://github.com/josephschorr))
- See if we can fix travis by enabling sudo for more memory [\#113](https://github.com/serulian/compiler/pull/113) ([josephschorr](https://github.com/josephschorr))
- Small query performance improvements [\#112](https://github.com/serulian/compiler/pull/112) ([josephschorr](https://github.com/josephschorr))
- Various small fixes [\#111](https://github.com/serulian/compiler/pull/111) ([josephschorr](https://github.com/josephschorr))
- Changes to make use of typed cayley graph values [\#110](https://github.com/serulian/compiler/pull/110) ([josephschorr](https://github.com/josephschorr))
- Nullable cast checking fix [\#109](https://github.com/serulian/compiler/pull/109) ([josephschorr](https://github.com/josephschorr))
- Fix for recent cobra changes [\#108](https://github.com/serulian/compiler/pull/108) ([josephschorr](https://github.com/josephschorr))
- Relative import support [\#107](https://github.com/serulian/compiler/pull/107) ([josephschorr](https://github.com/josephschorr))
- Fix for recent cobra changes [\#106](https://github.com/serulian/compiler/pull/106) ([josephschorr](https://github.com/josephschorr))
- Add support for the resolve statement [\#105](https://github.com/serulian/compiler/pull/105) ([josephschorr](https://github.com/josephschorr))
- Nominal shortcutting support [\#104](https://github.com/serulian/compiler/pull/104) ([josephschorr](https://github.com/josephschorr))
- Various smaller optimizations [\#103](https://github.com/serulian/compiler/pull/103) ([josephschorr](https://github.com/josephschorr))
- Any-related things [\#102](https://github.com/serulian/compiler/pull/102) ([josephschorr](https://github.com/josephschorr))
- Change to new namespace for Cayley [\#101](https://github.com/serulian/compiler/pull/101) ([josephschorr](https://github.com/josephschorr))
- Various small fixes [\#100](https://github.com/serulian/compiler/pull/100) ([josephschorr](https://github.com/josephschorr))
- Serulian Markup Language support [\#99](https://github.com/serulian/compiler/pull/99) ([josephschorr](https://github.com/josephschorr))
- Inline loop expression support [\#98](https://github.com/serulian/compiler/pull/98) ([josephschorr](https://github.com/josephschorr))
- Add a generator test using a resource [\#97](https://github.com/serulian/compiler/pull/97) ([josephschorr](https://github.com/josephschorr))
- Add support for generators [\#96](https://github.com/serulian/compiler/pull/96) ([josephschorr](https://github.com/josephschorr))
- Short circuiting fixes and improvements [\#95](https://github.com/serulian/compiler/pull/95) ([josephschorr](https://github.com/josephschorr))
- True casting [\#94](https://github.com/serulian/compiler/pull/94) ([josephschorr](https://github.com/josephschorr))
- Multiple import support [\#92](https://github.com/serulian/compiler/pull/92) ([josephschorr](https://github.com/josephschorr))
- Various small fixes [\#91](https://github.com/serulian/compiler/pull/91) ([josephschorr](https://github.com/josephschorr))
- Another set of small optimizations [\#90](https://github.com/serulian/compiler/pull/90) ([josephschorr](https://github.com/josephschorr))
- Disable GC when run as a single step binary [\#89](https://github.com/serulian/compiler/pull/89) ([josephschorr](https://github.com/josephschorr))
- Small fixes and optimization [\#88](https://github.com/serulian/compiler/pull/88) ([josephschorr](https://github.com/josephschorr))
- Various optimizations [\#87](https://github.com/serulian/compiler/pull/87) ([josephschorr](https://github.com/josephschorr))
- Change Iter on ordered\_map usage to UnsafeIter [\#86](https://github.com/serulian/compiler/pull/86) ([josephschorr](https://github.com/josephschorr))
- Make sure boxing of null returns null [\#85](https://github.com/serulian/compiler/pull/85) ([josephschorr](https://github.com/josephschorr))
- Fix void settlement error [\#84](https://github.com/serulian/compiler/pull/84) ([josephschorr](https://github.com/josephschorr))
- Fix handling of args to async functions [\#83](https://github.com/serulian/compiler/pull/83) ([josephschorr](https://github.com/josephschorr))
- Optimize generation of root nominal operator [\#82](https://github.com/serulian/compiler/pull/82) ([josephschorr](https://github.com/josephschorr))
- Add basic README about the various toolkit commands [\#81](https://github.com/serulian/compiler/pull/81) ([josephschorr](https://github.com/josephschorr))
- Start on README and Travis support [\#80](https://github.com/serulian/compiler/pull/80) ([josephschorr](https://github.com/josephschorr))
- Move Serulian main package into a cmd folder [\#79](https://github.com/serulian/compiler/pull/79) ([josephschorr](https://github.com/josephschorr))
- Refactor the entire ES5 generator [\#78](https://github.com/serulian/compiler/pull/78) ([josephschorr](https://github.com/josephschorr))
- Start on ES builder library for better code generation [\#77](https://github.com/serulian/compiler/pull/77) ([josephschorr](https://github.com/josephschorr))
- Initial work on testing plugin and framework support [\#76](https://github.com/serulian/compiler/pull/76) ([josephschorr](https://github.com/josephschorr))
- Small source mapping improvements [\#75](https://github.com/serulian/compiler/pull/75) ([josephschorr](https://github.com/josephschorr))
- Add basic source mapping support for all generated code [\#74](https://github.com/serulian/compiler/pull/74) ([josephschorr](https://github.com/josephschorr))
- Add commands for freezing and unfreezing VCS imports [\#73](https://github.com/serulian/compiler/pull/73) ([josephschorr](https://github.com/josephschorr))
- Fix formatting of template strings [\#72](https://github.com/serulian/compiler/pull/72) ([josephschorr](https://github.com/josephschorr))
- Fix formatting of imports [\#71](https://github.com/serulian/compiler/pull/71) ([josephschorr](https://github.com/josephschorr))
- Formatter command [\#70](https://github.com/serulian/compiler/pull/70) ([josephschorr](https://github.com/josephschorr))
- State machine optimizations [\#69](https://github.com/serulian/compiler/pull/69) ([josephschorr](https://github.com/josephschorr))
- Optimize handling of structs and nominals [\#68](https://github.com/serulian/compiler/pull/68) ([josephschorr](https://github.com/josephschorr))
- Various bug fixes and improvements [\#67](https://github.com/serulian/compiler/pull/67) ([josephschorr](https://github.com/josephschorr))
- Add binary numeric literal support [\#66](https://github.com/serulian/compiler/pull/66) ([josephschorr](https://github.com/josephschorr))
- Basic development mode for the compiler [\#65](https://github.com/serulian/compiler/pull/65) ([josephschorr](https://github.com/josephschorr))
- Require explicitly declared types on module and class-level vars [\#64](https://github.com/serulian/compiler/pull/64) ([josephschorr](https://github.com/josephschorr))
- Fix lazy checking of nominal fields in structs [\#63](https://github.com/serulian/compiler/pull/63) ([josephschorr](https://github.com/josephschorr))
- Fix bug where we were not properly transforming binary and unary op return types [\#62](https://github.com/serulian/compiler/pull/62) ([josephschorr](https://github.com/josephschorr))
- Mapping literal support [\#61](https://github.com/serulian/compiler/pull/61) ([josephschorr](https://github.com/josephschorr))
- Large set of fixes for bugs discovered recently [\#60](https://github.com/serulian/compiler/pull/60) ([josephschorr](https://github.com/josephschorr))
- Add missing finish node call [\#59](https://github.com/serulian/compiler/pull/59) ([josephschorr](https://github.com/josephschorr))
- WebIDL and slice improvements [\#58](https://github.com/serulian/compiler/pull/58) ([josephschorr](https://github.com/josephschorr))
- Better numeric literals [\#57](https://github.com/serulian/compiler/pull/57) ([josephschorr](https://github.com/josephschorr))
- Regenerate golden files and add --vcs-dev-dir flag [\#56](https://github.com/serulian/compiler/pull/56) ([josephschorr](https://github.com/josephschorr))
- Structural equality [\#55](https://github.com/serulian/compiler/pull/55) ([josephschorr](https://github.com/josephschorr))
- Better resolution error messages [\#54](https://github.com/serulian/compiler/pull/54) ([josephschorr](https://github.com/josephschorr))
- Disallow slicing and indexing on nullable values [\#53](https://github.com/serulian/compiler/pull/53) ([josephschorr](https://github.com/josephschorr))
- In operator support [\#52](https://github.com/serulian/compiler/pull/52) ([josephschorr](https://github.com/josephschorr))
- Change nominals to always be a single-level wrapping [\#51](https://github.com/serulian/compiler/pull/51) ([josephschorr](https://github.com/josephschorr))
- Add unexported member test [\#50](https://github.com/serulian/compiler/pull/50) ([josephschorr](https://github.com/josephschorr))
- Fix some possible infinite loops in the parser [\#49](https://github.com/serulian/compiler/pull/49) ([josephschorr](https://github.com/josephschorr))
- Fixes for serialization, async, nominals and the type system [\#48](https://github.com/serulian/compiler/pull/48) ([josephschorr](https://github.com/josephschorr))
- Async functions [\#47](https://github.com/serulian/compiler/pull/47) ([josephschorr](https://github.com/josephschorr))
- Basic serialization support for structs with JSON [\#46](https://github.com/serulian/compiler/pull/46) ([josephschorr](https://github.com/josephschorr))
- Adding mapping and generic nominal types [\#45](https://github.com/serulian/compiler/pull/45) ([josephschorr](https://github.com/josephschorr))
- Some small optimizations [\#44](https://github.com/serulian/compiler/pull/44) ([josephschorr](https://github.com/josephschorr))
- Structural type support [\#43](https://github.com/serulian/compiler/pull/43) ([josephschorr](https://github.com/josephschorr))
- Better Messaging [\#42](https://github.com/serulian/compiler/pull/42) ([josephschorr](https://github.com/josephschorr))
- Transactional changes [\#41](https://github.com/serulian/compiler/pull/41) ([josephschorr](https://github.com/josephschorr))
- Ensure we have non-nullable types where expected [\#40](https://github.com/serulian/compiler/pull/40) ([josephschorr](https://github.com/josephschorr))
- Add code generation for remaining literals [\#39](https://github.com/serulian/compiler/pull/39) ([josephschorr](https://github.com/josephschorr))
- Slice typeref support and some small fixes [\#38](https://github.com/serulian/compiler/pull/38) ([josephschorr](https://github.com/josephschorr))
- Fix panic when trying to resolve under 'any' constrained generic [\#37](https://github.com/serulian/compiler/pull/37) ([josephschorr](https://github.com/josephschorr))
- Some small fixes and root type operator [\#36](https://github.com/serulian/compiler/pull/36) ([josephschorr](https://github.com/josephschorr))
- Fix handling of nominal types [\#35](https://github.com/serulian/compiler/pull/35) ([josephschorr](https://github.com/josephschorr))
- Fix handling of generic constraints with subtypes and member resolution [\#34](https://github.com/serulian/compiler/pull/34) ([josephschorr](https://github.com/josephschorr))
- Null equality comparison operator [\#33](https://github.com/serulian/compiler/pull/33) ([josephschorr](https://github.com/josephschorr))
- Better messaging [\#32](https://github.com/serulian/compiler/pull/32) ([josephschorr](https://github.com/josephschorr))
- Ensure all promise paths resolve [\#31](https://github.com/serulian/compiler/pull/31) ([josephschorr](https://github.com/josephschorr))
- Smaller fixes [\#30](https://github.com/serulian/compiler/pull/30) ([josephschorr](https://github.com/josephschorr))
- Have WebIDL operators construct their types from the definitions [\#29](https://github.com/serulian/compiler/pull/29) ([josephschorr](https://github.com/josephschorr))
- Fix generation of loops over expressions [\#28](https://github.com/serulian/compiler/pull/28) ([josephschorr](https://github.com/josephschorr))
- Fix ReplaceType for unequal replacement strings [\#27](https://github.com/serulian/compiler/pull/27) ([josephschorr](https://github.com/josephschorr))
- Add support for Streamable to loops [\#26](https://github.com/serulian/compiler/pull/26) ([josephschorr](https://github.com/josephschorr))
- Fix handling of identifiers referring to aliased types [\#25](https://github.com/serulian/compiler/pull/25) ([josephschorr](https://github.com/josephschorr))
- Add tests and parsing for accessing a member under a literal value [\#24](https://github.com/serulian/compiler/pull/24) ([josephschorr](https://github.com/josephschorr))
- Fix parsing of numeric literal ranges [\#23](https://github.com/serulian/compiler/pull/23) ([josephschorr](https://github.com/josephschorr))
- Small code fix for a test [\#22](https://github.com/serulian/compiler/pull/22) ([josephschorr](https://github.com/josephschorr))
- Fix parsing issue around multiline comments [\#21](https://github.com/serulian/compiler/pull/21) ([josephschorr](https://github.com/josephschorr))
- WebIDL indexer support [\#20](https://github.com/serulian/compiler/pull/20) ([josephschorr](https://github.com/josephschorr))
- Small fixes and improvements [\#19](https://github.com/serulian/compiler/pull/19) ([josephschorr](https://github.com/josephschorr))
- Better handling of promises and arrows [\#18](https://github.com/serulian/compiler/pull/18) ([josephschorr](https://github.com/josephschorr))
- New generation of ES5 based on a CodeDOM model. [\#17](https://github.com/serulian/compiler/pull/17) ([josephschorr](https://github.com/josephschorr))
- Disallow inheritance from special type references \(any, nullable, str… [\#16](https://github.com/serulian/compiler/pull/16) ([josephschorr](https://github.com/josephschorr))
- Nominal type support [\#15](https://github.com/serulian/compiler/pull/15) ([josephschorr](https://github.com/josephschorr))
- WebIDL operator support [\#14](https://github.com/serulian/compiler/pull/14) ([josephschorr](https://github.com/josephschorr))
- WebIDL support [\#13](https://github.com/serulian/compiler/pull/13) ([josephschorr](https://github.com/josephschorr))
- Switch to using a github-based core library [\#12](https://github.com/serulian/compiler/pull/12) ([josephschorr](https://github.com/josephschorr))
- Initial implementation of ES5 code generation [\#11](https://github.com/serulian/compiler/pull/11) ([josephschorr](https://github.com/josephschorr))
- Full scoping and scope graph construction [\#10](https://github.com/serulian/compiler/pull/10) ([josephschorr](https://github.com/josephschorr))
- Full type graph construction and semantic checking [\#9](https://github.com/serulian/compiler/pull/9) ([josephschorr](https://github.com/josephschorr))
- Initial package loader merge PR [\#7](https://github.com/serulian/compiler/pull/7) ([josephschorr](https://github.com/josephschorr))
- Begin work on VCS import support [\#5](https://github.com/serulian/compiler/pull/5) ([josephschorr](https://github.com/josephschorr))
- Full implementation of the parser for the initial Serulian spec'ed language [\#4](https://github.com/serulian/compiler/pull/4) ([josephschorr](https://github.com/josephschorr))
- Merge parser into master [\#3](https://github.com/serulian/compiler/pull/3) ([josephschorr](https://github.com/josephschorr))
- Foundation of the Serulian parser [\#2](https://github.com/serulian/compiler/pull/2) ([josephschorr](https://github.com/josephschorr))
- Initial implementation of the Serulian compiler's lexer [\#1](https://github.com/serulian/compiler/pull/1) ([josephschorr](https://github.com/josephschorr))
\* *This Change Log was automatically generated by [github_changelog_generator](https://github.com/skywinder/Github-Changelog-Generator)* | code |
മേതില് ദേവികയുമായുള്ള വിവാഹ മോചനം: അടുത്ത കല്യാണം എന്നാണ്, എംഎല്എ സാറെന്ന് സോഷ്യല് മീഡിയ, പ്രതിഷേധം കൊല്ലം: നര്ത്തകി മേതില് ദേവികയുമായുള്ള വിവാഹ ബന്ധം വേര്പിരിയുന്നുവെന്ന റിപ്പോര്ട്ട് പുറത്തുവന്നതിന് പിന്നാലെ മുകേഷിനെതിരെ സോഷ്യല് മീഡിയയില് പ്രതിഷേധം ശക്തം. ഇന്നലെയാണ് വിവാഹ ബന്ധം വേര്പിരിയുന്നുവെന്ന വാര്ത്ത സമൂഹമാദ്ധ്യമങ്ങളില് പ്രചരിച്ചത്. എട്ടു വര്ഷം മുന്പാണ് മുകേഷും ദേവികയും വിവാഹിതരായത്. വാര്ത്തകള് പുറത്ത് വന്നതോടെ നിരവധി പ്രതിഷേധ കമന്റുകളാണ് മുകേഷിന്റെ ഫേസ്ബുക്ക് പേജില് വരുന്നത്. നടന് ആയത്. ജീവിതത്തില് പലരുടെയും വില്ലന് ആവാന് ആയിരുന്നുല്ലേ എന്നാണ് ഒരാള് കുറിച്ചത്. വധുവിനെ ആവശ്യമുണ്ട്, വിവാഹതട്ടിപ്പ് വീരന് കല്യാണ ഉണ്ണി മുകേഷ് കൊല്ലത്തിന്റെ ഐശ്വര്യം, അടുത്ത കല്യാണം ഇനി എന്നാണ്, എംഎല്എ സാര് എന്നിങ്ങനെ നിരവധി കമന്റുകളാണ് മുകേഷിന്റെ പേജില് പ്രത്യക്ഷപ്പെടുന്നത്. രണ്ടു പേരുടെയും ആശയങ്ങള് യോജിച്ച് പോകുന്ന സാഹചര്യമല്ലെന്ന് തോന്നിയതിനാലാണ് വിവാഹബന്ധം വേര്പെടുത്തുന്നതെന്നാണ് ദേവിക പ്രതികരിച്ചത്. ആദ്യ വിവാഹബന്ധം വേര്പിരിഞ്ഞ ശേഷമായിരുന്നു ദേവികയുമായുള്ള വിവാഹം. തെന്നിന്ത്യന് നടി സരിതയാണ് മുകേഷിന്റെ ആദ്യ ഭാര്യ. 1987ല് വിവാഹിതരായ മുകേഷും സരിതയും 25 വര്ഷത്തെ വിവാഹ ജീവിതത്തിനു ശേഷം 2011ലായിരുന്നു വേര്പിരിഞ്ഞത്. പ്രണയവിവാഹമായിരുന്നു മുകേഷിന്റേയും സരിതയുടേതും. ദേവികയുമായുളള വിവാഹത്തിന് ശേഷം മുകേഷിനെതിരെ സരിത രംഗത്ത് എത്തിയിരുന്നു. അന്ന് നടനെതിരെ നിരവധി ആരോപണങ്ങള് സരിത ഉന്നയിച്ചിരുന്നു. രണ്ടാമതും വിവാഹമോചനത്തെ കുറിച്ചുള്ള വാര്ത്തകള് പുറത്ത് വരുമ്ബോള് അന്ന് സരിത ഉന്നയിച്ച ആരോപണങ്ങള് വീണ്ടും സോഷ്യല് മീഡിയയില് ചര്ച്ചയാവുകയാണ്. | malyali |
झारखण्ड के एक्सप्रेसवे निर्माण कार्य मे आएगा तेजी, वन्दे भारत ट्रेन भी जल्दी चलेगी केंद्रीय वित्त मंत्री निर्मला सीतारमण ने मंगलवार को वित्तीय वर्ष 202223 का बजट पेश किया। इससे झारखंड में एक्सप्रवेस वे और सौर ऊर्जा को गति मिलेगी। इतना ही नहीं केंद्रीय योजनाओं में बजट प्रावधान से हिस्सेदारी से भी झारखंड को लाभ मिलेगा। झारखंड में उद्योग, आधारभूत संरचना, कृषि, आवास, निर्मल जल योजना के साथ शिक्षा के विकास में भी संभावनाओ को बल मिलेगा। झारखंड में नई सौर ऊर्जा नीति लाई जा रही है। इसमें अगले 5 वर्षों में 5000 मेगावाट सौर ऊर्जा उत्पादन का लक्ष्य है। केंद्र के आम बजट में सौर ऊर्जा के क्षेत्र में 2030 तक 280 गीगा वाट सौर ऊर्जा उत्पादन का लक्ष्य रखा गया है। इसके लिए अतिरिक्त 19500 करोड़ निवेश किया जाएगा। झारखंड को इससे काफी लाभ होगा। झारखंड में आवास योजना के तहत काम चल रहा है। अभी 60353 आवास का निर्माण होना है। कुल तीन लाख 14 हजार आवास योजना को स्वीकृति मिली है। इन आवासों के निर्माण की पहली किश्त की राशि मिलना बाकी है। इस बजट में भी पीएम आवास निर्माण के लिए बड़ी राशि का प्रावधान किया गया है। उम्मीद है कि इस बजट में आवास योजना के लिए झारखंड अपने हिस्से की राशि ले पाएगा। राज्य में एक्सप्रेसवे की प्रस्तावित योजनाएं इसके अलावा 25000 किलोमीटर राष्ट्रीय उच्च पथ परियोजना में झारखंड में एक्सप्रेसवे की चार योजनाएं शुरू होनेवाली हैं। इनका निर्माण कार्य पाइप लाइन में है। रांची से लिट्टीबेड़ा 3200 करोड़ सड़क में 119 किलोमीटर और रांची से कुड़ूबिंढमगंज 5000 करोड़ सड़क योजना पर काम होना है। इसी के तहत गढ़वा बाइपास सड़क का निर्माण शुरू हो चुका है। यह फोरलेन सड़क 202 किलोमीटर की है। रांचीचाईबासा जैंतगढ़ सीमा 203 किलोमीटर की फोर लेन परियोजना की लागत 4000 करोड़ की है। चौथीहजारीबाग से बगोदर 50 किमी की परियोजना 1300 करोड़ की है। इन परियोजनाओं को केंद्र से राशि मिलनी है। एनएचएआई इन सभी सड़कों का निर्माण कार्य करायेगा। वंदे भारत की भी सौगात रेलवे की नई परियोजनाओं व वंदे भारत रेलगाड़ियों का लाभ भी झारखंड को मिलना तय है। इनके अलावा झारखंड में भी नदियों को जोड़ने की योजना पर काम शुरू होनेवाला है। झारखंड जैविक कृषि का भी लाभ लेगा। एमएसएमई सेक्टर में भी झारखंड के उद्यमियों को बड़ी राशि मिलने की उम्मीद जतायी जा रही है। झारखंड को अब केंद्रीय योजनाओं में हिस्सेदारी ज्यादा से ज्यादा हासिल करनी होगी, तभी केंद्रीय बजट योजनाओं का लाभ राज्य को मिलेगा। The post झारखण्ड के एक्सप्रेसवे निर्माण कार्य मे आएगा तेजी, वन्दे भारत ट्रेन भी जल्दी चलेगी first appeared on Quick Joins. Punjab, Ludhiana, Jalandhar, Amritsar, Patiala, Sangrur, Gurdaspur, Pathankot, Hoshiarpur, Tarn Taran, Firozpur, Fatehgarh Sahib, Faridkot, Moga, Bathinda, Rupnagar, Kapurthala, Badnala, Ambala,Uttar Pradesh, Agra, Bareilly, Banaras, Kashi, Lucknow, Moradabad, Kanpur, Varanasi, Gorakhpur, Bihar, Muzaffarpur, East Champaran, Kanpur, Darbhanga, Samastipur, Nalanda, Patna, Muzaffarpur, Jehanabad, Patna, Nalanda, Araria, Arwal, Aurangabad, Katihar, Kishanganj, Kaimur, Khagaria, Gaya, Gopalganj, Jamui, Jehanabad, Nawada, West Champaran, Purnia, East Champaran, Buxar, Banka, Begusarai, Bhagalpur, Bhojpur, Madhubani, Madhepura, Munger, Rohtas, Lakhisarai, Vaishali, Sheohar, Sheikhpura, Samastipur, Saharsa, Saran, Sitamarhi, Siwan, Supaul,Gujarat, Ahmedabad, Vadodara, Surat, Rajkot, Vadodara, Junagadh, Anand, Jamnagar, Gir Somnath, Mehsana, Kutch, Sabarkantha, Amreli, Kheda, Rajkot, Bhavnagar, Aravalli, Dahod, Banaskantha, Gandhinagar, Bhavnagar, Jamnagar, Valsad, Bharuch , Mahisagar, Patan, Gandhinagar, Navsari, Porbandar, Narmada, Surendranagar, Chhota Udaipur, Tapi, Morbi, Botad, Dang, Rajasthan, Jaipur, Alwar, Udaipur, Kota, Jodhpur, Jaisalmer, Sikar, Jhunjhunu, Sri Ganganagar, Barmer, Hanumangarh, Ajmer, Pali, Bharatpur, Bikaner, Churu, Chittorgarh, Rajsamand, Nagaur, Bhilwara, Tonk, Dausa, Dungarpur, Jhalawar, Banswara, Pratapgarh, Sirohi, Bundi, Baran, Sawai Madhopur, Karauli, Dholpur, Jalore,Haryana, Gurugram, Faridabad, Sonipat, Hisar, Ambala, Karnal, Panipat, Rohtak, Rewari, Panchkula, Kurukshetra, Yamunanagar, Sirsa, Mahendragarh, Bhiwani, Jhajjar, Palwal, Fatehabad, Kaithal, Jind, Nuh, बिहार, मुजफ्फरपुर, पूर्वी चंपारण, कानपुर, दरभंगा, समस्तीपुर, नालंदा, पटना, मुजफ्फरपुर, जहानाबाद, पटना, नालंदा, अररिया, अरवल, औरंगाबाद, कटिहार, किशनगंज, कैमूर, खगड़िया, गया, गोपालगंज, जमुई, जहानाबाद, नवादा, पश्चिम चंपारण, पूर्णिया, पूर्वी चंपारण, बक्सर, बांका, बेगूसराय, भागलपुर, भोजपुर, मधुबनी, मधेपुरा, मुंगेर, रोहतास, लखीसराय, वैशाली, शिवहर, शेखपुरा, समस्तीपुर, सहरसा, सारण सीतामढ़ी, सीवान, सुपौल, बिहार, मुजफ्फरपुर, पूर्वी चंपारण, कानपुर, दरभंगा, समस्तीपुर, नालंदा, पटना, मुजफ्फरपुर, जहानाबाद, पटना, नालंदा, अररिया, अरवल, औरंगाबाद, कटिहार, किशनगंज, कैमूर, खगड़िया, गया, गोपालगंज, जमुई, जहानाबाद, नवादा, पश्चिम चंपारण, पूर्णिया, पूर्वी चंपारण, बक्सर, बांका, बेगूसराय, भागलपुर, भोजपुर, मधुबनी, मधेपुरा, मुंगेर, रोहतास, लखीसराय, वैशाली, शिवहर, शेखपुरा, समस्तीपुर, सहरसा, सारण सीतामढ़ी, सीवान, सुपौल, | hindi |
ಶ್ಯಾಮ್ಪ್ರಸಾದ್ ಮುಖರ್ಜಿ ಜಯಂತಿ ಆಚರಣೆ ಕೋಲಾರ: ಭಾರತೀಯ ಜನಸಂಘದ ಸಂಸ್ಥಾಪಕ ಶ್ಯಾಮ್ಪ್ರಸಾದ್ ಮುಖರ್ಜಿ ಅವರು ವೃತ್ತಿಯಲ್ಲಿ ವಕೀಲರಾಗಿ ಸಮಾಜ ಸುಧಾರಣೆಗೆ ದಾರಿದೀಪವಾಗಿದ್ದರು ಎಂದು ಬಿಜೆಪಿ ಕಾನೂನು ಪ್ರಕೋಷ್ಠ ರಾಜ್ಯ ಕಾರ್ಯದರ್ಶಿ ಎಸ್.ಕೇಶವಪ್ರಸಾದ್ ಅಭಿಪ್ರಾಯಪಟ್ಟರು. ಬಿಜೆಪಿ ಕಾನೂನು ಪ್ರಕೋಷ್ಠವು ಇಲ್ಲಿ ಮಂಗಳವಾರ ಹಮ್ಮಿಕೊಂಡಿದ್ದ ಶ್ಯಾಮ್ಪ್ರಸಾದ್ ಮುಖರ್ಜಿ ಜಯಂತಿಯಲ್ಲಿ ಮಾತನಾಡಿ, ಮುಖರ್ಜಿ ಅವರು ಸ್ವಾತಂತ್ರ್ಯೋತ್ತರದಲ್ಲಿ ಹಿಂದೂಗಳ ಮೇಲಿನ ಶೋಷಣೆ ವಿರುದ್ಧ ಧ್ವನಿ ಎತ್ತಿದರು ಎಂದರು. ಮುಖರ್ಜಿ ಅವರ ಪ್ರತಿ ಕಾರ್ಯದಲ್ಲೂ ದೇಶಾಭಿಮಾನ ಎದ್ದು ಕಾಣುತ್ತಿತ್ತು. ದುಷ್ಟ ಶಕ್ತಿಗಳಿಂದ ದೇಶ ವಿಭಜಿಸುವ ಹುನ್ನಾರ ನಡೆದಾಗ ಪ್ರತಿ ಹಂತದಲ್ಲೂ ಪ್ರಬಲವಾಗಿ ವಿರೋಧಿಸುತ್ತಾ ಬಂದರು. ದೇಶದ ಏಕತೆ ಅಖಂಡತೆಗೆ ಅವರಿಗಿದ್ದ ಪ್ರೇಮ, ಕಾಳಜಿಯು ಪ್ರತಿ ಭಾರತೀಯನಿಗೆ ಅವರ ಮೇಲೆ ಅಪಾರ ಗೌರವ ಮೂಡಿಸುತ್ತದೆ ಎಂದು ತಿಳಿಸಿದರು. ಬಿಜೆಪಿ ಕಾನೂನು ಪ್ರಕೋಷ್ಠ ರಾಜ್ಯ ಸಹ ಸಂಚಾಲಕ ಯೋಗನಾಥ್, ರಾಜ್ಯ ಸಮಿತಿ ಸದಸ್ಯ ಶ್ರೀನಿವಾಸ್, ಜಿಲ್ಲಾ ಸಂಚಾಲಕ ವೆಂಕಟರೆಡ್ಡಿ, ಸಹ ಸಂಚಾಲಕ ನಾಗೇಂದ್ರ, ಜಿಲ್ಲಾ ಸರ್ಕಾರಿ ಅಭಿಯೋಜಕ ನಾರಾಯಣಸ್ವಾಮಿ ಪಾಲ್ಗೊಂಡರು. | kannad |
Lata Mangeshkar RIP: झुमरी तिलैया और लता मंगेशकर का एक अटूट रिश्ता Lata Mangeshkar RIP: झुमरी तिलैया संगीत प्रेमियों के लिए एक जाना पहचाना नाम है. झारखंड में प्रकृति की गोद में बसे इस कस्बे ने नयनाभिराम झीलों के दृश्य से प्रकृति प्रेमियों के बीच एक अलग पहचान बनाई है. इस छोटे से कस्बाई शहर को 1950 के दशक की शुरुआत में रेडियो सीलोन और आकाशवाणी ने भी खासी प्रसिद्धि दिलाई. आकाशवाणी के विविध भारती पर उस समय फरमाइशी गानों के कार्यक्रम में झुमरी तिलैया से सबसे अधिक महान गायिका लता मंगेशकर द्वारा गाए गानों को बजाने की मांग की जाती थी. खबर में खास रेडियो सेट पर गाने सुनने के शौकीनपोस्टकार्ड भेजने को लेकर प्रेरितझुमरी तिलैया और लता का रिश्ता अटूट रेडियो सेट पर गाने सुनने के शौकीन पीढ़ियों से संगीत को परिभाषित करने वाली भारत कोकिला के निधन की खबर जैसे ही इस छोटे से शहर में पहुंची, देवी सरस्वती की पूजा के लिए बने पंडालों, दुकानों और अन्य प्रतिष्ठानों में लाउडस्पीकरों से लता दीदी के गाने बजाए जाने लगे. लोग सड़कों पर गमगीन नजर आ रहे थे. भारत के अधिकतर हिस्सों में भले ही आज रेडियो की जगह टेलीविजन ने ले ली हो लेकिन झारखंड की राजधानी रांची से लगभग 165 किलोमीटर दूर झुमरी तिलैया में आज भी सड़क किनारे ढाबों, छोटी दुकानों और अन्य प्रतिष्ठानों में रेडियो सेट पर गाने सुनने के शौकीन मिल जाएंगे. पोस्टकार्ड भेजने को लेकर प्रेरित झुमरी तिलैया के एक खदान मालिक, रामेश्वर प्रसाद बरनवाल ने रेडियो सीलोन पर अमीन सयानी के बिनाका गीतमाला कार्यक्रम के लिए फरमाइशों के साथ पोस्टकार्ड भेजना शुरू किया, जिससे शहर के अन्य लोग भी पोस्टकार्ड भेजने को लेकर प्रेरित हुए. जल्द ही झुमरी तिलैया में रेडियो प्रेमियों ने एक छोटा रेडियो श्रोताओं का क्लब बना लिया. रेडियो सीलोन पर अमीन सयानी और विविध भारती पर अन्य उद्घोषकों को अक्सर यह घोषणा करते सुना जाता थाअगली फरमाइश है झुमरी तिलैया से. जब रेडियो लोकप्रिय हो गए, तो इस कस्बे में पोस्टकार्ड पर गाने के लिए अनुरोध भेजना लोगों का एक प्रमुख शौक बन गया. झुमरी तिलैया और लता का रिश्ता अटूट क्षेत्र के एक प्रमुख उद्योगपति राहुल मोदी के दादा रामेश्वर मोदी उस समय अभ्रक के प्रमुख कारोबारी थे. राहुल ने बताया, मेरे दादा और अन्य लोग एकदूसरे के साथ शर्त लगाते थे, किसके नाम की घोषणा संगीत के फरमाइशी कार्यक्रम में की जाएगी. राहुल मोदी ने कहा कि ज्यादातर फरमाइशें लता मंगेशकर के गानों की होती थी. उन्होंने कहा, यह क्षेत्र के लिए एक दुखद दिन है. यह बहुत बड़ी क्षति है. झुमरी तिलैया और लता दीदी के गीत का रिश्ता अटूट है. संयुक्त राष्ट्र की संस्था के एक प्रमुख चिकित्सक डॉ. दीपक कुमार ने कहा, झुमरी तिलैया की पहचान रेडियो और लता जी के गीतों के कारण है. The post Lata Mangeshkar RIP: झुमरी तिलैया और लता मंगेशकर का एक अटूट रिश्ता first appeared on India Ahead Hindi. | hindi |
IPL Mega Auction: আইয়ার থেকে ওয়ার্নার, অধিনায়ক হিসাবে এই তারকাদের দিকে নজর ফ্র্যাঞ্চাইজিগুলির প্রথম কলকাতা আর মাত্র কয়েকদিনের অপেক্ষা আগামী ১২ ও ১৩ই ফেব্রুয়ারি বেঙ্গালুরুতে আয়োজিত হবে আইপিএল মহানিলাম নিলামে নাম লিখিয়েছেন একাধিক মহাতারকা যাদেরকে দলে নেওয়ার দৌড়ে ঝাঁপাবে ফ্র্যাঞ্চাইজিগুলি তবে, কয়েকটি দলকে তাদের অধিনায়কও বেছে নিতে হবে মহানিলাম থেকে রয়্যাল চ্যালেঞ্জার্স ব্যাঙ্গালোর, কলকাতা নাইট রাইডার্স ও পাঞ্জাব কিংসের মতো দলগুলির নজর কোন তারকাদের দিকে? দেখে নিন নেতৃত্ব ক্ষমতা রয়েছে এমন তারকাদের তালিকা শ্রেয়স আইয়ার: দিল্লি ক্যাপিটালস দলের প্রাক্তন অধিনায়ক শ্রেয়স আইয়ারের দিকে একাধিক ফ্র্যাঞ্চাইজি নজর রেখেছে শুরুর দিকে আহমেদাবাদের সঙ্গে শ্রেয়সের আলোচনার কথা শোনা গেলেও তারা আস্থা রেখেছ হার্দিকের উপর অন্যদিকে আইপিএল মহানিলামের হট প্রোপার্টি হয়ে উঠেছেন শ্রেয়স ডেভিড ওয়ার্নার: তারকা অজি ওপেনার ডেভিড ওয়ার্নারের সঙ্গে বেশ তিক্ত ভাবেই সম্পর্কচ্ছেদ করেছে সানরাইজার্স হায়দ্রাবাদ তবে, ওয়ার্নারের নেতৃত্বেই আইপিএল চ্যাম্পিয়ন হয়েছিল ফ্র্যাঞ্চাইজিটি ২০২১ সালে অধিনায়ক পর হারানোর পর, প্রথম একাদশ থেকেও বাদ পড়েছিলেন ওয়ার্নার মহানিলামে ওয়ার্নারকে ঘিরে বড়সড় নিলাম যুদ্ধের আভাস রয়েছে শিখর ধাওয়ান: দিল্লি ক্যাপিটালস থেকে রিলিজ পেয়েছেন তারকা ওপেনার শিখর ধাওয়ান ২০১৪ আইপিএলে সানরাইজার্স হায়দ্রাবাদ দলের অধিনায়কত্ব করা ধাওয়ানের অভিজ্ঞতা তার পক্ষে যেতে পারে ব্যাট হাতেও ধারাবাহিক ছন্দে রয়েছেন তিনি জেসন হোল্ডার: পেসার অলরাউন্ডার হিসাবে আন্তর্জাতিক ক্রিকেটের পাশাপাশি আইপিএলেও নজর কেড়েছেন হোল্ডার মহানিলামে বড় অঙ্কের দাম পেতে চলা ক্যারিবিয়ান অলরাউন্ডারকে অধিনায়ক হিসাবে নেওয়ার জন্যে নিলাম যুদ্ধে নামবে বেশ কয়েকটি দল ঈশান কিষান: মুম্বাই ইন্ডিয়ান্স থেকে রিলিজ পেয়ে যাওয়া প্রতিশ্রুতিমান ঈশানকে দলে নেওয়ার জন্যে ঝাঁপাবে ফ্র্যাঞ্চাইজিগুলি চোখধাঁধানো দাম পেতে চলা এই বাম হাতি ব্যাটার অনূর্ধ্ব ১৯ পর্যায়ে ভারতীয় দলের অধিনায়ক ছিলেন ব্যাট হাতে বিস্ফোরক ইনিংস খেলবার পাশাপাশি ব্যাটিং অর্ডারের বিভিন্ন পজিশনে ব্যাট করতে পারেন ঈশান | bengali |
43952 * 98356'] ", '43952 * 98356 \n '] | punjabi |
अंतरराज्यीय वाहन चोर गैंग के दो बदमाश गिरफ्तार नई दिल्लीटीम डिजिटल। दिल्ली.एनसीआर क्षेत्र से वाहन चुराकर मेरठ के कबाडिय़ों को बेचने वाले गैंग का सिहानी गेट पुलिस ने भंडाफोड़ किया है। इस मामले में पुलिस ने दो चोरों को गिरफ्तार कर उनके पास से चोरी की सेंट्रो कार और लॉक तोडऩे के औजार, तमंचा व कारतूस बरामद किए हैं। पूछताछ के आधार पर पुलिस आरोपियों के अन्य साथियों की भी तलाश कर रही है। एसएचओ सिहानी गेट सौरभ विक्रम सिंह ने बताया कि वीरवार रात को उनकी टीम गश्त पर थी। लोहियानगर में बिजली घर चौराहे के पास पहुंचने पर सेंट्रो कार सवार दो संदिग्ध युवक दिखाई दिए। पुलिस ने रुकने का इशारा किया तो वह कार मोडक़र भागने लगे। घेराबंदी कर उन्हें दबोच लिया गया। तलाशी में तमंचा व कारतूस बरामद होने पर दोनों को गिरफ्तार कर लिया गया। पूछताछ में दोनों की पहचान मेरठ के गांव किनौनी निवासी हरेन्द्र उर्फ हन्नी और अलीगढ़ के गांव सुनहैरा निवासी अमित के रूप में हुई। दोनों आरोपी वर्तमान में दिल्ली में किराए का कमरा लेकर रहते हैं। एसएचओ ने बताया कि आरोपियों के खिलाफ हापुड़, मेरठ, बुलंदशहर और दिल्ली के विभिन्न थानों में एक दर्जन मुकदमे दर्ज हैं। दोनों आरोपियों ने दिल्ली.एनसीआर से वाहन चोरी करके मेरठ के कबाडिय़ों को सस्ते दामों में बेचने की बात कबूली है। पुलिस का कहना है कि आरोपियों के पास से जो कार बरामद हुई है, वह दिल्ली से चोरी की गई थी। आरोपी कबाडिय़ों को वाहन बेचकर रकम आपस में बराबर बांट लेते थे। पुलिस का दावा है कि आरोपी 100 से अधिक वाहनों को चुराकर बेच चुके हैं। ऐसी ही और लोकल खबरों के लिए, डाउनलोड करें हमारा नया लोकल वीडियो ऐप पब्लिक वाइब | hindi |
خلا میں موجود سیاہ گڑھے یا بلیک ہول کی حقیقت کیا ہے ویسے تو دنیا عقل سے ماوراء کئی عجوبات سے بھری پڑی ہے اور انہیں میں سے ایک بلیک ہول بھی ہے بلیک ہول خلا میں پایا جانیوالا وہ خفیہ گڑھا ہے جوکہ اپنی بے انتہا کشش کے باعث بڑی سے بڑی شے کو اپنے اندر نگل جانے کی صلاحیت رکھتا ہےاسکی کشش کی قوت اتنی زیادہ ہوتی ہے کہ روشنی تیز ترین سفر کرنے کی صلاحیت رکھنے کے باوجود اس سے بچ نہیں سکتی بلیک ہول کے باہر گول شکل میں پایا جانیوالا دائرہ ایونٹ ہاریزن یا واقعہ افق کے نام سے جانا جاتا ہےاگر کوئی بھی اجسام بلیک کی سرحد کے قریب سے گزرتا ہے تو وہ کشش کی قوت کے باعث خودبخود بلیک ہول میں کھینچتا چلا جاتا ہے بلیک ہول کی جسامت مختلف ہوتی ہیںبلیک ہول کا سب سے پہلے انکشاف نیوٹن نے کیا اور دیگر سائنسدانوں نے اس پر ریسرچ کرکے اپنے مختلف نظریات پیش کیا تاہم اہم سوال یہ ہے کہ بلیک ہول کی تشکیل یا خلا میں یہ کیا کام سرانجام دیتے ہیںبلیک ہول خلا میں پائی جانیوالی ایک خفیہ تشکیل ہے جو کہ ناقابل مزاحمت کشش رکھتی ہے جس کی بدولت یہ کئی ستارے اپنے اندر کھینچ لیتی ہے لیکن انکی تخلیق اور موجودگی کا دارومدار خلا میں پائے جانیوالے دیگر اجسام فلکی پر ہےسائنس کے مطابق ایک طویل القامت بلیک ہول اس وقت خلا میں تخلیق پاتی ہے جب کوئی غیر معمولی جسامت کا ستارہ یا تو خلا میں گرتا ہے یا پھر مر جاتا ہے | urdu |
વરાછામાં સરકારી કોલેજ સહિતની માંગણીઓ પુરી કરવા કોંગ્રેસ ધારાસભ્યોને લોલિપોપ આપશે સુરત, 16 એપ્રિલ હિ. સ. વિધાનસભા ચુંટણી નજીક આવતા શહેરમાં પાટીદારોનું વર્ચસ્વ ધરાવતી કામરેજ વિધાનસભા બેઠક પર અલગ અલગ માંગ સાથે સુરત શહેર જિલ્લા કોંગ્રેસ સમિતિ દ્વારા જન આંદોલન નાં કાર્યક્રમો કરવામાં આવનાર છે. આગામી 18મી તારીખથી એક મહિના સુધી જનજાગૃત્તિ કાર્યક્રમો થકી કોંગ્રેસ દ્વારા સરકારને ઘેરવાનો પ્રયાસ કરવામાં આવશે. જેમાં વરાછા વિસ્તારમાં સરકારી કોલેજ અને અસંખ્ય કબ્જા રસીદવાળી મિલ્કતોને કાયદેસર કરવાની સાથે માલિકી હક્ક આપવા સહિતની માંગણીઓ પુરી કરવાની માગ સાથે ધારાસભ્યોની ઓફિસો પર શંકનાદના કાર્યક્રમોનું આયોજન પણ કરવામાં આવશે.શહેરના કામરેજ વિધાનસભા મત વિસ્તારમાં શહેર જિલ્લા કોંગ્રેસ દ્વારા છેલ્લા ત્રણ વર્ષથી સરકારી કોલેજ, કબ્જા રસીદવાળી મિલ્કતો કાયદેસર કરવા અને રસ્તા પર બંધ પડેલ હાઈટેન્શન લાઈનો દુર કરીને રસ્તા ખુલ્લા કરવા સાથે ખાડીઓમાં ગંદકી દુર કરીને તેને પેક કરવા સંદર્ભે છાશવારે શાસકો સમક્ષ રજુઆત કરવામાં આવી રહી છે. 18મી તારીખથી 24મી તારીખ સુધી દરેક ધારાસભ્યોની ઓફિસ પર શંખનાદ સાથે લોલિપોપનું વિતરણ કરવામાં આવશે.આ સિવાય 25મી એપ્રિલથી સામાજીક અને રાજકીય આગેવાનો સાથે ઓટલા બેઠક, ખાટલા બેઠક અને ગ્રુપ મીટિંગો કરીને લોકોને આંદોલનમાં મહત્તમ સંખ્યામાં જોડાવવા માટે પણ અપીલ કરવામાં આવશે. આ સિવાય વિધાનસભાના વિરોધ પક્ષના નેતાની આગેવાની સોસાયટીના પ્રમુખો, સામાજીક આગેવાનો અને વિદ્યાર્થી સાથે મુખ્યમંત્રીને પણ રૂબરૂમાં રજુઆત કરવામાં આવશે.હિન્દુસ્થાન સમાચાર | gujurati |
ಚಿತ್ರದುರ್ಗ: ನಗರಸಭೆ ಅಧ್ಯಕ್ಷ ಗಾದಿಗೆ ಚುನಾವಣೆ ಇಂದು ಚಿತ್ರದುರ್ಗ: ನಗರಸಭೆ ಚುನಾವಣೆ ನಡೆದ ಎರಡು ವರ್ಷ, ಎರಡು ತಿಂಗಳ ಬಳಿಕ ಕೌನ್ಸಿಲ್ ರಚನೆ ನಿಶ್ಚಿತವಾಗುತ್ತಿದೆ. ಅಧ್ಯಕ್ಷ, ಉಪಾಧ್ಯಕ್ಷ ಚುನಾವಣೆಗೆ ಈಗಾಗಲೇ ಕ್ಷಣಗಣನೆ ಆರಂಭವಾಗಿದ್ದು, ನ.1ರಂದು ಮುಹೂರ್ತ ನಿಗದಿಯಾಗಿದೆ. ನಗರಸಭೆ ಗದ್ದುಗೆ ಏರಲು ಬಿಜೆಪಿ ಉತ್ಸುಕವಾಗಿದೆ. ಪರಿಶಿಷ್ಟ ಪಂಗಡದ ಮಹಿಳೆಗೆ ಮೀಸಲಾದ ಅಧ್ಯಕ್ಷ ಗಾದಿಗೆ ಇಬ್ಬರು ಕಸರತ್ತು ನಡೆಸುತ್ತಿದ್ದಾರೆ. 1ನೇ ವಾರ್ಡ್ನಿಂದ ಆಯ್ಕೆಯಾದ ನಾಗಮ್ಮ ಹಾಗೂ 19ನೇ ವಾರ್ಡ್ನಿಂದ ಆಯ್ಕೆಯಾದ ತಿಪ್ಪಮ್ಮ ಅಧ್ಯಕ್ಷ ಸ್ಥಾನಕ್ಕೆ ಅರ್ಹತೆ ಪಡೆದಿದ್ದಾರೆ. ಅಧ್ಯಕ್ಷ ಸ್ಥಾನಕ್ಕೆ ತಿಪ್ಪಮ್ಮ ವೆಂಕಟೇಶ್, ಉಪಾಧ್ಯಕ್ಷೆ ಸ್ಥಾನಕ್ಕೆ 28ನೇ ವಾರ್ಡ್ನ ಶ್ವೇತಾ ವೀರೇಶ್ ಸ್ಪರ್ಧಿಸುವುದು ಬಹುತೇಕ ಖಚಿತವಾಗಿದೆ. ಉಳಿದ ಆಕಾಂಕ್ಷಿಗಳು ಕಣದಿಂದ ಹಿಂದೆ ಸರಿಯುವ ಸಾಧ್ಯತೆ ಇದೆ. 35 ವಾರ್ಡ್ ಹೊಂದಿರುವ ಚಿತ್ರದುರ್ಗ ನಗರಸಭೆಯಲ್ಲಿ ಬಿಜೆಪಿ 17, ಪಕ್ಷೇತರ 7, ಜೆಡಿಎಸ್ 6 ಹಾಗೂ ಕಾಂಗ್ರೆಸ್ 5 ಸದಸ್ಯ ಬಲ ಹೊಂದಿವೆ. ಸರಳ ಬಹುಮತಕ್ಕೆ 19 ಮತಗಳ ಅಗತ್ಯವಿದೆ. ಶಾಸಕ ಜಿ.ಎಚ್. ತಿಪ್ಪಾರೆಡ್ಡಿ ಹಾಗೂ ಸಂಸದ ಎ. ನಾರಾಯಣಸ್ವಾಮಿ ಅವರ ಮತಗಳು ಸೇರಿದರೆ ಬಿಜೆಪಿ ನಿರಾತಂಕವಾಗಿ ಅಧಿಕಾರಕ್ಕೆ ಏರಲಿದೆ. ಇಲ್ಲಿನ ನಗರಸಭೆಯ ಇತಿಹಾಸದಲ್ಲಿ ಇದೇ ಪ್ರಥಮ ಬಾರಿ ಅಧಿಕಾರದ ಚುಕ್ಕಾಣಿ ಹಿಡಿಯಲು ಉತ್ಸುಕವಾಗಿರುವ ಬಿಜೆಪಿಗೆ ಪಕ್ಷೇತರ ಮತ್ತು ಕಾಂಗ್ರೆಸ್ನ ತಲಾ ನಾಲ್ವರು ಸದಸ್ಯರು ಹಾಗೂ ಜೆಡಿಎಸ್ನಿಂದ ಒಬ್ಬರು ಬೆಂಬಲ ನೀಡಲು ಮುಂದಾಗಿದ್ದಾರೆ. ಈ ಮೂಲಕ ಶಾಸಕರು, ಸಂಸದರ ಮತ ಹೊರತುಪಡಿಸಿ ಬಿಜೆಪಿ ಸಂಖ್ಯಾಬಲ 26ಕ್ಕೆ ಏರಿಕೆಯಾಗುವಸಾಧ್ಯತೆ ಹೆಚ್ಚಿದೆ. ಅಧ್ಯಕ್ಷ ಸ್ಥಾನಕ್ಕೆ ತಿಪ್ಪಮ್ಮ ಅವರ ಹೆಸರು ಮೊದಲಿನಿಂದಲೂ ಮುಂಚೂಣಿಯಲ್ಲಿದೆ. ಉಪಾಧ್ಯಕ್ಷ ಸ್ಥಾನದ ಆಕಾಂಕ್ಷಿಗಳ ಪಟ್ಟಿ ಬಿಜೆಪಿಯಲ್ಲಿ ದೊಡ್ಡದಾಗಿದೆ. ಇವರಲ್ಲಿ 9 ಸದಸ್ಯರು ಮಹಿಳೆಯರಿದ್ದಾರೆ. ಹಿಂದುಳಿದ ವರ್ಗ ಎಗೆ ಮೀಸಲಾದ ಉಪಾಧ್ಯಕ್ಷ ಸ್ಥಾನದ ಮೇಲೆ ಹಲವರು ಕಣ್ಣಿಟ್ಟಿದ್ದಾರೆ. ಅಧ್ಯಕ್ಷ ಸ್ಥಾನ ಈಗಾಗಲೇ ಮಹಿಳೆಗೆ ಖಚಿತವಾಗಿದೆ. ಆದ್ದರಿಂದ ಉಪಾಧ್ಯಕ್ಷ ಸ್ಥಾನವನ್ನಾದರೂ ಪುರುಷರಿಗೆ ನೀಡಬೇಕು ಎಂಬ ಅಭಿಪ್ರಾಯವನ್ನು ಕೆಲ ಸದಸ್ಯರು ವ್ಯಕ್ತಪಡಿಸಿದ್ದರು. ಒಂದು ವೇಳೆ ಉಪಾಧ್ಯಕ್ಷ ಸ್ಥಾನವೂ ಮಹಿಳೆಗೆ ಸಿಕ್ಕರೆ ನಗರಸಭೆಯಲ್ಲಿ ಮಹಿಳಾ ಆಡಳಿತ ಪರ್ವ ಶುರುವಾಗಲಿದೆ. ಆಯ್ಕೆ ಸಂಬಂಧ ಶಾಸಕ ಜಿ.ಎಚ್. ತಿಪ್ಪಾರೆಡ್ಡಿ ಅವರ ಕಾಟನ್ ಮಿಲ್ ಆವರಣದಲ್ಲಿ ಶನಿವಾರ 20ಕ್ಕೂ ಹೆಚ್ಚು ನಗರಸಭೆ ಸದಸ್ಯರು ಚರ್ಚೆ ನಡೆಸಿದರು. ತಮ್ಮ ಅಭಿಪ್ರಾಯವನ್ನು ಶಾಸಕರ ಮುಂದಿಟ್ಟರು. ನಂತರ ಮಾಧ್ಯಮದವರೊಂದಿಗೆ ಮಾತನಾಡಿದ ಶಾಸಕರು, ಸದಸ್ಯರ ಮಧ್ಯೆ ಯಾವುದೇ ರೀತಿಯ ವ್ಯತ್ಯಾಸ ಬಾರದಂತೆ ನಗರದ ಸ್ವಚ್ಛತೆ, ರಸ್ತೆಗಳ ನಿರ್ಮಾಣ ಸೇರಿ ಸಾರ್ವಜನಿಕರಿಗೆ ಮೂಲ ಸೌಕರ್ಯ ಕಲ್ಪಿಸುವ ಕಾರ್ಯಗಳಿಗೆ ಪಕ್ಷಾತೀತವಾಗಿ ಎಲ್ಲರೂ ಒಗ್ಗಟ್ಟಿನಿಂದ ಕೆಲಸ ಮಾಡುವ ದೃಷ್ಟಿಯಿಂದ ಉತ್ತಮ ನಿರ್ಣಯ ಕೈಗೊಳ್ಳಲಿದ್ದೇವೆ. ಒಮ್ಮತದ ಅಭ್ಯರ್ಥಿಗಳ ಆಯ್ಕೆ ಖಚಿತ ಎಂದು ಸ್ಪಷ್ಟಪಡಿಸಿದರು. | kannad |
India Team: ইংল্যান্ডের বিরুদ্ধে টি২০ এবং একদিনের সিরিজের দল ঘোষণা বিসিসিআইয়ের, দলে অর্শদীপ ইংল্যান্ডের England বিরুদ্ধে তিনটি টি২০ T20 এবং তিনটি একদিনের ODI ম্যাচের দল ঘোষণা করল ভারতীয় বোর্ড BCCI দুটি সিরিজেই নেতা রোহিত শর্মা Rohit Sharma এক দিনের দলে জায়গা পেলেন অর্শদীপ সিং প্রথম টি২০ দলে নেই বিরাট কোহলি, ঋষভ পন্থ, যশপ্রীত বুমরাহ এবং রবীন্দ্র জাদেজা দ্বিতীয় এবং তৃতীয় টি২০ ম্যাচে ফিরবেন তারা টি২০ দলে নেই মহম্মদ শামি একদিনের ম্যাচে ফিরছেন তিনি অপরদিকে একদিনের সিরিজে রোহিত, বিরাট, বুমরাহদের রেখেই গড়া হয়েছে দল সেই দলে নেওয়া হয়েছে শিখর ধাওয়ানকে রয়েছেন মহম্মদ শামি এবং মহম্মদ সিরাজও সেই দলেই নেওয়া হয়েছে অর্শদীপকেও যদিও আয়ারল্যান্ডের বিরুদ্ধে অভিষেক ঘটা উমরান মালিককে রাখা হয়নি একদিনের ম্যাচে একনজরে ইংল্যান্ডের বিরুদ্ধে ভারতের টি২০ এবং একদিনের ম্যাচের দল প্রথম টি২০ দল রোহিত শর্মা অধিনায়ক, ঈশান কিষান, রুতুরাজ গায়কোয়াড, সঞ্জু স্যামসন, সূর্যকুমার যাদব, দীপক হুডা, রাহুল ত্রিপাঠী, দীনেশ কার্তিক, হার্দিক পান্ডিয়া, বেঙ্কটেশ আইয়র, যুজবেন্দ্র চ্যাহাল, অক্ষর প্যাটেল, রবি বিষ্ণোই, ভুবনেশ্বর কুমার, হর্ষল প্যাটেল, আবেশ খান, অর্শদীপ সিং এবং উমরান মালিক দ্বিতীয় এবং তৃতীয় টি২০ দল রোহিত শর্মা অধিনায়ক, ঈশান কিষান, বিরাট কোহলি, সূর্যকুমার যাদব, দীপক হুডা, শ্রেয়স আইয়র, দীনেশ কার্তিক, ঋষভ পন্থ, হার্দিক পান্ডিয়া, রবীন্দ্র জাদেজা, যুজবেন্দ্র চ্যাহাল, অক্ষর প্যাটেল, রবি বিষ্ণোই, যশপ্রীত বুমরাহ, ভুবনেশ্বর কুমার, হর্ষল প্যাটেল, আবেশ খান এবং উমরান মালিক একদিনের সিরিজের দল রোহিত শর্মা অধিনায়ক, শিখর ধাওয়ান, ঈশান কিষান, বিরাট কোহলি, সূর্যকুমার যাদব, শ্রেয়স আইয়র, ঋষভ পন্থ, হার্দিক পান্ডিয়া, রবীন্দ্র জাদেজা, শার্দূল ঠাকুর, যুজবেন্দ্র চ্যাহাল, অক্ষর প্যাটেল, যশপ্রীত বুমরাহ, প্রসিদ্ধ কৃষ্ণ, মহম্মদ শামি, মহম্মদ সিরাজ এবং অর্শদীপ সিং | bengali |
రైతులకు మద్దతుగా ఆకు పచ్చ కండువాతో సంజయ్ పాదయాత్ర రైతులకు మద్దతుగా ఆకు పచ్చ కండువాతో సంజయ్ పాదయాత్ర V6 Velugu Posted on Sep 01, 2021 ప్రజా సంగ్రామ యాత్రలో భాగంగా చేవెళ్లకు వెళ్తూ మధ్య మధ్యలో రోడ్డు పక్కన ఉన్న పంట పొలాల్లోని రైతులను కలిసి మాట్లాడారు. వారి సమస్యలను అడిగి తెలుసుకున్నారు బండిసంజయ్. రైతులు తమ సమస్యలను ఆయనకు చెప్పుకున్నారు. తాము పండించిన పంటలకు గిట్టుబాటు ధర లేదని..దీంతో పాటు పండిస్తున్న కూరగాయలకు, పూలకు అతి తక్కువ ధరలకు కొంటున్నారని రైతులు ఫిర్యాదు. TRS అధికారంలోకి రాకముందు కేసీఆర్ రైతులకు ఎన్నో హామీలు ఇచ్చారని..అవి అమలు పరచలేదని తెలిపారు. ముఖ్యంగా రైతు పండించే పంట రైతు ఇంటి దగ్గరనే అమ్ముకునే వెసులుబాటు కల్పిస్తామని హామీలు ఇచ్చిండని సంజయ్ కి తెలిపారు రైతులు. ఒక బాక్స్ టమాటా వంద రూపాయలకు.. ఒక సంచి వంకాయలు 70 రూపాయలకు అమ్ముకోవాల్సి వస్తుందని రైతులు ఆవేదన వ్యక్తం చేశారు.సజ్జన పిల్లి దగ్గర సిమెంట్ పోల్స్ కార్మికుల తో మాట్లాడిన తర్వాత గులాబి పూలు పండిస్తున్న రైతులతో మాట్లాడిన బండి సంజయ్.. రైతులతో కలిసి కలుపు తీశారు. | telegu |
/*
Copyright 2014 Google Inc. All Rights Reserved.
Licensed under the Apache License, Version 2.0 (the "License");
you may not use this file except in compliance with the License.
You may obtain a copy of the License at
http://www.apache.org/licenses/LICENSE-2.0
Unless required by applicable law or agreed to in writing, software
distributed under the License is distributed on an "AS IS" BASIS,
WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
See the License for the specific language governing permissions and
limitations under the License.
*/
package com.google.security.zynamics.reil.translators.arm;
import static org.junit.Assert.assertEquals;
import com.google.common.collect.Lists;
import com.google.security.zynamics.reil.OperandSize;
import com.google.security.zynamics.reil.ReilInstruction;
import com.google.security.zynamics.reil.TestHelpers;
import com.google.security.zynamics.reil.interpreter.CpuPolicyARM;
import com.google.security.zynamics.reil.interpreter.EmptyInterpreterPolicy;
import com.google.security.zynamics.reil.interpreter.Endianness;
import com.google.security.zynamics.reil.interpreter.InterpreterException;
import com.google.security.zynamics.reil.interpreter.ReilInterpreter;
import com.google.security.zynamics.reil.interpreter.ReilRegisterStatus;
import com.google.security.zynamics.reil.translators.InternalTranslationException;
import com.google.security.zynamics.reil.translators.StandardEnvironment;
import com.google.security.zynamics.reil.translators.arm.ARMShsubaddxTranslator;
import com.google.security.zynamics.zylib.disassembly.ExpressionType;
import com.google.security.zynamics.zylib.disassembly.IInstruction;
import com.google.security.zynamics.zylib.disassembly.MockInstruction;
import com.google.security.zynamics.zylib.disassembly.MockOperandTree;
import com.google.security.zynamics.zylib.disassembly.MockOperandTreeNode;
import org.junit.Before;
import org.junit.Test;
import org.junit.runner.RunWith;
import org.junit.runners.JUnit4;
import java.math.BigInteger;
import java.util.ArrayList;
import java.util.List;
@RunWith(JUnit4.class)
public class ARMShsubaddxTranslatorTest {
private final ReilInterpreter interpreter = new ReilInterpreter(Endianness.BIG_ENDIAN,
new CpuPolicyARM(), new EmptyInterpreterPolicy());
private final StandardEnvironment environment = new StandardEnvironment();
private final ARMShsubaddxTranslator translator = new ARMShsubaddxTranslator();
private final ArrayList<ReilInstruction> instructions = new ArrayList<ReilInstruction>();
final OperandSize dw = OperandSize.DWORD;
final OperandSize wd = OperandSize.WORD;
final OperandSize bt = OperandSize.BYTE;
@Before
public void setUp() {
}
@Test
public void testSimpleRegister() throws InternalTranslationException, InterpreterException {
interpreter.setRegister("R0", BigInteger.valueOf(0x0L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("R1", BigInteger.valueOf(0x8f4f4c86L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("R2", BigInteger.valueOf(0x01ffc9e0L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("C", BigInteger.ZERO, bt, ReilRegisterStatus.DEFINED);
final MockOperandTree operandTree1 = new MockOperandTree();
operandTree1.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree1.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R0"));
final MockOperandTree operandTree2 = new MockOperandTree();
operandTree2.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree2.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R1"));
final MockOperandTree operandTree3 = new MockOperandTree();
operandTree3.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree3.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R2"));
final List<MockOperandTree> operands =
Lists.newArrayList(operandTree1, operandTree2, operandTree3);
final IInstruction instruction = new MockInstruction("SHSUBADDX", operands);
translator.translate(environment, instruction, instructions);
interpreter.interpret(TestHelpers.createMapping(instructions), BigInteger.valueOf(0x100L));
assertEquals(BigInteger.valueOf(0xe2b72742L), interpreter.getVariableValue("R0"));
assertEquals(BigInteger.valueOf(0x8f4f4c86L), interpreter.getVariableValue("R1"));
assertEquals(BigInteger.valueOf(0x01ffc9e0L), interpreter.getVariableValue("R2"));
assertEquals(BigInteger.ZERO, interpreter.getVariableValue("C"));
assertEquals(BigInteger.ZERO, BigInteger.valueOf(interpreter.getMemorySize()));
assertEquals(5, TestHelpers.filterNativeRegisters(interpreter.getDefinedRegisters()).size());
}
@Test
public void testSimpleRegisterDifferentMnemonic() throws InternalTranslationException,
InterpreterException {
interpreter.setRegister("R0", BigInteger.valueOf(0x0L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("R1", BigInteger.valueOf(0x8f4f4c86L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("R2", BigInteger.valueOf(0x01ffc9e0L), dw, ReilRegisterStatus.DEFINED);
interpreter.setRegister("C", BigInteger.ZERO, bt, ReilRegisterStatus.DEFINED);
final MockOperandTree operandTree1 = new MockOperandTree();
operandTree1.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree1.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R0"));
final MockOperandTree operandTree2 = new MockOperandTree();
operandTree2.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree2.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R1"));
final MockOperandTree operandTree3 = new MockOperandTree();
operandTree3.root = new MockOperandTreeNode(ExpressionType.SIZE_PREFIX, "b4");
operandTree3.root.m_children.add(new MockOperandTreeNode(ExpressionType.REGISTER, "R2"));
final List<MockOperandTree> operands =
Lists.newArrayList(operandTree1, operandTree2, operandTree3);
final IInstruction instruction = new MockInstruction("SHSAX", operands);
translator.translate(environment, instruction, instructions);
interpreter.interpret(TestHelpers.createMapping(instructions), BigInteger.valueOf(0x100L));
assertEquals(BigInteger.valueOf(0xe2b72742L), interpreter.getVariableValue("R0"));
assertEquals(BigInteger.valueOf(0x8f4f4c86L), interpreter.getVariableValue("R1"));
assertEquals(BigInteger.valueOf(0x01ffc9e0L), interpreter.getVariableValue("R2"));
assertEquals(BigInteger.ZERO, interpreter.getVariableValue("C"));
assertEquals(BigInteger.ZERO, BigInteger.valueOf(interpreter.getMemorySize()));
assertEquals(5, TestHelpers.filterNativeRegisters(interpreter.getDefinedRegisters()).size());
}
}
| code |
कर्क, कन्या और ये राशि वाले लोग रहे सतर्क, जानें आपके ऊपर क्या रहेगा ग्रह का असर जोधपुर के पंडित रमेश भोजराज द्विवेदी से जानिए 8 फरवरी 2022 राशिफल मेष: भूमि, भवन, प्लॉट, भूखंड आदि का सौदा इस समय करेंगे. आपका पराक्रम श्रेष्ठ रहेगा. भाईयों और परिवार से सहयोग प्राप्त होगा. पढ़ाई कर रहे जातकों को इस समय मिश्रित फल की प्राप्ति होगी. स्वास्थ्य का पाया कमजोर रह सकता है. क्या करें: लाल वस्तुओं का सेवन अधिकाधिक करें. क्या नहीं करें: काला वस्त्र धारण नहीं करें. उपाय: गोमूत्र घर में छिड़के और घर में समुद्री नमक का पौछा लगाएं. वृषभ: आर्थिक स्थिति में सुधार होगा. मन में राहत मिलेगी. वातावरण में जो भी परिवर्तन हो रहे है, उसके अनुरूप स्वयं को रूपांतरित कर लेंगे. प्रत्येक अवसर जो आपको प्राप्त होगा, उसका पूरा फायदा उठाने में आप समर्थ रहेंगे. क्या करें: पालतू जानवरों की देखरेख अच्छे से करें. क्या ना करें: आर्थिक बजट को बिगड़ने नहीं दें. उपायः मक्खन, दूध और दही का दान करें. मिथुन: दवाईयों के बजाय योग, ध्यान और जीवनशैली में बदलाव से आपको शारीरिक पुष्टता मिलेगी. धन संबंधी मामलों में आप जो भी खरीदबेचान करेंगे, उसमें आपको लाभ की प्राप्ति होगी. क्या करें: उच्चाधिकारियों से संबंध मधुर रखें. क्या ना करें: अपने घर की बात बाहर नहीं आने दें. उपायः पूर्णतया धार्मिक आचरण रखें. कर्कः अपना अड़ियल रवैया त्यागकर ज्यादा घमंड न करें और दूसरों को नजरअंदाज नहीं करें. ज्यादा कमाने के चक्कर में थोड़े को हाथ से ना जाने दें. क्या करें: व्यापार की तरक्की के लिए नईनई तकनीक का उपयोग करें. क्या ना करें: ज्यादा ओवरस्मार्ट नहीं बने. उपायः पराई स्त्रियों को सम्मान की दृष्टि से देखें. प्रतिदिन मंदिर जाएं. सिंहः परिवार और जीवनसाथी में आपसी सामंजस्य और प्रेम का भाव विद्यमान रहेगा. दोनों एकदूसरे की भावनाएं देखकर आचरण करेंगे. आर्थिक लाभ के मार्ग प्रशस्त होंगे. कड़ी मेहनत और प्रतिष्ठा हासिल करने में समय लगेगा. क्या करेंः दोस्तों के मध्य गोपनीयता का ध्यान रखें. क्या ना करेंः दूसरों को परेशान नहीं करें. उपायः परिवार की कोई भी स्त्री आपके द्वारा परेशान ना हो. कन्याः जो काम भी आप सोच लेंगे, वह येनकेनप्रकारेण सम्पन्न हो ही जायेगा. संतान पक्ष की पढ़ाई और कैरियर की जबरदस्त चिंता रहेगी. समय पर काम पूरा नहीं कर पाने से परेशानी का सामना करना पड़ सकता है. क्या करेंः स्वास्थ की जांच समयसमय पर करवाते रहे. क्या ना करेंः भूलकर भी विषयों के विरूद्ध जाकर कोई काम नहीं करें. उपायः ब्राह्मण एवं जरूररतमंद व्यक्तियों की मदद करें. तुलाः भयंकर कष्ट का सामना करना पड़ सकता है. किसी प्रियजन के बिछोह का गम आपको मिलेगा, जिसे सहन कर पाना संभव नहीं होगा. इस समय परिवार का साथ रहेगा. धैर्य और संयम से काम लेना होगा. आर्थिक स्थिति सामान्य रहेगी. क्या करेंः अपने हित के लिए कोई भी निर्णय तुरंत करें. क्या ना करेंः रिश्तो में गलतफहमियों को पनपने नहीं दें. उपायः तवा या चिमटा ब्राम्हण स्त्री को दान करें. वृश्चिकः आप दूसरों की सहायता लिए हाथ बढ़ाएंगे. किसी पुराने मित्र से मुलाकात होगी, जिससे मन हल्का होगा और आप स्वयं को तरोताजा और हल्का महसूस करेंगे. क्या करेंः विद्यार्थियों को चाहिए कि वे अपना ध्यान पढ़ाई पर दें. क्या ना करेंः परिवार वालों की बातों का विरोध नहीं करें. उपायः हरी सब्जियों का दान करें. धनुः आय में कमी आयेगी. अनावश्यक व्यय होंगे. खर्च की स्थिति रहेगी, लेकिन बिना रूकावट व्यवस्था भी हो जाएगी. सावधान रहकर काम करने की आवश्यकता है. राजनीति में सफलता एवं सहयोग की प्राप्ति होगी. क्या करेंः मानसिक शांति हेतु किसी एकांत स्थान या मंदिर आदि सुकून वाली जगह पर जाए. क्या ना करेंः सहकर्मियों की चिकनी चुपड़ी बातों में ना आएं. उपायः गोमूत्र के अर्क का सेवन करने और गाय की सेवा करने से लाभ मिलेगा. मकरः परीक्षार्थी के लिए समय अच्छा है. खरीदबिक्री के योग बन रहे है. घर में बड़ेबुजुर्गों के स्वास्थ्य में गड़बड़ चलेगी. अस्पताल के चक्कर काटने पड़ सकते हैं. क्या करेंः समय रहते अपने सारे कार्य निपटा लें. क्या ना करेंः कोई आपसे मदद मांगे तो उसे मना नहीं करें. उपायः सरसों का दान करें. कुंभः आज आप बहुत प्रसन्नचित्त रहेंगे. मनपसंद कार्य होगा. जीवन में आगे बढ़ने के लिए आप दूसरों का सहारा लेंगे, लेकिन सावधानी भी रखें. कई बार ज्यादा भरोसा भी नुकसानप्रद होता है. क्या करेंः वाहन आदि की समयसमय पर सर्विसिंग कराते रहे. क्या ना करेंः काम के वक्त बिना जरूरत मोबाइल का प्रयोग नहीं करें. उपायः ससुराल पक्ष या माता द्वारा दिया हुआ चांदी का सिक्का अपने पास रखें. मीनः प्यार और रोमांस में पड़कर आप अपना कैरियर चैपट कर सकते है. अतः आप सावधानी रखें. आपकी कोई गोपनीय बात सबके सामने आ जाएगी, जिससे आपको बहुत शर्मिंदगी और दुःख होगा. क्या करेंः प्रतियोगिता के लिए हर वक्त स्वयं को तैयार रखें. क्या ना करेंः अचानक आए खर्च और कष्ट से घबराएं नहीं. उपायः गाय और अन्य पशु पक्षियों को चारा और अन्न पदार्थ खिलाएं. सभी 12 राशियों के लिए कॉमन उपाय कॉमन उपायः लाल मसूर की दाल पक्षियों को चुगावाएं. | hindi |
Cervical spine fusion surgery in India is generally performed by expert spine surgeons to relieve pain, numbness, tingling and weakness, restore nerve function and stop or prevent abnormal motion in the spine. It is one of the safest spine surgery procedures for the correction of all spine disorders and help in achieving a normal life. Your surgeon does this by removing a disc or a bone and fusing the vertebrae together with a bone graft either in front of or behind the spine. The bone graft may be one of two types: an auto graft or an allograft. Sometimes metal plates, screws or wires are also used to further stabilize the spine. These techniques are called instrumentation. When the vertebrae have been surgically stabilized, abnormal motion is stopped and function is restored to the spinal nerves. Cervical spine fusion surgery may be indicated for a variety of cervical spine problems. Generally, surgery may be performed for degenerative disorders, trauma or instability. These conditions may produce pressure on the spinal cord or on the nerves coming from the spine.
Being an Australian native you would require assistance of medical tourism in India for managing a successful cervical spine fusion surgery in India.
In degenerative disease the discs or cushion pads between your vertebrae shrink, causing wearing of the disc, which may lead to herniation. You may also have arthritic areas in your spine. This degeneration can cause pain, numbness, tingling and weakness from the pressure on the spinal nerves. With cervical spine fusion surgery from a reputed spine surgery center in India you can get rid of all these spine problems.
Medical tourism in India provides very good assistance to international patients for their comfortable spine treatment. Also, they take care of the entire things required to patients after their arrival in India for their spine surgery.
They will ask you to provide them all you previous case history, the case history provided by you will be forwarded to some of the best surgeons in India. Then the spine surgeons will suggest you appropriate spine surgery procedure depending on your health and condition requirement.
Cervical spine fusion surgery in India is an advanced surgical procedure used for the correction of all spine problems and is being generally used allover. It takes few hours for performing the surgery, but some time is required for the recovery of patients.
You need to come to India a day prior to your cervical spine fusion surgery, and have a meeting with the surgeon to discuss your case with the surgeon before going ahead for the surgery. So that everything about your case should be clear to your surgeon.
Now a day treatment facilities being used are most advanced, thus it takes very little time for the recovery of patients. Within one week patients start walking with some light support after their surgery. And the support requirement is not there as time passes. Click the link to see advanced treatment facilities provided in advanced spine treatment centers: Treatment Facilities.
Also, if you want detailed information of cervical spine fusion surgery in India, you can contact us by filling up an enquiry form here: https://www.dheerajbojwani.com/contact-dheerajbojwani.html. | english |
دھرم شالاسپورٹس ڈیسکجنوبی افریقن ٹیم کے مایہ ناز اوپننگ بلے باز ہاشم املہ نے کہا ہے کہ ورلڈ ٹی ٹونٹی کپ کے اگلے میچ میں افغانستان کو اسان حریف تصور کرنے کی غلطی نہیں کریں گے جبکہ افغانستان ٹیم کے کپتان اصغرسٹانکزئی نے کہا کہ جنوبی افریقہ ورلڈ کلاس ٹیم ہے ٹائم دینے کی کوشش کریں گے اپنے ایک بیان میں املہ نے کہا کہ افغانستان کی ٹیم اچھی کرکٹ کھیل رہی ہے اس نے سری لنکا کو بھی ٹف ٹائم دیا تھا ہمیں میچ میں کامیابی کیلئے سو فیصد کارکردگی دکھانا ہو گی اصغر سٹانکزئی نے کہا کہ پروٹیز سے خوفزدہ نہیں ڈٹ کر مقابلے کیلئے تیار ہیں انگلینڈ سے شکست کے بعد پروٹیز پر دبائو ہو گا اسلئے ہمارے پاس اسے ٹف ٹائم دینے کا بہترین موقع ہے تاہم کھلاڑیوں کو کھیل کے تینوں شعبوں میں صلاحیتوں سے بڑھ کر کارکردگی دکھانا ہو گی | urdu |
अभिषेक बनर्जी फिर से तृणमूल कांग्रेस के महासचिव नियुक्त स्टाफ रिपोर्टर, एएनएम न्यूज: तृणमूल कांग्रेस को लेकर बड़ी खबर सामने आ रही है। बताया जा रहा है कि अभिषेक बनर्जी को तृणमूल कांग्रेस के नए पदाधिकारियों की समिति का फिर से महासचिव नियुक्त किया गया है। पार्टी के वरिष्ठ नेता पार्थ चटर्जी ने इसकी पुष्टि भी कर दी है।इससे बंगाल की मुख्यमंत्री और टीएमसी प्रमुख ममता बनर्जी ने शनिवार को पार्टी की राष्ट्रीय पदाधिकारियों की समिति को भंग कर दिया था और पार्टी के अनुभवी नेताओं और अगली पीढ़ी के नेताओं के बीच बढ़ती दरार के बीच 20 सदस्यीय कार्य समिति का गठन कर दिया था। | hindi |
\begin{document}
\title{Splitting schemes for hyperbolic heat conduction equation}
\begin{abstract}
Rapid processes of heat transfer are not described by the standard heat conduction equation.
To take into account a finite velocity of heat transfer, we use the hyperbolic model of heat conduction,
which is connected with the relaxation of heat fluxes.
In this case, the mathematical model is based on a hyperbolic equation of second order
or a system of equations for the temperature and heat fluxes.
In this paper we construct for the hyperbolic heat conduction equation the additive schemes of splitting with respect to directions.
Unconditional stability of locally one-dimensional splitting schemes is established.
New splitting schemes are proposed and studied for a system of equations written in terms of the temperature and heat fluxes.
\end{abstract}
\begin{keywords}
hyperbolic heat conduction equation, finite difference schemes,
splitting schemes
\end{keywords}
\begin{AMS}
80A20, 65M06, 65M12
\end{AMS}
\pagestyle{myheadings}
\thispagestyle{plain}
\markboth{PETR N. VABISHCHEVICH}{SPLITTING SCHEMES FOR HYPERBOLIC HEAT}
\section{Introduction}
Linear parabolic theories of diffusion and heat conduction are based on the Fick and Fourier laws, respectively,
and predict an infinite speed of propagation \cite{crank1979mathematics,incropera1996fundamentals}.
In this case, the amplitude of propagating perturbations decreases exponentially
with the distance and the infinite speed of perturbations can often be ignored.
Nevertheless, in many applied problems the wave nature of heat transfer should be taken into account.
Since paper \cite{cattaneo1958forme}, various corrections have been proposed for parabolic heat conduction models in order
to eliminate the paradox of infinite speed of perturbation propagation
\cite{joseph1989heat,casas2010extended,shashkov2004wave}.
The standard parabolic heat conduction model is based on the explicit representation of the heat flux through the temperature gradient.
The hyperbolic heat conduction model includes an additional term with the time derivative for the heat flux
which is proportional to the relaxation tensor.
More general models (see \cite{joseph1989heat}) in addition includes the relaxation of the temperature gradient.
Separate attention should be given to the hyperbolic model of convection-diffusion for moving media
\cite{gomez2008hyperbolic,gomez2008mathematical}.
Two possibilities can be highlighted in constructing computational methods for
the approximate solution of hyperbolic heat transfer problems.
The first is connected with the transition from a system of the first order evolutionary
equations for the temperature and heat flux to a single hyperbolic equation of second order.
In contrast to the standard parabolic equation of heat conduction there does present a term with the second time derivative.
The second possibility is based on the usage of the initial scalar-vector system of equations.
To solve approximately the boundary value problem for the hyperbolic heat conduction equation,
classical numerical methods can be used including finite-difference approximations in space,
finite volume schemes or finite-element approximations. For instance, three-level difference schemes for the hyperbolic
heat conduction equation are constructed in \cite{ciegis-numerical,samarskii1995computational}.
To investigate the stability and convergence of difference schemes, the general theory of stability for operator-difference schemes
is used in \cite{samarskii2001theory,samarskii2002difference}.
Investigation of the stability on the basis of a priori estimates of the finite-difference solution for
the model with the relaxation of the temperature gradients is given in \cite{dai2004unconditionally,zhang2001unconditionally}.
An analysis of possibilities to use the simplest schemes of first and second order for integration
in time is given in \cite{moosaie2009comparative} for model one-dimensional problems
of hyperbolic heat conduction.
The system of equations governing thermal processes in terms of the temperature and heat flux
has a defined structure with conjugated to each other operators.
Such a structure of the mathematical model makes possible to use
this feature in the construction of computational algorithms
\cite{brezzi1991mixed,roberts1991mixed}.
In a number of papers (see, eg,
\cite{glass1985numerical,shen2003numerical,tamma1991hyperbolic,yang1990characteristics})
the hyperbolic nature of mathematical models with heat waves emphasizes
in using traditional technologies of compressible media dynamics.
Various finite element methods are applied in papers \cite{gomez2007discontinuous,gomez2007finite}
for equations of hyperbolic convection-diffusion theory.
Much attention is paid to the construction of the additive schemes (splitting schemes)
for approximate solving initial-boundary value problems for multi-dimensional partial differential equations
\cite{marchuk1990splitting,yanenko1971method}.
Transition to a sequence of more simple problems allows to construct, for example, economical
difference schemes - schemes based on the splitting with respect to spatial variables.
In some cases it is reasonable to perform splitting with respect to subproblems of different nature -
splitting in physical processes.
At present regionally-additive schemes (domain decomposition methods)
are actively discussed \cite{samarskii1999additive}. These schemes are oriented to the construction of
computational algorithms for parallel computers.
Additive difference schemes in general conditions of the splitting of the problem operator into a sum of noncommutative
non-selfadjoint operators are obtained in the most simple way for the case of two-component splitting.
In this case for the evolutionary equation of first order the classical alternative direction schemes, factorized
and predictor-corrector schemes are unconditionally stable at weak restrictions.
A more complicated situation takes place in the case of multi-component splitting (splitting into three and more operators).
For these problems the most interesting results are obtained on the basis of the concept of summarized approximation.
The initial problem at the transition from one time level to another is divided into several subproblems,
and each of these subproblems, in general, do not approximate the initial problem.
On this way, unconditionally stable schemes of componentwise splitting (locally one-dimensional schemes
of splitting with respect to spatial variables) are constructed.
A new class of operator-difference splitting schemes - vector additive schemes - was developed in
papers \cite{abrashin1990variant,vabishchevich1996vector}.
In this class of schemes we go from the initial scalar problem for one unknown function
to the problem for a vector, each component of which can be treated as the solution of the problem.
On this way we construct the full approximation schemes for evolutionary equations of the first
and second order based on a general multi-component splitting.
New additive difference schemes for differential-operator equations of the first and second order
for the general case of splitting with an arbitrary number of pairwise noncommutative
operator terms were constructed in \cite{samarskii1998regularized,samarskii1999additive}
using the principle of regularization.
The main theoretical results on the stability and convergence of the additive schemes
were obtained for scalar evolutionary equations of the first order and, in some cases, for second-order equations.
Splitting schemes for systems of evolutionary equations are of considerable interest for computational practice.
For standard parabolic and hyperbolic systems of equations with selfadjoint elliptic operators
additive schemes were constructed in \cite{samarskii2001theory}
using the principle of regularization for difference schemes.
The Cauchy problem for a special linear system of first order equations in the Hilbert
space with the conjugate operators (divergence and gradient) is considered in paper \cite{VabSys}.
Such a structure of equations is characteristic for the considering here problems of hyperbolic heat transfer.
In the present work there are constructed splitting schemes with respect to spatial variables for
the approximate solving the equation of hyperbolic heat conduction.
Unconditionally stable locally one-dimensional difference schemes are constructed here both
for a single heat conduction equation and for the system of equations based on the temperature and heat flux as unknowns.
This paper is organized as follows.
In section 2 the differential problem is formulated for the hyperbolic heat conduction.
Appropriate a priori estimates are obtained for the solution of the hyperbolic equation in the both
above mentioned formulations.
Approximation in space is discussed in Section 3 for a model problem in a rectangle.
It was shown that the grid operators of divergence and gradient are ajoint each other.
Standard three-level difference schemes for the hyperbolic heat conduction equation
are constructed in Section 4. The a priori estimates are derived for the difference solution.
Difference schemes for the system of equations based on the temperature and heat flux as unknowns are considered in Section 5.
Unconditionally stable schemes are derived via the regularization of explicit-implicit schemes.
Locally one-dimensional schemes for the hyperbolic heat conduction equation are studied in Section 6.
Splitting scheme for the system of hyperbolic heat conduction equations are proposed in Section 7.
\section{Differential problem}
Temperature $u(\mathbf{x},t)$ in bounded domain $\Omega$ with boundary $\partial\Omega$
is governed by the equation
\begin{equation}\label{2.1}
c \frac{\partial u}{\partial t} + \mathop{\rm div}\nolimits \mathbf{q} = f,
\quad \mathbf{x} \in \Omega,
\quad 0 < t \leq T,
\end{equation}
where $\mathbf{x} = (x_1, x_2, ..., x_n)$ is a point in space,
$t$ is the time ($T > 0$), and $\mathbf{q} = \mathbf{q}(\mathbf{x},t)$ is the heat flux.
In (\ ref (2.1)) $c = c(\mathbf{x}) \geq c_0 > 0$ is the specific heat capacity of a medium,
and $f = f(\mathbf{x},t)$ is the rate of volumetric heat sources.
The standard (parabolic) model of the heat conduction results from the following
representation for the heat flux (Fourier's law)
\begin{equation}\label{2.2}
\mathbf{q} + k \mathop{\rm grad}\nolimits u = 0,
\end{equation}
where $k = k(\mathbf{x}) \geq k_0 > 0$ is the thermal conductivity of the medium.
Substitution of (\ref{2.2}) in (\ref{2.1}) leads us to the parabolic heat conduction equation
\begin{equation}\label{2.3}
c \frac{\partial u}{\partial t} -
\mathop{\rm div}\nolimits( k \mathop{\rm grad}\nolimits u ) = f,
\quad \mathbf{x} \in \Omega,
\quad 0 < t \leq T,
\end{equation}
supplemented by appropriate boundary and initial conditions.
In the model of the hyperbolic heat conduction instead of (\ref{2.2}) we use the following relation
\begin{equation}\label{2.4}
\mathbf{q} + \nu \frac{\partial \mathbf{q}}{\partial t} +
k \mathop{\rm grad}\nolimits u = 0,
\end{equation}
where $\nu$ is the relaxation parameter for the heat flux.
From (\ref{2.1}) and (\ref{2.4}) we obtain the hyperbolic heat conduction equation
\begin{equation}\label{2.5}
\nu c \frac{\partial^2 u}{\partial t^2} +
c \frac{\partial u}{\partial t} -
\mathop{\rm div}\nolimits( k \mathop{\rm grad}\nolimits u ) =
f + \nu \frac{\partial f}{\partial t},
\quad \mathbf{x} \in \Omega,
\quad 0 < t \leq T .
\end{equation}
Consider a model boundary value problem for equation (\ref{2.5}) (system (\ref{2.1}) and (\ref{2.4})),
where the boundary conditions are as follows
\begin{equation}\label{2.6}
u(\mathbf{x},t) = 0,
\quad \mathbf{x} \in \partial \Omega,
\quad 0 < t \leq T .
\end{equation}
In addition, two initial conditions are prescribed
\begin{equation}\label{2.7}
u(\mathbf{x},0) = v_0(\mathbf{x}),
\quad \frac{\partial u}{\partial t} (\mathbf{x},0) = v_1(\mathbf{x}),
\quad \mathbf{x} \in \Omega.
\end{equation}
The simplest a priori estimates for problem (\ref{2.5})--(\ref{2.7}),
((\ref{2.1}), (\ref{2.4}), (\ref{2.6}), (\ref{2.7})) will be derived now in order
to be our guidelines in the investigation of grid problems.
Let $(\cdot, \cdot)$ and $\| \cdot \|$ be the scalar product and norm in $\mathcal{H} = L_2(\Omega)$, respectively.
Multiplying scalarly equation (\ref{2.5}) by $\partial u / \partial t$ in $\mathcal{H}$
we obtain
\begin{equation}\label{2.8}
\left ( c \frac{\partial u}{\partial t}, \frac{\partial u}{\partial t}\right ) +
\frac{\nu}{2} \frac{d}{d t}
\left ( c \frac{\partial u}{\partial t}, \frac{\partial u}{\partial t}\right ) +
\frac{1}{2} \frac{d}{d t} (k \mathop{\rm grad}\nolimits u, \mathop{\rm grad}\nolimits u) =
\left ( f + \nu \frac{\partial f}{\partial t}, \frac{\partial u}{\partial t}\right ) .
\end{equation}
The right hand side of (\ref{2.8}) is estimated as follows
\begin{equation}\label{2.9}
\left ( f + \nu \frac{\partial f}{\partial t}, \frac{\partial u}{\partial t}\right ) \leq
\left ( c \frac{\partial u}{\partial t}, \frac{\partial u}{\partial t}\right ) +
\frac{1}{4}\left (c^{-1} \left (f + \nu \frac{\partial f}{\partial t} \right ), f + \nu \frac{\partial f}{\partial t} \right ) .
\end{equation}
From (\ref{2.8}), (\ref{2.9}) we have the inequality
\begin{equation}\label{2.10}
\frac{d}{d t} S \leq \frac{1}{2} \left (c^{-1} f + \nu \frac{\partial f}{\partial t}, f + \nu \frac{\partial f}{\partial t} \right ) .
\end{equation}
Here
\begin{equation}\label{2.11}
S(t) = \nu
\left ( c \frac{\partial u}{\partial t}, \frac{\partial u}{\partial t}\right ) +
(k \mathop{\rm grad}\nolimits u, \mathop{\rm grad}\nolimits u)
\end{equation}
defines the squared norm for the solution of (\ref{2.5})--(\ref{2.7}) with boundary conditions (\ref{2.6}).
Applying to (\ref{2.10}) the Gronwall lemma, we obtain the desired estimate
\begin{equation}\label{2.12}
S(t) \leq S(0) +
\frac{1}{2} \int\limits_0^t
\left \|c^{-1/2} \left (f + \nu \frac{\partial f}{\partial t} \right )(\mathbf{x},\theta) \right \|^2 d \theta .
\end{equation}
At $\nu =0$ estimate (\ref{2.12}) degenerates into the corresponding estimate for the solution of parabolic heat equation (\ref{2.3}).
For the system of equations instead of initial conditions (\ref{2.7}) it is more natural to use
\begin{equation}\label{2.13}
u(\mathbf{x},0) = v_0(\mathbf{x}),
\quad \mathbf{q} (\mathbf{x},0) = \mathbf{g}_0(\mathbf{x}),
\quad \mathbf{x} \in \partial \Omega,
\end{equation}
ie instead of the rate of temperature variation we define the heat flux.
The transition from one to another initial conditions is provided by equation (\ref{2.3}).
To obtain a simple a priori estimate for system (\ref{2.1}), (\ref{2.4}) we scalarly multiply
equation (\ref{2.1}) by $u$, and (\ref{2.4}) - by $k^{-1} \mathbf{q}$ and sum them.
This gives
\begin{equation}\label{2.14}
\frac{1}{2} \frac{d}{d t} (c u,u) +
\frac{\nu}{2} \frac{d}{d t} ( k^{-1} \mathbf{q},\mathbf{q}) +
( k^{-1} \mathbf{q},\mathbf{q}) = (f,u) .
\end{equation}
For the right hand side we use the estimate
\[
(f,u) \leq \frac{1}{2} (c u,u) + \frac{1}{2} (c^{-1} f,f) .
\]
From (\ref{2.14}) we obtain
\begin{equation}\label{2.15}
\frac{d}{d t} G \leq G + (c^{-1} f,f) ,
\end{equation}
\begin{equation}\label{2.16}
G(t) = (c u,u) + \nu ( k^{-1} \mathbf{q},\mathbf{q}).
\end{equation}
From (\ref{2.15}) we derive the estimate
\begin{equation}\label{2.17}
G(t) \leq \exp(t) G(0) + \int\limits_0^t \exp(t-\theta)
\|c^{-1/2} f(\mathbf{x},\theta) \|^2 d \theta ,
\end{equation}
which ensures the stability of the solution of system (\ref{2.1}), (\ref{2.4}) with respect to initial data (\ref{2.13})
and the right hand side.
\section{Approximation in space}
Let us consider the 2D model problem of the hyperbolic heat conduction in the rectangle
\[
\Omega = \{ \ \mathbf{x} \ | \ \mathbf{x} = (x_1, x_2),
\quad 0 < x_{\alpha} < l_{\alpha}, \quad \alpha = 1,2 \} .
\]
Let $q_{\alpha}, \ \alpha = 1,2$ be the Cartesian components of heat flux $\mathbf{q} = (q_1,q_2)$.
The system of equations (\ref{2.1}), (\ref{2.4}) in the coordinate-wise representation
takes the form
\begin{equation}\label{3.1}
c \frac{\partial u}{\partial t} +
\sum_{\alpha =1}^{2} \frac{\partial q_{\alpha}}{\partial x_{\alpha}} = f,
\end{equation}
\begin{equation}\label{3.2}
q_{\alpha} + \nu \frac{\partial q_{\alpha}}{\partial t} +
k \frac{\partial u}{\partial x_{\alpha}} = 0,
\quad \alpha = 1,2 .
\end{equation}
On the set of functions $u$, satisfying homogeneous boundary conditions (\ref{2.6}),
we define the operators
\begin{equation}\label{3.3}
\mathcal{A}_{\alpha} u = \frac{\partial u}{\partial x_{\alpha}},
\quad \alpha = 1,2 .
\end{equation}
Taking into account that
\[
\int\limits_{\Omega} \frac{\partial u}{\partial x_{\alpha}} v d \mathbf{x} =
- \int\limits_{\Omega} u \frac{\partial v}{\partial x_{\alpha}} d \mathbf{x} ,
\]
we have
\begin{equation}\label{3.4}
\mathcal{A}^*_{\alpha} v = - \frac{\partial v}{\partial x_{\alpha}},
\quad \alpha = 1,2
\end{equation}
for the conjugate operators.
In view of (\ref{3.3}), (\ref{3.4}) the system of equations (\ref{3.1}), (\ref{3.2})
with boundary conditions (\ref{2.6}) can be written in the following operator form
\begin{equation}\label{3.5}
q_{\alpha} + \nu \frac{d q_{\alpha}}{d t} +
k \mathcal{A}_{\alpha} u = 0,
\quad \alpha = 1,2 ,
\end{equation}
\begin{equation}\label{3.6}
c \frac{d u}{d t} -
\sum_{\alpha =1}^{2} \mathcal{A}^*_{\alpha} q_{\alpha} = f .
\end{equation}
Thus, the system of equations governing the hyperbolic heat conduction does have the operator structure with conjugate operators.
For hyperbolic heat conduction equation (\ref{2.5}) the corresponding operator-differential
equation has the form
\begin{equation}\label{3.7}
\nu c \frac{d^2 u}{d t^2} +
c \frac{d u}{d t} + \mathcal{D} u =
f + \nu \frac{\partial f}{\partial t} ,
\end{equation}
\begin{equation}\label{3.8}
\mathcal{D} = \sum_{\alpha =1}^{2} \mathcal{D}_{\alpha},
\quad \mathcal{D}_{\alpha} u =
\mathcal{A}^*_{\alpha} k \mathcal{A}_{\alpha},
\quad \alpha = 1,2 .
\end{equation}
Operator $\mathcal{D}$, as well as its individual terms $\mathcal{D}_{\alpha}, \ \alpha = 1,2$,
is selfadjoint and positive definite in $L_2(\Omega)$ on the set of functions satisfying
boundary conditions (\ref{2.3}).
We want to preserve the above operator structure of the differential model for the hyperbolic heat conduction
after its approximation in space.
For simplicity, we will consider the simplest difference approximations on uniform grids.
In the considering problems it is natural to use for the scalar and vector unknowns staggered grids,
where scalar variables and vector components employ their own grids.
Such a technology is standard for problems of computational fluid dynamics \cite{versteeg2007introduction}
and electrodynamics \cite{taflove2000computational}.
The temperature is defined at the nodes of a uniform rectangular grid in $\Omega$:
\[
\bar{\omega} = \{ \mathbf{x} \ | \ \mathbf{x} = (x_1, x_2),
\quad x_\alpha = i_\alpha h_\alpha,
\quad i_\alpha = 0,1,...,N_\alpha,
\quad N_\alpha h_\alpha = l_\alpha, \quad \alpha = 1,2\}
\]
and let $\omega$ be a set of internal nodes ($\bar{\omega} = \omega \cup \partial \omega$).
The components of vector quantities are referred to the corresponding edges of the grid.
We define
\[
\bar{\omega}_1 = \{ \mathbf{x} \ |
\ x_1 = (i_1 + 0.5) h_1,
\ i_1 = 0,1,...,N_1-1,
\ x_2 = i_2 h_2,
\ i_2 = 0,1,...,N_2 \} ,
\]
\[
\bar{\omega}_2 = \{ \mathbf{x} \ |
\ x_1 = i_1 h_1,
\ i_1 = 0,1,...,N_1,
\ x_2 = (i_2 + 0.5) h_2,
\ i_2 = 0,1,...,N_2-1 \}
\]
and $\bar{\omega}_{\alpha} = \omega_{\alpha} \cup \partial \omega_{\alpha}, \ \alpha = 1,2$.
Component of heat flux $q_{\alpha}, \ \alpha = 1,2$ will be evaluated on the
grid $\bar{\omega}_{\alpha}, \ \alpha = 1,2$ (Fig.\ref{f-1}).
\begin{figure}
\caption{Ñåòêè: {\large $\bullet$}
\label{f-1}
\end{figure}
For grid functions $y(\mathbf{x}) = 0, \ \mathbf{x} \in \partial \omega$
we define the Hilbert space $H = L_2({\omega})$
with the scalar product and norm
\[
(y,w) \equiv \sum_{{\bf x} \in \omega}
y({\bf x}) w({\bf x}) h_1 h_2,
\quad \|y\| \equiv (y,y)^{1/2} .
\]
Similarly, for the grid functions defined on grid $\omega_{\alpha}, \ \alpha = 1,2$,
we define the Hilbert space $H_{\alpha}, \ \alpha = 1,2$, where
\[
(y,w)_{\alpha} \equiv \sum_{{\bf x} \in \omega_{\alpha}}
y({\bf x}) w({\bf x}) h_1 h_2,
\quad \|y\|_{\alpha} \equiv (y,y)_{\alpha}^{1/2},
\quad \alpha = 1,2.
\]
Let us construct the grid analogs of differential operators
$\mathcal{A}_{\alpha}, \ \mathcal{A}^*_{\alpha}, \ \alpha = 1,2$,
defined above according to (\ref{3.3}), (\ref{3.4}).
We will use the standard \cite{samarskii2001theory}
central-difference approximations for derivatives in space.
We set
\begin{equation}\label{3.9}
(A_1 y)(\mathbf{x}) = \frac{y(x_1+0.5h_1, x_2)-y(x_1-0.5h_1, x_2)}{h_1},
\quad \mathbf{x} \in \omega_1 ,
\end{equation}
so that $A_1: H \rightarrow H_1$.
Similarly, we define $A_2: H \rightarrow H_2$, where
\begin{equation}\label{3.10}
(A_2 y)(\mathbf{x}) = \frac{y(x_1, x_2+0.5h_2)-y(x_1, x_2-0.5h_2)}{h_2},
\quad \mathbf{x} \in \omega_2 .
\end{equation}
By the construction we have
\begin{equation}\label{3.11}
A_{\alpha} u =
\mathcal{A}_{\alpha} u + O(h_{\alpha}^2),
\quad \alpha = 1,2 .
\end{equation}
Direct calculations verify that for the adjoint operators
$A^*_{\alpha}: H_{\alpha} \rightarrow H, \ \alpha = 1,2$ we have the representation
\begin{equation}\label{3.12}
(A^*_1 y)(\mathbf{x}) = - \frac{y(x_1+0.5h_1, x_2)-y(x_1-0.5h_1, x_2)}{h_1},
\quad \mathbf{x} \in \omega ,
\end{equation}
\begin{equation}\label{3.13}
(A^*_2 y)(\mathbf{x}) = - \frac{y(x_1, x_2+0.5h_2)-y(x_1, x_2-0.5h_2)}{h_2},
\quad \mathbf{x} \in \omega .
\end{equation}
For sufficiently smooth functions $u$
\begin{equation}\label{3.14}
A^*_{\alpha} u =
\mathcal{A}^*_{\alpha} u + O(h_{\alpha}^2),
\quad \alpha = 1,2
\end{equation}
holds.
After approximation of system (\ref{3.5}), (\ref{3.6}) in space we obtain the system of evolutionary equations
\begin{equation}\label{3.15}
q_{\alpha} + \nu \frac{d q_{\alpha}}{d t} +
k A_{\alpha} u = 0,
\quad \mathbf{x} \in \omega_{\alpha} ,
\quad \alpha = 1,2 ,
\end{equation}
\begin{equation}\label{3.16}
c \frac{d u}{d t} -
\sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha} = f ,
\quad \mathbf{x} \in \omega .
\end{equation}
Similarly, equation (\ ref (3.7)) is associated with the evolutionary equation
\begin{equation}\label{3.17}
\nu c \frac{d^2 u}{d t^2} +
c \frac{d u}{d t} + D u =
f + \nu \frac{\partial f}{\partial t} ,
\quad \mathbf{x} \in \omega ,
\end{equation}
\begin{equation}\label{3.18}
D = \sum_{\alpha =1}^{2} D_{\alpha},
\quad D_{\alpha} = A^*_{\alpha} k A_{\alpha},
\quad \alpha = 1,2 .
\end{equation}
Taking into account (\ref{3.9}),(\ref{3.10}) and (\ref{3.12}),(\ref{3.13}),
for grid operators $D_{\alpha}: H \rightarrow H,$ $\alpha = 1,2$
we obtain
\begin{equation}\label{3.19}
(D_1 y)(\mathbf{x}) = \frac{1}{h_1} \left (
k(x_1+0.5h_1, x_2) \frac{y(x_1+h_1, x_2)-y(\mathbf{x})}{h_1}
\right .
\end{equation}
\[
\left .
- k(x_1-0.5h_1, x_2) \frac{y(\mathbf{x})- y(x_1-h_1, x_2)}{h_1}
\right ),
\quad \mathbf{x} \in \omega ,
\]
\begin{equation}\label{3.20}
(D_2 y)(\mathbf{x}) = \frac{1}{h_2} \left (
k(x_1, x_2+0.5h_2) \frac{y(x_1, x_2+h_2)-y(\mathbf{x})}{h_2}
\right .
\end{equation}
\[
\left .
- k(x_1, x_2-0.5h_2) \frac{y(\mathbf{x})- y(x_1, x_2-h_2)}{h_2}
\right ),
\quad \mathbf{x} \in \omega .
\]
Similarly to (\ref{3.11}),(\ref{3.14}), we have \cite{samarskii2001theory,samarskii1989numerical}
\begin{equation}\label{3.21}
D_{\alpha} u =
\mathcal{D}_{\alpha} u + O(h_{\alpha}^2),
\quad \alpha = 1,2
\end{equation}
in the class of sufficiently smooth coefficients $k$ and functions $u$.
In addition, in the space of grid functions $H$
\begin{equation}\label{3.22}
D_{\alpha} = D^*_{\alpha},
\quad k_0 \delta_{\alpha} E \leq D_{\alpha} \leq k_1 \Delta_{\alpha} E,
\end{equation}
\[
\delta_{\alpha} =
\frac{4}{h^2_{\alpha}} \sin^2 \frac{\pi h_{\alpha}}{2 l_{\alpha}} ,
\quad \Delta_{\alpha} =
\frac{4}{h^2_{\alpha}} \cos^2 \frac{\pi h_{\alpha}}{2 l_{\alpha}} ,
\quad \alpha = 1,2 ,
\]
where $ E $ is the unit (identity) operator and
$k(\mathbf{x}) \leq k_1, \ \mathbf{x} \in \Omega$.
\section{Difference schemes for the hyperbolic heat conduction equation}
We consider approximation in time for the approximate solution of differential-operator equation (\ref{3.17}),
which is supplemented by the initial conditions
\begin{equation}\label{4.1}
u(\mathbf{x},0) = v_0(\mathbf{x}),
\quad \frac{d u}{d t} (\mathbf{x},0) = v_1(\mathbf{x}),
\quad \mathbf{x} \in \omega.
\end{equation}
Let us define a uniform grid in time
\[
\overline{\omega}_\tau =
\omega_\tau\cup \{T\} =
\{t_n=n\tau,
\quad n=0,1,...,N,
\quad \tau N=T\}
\]
and denote $y^n = y(t_n), \ t_n = n \tau$.
Standard three-level difference schemes with the second-order approximation in time will be considered.
Equation (\ref{3.7}) is approximated by the difference scheme with weights
\begin{equation}\label{4.2}
\nu c \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c \frac{u^{n+1} - u^{n-1}}{2\tau}
\end{equation}
\[
+ D (\sigma u^{n+1} + (1- 2\sigma) u^{n} + \sigma u^{n-1}) =
\varphi^n ,
\quad n = 1,2, \ldots, N-1 ,
\]
where
\[
\varphi^n = f^n + \nu \frac{f^{n+1} - f^{n-1}}{2\tau},
\]
with the corresponding initial data
\begin{equation}\label{4.3}
u^0= v_0,
\quad \frac{u^{1} - u^{0}}{\tau} = w_0 .
\end{equation}
Scheme (\ref{4.2}), (\ref{4.3}) which belongs to the class of three-level operator-difference schemes
can be investigated on the basis of the Samarskii stability (correctness) theory of operator-difference schemes.
Coincident necessary and sufficient conditions of the stability of these schemes in various norms are obtained in \cite{samarskii2001theory,samarskii2002difference}.
With this in mind, we give here only the simplest a priori estimates of stability with respect to
the initial data and right hand side for scheme (\ref{4.2}), (\ref{4.3}).
\begin{theorem}
\label{t-1}
Difference scheme (\ref{4.2}), (\ref{4.3})
is unconditionally stable at $\sigma \geq 0.25$ and for the finite-difference solution we have the estimate
\begin{equation}\label{4.4}
S^{n+1} \leq S^n + \frac{\tau }{2} (c^{-1} \varphi^n,\varphi^n) ,
\end{equation}
where
\begin{equation}\label{4.5}
S^n = \left (\left (\nu c E + \left (\sigma - \frac{1}{4} \right )
\tau^2 D \right )\frac{u^n - u^{n-1}}{\tau },
\frac{u^n - u^{n-1}}{\tau } \right )
\end{equation}
\[
+ \left (D \frac{u^n + u^{n-1}}{2}, \frac{u^n + u^{n-1}}{2} \right ) .
\]
\end{theorem}
\begin{proof}
To prove this, we introduce the notation
\[
\zeta^n = \frac{u^n + u^{n-1}}{2} ,
\quad \eta^n = \frac{u^n - u^{n-1}}{\tau } .
\]
Taking into account the identities
\[
u^n = \frac{1}{4} ( u^{n+1} + 2 u^n + u^{n-1})
- \frac{1}{4} ( u^{n+1} - 2 u^n + u^{n-1}),
\]
\[
\sigma u^{n+1} + (1- 2\sigma) u^{n} + \sigma u^{n-1} =
u^n + \sigma ( u^{n+1} - 2 u^n + u^{n-1})
\]
we rewrite (\ref{4.2}) in the form
\begin{equation}\label{4.6}
\left (\nu c E + \left (\sigma - \frac{1}{4} \right ) \tau^2 D \right ) \frac{\eta^{n+1} - \eta^{n}}{\tau} +
c \frac{\eta^{n+1} + \eta^{n}}{2} +
D \frac{\zeta^{n+1} + \zeta^{n}}{2} = \varphi^n .
\end{equation}
We scalarly multiply in $H$ this equation by
\[
2(\zeta^{n+1} - \zeta^{n}) = \tau (\eta^{n+1} + \eta^{n}) .
\]
This gives
\begin{equation}\label{4.7}
\left (\left (\nu c E + \left (\sigma - \frac{1}{4} \right ) \tau^2 D \right )\eta^{n+1}, \eta^{n+1}\right ) +
(D \zeta^{n+1}, \zeta^{n+1})
\end{equation}
\[
- \left (\left (\nu c E + \left (\sigma - \frac{1}{4} \right ) \tau^2 D \right )\eta^{n}, \eta^{n}\right ) -
(D \zeta^{n}, \zeta^{n})
\]
\[
+ \frac{\tau }{2} (c (\eta^{n+1} + \eta^{n}), (\eta^{n+1} + \eta^{n})) =
\tau (\varphi^n,(\eta^{n+1} + \eta^{n})) .
\]
If $\sigma \geq 0.25$ then value
\[
S^n = \left (\left (\nu c E + \left (\sigma - \frac{1}{4} \right ) \tau^2 D \right )\eta^{n}, \eta^{n}\right ) +
(D \zeta^{n}, \zeta^{n})
\]
defines the squared norm of the difference solution.
With this notation we obtain the required estimate (\ref{4.5}).
\qquad\end{proof}
Estimate (\ref{4.5}) for the numerical solution is consistent with estimate (\ref{2.12})
for the solution of the differential problem.
Using this estimate it is easy to prove in the standard enough way \cite{samarskii2001theory,samarskii2002difference}
that the difference solution converges to the exact one with truncation error $O(\tau^2 +h_1^2 + h_2^2)$
(with the second order in time and space).
\section{Difference schemes for the hyperbolic heat conduction governed by the system of equations}
For the approximate solution of the Cauchy problem for system (\ref{3.5}), (\ref{3.6})
we use the simplest schemes with weights
\begin{equation}\label{5.1}
q_{\alpha}^{\sigma(n)} +
\nu \frac{q^{n+1}_{\alpha} - q^{n}_{\alpha} }{\tau} +
k A_{\alpha} u^{\sigma(n)} =
0, \quad \alpha =1,2,
\end{equation}
\begin{equation}\label{5.2}
c \frac{u^{n+1} - u^{n} }{\tau} -
\sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha}^{\sigma(n)} =
f^{n+1/2}, \quad n = 0,1, \ldots, N-1 ,
\end{equation}
where $\sigma$ is a numerical parameter (weight), which is usually $0 \le \sigma \le 1$.
We have used the notation
\[
u^{\sigma(n)} =
\sigma u^{n+1} + (1-\sigma) u^{n},
\quad q_{\alpha}^{\sigma(n)} =
\sigma q^{n+1}_{\alpha} + (1-\sigma) q^{n}_{\alpha},
\quad \alpha =1,2 .
\]
For simplicity, we restrict ourselves to the same weight for all equations of system (\ref{5.1}), (\ref{5.2}).
Taking into account (\ref{2.13}) we will supply (\ref{5.1}), (\ref{5.2}) with the initial conditions
\begin{equation}\label{5.3}
u^0 = v_0,
\quad q_{\alpha}^0 = g_0^{(\alpha)},
\quad \alpha =1,2 .
\end{equation}
We give the simplest estimates of stability for operator-difference scheme (\ref{5.1})--(\ref{5.3}).
Estimate (\ref{2.17}) is used to guide us.
\begin{theorem}
\label{t-2}
Difference scheme (\ref{5.1})--(\ref{5.3}) is unconditionally stable at
$\sigma \geq 0.5$ and for the numerical solution the following estimate holds
\begin{equation}\label{5.4}
G^{n+1} \leq \exp\left (\frac{4 \tau }{T} \right ) G^{n} +
\tau \, T \, \exp\left (\frac{2 \sigma -1 }{T} \tau \right )
(c^{-1} f^{n+1/2},f^{n+1/2}) ,
\end{equation}
where
\begin{equation}\label{5.5}
G^{n} =
(c u^{n}, u^{n}) +
\nu \sum_{\alpha =1}^{2} (k^{-1} q^{n}_{\alpha},q^{n}_{\alpha})_{\alpha} .
\end{equation}
\end{theorem}
\begin{proof}
Scalarly multiply in $H$ equation (\ref{5.2}) by $2 \tau \, u^{\sigma(n)}$,
and each separate equation (\ref{5.1}) scalarly multiply in $H_{\alpha}$ by $2 \tau \, k^{-1} q_{\alpha}^{\sigma(n)}, \ \alpha =1,2$
and sum them.
Taking into account that
\[
2 \tau \, u^{\sigma(n)} =
\tau (u^{n+1} + u^{n}) +
(2\sigma -1) \tau^2 \frac{u^{n+1} - u^{n}}{\tau} ,
\]
we obtain
\begin{equation}\label{5.6}
(c u^{n+1}, u^{n+1}) - (c u^{n}, u^{n}) +
\nu \sum_{\alpha =1}^{2} (k^{-1} q^{n+1}_{\alpha},q^{n+1}_{\alpha})_{\alpha} -
\nu \sum_{\alpha =1}^{2} (k^{-1} q^{n}_{\alpha},q^{n}_{\alpha})_{\alpha}
\end{equation}
\[
+ (2\sigma -1) \tau^2 \left (c \frac{u^{n+1} - u^{n}}{\tau},
\frac{u^{n+1} - u^{n}}{\tau} \right )
+ (2\sigma -1) \tau^2 \nu \sum_{\alpha =1}^{2}
\left (k^{-1} \frac{q^{n+1}_{\alpha} - q^{n}_{\alpha} }{\tau},
\frac{q^{n+1}_{\alpha} - q^{n}_{\alpha} }{\tau} \right )
\]
\[
+ 2 \tau \sum_{\alpha =1}^{2}
(k^{-1} q_{\alpha}^{\sigma(n)},q_{\alpha}^{\sigma(n)})_{\alpha}
= 2 \tau (f^{n+1/2}, u^{\sigma(n)}) .
\]
For terms in the right hand side of (\ref{5.6}) we have
\[
2 \tau (f^{n+1/2}, u^{\sigma(n)}) =
(2\sigma -1) \tau^2 \left(f^{n+1/2},
\frac{u^{n+1} - u^{n}}{\tau} \right) +
\tau (f^{n+1/2}, u^{n+1} + u^{n} ) .
\]
We restrict ourselves to schemes with $\sigma \geq 0.5$ and use the estimates
\[
(2\sigma -1) \tau^2 \left (f^{n+1/2},
\frac{u^{n+1} - u^{n}}{\tau} \right )
\]
\[
\leq
(2\sigma -1) \tau^2 \left ( c \frac{u^{n+1} - u^{n}}{\tau},
\frac{u^{n+1} - u^{n}}{\tau}\right ) +
\frac{(2\sigma -1)}{4} \tau^2 (c^{-1} f^{n+1/2},f^{n+1/2}) ,
\]
\[
\tau (f^{n+1/2}, u^{n+1} + u^{n} ) \leq
\frac{\tau}{2 T} (c (u^{n+1} + u^{n}),(u^{n+1} + u^{n}) ) +
\frac{\tau \, T}{2} (c^{-1} f^{n+1/2},f^{n+1/2}) ,
\]
\[
(c (u^{n+1} + u^{n}),(u^{n+1} + u^{n}) ) \leq
2 (c u^{n+1}, u^{n+1}) + 2 (c u^{n}, u^{n}) .
\]
Substitution in (\ref{5.6}) gives
\begin{equation}\label{5.7}
\left (1 - \frac{\tau}{T} \right ) (c u^{n+1}, u^{n+1}) +
\nu \sum_{\alpha =1}^{2} (k^{-1} q^{n+1}_{\alpha},q^{n+1}_{\alpha})_{\alpha}
\end{equation}
\[
\leq
\left (1 + \frac{\tau}{T} \right )
(c u^{n}, u^{n}) +
\nu \sum_{\alpha =1}^{2} (k^{-1} q^{n}_{\alpha},q^{n}_{\alpha})_{\alpha}
+ \frac{\tau \, T}{2} \left (1 + \frac{2 \sigma -1}{2T}\tau \right )
(c^{-1} f^{n+1/2},f^{n+1/2}) .
\]
Without loss of generality, we assume that $2 \tau \leq T$ and therefore
\[
\left (1 + \frac{\tau}{T} \right )
\left (1 - \frac{\tau}{T} \right )^{-1} \leq \exp\left (\frac{4 \tau }{T} \right ) .
\]
With this in mind, from (\ref{5.7}) we obtain timelevel-wise stability estimate (\ref{5.4}), (\ref{5.5}).
\qquad\end{proof}
A priori estimate (\ref{5.4}) is nothing but the grid analog of estimate (\ref{2.17}) and provides
unconditional stability of the difference scheme with weights (\ref{5.1}), (\ref{5.2})
under natural conditions $\sigma \geq 0.5$.
Considering the corresponding problem for the error \cite{samarskii2001theory,samarskii2002difference},
we prove the convergence of the solution of operator-difference problem (\ref{5.1})--(\ref{5.3})
to the solution of differential-difference problem (\ref{2.1}), (\ref{2.3}), (\ref{2.13})
at $\sigma \geq 0.5$ with order $\mathcal{O}((2 \sigma -1)\tau + \tau^2)$.
If $\sigma = 0.5$, we have the second order of convergence with respect to $\tau$.
The computational implementation of scheme (\ref{5.1}), (\ref{5.2}) requires
to solve the following grid problem at new time level $n+1$:
\begin{equation}\label{5.8}
\sigma \tau q_{\alpha}^{n+1} +
\nu q^{n+1}_{\alpha} +
\sigma \tau k A_{\alpha} u^{n+1} =
\chi^n_{\alpha}, \quad \alpha =1,2,
\end{equation}
\begin{equation}\label{5.9}
c u^{n+1} -
\sigma \tau \sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha}^{n+1} =
\phi^{n}
\end{equation}
for given $\chi^n_{\alpha}, \ \alpha =1,2$ and $\phi^{n}$.
Substituting $q^{n+1}_{\alpha}$ from equations (\ref{5.8}) in equation (\ref{5.8}), we obtain
\begin{equation}\label{5.10}
(\nu + \sigma \tau) c u^{n+1} + \sigma^2 \, \tau^2
\sum_{\alpha =1}^{2} A^*_{\alpha} A_{\alpha} u^{n+1} =
(\nu + \sigma \tau)\phi^{n} + \sigma \, \tau
\sum_{\alpha =1}^{2} A^*_{\alpha} \chi_{\alpha}^{n} .
\end{equation}
Other components of the approximate solution are evaluated after solving grid problem
(\ref{5.10}) via the explicit formulas of equations (\ref{5.8}).
To preserve the second order of approximation, different grids in time are often employed
for the individual components of the solution.
The following scheme for system (\ref{3.5}), (\ref{3.6}) provides an example
\begin{equation}\label{5.11}
\frac{q^{n+1/2}_{\alpha} + q^{n-1/2}_{\alpha} }{2} +
\nu \frac{q^{n+1/2}_{\alpha} - q^{n-1/2}_{\alpha} }{\tau} +
k A_{\alpha} u^{n} =
0, \quad \alpha =1,2,
\end{equation}
\begin{equation}\label{5.12}
c \frac{u^{n+1} - u^{n} }{\tau} -
\sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha}^{n+1/2} =
f^{n+1/2}, \quad n = 0,1, \ldots, N-1 .
\end{equation}
Such explicit schemes are widely used in computational practice.
A detailed discussion of such schemes in application to problems of electrodynamics is presented, for example, in
\cite{taflove2000computational}, with references to works of other authors.
The main drawback of such schemes is connected with restrictions on the time step (conditional stability).
Considering equation (\ ref (5.12)) at two time levels, we obtain the following equations
\[
c \frac{u^{n+1} - u^{n-1} }{2 \tau} -
\sum_{\alpha =1}^{2} A^*_{\alpha}
\frac{q^{n+1/2}_{\alpha} + q^{n-1/2}_{\alpha} }{2} =
\frac{f^{n+1/2} + f^{n-1/2}}{2} ,
\]
\[
c \frac{u^{n+1} - 2 u^{n} + u^{n-1} }{\tau^2} -
\sum_{\alpha =1}^{2} A^*_{\alpha}
\frac{q^{n+1/2}_{\alpha} - q^{n-1/2}_{\alpha} }{\tau } =
\frac{f^{n+1/2} - f^{n-1/2}}{\tau} .
\]
Taking into account equation (\ref{5.11}), we derive
\begin{equation}\label{5.13}
\nu c \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c \frac{u^{n+1} - u^{n-1}}{2\tau} +
\sum_{\alpha =1}^{2} A^*_{\alpha} k A_{\alpha} u^{n}
\end{equation}
\[
= \frac{f^{n+1/2} + f^{n-1/2}}{2} +
\nu \frac{f^{n+1/2} - f^{n-1/2}}{\tau} .
\]
Thus we have the explicit approximation of hyperbolic equation (\ref{4.2}) with $\sigma = 0$.
Unconditionally stable (at $\sigma \geq 0.25$) scheme with weights
\[
\nu c \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c \frac{u^{n+1} - u^{n-1}}{2\tau}
+ \sum_{\alpha =1}^{2} A^*_{\alpha} k A_{\alpha}
(\sigma u^{n+1} + (1- 2\sigma) u^{n} + \sigma u^{n-1})
\]
\[
= \frac{f^{n+1/2} + f^{n-1/2}}{2} +
\nu \frac{f^{n+1/2} - f^{n-1/2}}{\tau}
\]
is equivalent to the following scheme for system (\ref{3.5}), (\ref{3.6}),
if in scheme (\ref{5.11}), (\ref{5.12}) instead of (\ref{5.11}) we use
\begin{equation}\label{5.14}
\frac{q^{n+1/2}_{\alpha} + q^{n-1/2}_{\alpha} }{2} +
\nu \frac{q^{n+1/2}_{\alpha} - q^{n-1/2}_{\alpha} }{\tau}
\end{equation}
\[
+ k A_{\alpha} (\sigma u^{n+1} + (1- 2\sigma) u^{n} + \sigma u^{n-1}) =
0, \quad \alpha =1,2 .
\]
Scheme (\ref{5.12}), (\ref{5.14}) is not very convenient for the practical usage.
Its main drawback results from the explicit coupling of equations for the temperature and heat fluxes.
We must perform some preliminary work in order to obtain acceptable grid problems for evaluating
the individual components of the solution at the new time level.
Starting from explicit scheme (\ref{5.11}), (\ref{5.12}),
we can construct unconditionally stable implicit schemes.
We can do it in the most simple way using the Samarskii principle
of regularization for difference schemes \cite{samarskii2001theory,samarskii2002difference},
which is based on increasing the stability of a scheme via the perturbation of its operators.
Stability of scheme (\ref{5.11}), (\ref{5.12}) can be achieved in different ways.
The most interesting possibility is connected with the multiplicative \cite{SamVabSReg,samarskii1999additive}
perturbation (increasing) of the time derivative operator or perturbation (decreasing) of the spatial variables
operator for the individual equations of the system.
Consider the perturbation of equation (\ref{5.12}) in detail.
The implicit scheme can be written as
\begin{equation}\label{5.15}
c^{1/2} \left ( E + \sigma \tau^2
\sum_{\alpha =1}^{2} A^*_{\alpha} A_{\alpha} \right ) c^{1/2} \,
\frac{u^{n+1} - u^{n} }{\tau} -
\sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha}^{n+1/2} =
f^{n+1/2} .
\end{equation}
The perturbation has the order of $\mathcal{O}(\tau^2)$ and therefore
regularized scheme (\ref{5.11}), (\ref{5.15}) remains in the class of schemes with the second order approximation.
The operator at the time derivative is selfadjoint and positive definite.
\begin{theorem}
\label{t-3}
Difference scheme (\ref{5.11}), (\ref{5.15})
is unconditionally stable at $\nu c_0 \sigma \geq 0.25$ and for the difference solution
we have estimate (\ref{4.4}) where
\begin{equation}\label{5.16}
S^n = \left (\left (
\nu c^{1/2}(E + \sigma \tau^2 D)c^{1/2} -
\frac{\tau^2}{4} D \right )\frac{u^n - u^{n-1}}{\tau },
\frac{u^n - u^{n-1}}{\tau } \right )
\end{equation}
\[
+ \left (D \frac{u^n + u^{n-1}}{2}, \frac{u^n + u^{n-1}}{2} \right ) ,
\]
\[
\varphi^n = \frac{f^{n+1/2} + f^{n-1/2}}{2} +
\nu \frac{f^{n+1/2} - f^{n-1/2}}{\tau} .
\]
\end{theorem}
\begin{proof}
From (\ref{5.11}), (\ref{5.15}) in the standard way we obtain the following
three-level scheme
\begin{equation}\label{5.17}
\nu c^{1/2}(E + \sigma \tau^2 D)c^{1/2}
\frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2}
\end{equation}
\[
+ c^{1/2}(E + \sigma \tau^2 D) c^{1/2} \frac{u^{n+1} - u^{n-1}}{2\tau}
+ D u^{n} = \varphi^n ,
\quad n = 1,2, \ldots, N-1 .
\]
Further investigation is conducted similarly to the proof of Theorem~\ref{t-1}.
Under the above restrictions on the weight we have
\[
\nu c^{1/2}(E + \sigma \tau^2 D)c^{1/2} -
\frac{\tau^2}{4} D > 0
\]
and we associate the squared norm of the difference solution with $S^n$.
\qquad\end{proof}
The numerical implementation of scheme (\ref{5.11}), (\ref{5.15})
is based on inversion of the same grid elliptic operator $E + \sigma \tau^2 D$,
whereas the schemes with weights for the hyperbolic heat conduction equation
(\ref{4.2}) requires to invert $c(\nu + \tau) E + \sigma \tau^2 D$.
You can also obtain a grid analog of (\ref{2.16}), (\ref{2.17}) for difference scheme (\ref{5.11}), (\ref{5.15}).
However, it seems difficult to proof same analog of Theorem~\ref{t-2} in this.
\section{Splitting scheme for the hyperbolic heat conduction equation}
The above considered unconditionally stable operator-difference schemes --- (\ref{4.2})
for the hyperbolic heat equation and (\ref{5.11}), (\ref{5.15})
for the system of hyperbolic heat conduction, respectively, --- are not very convenient in the numerical implementation.
We construct the additive schemes for problem (\ref{2.5}), (\ref{2.7}),
where the transition to a new time level will be connected
with the solution of more simple problems related to the inversion of
individual operators $A^*_{\alpha} A_{\alpha}, \ \alpha =1,2$,
rather than their sum (operator $D$ in (\ref{4.2})).
Taking into account the nature of operators $A^*_{\alpha}, A_{\alpha}, \ \alpha =1,2$,
we are talking about locally one-dimensional schemes \cite{samarskii2001theory}.
We will focus on using regularized additive schemes of full approximation \cite{samarskii1998regularized,samarskii1992regularized}.
The principle of regularization of difference schemes is used traditionally widely
\cite{samarskii2001theory} to construct stable difference schemes for the
numerical solution of problems governed by partial differential equations.
Due to small perturbations of the problem operators we can control the growth of the norm
for the solution at the transition from one time level to another.
The construction of unconditionally stable difference schemes via
the principle of regularization is implemented as follows.
For the initial problem there is constructed some simple difference scheme
(producing difference scheme) which does not meet the necessary properties,
ie the scheme is conditionally stable or even absolutely unstable.
Then the quality of the difference scheme (its stability) is improved via
perturbations of the difference scheme operators.
It is natural to consider as the producing schemes the following explicit scheme
\begin{equation}\label{6.1}
\nu c \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c \frac{u^{n+1} - u^{n-1}}{2\tau} +
D u^{n} = \varphi^n ,
\end{equation}
which is complemented by initial conditions (\ref{4.3}).
The stability of this scheme (see (\ref{4.3}) at $\sigma = 0$) will be provided
if the following inequality holds
\begin{equation}\label{6.2}
R = \nu c E - \frac{\tau^2}{4} D > 0.
\end{equation}
In this case we have estimate (\ref{4.4}), in which
\begin{equation}\label{6.3}
S^n = \left (R \frac{u^n - u^{n-1}}{\tau },
\frac{u^n - u^{n-1}}{\tau } \right )
+ \left (D \frac{u^n + u^{n-1}}{2}, \frac{u^n + u^{n-1}}{2} \right ) .
\end{equation}
Taking into account (\ref{3.22}), from (\ref{6.2}) we obtain the condition
for stability of explicit scheme (\ref{6.1})
\[
\tau^2 \leq \frac{4 \nu c_0}{k_1 (\Delta_1 + \Delta_2)}
= \mathcal{O}(h_1^{-2} + h_2^{-2}) .
\]
To increase the stability limit (increase operator $R$),
we can employ the perturbation of both the first term in $R$ ($\nu c E $)
and second one ($D$).
In the case of perturbing the operator for the second time derivative we construct the regularized
scheme by analogy with (\ref{5.11}), (\ref{5.15}):
\begin{equation}\label{6.4}
\nu c^{1/2} Q c^{1/2} \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c^{1/2} Q c^{1/2} \frac{u^{n+1} - u^{n-1}}{2\tau} +
D u^{n} = \varphi^n ,
\end{equation}
where operator $Q = Q^* = E + \mathcal{O} (\tau^2)$.
In the construction of additive schemes, we need to take into account
the structure of the grid operator at new time level.
Assume that
\begin{equation}\label{6.5}
Q = \left (E + \frac{\sigma}{2} \tau^2 A_2^* A_2 \right )
\left (E + \sigma \tau^2 A_1^* A_1 \right )
\left (E + \frac{\sigma}{2} \tau^2 A_2^* A_2 \right ) ,
\end{equation}
so that $Q > E + \sigma \tau^2 D$.
Direct calculations verify that for scheme (\ref{6.4}), (\ref{6.5}) instead of (\ref{6.2}) we have
\begin{equation}\label{6.6}
R = \nu c^{1/2} Q c^{1/2} - \frac{\tau^2}{4} D
\end{equation}
and $R > 0$ at $\nu c_0 \sigma \geq 0.25$.
\begin{theorem}
\label{t-4}
Additive-difference scheme (\ref{6.4}), (\ref{6.5})
is unconditionally stable at $\nu c_0 \sigma \geq 0.25$
and estimate (\ref{4.4}), \ref{6.3}), (\ref{6.6})
is valid for the difference solution.
\end{theorem}
The second possibility of constructing unconditionally stable additive
operator-difference schemes is connected with the perturbation of operator $D$ in explicit scheme (\ref{6.1}).
Instead of operator $D$, which is defined according to (\ref{3.18}), we use
\begin{equation}\label{6.7}
C = \sum_{\alpha =1}^{2} C_{\alpha},
\quad C_{\alpha} = A^*_{\alpha}
\left (k^{-1} E + \sigma \tau^2 A_{\alpha} A^*_{\alpha} \right )^{-1}
A_{\alpha},
\quad \alpha = 1,2 .
\end{equation}
For these difference operators
\[
C_{\alpha} = C^*_{\alpha} < \frac{1}{\sigma \tau^2} E,
\quad \alpha = 1,2 .
\]
Because of this, for
\begin{equation}\label{6.8}
R = \nu c E - \frac{\tau^2}{4} C
\end{equation}
we have $R > 0$ at $\nu c_0 \sigma \geq 0.5$.
\begin{theorem}
\label{t-5}
Additive difference schemes
\begin{equation}\label{6.9}
\nu c \frac{u^{n+1} - 2u^{n} + u^{n-1}}{\tau^2} +
c \frac{u^{n+1} - u^{n-1}}{2\tau} +
C u^{n} = \varphi^n ,
\end{equation}
where operator $C$ is defined according to (\ref{6.7}),
is unconditionally stable at $\nu c_0 \sigma \geq 0.5$. Estimate (\ref{4.4}) is true
for the difference solution with
\begin{equation}\label{6.10}
S^n = \left (R \frac{u^n - u^{n-1}}{\tau },
\frac{u^n - u^{n-1}}{\tau } \right )
+ \left (C \frac{u^n + u^{n-1}}{2}, \frac{u^n + u^{n-1}}{2} \right )
\end{equation}
and $R$ corresponding to (\ref{6.8}).
\end{theorem}
The main computational cost in the constructed splitting schemes results from the inversion
of one-dimensional grid operators $k^{-1} E + \sigma \tau^2 A_{\alpha} A^*_{\alpha}, \ \alpha = 1,2$.
The potential advantage of additive scheme (\ref{6.7}), (\ref{6.9})
in compare with scheme (\ref{6.4}), (\ref{6.5}) is connected primarily with lower computational
cost during the transition to a new time level.
This advantage is more impressive for the three-dimensional problems (splitting in three directions).
\section{Additive schemes for the hyperbolic heat conduction governed by the system of equations}
In the construction of splitting schemes for the system of equations
governing the hyperbolic heat conduction, the theory and practice of using additive
schemes for first order evolutionary equations will be employed.
We treat the system of equations (\ref{3.15}), (\ref{3.16})
as a single evolutionary equation for the vector
$\mathbf{u} \equiv [u_1, u_2, u _3]^T = [q_1, q_2, u ]^T$:
\begin{equation}\label{7.1}
\mathbf{B}\frac{d \mathbf{u}}{d t} + \mathbf{A} \mathbf{u} = \mathbf{f}(t),
\end{equation}
where $\mathbf{f} = [0, 0, f]^T$.
For the elements of operator matrices $\mathbf{B}$ and $\mathbf{A}$ we have the representation
\begin{equation}\label{7.2}
\mathbf{B} =
\left[
\begin{array}{ccc}
\nu k^{-1} & 0 & 0 \\
0 & \nu k^{-1} & 0 \\
0 & 0& c
\end{array}
\right],
\qquad
\mathbf{A} =
\left[
\begin{array}{ccc}
k^{-1} & 0 & A_1 \\
0 & k^{-1} & A_2 \\
- A_1^* & - A_2^* & 0
\end{array}
\right] .
\end{equation}
For the direct sum of spaces $\mathbf{H} = H_1 \oplus H_2 \oplus H$, we set
\[
(\mathbf{u}, \mathbf{v}) = \sum_{\alpha =1}^{p} (u_{\alpha},v_{\alpha})_{\alpha} ,
\quad \|\mathbf{u} \|^2 = \sum_{\alpha =1}^{p} \|u_{\alpha}\|^2_{\alpha} .
\]
In this case, $\mathbf{A} > 0$ in $\mathbf{H}$ and estimate (\ref{2.16}), (\ref{2.17})
can be rewritten as
\begin{equation}\label{7.3}
\left (\mathbf{B} \mathbf{u}(t),\mathbf{u}(t) \right ) \leq
\exp(t)\left (\mathbf{B} \mathbf{u}(0),\mathbf{u}(0) \right ) +
\int\limits_0^t \exp(t-\theta)
\left (\mathbf{B}^{-1} \mathbf{f}(\theta), \mathbf{f}(\theta)
\right )^2 d \theta .
\end{equation}
To construct locally one-dimensional schemes for the Cauchy problem for (\ref{7.1}), (\ref{7.2}),
we use the additive representation of operator $\mathbf{A}$ in the form
\begin{equation}\label{7.4}
\mathbf{A} = \sum_{\alpha =1}^{p} \mathbf{A}^{(\alpha)} .
\end{equation}
The first variant of decomposition (\ref{7.4}) corresponds to the selection $p=3$ and
\begin{equation}\label{7.5}
\mathbf{A}^{(1)} =
\left[
\begin{array}{ccc}
0 & 0 & A_1 \\
0 & 0 & 0 \\
- A_1^* & 0 & 0
\end{array}
\right] ,
\quad
\mathbf{A}^{(2)} =
\left[
\begin{array}{ccc}
0 & 0 & 0 \\
0 & 0 & A_2 \\
0 & - A_2^* & 0
\end{array}
\right] ,
\end{equation}
\[
\mathbf{A}^{(3)} =
\left[
\begin{array}{ccc}
k^{-1} & 0 & 0 \\
0 & k^{-1} & 0 \\
0 & 0 & 0
\end{array}
\right] .
\]
Thus, we separate the individual terms with operators
$A_{\alpha}, \ A^*_{\alpha}, \ \alpha =1,2$.
The main properties of these operators are connected with their non-negativity
\[
\mathbf{A}^{(\alpha)} = - (\mathbf{A}^{(\alpha )})^*,
\quad \alpha = 1,2,
\quad \mathbf{A}^{(3)} = (\mathbf{A}^{(3)})^* \geq 0 .
\]
We can consider the splitting of (\ref{7.4}) so that $p=2$ where
\begin{equation}\label{7.6}
\mathbf{A}^{(1)} =
\left[
\begin{array}{ccc}
0.5 k^{-1} & 0 & A_1 \\
0 & 0.5 k^{-1} & 0 \\
- A_1^* & 0 & 0
\end{array}
\right] ,
\quad
\mathbf{A}^{(2)} =
\left[
\begin{array}{ccc}
0.5 k^{-1} & 0 & 0 \\
0 & 0.5 k^{-1} & A_2 \\
0 & - A_2^* & 0
\end{array}
\right] .
\end{equation}
In this case $\mathbf{A}^{(\alpha)} \geq 0, \ \alpha = 1,2$
in space $\mathbf{H}$.
This non-negative property of operators $\mathbf{A}^{(\alpha)}, \ \alpha = 1,2,...,p$
in splitting (\ref{7.4}) allows to use for the approximate solution of the Cauchy
problem for equation (\ref{7.1}), (\ref{7.2}) different classes of unconditionally stable
additive operator-difference schemes \cite{marchuk1990splitting,samarskii1999additive}.
With regard to our problem of the hyperbolic heat conduction, we can employ the schemes of second-order approximation in time.
For the general case (in (\ref{7.4}) $p>2$)
the standard additive schemes are based on the concept
of summarized approximation.
To construct the schemes of second order, we arrange computations via the algorithm
\[
\frac12 \mathbf{A}^{(1)} \rightarrow \frac12 \mathbf{A}^{(2)}
\rightarrow \cdots \rightarrow \frac12 \mathbf{A}^{(p)}
\rightarrow \frac12 \mathbf{A}^{(p)}
\rightarrow \frac12 \mathbf{A}^{(p-1)} \rightarrow \cdots \rightarrow
\frac12 \mathbf{A}^{(1)} .
\]
The corresponding additive scheme of component-wise splitting seems like this:
\begin{equation}\label{7.7}
\mathbf{B}\frac{\mathbf{u}^{n+\alpha /(2p)} - \mathbf{u}^{n+(\alpha-1) /(2p)} }{\tau} +
\frac{1}{2} \tilde{\mathbf{A}}^{(\alpha )}
(\mathbf{u}^{n+\alpha /(2p)} + \mathbf{u}^{n+(\alpha-1) /(2p)})
\end{equation}
\[
= \mathbf{f}_{\alpha }^{n+1/2}, \quad \alpha = 1,2,\dots,2p,
\]
where
$\tilde{\mathbf{A}}^{(\alpha )} = \mathbf{A}^{(\alpha )}, \ \alpha = 1,2,...,p$,
$\tilde{\mathbf{A}}^{(\alpha )} = \mathbf{A}^{(2p+1-\alpha )}, \ \alpha = p+1,p+2,...,2p$ è
\[
\mathbf{f}^{n+1/2} = \sum_{\alpha =1}^{2p-2}
\mathbf{f}_{\alpha }^{n+1/2} .
\]
The proof of stability and convergence is conducted in the standard way, the technical details can be found, for example, in \cite{samarskii1999additive}.
Scalarly multiplying the equations of scheme (\ref{7.7}) by $\mathbf{y}^{n+\alpha /(2p)} + \mathbf{y}^{n+(\alpha-1) /(2p)}$,
we obtain the corresponding analog of a priori estimate (\ref{7.3}).
\begin{theorem}
\label{t-6}
Additive operator-difference scheme of summarized approximation (\ref{7.7}) is
unconditionally stable and approximates the system of equations (\ref{7.1}) with the second order relative to $\tau$.
\end{theorem}
The computational implementation of the considered additive schemes is much
simpler than for schemes (\ref{5.1}), (\ref{5.2}) or (\ref{5.11}), (\ref{5.15}).
To explain this fact, we consider, for example, the first step in scheme (\ref{7.7})
with splitting (\ref{7.5}), where
\[
\left ( \mathbf{B} + \frac{\tau }{2} \mathbf{A}^{(1)}\right )
\mathbf{u}^{n+1/6}
= \mathbf{r}^{n+1/2}
\]
for given right hand side $\mathbf{r}^{n+1/2}$.
In the coordinate-wise form of this equation we have the system of equations
\[
\nu u_1^{n+1/6} + \frac{\tau }{2} k A_1 y_3^{n+1/6} = k r_1^{n+1/2},
\quad \nu u_2^{n+1/6} = k r_2^{n+1/2},
\]
\[
c u_3^{n+1/6} - \frac{\tau }{2} A_1^* u_1^{n+1/6} = r_3^{n+1/2}.
\]
Substituting $u_1^{n+1/6}$ from the first equation into the third one, we obtain
\[
\nu c u_3^{n+1/6} + \frac{\tau^2 }{4} A_1^* k A_1 u_3^{n+1/6} =
\nu r_3^{n+1/2} + \frac{\nu \tau }{2} A_1^* k r_1^{n+1/2}.
\]
Thus we must solve the one-dimensional grid problems with a single operator,
which are connected with operators $A_1, A^*_1$.
We have a similar realization for splitting (\ref{7.6}).
Among shortcomings of the constructed locally one-dimensional schemes (\ref{7.7})
it should be noted the lack of transparency (each individual equation does not
approximate the differential problem) as well as the relative difficulty of obtaining
and studying schemes of increased approximation order.
It is possible to construct for system (\ref{3.15}), (\ref{3.16}) another splitting schemes
which belong to the class of regularized additive operator-difference schemes \cite{samarskii1999additive}.
Regularized additive schemes can be constructed on the basis of scheme (\ref{5.11}), (\ref{5.15}).
Instead of (\ref{5.15}) we use the difference equation
\begin{equation}\label{7.8}
c^{1/2} Q c^{1/2} \,
\frac{u^{n+1} - u^{n} }{\tau} -
\sum_{\alpha =1}^{2} A^*_{\alpha} q_{\alpha}^{n+1/2} =
f^{n+1/2} ,
\end{equation}
where the factorized operator $Q$ is defined according to (\ref{6.5}).
Similar to Theorem~\ref{t-3}, we can prove the following statement involving estimate $Q > E + \sigma \tau^2 D$.
\begin{theorem}
\label{t-7}
Additive-difference scheme (\ref{5.11}), (\ref{6.5}), (\ref{7.8}) is unconditionally stable at $\nu c_0 \sigma \geq 0.25$,
and estimate (\ref{4.4}) holds for the difference solution, where
\[
S^n = \left (\left (
\nu c^{1/2}Q c^{1/2} -
\frac{\tau^2}{4} D \right )\frac{u^n - u^{n-1}}{\tau },
\frac{u^n - u^{n-1}}{\tau } \right )
\]
\[
+ \left (D \frac{u^n + u^{n-1}}{2}, \frac{u^n + u^{n-1}}{2} \right ) .
\]
\end{theorem}
Additive scheme (\ref{5.11}), (\ref{6.5}), (\ref{7.8})
is based on the perturbation of the operator at the time derivative in the last equation of system (\ref{5.11}), (\ref{5.12})
(difference derivative of the temperature).
It is interesting to consider the schemes with the perturbation of difference derivatives in time for heat fluxes.
Instead of (\ref{5.11}) we use the difference equations
\begin{equation}\label{7.9}
(k^{-1} E + \sigma \tau^2 A_{\alpha} A^*_{\alpha} )
\frac{q^{n+1/2}_{\alpha} + q^{n-1/2}_{\alpha} }{2}
\end{equation}
\[
+ \nu (k^{-1} + \sigma \tau^2 A_{\alpha} A^*_{\alpha} )
\frac{q^{n+1/2}_{\alpha} - q^{n-1/2}_{\alpha} }{\tau}
+ A_{\alpha} u^{n} =
0, \quad \alpha =1,2 .
\]
Difference equations (\ref{7.9}) can be written in the form
\[
\frac{q^{n+1/2}_{\alpha} + q^{n-1/2}_{\alpha} }{2}
+ \nu \frac{q^{n+1/2}_{\alpha} - q^{n-1/2}_{\alpha} }{\tau}
\]
\[
+ (k^{-1} + \sigma \tau^2 A_{\alpha} A^*_{\alpha} )^{-1}
A_{\alpha} u^{n} = 0,
\quad \alpha =1,2
\]
with treating them as the multiplicative perturbation of operators
$A_{\alpha}, \ \alpha =1,2$ â (\ref{5.11}).
It is easy to see by direct calculations that additive scheme (\ref{5.12})(\ref{7.9})
corresponds to scheme (\ref{6.7})(\ref{6.9}).
\begin{theorem}
\label{t-8}
Additive-difference scheme (\ref{5.12}), (\ref{7.9}) is unconditionally stable at $\nu c_0 \sigma \geq 0.5$,
and estimate (\ref{6.10}) holds for the difference solution under the condition of
setting operators $C$ and $R$ according to (\ref{6.7}) and (\ref{6.8}), respectively.
\end{theorem}
In contrast to the scheme of summarized approximation (\ref{7.7}), operator-difference splitting scheme (\ref{5.11}), (\ref{7.9})
has a clear and transparent structure.
\end{document} | math |
मोदी, योगी के चेहरे वाली पतंगों का सबसे ज्यादा क्रेज शनिवार को लक्सर में वसंत पंचमी का त्योहार मनाया गया। चुनावी बयार के बीच वसंत पंचमी पर भी सियासी असर साफ दिखाई दिया। युवाओं में नेताओं के चेहरे वाली पतंग खरीदने का खासा क्रेज रहा। मौसम साफ होने के बाद लोग दिन भर पतंग उड़ाने में व्यस्त रहे।फिलहाल हर तरफ विधानसभा चुनाव का शोर मच रहा है। चुनावी मौसम के बीच शनिवार को लक्सर में पूरे जोशोखरोश के साथ वसंत पंचमी का त्योहार मनाया गया। इस बार वसंत पंचमी पर लक्सर ही नहीं बल्कि आसपास के रायसी, गोवर्धनपुर, खानपुर, सुल्तानपुर, बसेड़ी, बहादरपुर, भिक्कमपुर आदि कस्बों में नेताओं का चेहरा बनी हुई पतंगों की सबसे ज्यादा बिक्री हुई। खासकर युवाओं ने इन पतंगों में ज्यादा दिलचस्पी ली। युवा शुभम, अनिरुद्ध, वंश, निखिल ने बताया कि शनिवार को बहुत ढूंढने पर ही उनके पसंद नेताओं के चेहरे वाली पतंग मिल सकी हैं। युवाओं ने कार्टून बनी हुई पतंगें खरीदकर भी उड़ाई। शनिवार को कई दिन बाद मौसम सामान्य होने का भी युवाओं को खासा फायदा मिला। युवा पूरे दिन अपने घरों की छत पर पतंगे उड़ाने में मशगूल दिखाई दिए। पूरे दिन आसमान पतंगों से पटा रहा। बच्चे कटी हुई पतंगें पकड़ते दिखाई दिए। For Hindustan : हिन्दुस्तान ईसमाचार पत्र के लिए क्लिक करें epaper.livehindustan.com | hindi |
अंग्रेजी का पेपर बिगड़ने पर छात्र ने फांसी लगाकर दी जान सीहोर। अंग्रेजी विषय का पेपर बिगड़ने पर छात्र ने फांसी लगाकर खुदकुशी कर ली। कोतवाली थाना पुलिस ने मर्ग कायम कर जांच शुरू कर दी है। पोस्टमार्टम के बाद छात्र का शव पुलिस ने परिजनों को सौंप दिया है। कोतवाली थाना प्रभारी नलिन बुधोलिया के मुताबिक स्टेशन रोड सुभाष नगर कालोनी में निवासरत 17 वर्षीय छात्र अजय गौर पुत्र कमलेश गौर गंज में संचालित स्टार पब्लिक स्कूल की कक्षा 12 का विद्यार्थी था। अजय के पिता श्यामपुर तहसील के मोती नगर में कृषि कार्य करते है। पूरा परिवार गांव पर ही रहता है। सुभाष नगर कालोनी में स्थित उनके निजी मकान में केवल अजय ही रहकर पढ़ाई कर रहा था। अजय के द्वारा 17 फरवरी को दिया गया, अंग्रेजी विषय का पेपर बिगड़ गया था। जिस कारण अजय काफी परेशान था, शुृक़्रवार की रात वह अपने कमरे में पढ़ाई कर शनिवार को होने वाले हिन्दी विषय के पेपर की तैयारी कर रहा था। इस दौरान रात में छत पर लगे पंखें से गले में गमझा बांधकर लटक गया। सुबह देर तक घर का दरवाजा नहीं खुलने पर पड़ोसियों को शंका हुई। पड़ोसियों ने अजय के नहीं जागने और घर में कुछ भी हलचल नहीं होने पर मोतीनगर गांव स्थित उसके परिजनों को जानकारी दी। कोतवाली पुलिस को भी घटना की जानकारी दी गई। परिजनों के पहुुंचने पर कमरा खोला गया। कोतवाली पुलिस के द्वारा मौके पर पहुंचकर पंचनामा बना कर मर्ग कायम किया गया है। पुलिस को किसी भी प्रकार का सुसाईट नोट नहीं मिला है। मोती नगर में छात्र का अंतिम संस्कार कर किया गया। | hindi |
The Lee’s Summit Chamber Board of Directors unanimously endorsed the continuation of the out-of-state vehicle sales tax at its July 20 board meeting. It was determined by leaders serving on the board that an endorsement was in the long-term best interest of our community. Voters headed to the polls on August 2 will be asked to vote on the continuation, and the Chamber encourages voters to support this important initiative.
The Vehicle Sales Tax Renewal is not a new tax, but simply a continuation of the city’s collection of this tax for several years. The tax only applies to motor vehicles, boats, trailers and outboard motors purchased out of the state. Money generated has long been a part of Lee’s Summit’s General Fund, which provides for a number of basic city services, such as road maintenance and repairs.
Without passage, the total revenue loss to the city is estimated at $880,000, based upon past purchases from out-of-state car dealerships. More importantly, the existence of this tax creates a level playing field for Missouri dealerships. The city of Lee’s Summit is home to 10 major car dealerships and a number of smaller dealerships employing approximately 542 full-time and 38 part-time employees. The loss of revenue could put these businesses at a competitive disadvantage versus out of state businesses and be damaging to employees affected by the loss.
The Lee’s Summit Chamber encourages members, community leaders and residents to vote “no” on question one to keep our city economically strong. Every vote counts. For more information about the ballot issue on August 2, visit cityofls.net. | english |
सालों बाद हुआ खुलासा की आखिर क्यों Raveena Tandon ने चोरी चुप्पे गोद ली थी बेटियां, सच्चाई का हुआ पर्दाफाश रवीना टंडन चार बच्चों की देखभाल करने वाली मां हैं दो दत्तक और दो जैविक। अनजान लोगों के लिए, अभिनेत्री ने पूजा और छाया को तब गोद लिया जब वह 1995 में सिर्फ 21 साल की थीं। उनकी दोनों दत्तक बेटियाँ अब शादीशुदा हैं और रवीना ने छाया को उनकी शादी की सालगिरह पर बधाई देने के लिए इंस्टाग्राम का सहारा लिया। अपनी बेटी छाया की शादी की तस्वीरों की एक श्रृंखला साझा करते हुए, रवीना ने लिखा, जब जीवन खूबसूरत यादों और क्षणों से भरा हो.. हमेशा के लिए पोषित और मनाया जाना! हैप्पी एनिवर्सरी माय बेबी! हमेशा खुशी और प्यार! chaya.mm शॉन सदा सौभाग्यवती भव। आरजे सिद्धार्थ कानन के साथ हाल ही में एक साक्षात्कार में, रवीना ने अपनी बेटियों को गोद लेने और उन पर ध्यान न देने के बारे में खोला। यह खुलासा करते हुए कि उन्होंने 90 के दशक में गोद लेने को कम क्यों रखा, रवीना ने कहा, शुरुआत में, यह टैब्लॉइडिज्म और पीली गंदी पत्रकारिता का युग था। ये कट्टर लेखक थे जो सिर्फ गंदा सामान लिख रहे थे और सुर्खियों में थे। उन में दिनों में, किसी भी चीज़ का घोटाला हो सकता है। जब मैंने लड़कियों को गोद लिया था, शुरू में, मैंने उनके बारे में बिल्कुल भी बात नहीं की, जब तक कि वे अपनी 10 वीं कक्षा पास नहीं कर लेते और उसके बाद, वे मेरे साथ मेरी शूटिंग पर घूमने लगे। फिर , सभी पूछने लगे, ये लड़कियां कौन हैं? और मैं उन्हें बताना शुरू कर देता कि यह कैसा है। उन्होंने आगे कहा, आप इतने डरे हुए थे कि अगर आप कुछ भी कहते हैं, तो ये लोग इसे क्या बना देंगे? मैगज़ीन वाले बोल देंगे की इसे सीक्रेटली बेबी हो गया, किसका बेबी है पत्रिकाएं कहती हैं कि मैंने चुपके से ए बेबी और पिता के बारे में अटकलें। यह वह युग था। इतना गंध था उनके दिमाग में उनके दिमाग इतने गंदे थे। ऐसी कहानियों से बचने के लिए, मैंने इसे बहुत चुपचाप किया। रवीना ने अपने डिजिटल डेब्यू अरण्यक के साथ अभिनय में वापसी की, जो पिछले साल नेटफ्लिक्स पर रिलीज़ हुई थी। | hindi |
ಹಂಪಿ ಉತ್ಸವ ನವೆಂಬರ್ 13 ರ ಶುಕ್ರವಾರ ಒಂದು ದಿನ ಮಾತ್ರ ಬಳ್ಳಾರಿ, ನ. 05, ಹಿ.ಸ ವಿಶ್ವವಿಖ್ಯಾತ ಹಂಪೆಯಲ್ಲಿ ಪ್ರತೀ ವರ್ಷ ಮೂರು ದಿನಗಳ ಕಾಲ ನಡೆಸುತ್ತಿದ್ದ ಹಂಪಿ ಉತ್ಸವವನ್ನು ಕೋವಿಡ್19ರ ಹಿನ್ನಲೆಯಲ್ಲಿ ಸಾಂಕೇತಿಕವಾಗಿ ನವೆಂಬರ್ 13 ರ ಶುಕ್ರವಾರ ಒಂದು ದಿನ ಮಾತ್ರ ನಡೆಸಲು ಸರ್ಕಾರ ನಿರ್ಧರಿಸಿದೆ. ಜಿಲ್ಲಾ ಉಸ್ತುವಾರಿ ಸಚಿವ ಬಿ.ಎಸ್. ಆನಂದ್ ಸಿಂಗ್ ಅವರು ಮಾಹಿತಿ ನೀಡಿದ್ದು, ಕೋವಿಡ್ನ ಹಿನ್ನಲೆಯಲ್ಲಿ ಈ ಬಾರಿಯ ಹಂಪಿ ಉತ್ಸವವನ್ನು ನವೆಂಬರ್ 13 ರಂದು ಒಂದು ದಿನ ಮಾತ್ರ ನಡೆಸಲು ಸರ್ಕಾರ ನಿರ್ಧರಿಸಿದೆ ಎಂದು ತಿಳಿಸಿದ್ದಾರೆ. ಪ್ರತಿ ವರ್ಷ ಮೂರು ದಿನಗಳ ಕಾಲ ಹಂಪಿ ಉತ್ಸವ ನಡೆಯುತ್ತಿತ್ತು. ಪ್ರತಿ ವರ್ಷದಂತೆ ಮೂರು ದಿನ ಹಂಪಿ ಉತ್ಸವ ನಡೆಸಬೇಕು ಎಂದು ಹಲವು ಸಂಘಟನೆಗಳು ಆಗ್ರಹಿಸಿದ್ದವು. ಆದರೆ, ಕೋವಿಡ್19 ಇರುವ ಕಾರಣ ಒಂದು ದಿನ ಸಾಂಕೇತಿಕವಾಗಿ ಆಚರಿಸಲು ಸರ್ಕಾರ ನಿರ್ಧರಿಸಿದೆ.ಕನ್ನಡ ಮತ್ತು ಸಂಸ್ಕೃತಿ ಇಲಾಖೆಯ ಮೂಲಗಳ ಪ್ರಕಾರ, ಸಾಂಕೇತಿಕವಾಗಿ ಒಂದೇ ದಿನ ಹಂಪಿ ಉತ್ಸವ ನಡೆಯುತ್ತಿರುವ ಕಾರಣ ಜಿಲ್ಲೆಯ ಕಲಾವಿದರ ತಂಡಗಳನ್ನು ಸೇರಿಸಿಕೊಂಡು ಜನಪದ ಕಲಾತಂಡಗಳ ಶೋಭಾಯಾತ್ರೆಯು ಮಾತ್ರ ನಡೆಯಲಿದೆ. ಉದ್ದಾನ ವೀರಭದ್ರೇಶ್ವರ ದೇವಸ್ಥಾನದಿಂದ ಶ್ರೀ ವಿರೂಪಾಕ್ಷೇಶ್ವರ ದೇವಸ್ಥಾನದವರೆಗೆ ಶೋಭಾಯಾತ್ರೆ ಸಾಗಲಿದೆ. ನಂತರ, ತುಂಗಾ ಆರತಿ ನಡೆಯಲಿದೆ ಎಂದು ವಿವರಿಸಿವೆ. | kannad |
“The velvet tyranny of political correctness.” » « Time for Goop.
Very annoyed. Bad thread day.
I hope Tuesday was better. | english |
[adj] clear and distinct to the senses; easily perceptible; "as clear as a whistle"; "clear footprints in the snow"; "the letter brought back a clear image of his grandfather"; "a spire clean-cut against the sky"; "a clear-cut pattern"
[adj] having had all the trees removed at one time; "clear-cut hillsides subject to erosion"
CLEAR-CUT is a 9 letter word that starts with C.
1. Having a sharp, distinct outline, like that of a cameo.
She has . . . a cold and clear-cut face. --Tennyson.
2. Concisely and distinctly expressed. | english |
என்னய்யா பட்டுன்னு இப்படி சொல்லிட்டிங்க.. தாமரை கேரக்டரை தவறாக பேசிய சிபி.. கொதிக்கும் நெட்டிசன்ஸ்! சென்னை: பிக்பாஸ் நிகழ்ச்சிக்கான இன்றைய மூன்றாவது புரமோவை பார்த்த நெட்டிசன்கள் சிபியை விளாசி வருகின்றனர். பிக்பாஸ் நிகழ்ச்சியின் இன்றைய எபிசோடுக்கான மூன்றாவது புரமோவில் பிக்பாஸ் வீட்டில் பட்டணமா பட்டிக்காடா என்ற தலைப்பில் விவாத மேடை நடைபெறுகிறது. தாயுமான நடிகர்... புடவை அணிந்து குழந்தையை சமாளிக்கும் சதீஷ்! இதில் பேசும் தாமரை, ஸ்ருதியின் ஆடையை குறித்து பேசினார். ஸ்ருதி அணியும் உடை தனக்கு பிடிக்கவில்லை என கூறினார். நீங்க அடக்கமா இருக்கீங்களா? இதனை பார்த்த சிபி , நீங்க அடக்கமா இருக்கீங்களா என கேட்கிறார். இதனால் நொந்துப்போன தாமரை நான் என்ன அப்படி அடக்கம் இல்லாமல் இருக்கிறேன் என்று கேட்டு வேதனைப்படுகிறார். இந்த புரமோவை பார்த்த நெட்டிசன்கள் சிபியை விளாசி வருகின்றனர். ஸ்ருதிக்கு பதில் புரமோவை பார்த்த இந்த நெட்டிசன், நீங்க போடுற டிரெஸ் புடிக்கலன்னு சொல்றது அவங்க ஒபினியன்.. எடுத்துக்குங்க இல்ல விட்டுடுங்க.. ஆனா சொல்லவே கூடாதுன்னு சொல்ல முடியாது.. சிலர் ஐக்கி முடியின் கலர் குறித்து பேசினார்கள். எடுத்துக்கொள்ளுங்கள் இல்லையெனில் விட்டுவிடுங்கள்... ஒப்பினியன் சொல்லப்பட்டது. ஆனால் ஸ்ருதிக்கு பதில் சிபி பேசுகிறார்.. நிகழ்ச்சியை பார்ப்போம் என கூறியுள்ளார். பெரிய வார்த்தை புரமோவை பார்த்த இந்த நெட்டிசன், தாமரை ஸ்ருதியுடன் ஏற்கனவே பிரச்சனை உள்ளது. இந்த நேரத்தில் அவருடைய உடையை பத்தி பேச வேண்டிய தேவை இல்லை. அவர் தன்னுடைய வெறுப்பை வெளிப்படுத்துவது போல் உள்ளது. ஆனால் சிபி அடக்கம் என்ற பெரிய வார்த்தையை பயன்படுத்தியுள்ளார். அவரை குறை சொல்ல முடியாது தாமரை செல்விதான் முதலில் பர்சனலாக பேசினார் என பதிவிட்டுள்ளார். கேரக்டர் குறித்து விமர்சனம் புரமோவை பார்த்த இந்த நெட்டிசன், தாமரை ஸ்ருதியின் உடையை பிடிக்கவில்லை என்று கூறினார். ஆனால் சிபி தாமரை செல்வியின் கேரக்டரை விமர்சித்துள்ளார். விவாதத்துக்கு உடை, உணவு என எதைப்பற்றி வேண்டுமானாலும் பேசலாம்.. அது அவருடைய உரிமை.. பெரும் வித்தியாசம்.. அவருக்கு டிரெஸ் பிடிக்கவில்லை.. அவருக்கு பீஸாவும் பிடிக்காமல் போகலாம்.. என பதிவிட்டுள்ளார். ஏன் அடக்கம்? புரமோவை பார்த்த இந்த நெட்டிசன், தாமரை செல்வி புடிக்கலன்னு சொன்னது ஒன்னும் தப்பில்ல. அது அவருடைய ஓபினியன். ஆனால் ஏன் சிபி அடக்கம் என்ற வார்த்தையை பயன்படுத்தினார்? ஆகையால் அவர் வேறு ஏதோ பேசியிருக்கிறார்? எப்படியோ இது விவாதம், விவாதத்தை டைனிங் டேபிள் வரை கொண்டு செல்வது சரியல்ல.. எபிசோடை பார்க்கலாம் என பதிவிட்டுள்ளர். இரண்டு பேருமே தவறு புரமோவை பார்த்த இந்த நெட்டிசன், துணிக்கும் அடக்கத்துக்கும் என்ன சம்பந்தம்? சிபி என்னய்யா இப்படி கேட்டுட்ட.. தாமரை சிபியிடம் அடக்கம் பற்றி ஏதாவது பேசினாரா? எடிட்டர் விளையாடுறனோ ஆனால் தாமரை ஒருவரின் உடை குறித்து பேசக்கூடாது. இந்த புரமோவின் படி இரண்டு பேருமே தவறுதான் என குறிப்பிட்டுள்ளார். கேள்வி கேட்பது தவறு.. புரமோவை பார்த்த இந்த நெட்டிசன், சிபி ஓபினியன் சொல்லவில்லை. தாமரையின் கேரக்டர் குறித்து அவர் கேள்வி கேட்கிறார். ஒருவரின் கேரக்டரை கேள்வி கேட்பது தவறு.. சிபி பேசியது தவறு என பதிவிட்டுள்ளார் இந்த நெட்டிசன். source: filmibeat.com | tamil |
क्या आपके घर में भी है किसी को घातक बीमारी, टॉयलेट हो सकता है इसकी वजह Written by Ashish Maheshwari Edited by: Ashish Maheshwari Last Updated: Jan 30, 2022, 02:12 PM IST Jaipur: मनुष्य के स्वास्थ्य पर भी वास्तु का प्रभाव पड़ता है. कई बार ऐसा होता है कि दवापरहेज करने के बाद भी बीमारी व्यक्ति का पीछा ही नहीं छोड़ती. शास्त्रों में कहा जाता है कि प्रथम सुख निरोगी काया अर्थात शरीर को स्वस्थ रखना व्यक्ति का प्रथम लक्ष्य है. यदि मनुष्य का शरीर स्वस्थ होता है तो वे अपने दैनिक कार्यों को सुचारू ढंग से करने में सक्षम रहता है. पाल बालाजी ज्योतिष संस्थान जयपुर जोधपुर के निदेशक ज्योतिषाचार्य डॉ. अनीष व्यास ने बताया कि वास्तु शास्त्र के मुताबिक घरमकान से जुड़े दोष व्यक्ति के जीवन को प्रभावित करते हैं. यदि घर के ईशान कोण यानी उत्तरपूर्व दिशा में टॉयलेट या फिर सीढ़ियां बनी होती हैं तो घर की मुख्य महिला ही नहीं, बल्कि अन्य सदस्यों को भी मानसिक तनाव या मस्तिष्क से जुड़ी समस्याएं हो सकती हैं. ज्योतिषाचार्य ने बताया कि घर की उत्तर एवं उत्तरपूर्व दिशा का बंद होना और दक्षिण और दक्षिणपश्चिम दिशा का खुला होना भी एक गंभीर वास्तु दोष है. ऐसा होने पर घर के भीतर बीमारी और खर्च दोनों ही अत्यधिक बढ़ जाते हैं. किचन के चूल्हे पर खाना बनाते समय घर की महिला का मुंह भूलकर भी दक्षिण दिशा की तरफ नहीं होना चाहिए. ऐसी सूरत में कमर दर्द, सर्वाइकल, जोड़ों का दर्द जैसी समस्याओं का सामना करना पड़ सकता है. पूर्व दिशा में सिर यानी पश्चिम दिशा में पैर करके सोना स्वास्थ्य के लिहाज से अच्छा होता है. दरअसल, सूरज पूर्व दिशा की ओर से निकलता है. सूरज की पहली किरण पूर्व दिशा में ही देखने को मिलती है. ज्योतिषाचार्य ने बताया कि ईशान कोण में बना शौचालय बहुत बड़ा वास्तु दोष माना जाता है. देवस्थान पर बना टॉयलेट घर की महिलाओं को न सिर्फ बीमार बनाता है बल्कि संतान सुख में भी कमी आती है. घर का ईशान कोण ऊंचा हो और बाकी सभी दिशाएं उससे नीची हों तो घर की महिलाओं को गंभीर बीमारी होने की आशंका बनी रहती है. उत्तर दिशा की तरफ सिर करके सोने से माइग्रेन, साइनस, सिर दर्द जैसी बीमारियां हो सकती हैं. बेड के सामने शीशा होने से सोते समय छवि दर्पण में नज़र आने से व्यक्ति धीरेधीरे बीमार होने लगता है. जी राजस्थान के स्टिंग ऑपरेशन पेंशन से सिस्टम में खलबली, गुनहगारों को दबोचा अनिद्रा वास्तुशास्त्र में पूर्व तथा उत्तर दिशा का हल्का और नीचा होना तथा दक्षिण व पश्चिम दिशा का भारी व ऊंचा होना अच्छा माना गया है. यदि पूर्व दिशा में भारी निर्माण हो तथा पश्चिम दिशा एकदम खाली और निर्माण रहित हो तो अनिद्रा का शिकार होना पड़ सकता है. उत्तर दिशा में भारी निर्माण हो परन्तु दक्षिण और पश्चिम दिशा निर्माण रहित हो तो भी ऐसी स्थिति उत्त्पन्न होती है. अनिद्रा से आपको कई तरह की बीमारियां घेर सकती हैं तो इस वास्तु का ध्यान रखकर आप स्वस्थ जीवन जी सकते हैं. चक्कर, बेचैनी और सिरदर्द वास्तु विशेषज्ञ ने बताया कि गृहस्वामी अग्निकोण या वायव्य कोण में शयन करें या उत्तर में सिर और दक्षिण में पैर करके सोए तब भी अनिद्रा या बेचैनी, सिरदर्द और चक्कर जैसी परेशानी हो सकती है, जिसके कारण दिन भर थकान की समस्या हो सकती है. धन आगमन और स्वास्थ्य की दृष्टि से दक्षिण या पूर्व की ओर पैर करना अच्छा माना गया है. हार्ट अटैक, लकवा, हड्डी और स्नायु रोग वास्तु विशेषज्ञ ने बताया कि वास्तु शास्त्र के अनुसार, दक्षिणपश्चिम दिशा में प्रवेश द्वार या हल्की चाहरदीवारी अथवा खाली जगह होना शुभ नहीं है. ऐसा होने से हार्ट अटैक, लकवा हड्डी एवं स्नायु रोग संभव हैं. अतः यहां प्रवेश द्वार या खाली जगह छोड़ने से बचना चाहिए. हड्डी के रोग वास्तु विशेषज्ञ ने बताया कि रसोई घर में भोजन बनाते समय यदि गृहणी का मुख दक्षिण दिशा की ओर है तो त्वचा एवं हड्डी के रोग हो सकते हैं. दक्षिण दिशा की ओर मुख करके भोजन पकाने से पैरों में दर्द की संभावना भी बनती है. इसी तरह पश्चिम की ओर मुख करके खाना पकने से आंख, नाक, कान एवं गले की समस्याएं हो सकती है. पूर्व दिशा की ओर चेहरा करके रसोई में भोजन बनाना स्वास्थ्य के लिए श्रेष्ठ माना गया है. वायु रोग और रक्त विकार वास्तु विशेषज्ञ ने बताया कि दीवारों पर रंगरोगन भी ध्यान से करवाना चाहिए. काला या गहरा नीला रंग वायु रोग, पेट में गैस, हाथपैरों में दर्द, नारंगी या पीला रंग ब्लड प्रेशर, गहरा लाल रंग रक्त विकार या दुर्घटना का कारण बन सकता है. अच्छे स्वास्थ्य के लिए दीवारों पर दिशा के अनुरूप हल्के एवं सात्विक रंगों का प्रयोग करना चाहिए. जोड़ों का दर्द और गठिया वास्तु विशेषज्ञ ने बताया कि वास्तु शास्त्र के अनुसार, ध्यान रहे कि आपके भवन की दीवारें एकदम सही सलामत हों, उनमें कहीं भी दरार या रंग रोगन उड़ा हुआ या फिर दागधब्बे आदि न हों वरना वहां रहने वालों में जोड़ों का दर्द, गठिया, कमर दर्द, सायटिका जैसे समस्याएं हो सकती हैं. | hindi |
एक साल में यहां एक लाख रुपये बने करीब 3 लाख रुपये, जानिये कहां मिला इतना रिटर्न स्टॉक मार्केट में ऐसे स्टॉक्स की कोई कमी नहीं है जो निवेशकों को बेहद ऊंचे रिटर्न दें. हालांकि इनमें से कुछ ही कंपनियां ऐसी होती हैं जिनका अपना प्रदर्शन भी वैसा होता है जैसा स्टॉक प्रदर्शन stock performance कर रहा हो. ऐसी मजबूत कंपनियों में समय पर किया गया निवेशकों की मामूली रकम को बेहद ऊंची रकम High Return stock में बदल सकता है. आज हम आपको ऐसी ही एक कंपनी के बारे में जानकारी दे रहे हैं, जिसका न केवल स्टॉक लगातार तेजी दर्ज कर रहा है साथ ही कंपनी का अपना प्रदर्शन भी शानदार रहा है , ये स्टॉक है ट्रांसपोर्ट कॉर्पोरेशन ऑफ इंडिया transport corporation of India का जहां निवेशकों की रकम एक साल में ही 3 गुना के करीब पहुंच गई है. कैसा रहा स्टॉक का प्रदर्शन एक साल पहले ट्रांसपोर्ट कॉर्पोरेशन ऑफ इंडिया का स्टॉक 250 के स्तर से नीचे था. शुक्रवार के कारोबार में स्टॉक 712 के स्तर से ऊपर बंद हुआ है. यानि एक साल पहले अगर किसी निवेशक ने इस स्टॉक में एक लाख रुपये लगाये होते तो कुल रकम बढ़कर अब तक 2.9 लाख रुपये हो चुकी होती. खास बात ये है कि साल के दौरान स्टॉक ने 858 का उच्चतम स्तर हासिल किया है. यानि स्टॉक में निवेश का अधिकतम रिटर्न 3 गुने से भी कहीं ज्यादा है. कंपनी स्मॉलकैप इंडेक्स में शामिल है और इसका फ्रीफ्लोट मार्केट कैप 18 सौ करोड़ रुपये के करीब है. वहीं कंपनी के छोटी अवधि के प्रदर्शन को देखें तो बीते 6 महीने में स्टॉक 416 के स्तर से बढ़कर 712 के स्तर पर पहुंच गया है यानि सिर्फ 6 महीने में स्टॉक में 71 प्रतिशत से ज्यादा की बढ़त देखने को मिली है. वहीं एक महीने के दौरान स्टॉक 10 प्रतिशत से ज्यादा रिटर्न अपने निवेशकों को दे चुका है. कैसे रहे हैं कंपनी के नतीजे कंपनी के स्टॉक में तेजी कंपनी के अपने प्रदर्शन के मुताबिक रही है. कंपनी के तीसरी तिमाही के नतीजे अनुमानों से बेहतर रहे हैं. तिमाही के दौरान कंपनी का मुनाफा करीब दोगुना हो गया है. वही एबिटडा पिछले साल के मुकाबले 36 प्रतिशत बढ़ा है.कारोबार से आय 6 प्रतिशत बढ़कर 759 करोड़ रुपये रही है. तिमाही के दौरान एबिटडा 84 करोड़ रुपये से बढ़कर 114 करोड़ रुपये पर पहुंच गया है. वहीं वित्त वर्ष के पहले 9 महीने में कंपनी की कारोबार से आय पिछले साल के मुकाबले 23 प्रतिशत की बढत के साथ 2359 करोड़ रुपये रही है. वहीं इस दौरान प्रॉफिट 143 प्रतिशत बढ़ा है. यह निवेश सलाह नहीं है. यहां कंपनी के स्टॉक के प्रदर्शन को दर्शाया गया है. बाजार में निवेश के अपने जोखिम है, कृपया कोई भी फैसला सोच समझ कर लें : बेहतर नतीजे के बाद इन कंपनियों पर बढ़ा जानकारों का भरोसा, स्टॉक में 33 प्रतिशत तक बढ़त का अनुमान : एक हफ्ते में 28 प्रतिशत टूटा जोमैटो का शेयर, निवेश का है मौका या रहें दूर | hindi |
ബ്രേക്ക് ഫാസ്റ്റിന് ചൂട് അട ദോശ കഴിച്ചാലോ...? അട ദോശയെ കുറിച്ച് നിങ്ങള് കേട്ടിട്ടുണ്ടാകും. അരച്ച ഉടനെ പെട്ടെന്ന് ചുട്ടെടുക്കാന് പറ്റുന്ന ഒരു അടിപൊളി ദോശയാണിത്. അരി കുറച്ചും പരിപ്പ് കൂടുതലും ആയതുകൊണ്ട് ഒരു ഹെല്ത്തി ദോശയാണ്. എങ്ങനെയാണ് അട ദോശ തയ്യാറാക്കുന്നതെന്ന് നോക്കാം... വേണ്ട ചേരുവകള്... ഇഡ്ഡലിക്കുപയോഗിക്കുന്ന അരി 1 കപ്പ് ഉഴുന്നുപരിപ്പ് ¼ കപ്പ് തുവര പരിപ്പ് ¼ കപ്പ് ചെറിയ ഉള്ളി ചെറുതായി അരിഞ്ഞത് ¼ കപ്പ് പച്ചമുളക് ചെറുതായി അരിഞ്ഞത് 1 ടീസ്പൂണ് ഇഞ്ചി ചെറുതായി അരിഞ്ഞത് 1 ടീസ്പൂണ് കറിവേപ്പില ചെറുതായി അരിഞ്ഞത് കുറച്ച് വറ്റല്മുളക് 5 എണ്ണം ഉപ്പ് ആവശ്യത്തിന് എണ്ണ ആവശ്യത്തിന് കായം ¼ ടീസ്പൂണ് തയ്യാറാക്കുന്ന വിധം... തലേദിവസം തന്നെ അരി, ഉഴുന്ന്, പരിപ്പ് ഇവ നല്ലപോലെ കഴുകിയശേഷം കുതിരാനായി വെള്ളം ഒഴിച്ച് വയ്ക്കുക. അടുത്ത ദിവസം ഇതില് വറ്റല് മുളക് കൂടി ചേര്ത്ത് അരയ്ക്കുക. നല്ലപോലെ അരയ്ക്കണ്ട. കുറച്ച് തരിയുള്ള തരത്തിലാവണം അരയ്ക്കേണ്ടത്. മിക്സിയില് നിന്നും മാറ്റുന്നതിനു മുമ്ബായി പച്ചമുളക്, ഇഞ്ചി, കറിവേപ്പില, കായം, ഉപ്പ് ഇവ ചേര്ത്ത് ഒരു മിനിട്ടുകൂടി അരയ്ക്കുക. അട ദോശയ്ക്കുള്ള മാവ് തയ്യാറായി. ദോശകല്ലുവച്ച് ഓരോ തവി മാവ് പരത്തി ഒഴിക്കുക. രണ്ട് വശവും മൊരിഞ്ഞുവരുമ്ബോള് ഒരു സ്പൂണ് നല്ലെണ്ണ ചുറ്റിനും തൂകികൊടുക്കുക. ചട്ണിക്കൊപ്പമോ സാമ്ബാറിനൊപ്പമോ കഴിക്കാവുന്നതാണ്. പഴുത്ത ചക്കയും ഈന്തപ്പഴവും കൊണ്ട് ഹെല്ത്തി ഷേക്ക് റെസിപ്പി Last Updated Jul 28, 2021, 8:58 AM IST ada dosa ബ്രേക്ക് ഫാസ്റ്റിന് recipe | malyali |
റസ്റ്റ് ഹൗസുകളില് ഓണ്ലൈന് ബുക്കിങ് ആരംഭിക്കും: മന്ത്രി പി.എ മുഹമ്മദ് റിയാസ്, ഹരിതാഭമാകാന് റസ്റ്റ് ഹൗസുകള് പദ്ധതി ഉദ്ഘാടനം ചെയ്തു കോഴിക്കോട് : കേരളത്തിലെ എല്ലാ റസ്റ്റ് ഹൗസുകളിലും പൊതുജനങ്ങള്ക്ക് ഓണ്ലൈന് ബുക്കിങ് ആരംഭിക്കുമെന്ന് പൊതുമരാമത്ത് വകുപ്പ് മന്ത്രി പി.എ മുഹമ്മദ് റിയാസ്. പൊതുമരാമത്ത് വകുപ്പിന്റെ കീഴിലുള്ള വിശ്രമ മന്ദിരങ്ങളുടെ പരിസരങ്ങള് മനോഹരമായും ഹരിതാഭമായും പരിപാലിക്കുവാനുള്ള പദ്ധതി ഉദ്ഘാടനം കോഴിക്കോട് റസ്റ്റ് ഹൗസ് പരിസരത്തു വൃക്ഷത്തൈ നട്ടുകൊണ്ട് നിര്വഹിക്കുകയായിരുന്നു മന്ത്രി. റസ്റ്റ് ഹൗസുകളില് ഉദ്യോഗസ്ഥര്ക്കുള്ള റിസര്വേഷന് നിലനിര്ത്തിക്കൊണ്ട് ബാക്കിയുള്ള ഇടങ്ങള് പൊതുജനങ്ങള്ക്ക് എപ്പോള് വേണമെങ്കിലും ലഭ്യമാക്കുവാനുള്ള സാഹചര്യം ഒരുക്കുക എന്ന തീരുമാനത്തിന്റെ ഭാഗമായാണ് ഓണ്ലൈന് ബുക്കിങ് സൗകര്യം ആരംഭിക്കുന്നത്. ഇത് സംബന്ധിച്ച് ഉദ്യോഗസ്ഥരുടെ യോഗം ചേരുകയും നോഡല് ഓഫീസറെ തീരുമാനിച്ചതായും മന്ത്രി പറഞ്ഞു. ആദ്യപടിയായി ജില്ലയിലുള്പ്പടെ 32 റസ്റ്റ് ഹൗസുകള് നവീകരിക്കാന് തീരുമാനിച്ചിട്ടുണ്ട്. പൊതുജന പങ്കാളിത്തത്തോടെ വിശ്രമമന്ദിരങ്ങളെ വൃത്തിയായും മനോഹരമായും പരിപാലിക്കുകയെന്ന പദ്ധതി കുടുംബശ്രീ പ്രവര്ത്തകര്, തൊഴിലുറപ്പു തൊഴിലാളികള് എന്നിവരുടെ സഹകരണത്തോടെയാണ് നടപ്പാക്കുന്നത്. ഹരിതാഭമാക്കുന്നതിന്റെ ഭാഗമായി ആദ്യഘട്ടത്തില് 14 വിശ്രമ കേന്ദ്രങ്ങളാണ് തെരഞ്ഞെടുത്തിട്ടുള്ളതെന്നും മന്ത്രി പറഞ്ഞു. ആകെ 156 വിശ്രമമന്ദിരങ്ങളാണ് പൊതുമരാമത്ത് വകുപ്പിന്റെ കീഴിലുള്ളത്. ഘട്ടംഘട്ടമായി എല്ലാ മന്ദിരങ്ങളിലും പദ്ധതി നടപ്പിലാക്കും. ഉദ്ഘാടനത്തോടനുബന്ധിച്ച് വിശ്രമമന്ദിരത്തിനുചുറ്റുമുള്ള രണ്ട് ഏക്കറില് അധികം വരുന്ന ഭൂമിയില് വേപ്പ്, ജാതിക്ക, ലക്ഷ്മിതരു തുടങ്ങിയ ഔഷധ സസ്യങ്ങളും മറ്റ് ഫലവൃക്ഷങ്ങളും നടുവാനാണ് പദ്ധതി. പതിനഞ്ചോളം തൊഴിലുറപ്പ് തൊഴിലാളികളുടെ സേവനം ഇതിനായി ഉറപ്പുവരുത്തിയിട്ടുണ്ട്. ചടങ്ങില് തോട്ടത്തില് രവീന്ദ്രന് എംഎല്എ, കൗണ്സിലര് സത്യഭാമ തുടങ്ങിയവര് പങ്കെടുത്തു. | malyali |
## INDICATION:
y/o man with history of antiphospholipid syndrome, multiple
embolic strokes, paraoxysmal atrial fibrillation (on warfarin), IDDMII, COPD,
nonverbal and wheel-chair bound with indwelling suprapubic catheter a G-tube
feeding dependent who presented on with altered mental status and
hypothermic, and was found to have sepsis secondary to UTI, PNA, and has had
respiratory failure and intermittent hypotension c/b hematuria from suprapubic
Foley.// ? parenchymal pathology ? parenchymal pathology
## IMPRESSION:
In comparison with the study of , there is increased obliquity of the
patient. The monitoring and support devices appear stable. Continued low
lung volumes that accentuate the transverse diameter of the heart. Again
there is elevation of pulmonary venous pressure with substantial retrocardiac
opacification consistent with volume loss in the left lower lobe and pleural
effusion. Less prominent effusion atelectasis are seen at the right base.
| medical |
ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ਚਿੱਤਰ ", 'ਇੱਕ ਅੰਗਰੇਜ਼ੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, ਇੱਕ ਚੀਨੀ ਭਾਸ਼ਾ ਵਿੱਚ, \n '] | punjabi |
ಕ್ಷೀಣಿಸಿದ ಗಣೇಶ ಮೂರ್ತಿ ಮಾರಾಟ ದೇವನಹಳ್ಳಿ: ಈ ಬಾರಿ ಗಣೇಶ ಚತುರ್ಥಿಗೆ ಕೋವಿಡ್ ಹಿನ್ನೆಲೆಯಲ್ಲಿ ಮಂಕು ಕವಿದಿದೆ. ಗಣೇಶ ಹಬ್ಬಕ್ಕೆ ವ್ಯಾಪಾರದ ಭರಾಟೆ ಪ್ರಾರಂಭವಾಗಿದ್ದು, ಹೂ, ಹಣ್ಣುಗಳ ಬೆಲೆಗಳು ಗಗನಕ್ಕೇರಿದ್ದು, ಗ್ರಾಹಕರಿಗೆ ಹೊರೆಯಾಗಿದೆ. ಸರ್ಕಾರದ ಹಲವು ನಿಬಂಧನೆ: ಪ್ರತಿ ವರ್ಷ ಗಣೇಶ ಮೂರ್ತಿಗಳನ್ನು ಪ್ರತಿ ಬೀದಿ, ಗಲ್ಲಿಗಲ್ಲಿಗಳಲ್ಲಿ ಪ್ರತಿಷ್ಠಾಪಿಸಿ ಹಬ್ಬ ಆಚರಣೆ ಮಾಡಲಾಗುತ್ತಿತ್ತು. ಆದರೆ ಈ ಬಾರಿ ಕೊರೊನಾ ಇರು ವುದರಿಂದ ಗಣಪತಿ ಪ್ರತಿಷ್ಠಾಪನೆಗೆ ಸರ್ಕಾರ ಮನೆಗಳಲ್ಲಿ, ದೇವಾಲಯಗಳಲ್ಲಿ ಪೂಜಿಸುವಂತೆ ಆದೇಶ ನೀಡಿದೆ. 20 ಜನರಿ ಗಿಂತ ಹೆಚ್ಚು ಇರಬಾರದು. ಪ್ರತಿಯೊಬ್ಬರು ಮಾಸ್ಕ್ ಧರಿಸಬೇಕು. ಸರ್ಕಾರ ಹಲವಾರು ನಿಯಮಗಳನ್ನು ವಿಧಿಸಿ ರುವುದರಿಂದ ಹಬ್ಬ ಆಚರಣೆಗೆ ಕೋವಿಡ್ ಅಡ್ಡಿಯನ್ನುಂಟು ಮಾಡಿದೆ. ಗಣೇಶ ಪ್ರತಿಷ್ಠಾಪಿಸಿದರೂ ಮೆರವಣಿಗೆ, ಸಾಂಸ್ಕೃತಿಕ ಕಾರ್ಯಕ್ರಮ ಗಳನ್ನು ನಿಷೇಧಿಸಿರುವ ಹಿನ್ನೆಲೆಯಲ್ಲಿ ಗಣೇಶ ಮೂರ್ತಿಗಳನ್ನು ಪ್ರತಿಷ್ಠಾಪಿಸಲು ಆಯೋಜಕರು ಹಿಂದೇಟು ಹಾಕಿದ್ದಾರೆ. ತಯಾರಕರಿಗೆ ಹೊಡೆತ: ಗಣಪತಿ ತಯಾರಿಸಲು, ಜೇಡಿ ಮಣ್ಣು ತರಬೇಕು, ಕಾರ್ಮಿಕರು, ವಾಹನ ಬೇಕು. ಮೂರ್ತಿಗೆಬಣ್ಣ ಹಚ್ಚಲು ಬಣ್ಣದಂಗಡಿ, ಕಾರ್ಖಾನೆ ಕಾರ್ಮಿಕರು ಬೇಕು. ಹೀಗೆ ಹಲವಾರು ಹೊಡೆತಗಳನ್ನು ಗಣೇಶ ತಯಾರಿಕೆ ಮಾಡು ವವರು ಅನುಭವಿಸುತ್ತಿದ್ದಾರೆ. ಗಣೇಶ ಮೂರ್ತಿ ನಿರ್ಮಾಣ ಮಾಡುತ್ತಿರುವ ಕುಟುಂಬಗಳು ಪ್ರತಿ ವರ್ಷ ಸುಮಾರು 500 ರಿಂದ 2000 ವರೆಗೆ ಮೂರ್ತಿಗಳ ತಯಾರಿಕೆಯಲ್ಲಿ ತೊಡಗಿಕೊಂಡು ಮಾರಾಟ ಮಾಡುತ್ತಿದ್ದರು. ಆದರೆ ಕೋವಿಡ್ನಿಂದಾಗಿ 100 ರಿಂದ 200 ಮೂರ್ತಿಗಳನ್ನು ತಯಾರಿಸಲು ಮುಂದಾಗದಿರುವುದರಿಂದ ನಷ್ಟ ಅನುಭವಿಸುವಂತೆ ಆಗಿದೆ ಎಂದು ಗಣೇಶ ಮೂರ್ತಿ ಮಾರಾಟಗಾರರು ಅಳಲನ್ನು ತೋಡಿಕೊಂಡಿದ್ದಾರೆ.ಬೆಲೆಗಳಲ್ಲಿ ಇಳಿಕೆ ಇಲ್ಲ: ನಗರದ ಬಜಾರ್ ರಸ್ತೆ, ಇತರೆ ಕಡೆ ಗಳಲ್ಲಿ ಬಾಳೆ ಕಂದು, ಮಾವಿನ ಸೊಪ್ಪು, ಹೂವು, ಹಣ್ಣು, ಇತರೆ ವಸ್ತುಗಳ ಖರೀದಿ ಭರಾಟೆ ಜೋರಾಗಿ ನಡೆಯುತ್ತಿತ್ತು. ಕೋವಿಡ್ ಭೀತಿ ನಡುವೆಯೂ ಬೆಲೆ ಏರಿಕೆಯಾಗಿತ್ತು. ಹಬ್ಬ ಅದ್ಧೂರಿ ಆಚರಣೆ ಇಲ್ಲದಿದ್ದರೂ ಬೆಲೆಗಳಲ್ಲಿ ಮಾತ್ರ ಇಳಿಕೆ ಇರಲಿಲ್ಲ. ಮಲ್ಲಿಗೆ 500 ರಿಂದ 600 ರೂ., ಕನಕಾಂಬರ 1500 ರೂ., ಸೇವಂತಿಗೆ 250300 ರೂ., ಗುಲಾಬಿ 150 ರಿಂದ 200 ರೂ., ಮಳ್ಳೆ ಹೂ 600 ರೂ., ಯಾಲಕ್ಕಿ ಬಾಳೆಹಣ್ಣು ರೂ.70, ಸೇಬು 200 ರೂ., ದಾಳಿಂಬೆ 180 ರಿಂದ 200 ರೂ., ಸೀಬೆ 60 ರೂ., ಮೋಸಂಬಿ 7080 ರೂ., ಪಚ್ಚಬಾಳೆ ಕೆ.ಜಿ.ಗೆ 25ರೂ., ತೆಂಗಿನ ಕಾಯಿ 1ಕ್ಕೆ 25 ರಿಂದ 30 ರೂ.ಮಾರಾಟವಾಗುತ್ತಿತ್ತು. ಈ ವರ್ಷ ಗಣೇಶ ಮೂರ್ತಿ ಬೇಕು ಎಂದು ಮುಂಗಡ ಕೊಟ್ಟು ಆದೇಶ ನೀಡಿದವರಿಗೆ ಮಾತ್ರ ಮೂರ್ತಿ ಸಿದ್ಧಪಡಿಸಿ ಕೊಡಲಾಗುತ್ತಿದೆ. ಗಣೇಶ ಮೂರ್ತಿ ಮಾರಾಟದಲ್ಲಿ ಲಾಭ ಇಲ್ಲದಿದ್ದರೂ ನಮ್ಮ ಪೂರ್ವಜರು ಮಾಡಿಕೊಂಡು ಬಂದಿರುವ ಮೂಲ ವೃತ್ತಿ ಬಿಡಬಾರದೆಂಬ ಕಾರಣಕ್ಕೆ ಕಾಯಕ ಮಾಡಿಕೊಂಡು ಬರುತ್ತಿದ್ದೇವೆ. ರಾಮಪ್ಪ, ಗಣಪತಿ ಮಾರಾಟಗಾರ ಎಸ್.ಮಹೇಶ್ | kannad |
केंद्र का बड़ा फैसला वर्क फ्रॉम होम खत्म, कल से सभी केंद्रीय कर्मचारियों को आना होगा ऑफिस देश में कोरोनावायरस coronavirusके मामलों में कमी को देखते हुए केंद्र सरकारcentral government ने एक बड़ा फैसला लिया है.यहां कोविड से संबधित सभी नए अपडेट पढ़ें सरकार ने कहा है कि 7 फरवरी 2022 से यानि सोमवार से सभी सरकारी कार्यालयों के सभी अधिकारियों को दफ्तर आना होगा. इस बात की घोषणा रविवार को केंद्रीय मंत्री डॉ जितेंद्र सिंह Dr Jitendra Singh ने की. घोषणा के मुताबिक, कोरोनो संक्रमण के केस में गिरावट को देखते हुए यह निर्णय लिया गया. कल से पूर्ण कार्यालय उपस्थिति फिर से शुरू की जाएगी और सभी स्तरों के कर्मचारी बिना किसी छूट के नियमित आधार पर कार्यालय में उपस्थित होंगे. UP Election: सीएम योगी के खिलाफ BSP ने किसे दिया टिकट? देखें 54 बसपा उम्मीदवारों की नई लिस्टजितेंद्र सिंह ने कहा कि कोरोना महामारी की स्थिति की समीक्षा करने के बाद ये फैसला लिया गया है. सोमवार से सभी केंद्रीय कर्मचारियों के लिए वर्क फ्रॉम होम Work from home की सुविधा खत्म की जा रही है और सभी कर्मचारियों को कल से रेगुलर दफ्तर आना होगा.उन्होंने आगे कहा कि दफ्तरों के अंदर कर्मचारियों को कोरोना के नियमों का पालन करना होगा. आपको जानकारी के लिए बता दें कि कार्मिक मंत्रालय ने 3 जनवरी को जारी अपने आदेश में कहा था कि सचिव स्तर से नीचे के 50 प्रतिशत कर्मचारियों को 31 जनवरी तक घर से काम करने की अनुमति दी गई है. ये इसलिए किया गया था क्योंकि देश में कोरोना संक्रमण के मामले तेजी से बढ़ रहे थे. CM योगी आदित्यनाथ किस कंपनी का मोबाइल करते हैं इस्तेमाल? जानें इसकी कीमतपिछले 1 दिन के दौरान संक्रमण से हुई इतनी मौतबता दे कि देश में रविवार को पिछले 24 घंटे में कोरोना वायरस के 1,07,474 नए मामले सामने आए हैं. 2,13,246 लोग कोरोना से ठीक होकर घर जा चुके हैं. वहीं 865 मरीजों की मौत हुई है.देश में कोरोना के एक्टिव मामलों की संख्या 12,25,011 है. मौत का कुल आंकड़ा बढ़कर 5,01,979 हो गया है. डेली पॉजिटिविटी रेट 7.42 प्रतिशत है. वैक्सीनेशन की बात करें तो अब तक देश में कोरोना वैक्सीन की कुल 1,69,46,26,697 डोज लगाई जा चुकी हैं. भारत रत्न लता मंगेशकर को किसने दिया मुखाग्नि? जानें कैसे हुआ अंतिम संस्कार | hindi |
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हड़िया चौराहे के पास गांजा बेच रहा युवक गिरफ्तार जागरण संवाददाता, बस्ती : पुरानी बस्ती पुलिस ने बुधवार को हड़िया चौराहे के निकट स्थित एक स्कूल के पास चुपके से गांजा बेच रहे युवक को गिरफ्तार कर लिया। उसके पास से बरामद झोले में 9.760 किलोग्राम गांजा बरामद हुआ। झोले में गांजा की पुड़िया व खुला गांजा रखा हुआ था। पूछताछ के दौरान उसकी पहचान शिवकुमार जायसवाल निवासी गोकरननाथ शिवाला कलवार पुरवा थाना खरगूपुर जनपद गोंडा के रूप में हुई। थानाध्यक्ष पुरानी बस्ती आलोक श्रीवास्तव के अनुसार पुलिस को सूचना मिली कि एक युवक हड़िया चौराहे के पास स्थित एक स्कूल के निकट चोरी चुपके पुड़िया में लोगों को गांजा बेच रहा है। सूचना को गंभीरता से लेते हुए पुलिस ने उस स्थान पर दबिश दी। मौके पर शिवकुमार जायसवाल झोले में गांजा के साथ पकड़ा गया। पूछताछ में उसने बताया कि वह गांजा बाराबंकी से ले आया था। बस्ती में बेचते समय पुलिस ने उसे पकड़ लिया। थानाध्यक्ष ने बताया कि पूछताछ के दौरान कुछ और महत्वपूर्ण जानकारियां मिलीं हैं, उन पर काम किया जा रहा है। आरोपित के विरुद्ध एनडीपीएस एक्ट के तहत मुकदमा दर्ज कर रहे हैं। आरोपित को गिरफ्तार करने वाली पुलिस टीम में स्वाट टीम प्रभारी उपनिरीक्षक राजकुमार पांडेय, चौकी प्रभारी दक्षिण दरवाजा जितेंद्र सिंह, प्रधान आरक्षी दुर्बल यादव, राकेश कुमार, ज्ञान प्रताप सिंह, आरक्षी उमेश कुमार, रविशंकर शाह, धीरज कुमार आदि मौजूद रहे। महिला अस्पताल से चोरों ने उड़ाई बाइक बस्ती: जिला महिला अस्पताल परिसर से चोरों ने एक बाइक पर हाथ साफ कर दिया। बढ़या राजा निवासी राजकुमार ने तहरीर देकर बताया है कि मंगलवार की सुबह बाइक महिला अस्पताल परिसर में खड़ी की थी। थोड़ी देर बाद लौटे तो वाहन गायब था। उन्होंने सूचना पुलिस को दी। चोरी का केस दर्ज कर कोतवाली पुलिस ने जांच चौकी प्रभारी गांधीनगर मनीष कुमार जायसवाल को सौंपी है। | hindi |
ബംഗളൂരുവിന് തിലകക്കുറിയായി ഗ്ലോബല് മാള്സ് ലുലു ഹൈപ്പര്മാര്ക്കറ്റ് ബംഗളൂരില് പ്രവര്ത്തനമാരംഭിച്ചു മുന് കര്ണ്ണാടക മുഖ്യമന്ത്രിയും കേന്ദ്രമന്ത്രിയുമായിരുന്ന എസ്. എം. കൃഷ്ണയാണ് ഹൈപ്പര്മാര്ക്കറ്റ് ഉദ്ഘാടനം ചെയ്തത് ബംഗളൂരു: ലുലു ഗ്രൂപ്പിന്റെ ഇന്ത്യയിലെ മൂന്നാമത്തെ ഹൈപ്പര്മാര്ക്കറ്റ് ബംഗളൂരില് ഉദ്ഘാടനം ചെയ്തു. മുന് കര്ണ്ണാടക മുഖ്യമന്ത്രിയും കേന്ദ്രമന്ത്രിയുമായിരുന്ന എസ്. എം. കൃഷ്ണയാണ് ഹൈപ്പര്മാര്ക്കറ്റ് ഉദ്ഘാടനം ചെയ്തത്. കര്ണ്ണാടക മുന്മന്ത്രി ഡി. ശിവകുമാര്, ലുലു ഗ്രൂപ്പ് ചെയര്മാന് എം.എ.യൂസഫലി, എക്സിക്യൂട്ടി ഡയറക്ടര് എം.എ. അഷ് റഫലി, ലുലു ഇന്ത്യ ഡയറക്ടര് ഏ.വി. ആനന്ദ് ഉള്പ്പെടെ നിരവധി പ്രമുഖര് ചടങ്ങില് സന്നിഹിതരായിരുന്നു,. ബംഗളൂരു രാജാജി നഗറില് പുതുതായി ആരംഭിച്ച ഗ്ലോബല് മാളിലാണ് 2 ലക്ഷം ചതുരശ്ര അടിയില് ലുലു ഹൈപ്പര്മാര്ക്കറ്റ് പ്രവര്ത്തിക്കുന്നത്. ലോകത്തിന്റെ വിവിധ ഭാഗങ്ങങ്ങളില് നിന്നായി ഉപഭോക്താക്കളുടെ താത്പര്യത്തിനനുസരിച്ചുള്ള ഉന്നത ഗുണനിലവാരമുള്ള വൈവിധ്യമാര്ന്ന ഉല്പ്പന്നങ്ങള് ഏറ്റവും ആകര്ഷകമായ വിലയില് ഹൈപ്പര്മാര്ക്കറ്റില് ലഭ്യമാക്കിയിട്ടുണ്ട്. ഇന്ത്യയിലെ ഗാര്ഡന് സിറ്റി എന്നറിയപ്പെടുന്ന ബംഗളൂരില് ലുലു ഹൈപ്പര്മാര്ക്കറ്റ് ആരംഭിക്കാനായതില് ഏറെ സന്തോഷമുണ്ടെന്ന് ലുലു ഗ്രൂപ്പ് ചെയര്മാന് എം.എ.യൂസഫലി പറഞ്ഞു. അത്യാധുനിക സൗകര്യങ്ങളോടെയുള്ള സൗകര്യപ്രദമായ ഷോപിംഗ് അനുഭവം ബംഗളൂരിലെ ജനങ്ങള്ക്കും ലഭ്യമായിരിക്കുകയാണ്. നേരിട്ടും അല്ലാതെയും ഏകദേശം അയ്യായിരത്തിലധികം ആളുകള്ക്ക് തൊഴില് ലഭ്യമാകും. കര്ണ്ണാടകയുടെ വിവിധ ഭാഗങ്ങളില് പത്ത് ഹൈപ്പര്മാര്ക്കറ്റുകള് കൂടി ആരംഭിക്കാന് പദ്ധതിയുണ്ടെന്നും യൂസഫലി പറഞ്ഞു. തിരുവനന്തപുരത്തെ ലുലു മാള് ഈ വര്ഷാവസാനവു ലക്നോവിലെത് അടുത്ത മാര്ച്ചിലും പ്രവര്ത്തനമാരംഭിക്കാന് ഉദ്ദേശിക്കുന്നതായും യൂസഫലി കൂട്ടിച്ചേര്ത്തു. അന്താരാഷ്ട് ബ്രാന്ഡുകള് ഉള്പ്പെടുന്ന 132 സ്റ്റോറുകള്, അക്സസറികള്, ജൂവലറി, ഫുഡ് കോര്ട്ട്, റസ്റ്റോറന്റ് , കഫേ, 60,000 ചതുരശ്ര അടിയിലേറേ വ്യാപിച്ചുകിടക്കുന്ന ഫണ്ടൂറ ഒരു റോളര് ഗ്ലൈഡര്, ടാഗ് അറീന, ഒരു അഡ്വഞ്ചര് കോഴ്സും ട്രാമ്ബൊലിനും, ഏറ്റവും പുതിയ വി.ആര് റൈഡുകള്, 9ഡി തിയേറ്റര്, ബമ്ബര് കാറുകള്, എന്നിങ്ങനെ നിരവധി സവിശേഷതകളുള്ളതാണ് ബംഗളൂരു ഗ്ലോബല് മാള്സ് | malyali |
రూ.20 కోట్లు విలువైన మార్కెట్ యార్డ్ స్థలం కబ్జా చిన్నమండెం రూ.20 కోట్ల విలువ చేసే మార్కెట్ యార్డ్ స్థలం కబ్జా గురైంది. కొంత మంది అక్రమార్కులు ఆ స్థలాన్ని ఆన్లైన్లో తమ పేరుపై మార్చుకున్నారు. యార్డ్ భవనాలు అధ్వాన్న స్థితికి చేరడమే అన్యాక్రాంతమైందనే చర్చ సాగుతోంది. 40 ఏళ్ల క్రితం మాజీ ఎమ్మెల్యే మండపల్లి నాగిరెడ్డి హయాంలో సర్వే నెంబర్ 102లో మార్కెట్ యార్డ్కు 10 ఎకరాల స్థలం కేటాయించారు. అందులో షెడ్లు నిర్మించి ప్రహరీ ఏర్పాటు చేశారు. ఎన్నో ఏళ్లుగా రైతులకు సౌకర్యవంతంగా ఉన్న మార్కెట్ స్థలం అన్యాక్రాంతమైంది. మండల కేంద్రానికి సమీపంలోని మార్కెట్ యార్డు మండలంలోని 13 గ్రామాలతో పాటు నియోజకవర్గంలోని సంబేపల్లి, దేవపట్ల, చిత్తూరు జిల్లాలోని పెద్దమండ్యం, గుర్రంకొండ మండలాల రైతులకు అనుకూలంగా ఉంది. ఆయా ప్రాంతాల్లోని రైతులు వ్యవసాయంపైనే ఆధారపడి వేరుశనగ, టమోటా తదితర పంటలను సాగు చేసుకుంటూ జీవనం సాగిస్తున్నారు. రైతులు తాము పండించిన పంటలను యార్డులో నిల్వచేసుకుని మార్కెటింగ్ చేసుకునేవారు. కొన్నేళ్లుగా మార్కెట్ యార్డును, స్థలాన్ని సంబంధిత అధికారులు గానీ, పాలకులు గానీ పట్టించుకున్న పాపాన పోలేదు. కొంత మంది అక్రమార్కులు మార్కెట్ యార్డును తమకు అనుకూలంగా మార్చుకొని తహశీల్దార్ కార్యాలయంలో పనిచేసే ఓ ఉద్యోగి సాయంతో పట్టాదారు పాస్పుస్తకం చేయించకున్నారు. ఏకంగా బ్యాంకు లావాదేవీలు సైతం జరిపి ప్రభుత్వం నుంచి వచ్చే అన్ని రకాల పథకాల లబ్ది పొందుతున్నారు. ఇంత జరుగుతున్నా మార్కెట్ యార్డు అధికారులు గానీ, పాలకులు గానీ పట్టించుకోకపోవడంతో మార్కెట్ యార్డులో నిర్మించిన షెడ్లపై కప్పులు సైతం మాయమయ్యాయి. ఇప్పటికైనా సంబంధిత అధికారులు చర్యలు తీసుకొని రైతులకు ఉపయోగపడే మార్కెట్ యార్డ్ స్థలం అన్యాక్రాంతం కాకుండా కాపాడాల్సిన అవసరం ఎంతైనా ఉంది. కాగా ఈ విషయమై మార్కెట్ యార్డ్ చైర్మన్ అన్వర్బాషాను ప్రజాశక్తి వివరణ కోరగా భూమి విషయాలు మీడియాకరు చెప్పే అవసరం లేదని తెలిపారు. | telegu |
कोरोना से राहत: स्वस्थ होने वालों का बढ़ा आंकड़ा, 465 रह गए एक्टिव केस नई दिल्लीटीम डिजीटल। जनपद गाजियाबाद में कोरोना संक्रमण का प्रभाव लगातार कम हो रहा है। राहत की बात यह रही कि मतदान होने के दो दिन बाद भी संक्रमण का फैलाव देखने को नहीं मिला है।यहां कोविड से संबधित सभी नए अपडेट पढ़ें जबकि मतदान के दौरान कोविड प्रोटोकॉल की धज्ज्यिां उड़ती देखी गई थी। शनिवार को केवल 40 नए मरीजों की पुष्टि हुई है। इसके अलावा 99 मरीज कोरोना को मात देकर स्वस्थ हुए है। वहीं, एक्टिव केस भी 500 से कम 465 शेष रह गए है। कोरोना संक्रमण की तीसरी लहर समाप्त होती दिख रही है। संक्रमित मरीजों की संख्या में कमी आने के साथ ही स्वस्थ होने वालों का आंकड़ा बढ़ा है। जिसके बाद जिले में एक्टिव केस भी कम हो रहे है। बीते जनवरी माह में एक्टिव केस की संख्या 10 हजार से अधिक पहुंच गई थी। जिसके बाद जनवरी में संक्रमण दर भी 9.96 पर पहुंच गई थी। प्रतिदिन एक हजार से अधिक केस सामने आए। 31 दिन में ही 27 हजार केस मिले थे। जनवरी माह ने बीते कोरोना संक्रमण की दोनों लहर को भी पीछे छोड़ दिया था। जिले में मौत का सिलसिला भी जनवरी माह से शुरू हुआ था। जनवरी माह में 10 संक्रमित मरीजों की जान जा चुकी है। हालांकि जनवरी अंत में कोरोना से राहत मिलना शुरू हो गया। जो फरवरी माह में भी लगातार जारी है। इस माह अभी तक 1235 केस मिले है। हालांकि विधानसभा चुनाव के पहले चरण में हुए मतदान के दौरान कई मतदान केंद्रों पर कोविड नियमों का पालन नहीं होने पर कोरोना संक्रमण फैलने की आशंका थी। लेकिन, मतदान हुए दो दिन बाद भी संक्रमित मरीजों की संख्या में तेजी नहीं देखी गई है। शनिवार को भी केवल 40 नए मरीज मिलें। इसके अलावा 99 मरीजों ने कोरोना को मात दी है। शनिवार को एक्टिव केस का आंकड़ा भी 500 से नीचे गिरकर 465 पर पहुंच गया। एक्टिव केस में 388 मरीजों का उपचार होम आइसोलेशन में चल रहा है। जिले में मार्च 2020 से अब तक 84201 मरीजों की पुष्टि हो चुकी है। इसमें 82673 मरीज स्वस्थ हो चुके है।जिला सर्विलांस अधिकारी डॉ. आरके गुप्ता का कहना है कि कोरोना संक्रमण का फैलाव कम हो रहा है। वहीं, मरीजों के स्वास्थ्य में भी तेजी से सुधार हो रहा है। हालंाकि, कोविड प्रोटोकॉल का पालन लगातार जारी रखना होगा। 6 दिन से कोई मौत नहीं कोरोना संक्रमण की रफ्तार धीमी होने के साथ ही जिले में कोरोना संक्रमित मरीजों की मौत का सिलसिला भी थम गया है। बीते 6 दिन से जिले में किसी संक्रमित मरीज की मौत की पुष्टि नहीं की गई है। 16 जनवरी को कोरोना संक्रमित हुए 55 वर्षीय मरीज की मौत हुई थी। जो कोरोना संक्रमण की तीसरी लहर में पहली मौत थी। जिले में अब तक 17 वर्षीय एक किशोर समेत 12 मौत हो चुकी है। | hindi |
పుష్ప సెకండ్ సింగిల్ ముహూర్తం ఫిక్స్ .. క్రియేటివ్ డైరెక్టర్ సుకుమార్, ఐకాన్ స్టార్ అల్లు అర్జున్ కాంబినేషన్లో రూపొందుతున్న పాన్ ఇండియా చిత్రం పుష్ప. ఎర్రచందనం స్మగ్లింగ్ నేపథ్యంలో తెరకెక్కుతున్న ఈ సినిమాలో లారీ డ్రైవర్ అల్లు అర్జున్ కనిపించబోతున్నాడు. ఈ మూవీలో హీరోయిన్గా రష్మిక మందన్నా నటిస్తోంది. ఇటీవల రష్మిక ఫస్ట్లుక్ విడుదల చేసిన చిత్ర బృందం, తాజాగా ఆమెకు సంబంధించిన మరో అప్డేట్ను ప్రకటించింది. దసరా సందర్భంగా సెకండ్ సింగిల్ రష్మిక సంబంధించిన ఆసక్తికర అప్డేట్ ఇవ్వనున్నట్లు స్పష్టం చేసింది. ఈ సందర్భంగా పుష్ప టీం ట్వీట్ చేస్తూ.. ఆమె మన భయంకరమైన పుష్పరాజ్ హృదయాన్ని దొంగిలించింది. ఇప్పడు మన ఊపిరి తీసేందుకు వస్తోంది. పుష్ప నుంచి అక్టోబర్ 13న శ్రీవల్లి రాబోతోంది అంటూ తమ ట్వీట్లో రాసుకొచ్చారు. ఈ చిత్రంలో రష్మిక.. శ్రీవల్లి పాత్ర పోషిస్తున్న సంగతి తెలిసిందే. ఇప్పటికే ఈ సినిమాకు సంబంధించి విడుదలైన టీజర్ ప్రేక్షకులను విశేషంగా ఆకట్టుకుంది. అలాగే ఫస్ట్ సింగిల్ దాక్కో దాక్కో మేక సాంగ్ కూడా ప్రేక్షకులను ఆకట్టుకుంది. ప్రస్తుతం ఈ సినిమా షూటింగ్ శరవేగంగా సాగుతోంది. డిసెంబర్ 17న పుష్ప ఫస్ట్ పార్ట్ ప్రేక్షకుల ముందుకు తీసుకురాబోతున్నట్లు మూవీ టీం ఇటీవల అధికారిక ప్రకటన చేసింది. The post పుష్ప సెకండ్ సింగిల్ ముహూర్తం ఫిక్స్ .. appeared first on telugu navyamedia. | telegu |
Aadhaar Card Update: आधार कार्ड के मोबाइल नंबर को चाहते हैं बदलना, इन Steps को फॉलो कर आसानी से करें चेंज Update of Aadhaar Card: आधार कार्ड के लिंक्ड नंबर को भी चेंज करना बहुत जरूरी है. बिना लिंक्ड मोबाइल नंबर के आप आधार कार्ड में किसी तरह का बदलाव नहीं कर सकते हैं. इसलिए इसे अपडेट करना बहुत जरूरी है. Aadhaar Card: आधार कार्ड देश के सबसे जरूरी पहचान पत्रों में से एक हैं. देश में सरकार ने नागरिकों की पहचान के लिए कुछ डॉक्यूमेंट्स निर्धारित कर रखे हैं. इनमें से आजकल सबसे जरूरी डॉक्यूमेंट बन गया है आधार कार्ड और पैन कार्ड. पैन कार्ड PAN Card का ज्यादातर इस्तेमाल वित्तीय लेनदेन Financial Transaction के लिए किया जाता है. पैन कार्ड का इस्तेमाल केवल वहीं लोग ज्यादा करते हैं जिनका बैंकिंग ट्रांजैक्शन Banking Transaction से संबंधित कोई काम रहता है. वहीं आधार कार्ड बच्चों से लेकर बड़ों तक के लिए बहुत जरूरी है. स्कूल में बच्चों के एडमिशन School Admission लेकर यात्रा के दौरान, बैंक अकाउंट Bank account खोलने से लेकर होटल बुकिंग तक हर जगह आधार कार्ड की जरूरत पड़ती है. ऐसे में आधार कार्ड का इस्तेमाल हर जगह आईडी प्रूफ Aadhaar Card Used as ID Proof के लिए किया जाता है. आधार कार्ड को बाकि पहचान पत्र से अलग इसलिए माना जाता है क्योंकि इसमें आपकी केवाईसी की जानकारी भी मौजूद होती है. इसलिए इसका अपडेट रहना बहुत जरूरी है. ऐसा कई बार होता है कि आधार कार्ड बनवाते समय हमारा नंबर कुछ और होता है और बाद में हम अपना मोबाइल नंबर चेंज कर लेते हैं. ऐसे में आधार कार्ड के लिंक्ड नंबर को भी चेंज करना बहुत जरूरी है. बिना लिंक्ड मोबाइल नंबर के आप आधार कार्ड में किसी तरह का बदलाव नहीं कर सकते हैं. इसलिए इसे अपडेट करना बहुत जरूरी है. तो चलिए हम आपको उस प्रोसेस के बारे में बताते हैं जिससे आप आसानी से अपने आधार के लिंक्ड मोबाइल नंबर में बदलाव कर सकते हैं इस तरह करें लिंक्ड मोबाइल नंबर में बदलाव Step 1: आधार कार्ड में किसी तरह का भी बदलाव करने के लिए आपको सबसे पहले UIDAI की ऑफिशियल वोबसाइट https:appointments.uidai.gov.inbookappointment.aspx पर क्लिक करना होगा. Step 2: यहां आपको Aadhaar Update के ऑप्शन का चुनाव करना होगा. Step 3: इसके बाद आपके सामने एक फॉर्म खुलेगा जिसे पूरी फील करना होगा. Step 4: फिर इस फार्म को आपको आधार एनरोलमेंट सेंटर में जमा करना पड़ेगा. Step 5: इसके बाद मोबाइल नंबर में बदलाव के लिए आपको शुल्क देना होगा. Step 6: इसके बाद 3 महीने के अंदर आधार कार्ड का लिंक्ड मोबाइल नंबर चेंज हो जाएगा. PM Kisan Samman Nidhi Yojana: क्या परिवार के दो लोग ले सकते हैं पीएम किसान सम्मान निधि योजना का लाभ? ये है स्कीम से जुड़े Rules | hindi |
انسٹاگرام نے اپنی حریف اپلیکشن اسنیپ چیٹ کے خلاف ایک اور اقدام کرتے ہوئے ایک نئی کیمرا فرسٹ ایپ تھریڈز متعارف کرادی ہے جس کا مقصد مواد قریبی دوستوں کے ساتھ شیئر کرنا ہےجمعرات کو ایک بلاگ پوسٹ میں فیس بک کی زیرملکیت انسٹاگرام نے اس نئی ایپ کے بارے میں بتاتے ہوئے کہا کہ یہ قریبی دوستوں کے لیے ایک مختص اور نجی جگہ پر رابطے کا نیا ذریعہ ہے اگر صارفین تھریڈز کو استعمال کرتا ہے تو وہ میسجز فوٹوز ویڈیو اور اسٹوریز کو اپنے دوستوں کی مخصوص فہرست کے ساتھ شیئر کرسکے گا اور ضروری نہیں کہ وہ مواد ہر ایک کے ساتھ شیئر کیا جائےاس ایپ میں اسٹیٹس یا ٹو اسٹیٹس کی سہولت بھی دی جارہی ہے جو دوستوں کو بتاسکے گا کہ اس وقت کہاں ہیں جیسے گھر میں موجود ہیں اور یہ عمل خودکار ہوگا جس کے لیے خود انگلیوں کو زحمت دینے کی ضرورت نہیں ہوگیصارفین اس میں اپنے قریبی دوستوں کے ساتھ لوکیشن اسپیڈ اور بیٹری لائف شیئر کرسکیں گے جبکہ ٹیکسٹ فوٹو اور ویڈیو پیغامات کی سہولت تو موجود ہوگی ہیاس ایپ کو اس وقت متعارف کرایا گیا ہے جب انسٹاگرام نے اپنی میسجنگ ایپ ڈائریکٹ کو مئی میں ختم کرنے کا اعلان کیا تھا جس پر 2017 سے کام ہورہا تھاکمپنی کا کہنا تھا کہ بیٹا ٹیسٹرز کی جانب سے میسج بھیجنے کے لیے انسٹاگرام سے دوسری ایپ پر جانا پسند نہیں تھا مگر کمپنی نے اس وقت بھی ایک نئی میسجنگ ایپ کی تیاری میں دلچسپی ضرور ظاہر کی تھیکمپنی کے خیال میں ایسی میسجنگ ایپ جو صرف قریبی دوستوں سے رابطے کے لیے استعمال ہو زیادہ مقبول ثابت ہوگی جبکہ اس طرح وہ اپنے صارفین کو اسنیپ چیٹ کے مقابلے میں زیادہ وقت ایپ کے اندر گزارنے پر بھی مجبور کرسکے گیفیس بک اور انسٹاگرام کئی برس سے اسنیپ چیٹ کی نوجوانوں کی مقبولیت ختم کرنے کی کوشش کررہے ہیں اور تھریڈز مخالف ایپ کی کشش ختم کرنے کی ایک اور کوشش ہےاسنیپ چیٹ کی طرح تھریڈز بھی صارفین کو خودکار طور پر کلوز فرینڈز لسٹ کو اپ ڈیٹس ارسال کرے گیاسنیپ چیٹ میں رواں سال اسنیپ میپ فیچر متعارف کرایا گیا تھا جس کی مدد سے صارفین دیکھ سکتے تھے کہ ان کے دوست کہاں ہے اور اس سے ملتا جلتا فیچر تھریڈز میں بھی دیا جارہا ہے | urdu |
>What kind of calendar does your conlang and/or conculture include?
Oh goodness, a conculturing post... let's do this for Nrit.
"feminine", after the 12 gods and 18 goddesses that make up the months.
calendar is linked to ecclesiastical feast days.
>What set of monomorphemic color terms does your conlang and/or conculture include?
for coining new tincture terms.
the Nrit culture is very visually-oriented.
out from them are not distinct from blood-kin relation.
as taught by P'an Ku's axe. | english |
సామాన్య రైతు బతికేదెలా..! సరిహద్దుల్లో సైనికులు నాడు యుద్ధాలు చేసి జై జవాన్ నినాదాన్ని నిజం చేశారు. అదే స్ఫూర్తితో భారత రైతాంగం కూడా ఎల్లలులేని వ్యవసాయ స్ఫూర్తితో దేశానికి ఆహారాన్ని అందించారు. భారత యొక్క సార్వభౌమాధికారాన్ని హేళన చేసినటువంటి పాశ్చాత్య దేశాల అహంకారాన్ని తమ కృషితో రైతులు ప్రశ్నించే ఆత్మగౌరవాన్ని, సాధికారతను సాధించారు. దీంతో ఆదాయపరంగా రైతులకు మాత్రం ఒరిగింది ఏమీ లేదు అని చెప్పవచ్చు. దేశంలో ఇప్పటి వరకు వచ్చినటువంటి పార్టీలు మరియు ప్రభుత్వాలు రూపొందించినటువంటి అన్ని ప్రణాళికలు వ్యవసాయ ఉత్పత్తులను పెంచడమే కాకుండా ఆహారభద్రత సాధించడం పైన దృష్టి సారించాలి. పంటల బాగు కోసం ఆధునిక పరిజ్ఞానాన్ని వినియోగించడం మరియు కొత్త వంగడాలు వాడడం ద్వారా ఉత్పాదకతను పెంచడమనేది బాగానే ఉన్నాయి. ఈ ఉత్పాదకాలకు సబ్సిడీలు, అనేవి కొద్ది పంటలకు కనీస మాత్రంగానే గిట్టుబాటు ధరలు అనేవి లభించాయి. దేశవ్యాప్తంగా అన్ని రాష్ట్రాలలో వ్యవసాయ విశ్వవిద్యాలయాలు, అలాగే ఇబ్బడిముబ్బడిగా పరిశోధనా సంస్థలు స్థాపించడంతో వ్యవసాయ రంగంలో పెట్టుబడుల శాఖ కూడా గణనీయంగా పెరిగింది అని చెప్పవచ్చు. అయినా రైతుల బాధలు తీరలేదు. వారి యొక్క కష్టాలు కడగంళ్లు తీర్చే నాథుడే లేకుండా పోయారు. స్వాతంత్రానికి ముందు మరియు ఆ తర్వాత దేశంలో ఆహార ధాన్యాల కొరత వల్ల బెంగాల్, ఇతర ప్రాంతాల్లో తీవ్రంగా కరువు సంభవించి లక్షలాది మంది ప్రజలు ఆకలితో మరణించడం, ఈ విపత్తు నుంచి దేశాన్ని రక్షించడం కొరకు ఆరుగాలం పంటలు పండించిన రైతులు ఇక దిగుమతుల అవసరం లేకుండానే ఆహార ఉత్పత్తులు రంగంలో స్వయంగా సమృద్ధిని సాధించి నిజమైనటువంటి దేశభక్తిని నిరూపించు కున్నారు. అమెరికా సంయుక్త రాష్ట్రాల నుంచి పిఎల్ 431 నిబంధనతో కింద గోధుమలు, బియ్యాన్ని దిగుమతులు చేసుకొని, భారతు ఆత్మగౌరవాన్ని తాకట్టు పెట్టే విధముగా ఉన్న షరతులను పాటించవలసిన అవసరం లేకుండా మువ్వన్నెల జెండాను అత్యున్నతంగా నిలబెట్టినది మన భారత రైతులే అని చెప్పవచ్చు. భారత ప్రభుత్వం కానీ, సమాజం కానీ, రైతులకు మిగిల్చింది చివరికి ఆకలి మంటలే. 1990 దశకం నుండి మొదలు వ్యవసాయం పరాధీనమై, విత్తనంపై రైతులు సాధికారతను కోల్పోయి ఋణం భారాలతో, గిట్టుబాటు ధరలు లేక అమ్ముకోవడానికి కూడా మార్కెట్ సౌకర్యాలు కరువై, బతుకంతా బరువై, తమ ఆత్మగౌరవాన్ని కూడా కోల్పోయి తీవ్రమైన నిరాశకు గురై చివరికి పురుగుల మందునే పెరుగన్నంగా భావించి నటువంటి రైతులు ఆత్మహత్యలకు పాల్పడుతున్న దుస్థితి సాక్షాత్తుగా మన పాలకులే తెచ్చిపెట్టారని చెప్పవచ్చు. 1990 నుంచి ఇప్పటివరకు పవిత్రమైన భారతదేశంలో అన్నదాతలు మూడు లక్షలకు పైగానే ఆత్మహత్య చేసుకున్నారు అంటే ఎంత అమానవీయ స్థితి నెలకొన్నదో మనం అర్థం చేసుకోవచ్చు. దేశానికి స్వాతంత్రం సిద్ధించినప్పటి నుంచి ఆరున్నర దశాబ్దాలు దాటినా అనంతరం కూడా రైతులు నిరంతరం ప్రకృతిపైనే ఆధారపడి వ్యవసాయం చేయడ మనేది ఒక జూదంగా మారింది. ప్రతికూల పరిస్థితులలో వ్యవసాయాన్ని కొనసాగిస్తూనే రైతులు చేస్తున్న కృషితో ఆహారధాన్యాల దిగుబడులు తగ్గిపోయి, ఎగుమతులకు మన దేశం ఏదగగలిగింది అని గర్వంగా చెప్పొచ్చు.హుస్సేన్ సాగర్ నిమజ్జనం: సుప్రీం కోర్ట్ లో బంతి వైసీపీకే వినాయకుడు... జనసేనకు కాదా...? పిల్లల్లో పెరుగుతున్న కరోనా కేసులు.. ఓవర్ టు మోడీ : పెద్దన్న రాజ్యానికి పెద్దాయన ! కాస్టింగ్ కౌచ్ పై సంచలన వ్యాఖ్యలు చేసిన ఇంద్రజ...? బిగ్ బాస్ పై నాని సంచలన కామెంట్ ? జీవితంలో మర్చిపొడట ! డ్రగ్స్ కేసు విచారణకు ఎమ్మెల్యేలు? మంచిమాట: మనలో అప్రమత్తత లేనినాడు నష్టపోక తప్పదు..! వైద్య సీట్లు .. రెండింతలు .. ! సోర్స్: ఇండియాహెరాల్డ్.కామ్ MOHAN BABU | telegu |
Sri Lalitha Vidya: ఘనంగా శ్రీ లలితావిద్య ఆవిష్కరణ Sri Lalitha Vidya: శ్రీ లలితా సహస్ర నామ స్తోత్రం అనేది ఓ శాస్త్రమని, ఉపాసనా రహస్యాలతో కూడుకున్న ఉపనిషద్విజ్ఞానమని సమన్వయసరస్వతి, బ్రహ్మశ్రీ సామవేదం షణ్ముఖ శర్మ చెప్పారు. అనేక భాష్యాలను, శాస్త్రాలను అధ్యయనం చేసి తాను శ్రీ లలితావిద్యను రచించానని తెలిపారు. ఈ పుస్తక రచనకు గురువుల కృపతోనూ, దేవీ ప్రేరణతోనూ స్ఫురించిన భావాలను మేళవించినట్లు తెలిపారు. ధర్మ, భక్తి, జ్ఞాన సంస్కారాలతో అమ్మవారి వైభవాన్ని ఆవిష్కరించినట్లు వివరించారు. ఈ పుస్తకాన్ని కింగ్ కోఠిలోని భారతీయ విద్యా భవన్లో ఆవిష్కరించారు. ఋషిపీఠం ఆధ్వర్యంలో జరిగిన ఈ పుస్తకావిష్కరణ సభ జ్యోతి ప్రజ్వలన, గ్రంధ పూజతో ప్రారంభమైంది. అనంతరం నూతి లక్ష్మిప్రసూన బృందం వందే శ్రీ మాతరం నృత్య రూపకాన్ని ప్రదర్శించింది. రాజమహేంద్రవరానికి చెందిన భాగవత విరించి డాక్టర్ టీవీ నారాయణ రావు ఈ కార్యక్రమానికి అధ్యక్షత వహించారు. కాంచీపురం కంచి కామకోటిపీఠం శ్రీకార్యం ఏజెంట్, చల్లా విశ్వనాథశాస్త్రి ముఖ్య అతిథిగా పాల్గొని శ్రీ లలితావిద్యను ఆవిష్కరించారు. చెన్నైకి చెందిన జ్ఞానానందనాథ గోటేటి శ్రీనివాసరావు ప్రథమ ప్రతిని స్వీకరించారు. డా. టి.వి. నారాయణరావు మాట్లాడుతూ.. శ్రీ లలితావిద్య అందుబాటులోకి రావడం తెలుగువారి అదృష్టమని చెప్పారు. శ్రీ లలిత సహస్ర నామాలకు అనేక మంది అనేక భాష్యాలను రచించారని చెప్పారు. లలితా దేవీ వైభవాన్ని వివరించడంలో అవన్నీ వేటికవే ప్రత్యేకమైనవని, బ్రహ్మశ్రీ సామవేదం షణ్ముఖ శర్మ రాసిన శ్రీ లలితావిద్య విలక్షణమైనదని తెలిపారు. వాగ్దేవతలు పలికిన రహస్య నామాలలోని గూఢార్థాలు మస్తిష్కంలోకి వెళ్లి, హృదయాలను తాకి, అమ్మవారి భావనలో లీనమయ్యే విధంగా ఈ పుస్తకం ఉందన్నారు. విశ్వనాథ శాస్త్రి మాట్లాడుతూ.. ఈ గ్రంథాన్ని సాక్షాత్తూ లలితా దేవికి అక్షర రూపంగా వర్ణించారు. అమ్మవారి అరుణ ప్రభలు పుస్తకంపై ముఖచిత్రంలోనూ, ప్రతి అక్షరంలోనూ కనిపిస్తున్నట్లు తెలిపారు. ఒక్కొక్క నామాన్ని చదువుతూ, భావిస్తూ అమ్మవారి భక్తిలో ఓలలాడవచ్చునన్నారు. ప్రథమ ప్రతి స్వీకర్త శ్రీనివాసరావు మాట్లాడుతూ, ఈ గ్రంథం లలితా దేవి చరిత్ర, నామ వైశిష్ట్యంతో అద్భుతంగా ఉందన్నారు. లలితా దేవి రహస్య నామాలలో దాగి ఉన్న శ్రీ విద్య రహస్యాలు సామాన్యుడి నుంచి పండితుల వరకు అర్థం చేసుకుని, దివ్యత్వాన్ని అనుభవించేలా ఉన్నట్లు తెలిపారు. | telegu |
مےٚ سَپُد واریَہہ کٲلۍ دربارس منٛز چون یہِ تکجار تہٕ تیزر وُچھِتھ سخت دۄکھ | kashmiri |
कोरोना वारियर के रूप सम्मानित हुए अधीक्षक कोरोना महामारी को लेकर सीएचसी करछना में लगातार चल रहे टीकाकरण और सराहनीय कार्य को लेकर सामुदायिक स्वास्थ केंद्र करछना के अधीक्षक डॉ.नवीन कुमार गिरी को प्रयागराज में आयोजित एक कार्यक्रम के दौरान कमिश्नर संजय गोयल ने सम्मानित किया। इसी तरह शत प्रतिशत कोरोना टीकाकरण को लेकर कचरी ग्राम प्रधान धर्मराज सिंह और चनैनी प्रधान को धीतेश तिवारी धीरू को संगम सभागार में प्रभारी जिलाधिकारी एवं सीडीओ शीपू गिरी ने सम्मानित किया। For Hindustan : हिन्दुस्तान ईसमाचार पत्र के लिए क्लिक करें epaper.livehindustan.com | hindi |
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<td><img src="../Img/Img_edit.png" /><span>555577894</span></td>
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তরুণ প্রজন্মের মধ্যে ইউটিউবার হওয়ার ঝোঁক বাড়ছে কেন? বিস্তারিত জানতে পড়ুন ইন্টারনেট ও মোবাইল নয়া প্রজন্মের এই দুই অপরিহার্য জিনিসকে হাতিয়ার করে বিশ্ব জুড়ে বিনোদনের সংজ্ঞাই বদলে দিয়েছে ইউটিউব ক্রমশ লাফিয়ে বাড়ছে ইউটিউবের দর্শক সংখ্যা শুধু দর্শক হিসেবেই নয়, নতুন প্রজন্ম ইউটিউবকে উপার্জনের মাধ্যম হিসেবেও ব্যবহার করছে এখন অনেকেই অন্য পেশা ছেড়ে ইউটিউবার হতে চান নিজস্ব একটি ইউটিউব চ্যানেল তৈরি করেন সেখানেই বিভিন্ন ধরনের বিনোদনমূলক কর্মকাণ্ড করে থাকেন যারা ইউটিউবের নিয়মিত দর্শক, এই ধরনের বিনোদন পেতে নিজেদের সেই চ্যানেলের সদস্য করে নেন চ্যানেলের সদস্য সংখ্যা অনুযায়ী প্রতি মাসে আয়ের পরিমাণ নির্ভর করে অর্থাত্ সাবস্ক্রাইবার বেশি হলে ইউটিউব সংস্থাও সংশ্লিষ্ট চ্যানেলটিকে বেশি টাকা দেয় টেলিভিশনের জনপ্রিয়তাকে প্রায় ছুঁয়ে ফেলেছে ইউটিউব কোনো কোনো ক্ষেত্রে টেলিভিশনকেও ছাপিয়ে গিয়েছে এমনকি, প্রতিদিন টেলিভিশনের পর্দায় যারা ফুটে ওঠেন, তাদের অনেকেই একএকটি করে ইউটিউব চ্যানেল চালান নিজেদের জীবনের রোজনামচা ইউটিউব চ্যানেলের দর্শকদের সামনে তুলে ধরেন দ্রুত জনপ্রিয়তা পেতে এবং আর্থিক ভাবে স্বচ্ছল হতে এই প্রজন্মের অনেকেই তাই পেশা হিসেবে বেছে নিচ্ছেন ইউটিউব ঝাঁপ দিচ্ছেন অনিশ্চিতের দিকে বিশেষজ্ঞদের মতে, সব জিনিসেরই ভালো এবং খারাপ, দুটি দিক আছে ইউটিউবার হতে চাওয়ার ইচ্ছা খারাপ নয় কিন্তু তার উপরেই যেন জীবন নির্ভরশীল না হয়ে পড়ে বিকল্প কোনো ভাবনা ভেবে রাখা জরুরি কিংবা অন্য কোনো কাজ করার পাশাপাশি, এটি করা যেতে পারে তাতে মানসিক চাপ কিছুটা হলেও নিয়ন্ত্রণে থাকে নেটমাধ্যমে প্রভাবী ইনফ্লুয়েন্সার হওয়ার স্বপ্ন দেখেন তরুণ প্রজন্মের অনেকেই প্রচুর পরিশ্রমও দরকার হয় দশর্কের পছন্দ বুঝে সেই মতো অনুষ্ঠান তাদের সামনে নিয়ে আসে একেবারেই সহজসাধ্য নয় সেই সঙ্গে তো রয়েছে পারস্পরিক প্রতিযোগিতা এর ফলে মাঝেমাঝেই শিল্পের মান কিছুটা হলেও কমে যায় তাছাড়া সাফল্যব্যর্থতা তো লেগেই আছে সমীক্ষা বলছে, ১৮ থেকে ২৬ বছর বয়সিদের মধ্যে ইউটিউবকে পেশা হিসেবে বেছে নেয়ার প্রবণতা সবচেয়ে বেশি সোশ্যাল মিডিয়ার অর্থনীতি বিশেষজ্ঞরা বলছেন, কম সময়ে বিপুল জনপ্রিয়তা এবং অর্থ এই দুইয়ের কারণেই মূলত এই পেশার প্রতি ঝোঁক বাড়ছে সেই সঙ্গে বাড়ছে প্রতিযোগিতা ও মানসিক চাপ এর প্রভাব পড়ছে ব্যক্তিগত জীবনেও তাদের মতে, ইউটিউবকে আয়ের একমাত্র উত্স হিসেবে না দেখে নিজেদের শিল্পী সত্ত্বা প্রকাশের মাধ্যম হিসেবে দেখাই শ্রেয় | bengali |
ദക്ഷിണ കൊറിയയില് ഉള്ളികൃഷി ജോലി, കേരളത്തില് നിന്ന് തള്ളിക്കയറ്റം : 100 ഒഴിവിലേയ്ക്ക് 5000 അപേക്ഷകള് കവിഞ്ഞു കൊച്ചി: ദക്ഷിണ കൊറിയയില് ഉള്ളികൃഷി ജോലിക്കായി കേരളത്തില് നിന്നും തള്ളിക്കയറ്റം. പത്താം ക്ലാസ് മാത്രം യോഗ്യത നിശ്ചയിച്ചിട്ടുള്ള ജോലിക്ക് ആകെ 100 ഒഴിവാണുള്ളത്. എന്നാല് ജോലിക്കായി അപേക്ഷിച്ചിരിക്കുന്നത് ബിരുദവും ബിരുദാനന്തര ബിരുദവും അടക്കം യോഗ്യതയുള്ളവരും. അപേക്ഷകരുടെ എണ്ണം 5000 കടന്നുവെന്നാണ് റിക്രൂട്ടിങ് ഏജന്സിയായ ഒഡെപെക് വ്യക്തമാക്കുന്നത്. ഇതോടെ രജിസ്ട്രേഷന് നിര്ത്തിവച്ചു. ആദ്യഘട്ടത്തില് നൂറു പേര്ക്കാണ് അവസരം നല്കുക. അടുത്ത ഘട്ടത്തില് കൂടുതല് പേരെ റിക്രൂട്ട് ചെയ്യുമെന്നും സംസ്ഥാന സര്ക്കാര് സ്ഥാപനമായ ഒഡെപെക് അറിയിച്ചു. ദക്ഷിണ കൊറിയന് സര്ക്കാര് പദ്ധതിയുടെ ഭാഗമായാണ് കരാറടിസ്ഥാനത്തില് കേരളത്തില് നിന്ന് ആളുകളെ റിക്രൂട്ട് ചെയ്യുന്നത്. 1500 ഡോളര് ഏകദേശം ഒരു ലക്ഷം രൂപ ശമ്ബളമാണ് വാഗ്ദാനം. റിക്രൂട്ട്മെന്റിന്റെ ഭാഗമായി ബുധനാഴ്ച തിരുവനന്തപുരത്തും, വെള്ളിയാഴ്ച എറണാകുളത്തും സെമിനാര് സംഘടിപ്പിക്കുന്നുണ്ട്. ജോലി സംബന്ധമായ വിവരങ്ങളും കൊറിയയിലെ സാഹചര്യങ്ങളും വിശദീകരിക്കാനാണ് സെമിനാര്. രജിസ്റ്റര് ചെയ്തവരെല്ലാം സെമിനാറില് സംബന്ധിക്കണം. ഇതിനു ശേഷമായിരിക്കും യോഗ്യരെ തെരഞ്ഞെടുക്കുക. വളരെ തണുപ്പേറിയ കാലാവസ്ഥയാണ് കൃഷി മേഖലയിലേത്. കാലാവസ്ഥ, ജീവിതരീതി തുടങ്ങിയ കാര്യങ്ങളെല്ലാം ഉദ്യോഗാര്ഥികളെ ബോധ്യപ്പെടുത്തുമെന്നും ഒഡെപെക് എം.ഡി കെ.എ അനൂപ് പറഞ്ഞു. Tags Kerala South Korea Onion cultivation shortlink | malyali |
આઠ મ્યુ. કોર્પોરેશનમાં OBC માત્ર 20 ટકા, ડાંગ, તાપી જિલ્લામાં અનામત નહી સરકારમાં પાલિકા પંચાયતો દીઠ OBCની વસતીના આંકડા તૈયારવર્ષ 2011ના ડેટામાં જ્યાં પાંચ ટકા કે તેથી વધુ વસતી હશે ત્યાં જ બેઠકો અનામત થશે રાજ્યભરની પાલિકા પંચાયતોમાં 27 ટકા સુધી અનામત વધશે સુપ્રિમ કોર્ટના આદેશ મુજબ સ્થાનિક સ્વરાજ્યની સંસ્થાઓમાં અન્ય પછાત વર્ગો OBC અનામતના અમલ માટે ગુજરાત સરકારે પાલિકા પંચાયત દિઠ આંકડાનો ડેટા તૈયાર કરી લીધો છે. સામાજીક ન્યાય અધિકારીતા વિભાગે પંચાયત અને શહેરી વિકાસ વિભાગ પાસેથી વર્ષ 2011ની વસ્તી ગણતરીને આધારે મેળેલા આ ડેટામાં આઠ મ્યુનિસિપલ કોર્પોરેશનમાં OBCની વસ્તી માત્ર 20.5 ટકા જ હોવાનું કહેવાયુ છે. જ્યારે ડાંગ અને તાપી જેવા જિલ્લાઓમાં OBCની સંખ્યા અનુક્રમે માત્ર 1.53 ટકા તેમજ 3.01 ટકા છે. પંચાયત અધિનિયમ મુજબ પાંચ ટકાથી ઓછી OBC વસ્તી હોય ત્યાં અનામત મળી શકે નહી. વર્ષ 2010માં સુપ્રિમે આપેલા ચૂકાદાના અમલ માટે છેક 12 વર્ષે રચાયેલા જસ્ટીસ કે.એસ. ઝવેરીના કમિશન આગામી સપ્તાહે કામગીરી શરૂ કરશે. કમિશન સમક્ષ એક એક સંસ્થા દિઠ OBCની વસ્તીની સંખ્યા નક્કી કરવા માટે સરકારે વર્ષ 2011ની વસ્તી ગણતરીનો આધાર આપવા નિર્ણય કર્યો છે. વર્ષ 2015માં અને છેલ્લે 2021માં સ્થાનિક સ્વરાજ્યની સંસ્થાઓની ચૂંટણી આ ડેટાને આધારે યોજાઈ હતી. કોરોનાને કારણે વર્ષ 2021માં વસ્તી ગણતરી થઈ નથી આ સંજોગોમાં અમદાવાદ, ગાંધીનગર, સુરત, વડોદરા જેવા મ્યુનિસિપલ કોર્પોરેશન તેમજ કેટલીક નગરપાલિકાઓમાં છેલ્લી ચૂંટણીમાં નવા ઉમેરાયેલા વિસ્તારો સાથે OBC વસ્તીને અપડેટ કરવા શનિવારે પણ સચિવાલયમાં ધમધમાટ જોવા મળ્યો હતો. સ્થાનિક સ્વરાજ્યની સંસ્થાઓમાં OBC પ્રતિનિધિત્વ માટે જે તે સંસ્થામાં આ વસ્તીનું પ્રમાણ પાંચ ટકા કે તેથી વધુ હોવુ અનિવાર્ય છે. આ સ્થિતિમાં ગુજરાતમાં 4 જેટલા આદિવાસી જિલ્લા અને તે સિવાયના જિલ્લાના 14 તાલુકાઓમાં આ પ્રમાણ નહિ જળવાતુ નથી. આથી, 95 ટકાથી વધુ આદિવાસી અથવા બિન OBC વસ્તી ધરાવતી સંસ્થાઓમાં OBC અનામત મળી શકશે નહી. અલબત્ત તે સિવાય રાજ્યભરની પાલિકા પંચાયતોમાં 27 ટકા સુધી અનામત વધશે. | gujurati |
# -*- coding: utf-8 -*-
from datetime import datetime
from sqlalchemy import Table, ForeignKey, Column, Integer, String, DateTime, Date, SmallInteger
from sqlalchemy.ext.declarative import declarative_base
Base = declarative_base()
class User(Base):
__tablename__ = "users"
id = Column("id", Integer, primary_key=True, autoincrement=True)
email = Column("email", String(64), unique=True)
name = Column("name", String(64), unique=True)
passwd = Column("password", String(64))
role = Column("role", SmallInteger, default=10)
gold = Column("gold", Integer, default=0)
sudate = Column("sign_up_date", DateTime, default=datetime.now)
sidate = Column("sign_in_date", DateTime, default=datetime.now)
sodate = Column("sign_out_date", DateTime, default=datetime.now)
def __repr__(self):
return "<User(name='%s', password='%s')>" % (self.name, self.passwd)
class Animt(Base):
__tablename__ = "animations"
id = Column("id", Integer, primary_key=True, autoincrement=True)
name = Column("name", String(64), unique=True)
sdate = Column("start_date", Date)
odate = Column("over_date", Date)
pid = Column("provider_id", Integer)
def __repr__(self):
return "<Animt(name='%s')>" % self.name
class TmpAnimt(Base):
__tablename__ = "tmp_animations"
id = Column("id", Integer, primary_key=True, autoincrement=True)
name = Column("name", String(64), unique=True)
sdate = Column("start_date", Date)
odate = Column("over_date", Date)
pid = Column("provider_id", Integer)
cdate = Column("commit_date", DateTime, default=datetime.now)
def __repr__(self):
return "<TmpAnimt(name='%s')>" % self.name
class UserAnimt(Base):
__tablename__ = "user_animt"
#id = Column("id", Integer, primary_key=True, autoincrement=True)
uid = Column("user_id", Integer, ForeignKey("users.id"), primary_key=True)
aid = Column("animation_id", Integer, ForeignKey("animations.id"),
primary_key=True)
score = Column("score", SmallInteger, default=0)
#__table_args__ = (
# UniqueConstraint("user_id", "animation_id", name="_user_animation_uc"),
#)
def __repr__(self):
return "<UserAnimt(...)>"
class UserVerify(Base):
__tablename__ = "user_verify"
uid = Column("user_id", Integer, ForeignKey("users.id"), primary_key=True)
hash = Column("hash", String(32), nullable=False)
date = Column("date", DateTime, default=datetime.now)
def __repr__(self):
return "<UserVerify(...)>"
# vim: set ts=4 sw=4 sts=4 et:
| code |
04022021 8 AM: இந்தியாவில் கொரோனா பாதிப்பு 1,07,77,284 ஆக உயர்வு. டெல்லி: இந்தியாவில் கொரோனா பாதிக்கப்பட்டோர் மொத்த எண்ணிக்கை 1,07,77,284 ஆக உயர்ந்துள்ளது. இதுவரை உயிரிழந்தோர் மொத்த எண்ணிக்கை 1,53,221 ஆக அதிகரித்துள்ளது. இந்தியாவில் கொரோனா தொற்று பரவல் கட்டுக்குள் உள்ளது. தற்போது, நாடு முழுவதும் முதல்கட்டமாக முன்களப் பணியாளர்களுக்கு கொரோனா தடுப்பூசி போடும் பணிகள் நடைபெற்று வருகின்றன. நாடு முழுவதும கடந்த 24 மணிநேரத்தில் 11 ஆயிரத்து 39 பேருக்கு கொரோனா தொற்று உறுதி செய்யப்பட்டுள்ளது. இதன் காரணமாக, தொற்று பாதிக்கப்பட்டோர் மொத்த எண்ணிக்கை 1,07,77,284 ஆக உயர்நதுள்ளது. இந்த எண்ணிக்கை நேற்றும் நீடித்தது. நேற்று காலை 8 மணி வரையிலான முந்தைய 24 மணி நேரத்தில் நாடு முழுவதும் மேலும் 11 ஆயிரத்து 39 பேர் புதிதாக கொரோனாவிடம் சிக்கி உள்ளனர். இவர்களையும் சேர்த்து நாட்டின் ஒட்டுமொத்த பாதிப்பு எண்ணிக்கை 1 கோடியே 7 லட்சத்து 77 ஆயிரத்து 284 ஆகியிருக்கிறது. கடந்த 24 மணி நேரத்தில் 110 பேர் கொரோனாவுக்கு பலியாகி உள்ளனர். இதன்மூலம் நாட்டின் கொரோனா பலி எண்ணிக்கை 1 லட்சத்து 54 ஆயிரத்து 596 ஆகியிருக்கிறது. மரண விகிதம் 1.43 ஆக குறைந்துள்ளது. அதிகபட்சமாக மராட்டியத்தில் 30 பேர், கேரளாவில் 16 பேர், பஞ்சாப்பில் 12 பேர் உயிரிழந்துள்ளனர். 14 மாநிலங்கள் மற்றும் யூனியன் பிரதேசங்களில் மேற்படி 24 மணி நேரத்தில் மரணம் எதுவும் நிகழவில்லை என்பது மகிழ்ச்சிக்குரிய அம்சமாகும். நாடு முழுவதும் கடந்த 24 மணி நேரத்தில் குணமடைந்த 14 ஆயிரத்து 225 பேர். இதுவரை 1 கோடியே 4 லட்சத்து 62 ஆயிரத்து 631 பேர் குணமடைந்து உள்ளனர். குணமடைந்தோர் சதவிகிதம் 97.08 ஆக உள்ளது. நாடு முழுவதும் இதுவரை, 19 கோடியே 84 லட்சத்து 73 ஆயிரத்து 178 கொரோனா பரிசோதனைகள் நடந்துள்ளன. 163 நாட்களுக்கு பிறகு 3லட்சத்துக்கு கீழே குறைந்த கொரோனா தொற்று! சுகாதாரத்துறை செயலர் ராஜேஷ் பூஷன் தகவல் 31122020 6AM: இந்தியாவில் கொரோனா பாதிப்பு 1,02,67,283 ஆக உயர்வு 23012021 9 AM: இந்தியாவில் கொரோனா பாதிப்பு 1,06,40,544ஆக உயர்வு, உயிரிழப்பு 1,53,221 ஆக அதிகரிப்பு , , , , , , , , , , , , , , , , , , , , , 04022021 8 AM: உலக அளவில் கொரோனா பாதிப்பு 10.49 கோடியாக உயர்வு Next அதிகாரிகள் கூட்டத்தில் தண்ணீர் என நினைத்து சானிடைசரை குடித்த மும்பை மாநகராட்சி அதிகாரி வைரல் வீடியோ | tamil |
உள்ளாட்சித் தேர்தல் பாதுகாப்பு ஏற்பாடுகள் குறித்து பதிலளிக்க தேர்தல் ஆணையத்துக்கு உயர்நீதிமன்றம் உத்தரவு உள்ளாட்சித் தேர்தல் பாதுகாப்பு ஏற்பாடுகள் குறித்து மாநில தேர்தல் ஆணையம் பதிலளிக்க சென்னை உயர்நீதிமன்றம் உத்தரவிட்டுள்ளது. தமிழ்நாட்டில் ஒன்பது மாவட்டங்களில் நடைபெற உள்ள ஊரக உள்ளாட்சித் தேர்தலுக்கு மத்திய அரசுப் பணியாளர்களை தேர்தல் பார்வையாளர்களாக நியமிக்க மாநில தேர்தல் ஆணையத்திற்கு உத்தரவிட வேண்டும் என சென்னை உயர் நீதிமன்றத்தில் அதிமுக தேர்தல் பிரிவு துணைச்செயலாளர் இன்பதுரை வழக்கு தொடர்ந்திருந்தார். மேலும், பணப்பட்டுவாடாவை தடுக்க மத்திய ரிசர்வ் படையை அமைக்க வேண்டும் உள்ளிட்ட கோரிக்கையை நடைமுறைப்படுத்த மாநில தேர்தல் ஆணையத்திற்கு உத்தரவிட வேண்டும் என மனுவில் குறிப்பிட்டிருந்தார். இந்த வழக்கு இன்று விசாரணைக்கு வந்தது. அப்போது, தொழில் நுட்பம் வளர்ந்துள்ள நிலையில் தேர்தலில் எந்த புகாரும் வராத வகையில் நடவடிக்கைகள் எடுக்க வேண்டும் என்றும் கண்காணிப்பு கேமராக்கள் அமைக்கப்படுவது அவசியம் என்றும் தலைமை நீதிபதி குறிப்பிட்டார். ஊரக உள்ளாட்சித் தேர்தலில் மேற்கொள்ளப்பட்டுள்ள பாதுகாப்பு ஏற்பாடுகள் குறித்து மாநில தேர்தல் ஆணையம் நாளை பதில் அளிக்க வேண்டும் என்று தலைமை நீதிபதி உத்தரவிட்டு, வழக்கு விசாரணையை ஒத்தி வைத்தார். Advertisement: SHARE | tamil |
\begin{document}
\begin{abstract}
The infinite pigeonhole principle for 2-partitions ($\rt^1_2$) asserts the existence, for every set $A$, of an infinite subset of $A$ or of its complement. In this paper, we study the infinite pigeonhole principle from a computability-theoretic viewpoint. We prove in particular that $\rt^1_2$ admits strong cone avoidance for arithmetical and hyperarithmetical reductions. We also prove the existence, for every $\Delta^0_n$ set, of an infinite low${}_n$ subset of it or its complement. This answers a question of Wang. For this, we design a new notion of forcing which generalizes the first and second-jump control of Cholak, Jockusch and Slaman.
\end{abstract}
\title{The weakness of the pigeonhole principle under hyperarithmetical reductions}
\section{Introduction}
In this paper, we study the infinite pigeonhole principle ($\rt^1_k$) from a computability-theoretic viewpoint. The infinite pigeonhole principle asserts that every finite partition of $\omega$ admits an infinite part. More formally, $\rt^1_k$ is the problem whose instances are colorings $f : \omega \tildeo k$. An $\rt^1_k$-solution to $f$ is an infinite set $H \subseteq \omega$ such that $|f[H]| = 1$. The general question we aim to address is the following:
\begin{question}
Does every instance of $\rt^1_k$ admit a \qt{weak} solution?
\end{question}
We consider various notions of weakness, among which the inability to bound a fixed non-zero degree for the $\Delta^0_n$, arithmetical and hyperarithmetical reduction. This property is known as \emph{strong cone avoidance}. With respect to $\Delta^0_n$ and arithmetical reductions, our main theorems are:
\begin{theorem}[Main theorem 1] \label{maintheorem1}
Fix $n \geq 0$. Let $B$ be non $\emptyset^{(n)}$-computable. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not $H^{(n)}$-computable.
\end{theorem}
\begin{theorem}[Main theorem 2] \label{maintheorem2}
Let $B$ be non arithmetical. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not arithmetical in $H$.
\end{theorem}
We also study restrictions of the infinite pigeonhole principle to $\Delta^0_n$ instances. With that respect, our main theorem is:
\begin{theorem}[Main theorem 3] \label{maintheorem3}
Fix $n \geq 0$. Every $\halt^{(n+1)}$-computable set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ of low${}_{n+2}$ degree.
\end{theorem}
Finally our main theorem with respect to hyperarithmetic reductions is:
\begin{theorem}[Main theorem 4] \label{maintheorem4}
Let $B$ be non hyperarithmetical. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not hyperarithmetical in $H$, in particular with $\omega_1^H = \omega_1^{ck}$.
\end{theorem}
Our motivation comes from \emph{reverse mathematics}.
Reverse mathematics is a foundational program which aims to find the weakest axioms needed to prove ordinary theorems. The early reverse mathematics showed the existence of an empirical structural phenomenon, in that most theorems are provably equivalent to one among five main systems of axioms, linearly ordered by the logical implication. See Simpson's book~\cite{Simpson2009Subsystems} for a reference on reverse mathematics. However, some natural statements escape this structural phenomenon, the most famous one being \emph{Ramsey's theorem for pairs} ($\rt^2_2$). Given a set $X$, let $[X]^n$ denote the set of unordered $n$-tuples over $X$. Ramsey's theorem for $n$-tuples and $k$-colors ($\rt^n_k$) asserts the existence, for every coloring $f : [\omega]^n \tildeo k$, of an infinite set $H \subseteq \omega$ such that $|f[\omega]^n| = 1$. In particular, $\rt^1_k$ is the infinite pigeonhole principle.
Ramsey's theorem for pairs and two colors received a lot of attention from the computability community as it was historically the first example of statement escaping the structural phenomenon of reverse mathematics. The study of $\rt^2_k$ revealed a deep connection between the computability-theoretic features of $\rt^2_k$ and the combinatorial features of $\rt^1_k$. More precisely, almost every proof of a statement of the form \qt{Every computable instance of $\rt^2_k$ admits a weak solution} can be obtained by a proof of the statement \qt{every (arbitrary) instance of $\rt^1_k$ admits a weak solution}, with the help of very weak computability-theoretic notion called \emph{cohesiveness}. This is in particular the case for cone avoidance~\cite{Seetapun1995strength,Dzhafarov2009Ramseys}, PA avoidance~\cite{Liu2012RT22}, constant-bound trace avoidance~\cite{Liu2015Cone}, preservation of hyperimmunity~\cite{Patey2017Iterative}, and preservation of non-c.e.\ definitions~\cite{Wang2014Definability,Patey2016weakness}, among others.
In many cases, the combinatorial features of $\rt^1_k$ and the computability-theoretic features of $\rt^2_k$ can be proven to be equivalent. See Cholak and Patey~\cite[Theorem 1.5]{Cholak2019Thin} for an equivalence in the case of cone avoidance. It therefore seems essential to obtain a good understanding of the infinite pigeonhole principle in order to better understand why Ramsey's theorem for pairs escapes the structural phenomenon of reverse mathematics.
\subsection{Strong cone avoidance}
Given a partial order $\leq_r$ on $2^\omega$ and a set $X$, we let $\deg_r(X) = \{ Y : X \equiv_r Y \}$ be the \emph{degree} of $X$, where $X \equiv_r Y$ if $X \leq_r Y$ and $Y \leq_r X$. We are in particular interested in the case where $\leq_r$ is among the $\Delta^0_n$ reduction $\leq_n$, the arithmetical reduction $\leq_{arith}$ and the hyperarithmetical reduction $\leq_{hyp}$.
Given a mathematical problem $\mathsf{P}$ formulated in terms of instances and solutions, it is natural to ask which sets are \emph{$\mathsf{P}$-encodable}. Here, we say that a set $X$ is $\mathsf{P}$-encodable if there is an instance $I$ of $\mathsf{P}$ such that for every $\mathsf{P}$-solution $Y$ to $I$, $X \leq_r Y$. Some problems are very weak with respect to the order $\leq_r$, and satisfy the following property:
\begin{definition}[Strong cone avoidance]
A problem $\mathsf{P}$ \emph{admits strong cone avoidance} for $\leq_r$ if for every pair of sets $Z$ and $C$ such that $C \not \leq_r Z$, every instance $X$ of $\mathsf{P}$ admits a solution $Y$ such that $C \not \leq_r Z \oplus Y$.
\end{definition}
Dzhafarov and Jockusch~\cite{Dzhafarov2009Ramseys} proved that $\rt^1_2$ admits strong cone avoidance of the Turing reduction. Their theorem has practical applications, and yield a simpler proof of Seetapun's theorem~\cite{Seetapun1995strength}. We prove a similar result for $\Delta^0_n$ and arithmetical reductions.
\begin{theoremnonumber}[Reformulation of Main theorem 1 (Theorem \ref{maintheorem1})]
$\rt^1_2$ admits strong cone avoidance for $\Delta^0_n$ reductions.
\end{theoremnonumber}
\begin{theoremnonumber}[Reformulation of Main theorem 2 (Theorem \ref{maintheorem2})]
$\rt^1_2$ admits strong cone avoidance for arithmetical reductions.
\end{theoremnonumber}
We finally prove in the last section strong cone avoidance for hyperarithmetical reductions, the main difficulty being to show that a non-computable ordinal is never $\rt^1_2$-encodable. This gives us the following theorem:
\begin{theoremnonumber}[Reformulation of Main theorem 4 (Theorem \ref{maintheorem4})]
$\rt^1_2$ admits strong cone avoidance for hyperarithmetical reductions.
\end{theoremnonumber}
These theorems show the combinatorial weakness of the pigeonhole principle with respect $\rt^1_2$-encodability. To prove this, we designed a new notion of forcing with an iterated jump control generalizing the first and second jump control of Cholak, Jockusch and Slaman~\cite{Cholak2001strength}.
\subsection{Lowness and hierarchies}
The computability-theoretic study of the pigeonhole principle is also motivated by questions on the strictness of hierarchies in reverse mathematics. Some consequences of Ramsey's theorem form hierarchies of statements, parameterized by the size of the colored tuples. A first example is Ramsey's theorem itself. Indeed, $\rt^{n+1}_k$ implies $\rt^n_k$ for every $n, k \geq 1$. By the work of Jockusch~\cite{Jockusch1972Ramseys}, this hierarchy collapses starting from the triples, and by Seetapun~\cite{Seetapun1995strength}, Ramsey's theorem for pairs is strictly weaker than Ramsey's theorem for triples. We therefore have
$$
\rt^1_k < \rt^2_k < \rt^3_k = \rt^4_k = \dots
$$
Some other hierarchies have been considered in reverse mathematics.
Friedman~\cite{FriedmanFom53free} introduced the free set ($\fs^n$) and thin set theorems ($\tildes^n$), while Csima and Mileti~\cite{Csima2009strength} introduced and studied the rainbow Ramsey theorem ($\rrt^n_k$). These statements are all of the form $\mathsf{P}^n$: \qt{For every coloring $f : [\omega]^n \tildeo \omega$, there is an infinite set $H \subseteq \omega$ such that $f {\upharpoonright} [H]^n$ avoids some set of forbidden patterns}.
The reverse mathematics of these statements were extensively studied in the literature~\cite{Cholak2001Free,Csima2009strength,Kang2014Combinatorial,PateyCombinatorial,Patey2015Somewhere,Patey2016weakness,RiceThin,WangSome,Wang2013Rainbow,Wang2014Cohesive,Wang2014Definability,Wang2014Some}. In particular, these theorems form hierarchies which are not known to be strictly increasing.
\begin{question}\label{quest:strictness-hiearchies}
Are the hierarchies of the free set, thin set, and rainbow Ramsey theorem strictly increasing?
\end{question}
Partial results were however obtained. All these statements admit lower bounds of the form “For every $n \geq 2$, there is a computable instance of $\mathsf{P}^n$ with no $\Sigma^0_n$ solution", where $\mathsf{P}^n$ denotes any of $\rt^n_k$ (Jockusch~\cite{Jockusch1972Ramseys}), $\rrt^n_k$ (Csima and Mileti~\cite{Csima2009strength}), $\fs^n$, or $\tildes^n$ (Cholak, Giusto, Hirst and Jockusch~\cite{Cholak2001Free}). From the upper bound viewpoint, all these statements follow from Ramsey's theorem. Therefore, by Cholak, Jockusch and Slaman~\cite{Cholak2001strength}, every computable instance of $\mathsf{P}^1$ admits a computable solution, and every computable instance of $\mathsf{P}^2$ admits a low${}_2$ solution. These results are sufficient to show that $\mathsf{P}^1 < \mathsf{P}^2 < \mathsf{P}^3$ in reverse mathematics. This upper bound becomes too coarse for triples. Wang~\cite{Wang2014Cohesive} proved that every computable instance of $\rrt^3_k$ admits a low${}_3$ solution. The following question is still open. A positive answer would also answer positively Question~\ref{quest:strictness-hiearchies}.
\begin{question}\label{quest:instances-hierarchies-lown}
Does every computable instance of $\fs^n$, $\tildes^n$, and $\rrt^n_k$ admit a low${}_n$ solution?
\end{question}
Indeed, suppose \Cref{quest:instances-hierarchies-lown} is answered positively for some $\mathsf{P} \in \{\rrt_2, \fs, \tildes\}$. For every $n$, one can iterate a relativization of \Cref{quest:instances-hierarchies-lown} to build a model $\mathcal{M}$ of $\mathsf{P}^n$ containing only sets of low${}_n$ degree. In particular, any set in $\mathcal{M}$ is $\Sigma^0_{n+1}$ , while by the lower bounds mentioned above, there is a computable instance of $\mathsf{P}^{n+1}$ with no $\Sigma^0_{n+1}$ solution. Thus, $\mathsf{P}^{n+1}$ fails in $\mathcal{M}$, hence $\mathsf{P}^n$ does not imply $\mathsf{P}^{n+1}$ over $\rca$.
Upper bounds to $\fs^n$, $\tildes^n$, and $\rrt^n_k$,
are usually proven inductively over $n$~\cite{Wang2014Some,PateyCombinatorial,Patey2017Iterative}, starting with the infinite pigeonhole principle for $n = 1$. In this paper, we therefore prove the following theorem, which introduces the machinery that hopefully will serve to answer positively Question~\ref{quest:instances-hierarchies-lown}.
\begin{theoremnonumber}[Main theorem 3 (\Cref{maintheorem3})]
Fix $n \geq 0$. Every $\halt^{(n+1)}$-computable set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ of low${}_{n+2}$ degree.
\end{theoremnonumber}
In particular, we fully answer two questions of Wang~\cite[Questions 6.1 and 6.2]{Wang2014Cohesive}, also asked by the second author~\cite[Question 5.4]{Patey2016Open}. The cases $n = 2$ and $n = 3$ were proven by Cholak, Jockusch and Slaman~\cite{Cholak2001strength} and by the authors~\cite{Monin2018Pigeons}, respectively.
\subsection{Definitions and notation}
A \emph{binary string} is an ordered tuple of bits $a_0, \dots, a_{n-1} \in \{0, 1\}$.
The empty string is written $\epsilon$. A \emph{binary sequence} (or a \emph{real}) is an infinite listing of bits $a_0, a_1, \dots$.
Given $s \in \omega$,
$2^s$ is the set of binary strings of length $s$ and
$2^{<s}$ is the set of binary strings of length $<s$. As well,
$2^{<\omega}$ is the set of binary strings
and $2^{\omega}$ is the set of binary sequences.
Given a string $\sigma \in 2^{<\omega}$, we use $|\sigma|$ to denote its length.
Given two strings $\sigma, \tildeau \in 2^{<\omega}$, $\sigma$ is a \emph{prefix}
of $\tildeau$ (written $\sigma \preceq \tildeau$) if there exists a string $\rho \in 2^{<\omega}$
such that $\sigma^\frown \rho = \tildeau$. Given a sequence $X$, we write $\sigma \prec X$ if
$\sigma = X {\upharpoonright} n$ for some $n \in \omega$.
A binary string $\sigma$ can be interpreted as a finite set $F_\sigma = \{ x < |\sigma| : \sigma(x) = 1 \}$. We write $\sigma \subseteq \tildeau$ for $F_\sigma \subseteq F_\tildeau$.
We write $\#\sigma$ for the size of $F_\sigma$. Given two strings $\sigma$ and $\tildeau$, we let $\sigma \cup \tildeau$ be the unique string $\rho$ of length $\max(|\sigma|, |\tildeau|)$ such that $F_\rho = F_\sigma \cup F_\tildeau$.
A \emph{binary tree} is a set of binary strings $T \subseteq 2^{<\omega}$ which is closed downward under the prefix relation. A \emph{path} through $T$ is a binary sequence $P \in 2^\omega$ such that every initial segment belongs to $T$.
A \emph{Turing ideal} $\mathcal{I}$ is a collection of sets which is closed downward under the Turing reduction and closed under the effective join, that is, $(\forall X \in \mathcal{I})(\forall Y \leq_T X) Y \in \mathcal{I}$ and $(\forall X, Y \in \mathcal{I}) X \oplus Y \in \mathcal{I}$, where $X \oplus Y = \{ 2n : n \in X \} \cup \{ 2n+1 : n \in Y \}$. A \emph{Scott set} is a Turing ideal $\mathcal{I}$ such that every infinite binary tree $T \in \mathcal{I}$ has a path in $\mathcal{I}$. In other words, a Scott set is the second-order part of an $\omega$-model of $\rca + \wkl$.
A Turing ideal $\mathcal{M}$ is \emph{countable coded} by a set $X$
if $\mathcal{M} = \{ X_n : n \in \omega \}$ with $X = \bigoplus_n X_n$.
A formula is $\Sigma^0_1(\mathcal{M})$ (resp.\ $\Pi^0_1(\mathcal{M})$) if it is $\Sigma^0_1(X)$ (resp.\ $\Pi^0_1(X)$) for some $X \in \mathcal{M}$.
Given two sets $A$ and $B$, we denote by $A < B$ the formula
$(\forall x \in A)(\forall y \in B)[x < y]$.
We write $A \subseteq^{*} B$ to mean that $A - B$ is finite, that is,
$(\exists n)(\forall a \in A)(a \not \in B \rightarrow a < n)$.
A \emph{$k$-cover} of a set $X$ is a sequence of sets $Y_0, \dots, Y_{k-1}$ such that $X \subseteq Y_0 \cup \dots \cup Y_{k-1}$.
\section{Preliminary tools}
We start by introduce the central tools used in the various forcings to come : the largeness and partition regular classes. They were introduced by the authors in~\cite{Monin2018Pigeons} to design a notion of forcing controlling the second jump of solutions to the pigeonhole principle. In this paper we push their use further, with the introduction of $\mathcal{M}$-cohesive and $\mathcal{M}$-minimal largeness classes, which are necessary for the third jump control and beyond.
\subsection{Largeness classes}
\begin{definition}
A \emph{largeness class} is a non-empty collection of sets $\mathcal{A} \subseteq 2^\omega$ such that
\begin{itemize}
\item[(a)] If $X \in \mathcal{A}$ and $Y \supseteq X$, then $Y \in \mathcal{A}$
\item[(b)] For every $k$-cover $Y_0, \dots, Y_{k-1}$ of $\omega$, there is some $j < k$ such that $Y_j \in \mathcal{A}$.
\end{itemize}
\end{definition}
For example, the collection of all the infinite sets is a largeness class. Moreover, any superclass of a largeness class is again a largeness class.
\begin{lemma}\label{lem:decreasing-largeness-yields-largeness}
Suppose $\mathcal{A}_0 \supseteq \mathcal{A}_1 \supseteq \dots$ is a decreasing sequence of largeness classes.
Then $\bigcap_s \mathcal{A}_s$ is a largeness class.
\end{lemma}
\begin{proof}
If $X \in \bigcap_s \mathcal{A}_s$ and $Y \supseteq X$, then for every $s$, since $\mathcal{A}_s$ is a largeness class, $Y \in \mathcal{A}_s$, so $Y \in \bigcap_s \mathcal{A}_s$.
Let $Y_0, \dots, Y_{k-1}$ be a $k$-cover of $\omega$. For every $s \in \omega$, there is some $j < k$
such that $Y_j \in \mathcal{A}_s$. By the infinite pigeonhole principle, there is some $j < k$ such that $Y_j \in \mathcal{A}_s$ for infinitely many $s$. Since $\mathcal{A}_0 \supseteq \mathcal{A}_1 \supseteq$ is a decreasing sequence, $Y_j \in \bigcap_s \mathcal{A}_s$.
\end{proof}
\begin{lemma}\label{lem:largeness-class-complexity}
Let $\mathcal{A}$ be a $\Sigma^0_1$ class.
The sentence “$\mathcal{A}$ is a largeness class" is $\Pi^0_2$.
\end{lemma}
\begin{proof}
Say $\mathcal{A} = \{ X : (\exists \sigma \preceq X)\varphi(\sigma) \}$ where $\varphi$ is a $\Sigma^0_1$ formula.
By compactness, $\mathcal{A}$ is a largeness class iff for every $\sigma$ and $\tildeau$ such that $\sigma \subseteq \tildeau$ and $\varphi(\sigma)$ holds, $\varphi(\tildeau)$ holds, and for every $k$, there is some $n \in \omega$ such that for every $\sigma_0 \cup \dots \cup \sigma_{k-1} = \{0, \dots, n\}$, there is some $j < k$ such that $\varphi(\sigma_j)$ holds.
\end{proof}
\subsection{Partition regular classes}
\begin{definition}
A \emph{partition regular class} is a collection of sets $\mathcal{L} \subseteq 2^\omega$ such that
\begin{itemize}
\item[(a)] $\mathcal{L}$ is a largeness class
\item[(b)] For every $X \in \mathcal{L}$ and $Y_0 \cup \dots \cup Y_{k-1} \supseteq X$, there is some $j < k$ such that $Y_j \in \mathcal{L}$.
\end{itemize}
\end{definition}
In particular, the class of all infinite sets is partition regular.
\begin{lemma}\label{lem:decreasing-largeness-yields-pr}
Suppose $\mathcal{A}_0 \supseteq \mathcal{A}_1 \supseteq \dots$ is a decreasing sequence of partition regular classes.
Then $\bigcap_s \mathcal{A}_s$ is a partition regular class.
\end{lemma}
\begin{proof}
The proof is easy, similar to the one of \Cref{lem:decreasing-largeness-yields-largeness} and left to the reader.
\end{proof}
\begin{definition}
Let $\mathcal{A}$ be a largeness class. Define
$$
\mathcal{L}(\mathcal{A}) = \{ X \in 2^\omega : \forall k\ \forall X_0 \cup \dots \cup X_{k-1} \supseteq X\ \exists i < k\ X_i \in \mathcal{A} \}
$$
\end{definition}
Note that a superset of a partition regular class need not to be partition regular, it is however always a largeness class. Note also that if $\mathcal{U}$ is a $\Sigma^0_1(X)$ class, then by compactness $\mathcal{L}(\mathcal{U})$ is a $\Pi^0_2(X)$ class.
\begin{lemma}\label{lem:lcal-of-largeness-is-partition-regular}
Let $\mathcal{A}$ be a largeness class. Then $\mathcal{L}(\mathcal{A})$ is the largest partition regular subclass of $\mathcal{A}$.
\end{lemma}
\begin{proof}
We first prove that $\mathcal{L}(\mathcal{A})$ is a partition regular subclass of $\mathcal{A}$.
By definition of $\mathcal{A}$ being a largeness class, $\omega \in \mathcal{L}(\mathcal{A})$.
Let $X \in \mathcal{L}(\mathcal{A})$ and $X_0 \cup \dots \cup X_{k-1} \supseteq X$. Suppose for the sake of absurd that $X_i \not \in \mathcal{L}(\mathcal{A})$ for every $i < k$. Then for every $i < k$, there is some $k_i \in \omega$ and some $Y^0_i \cup \dots \cup Y^{k_i-1}_i \supseteq X_i$ such that $Y^j_i \not \in \mathcal{A}$ for every $j < k_i$. Then $\{Y^j_i : i < k, j < k_i \}$ is a cover of $X$ contradicting $X \in \mathcal{L}(\mathcal{A})$. Therefore $\mathcal{L}(\mathcal{A})$ is a partition regular class.
Moreover, $\mathcal{L}(\mathcal{A}) \subseteq \mathcal{A}$ as witnessed by taking the trivial cover of $X$ by $X$ itself.
We now prove that $\mathcal{L}(\mathcal{A})$ is the largest partition regular subclass of $\mathcal{A}$.
Indeed, let $\mathcal{B}$ be a partition regular subclass of $\mathcal{A}$. Then for every $X \in \mathcal{B}$, every $X_0 \cup \dots \cup X_{k-1} \supseteq X$, there is some $j < k$ such that $X_j \in \mathcal{B} \subseteq \mathcal{A}$. Thus $X \in \mathcal{L}(\mathcal{A})$, so $\mathcal{B} \subseteq \mathcal{L}(\mathcal{A})$.
\end{proof}
\subsection{$\mathcal{M}$-cohesive classes}
We now introduce the notion of $\mathcal{M}$-cohesive largeness classes for a countable Scott set $\mathcal{M}$. One would ideally need $\mathcal{M}$-minimal largeness classes instead for the upcoming forcing (see \Cref{def_minmalclasses}). Unfortunately these classes are definitionally too complex for us. We use instead $\mathcal{M}$-cohesive largeness classes, which are definitionally simpler and can be seen as a way to ``almost'' build a minimal largeness class. The key property of these classes lies in \Cref{lem:cohesive-largeness-compatibility}, which is later used to show that an $\mathcal{M}$-cohesive largeness class contains a \emph{unique} $\mathcal{M}$-minimal largeness class.
Given an infinite set $X$, we let $\mathcal{L}_X$ be the $\Pi^0_2(X)$ largeness class of all sets having an infinite intersection with~$X$.
\begin{definition}
A class $\mathcal{A}$ is \emph{$\mathcal{M}$-cohesive}
if for every $X \in \mathcal{M}$, either $\mathcal{A} \subseteq \mathcal{L}_X$ or $\mathcal{A} \subseteq \mathcal{L}_{\overline{X}}$.
\end{definition}
In what follows, fix an effective enumeration $\mathcal{U}^Z_0, \mathcal{U}^Z_1, \dots$ of all the $\Sigma^{0,Z}_1$ classes upward-closed under the superset relation, that is, if $X \in \mathcal{U}^Z_e$ and $Y \supseteq X$, then $Y \in \mathcal{U}^Z_e$. Fix also a Scott set $\mathcal{M} = \{X_0, X_1, \dots \}$ countable coded by a set $M$. Given a set $C \subseteq \omega^2$, we write
$$
\mathcal{U}^\mathcal{M}_C = \bigcap_{\langle e, i \rangle \in C} \mathcal{U}^{X_i}_e
$$
\begin{lemma}\label{lem:cohesive-largeness-compatibility}
Let $\mathcal{U}^{\mathcal{M}}_C$ be an $\mathcal{M}$-cohesive class.
Let $\mathcal{U}^{\mathcal{M}}_D$ and $\mathcal{V}^{\mathcal{M}}_E$ be such that $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{\mathcal{M}}_D$ and $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{\mathcal{M}}_E$ are both largeness classes. Then $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{\mathcal{M}}_D \cap \mathcal{U}^{\mathcal{M}}_E$ is a largeness class.
\end{lemma}
\begin{proof}
Suppose for contradiction that $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{\mathcal{M}}_D \cap \mathcal{U}^{\mathcal{M}}_E$ is not a largeness class. Then by Lemma~\ref{lem:decreasing-largeness-yields-largeness}, there is some finite $C_1 \subseteq C$, $D_1 \subseteq D$ and $E_1 \subseteq E$ such that $\mathcal{U}^{\mathcal{M}}_{C_1} \cap \mathcal{U}^{\mathcal{M}}_{D_1} \cap \mathcal{U}^{\mathcal{M}}_{E_1}$ is not a largeness class. Since $\mathcal{U}^{\mathcal{M}}_{C_1} \cap \mathcal{U}^{\mathcal{M}}_{D_1} \cap \mathcal{U}^{\mathcal{M}}_{E_1}$ is $\Sigma^0_1(\mathcal{M})$, the collection $\mathcal{C}$ of all sets $Y_0 \oplus \dots \oplus Y_{k-1}$ such that $Y_0 \sqcup \dots \sqcup Y_{k-1} = \omega$ and for every $i < k$, $Y_i \not \in \mathcal{U}^{\mathcal{M}}_{C_1} \cap \mathcal{U}^{\mathcal{M}}_{D_1} \cap \mathcal{U}^{\mathcal{M}}_{E_1} \supseteq \mathcal{U}^{\mathcal{M}}_{C} \cap \mathcal{U}^{\mathcal{M}}_{D} \cap \mathcal{U}^{\mathcal{M}}_{E}$, is a non-empty $\Pi^0_1(\mathcal{M})$ class.
Since $\mathcal{M}$ is a Scott set, $\mathcal{C} \cap \mathcal{M} \neq \emptyset$, so fix such a set $Y_0 \oplus \dots \oplus Y_{k-1} \in \mathcal{C} \cap \mathcal{M}$.
Since $\mathcal{U}^{\mathcal{M}}_C$ is $\mathcal{M}$-cohesive, there must be some $i < k$ such that $\mathcal{U}^{\mathcal{M}}_C \subseteq \mathcal{L}_{Y_i}$. In particular, $Y_i \in \mathcal{U}^{\mathcal{M}}_C$, so $Y_i \not \in \mathcal{U}^{\mathcal{M}}_{D}$ or $Y_i \not \in \mathcal{U}^{\mathcal{M}}_{E}$. Suppose $Y_i \not \in \mathcal{U}^{\mathcal{M}}_{D}$, as the other case is symmetric. Since $Y_j \cap Y_i = \emptyset$ for every $j \neq i$, then $Y_j \not \in \mathcal{U}^{\mathcal{M}}_C \subseteq \mathcal{L}_{Y_i}$ for every $j \neq i$. It follows that $Y_0, \dots, Y_{k-1}$ witnesses that $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{\mathcal{M}}_{D}$ is not a largeness class. Contradiction.
\end{proof}
\subsection{$\mathcal{M}$-minimal classes}
\begin{definition} \label{def_minmalclasses}
A class $\mathcal{A}$ is \emph{$\mathcal{M}$-minimal} if for every $X \in \mathcal{M}$ and $e \in \omega$, either $\mathcal{A} \subseteq \mathcal{U}^X_e$ or $\mathcal{A} \cap \mathcal{U}^X_e$ is not a largeness class.
\end{definition}
The following is a corollary of \cref{lem:cohesive-largeness-compatibility} and informally says that an $\mathcal{M}$-cohesive largeness class contains a unique $\mathcal{M}$-minimal largeness class, which can be build with a greedy algorithm.
\begin{lemma}
Given an $\mathcal{M}$-cohesive largeness class $\mathcal{U}^{\mathcal{M}}_C$,
the collection of sets
$$
\langle \mathcal{U}^{\mathcal{M}}_C \rangle = \bigcap_ {e \in \omega, X \in \mathcal{M}} \{ \mathcal{U}_e^X : \mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}_e^X \mbox{ is a largeness class}\}
$$
is an $\mathcal{M}$-minimal largeness class contained in $\mathcal{U}^{\mathcal{M}}_C$.
\end{lemma}
\begin{proof}
We first prove that $\langle \mathcal{U}^{\mathcal{M}}_C \rangle$ is a largeness class.
Let $e_0, e_1, \dots$ and $X_0, X_1, \dots$ be an enumeration of all pairs $(e, X) \in \omega \tildeimes \mathcal{M}$ such that $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}_e^X$ is a largeness class.
By induction on $n$ using Lemma~\ref{lem:cohesive-largeness-compatibility},
$\bigcap_{i < n} \mathcal{U}_{e_i}^{X_i}$ is a largeness class for every $n \in \omega$. Thus, by Lemma~\ref{lem:decreasing-largeness-yields-largeness}, $\langle \mathcal{U}^{\mathcal{M}}_C \rangle = \bigcap_i \mathcal{U}_{e_i}^{X_i}$ is a largeness class. By construction $\langle \mathcal{U}^{\mathcal{M}}_C \rangle$ is $\mathcal{M}$-minimal.
\end{proof}
Note that we clearly have $\langle \mathcal{U}^{\mathcal{M}}_C \rangle \subseteq \mathcal{U}^{\mathcal{M}}_C$. The notation $\langle \mathcal{U}^{\mathcal{M}}_C \rangle$ for an $\mathcal{M}$-cohesive largeness class will be used all along this document. Note that $\langle \mathcal{U}^{\mathcal{M}}_C \rangle = \mathcal{U}^{\mathcal{M}}_D$ where $D$ is the set of all $\langle e, i\rangle$ such that $\mathcal{U}^{\mathcal{M}}_C \cap \mathcal{U}^{X_i}_e$ is a largeness class.
\begin{lemma}
Let $\mathcal{U}^{\mathcal{M}}_C$ be a largeness class. Then $\mathcal{L}(\mathcal{U}^{\mathcal{M}}_C) = \mathcal{U}^{\mathcal{M}}_D$ for some $D \subseteq \omega^2$. Furthermore $D$ is computable from $C$.
\end{lemma}
\begin{proof}
Let $\mathcal{U}^{\mathcal{M}}_C$ be a largeness class. Note that $\mathcal{L}(\mathcal{U}^{\mathcal{M}}_C) \subseteq \bigcap_{\langle e, i \rangle}\mathcal{L}(\mathcal{U}^{X_i}_e)$. By \cref{lem:decreasing-largeness-yields-pr} the class $\bigcap_{\langle e, i \rangle}\mathcal{L}(\mathcal{U}^{X_i}_e)$ is partition regular. By \cref{lem:lcal-of-largeness-is-partition-regular} we then must have $\mathcal{L}(\mathcal{U}^{\mathcal{M}}_C) = \bigcap_{\langle e, i \rangle}\mathcal{L}(\mathcal{U}^{X_i}_e)$. Also we have by definition of $\mathcal{L}(\mathcal{U})$ for a class $\mathcal{U}$ that $\mathcal{L}(\mathcal{U}^{X_i}_e)$ is a $\Pi^0_2(X_i)$ class whose indices are computable uniformly in $e$.
Thus we have that $\mathcal{L}(\mathcal{U}^{\mathcal{M}}_C) = \mathcal{U}^{\mathcal{M}}_D$ for some $D \subseteq \omega^2$. Furthermore $D$ is computable from $C$.
\end{proof}
\begin{corollary}\label{lem:minimal-is-partition-regular}
Suppose $\mathcal{U}^{\mathcal{M}}_C$ is an $\mathcal{M}$-minimal largeness class. Then $\mathcal{U}^{\mathcal{M}}_C$ is partition regular.
\end{corollary}
\begin{proof}
Let $D$ be such that $\mathcal{U}^{\mathcal{M}}_D = \mathcal{L}(\mathcal{U}^{\mathcal{M}}_C)$.
By Lemma~\ref{lem:lcal-of-largeness-is-partition-regular}, $\mathcal{U}^{\mathcal{M}}_D \subseteq \mathcal{U}^{\mathcal{M}}_C$. By $\mathcal{M}$-minimality of $\mathcal{U}^{\mathcal{M}}_C$, $\mathcal{U}^{\mathcal{M}}_C \subseteq \mathcal{U}^{\mathcal{M}}_D$. It follows that $\mathcal{U}^{\mathcal{M}}_C = \mathcal{U}^{\mathcal{M}}_D$. Since $\mathcal{U}^{\mathcal{M}}_D$ is partition regular, then so is $\mathcal{U}^{\mathcal{M}}_C$.
\end{proof}
It follows that if $\mathcal{U}^{\mathcal{M}}_C$ is an $\mathcal{M}$-cohesive largeness class, then the $\mathcal{M}$-minimal class $\langle \mathcal{U}^{\mathcal{M}}_C \rangle$ is a partition regular class.
\subsection{The framework}
We now build a sequence of sets $\{\mathcal{U}^{\mathcal{M}_n}_{C_n}\}_{n \in \omega}$ which will be used for the forcing in the next section.
\begin{proposition} \label{prop-hyp-scott}
There is a sequence of sets $\{M_n\}_{n < \omega}$ such that:
\begin{enumerate}
\item $M_n$ codes for a countable Scott set $\mathcal{M}_n$
\item $\halt^{(n)}$ is uniformly coded by an element of $\mathcal{M}_n$
\item Each $M_n'$ is uniformly computable in $\halt^{(n+1)}$
\end{enumerate}
\end{proposition}
\begin{proof}
Let us show the following: there is a functional $\Phi : 2^\omega \rightarrow 2^\omega$ such that for any oracle $X$, we have that $M' = \Phi(X')$ is such that $M = \oplus_{n \in \omega} X_n$ codes for a Scott set $\mathcal{M}$ with $X_0 = X$.
Fix a uniformly computable enumeration $\mathcal{C}^Y_0, \mathcal{C}^Y_1, \dots$ of all non-empty $\Pi^0_1(Y)$ classes.
Let $\mathcal{D}_X$ be the $\Pi^0_1(X)$ class of all $\bigoplus_n Y_n$ such that $Y_0 = X$ and for every $n = \langle a, b \rangle \in \omega$, $Y_{n+1} \in \mathcal{C}_a^{\bigoplus_{j \leq b} Y_j}$. Note that this $\Pi^0_1(X)$ class is uniform in $X$ and any member of $\mathcal{D}_X$ is a code of a Scott set whose first element is~$X$. Using the Low basis theorem~\cite{Jockusch197201}, there is a Turing functional $\Phi$ such that $\Phi(X')$ is the jump of a member of $\mathcal{D}_X$ for any $X$.
Using this function $\Phi$, it is clear that uniformly in $\halt^{(n+1)}$ one can compute the jump of a set $M_n$ coding for a Scott set $\mathcal{M}_n$ and containing $\halt^{(n)}$ as its first element.
\end{proof}
Let us assume that $\{\mathcal{M}_n\}_{n < \omega}$ is a sequence which verifies \Cref{prop-hyp-scott}. Recall the notation $\langle \mathcal{U}^\mathcal{M}_C \rangle$ : the unique minimal largeness subclass of an $\mathcal{M}$-cohesive largeness class.
\begin{proposition} \label{prop-hyp-cohesiveclassa}
There is a sequence of sets $\{C_n\}_{n \in \omega}$ such that:
\begin{enumerate}
\item $\mathcal{U}_{C_n}^{\mathcal{M}_{n}}$ is an $\mathcal{M}_n$-cohesive largeness class
\item $\mathcal{U}_{C_{n+1}}^{\mathcal{M}_{n+1}} \subseteq \langle \mathcal{U}_{C_n}^{\mathcal{M}_{n}} \rangle$
\item Each $C_n$ is coded by an element of $\mathcal{M}_{n + 1}$ uniformly in $n$ and $M_{n + 1}$.
\end{enumerate}
\end{proposition}
In order to prove \Cref{prop-hyp-cohesiveclassa} we use the two following uniformity lemmas, which will also be helpful later to continue the sequence of \Cref{prop-hyp-cohesiveclassa} through the computable ordinals (see \Cref{prop-hyp-cohesiveclass}).
\begin{lemma} \label{lem-hyp-cohesiveclass1}
There is a functional $\Phi : 2^\omega \tildeimes \omega \rightarrow 2^\omega$ such that for any set $M$ coding for a Scott set $\mathcal{M}$, for any $e$ such that $C = \Phi_e(M'')$ is such that $\mathcal{U}_C^{\mathcal{M}}$ is an $\mathcal{M}$-cohesive largeness class, $D = \Phi(M'', e)$ is such that $C \subseteq D$ and $\mathcal{U}_D^{\mathcal{M}} = \langle \mathcal{U}_C^{\mathcal{M}} \rangle$.
\end{lemma}
\begin{proof}
Say $\mathcal{M} = \{X_0, X_1, \dots \}$ with $M = \bigoplus_i X_i$.
Let $\{\langle e_t, i_t \rangle\}_{t \in \omega}$ be an enumeration of $\omega \tildeimes \omega$. Suppose that at stage $t$ a finite set $D^t \subseteq \{\langle e_0, i_0 \rangle, \dots, \langle e_t, i_t \rangle\}$ has been defined such that $\mathcal{U}_{D^t}^{\mathcal{M}} \cap \mathcal{U}_C^{\mathcal{M}}$ is a largeness class and such that for any $s \leq t$, $\langle e_s, i_s \rangle \notin D^t$ implies that $\mathcal{U}_{e_{s}}^{X_{i_{s}}} \cap \mathcal{U}_{D^t}^{\mathcal{M}} \cap \mathcal{U}_C^{\mathcal{M}}$ is not a largeness class.
Then at stage $t+1$, we ask $M''$ if $\mathcal{U}_{e_{t+1}}^{X_{i_{t+1}}} \cap \mathcal{U}_{D^t}^{\mathcal{M}} \cap \mathcal{U}_C^{\mathcal{M}}$ is a largeness class. If so we define $D^{t+1} = D^t \cup \{\langle e_{t+1}, i_{t+1} \rangle\}$. Otherwise we define $D^{t+1} = D^t$. Then $D = C \cup \bigcup_t D^t$ is uniformly $M''$-computable and $\mathcal{U}_D^{\mathcal{M}}$ equals $\langle \mathcal{U}_C^{\mathcal{M}} \rangle$.
\end{proof}
\begin{lemma} \label{lem-hyp-cohesiveclass2}
There is a functional $\Phi : 2^\omega \tildeimes \omega \tildeimes \omega \rightarrow \omega$ such that for any set $M$ coding for a Scott set $\mathcal{M}$, for any set $N$ coding for a Scott set $\mathcal{N}$ such that $M' \in \mathcal{N}$ with $N$-index $i_M$, for any $C \in \mathcal{N}$ with $N$-index $i_C$, such that $\mathcal{U}_C^{\mathcal{M}}$ is a partition regular class, $\Phi(N, i_M, i_C)$ is an $N$-index for $D \supseteq C$ such that $\mathcal{U}_D^{\mathcal{M}}$ is an $\mathcal{M}$-cohesive largeness class.
\end{lemma}
\begin{proof}
The functional $\Phi$ does the following : It looks for $M'$ at index $i_M$ inside $\mathcal{N}$. From $M'$ it computes $M = \oplus_{n} X_n$. It then computes with $M' + C$ the tree $T$ containing all the elements $\sigma$ such that
$$\left(\bigcap_{\sigma(i) = 0} 2^\omega - X_i\right) \cap \left(\bigcap_{\sigma(i) = 1} X_i \right) \in \bigcap_{\langle e, j \rangle \in C \upharpoonright {|\sigma|}} \mathcal{U}_e^{X_j}$$
Clearly $[T]$ is not empty. The functional $\Phi$ then finds an $N$-index for an element $Y \in [T]$. For $\sigma \prec Y$ let $X_\sigma = (\bigcap_{\sigma(i) = 0} (2^\omega - X_i)) \cap (\bigcap_{\sigma(i) = 0} X_i)$. We must have for every $\sigma \prec Y$ that $X_\sigma \in \mathcal{U}_C^{\mathcal{M}}$. It follows as $\mathcal{U}_C^{\mathcal{M}}$ is partition regular, that for every $\sigma \prec Y$, $\L_{X_\sigma} \cap \mathcal{U}_C^{\mathcal{M}}$ is a largeness class. Thus $\bigcap_{\sigma \prec Y} \L_{X_\sigma} \cap \mathcal{U}_C^{\mathcal{M}}$ is an $\mathcal{M}$-cohesive largeness class. Also $M \oplus Y \oplus C$ uniformly computes a set $D$ such that $\mathcal{U}_D^{\mathcal{M}} = \bigcap_{\sigma \prec Y} \L_{X_\sigma} \cap \mathcal{U}_C^{\mathcal{M}}$. The function $\Phi$ then returns an $N$-index for $D$.
\end{proof}
\begin{proof}[Proof of \Cref{prop-hyp-cohesiveclassa}]
Suppose that stage $n$ we have defined $C_n$ verifying $(1) (2)$ and $(3)$. Let us define $C_{n+1}$.
Note that the set $C_{n}$ is coded by an element of $\mathcal{M}_{n + 1}$, and thus that $C_{n}$ is computable in $\halt^{(n+2)}$ and then computable in $M_n''$. Using \Cref{lem-hyp-cohesiveclass1} we define $D_n \supseteq C_n$ to be such that $\mathcal{U}_{D_n}^{\mathcal{M}_n} = \langle \mathcal{U}_{C_n}^{\mathcal{M}_n} \rangle$ and such that $D_n$ is uniformly $M_{n}''$-computable. We define $E_{n+1}$ to be the transfer of the $M_n$-indices constituting $D_n$ into $M_{n+1}$-indices, using that $M_{n}$ is an element of $M_{n+1}$. So we have $\mathcal{U}_{E_{n+1}}^{\mathcal{M}_{n+1}} = \mathcal{U}_{D_n}^{\mathcal{M}_{n}}$.
Note that as $E_{n+1}$ is computable in $M_{n}'' \oplus M_{n+1}$ and thus in $\halt^{((n+1)+1)}$. It is then coded by an element of $\mathcal{M}_{(n+1)+1}$. Note also that $\mathcal{U}_{E_{n+1}}^{\mathcal{M}_{n+1}}$ is partition regular as it equals $\langle \mathcal{U}_{C_n}^{\mathcal{M}_n} \rangle$. Using \Cref{lem-hyp-cohesiveclass2} we uniformly find an $\mathcal{M}_{(n+1)+1}$-index of $C_{n+1} \supseteq E_{n+1}$ to be such that $\mathcal{U}_{C_{n+1}}^{\mathcal{M}_{n+1}}$ is an $\mathcal{M}_{n+1}$-cohesive largeness class.
\end{proof}
\section{Generalized Pigeonhole forcing}
The notion of forcing used to build solutions to the pigeonhole principle while controlling the first jump is a variant of Mathias forcing. In this section, we extend Mathias forcing to a more general notion of forcing while controlling iterated jumps, that is while tightly controlling the truth of $\Sigma^0_n$ and $\Pi^0_n$ formulas.
Let $\mathcal{M}_0, \mathcal{M}_1, \dots, \mathcal{M}_n$ be countable Scott sets coded by sets $M_0, M_1, \dots, M_n$, respectively, satisfying (1)(2) and (3) of \cref{prop-hyp-scott}. Let $C_0, C_1, C_2$ be sequence of sets satisfying (1)(2) and (3) of \cref{prop-hyp-cohesiveclassa}, that is, $\mathcal{U}^{\mathcal{M}_n}_{C_n}$ is an $\mathcal{M}$-cohesive largeness class, $\mathcal{U}^{\mathcal{M}_{n+1}}_{C_{n+1}} \subseteq \langle \mathcal{U}^{\mathcal{M}_n}_{C_n} \rangle$ and each $C_n$ is coded by an element of $\mathcal{M}_{n+1}$.
\subsection{The forcing conditions}
\begin{definition}
For each $n \geq 0$ let $\mathbb{P}_n$ be the set of pairs $(\sigma, X)$ such that
\begin{itemize}
\item[(a)] $X \cap \{ 0, \dots, |\sigma|\} = \emptyset$
\item[(b)] $X \in \langle \mathcal{U}^{\mathcal{M}_{n}}_{C_{n}} \rangle$
\end{itemize}
\end{definition}
Note that $X$ is infinite for $(\sigma, X) \in \mathbb{P}_n$ since $\mathcal{U}^{\mathcal{M}_{n}}_{C_{n}}$ contains only infinite sets. Mathias forcing builds a single object $G$ by approximations (conditions) which consist in an initial segment $\sigma$ of $G$, and an infinite reservoir of integers. The purpose of the reservoir is to restrict the set of elements we are allowed to add to the initial segment. The reservoir therefore enriches the standard Cohen forcing by adding an infinitary negative restrain.
\begin{definition}
The partial order on $\mathbb{P}_n$ is defined by $(\tildeau, Y) \leq (\sigma, X)$
if $\sigma \preceq \tildeau$, $Y \subseteq X$ and $\tildeau - \sigma \subseteq X$.
\end{definition}
Given a collection $\mathcal{F} \subseteq \mathbb{P}_n$, we let $G_\mathcal{F} = \bigcup \{ \sigma : (\sigma, X) \in \mathcal{F} \}$.
\subsection{The forcing question}
We now define what we call ``the forcing question'' : a relation between forcing conditions $p \in \mathbb{P}_n$ and $\Sigma^0_{m+1}$ formulas for $m \leq n$. The goal of the forcing question is to be definitionally not too complex, while being able to find extensions of conditions forcing formulas or their negation. The forcing question will also be used in the definition of the forcing relation, which is why it is introduced first.
\begin{definition} \label{def-hyp-forcingqua}
Let $\sigma \in 2^{<\omega}$. Let $(\exists x) \Phi_e(G, x)$ be a $\Sigma^0_1$ formula. Let $\sigma \qvdash (\exists x) \Phi(G, x)$ holds if
$$\{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\})(\exists x) \Phi_e(\sigma \cup \tildeau, x)\} \cap \mathcal{U}_{C_{0}}^{\mathcal{M}_{0}}$$
is a largeness class. Then inductively, given a $\Sigma^0_{m+1}$ formula $(\exists x) \Phi_e(G, x)$ with free variable $x$ for $1 \leq m < \omega$, we let $\sigma \qvdash (\exists x) \Phi_e(G, x)$ holds if
$$\{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}) (\exists x) \sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)\} \cap \mathcal{U}_{C_{m}}^{\mathcal{M}_{m}}$$
is a largeness class.
For a condition $p = (\sigma, X) \in \mathbb{P}_{n}$ for some $n < \omega$ and a $\Sigma^0_{m+1}$ formula $(\exists x) \Phi_e(G, x)$ with free variable $x$ for some $m \leq n$, we write $p \qvdash (\exists x) \Phi_e(G, x)$ if $\sigma \qvdash (\exists x) \Phi_e(G, x)$.
\end{definition}
\begin{proposition} \label{prop-hyp-effectivalla}
Let $\sigma \in 2^{<\omega}$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula for $m \geq 0$
\begin{enumerate}
\item The set
$$\{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\})(\exists x) \Phi_e(\sigma \cup \tildeau, x)\}$$
is an upward-closed $\Sigma^0_1$ open set if $m = 0$. The set
$$\{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}) (\exists x) \sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)\}$$
is an upward-closed $\Sigma^0_1(C_{m-1} \oplus \halt^{(m)})$ open set if $m>0$.
\item The relation $\sigma \qvdash (\exists x) \Phi_e(G, x)$ is $\Pi^0_1(C_{m} \oplus \halt^{(m+1)})$.
\end{enumerate}
This is uniform in $\sigma$ and $e$.
\end{proposition}
\begin{proof}
This is done by induction on $m$. We start with $m = 0$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{1}$ formula and $\sigma \in 2^{<\omega}$. It is clear that
$$\mathcal{U}(e, \sigma) = \{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\})(\exists x) \Phi_e(\sigma \cup \tildeau, x)\}$$
is an upward closed $\Sigma^0_1$ class. Then $\sigma \qvdash (\exists x) \Phi_e(G, x)$ iff $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_{0}}^{\mathcal{M}_{0}}$ is a largeness class, that is, iff for every finite set $F \subseteq C_{0}$, the class $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_{0}}$ is a largeness class. By \Cref{lem:largeness-class-complexity}, for each $F \subseteq C_0$, the statement is $\Pi^0_2(M_0)$ uniformly in $F$, and thus $\Pi^0_1(M_0')$ uniformly in $F$. It is then $\Pi^0_1(\halt')$ uniformly in $F$. Thus the whole statement is $\Pi^0_1(C_0 \oplus \halt')$.
Suppose (1) and (2) are true for $m-1$, every $\Sigma^0_{m}$ formula and every $\sigma$. Let $\sigma \in 2^{<\omega}$ and let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula. Let
$$\mathcal{U}(e, \sigma) = \{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}) (\exists x) \sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)\}$$
Let us show (1). For each $x \in \omega$, the formula $\neg \Phi_e(G, x)$ is $\Sigma^0_{m}$ uniformly in $x$ and in $e$. By induction hypothesis, the relation $\sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)$ is $\Sigma^0_1(C_{m-1} \oplus \halt^{(m)})$ uniformly in $\sigma \cup \tildeau$ in $x$ and in $e$. It follows that $\mathcal{U}(e, \sigma)$ is an upward closed $\Sigma^0_1(C_{m-1} \oplus \halt^{(m)})$ class.
Let us now show (2). We have $\sigma \qvdash (\exists x) \Phi_e(G, x)$ iff $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_{m}}^{\mathcal{M}_{m}}$ is a largeness class. Also $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_{m}}^{\mathcal{M}_{m}}$ is a largeness class if for all $F \subseteq C_{m}$, the class $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_{m}}$ is a largeness class. By \Cref{lem:largeness-class-complexity}, it is a $\Pi^0_2(M_{m})$ statement uniformly in $F$ and then a $\Pi^0_1(M_{m}')$ statement uniformly in $F$ and then a $\Pi^0_1(\halt^{(m+1)})$ statement uniformly in $F$. It follows that the statement \qt{$\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_{m}}^{\mathcal{M}_{m}}$ is a largeness class} is $\Pi^0_1(C_{m} \oplus \halt^{(m+1)})$.
\end{proof}
\subsection{The forcing relation}
The relation $\qvdash$ is now used to define the forcing relation.
\begin{definition} \label{def:qb2-forcing-relation}
Let $n \in \omega$. Let $p = (\sigma, X) \in \mathbb{P}_n$.
Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_1$ formula. We define
\begin{itemize}
\item[(a)] $p \Vdash (\exists x)\Phi_e(G, x)$ if $(\exists x)\Phi_e(\sigma, x)$
\item[(b)] $p \Vdash (\forall x)\Phi_e(G, x)$ if $(\forall \tildeau \subseteq X)(\forall x)\Phi_e(\sigma \cup \tildeau, x)$
\end{itemize}
Then inductively for $1 \leq m \leq n$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma_{m+1}$ formula. We define
\begin{itemize}
\item[(a)] $p \Vdash (\exists x) \Phi_e(G, x)$ if there is some $x \in \omega$ such that $p \Vdash \Phi_e(G, x)$
\item[(b)] $p \Vdash (\forall x) \neg \Phi_e(G, x)$ if for every $\tildeau \subseteq X$ and every $x \in \omega$, $\sigma \cup \tildeau \qvdash \neg \Phi_e(G, x)$
\end{itemize}
\end{definition}
\begin{lemma} \label{lem-hyp-forcepia}
Fix $0 \leq m \leq n$. Let $p \in \mathbb{P}_{n}$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula. Then $p \Vdash (\forall x) \neg \Phi_e(G, x)$ iff $q \qvdash \neg \Phi_e(G, x)$ for every $x \in \omega$ and every $q \leq p$.
\end{lemma}
\begin{proof}
Suppose $p \Vdash (\forall x) \neg \Phi_e(G, x)$ with $p = (\sigma, X)$. By definition of the forcing relation and forcing extensions it is clear that $q \qvdash \neg \Phi_e(G, x)$ for every $x$ and every $q \leq p$. Suppose now $q \qvdash \neg \Phi_e(G, x)$ for every $x$ and every $q \leq p$. Given any $\tildeau \subseteq X$ we have that $(\sigma \cup \tildeau, X - \{0, \dots, |\sigma \cup \tildeau|\})$ is a valid extension of $p$ for which we have $\sigma \cup \tildeau \qvdash \neg \Phi_e(G, x)$ for every $x$. It follows that $p \Vdash (\forall x) \neg \Phi_e(G, x)$.
\end{proof}
\begin{lemma}\label{lem:qb2-forcing-closed-under-extension}
Fix $0 \leq m \leq n$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula. Let $p, q \in \mathbb{P}_n$ be such that $q \leq p$.
\begin{itemize}
\item[(a)] If $p \Vdash (\exists x)\Phi_e(G, x)$ then so does $q$.
\item[(b)] If $p \Vdash (\forall x)\neg \Phi_e(G, x)$ then so does $q$.
\end{itemize}
\end{lemma}
\begin{proof}
We proceed by induction on $m$. It is clear for $\Sigma^0_1$ formulas. For $m > 0$ let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula.
For (a), by definition, there is some $x \in \omega$ such that $p \Vdash \Phi_e(G, x)$. As $\Phi_e(G, x)$ is a $\Pi^0_{m}$ formula, by induction hypothesis, $q \Vdash \Phi_e(G, x)$ and thus $q \Vdash (\exists x)\Phi_e(G, x)$.
For (b), by \Cref{lem-hyp-forcepia}, for all $x \in \omega$ and all $r \leq p$, $r \qvdash \neg \Phi_e(G, x)$. Thus if $q \leq p$, also for all $x$ and all $r \leq q$, $r \qvdash \neg \Phi_e(G, x)$. It follows still by \Cref{lem-hyp-forcepia} that $q \Vdash (\forall x)\neg \Phi_e(G, x)$.
\end{proof}
\subsection{The core lemmas}
We now show the core lemmas. The first one shows how to find extensions to force formulas, while the second one is the classic ``forcing imply truth'' whenever we work with generic enough filters.
\begin{lemma} \label{prop-hyp-forcexta}
Let $p \in \mathbb{P}_{n}$ with $p = (\sigma, X)$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula for $0 \leq m \leq n$.
\begin{enumerate}
\item Suppose $p \qvdash (\exists x)\Phi_e(G, x)$. Then there exists $q \leq p$ with $q \in \mathbb{P}_n$ such that $q \Vdash (\exists x)\Phi(G, x)$.
\item Suppose $p \nqvdash (\exists x)\Phi_e(G, x)$. Then there exists $q \leq p$ with $q \in \mathbb{P}_n$ such that $q \Vdash (\forall x)\neg \Phi(G, x)$.
\end{enumerate}
\end{lemma}
\begin{proof}
Let $p \in \mathbb{P}_{n}$. We start with $m = 0$. Suppose $p \qvdash (\exists x) \Phi_e(G, x)$. Let
$$\mathcal{U}(e, \sigma) = \{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\})(\exists x) \Phi_e(\sigma \cup \tildeau, x)\}$$
The class $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_0}^{\mathcal{M}_0}$ is a largeness class. As $\mathcal{U}_{C_0}^{\mathcal{M}_0}$ is $\mathcal{M}_0$-cohesive, then $\langle \mathcal{U}_{C_0}^{\mathcal{M}_0} \rangle \subseteq \mathcal{U}(e, \sigma)$. As $X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle \subseteq \langle \mathcal{U}_{C_0}^{\mathcal{M}_0} \rangle \subseteq \mathcal{U}(e, \sigma)$, there is $\tildeau \subseteq X$ such that $(\exists x)\Phi_e(\sigma \cup \tildeau, x)$ holds. As $\langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$ contains only infinite sets and is partition regular, $X - \{0, \dots, |\sigma \cup \tildeau|\} \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$. Then $(\sigma \cup \tildeau, X - \{0, \dots, \sigma \cup \tildeau\})$ is a valid extension of $(\sigma, X)$ such that $(\sigma \cup \tildeau, X - \{0, \dots, |\sigma \cup \tildeau|\}) \Vdash (\exists x) \Phi_e(G, x)$.
Suppose now $p \nqvdash (\exists x) \Phi_e(G, x)$. Then the class $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_0}^{\mathcal{M}_0}$ is not a largeness class. It follows that there is a finite set $F \subseteq C_0$ such that $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_0}$ is not a largeness class. For $k$ let $\mathcal{P}_k$ be the $\Pi^0_1(Z)$ class for some $Z \in \mathcal{M}_0$ of covers $Y_0 \cup \dots \cup Y_{k} \supseteq \omega$ such that $Y_i \notin \mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_0}$ for each $i \leq k$. As $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_0}$ is not a largeness class there must be some $k$ such that $\mathcal{P}_k$ is not empty. Then there are sets $Y_0 \oplus \dots \oplus Y_{k} \in \mathcal{M}_0 \cap \mathcal{P}_k$. As $\langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$ is partition regular and as $X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$ we have some $i \leq k$ such that $Y_i \cap X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle \subseteq \mathcal{U}_{C_0}^{\mathcal{M}_0}$. Thus $(\sigma, Y_i \cap X)$ is a valid extension of $(\sigma, X)$ for which $(\sigma, Y_i \cap X) \Vdash (\forall x) \neg \Phi(G, x)$.
Suppose now $m > 0$. Suppose $p \qvdash (\exists x) \Phi_e(G, x)$. Let
$$\mathcal{U}(e, \sigma) = \{Y\ :\ (\exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}) (\exists x) \sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)\}$$
By definition, the class $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_m}^{\mathcal{M}_m}$ is a largeness class. As $\mathcal{U}_{C_m}^{\mathcal{M}_m}$ is $\mathcal{M}_m$-cohesive and as, by \Cref{prop-hyp-effectivalla}, the set $\mathcal{U}(e, \sigma)$ is a $\Sigma^0_1(Y)$ for some $Y \in \mathcal{M}_m$, then $\langle \mathcal{U}_{C_m}^{\mathcal{M}_m} \rangle \subseteq \mathcal{U}(e, \sigma)$. As $X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle \subseteq \langle \mathcal{U}_{C_m}^{\mathcal{M}_m} \rangle \subseteq \mathcal{U}(e, \sigma)$, there is $\tildeau \subseteq X$ such that $\sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)$ for some $x$. Note that as $\langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$ contains only infinite sets and is partition regular we have $X - \{0, \dots, |\sigma \cup \tildeau|\} \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$. Also $(\sigma \cup \tildeau, X - \{0, \dots, |\sigma \cup \tildeau|\})$ is a valid extension of $(\sigma, X)$ such that $(\sigma \cup \tildeau, X - \{0, \dots, |\sigma \cup \tildeau|\}) \nqvdash \neg \Phi_e(G, x)$. Now by induction hypothesis we have some $Y \in \mathcal{M}_m$ with $(\sigma \cup \tildeau, X \cap Y) \leq (\sigma, X)$ and such that $(\sigma \cup \tildeau, X \cap Y) \Vdash \Phi_e(G, x)$. It follows that $(\sigma \cup \tildeau, X \cap Y) \Vdash (\exists x)\Phi_e(G, x)$.
Suppose now $p \nqvdash (\exists x)\Phi_e(G, x)$. Then $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{C_m}^{\mathcal{M}_m}$ is not a largeness class. It follows that there is a finite set $F \subseteq C_0$ such that $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_m}$ is not a largeness class. For $k$ let $\mathcal{P}_k$ be the $\Pi^0_1(Z)$ class for some $Z \in \mathcal{M}_m$ of covers $Y_0 \cup \dots \cup Y_{k} \supseteq \omega$ such that $Y_i \notin \mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_m}$ for each $i \leq k$. As $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_m}$ is not a largeness class there must be some $k$ such that $\mathcal{P}_k$ is not empty. There are sets $Y_0 \oplus \dots \oplus Y_{k} \in \mathcal{M}_m \cap \mathcal{P}_k$. As $\langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$ is partition regular and as $X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$, there is some $i \leq k$ such that $Y_i \cap X \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle \subseteq \mathcal{U}_{C_m}^{\mathcal{M}_m}$. It follows that $Y_i \cap X \notin \mathcal{U}(e, \sigma)$. It means that for every $\tildeau \subseteq Y_i \cap X$ and every $x \in \omega$, $\sigma \cup \tildeau \qvdash \neg\Phi_e(G, x)$. It follows that $(\sigma, Y_i \cap X) \Vdash (\forall x)\neg\Phi_e(G, x)$.
\end{proof}
We now sow that forcing implies truth. We define first for that the precise level of genericity that we need.
\begin{definition} \label{def:genericenough}
Let $\mathcal{F} \subseteq \mathbb{P}_n$ be a filter. The set $\mathcal{F}$ is $m$-generic if for every $k \leq m$ and every $\Sigma^0_{k+1}$ formula $(\exists x) \Phi_e(G, x)$ there is a condition $p \in \mathcal{F}$ such that $p \Vdash (\exists x) \Phi_e(G, x)$ or $p \Vdash (\forall x) \neg \Phi_e(G, x)$.
\end{definition}
Note that if a filter is $n$-generic, then it is $m$-generic for every $m < n$.
\begin{lemma} \label{lem:forcing-implytruth}
Let $\mathcal{F} \subseteq \mathbb{P}_{n}$ be an $n-1$-generic filter. Let $p \in \mathcal{F}$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ class for $0 \leq m \leq n$.
\begin{itemize}
\item[(a)] Suppose $p \Vdash (\exists x) \Phi_e(G, x)$. Then $(\exists x) \Phi_e(G_\mathcal{F}, x)$ holds.
\item[(b)] Suppose $p \Vdash (\forall x) \neg \Phi_e(G, x)$. Then $(\forall x) \neg \Phi_e(G_\mathcal{F}, x)$ holds.
\end{itemize}
\end{lemma}
\begin{proof}
The proof is done by induction on $m$. Let $p \in \mathbb{P}_{n}$ with $p = (\sigma, X)$. The result is clear and well-known for $m=0$. Suppose now $m>0$ and let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula. Suppose $p \Vdash (\exists x) \Phi_e(G, x)$. Then there exists $x$ such that $p \Vdash \Phi(G, x)$. By induction hypothesis $\Phi(G_\mathcal{F}, x)$ hods and then $\exists x\ \Phi(G_\mathcal{F}, x)$ holds.
Suppose $p \Vdash (\forall x) \neg \Phi_e(G, x)$. Then by \Cref{lem-hyp-forcepia} for every $x$ and every $q \leq p$, $q \qvdash \neg \Phi_e(G, x)$. From \Cref{prop-hyp-forcexta}, for every $x \in \omega$ and every $q \leq p$, there is some $r \leq q$ such that $r \Vdash \neg \Phi_e(G, x)$. It follows that for every $x$, the set $\{r \in \mathbb{P}_n\ :\ r \Vdash \neg \Phi_e(G, x)\}$ is dense below $p$. As $\mathcal{F}$ is $n-1$-generic and $p \in \mathcal{F}$ there must be for every $x$ some $q \in \mathcal{F}$ such that $q \Vdash \neg \Phi_{e}(G, x)$. By induction hypothesis $\neg \Phi_e(G_{\mathcal{F}}, x)$ for every $x$ and then $(\forall x) \neg \Phi_e(G_{\mathcal{F}}, x)$ holds.
\end{proof}
\section{Cone avoidance under $\Delta^0_n$ reductions}
We show in this section the first and third theorems of the introduction --- \Cref{maintheorem1} and \Cref{maintheorem3}. The proof of \Cref{maintheorem2} will be postponed to the next section, where it will be achieved together with hyperarithmetic cone avoidance. We fix a set $A^0 \sqcup A^1 = \omega$. We sometimes write $A$ for $A^0$ (with then $\omega - A = A^1$).
Unfortunately the above forcing is definitionally a bit too complex : the forcing question for $\Sigma^0_{m+1}$ statements is $\Pi^0_{m+2}$, whereas we would need it to be $\Sigma^0_{m+1}$.
For this reason, we need to plug upon the previous forcing another forcing notion, used only for ``the last step'' in formula induction. The drawbacks of this other forcing notions is that we are compelled to build two generic objects : one inside $A^0$ and one inside $A^1$. We then used the pairing argument first designed by Dzhafarov and Jockusch \cite{Dzhafarov2009Ramseys} to show that one of the object we build is sufficiently generic in the sense of \Cref{def:genericenough}.
\subsection{Another forcing on the top}
\begin{definition}\label{def:pb2-forcing-conditions}
Fix $n \geq 0$. Let $\mathbb{Q}_n$ denote the set of conditions $(\sigma^0, \sigma^1, X)$ such that
\begin{itemize}
\item[(a)] $\sigma^i \subseteq A^i$ for every $i < 2$
\item[(b)] $X \cap \{ 0, \dots, \max_i |\sigma^i|\} = \emptyset$
\item[(c)] $X \in \mathcal{M}_n$
\item[(d)] $X$ is infinite if $n = 0$ and $X \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ if $n \geq 1$.
\end{itemize}
A forcing condition $(\sigma^0, \sigma^1, X) \in \mathbb{Q}_n$ is \emph{valid for side} $i$ if $X \cap A^i \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ for $n>0$ and if $X \cap A^i$ is infinite for $n = 0$.
\end{definition}
By definition of a Turing ideal $\mathcal{M}$ countable coded by a set $M$, then $\mathcal{M}$ can be written as $\{Z_0, Z_1, \dots \}$ with $M = \bigoplus_i Z_i$.
We then say that $i$ is an \emph{$M$-index} of $Z_i$. Thanks to the notion of index, any $\mathbb{Q}_n$-condition can be finitely presented as follows. An \emph{index} of a $\mathbb{Q}_n$-condition $c = (\sigma^0, \sigma^1, X)$ is a tuple $(\sigma^0, \sigma^1, a)$ where $a$ is an $M_n$-index for $X$.
\begin{definition}
The partial order on $\mathbb{Q}_n$ is defined by
$$
(\tildeau^0, \tildeau^1, Y) \leq (\sigma^0, \sigma^1, X)
$$
if for every $i < 2$, $(\tildeau^i, Y) \leq (\sigma^i, X)$.
\end{definition}
Given a condition $c = (\sigma^0, \sigma^1, X)$ and $i < 2$, we write $c^{[i]} = (\sigma^i, X)$.
Each $\mathbb{Q}_n$-condition $c$ represents two $\mathbb{P}_{n-1}$-conditions $c^{[0]}$ and $c^{[1]}$.
We now design a disjunctive forcing question which builds upon the forcing question of $\mathbb{P}_n$ conditions. The difference is that it is only used at the last step of the induction of formulas.
\begin{definition}
Let $c = (\sigma^0, \sigma^1, X) \in \mathbb{Q}_0$ and let $(\exists x) \Phi_{e_0}(G, x)$ and $(\exists x) \Phi_{e_1}(G, x)$ be two $\Sigma_1$ formulas. Define the relation
$$
c \qvdash (\exists x)\Phi_{e_0}(G^0, x) \vee (\exists x)\Phi_{e_1}(G^1, x)
$$
to hold if for every 2-cover $Z^0 \cup Z^1 = X$, there is some side $i < 2$, some finite set $\rho \subseteq Z^i$ and some $x \in \omega$ such that $\Phi_{e_i}(\sigma^i \cup \rho, x)$ holds.
Let $n>0$. Let $c = (\sigma^0, \sigma^1, X) \in \mathbb{Q}_n$ and let $(\exists x) \Phi_{e_0}(G, x)$ and $(\exists x) \Phi_{e_1}(G, x)$ be two $\Sigma^0_{n+1}$ formulas. Define the relation
$$
c \qvdash (\exists x)\Phi_{e_0}(G^0, x) \vee (\exists x)\Phi_{e_1}(G^1, x)
$$
to hold if for every 2-cover $Z^0 \cup Z^1 = X$, there is some side $i < 2$, some finite set $\rho \subseteq Z^i$ and some $x \in \omega$ such that $\sigma^i \cup \rho \nqvdash \neg \Phi_{e_i}(G, x)$ holds.
\end{definition}
\subsection{The complexity aspects of the $\mathbb{Q}_n$ forcing}
This new forcing question now has the right definitional complexity
\begin{lemma}\label{lem:pb2-forcing-question-complexity}
Let $n \in \omega$. Let $c \in \mathbb{Q}_n$ and let $(\exists x) \Phi_{e_0}(G, x)$ and $(\exists x) \Phi_{e_1}(G, x)$ be two $\Sigma^0_{n+1}$ formulas. The relation
$$
c \qvdash (\exists x)\Phi_{e_0}(G^0, x) \vee (\exists x)\Phi_{e_1}(G^1, x)
$$
is $\Sigma^0_1(Y)$ for some $Y \in \mathcal{M}_n$. Moreover an $M_n$-index for $Y$ can be found uniformly in an index for $c$.
\end{lemma}
\begin{proof}
By compactness, for $n = 0$ the relation holds if there is a finite set $E \subseteq X$ such that for every $E_0 \cup E_1 = E$, there is some $i < 2$, some $\rho \subseteq E_i$ and $x_n \in \omega$ such that $\Phi_{e_i}(\sigma^i \cup \rho, x)$ holds, which is a $\Sigma^0_1(X)$ event for $X \in \mathcal{M}_0$.
For $n > 0$ the relation holds if there is a finite set $E \subseteq X$ such that for every $E_0 \cup E_1 = E$, there is some $i < 2$, some $\rho \subseteq E_i$ and $x_n \in \omega$ such that $\sigma^i \cup \rho \nqvdash \neg \Phi_{e_i}(G, x)$ holds.
By \Cref{prop-hyp-effectivalla}, this statement is $\Sigma^0_1(X \oplus C_{n-1} \oplus \halt^{(n)})$ and then $\Sigma^0_1(Y)$ for some $Y \in \mathcal{M}_n$.
\end{proof}
Before we continue, we need to study the effectivness of \Cref{prop-hyp-forcexta} about the forcing question for the $\mathbb{P}_n$ forcing.
\begin{lemma} \label{prop-hyp-forcexta-effect}
Let $n>0$. Let $c \in \mathbb{Q}_{n}$ with $c = (\sigma_0, \sigma_1, X)$. Let $(\exists x)\Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula for $0 \leq m < n$. Let $p = c^{[i]}$ for some $i < 2$ with $p = (\sigma, X)$.
\begin{enumerate}
\item Suppose $p \qvdash (\exists x)\Phi_e(G, x)$. The forcing condition $q \leq p$ of \Cref{prop-hyp-forcexta} which forces $(\exists x)\Phi_e(G, x)$ can always be of the form $(\sigma \cup \tildeau, X \cap Y)$ for $Y \in \mathcal{M}_m$ where $\tildeau$ and an $M_m$-index for $Y$ can be found uniformly in any PA over $\halt^{(n+1)}$. If furthermore $c$ is valid on side $i$ one can ensure $\tildeau \subseteq A^i \cap X$ uniformly in $A \oplus P$ for any $P$ which is PA over $\halt^{(n+1)}$.
\item Suppose $p \nqvdash (\exists x)\Phi_e(G, x)$. The forcing condition $q \leq p$ of \Cref{prop-hyp-forcexta} which forces $(\forall x) \neg \Phi_e(G, x)$ can always be of the form $(\sigma, X \cap Y)$ for $Y \in \mathcal{M}_m$ where an $M_m$ index for $Y$ can be found uniformly in any PA over $\halt^{(n+1)}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Suppose $p \qvdash (\exists x) \Phi_e(G, x)$. By the proof of \Cref{prop-hyp-forcexta} there is $\tildeau \subseteq X$ such that $(\exists x)\Phi_e(\sigma \cup \tildeau, x)$ holds if $m=0$ and such that $\sigma \cup \tildeau \nqvdash \neg \Phi_e(G, x)$ for some $x$ if $m>0$. Note that if $X \cap A^i \in \langle \mathcal{U}_{C_{n}}^{\mathcal{M}_{n}}\rangle$, still refering to the proof of \Cref{prop-hyp-forcexta} we can ensure $\tildeau \subseteq X \cap A^i$. Also finding $\tildeau$ is a $\Sigma^0_1(X)$ event if $m=0$ and a $\Sigma^0_1(X \oplus C_{m-1} \oplus \halt^{m})$ if $m>0$ (resp. a $\Sigma^0_1(X\oplus A)$ if $m=0$ and a $\Sigma^0_1(A \oplus X \oplus C_{m-1} \oplus \halt^{m})$ if $m>0$). As $X \in \mathcal{M}_{n}$ we can then find $\tildeau$ uniformly in $\halt^{n+1}$ (resp. in $\halt^{n+1} \oplus A$).
Suppose now $p \nqvdash (\exists x)\Phi_e(G, x)$. By the proof of \Cref{prop-hyp-forcexta} there is a finite set $F \subseteq C_{m}$ such that $\mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_0}$ is not a largeness class. Note that finding $F$ is a $\Sigma^0_1(\halt^{(m+2)})$ event. It can then be found uniformly in $\halt^{(m+2)}$ and then uniformly in $\halt^{(n+1)}$. Still by the proof of \Cref{prop-hyp-forcexta} there must be some $k$ such that $\mathcal{P}_k$ is not empty where $\mathcal{P}_k$ is the $\Pi^0_1(Z)$ class for some $Z \in \mathcal{M}_m$ of covers $Y_0 \cup \dots \cup Y_{k} \supseteq \omega$ such that $Y_i \notin \mathcal{U}(e, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_m}$ for each $i \leq k$. Searching for the first such $k$ is a $\Sigma^0_1(\halt^{m+1})$ event. Once found, one also compute uniformly in $M_m$ an index for $Y_0 \oplus \dots \oplus Y_{k} \in \mathcal{M}_m \cap \mathcal{P}_k$. As $\langle \mathcal{U}_{C_{n-1}}^{\mathcal{M}_{n-1}}\rangle$ is partition regular and as $X \in \langle \mathcal{U}_{C_{n-1}}^{\mathcal{M}_{n-1}}\rangle$, there is some $i \leq k$ such that $Y_i \cap X \in \langle \mathcal{U}_{C_{n-1}}^{\mathcal{M}_{n-1}}\rangle \subseteq \mathcal{U}_{C_m}^{\mathcal{M}_m}$. Finding the right $Y_i$ for $i \leq k$ is a $\Pi^0_1(C_{n-1} \oplus (Y_i \cap X \oplus M_{n-1})')$ event. As $X \in \mathcal{M}_{n}$ it can then be found in any PA over $\halt^{(n+1)}$.
\end{proof}
We shall now show the extension of \Cref{prop-hyp-forcexta} for the $\mathbb{Q}_n$ forcing conditions.
\begin{lemma}\label{lem:pb2-forcing-question-spec}
Let $n \in \omega$. Let $c \in \mathbb{Q}_n$ and let $(\exists x)\Phi_{e_0}(G, x)$ and $(\exists x)\Phi_{e_1}(G, x)$ be two $\Sigma^0_{n+1}$ formulas.
\begin{itemize}
\item[(a)] If $c \qvdash (\exists x)\Phi_{e_0}(G^0, x) \vee (\exists x)\Phi_{e_1}(G^1, x)$, then there is some $d \leq c$ and some $i < 2$ such that $$d^{[i]} \Vdash (\exists x)\Phi_{e_i}(G^i, x)$$
\item[(b)] If $c \nqvdash (\exists x)\Phi_{e_0}(G^0, x) \vee (\exists x)\Phi_{e_1}(G^1, x)$, then there is some $d \leq c$ and some $i < 2$ such that $$d^{[i]} \Vdash (\forall x)\neg \Phi_{e_i}(G^i, x)$$
\end{itemize}
Moreover an index of $d$ can be found in $A \oplus P$ for any set $P$ which is PA over $\halt^{(n+1)}$ uniformly in an index of $c$, $e_0$ and $e_1$.
\end{lemma}
\begin{proof}
Say $c = (\sigma^0, \sigma^1, X)$. Both (a) and (b) are trivial in the case $n=0$. We treat the case $n > 0$.
(a) Let $Z^0 = X \cap A^0$ and $Z^1 = X \cap A^1$. Unfolding the definition of the forcing question, there is some $i < 2$, some $\rho \subseteq Z^i$ and $x \in \omega$ such that $\sigma_i \cup \rho \nqvdash \neg \Phi_{e_i}(G^i, x)$. By \Cref{prop-hyp-forcexta-effect} we have a set $Y \in \mathcal{M}_{n}$ such that $(\sigma_i \cup \rho, X \cap Y) \leq (\sigma_i \cup \rho, X)$ and $(\sigma_i \cup \rho, X \cap Y) \Vdash (\exists x)\Phi_{e_i}(G^i, x)$. Note that $d = (\sigma_i \cup \rho, \sigma_{i-1}, X \cap Y)$ is a valid extension of $c$. From \Cref{prop-hyp-effectivalla} finding $\rho$ is a $\Sigma^0_1(A \oplus X \oplus C_{n-1} \oplus \halt^{(n)})$ event. From \Cref{prop-hyp-forcexta-effect} one can then find and $M_n$-index of $Y$ in any set $P$ which is PA over $\halt^{(n+1)}$. Overall an index for $d$ can be found in $A \oplus P$ for any set $P$ which is PA over $\halt^{(n+1)}$, uniformly in an index of $c$, $e_0$ and $e_1$.
(b) Let $\mathcal{D}$ be the $\Pi^0_1(\mathcal{M}_n)$ class of all $Z^0 \oplus Z^1$ with $Z^0 \cup Z^1 = X$, such that for every $i < 2$, every $\rho \subseteq Z^i$, and every $x \in \omega$ we have $\sigma \cup \rho \qvdash \neg \Phi_{e_i}(G^i, x)$. Let $Z^0 \oplus Z^1 \in \mathcal{D}$ such that $Z^0 \oplus Z^1 \in \mathcal{M}_n$. Since $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ is a partition regular class containing $X$, there is some $i < 2$ such that $Z^i \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$. Define the $\mathbb{Q}_n$-condition $d = (\sigma^0, \sigma^1, Z^i)$. Then $d^{[i]} \Vdash (\forall x)\neg \Phi_{e_i}(G, x)$. Finding the right $Z_i$ is a $\Pi^0_1(C_{n-1} \oplus (X \oplus Z_i \oplus M_{n-1})')$ event. It can the be found uniformly in any set $P$ which is PA over $\halt^{(n+1)}$. This completes the proof of the lemma.
\end{proof}
\subsection{The degenerate forcing question}
The forcing question will be used with a disjunctive argument. Doing so we will build two generics, one in $A^0$ and one in $A^1$. Possibly only one of them will force every $\Sigma^0_n$ statement or their negation. The challenge is to ensure in the same time that the same generic also forces every $\Sigma^0_m$ statement or their negation for $m < n$, so that we can then apply \Cref{lem:forcing-implytruth} saying that forcing implies truth. It is only possible to do so on side $i$ under the assumption that our current forcing condition is valid on side $i$:
\begin{lemma}
Let $n \geq 0$. Let $c \in \mathbb{Q}_n$ be valid for side $i$. Let $m < n$ and let $(\exists x) \Phi_e(G, x)$ be a $\Sigma^0_{m+1}$ formula. Then one can find uniformly in $A \oplus P$ for any $P$ which is PA over $\halt^{(n+1)}$, a condition $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_e(G, x)$ or $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x)$
\end{lemma}
\begin{proof}
We ask if $(\sigma_i, X) \qvdash (\exists x)\Phi_{e}(G, x)$. From \Cref{prop-hyp-forcexta-effect} if the answer is yes there is a $\tildeau \subseteq X \cap A^i$ and a set $Y \in \mathcal{M}_n$ such that $d = (\sigma_i \cup \tildeau, Y \cap X \cap A^i) \qvdash (\exists x) \Phi_{e}(G, x)$. If no then there is $Y \in \mathcal{M}_n$ such that $d = (\sigma_i \cup \tildeau, Y \cap X \cap A^i) \qvdash (\forall x) \neg \Phi_{e}(G, x)$.
In any case from \Cref{prop-hyp-forcexta-effect} an index for $d$ can be found uniformly in $A \oplus P$ for any $P$ which is PA over $\halt^{(n+1)}$.
\end{proof}
The difficulty is now to make sure that the side $i$ which turns out to be the right one, is also always a valid one. To do so we need a ``degenerate forcing question''.
\begin{definition}
Let $n > 0$. Let $c \in \mathbb{Q}_n$. Let $\mathcal{U}$ be $\Sigma^0_1(\halt^{(n)})$ large open set. Let $(\exists x)\Phi_{e}(G, x)$ be a $\Sigma^0_{n+1}$ formula. We define
$$
c \qvdash^\mathcal{U} (\exists x)\Phi_{e}(G, x)
$$
to hold if for every $Z^0 \cup Z^1 = X$, there exists $i<2$ such that $Z^i \in \mathcal{U}$ and such that there is some $\rho \subseteq Z^i$ and $x_n \in \omega$ for which $\sigma^i \cup \rho \nqvdash \neg \Phi_{e}(G, x)$ holds.
\end{definition}
\begin{lemma} \label{noideaeffploufplouf}
Let $n > 0$. Let $(\sigma_0, \sigma_1, X) \in \mathbb{Q}_n$. Let $\mathcal{U} \supseteq \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ be $\Sigma^0_1(\halt^{(n)})$ large open set such that $X \cap A^{i-1} \notin \mathcal{U}$. Let $(\exists x)\Phi_e(G,x)$ be a $\Sigma^0_{n+1}$ formula. The statement
$$
c \qvdash^\mathcal{U} (\exists x)\Phi_{e}(G, x)
$$
is $\Sigma^0_1(Y)$ for some $Y \in \mathcal{M}_n$. Moreover an $M_n$-index for $Y$ can be found uniformly in an index for $c$.
\end{lemma}
\begin{proof}
The relation holds if there is a finite set $E \subseteq X$ such that for every $E_0 \cup E_1 = E$, there is some $i < 2$, some $\rho \subseteq E_i$ and $x_n \in \omega$ such that $[E_i] \subseteq \mathcal{U}$ and $\sigma^i \cup \rho \nqvdash \neg \Phi_{e_i}(G, x)$ holds.
By \Cref{prop-hyp-effectivalla}, this statement is $\Sigma^0_1(X \oplus C_{n-1} \oplus \halt^{(n)})$ and then $\Sigma^0_1(Y)$ for some $Y \in \mathcal{M}_n$.
\end{proof}
\begin{lemma} \label{noideaeffploufplouf2}
Let $n > 0$. Let $(\sigma_0, \sigma_1, X) \in \mathbb{Q}_n$. Let $\mathcal{U} \supseteq \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ be $\Sigma^0_1(\halt^{(n)})$ large open set such that $X \cap A^{i-1} \notin \mathcal{U}$. Let $(\exists x)\Phi_e(G,x)$ be a $\Sigma^0_{n+1}$ formula.
\begin{itemize}
\item[(a)] Suppose $(\sigma_0, \sigma_1, X) \qvdash^{\mathcal{U}} (\exists x)\Phi_{e}(G, x)$.
\noindent Then there exists $d \leq c$ such that $d^{[i]} \Vdash (\exists x)\Phi_{e}(G, x)$
\item[(b)] Suppose $(\sigma_0, \sigma_1, X) \nqvdash^{\mathcal{U}} (\exists x)\Phi_{e}(G, x)$.
\noindent Then there exists $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_{e}(G, x)$
\end{itemize}
Furthermore an index for $d$ can be found in $A \oplus P$ for any set $P$ which is PA over $\halt^{(n+1)}$, uniformly in an index for $c$.
\end{lemma}
\begin{proof}
Say $c = (\sigma^0, \sigma^1, X)$.
(a) Let $Z^0 = X \cap A^0$ and $Z^1 = X \cap A^1$. Unfolding the definition of the forcing question, there is some $j < 2$ such that $Z_j \in \mathcal{U}$ and such that for some $\rho \subseteq Z^j$ and $x \in \omega$ we have $\sigma_j \cup \rho \nqvdash \neg \Phi_{e_i}(G^i, x)$. By hypothesis $Z^{i-1} \notin \mathcal{U}$. Thus $i=j$ and by \Cref{prop-hyp-forcexta-effect} we have a set $Y \in \mathcal{M}_{n}$ such that $(\sigma_i \cup \rho, X \cap Y) \leq (\sigma_i \cup \rho, X)$ and $(\sigma_i \cup \rho, X \cap Y) \Vdash (\exists x)\Phi_{e_i}(G^i, x)$. Note that $(\sigma_i \cup \rho, \sigma_{i-1}, X \cap Y)$ is a valid extension of $(\sigma_0, \sigma_1, X)$. From \Cref{prop-hyp-effectivalla} finding $\rho$ is a $\Sigma^0_1(A \oplus X \oplus C_{n-1} \oplus \halt^{(n)})$ event. From \Cref{prop-hyp-effectivalla} finding $\rho$ is a $\Sigma^0_1(A \oplus X \oplus C_{n-1} \oplus \halt^{(n)})$ event. From \Cref{prop-hyp-forcexta-effect} one can then find and $M_n$-index of $Y$ in any set $P$ which is PA over $\halt^{(n+1)}$. Overall an index for $d$ can be found in $A \oplus P$ for any set $P$ which is PA over $\halt^{(n+1)}$, uniformly in an index of $c$, $e_0$ and $e_1$.
(b) Let $\mathcal{D}$ be the $\Pi^0_1(\mathcal{M}_n)$ class of all $Z^0 \oplus Z^1$ with $Z^0 \cup Z^1 = X$, such that for every $i < 2$, $Z^i \notin \mathcal{U}$ or for every $\rho \subseteq Z^i$, and every $x \in \omega$ we have $\sigma \cup \rho \qvdash \neg \Phi_{e}(G, x)$. Let $Z^0 \oplus Z^1 \in \mathcal{D}$ be such that $Z^0 \oplus Z^1 \in \mathcal{M}_n$. Since $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ is a partition regular class containing $X$, there is some $i < 2$ such that $Z^i \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$. Since $\mathcal{U} \supseteq \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ we must have $Z^i \in \mathcal{U}$ and thus $d = (\sigma^0, \sigma^1, Z^i)$ is a $\mathbb{Q}_n$ extension of $(\sigma_0, \sigma_1, X)$ such that $d^{[i]} \Vdash (\forall x)\neg \Phi_{e_i}(G, x)$. Finding the right $Z_i$ is a $\Pi^0_1(C_{n-1} \oplus (X \oplus Z_i \oplus M_{n-1})')$ event. It can the be found uniformly in any set $P$ which is PA over $\halt^{(n+1)}$. This completes the proof of the lemma.
\end{proof}
We are now ready to derive our main theorems
\subsection{Preservation of non-$\Sigma^0_n$ definitions}
Our first application shows the existence, for every instance of the pigeonhole principle, of a solution which does not collapse the definition of a non-$\Sigma^0_n$ set into a $\Sigma^0_n$ one. This corresponds to preservation of one non-$\Sigma^0_n$ definition, following the terminology of Wang who showed that given $A$ non $\Sigma^0_n$, any non-empty $\Pi^0_1$ class contains an element $X$ such that $A$ is not $\Sigma^0_n(X)$ ~\cite{Wang2014Definability}.
\begin{theorem}\label{thm:rt12-preservation-non-sigma2}
Fix $n \geq 0$ and let $B$ be a non-$\Sigma^0_{n+1}$ set. For every set $A$, there is an infinite set $G \subseteq A$ or $G \subseteq \overline{A}$ such that $B$ is not $\Sigma^{0}_{n+1}(G)$.
\end{theorem}
\begin{proof}
We let $A^0 = \overline{A}$ and $A^1 = A$. We work with the $\mathbb{Q}_n$ forcing. By Wang~\cite[Theorem 3.6.]{Wang2014Definability}, we can also assume that $B$ is not $\Sigma^0_1(\mathcal{M}_n)$. We also suppose $n>0$, the case $n=0$ was proved by Dzhafarov and Jockusch \cite{Dzhafarov2009Ramseys}.
\tildeextbf{The asymmetric case:}
Suppose first there exists a $\mathbb{Q}_n$-condition $b = (\sigma_0, \sigma_1, X)$ which is invalid for some side $i-1 < 2$. Let $\mathcal{U} \subseteq \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ be a $\Sigma^0_1(\halt^{(n)})$ largeness class such that $X \cap A^i \notin \mathcal{U}$. Note that every condition $c \leq b$ must be valid for side $i$ as otherwise we would have $Y \cap A^0$ and $Y \cap A^1$ both not in $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ for some $Y \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ which would contradict that $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ is partition regular.
We then work below $b = (\sigma_0, \sigma_1, X)$. Given $c \leq b$ and a $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_e(G, x, y)$ with one free variable $y$, the set $\{y\ :\ c \qvdash^\mathcal{U} (\exists x) \Phi_e(G, x, y)\}$ is $\Sigma^0_1(M_n)$ from \Cref{noideaeffploufplouf}. As $B$ is not $\Sigma^0_1(M_n)$ there exists $y \in B$ such that $d \nqvdash^\mathcal{U} (\exists x) \Phi_e(G, x, y)$ or there exists $y \notin B$ such that $c \qvdash^\mathcal{U} (\exists x) \Phi_e(G, x, y)$. In the first case using \Cref{noideaeffploufplouf2} we find an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x, y)$ and in the second case an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_e(G, x, y)$.
Now given $c \leq b$ and a $\Sigma^0_{m+1}$ statement $(\exists x) \Phi_e(G, x)$ for $m < n$ we ask if $c^{[i]} \qvdash (\exists x) \Phi_e(G, x)$. Using \Cref{prop-hyp-forcexta-effect} if the answer is positive we find an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_e(G, x)$ and otherwise we find an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x)$.
In the end we build a $n-1$-generic filter $\mathcal{F} \subseteq \mathbb{P}_{n-1}$ such that $G_\mathcal{F} \subseteq A^i$ with in addition that some $p$ forces $B \neq \{y\ :\ (\exists x) \Phi_e(G, x, y)\}$ for every $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_e(G, x, y)$. By \Cref{lem:forcing-implytruth} we then have that $B \neq \{y\ :\ (\exists x) \Phi_e(G_\mathcal{F}, x, y)\}$ for every $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_e(G, x, y)$. Thus $B$ is not $\Sigma^{0}_{n+1}(G_\mathcal{F})$.\\
\tildeextbf{The symmetric case:}
Suppose now that every $\mathbb{Q}_n$-condition $c = (\sigma_0, \sigma_1, X)$ is valid for both sides. Given a condition $c$ and two $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_{e_0}(G, x, y), (\exists x) \Phi_{e_1}(G, x, y)$ with one free variable $y$, the set $\{y\ :\ c \qvdash (\exists x) \Phi_{e_0}(G, x, y) \vee (\exists x) \Phi_{e_1}(G, x, y)\}$ is $\Sigma^0_1(M_n)$ from \Cref{lem:pb2-forcing-question-complexity}. As $B$ is not $\Sigma^0_1(M_n)$ there exists $y \in B$ such that $c \nqvdash (\exists x) \Phi_{e_0}(G, x, y) \vee (\exists x) \Phi_{e_1}(G, x, y)$ or there exists $y \notin B$ such that $c \qvdash (\exists x) \Phi_{e_0}(G, x, y) \vee (\exists x) \Phi_{e_1}(G, x, y)$. In the first case using \Cref{lem:pb2-forcing-question-spec} we find an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_{e_i}(G, x, y)$ for some $i < 2$ and in the second case an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_{e_i}(G, x, y)$ for some $i < 2$.
Now given $c$ and a $\Sigma^0_{m+1}$ statement $(\exists x) \Phi_e(G, x)$ for $m < n$ we find using \Cref{prop-hyp-forcexta-effect} an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \Phi_e(G, x)$ or $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x)$ for both $i=0$ and $i=1$.
In the end we have one filter $\mathcal{F} \subseteq \mathbb{Q}_n$ giving two filters $\mathcal{F}_0, \mathcal{F}_1 \subseteq \mathbb{P}_{n-1}$ corresponding to side $0$ and $1$, which are both $n-1$-generic and such that $G_{\mathcal{F}_0} \subseteq A^0$ and $G_{\mathcal{F}_1} \subseteq A^1$. Also for every $\Sigma^0_{n+1}$ formulas $(\exists x) \Phi_{e_0}(G, x, y), (\exists x) \Phi_{e_1}(G, x, y)$ we have $d \in \mathcal{F}$ such that $d^{[0]}$ forces $B \neq \{y\ :\ (\exists x) \Phi_{e_0}(G, x, y)\}$ or $d^{[1]}$ forces $B \neq \{y\ :\ (\exists x) \Phi_{e_1}(G, x, y)\}$. By a usual pairing argument, there must be $i<2$ such that for every $\Sigma^0_{n+1}$ formula $(\exists x) \Phi_{e}(G, x, y)$ we have $d \in \mathcal{F}$ such that $d^{[i]}$ forces $B \neq \{y\ :\ (\exists x) \Phi_{e}(G, x, y)\}$. By \Cref{lem:forcing-implytruth} we then have that $B \neq \{y\ :\ (\exists x) \Phi_{e}(G_{\mathcal{F}_i}, x, y)\}$ for every such formula and then that $B$ is not $\Sigma^{0}_{n+1}(G_\mathcal{F})$.
\end{proof}
The following corollary would correspond to strong iterated jump cone avoidance of $\rt^1_2$,
following the terminology of Wang~\cite{Wang2014Some}.
\begin{theoremnonumber}[Main Theorem 1 (Theorem \ref{maintheorem1})]
Fix $n \geq 0$. Let $B$ be non $\emptyset^{(n)}$-computable. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not $H^{(n)}$-computable.
\end{theoremnonumber}
\begin{proof}
Given a set $B$ which is not $\emptyset^{(n)}$-computable, either $B$ or $\overline{B}$ is not $\Sigma^0_{n+1}$. By Theorem~\ref{thm:rt12-preservation-non-sigma2}, for every set $A$, there is an infinite set $H \subseteq A$ or $H \subseteq \overline{A}$ such that either $B$ or $\overline{B}$ is not $\Sigma^{0}_{n+1}(H)$, hence such that $B$ is not $H^{(n)}$-computable.
\end{proof}
\subsection{Preservation of $\Delta^0_n$ hyperimmunities}
Our second application concerns the ability to prevent solutions from computing fast-growing functions. Recall the definition of hyperimmunity.
\begin{definition}
A function $f$ \emph{dominates} a function $g$ if $f(x) \geq g(x)$
for every $x$. A function $f$ is \emph{$X$-hyperimmune} if it is not dominated by any $X$-computable function.
\end{definition}
The following lemma is proven by Downey et al.~\cite[Lemma 3.3]{Downey2019Relationships}.
\begin{lemma}[\cite{Downey2019Relationships}]\label{lem:k-non-ce-to-k-hyperimmune-preservation-1}
For every $k \le \omega$ and every $Z$, for any nondecreasing functions $(f_i)_{i < k}$ which are $Z$-hyperimmune, there is a $G$ and sets $(A_i)_{i < k}$ such that none of the $A_i$ is $\Sigma^0_1(Z\oplus G)$, but for any $i$ and any function $h$ dominating $f_i$, $A_i$ is $\Sigma^0_1(Z\oplus G \oplus h)$.
\end{lemma}
\begin{theorem}\label{thm:rt12-preservation-delta2-hyperimmunity}
Fix a $\emptyset^{(n)}$-hyperimmune function $f$. For every set $A$, there is an infinite set $H \subseteq A$
or $H \subseteq \overline{A}$ such that $f$ is $H^{(n)}$-hyperimmune.
\end{theorem}
\begin{proof}
By Lemma~\ref{lem:k-non-ce-to-k-hyperimmune-preservation-1}, letting $Z = \emptyset^{(n)}$, there is a set $G$ and a set $B$ such that $B$ is not $\Sigma^0_1(\emptyset^{(n)} \oplus G)$ but for any function $h$ dominating $f$, $B$ is $\Sigma^0_1(\emptyset^{(n)} \oplus G \oplus h)$.
By the jump inversion theorem, there is a set $Q$ such that $Q^{(n)} \equiv_T \emptyset^{(n)} \oplus G$.
In particular, $B$ is not $\Sigma^0_1(Q^{(n)})$, so it is not $\Sigma^0_{n+1}(Q)$.
By Theorem~\ref{thm:rt12-preservation-non-sigma2}, there is an infinite set $H \subseteq H$ or $H \subseteq \overline{A}$ such that $B$ is not $\Sigma^0_{n+1}(H \oplus Q)$. In particular $B$ is not $\Sigma^0_1((H \oplus Q)^{(n)})$ and therefore not $\Sigma^0_1(H^{(n)} \oplus G)$. Suppose for the contradiction that $f$ is dominated by an $H^{(n)}$-computable function $h$. Then $B$ is $\Sigma^0_1(\emptyset^{(n)} \oplus G \oplus h)$, hence $B$ is $\Sigma^0_1(H^{(n)} \oplus G)$. Contradiction.
\end{proof}
\subsection{Low${}_n$ solutions}
An effectivization of the forcing construction enables us to obtain lowness results for the infinite pigeonhole principle. The existence of low${}_2$ solutions for $\Delta^0_2$ sets, and of low${}_2$ cohesive sets for computable sequences of sets, was proven by Cholak, Jockusch and Slaman~\cite[sections 4.1 and 4.2]{Cholak2001strength}. The existence of low${}_3$ cohesive sets for $\Delta^0_2$ sequences of sets was proven by Wang~\cite[Theorem 3.4]{Wang2014Cohesive}. Wang~\cite[Questions 6.1 and 6.2]{Wang2014Cohesive} and the second author~\cite[Question 5.4]{Patey2016Open} asked whether such results can be generalized for every $\Delta^0_{n+1}$ instances of the pigeonhole and every $\Delta^0_n$ instances of cohesiveness. We answer positively both questions.
\begin{theorem}\label{thm:rt12-delta3-PA-double-jump-solution}
Let $n \geq 0$. For every $\halt^{(n+1)}$-computable set $A$ and every $P$ PA over $\emptyset^{(n+1)}$, there is an infinite set $G \subseteq A$ or $G \subseteq \overline{A}$ such that $G^{(n+1)} \leq_T P$.
\end{theorem}
\begin{proof}
The case $n = 0$ is proven by Cholak, Jockusch and Slaman~\cite[sections 4.1 and 4.2]{Cholak2001strength}. Suppose $n > 0$. Fix $P$ and $A$, and let $A^0 = \overline{A}$ and $A^1 = A$. We work with the $\mathbb{Q}_n$ forcing. We again have two constructions, based on whether every condition have both valid sides or not.\\
\tildeextbf{Asymmetric case:}
Suppose first there exists a $\mathbb{Q}_n$-condition $b = (\sigma_0, \sigma_1, X)$ which is invalid for some side $i-1 < 2$. Let $\mathcal{U} \subseteq \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ be a $\Sigma^0_1(\halt^{(n)})$ largeness class such that $X \cap A^i \notin \mathcal{U}$. Note that every condition $c \leq b$ must be valid for side $i$ as otherwise we would have $Y \cap A^0$ and $Y \cap A^1$ both not in $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ for some $Y \in \langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ which would contradict that $\langle \mathcal{U}^{\mathcal{M}_{n-1}}_{C_{n-1}} \rangle$ is partition regular. We then work below $b = (\sigma_0, \sigma_1, X)$.
Given $c \leq b$ and a $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_e(G, x, y)$ with one free variable $y$, we ask if $c \qvdash^\mathcal{U} (\exists x) \Phi_e(G, x, y)$. From \Cref{noideaeffploufplouf} we obtain the answer uniformly in $\halt^{(n+1)}$ and thus uniformly in $P$. If the answer is yes, from \Cref{noideaeffploufplouf2} we find uniformly in $P$ an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x, y)$ and in the second case an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_e(G, x, y)$.
Now given $c \leq b$ and a $\Sigma^0_{m+1}$ statement $(\exists x) \Phi_e(G, x)$ for $m < n$ we ask if $c^{[i]} \qvdash (\exists x) \Phi_e(G, x)$. From \Cref{prop-hyp-effectivalla} we obtain the answer uniformly in $\halt^{(n+1)}$ and thus uniformly in $P$. Using \Cref{prop-hyp-forcexta-effect} if the answer is positive we find uniformly in $P$ an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_e(G, x)$ and otherwise we find an extension $d \leq c$ such that $d^{[i]} \Vdash (\forall x) \neg \Phi_e(G, x)$.
In the end we build effectively in $P$ a $n$-generic filter $\mathcal{F} \subseteq \mathbb{P}_{n-1}$ such that $G_\mathcal{F} \subseteq A^i$. Using \cref{lem:forcing-implytruth} and by construction, $P$ can also decide every $\Sigma^0_{n+1}(G_\mathcal{F})$ statement. Thus $(G_\mathcal{F})^{(n+1)} \leq_T P$.
\tildeextbf{The symmetric case:}
Suppose now that every $\mathbb{Q}_n$-condition $c = (\sigma_0, \sigma_1, X)$ is valid for both sides. Given a condition $c$ and two $\Sigma^0_{n+1}$ statement $(\exists x) \Phi_{e_0}(G, x, y), (\exists x) \Phi_{e_1}(G, x, y)$ with one free variable $y$, we ask if $c \qvdash (\exists x) \Phi_{e_0}(G, x, y) \vee (\exists x) \Phi_{e_1}(G, x, y)$. From \Cref{lem:pb2-forcing-question-complexity} we obtain the answer uniformly in $\halt^{(n+1)}$ and then uniformly in $P$. Using \Cref{lem:pb2-forcing-question-spec} we find uniformly in $P$ an extension $d \leq c$ such that $d^{[i]} \Vdash (\exists x) \Phi_{e_i}(G, x, y)$ or $d^{[i]} \Vdash (\forall x) \neg \Phi_{e_i}(G, x, y)$ for some $i<2$.
Now given $c$ and a $\Sigma^0_{m+1}$ statement $(\exists x) \Phi_e(G, x)$ for $m < n$ we ask if $c^{[0]} \qvdash (\exists x) \Phi_e(G, x)$. From \Cref{prop-hyp-effectivalla} we obtain the answer uniformly in $\halt^{(n+1)}$ and thus uniformly in $P$. Using \Cref{prop-hyp-forcexta-effect} if the answer is positive we find uniformly in $P$ an extension $d \leq c$ such that $d^{[0]} \Vdash (\exists x) \Phi_e(G, x)$ and otherwise we find an extension $d \leq c$ such that $d^{[0]} \Vdash (\forall x) \neg \Phi_e(G, x)$. We then ask whether $d^{[1]} \qvdash (\exists x) \Phi_e(G, x)$ and find similarly an extension $h \leq d$ such that $h^{[1]} \Vdash (\forall x) \neg \Phi_e(G, x)$ or $h^{[1]} \qvdash (\exists x) \Phi_e(G, x)$.
In the end we build effectively in $P$ a filter $\mathcal{F} \subseteq \mathbb{Q}_n$ giving two filters $\mathcal{F}_0, \mathcal{F}_1 \subseteq \mathbb{P}_{n-1}$ corresponding to side $0$ and $1$, which are both $n-1$-generic and such that $G_{\mathcal{F}_0} \subseteq A^0$ and $G_{\mathcal{F}_1} \subseteq A^1$. By a pairing argument there must be $i<2$ such that $\mathcal{F}_i$ is $n$-generic. Using \cref{lem:forcing-implytruth} and by construction, $P$ can decide every $\Sigma^0_{n+1}(G_{\mathcal{F}_i})$ statement. Thus $(G_{\mathcal{F}_i})^{(n+1)} \leq_T P$.
\end{proof}
\begin{theoremnonumber}[Main theorem 3 (Theorem \ref{maintheorem3})]
Fix $n \geq 0$. Every $\halt^{(n+1)}$-computable set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ of low${}_{n+2}$ degree.
\end{theoremnonumber}
\begin{proof}
By the relativized low basis theorem~\cite{Jockusch197201}, there is some $P$ PA over $\emptyset^{(n+1)}$ such that $P' \leq_T \emptyset^{(n+2)}$. By Theorem~\ref{thm:rt12-delta3-PA-double-jump-solution}, there is an infinite set $G \subseteq A$ or $G \subseteq \overline{A}$ such that $G^{(n+1)} \leq_T P$. In particular, $G^{(n+2)} \leq_T P' \leq_T \emptyset^{(n+2)}$. Thus $G$ is of low${}_{n+2}$ degree.
\end{proof}
\section{Arithmetic and Hyperarithmetic cone avoidance}\label{sect:hypreduction}
In this section, we extend the jump control of solutions to the pigeonhole principle to ordinal iterations of the jump. We then derive a proof of strong cone avoidance for arithmetic and hyperarithmetic reductions. We prove in the mean time cone avoidance for arithmetical reductions. The reader already familiar with higher recursion theory may jump directly to \cref{sec_strategy_coneavoid} where we give the general strategy which will be used to show hyperarithmetic cone avoidance.
\subsection{Background on higher recursion theory}
\subsubsection{Computable ordinals}
We let $\omega_1^{ck}$ denote the first non-computable ordinal. There is a $\Pi^1_1$ set $\mathcal{O}_1 \subseteq \omega$ such that each $o \in \mathcal{O}_1$ codes for an ordinal $\alpha < \omega_1^{ck}$ and each ordinal $\alpha < \omega_1^{ck}$ has a unique code in $\mathcal{O}_1$. Furthermore given that $o \in \mathcal{O}_1$, one can computably recognize if $o$ codes for $0$, if $o$ codes for a successor ordinal $\alpha+1$, in which case we can uniformly and computably produce a code in $\mathcal{O}_1$ for $\alpha$, and if $o$ codes for a limit ordinal $\sup_n \beta_n$, in which case we can uniformly and computably produce for each $n$ codes in $\mathcal{O}_1$ for $\beta_n$. See \cite{Sacks1990Higher} for more details about $\mathcal{O}_1$. In this section, we manipulate each ordinal $\alpha < \omega_1^{ck}$ via its respective code in $\mathcal{O}_1$. To simplify the reading, we use the notation $\alpha$ instead of the code for $\alpha$.
\subsubsection{The effective Borel sets}
We also use codes for effective Borel subsets of $\omega$ or of $2^\omega$ : For $\alpha < \omega_1^{ck}$ a code for a $\Sigma^0_{\alpha+1}$ set $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{n}$ is the code of a function that effectively enumerate codes for each $\Pi^0_{\alpha}$ set $\mathcal{B}_{n}$. A code for a $\Pi^0_{\alpha+1}$ set $\mathcal{B} = \bigcap_{n < \omega} \mathcal{B}_{n}$ is the code of a function that effectively enumerate codes for each $\Sigma^0_{\alpha}$ set $\mathcal{B}_{n}$. For $\alpha = \sup_n \beta_n$ limit a code of a $\Sigma^0_{\alpha}$ set $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ is the code of a function that effectively enumerate codes for each $\Pi^0_{\beta_n}$ set $\mathcal{B}_{\beta_n}$ with $\sup_n \beta_n = \alpha$. The code of a $\Pi^0_{\alpha}$ set $\mathcal{B} = \bigcap_{n < \omega} \mathcal{B}_{\beta_n}$ is the code of a function that effectively enumerate codes for each $\Sigma^0_{\beta_n}$ set $\mathcal{B}_{\beta_n}$ with $\sup_n \beta_n = \alpha$. We also assume the codes for effective Borel sets include some information so that we can computably distinguish $\Pi^0_\alpha$ from $\Sigma^0_\alpha$ codes as well as distinguish if $\alpha=1$, if $\alpha$ is successor or if it is limit.
\subsubsection{The iterated jumps}
We use such codes to iterate the jump through the ordinals:
\begin{enumerate}
\item $\halt^{(0)} = \emptyset$
\item $\halt^{(\alpha+1)} = (\halt^{(\alpha)})'$
\item $\halt^{(\sup_n \alpha_n)} = \oplus_{n \in \omega} \halt^{(\alpha_n)}$
\end{enumerate}
Note that for $n < \omega$ the set $\halt^{(n)}$ is $\Sigma^0_n$ and complete for $\Sigma^0_n$ questions. Above the first limit ordinal the situation is slightly different : $\halt^{(\omega)}$ is $\Delta^0_{\omega}$ and not $\Sigma^0_{\omega}$. Also given $\alpha \geq \omega$ we have that $\halt^{(\alpha+1)}$ is $\Sigma^0_{\alpha}$ and complete for $\Sigma^0_{\alpha}$ questions.
\begin{proposition} \label{lemma-hyp-eff4}
Let $n \in \omega$.
\begin{enumerate}
\item Let $m > 0$. The set $\{X\ :\ n \in X^{(m)}\}$ is a $\Sigma^0_{m}$ class.
\item Let $\alpha$ be limit. The set $\{X\ :\ n \in X^{(\alpha)}\}$ is a $\Delta^0_{\beta}$ class for some $\beta < \alpha$.
\item Let $\alpha = \beta + 1$ with $\beta \geq \omega$. The set $\{X\ :\ n \in X^{(\alpha)}\}$ is a $\Sigma^0_{\beta}$ class.
\end{enumerate}
\end{proposition}
\begin{proof}
The set $\{X\ :\ n \in X'\}$ is clearly $\Sigma^0_1$. Let $m > 1$. the set $\{X\ :\ n \in X^{(m)}\}$ equals
$$\bigcup_{\{\sigma\ :\ \Phi_n(\sigma, n)\downarrow\}} \bigcap_{\{i\ :\ \sigma(i) = 0\}} \{X\ :\ i \notin X^{(m-1)}\} \cap \bigcap_{\{i\ :\ \sigma(i) = 1\}} \{X\ :\ i \in X^{(m-1)}\}$$
This is by induction a $\Sigma^0_{m}$ set.
Let $\alpha$ be limit. Let $p_1,p_2$ be projections of the pairing function, that is, $x = \langle p_1(x), p_2(x)\rangle$. Then $\{X\ :\ n \in X^{(\alpha)}\}$ equals $\{X\ :\ p_1(n) \in X^{(p_2(n))}\}$, which is a $\Delta^0_\beta$ set for $\beta < \alpha$.
Let $\alpha = \beta+1$. The set $\{X\ :\ n \in X^{(\beta+1)}\}$ equals
$$\bigcup_{\{\sigma\ :\ \Phi_n(\sigma, n)\downarrow\}} \bigcap_{\{i\ :\ \sigma(i) = 0\}} \{X\ :\ i \notin X^{(\beta)}\} \cap \bigcap_{\{i\ :\ \sigma(i) = 1\}} \{X\ :\ i \in X^{(\beta)}\}$$
This is by induction a $\Sigma^0_{\beta}$ class.
\end{proof}
\begin{proposition} \label{lemma-hyp-eff3}
Let $\Phi$ be a functional. Let $n,i \in \omega$.
\begin{enumerate}
\item Let $m > 0$. The set $\{X\ :\ \exists t\ \Phi(X^{(m)}, n)[t]\downarrow = i\}$ is a $\Sigma^0_{m+1}$ class.
\item Let $\alpha \geq \omega$. The set $\{X\ :\ \exists t\ \Phi(X^{(\alpha)}, n)[t]\downarrow = i\}$ is a $\Sigma^0_{\alpha}$ class.
\end{enumerate}
\end{proposition}
\begin{proof}
Trivial using \Cref{lemma-hyp-eff4}
\end{proof}
\subsubsection{$\Pi^1_1$ and $\Sigma^1_1$ sets of integers}
We previously mentioned a $\Pi^1_1$ set $\mathcal{O}_1$ of unique notations for ordinals. This set is included in Kleene's $\mathcal{O}$, the set of all the constructible codes for the computable ordinals. Given an ordinal $\alpha < \omega_1^{ck}$, let $\mathcal{O}_{< \alpha}$ denote the elements of $\mathcal{O}$ which code for an ordinal strictly smaller than $\alpha$. Each $\mathcal{O}_{<\alpha}$ is $\Delta^1_1$ uniformly in $\alpha$ (it actually is always a $\Sigma^0_{\alpha+1}$ set \cite{monin2014higherthesis}). It is well-known that $\mathcal{O}$ is a $\Pi^1_1$-complete set \cite{Sacks1990Higher}, that is, for any $\Pi^1_1$ set $B \subseteq \omega$ there is a computable function $f:\omega \rightarrow \omega$ such that $n \in B \leftrightarrow f(n) \in \mathcal{O}$. Let us define $B_\alpha = \{n\ :\ f(n) \in \mathcal{O}_{<\alpha}\}$. In particular, each $B_{\alpha}$ is $\Delta^1_1$ uniformly in $\alpha$ and $B = \bigcup_{\alpha < \omega_1^{ck}} B_\alpha$. In particular $B$ is a $\Sigma^0_{\omega_1^{ck}}$ set. Note that contrary to $\Sigma^0_\alpha$ sets for $\alpha < \omega_1^{ck}$, the $\Sigma^0_{\omega_1^{ck}}$ are not described with a computable code, but rather with a $\Pi^1_1$ set of codes for all the $\Pi^0_{\alpha}$ that constitutes the $\Sigma^0_{\omega_1^{ck}}$ set $B$. With a little hack, we can even make sure that at most one new element appears in each $B_\alpha$. For this reason, we often see $\Pi^1_1$ sets as enumerable along the computable ordinals.
By complementation a $\Sigma^1_1$ set $B \subseteq \omega$ can be seen as co-enumerable along the computable ordinals and we have $B = \bigcap_{\alpha < \omega_1^{ck}} B_\alpha$ where each $B_\alpha$ is $\Delta^1_1$ uniformly in $\alpha$. We also say in this case that $B$ is $\Pi^0_{\omega_1^{ck}}$.
\subsubsection{$\Sigma^1_1$-boundedness}
A central theorem when working with $\Sigma^1_1$ and $\Pi^1_1$ sets is $\Sigma^1_1$-boundedness:
\begin{theorem}[$\Sigma^1_1$-boundedness \cite{spector1955recursive}]
Let $B$ be a $\Sigma^1_1$ set of codes for ordinals, then the supremum of the ordinals coded by elements of $B$ is strictly smaller than $\omega_1^{ck}$.
\end{theorem}
We mostly here use the following corollary:
\begin{corollary}
Let $f:\omega \tildeo \omega_1^{ck}$ be a total $\Pi^1_1$ function. Then $\sup_n f(n) = \alpha < \omega_1^{ck}$.
\end{corollary}
Note that $f:\omega \tildeo \omega_1^{ck}$ means the range of $f$ is a subset of $\mathcal{O}_1$. The corollary comes from the fact that if $f$ is total, then it becomes $\Delta^1_1$ and its range is then a $\Sigma^1_1$ set of codes for ordinals. As an example we apply here $\Sigma^1_1$-boundedness to show a simple fact that will be needed later : adding an $\omega$-bounded quantifier to a $\Sigma^0_{\omega_1^{ck}}$ or a $\Pi^0_{\omega_1^{ck}}$ set does not change its complexity.
\begin{lemma}
Every $\Sigma^0_{\omega_1^{ck} + 1}$ set of integers is $\Pi^0_{\omega_1^{ck}}$.
\end{lemma}
\begin{proof}
Let $B$ be $\Sigma^0_{\omega_1^{ck} + 1}$, that is, $B = \bigcup_{n \in \omega} \bigcap_{\alpha \in \omega_1^{ck}} B_{n, \alpha}$ where each $B_{n, \alpha}$ is $\Sigma^0_{\alpha}$ uniformly in $\alpha$. Then $B$ is $\Pi^0_{\omega_1^{ck}}$ via the following equality : $\bigcup_{n \in \omega} \bigcap_{\alpha \in \omega_1^{ck}} B_{n, \alpha} = \bigcap_{\alpha \in \omega_1^{ck}} \bigcup_{n \in \omega} \bigcap_{\beta \in \alpha} B_{n, \beta}$.
\end{proof}
It is clear that if $m$ is in the leftmost set it is also in the rightmost set. The reader should have no trouble to apply $\Sigma^1_1$-boundedness to show that if $m$ is not in the leftmost set, then it is not in the rightmost one.
\subsubsection{$\Pi^1_1$ and $\Sigma^1_1$ sets of reals}
Given $X \in 2^{\omega}$ we let $\mathcal{O}^X$ be the set of $X$-constructible codes for $X$-computable ordinals. We let $\omega_1^X \geq \omega_1^{ck}$ be the smallest non $X$-computable ordinal. For $\alpha < \omega_1^X$, we let $\mathcal{O}^X_{<\alpha}$ be the elements of $\mathcal{O}^X$ coding for an ordinal strictly smaller than $\alpha$.
One can show that a set $\mathcal{B} \subseteq 2^{\omega}$ is $\Pi^1_1$ iff there exists some $e \in \omega$ such that $\mathcal{B} = \{X\ :\ e \in \mathcal{O}^X\}$, that is, $\mathcal{B}$ is the set of elements relative to which $e$ codes for an $X$-computable ordinal. In particular, $\mathcal{B} = \bigcup_{\alpha < \omega_1} \{X\ :\ e \in \mathcal{O}_{<\alpha}^X\}$. Note that the union may go up to $\omega_1$, indeed, $\Pi^1_1$ sets of reals are not necessarily Borel.
A $\Pi^1_1$ set of particular interest is the set of element $X$ such that $\omega_1^X > \omega_1^{ck}$. The set is Borel, but not effectively. One can even prove that it contains no non-empty $\Sigma^1_1$ subset : this is known as the Gandy Basis theorem (see Sacks \cite[III.1.5]{Sacks1990Higher}):
\begin{theorem}[Gandy Basis theorem]
Let $\mathcal{B} \subseteq 2^{\omega}$ be a non-empty $\Sigma^1_1$ set. Then there exists $X \in \mathcal{B}$ such that $\omega_1^X = \omega_1^{ck}$.
\end{theorem}
\subsubsection{The general strategy to show hyperarithmetic cone avoidance} \label{sec_strategy_coneavoid}
Let $Z$ be non $\Delta^1_1$. Our goal is to build a generic $G \subseteq A$ or $G \subseteq \omega - A$ such that $Z$ is not $\Delta^1_1(G)$. This is done in two steps: first show that $Z$ is not $G^{(\alpha)}$-computable for any $\alpha < \omega_1^{ck}$ and second show that $\omega_1^{G} = \omega_1^{ck}$, so in particular we cannot have that $Z$ is $G^{(\alpha)}$-computable for $\omega_1^{ck} \leq \alpha < \omega_1^{G}$.
The first part is simply an iteration of the forcing through the computable ordinals, and raises no particular issue. This is done in \Cref{sec-hyp-theforcing}.
The second part is a little bit trickier but still follows a canonical technique, which has often been used, up to some cosmetic changes in its presentation, to show this kind of preservation theorem (see for instance \cite{greenberg2017higher}, \cite{sacks1969measure} or \cite{tanaka1967basis}) : Suppose $\omega_1^G > \omega_1^{ck}$, in particular there is an element $e \in \mathcal{O}^G$ which codes for $\omega_1^{ck}$, that is $e$ is the code of a functional with $\forall n\ \Phi_e(G, n) \downarrow \in \mathcal{O}_{<\omega_1^{ck}}^G$ with $\sup_n |\Phi_e(G,n)| = \omega_1^{ck}$ where $|\Phi_e(G,n)|$ is the ordinal coded by $\Phi_e(G,n)$. All we have to do is to show that such a code $e$ does not exist. Given $e$ we show that one of the following holds:
\begin{enumerate}
\item $\exists n\ \forall \alpha < \omega_1^{ck}\ \Phi_e(G,n) \notin \mathcal{O}^G_{<\alpha}$
\item $\exists \alpha < \omega_1^{ck}\ \forall n\ \Phi_e(G,n) \in \mathcal{O}^G_{<\alpha}$
\end{enumerate}
Each set $\{X\ :\ \Phi_e(X,n) \notin \mathcal{O}^X_{<\alpha}\}$ is $\Delta^1_1$ uniformly in $\alpha$. It follows that the set $\{X\ :\ \exists n\ \forall \alpha < \omega_1^{ck}\ \Phi_e(X,n) \notin \mathcal{O}^X_{<\alpha}\}$ is a $\Sigma^0_{\omega_1^{ck} + 1}$ set of reals. Contrary to $\Sigma^0_{\omega_1^{ck} + 1}$ sets of integers, such sets cannot be simplified. We are then required to extend our forcing questions in order to control the truth of $\Sigma^0_{\omega_1^{ck} + 1}$-statements. This is what will be done in \Cref{sec-hyp-preserv}.
\subsection{Preliminaries}
We now design a notion of forcing for controlling the $\alpha$-jump of solutions
to the pigeonhole principle. Unlike the notion of forcing for controlling finite iterations of the jump, this notion is non-disjunctive and initially fixes the side of the instance $A$ from which we will construct a solution. This is at the cost of a forcing question whose definitional complexity is higher than the question it asks.
\begin{proposition} \label{prop-hyp-scott2}
There is a sequence of sets $\{M_\alpha\}_{\alpha < \omega_1^{ck}}$ such that:
\begin{enumerate}
\item $M_\alpha$ codes for a countable Scott set $\mathcal{M}_\alpha$
\item $\halt^{(\alpha)}$ is uniformly coded by an element of $\mathcal{M}_\alpha$
\item Each $M_\alpha'$ is uniformly computable in $\halt^{(\alpha+1)}$
\end{enumerate}
\end{proposition}
\begin{proof}
In the proof of \Cref{prop-hyp-scott2} we show how to build a functional $\Phi : 2^\omega \rightarrow 2^\omega$ such that for any oracle $X$, we have that $M' = \Phi(X')$ is such that $M = \oplus_{n \in \omega} X_n$ codes for a Scott set $\mathcal{M}$ with $X_0 = X$.
We simply use here this functionnal with any $\halt^{(\alpha+1)}$ for $\alpha < \omega_1^{ck}$.
\end{proof}
Note $\halt^{(\beta)}$ is computable in $\halt^{(\alpha)}$ for $\beta < \alpha$ in a uniform way : there is a unique computable function $f(\halt^{(\alpha)}, \alpha, \beta)$ which outputs $\halt^{(\beta)}$ for every $\beta < \alpha$. Also \Cref{prop-hyp-scott2} implies that $M_\beta$ is computable in $\halt^{(\alpha)}$ for $\beta < \alpha$ and similarly, the computation is uniform in $\beta, \alpha$.
We now turn to an extention of \cref{prop-hyp-cohesiveclassa} to the computable ordinals, for which we reuse \cref{lem-hyp-cohesiveclass1} and \cref{lem-hyp-cohesiveclass2}.
\begin{proposition} \label{prop-hyp-cohesiveclass}
There is a sequence of sets $\{C_\alpha\}_{\alpha < \omega_1^{ck}}$ such that:
\begin{enumerate}
\item $\mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is an $\mathcal{M}_\alpha$-cohesive largeness class
\item $\beta < \alpha$ implies $\mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \langle \mathcal{U}_{C_\beta}^{\mathcal{M}_{\beta}} \rangle$
\item Each $C_\alpha$ is coded by an element of $\mathcal{M}_{\alpha + 1}$ uniformly in $\alpha$ and $M_{\alpha + 1}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Let $X_i^\alpha$ be the element of $\mathcal{M}_\alpha$ of code $i$, so that each $M_\alpha = \oplus_{i} X_i^\alpha$. Let us argue that there is a computable function $f:\omega_1^{ck} \tildeimes \omega_1^{ck} \tildeimes \omega$ such that whenever $\beta < \alpha$, then $X_{i}^\beta = X_{f(\alpha, \beta, i)}^\alpha$: Given an ordinal $\alpha$ the function $f$ considers the $M_\alpha$-code of $\halt^{(\alpha)}$ (which is uniformly coded in $\mathcal{M}_{\alpha}$) and uses it produce an $M_\alpha$-code of $M_\beta = \oplus_{i} X_i^\beta$ (as $M_\beta$ is computable in $\halt^{(\alpha)}$, uniformly in $\beta,\alpha$) and then returns an $M_\alpha$-code of $X_i^\beta$. Given $\alpha < \beta$ and $C \subseteq \omega^2$, we then let $g(\alpha, \beta, C) = \{\langle e, f(\alpha, \beta, i)\rangle : \langle e, i \rangle \in C \}$. In particular, $\mathcal{U}^{\mathcal{M}_{\alpha}}_{g(\alpha, \beta, C)} = \mathcal{U}^{\mathcal{M}_\beta}_C$.
Suppose that stage $\alpha$ we have defined by induction sets $C_\beta$ for each $\beta < \alpha$, verifying $(1) (2)$ and $(3)$. Let us proceed and define $C_\alpha$.
Suppose first that $\alpha = \beta + 1$ is successor. Note that the set $C_{\beta}$ is coded by an element of $\mathcal{M}_{\beta + 1}$ uniformly in $\beta$, and thus that $C_{\beta}$ is uniformly computable in $\halt^{(\beta+2)}$ and then uniformly computable in $M_\beta''$. Using \Cref{lem-hyp-cohesiveclass1} we define $D_\beta \supseteq C_\beta$ to be such that $\mathcal{U}_{D_\beta}^{\mathcal{M}_\beta} = \langle \mathcal{U}_{C_\beta}^{\mathcal{M}_\beta} \rangle$ and such that $D_\beta$ is uniformly $M_{\beta}''$-computable. We define $E_\alpha$ to be $g(\alpha, \beta, D_{\beta})$, so that $\mathcal{U}_{E_\alpha}^{\mathcal{M}_\alpha} = \mathcal{U}_{D_\beta}^{\mathcal{M}_\beta}$. Note that as $E_\alpha$ is uniformly computable in $M_{\beta}''$ and thus in $\halt^{(\alpha+1)}$, it is uniformly coded by an element of $\mathcal{M}_{\alpha+1}$. Note also that $\mathcal{U}_{E_\alpha}^{\mathcal{M}_\alpha}$ is partition regular as it equals $\langle \mathcal{U}_{C_\beta}^{\mathcal{M}_\beta} \rangle$. Using \Cref{lem-hyp-cohesiveclass2} we uniformly find an $\mathcal{M}_{\alpha+1}$-index of $C_\alpha \supseteq E_\alpha$ to be such that $\mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is an $\mathcal{M}_\alpha$-cohesive largeness class.
At limit stage $\alpha = \sup_n \beta_n$, each set $C_{\beta_n}$ is coded by an element of $\mathcal{M}_{\beta_n + 1}$ uniformly in $\beta_n$ and that $\mathcal{M}_{\beta_n + 1}$ is uniformly computable in $\halt^{(\alpha)}$. It follows that $\bigcup_{n} C_{\beta_n}$ is uniformly computable in $\halt^{(\alpha)}$. We define $D_\alpha$ to be $\bigcup_n g(\alpha, \beta_n, C_{\beta_n})$. Note that $D_\alpha$ is uniformly computable in $\halt^{(\alpha)}$ and thus coded by an element of $\mathcal{M}_\alpha$ uniformly in $\alpha$. Note also that $\mathcal{U}_{D_\alpha}^{M_\alpha} = \bigcap_{n \in \omega} \mathcal{U}_{C_{\beta_n}}^{M_{\beta_n}} = \bigcap_{n \in \omega} \langle \mathcal{U}_{C_{\beta_n}}^{M_{\beta_n}} \rangle$. As an intersection of partition regular class, $\mathcal{U}_{D_\alpha}^{M_\alpha}$ is partition regular. Using \Cref{lem-hyp-cohesiveclass2} there is a set $C_\alpha \supseteq D_\alpha$ such that $\mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha}$ is $\mathcal{M}_\alpha$-cohesive and such that $C_\alpha$ is uniformly coded by an element of $\mathcal{M}_{\alpha+1}$.
\end{proof}
\subsection{The forcing} \label{sec-hyp-theforcing}
From now on, fix sequences $\{\mathcal{M}_\alpha\}_{\alpha < \omega_1^{ck}}$ and $\{C_\alpha\}_{\alpha < \omega_1^{ck}}$ which verify \Cref{prop-hyp-scott2} and \Cref{prop-hyp-cohesiveclass}, respectively. Assume also that we have a class $\mathcal{S} \subseteq \bigcap_{\beta < \omega_1^{ck}} \mathcal{U}_{C_\beta}^{\mathcal{M}_{\beta}}$ which is partition regular and that will be detailed later.
Let $A^0 \cup A^1 = \omega$. Note that there must be $i< 2$ such that $A^i \in \mathcal{S}$. Let then $A = A^i$ for some $i$ such that $A^i \in \mathcal{S}$.
\begin{definition}
Let $\mathbb{P}_{\omega_1^{ck}}$ be the set of conditions $(\sigma, X)$ such that:
\begin{enumerate}
\item $\sigma \subseteq A$
\item $X \subseteq A$
\item $X \cap \{0, \dots, |\sigma|\} = \emptyset$.
\item $X \in \mathcal{S}$
\end{enumerate}
Given two conditions $(\sigma, X), (\tildeau, Y) \in \mathbb{P}_{\omega_1^{ck}}$ we let $(\sigma, X) \leq (\tildeau, Y)$ be the usual Mathias extension, that is, $\sigma \succeq \tildeau$, $X \subseteq Y$ and $\sigma - \tildeau \subseteq Y$.
\end{definition}
We now define an abstract forcing question for $\Sigma^0_\alpha$ sets, which is merely an extension of the forcing question of the $\mathbb{P}_n$ forcing for $\Sigma^0_{n+1}$ sets : when $\alpha < \omega$, the definition below is merely a reformulation of \Cref{def-hyp-forcingqua} with the use of effective Borel sets instead of formulas.
\begin{definition} \label{def-hyp-forcingqu}
Let $\sigma \in 2^{<\omega}$. Given a $\Sigma^0_1$ class $\mathcal{U}$, let $\sigma \qvdash \mathcal{U}$ hold if
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ [\sigma \cup \tildeau] \subseteq \mathcal{U}\} \cap \mathcal{U}_{C_{0}}^{\mathcal{M}_{0}}$$
is a largeness class. Then inductively, given a $\Sigma^0_m$ class $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{n}$ with $1 < m < \omega$, we let $\sigma \qvdash \mathcal{B}$ hold if
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n}\} \cap \mathcal{U}_{C_{m-1}}^{\mathcal{M}_{m-1}}$$
is a largeness class. Then inductively, given a $\Sigma^0_\alpha$ class $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ with $\omega \leq \alpha < \omega_1^{ck}$, we define $\sigma \qvdash \mathcal{B}$ if
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\beta_n}\} \cap \mathcal{U}_{C_{\alpha}}^{\mathcal{M}_{\alpha}}$$
is a largeness class.
For a condition $p = (\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$ and an effectively Borel set $\mathcal{B}$, we write $p \qvdash \mathcal{B}$ if $\sigma \qvdash \mathcal{B}$.
\end{definition}
We shall now study the effectivity of the relation $\qvdash$. To do so we introduce the following notation.
\begin{definition}
Let $\sigma \in 2^{<\omega}$. Given a $\Sigma^0_1$ class $\mathcal{B}$, we write $\mathcal{U}(\mathcal{B}, \sigma)$ for the open set:
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ [\sigma \cup \tildeau] \subseteq \mathcal{B}\}$$
Given a $\Sigma^0_\alpha$ class $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ for $1 < \alpha < \omega_1^{ck}$ we write $\mathcal{U}(\mathcal{B}, \sigma)$ for the open set:
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\beta_n}\}$$
\end{definition}
\Cref{prop-hyp-effectivalla} settled the complexity of the relation $\qvdash$ by showing that it is $\Pi^0_{1}(C_{m-1} \oplus \halt^{(m)})$ for a $\Sigma^0_m$ class. We extend here the proposition for $\Sigma^0_\alpha$ classes. Note that in the following one might have the false impression that we loose one jump compare to \cref{prop-hyp-effectivalla}. This is due to the fact that for $\alpha \geq \omega$ the $\Sigma^0_\alpha$-complete set is $\halt^{(\alpha+1)}$ and not $\halt^{(\alpha)}$.
\begin{proposition} \label{prop-hyp-effectivall}
Let $\sigma \in 2^{<\omega}$.
\begin{enumerate}
\item Let $\mathcal{B}$ be a $\Sigma^0_m$ class for $0 < m < \omega$
\begin{enumerate}
\item The set $\mathcal{U}(\mathcal{B}, \sigma)$ is an upward-closed $\Sigma^0_1(C_{m-2} \oplus \halt^{(m-1)})$ open set if $m>1$ and an upward-closed $\Sigma^0_1$ open set if $m = 1$.
\item The relation $\sigma \qvdash \mathcal{B}$ is $\Pi^0_1(C_{m-1} \oplus \halt^{(m)})$.
\end{enumerate}
\item Let $\mathcal{B}$ be a $\Sigma^0_\alpha$ class for $\alpha \geq \omega$.
\begin{enumerate}
\item The set $\mathcal{U}(\mathcal{B}, \sigma)$ is an upward closed $\Sigma^0_1(C_{\alpha-1} \oplus \halt^{(\alpha)})$ open set if $\alpha$ is successor and an upward closed $\Sigma^0_1(\halt^{(\alpha)})$ open set if $\alpha$ is limit.
\item The relation $\sigma \qvdash \mathcal{B}$ is $\Pi^0_1(C_\alpha \oplus \halt^{(\alpha+1)})$.
\end{enumerate}
\end{enumerate}
This is uniform in $\sigma$ and a code for the class $\mathcal{B}$.
\end{proposition}
\begin{proof}
(1) was already proved in \Cref{prop-hyp-effectivalla}. We then only prove (2). This is done by induction on the effective Borel codes. Let $\omega \leq \alpha < \omega_1^{ck}$. Suppose (a) and (b) are true for any $\omega \leq \beta < \alpha$. Let $\sigma \in 2^{<\omega}$ and let $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ be a $\Sigma^0_\alpha$ class. Let
$$\mathcal{U}(\mathcal{B}, \sigma) = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\beta_n}\}$$
Let us show (a). Suppose first $\alpha$ is limit. For each $n \in \omega$, the class $2^\omega - \mathcal{B}_{\beta_n}$ is a $\Sigma^0_{\beta_n}$ class uniformly in $\sigma \cup \tildeau$ and in a code for $\mathcal{B}_{\beta_n}$. By induction hypothesis, or by \cref{prop-hyp-effectivalla} in case $\alpha = \omega$, the relation $\sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\beta_n}$ is, in any case, $\Sigma^0_1(\halt^{(\beta_n+2)})$ and thus $\Sigma^0_1(\halt^{(\alpha)})$. It follows that $\mathcal{U}(\mathcal{B}, \sigma)$ is an upward-closed $\Sigma^0_1(\halt^{(\alpha)})$ open set.
Suppose now $\alpha \geq \omega$ with $\alpha = \beta+1$. For each $n$ we have that $2^\omega - \mathcal{B}_{\beta_n}$ is a $\Sigma^0_{\beta}$ class uniformly in $n$. By induction hypothesis, the relation $\sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\beta_n}$ is $\Sigma^0_1(C_\beta \oplus \halt^{(\beta+1)})$. It follows that $\mathcal{U}(\mathcal{B}, \sigma)$ is an upward closed $\Sigma^0_1(C_{\alpha-1} \oplus \halt^{(\alpha)})$ class.
Let us now show (b). Suppose $\alpha \geq \omega$ successor or limit. Then $\mathcal{U}(\mathcal{B}, \sigma) \cap \mathcal{U}_{C_{\alpha}}^{\mathcal{M}_{\alpha}}$ is a largeness class if for all $F \subseteq C_{\alpha}$, the class $\mathcal{U}(\mathcal{B}, \sigma) \cap \mathcal{U}_{F}^{\mathcal{M}_{\alpha}}$ is a largeness class. It is a $\Pi^0_2(M_\alpha)$ statement uniformly in $F$ and then a $\Pi^0_1(M_\alpha')$ statement uniformly in $F$ and then a $\Pi^0_1(\halt^{(\alpha+1)})$ statement uniformly in $F$. It follows that the statement $\mathcal{U}(\mathcal{B}, \sigma) \cap \mathcal{U}_{C_{\alpha}}^{\mathcal{M}_{\alpha}}$ is a largeness class is $\Pi^0_1(C_{\alpha} \oplus \halt^{(\alpha+1)})$.
\end{proof}
We finally extend the forcing relation of \Cref{def:qb2-forcing-relation} to the transfinite.
\begin{definition}
Let $(\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$. Let $\mathcal{U}$ be a $\Sigma^0_1$ class. We define
$$
\begin{array}{rcccl}
(\sigma, X)&\Vdash&\mathcal{U}&\leftrightarrow&[\sigma] \subseteq \mathcal{U}\\
(\sigma, X)&\Vdash&2^\omega - \mathcal{U}&\leftrightarrow&\forall \tildeau \subseteq X\ [\sigma \cup \tildeau] \nsubseteq \mathcal{U}\\
\end{array}
$$
Then inductively for $\Sigma^0_\alpha$ classes $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$, we define:
$$
\begin{array}{rcccl}
(\sigma, X)&\Vdash&\mathcal{B}&\leftrightarrow&\exists n\ (\sigma, X) \Vdash \mathcal{B}_{\beta_n}\\
(\sigma, X)&\Vdash&2^\omega - \mathcal{B}&\leftrightarrow&\forall n\ \forall \tildeau \subseteq X\ \sigma \cup \tildeau \qvdash 2^\omega - \mathcal{B}_{\beta_n}\\
\end{array}
$$
\end{definition}
Note that the relation $\Vdash$ does not change compare to the arithmetical case : the definition goes through exactly the same way in the transfinite. It is the same for the relation $\qvdash$. For these reasons the following lemmas and propositions and theorems are all proved exactly the same way as for the arithmetical case, only now our set $\mathcal{S}$ is included in $\bigcap_{\beta < \omega_1^{ck}} \mathcal{U}_{C_\beta}^{\mathcal{M}_{\beta}}$ and not just in $\bigcap_{m < \omega} \mathcal{U}_{C_m}^{\mathcal{M}_{m}}$.
\begin{lemma} \label{lem-hyp-forcepi}
Let $p \in \mathbb{P}_{\omega_1^{ck}}$. Let $\mathcal{B} = \bigcap_{n < \omega} \mathcal{B}_{\beta_n}$ be a $\Pi^0_\alpha$ class. Then $p \Vdash \bigcap_{n < \omega} \mathcal{B}_{\beta_n}$ iff for every $n \in \omega$ and every $q \leq p$, $q \qvdash \mathcal{B}_{\beta_n}$.
\end{lemma}
\begin{proof}
Same as \Cref{lem-hyp-forcepia}.
\end{proof}
\begin{proposition}
Let $p \in \mathbb{P}_{\omega_1^{ck}}$. Let $\mathcal{B}$ be an effectively Borel set. If $p \Vdash \mathcal{B}$ and $q \leq p$ then $q \Vdash \mathcal{B}$.
\end{proposition}
\begin{proof}
Same as \Cref{lem:qb2-forcing-closed-under-extension}.
\end{proof}
\begin{proposition} \label{prop-hyp-forcext}
Let $p \in \mathbb{P}_{\omega_1^{ck}}$. Let $\mathcal{B} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ be a $\Sigma^0_\alpha$ class for $0 < \alpha < \omega_1^{ck}$.
\begin{enumerate}
\item Suppose $p \qvdash \mathcal{B}$. Then there exists $q \leq p$ such that $q \Vdash \mathcal{B}$.
\item Suppose $p \nqvdash \mathcal{B}$. Then there exists $q \leq p$ such that $q \Vdash 2^\omega - \mathcal{B}$.
\end{enumerate}
\end{proposition}
\begin{proof}
Same as \Cref{prop-hyp-forcexta}.
\end{proof}
\begin{definition}
Let $\mathcal{F} \subseteq \mathbb{P}_{\omega_1^{ck}}$ be a sufficiently generic filter. Then there is a unique set $G_\mathcal{F} \in 2^{\omega}$ such that for every $(\sigma, X) \in \mathcal{F}$ we have $\sigma \prec G_\mathcal{F}$.
\end{definition}
\begin{theorem} \label{th-hyp-forcingimpliestruth}
Let $\mathcal{F} \subseteq \mathbb{P}_{\omega_1^{ck}}$ be a generic enough filter. Let $p \in \mathcal{F}$. Let $\mathcal{B}_{\alpha} = \bigcup_{n < \omega} \mathcal{B}_{\beta_n}$ be a $\Sigma^0_\alpha$ class for $0 < \alpha < \omega_1^{ck}$. Suppose $p \Vdash \mathcal{B}_{\alpha}$. Then $G_\mathcal{F} \in \mathcal{B}_{\alpha}$. Suppose $p \Vdash 2^\omega - \mathcal{B}_{\alpha}$. Then $G_\mathcal{F} \in 2^\omega - \mathcal{B}_{\alpha}$.
\end{theorem}
\begin{proof}
Same as \Cref{lem:forcing-implytruth}.
\end{proof}
We now have all the necessary parts to show arithmetic strong cone avoidance, and more generally $\alpha$ cone avoidance for a limit ordinal $\alpha$.
\begin{theorem} \label{th-alphaconeavoid}
Let $\alpha \leq \omega_1^{ck}$ be a limit ordinal. Suppose $Z$ is not $\Delta^0_1(\halt^{(\beta)})$ for every $\beta < \alpha$. Let $\mathcal{F}$ be a sufficiently generic filter. Then for every $\beta < \alpha$, $Z$ is not $\Delta^0_1(G_{\mathcal{F}}^{(\beta)})$.
\end{theorem}
\begin{proof}
Let $\Phi$ be a functional and $\beta < \alpha$. Let $\mathcal{B}^{n} = \{X\ : \Phi(X^{(\beta)}, n) \downarrow\}$. We want to show that $Z \neq \{n\ : G_\mathcal{F}^{(\beta)} \in \mathcal{B}^{n}\}$. From \Cref{lemma-hyp-eff3}, $\mathcal{B}^{n}$ is a $\Sigma^0_{\beta+1}$ set for each $n \in \omega$ ($\Sigma^0_{\beta}$ if $\beta \geq \omega$ and $\Sigma^0_{\beta+1}$ if $\beta < \omega$).
Let $p \in \mathbb{P}_{\omega_1^{ck}}$ be a condition. From \Cref{prop-hyp-effectivall}, the set $\{n\ :\ p \qvdash \mathcal{B}^{n}\}$ is $\Pi^0_{1}(\halt^{(\beta+3)})$. As $Z$ is not $\Pi^0_{1}(\halt^{(\beta+3)})$, then there is some $n \in Z$ such that $p \nqvdash \mathcal{B}^{n}$ or some $n \notin Z$ such that $p \qvdash \mathcal{B}^{n}$. In the first case, there is an extension $q \leq p$ such that $q \Vdash 2^\omega - \mathcal{B}^{n}$ for some $n \in Z$. In the second case, there is an extension $q \leq p$ such that $q \Vdash \mathcal{B}^{n}$ for some $n \notin Z$. By \Cref{th-hyp-forcingimpliestruth}, in the first case $\Phi(G_{\mathcal{F}}^{(\beta)}, n) \uparrow$ holds for some $n \in Z$, and in the second case, $\Phi(G_{\mathcal{F}}^{(\beta)}, n) \downarrow$ holds for some $n \notin Z$.
If $\mathcal{F}$ is sufficiently generic, this is true for any $\beta < \alpha$ and any functional $\Phi$. It follows that for any ordinal $\beta$ the set $Z$ is not $\Sigma^0_1(G_{\mathcal{F}}^{(\beta)})$ and thus not $\Delta^0_1(G_{\mathcal{F}}^{(\beta)})$.
\end{proof}
This shows in particular cone avoidance for arithmetic degrees.
\begin{theoremnonumber}[Main theorem 2 (Theorem \ref{maintheorem2})]
Let $B$ be non arithmetical. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not arithmetical in $H$.
\end{theoremnonumber}
\begin{proof}
A direct corollary of the above theorem with $\alpha = \omega$.
\end{proof}
In order to show cone avoidance for hyperarithmetic degrees, one should additionally argue that if $\mathcal{F}$ is sufficiently generic, then $\omega_1^{G_{\mathcal{F}}} = \omega_1^{ck}$. The remainder of this section is devoted to the proof of this fact.
\subsection{Preservation of hyperarithmetic reductions} \label{sec-hyp-preserv}
We now prove that the infinite pigeonhole principle admits strong cone avoidance for hyperarithmetic reductions.
\begin{definition}
A largeness class $\mathcal{A}$ is $\Gamma$-minimal, where $\Gamma$ is a class of complexity, if for every $\Gamma$-open set $\mathcal{U}$ we have $\mathcal{A} \cap \mathcal{U}$ large implies $\mathcal{A} \subseteq \mathcal{U}$.
\end{definition}
\begin{proposition} \label{prop-hyp-delta-min}
The class $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is $\Delta^1_1$-minimal.
\end{proposition}
\begin{proof}
For every $\alpha < \omega_1^{ck}$ we have that $\halt^{(\alpha)} \in \mathcal{M}_{\alpha}$ and $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \langle \mathcal{M}_{\alpha} \rangle$ where $\langle \mathcal{M}_{\alpha} \rangle$ is $\mathcal{M}_\alpha$-minimal. As $\halt^{(\alpha)} \in \mathcal{M}_{\alpha}$ we also have that $\langle \mathcal{M}_{\alpha} \rangle$ is minimal for $\Sigma^0_1(\halt^{(\alpha)})$ open sets. It follows that $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is $\Delta^1_1$-minimal.
\end{proof}
\begin{proposition}
There is a set $C \in \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ such that $C$ is $\Delta^1_1$-cohesive and $\omega_1^C = \omega_1^{ck}$
\end{proposition}
\begin{proof}
Let us argue that for any upward closed partition regular class $\bigcap_{n < \omega} \mathcal{U}_n$ where each $\mathcal{U}_n$ is open, not necessarily effectively of uniformly, there is a $\Delta^1_1$-cohesive $C$ in $\bigcap_{n < \omega} \mathcal{U}_n$. This is done by Mathias forcing with conditions $(\sigma, X)$ such that $X \cap \{0, \dots, |\sigma|\} = \emptyset$ and such that $X$ is $\Delta^1_1$ with $X \in \bigcap_{n < \omega} \mathcal{U}_n$. Given a condition $(\sigma, X)$ and $n$ we can force the generic to be in $\mathcal{U}_n$ as follows : As $X \in \mathcal{U}_{n}$ we must have that $\sigma \cup X \in \mathcal{U}_{n}$ because $\mathcal{U}_n$ is upward closed. Thus there must be $\tildeau \subseteq X \cap \{0, \dots, |\sigma|\}$ such that $[\sigma \cup \tildeau] \subseteq \mathcal{U}_{n}$. As $\bigcap_{n < \omega} \mathcal{U}_n$ contains only infinite set we must have $X - \{0, \dots, \sigma \cup \tildeau\} \in \bigcap_{n < \omega} \mathcal{U}_n$. Thus $(\sigma \cup \tildeau, X - \{0, \dots, \sigma \cup \tildeau\})$ is a valid extension. Let now $Y$ be $\Delta^1_1$. We can force the generic to be included in $Y$ or $\omega - Y$ up to finitely many elements as follow : We have $X \cap Y \in \bigcap_{n < \omega} \mathcal{U}_n$ or $X \cap (\omega - Y) \in \bigcap_{n < \omega} \mathcal{U}_n$. Then $(\sigma, X \cap Y)$ or $(\sigma, X \cap (\omega - Y))$ is a valid extension.
We have that the set $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is a $\Sigma^1_1$ class which is also upward closed and partition regular. We also have that the class of $\Delta^1_1$-cohesive sets is a $\Sigma^1_1$ class. By the previous argument their intersection is non-empty. By the $\Sigma^1_1$-basis theorem it must contains $C$ with $\omega_1^C = \omega_1^{ck}$.
\end{proof}
\begin{lemma} \label{hyp-lemma-minC}
Suppose $C$ is $\Delta^1_1$-cohesive with $C \in \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$. Let $\mathcal{U}$ be a $\Delta^1_1$ open set. If $\mathcal{L}_C \cap \mathcal{U}$ is a largeness class, then $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \mathcal{U}$
\end{lemma}
\begin{proof}
Suppose $\mathcal{L}_C \cap \mathcal{U}$ is a largeness class. Let us show that $\mathcal{U} \cap \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is a largeness class. Suppose first for contradiction that it is not. Then there is a $\Delta^1_1$ cover $Y_0 \cup \dots \cup Y_{k-1} \supseteq \omega$ together with a $\Delta^1_1$ open largeness class $\mathcal{V} \supseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ such that $Y_i \notin \mathcal{U} \cap \mathcal{V}$ for every $i < k$. As each $Y_i$ is $\Delta^1_1$, there is some $i < k$ such that $C \subseteq^* Y_i$. Note also that since $C \in \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$, then $C \in \mathcal{L}(\mathcal{V})$ and thus $\mathcal{L}_C \cap \mathcal{V}$ is a largeness class. It follows that $Y_j \in \mathcal{L}_C \cap \mathcal{V}$ for some $j < k$. As $j \neq i$ implies $|Y_j \cap C| < \infty$, then $Y_i \in \mathcal{L}_C \cap \mathcal{V}$ and thus $Y_i \in \mathcal{V}$. As $\mathcal{L}_C \cap \mathcal{U}$ is a largeness class then by a similar argument, $Y_i \in \mathcal{L}_C \cap \mathcal{U}$ and thus $Y_i \in \mathcal{U}$. It follows that $Y_i \in \mathcal{U} \cap \mathcal{V}$, contradicting our hypothesis. Thus $\mathcal{U} \cap \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is a largeness class.
Now from \Cref{prop-hyp-delta-min} we have that $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}}$ is minimal for $\Delta^1_1$ open sets, then $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \mathcal{U}$.
\end{proof}
\begin{definition}
Let $\mathcal{B} = \bigcup_{\alpha < \omega_1^{ck}} \mathcal{B}_{\alpha}$ be a $\Sigma^0_{\omega_1^{ck}}$ class. Let $p = (\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$. We define $p \qvdash \mathcal{B}$ if the set
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists \alpha < \omega_1^{ck}\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\alpha} \} \cap \mathcal{L}_C$$
is a largeness class.
\end{definition}
Given a $\Sigma^0_{\omega_1^{ck}}$ class $\mathcal{B} = \bigcup_{\alpha < \omega_1^{ck}} \mathcal{B}_{\alpha}$ the following set
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists \alpha < \omega_1^{ck}\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\alpha} \}$$
is a $\Pi^1_1$ open set, that is an open set $\bigcup_{\sigma \in B} [\sigma]$ where $B = \bigcup_{\alpha < \omega_1^{ck}} B_\alpha$ is a $\Pi^1_1$ set of strings. We also suppose that each $B_\alpha$ is $\halt^{(\alpha)}$-computable and that $\{B_\alpha\}_{\alpha <\omega_1^{ck}}$ is increasing. Given such sets we write $\mathcal{U}_\alpha$ for the $\Delta^1_1$ open set $\bigcup_{\sigma \in B_\alpha} [\sigma]$.
\begin{proposition}
Let $\mathcal{U}$ be an upward-closed $\Pi^1_1$ open set. The class $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class iff there exists some $\alpha < \omega_1^{ck}$ such that $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is a largeness class.
\end{proposition}
\begin{proof}
Suppose $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is a largeness class. Then clearly $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class. Suppose now that $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class. For each $n$ let $\mathcal{U}_n^C$ be the $\Sigma^0_1(C)$ open set such that $\mathcal{L}_C = \bigcap_{n} \mathcal{U}_n^C$. We have
$$\forall n\ \forall k\ \exists \alpha\ \forall Y_0 \cup \dots \cup Y_{k-1}\ \exists i < k\ \exists \sigma \subseteq Y_i\ [\sigma] \subseteq \mathcal{U}_\alpha \cap \mathcal{U}_n^C$$
Note that given $k$ and $\alpha$ the predicate $P^{n, k}_\alpha \equiv \forall Y_0 \cup \dots \cup Y_{k-1}\ \exists i < k\ \exists \sigma \subseteq Y_i\ [\sigma] \subseteq \mathcal{U}_\alpha \cap \mathcal{U}_n^C$ is $\Sigma^0_1(C \oplus \halt^{(\alpha+1)})$ uniformly in $n,k$ and $\alpha$. Thus the function $f:\omega^2 \rightarrow \omega_1^{ck}$ which to $n,k$ associates the smallest $\alpha$ such that $P^{n,k}_\alpha$ is true is a total $\Pi^1_1(C)$ function. By $\Sigma^1_1$-boundedness we have $\beta = \sup_{n,k} f(n,k) < \omega_1^C = \omega_1^{ck}$. It follows that
$$\forall n\ \forall k\ \forall Y_0 \cup \dots \cup Y_{k-1}\ \exists i < k\ \exists \sigma \subseteq Y_i\ [\sigma] \subseteq \mathcal{U}_\beta \cap \mathcal{U}_n^C$$
Also $\mathcal{U}_\beta \subseteq \mathcal{U}$ is such that $\mathcal{U}_\beta \cap \mathcal{L}_C$ is a largeness class.
\end{proof}
\begin{corollary} \label{cor-hyp-forcepii}
Let $\mathcal{B} = \bigcup_{\alpha < \omega_1^{ck}} \mathcal{B}_{\alpha}$ be a $\Sigma^0_{\omega_1^{ck}}$ class. Let $(\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$. The relation $p \qvdash \mathcal{B}$ is $\Sigma^0_{\omega_1^{ck}}(C)$
\end{corollary}
\begin{proof}
The relation $p \qvdash \mathcal{B}$ is equivalent to
$$\exists \alpha < \omega_1^{ck}\ \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{\alpha} \} \cap \mathcal{L}_C$$
is a largeness class
\end{proof}
\begin{corollary} \label{hyp-corpi11accclass}
The class $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha}$ is minimal for $\Pi^1_1$ open sets $\mathcal{U}$ such that $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class.
\end{corollary}
\begin{proof}
Given a $\Pi^1_1$-open set $\mathcal{U}$ such that $\mathcal{U} \cap \mathcal{L}_C$, there must be $\alpha < \omega_1^{ck}$ such that $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is a largeness class. By \Cref{hyp-lemma-minC} it must be that $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \mathcal{U}_\alpha$.
\end{proof}
\begin{definition}
Let $\mathcal{B} = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{\alpha}$ be a $\Pi^0_{\omega_1^{ck}}$ class. Let $p = (\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$. We define $p \Vdash \mathcal{B}$ if for every $\tildeau \subseteq X - \{0, \dots, |\sigma|\}$ and for every $\alpha < \omega_1^{ck}$ we have $\sigma \cup \tildeau \qvdash \mathcal{B}_{\alpha}$
\end{definition}
\begin{proposition} \label{hyp-sideprop-truth}
Let $\mathcal{B} = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{\alpha}$ be a $\Pi^0_{\omega_1^{ck}}$ class. Let $\mathcal{F}$ be sufficiently generic with $p \in \mathcal{F}$. If $p \Vdash \mathcal{B}$, then $G_{\mathcal{F}} \in \mathcal{B}$.
\end{proposition}
\begin{proof}
Using \Cref{prop-hyp-forcext}, for every $\alpha$ and every $q \leq p$, there is some $r \leq q$ such that $r \Vdash \mathcal{B}_{\alpha}$. Thus for every $\alpha$ the set $\{r\ :\ r \Vdash \mathcal{B}_{\alpha}\}$ is dense below $p$. It follows from \Cref{th-hyp-forcingimpliestruth} that if $\mathcal{F}$ is sufficiently generic, $G_{\mathcal{F}} \in \mathcal{B}$.
\end{proof}
\begin{definition}
Let $\mathcal{B} = \bigcup_{n \in \omega} \mathcal{B}_{n}$ be a $\Sigma^0_{\omega_1^{ck}+1}$ class where each $\Pi^0_{\omega_1^{ck}}$ set $\mathcal{B}_n = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$. We define
$p \qvdash \mathcal{B}$ if the set
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n} \} \cap \mathcal{L}_C$$
is a largeness class.
\end{definition}
Given a $\Sigma^0_{\omega_1^{ck}+1}$ class $\mathcal{B} = \bigcup_{n \in \omega} \mathcal{B}_{n}$ with $\mathcal{B}_n = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$, the following set
$$\mathcal{U} = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n} \}$$
is a $\Sigma^1_1(C)$ open set, that is an open set $\mathcal{U} = \bigcup_{\sigma \in B} [\sigma]$ where $B = \bigcap_{\alpha < \omega_1^{ck}} B_\alpha$ is a $\Sigma^1_1(C)$ set of strings. We furthermore assume that $\{B_\alpha\}_{\alpha < \omega_1^{ck}}$ is decreasing. We then write $\mathcal{U}_\alpha$ for the $\Delta^1_1(C)$-open set $\bigcup_{\sigma \in B_\alpha} [\sigma]$.
Computability theorists have a strong habits of working with enumerable open sets. With that respect, $\Sigma^1_1$-open sets, that is, co-enumerable along the computable ordinals, are strange objects to consider. Note that given such an open set we have $\mathcal{U} \subseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_\alpha$, but not necessarily equality. However the elements $X$ of $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_\alpha - \mathcal{U}$ are all such that $\omega_1^X > \omega_1^{ck}$. It is in particular a meager and nullset.
Let us detail a little bit the set $B = \bigcap_{\alpha < \omega_1^{ck}} B_\alpha$ that we can consider so that $\mathcal{U} = \bigcup_{\sigma \in B} [\sigma]$. To ease the notation we introduce the following definition, in the same spirit as $\mathcal{U}(\mathcal{B}, \sigma)$ defined above:
\begin{definition} \label{def-hyp-mainstuff}
Let $\mathcal{B}$ be a $\Sigma^0_{\alpha}$ class. We define
$\mathcal{V}(\mathcal{B}, \sigma)$ to be the set
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \sigma \cup \tildeau \nqvdash \mathcal{B} \}$$
\end{definition}
Given a $\Sigma^0_{\omega_1^{ck}+1}$ class $\mathcal{B} = \bigcup_{n \in \omega} \mathcal{B}_{n}$ with $\mathcal{B}_n = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$, given
$$\mathcal{U} = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n} \}$$
we have by \Cref{cor-hyp-forcepii} that $\mathcal{U}$ equals:
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \forall \alpha < \omega_1^{ck}\ \mathcal{V}(2^\omega - \mathcal{B}_{n, \alpha}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class}\}$$
Let
$$B = \{\tildeau\ :\ \exists n\ \forall \alpha < \omega_1^{ck}\ \mathcal{V}(2^\omega - \mathcal{B}_{n, \alpha}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class} \}$$
Let
$$B_\alpha = \{\tildeau\ :\ \exists n\ \forall \beta < \alpha\ \mathcal{V}(2^\omega - \mathcal{B}_{n, \beta}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class} \}$$
By $\Sigma^1_1$-boundedness we have that $B = \bigcap_{\alpha} B_\alpha$. We also have $\mathcal{U} = \bigcup_{\sigma \in B} [\sigma]$.
We now show the core lemma that will be used to show $\omega_1^{G_{\mathcal{F}}} = \omega_1^{ck}$ for $\mathcal{F}$ a sufficiently generic filter:
\begin{lemma} \label{hyp-lem-sigma1}
Let $B = \bigcap_{\alpha < \omega_1^{ck}} B_\alpha$ be a $\Sigma^1_1(C)$ set of strings where each $B_\alpha$ is $\Delta^1_1(C)$ uniformly in $\alpha$ and where $\beta < \alpha$ implies $B_\alpha \subseteq B_\beta$. Let $\mathcal{U} = \bigcup_{\sigma \in B} [\sigma]$ be a $\Sigma^1_1(C)$ upward closed open set with $\mathcal{U}_\alpha = \bigcup_{\sigma \in B_\alpha} [\sigma]$ be a $\Delta^1_1(C)$ upward closed open set. We have $\mathcal{U} \subseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_\alpha$. Furthermore, $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class iff for every $\alpha < \omega_1^{ck}$ , $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is a largeness class.
\end{lemma}
\begin{proof}
It is clear that $\mathcal{U} \subseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_\alpha$. Also it is clear that if $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class, then also $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{\alpha} \cap \mathcal{L}_C$ is a largeness class.
Suppose $\mathcal{U} \cap \mathcal{L}_C$ is not a largeness class. Then there is a cover $Y_0 \cup \dots \cup Y_{k - 1} \supseteq \omega$ with $Y_i \notin \mathcal{U} \cap \mathcal{L}_C$ for every $i < k$. There must be a $\Sigma^0_1(C)$ open set $\mathcal{V}$ such that $Y_i \notin \mathcal{U} \cap \mathcal{V}$ for every $i \leq k$.
Let $f:\omega \rightarrow \omega_1^{ck}$ be the function which on $n$ finds a cover $\sigma_0 \cup \dots \cup \sigma_k \supseteq \{0, \dots, n\}$ and $\alpha$ such that for $i < k$ and every $\tildeau \preceq \sigma_i$ we have $[\tildeau] \subseteq \mathcal{V}$ implies $\tildeau \notin B_\alpha$. As $\mathcal{U} \cap \mathcal{V}$ is not a largeness class, $f$ is a total $\Pi^1_1(C)$ function. By $\Sigma^1_1$-boundedness, $\beta = \sup_n f(n) < \omega_1^C = \omega_1^{ck}$. By compactness, there is a cover $Y_0 \cup \dots \cup Y_{k-1}$ such that for every $i < k$ if $Y_i \in \mathcal{V}$ then for every $\tildeau \prec Y_i$, $\tildeau \notin B_\beta$ and thus $Y_i \notin \mathcal{U}_\beta$.
It follows that $\mathcal{U}_\beta \cap \mathcal{L}_C$ is not a largeness class.
\end{proof}
\begin{corollary}
$\mathcal{L}_C$ contains a unique largeness subclass, which is minimal for both $\Pi^1_1$ and $\Sigma^1_1(C)$-open sets $\mathcal{U}$.
\end{corollary}
\begin{proof}
Suppose $\mathcal{U}_0, \mathcal{U}_1$ are two $\Sigma^1_1(C)$ open sets with $\mathcal{U}_i = \bigcup_{\sigma \in B_i} [\sigma]$ and $\mathcal{U}_{i, \alpha} = \bigcup_{\sigma \in B_{i, \alpha}} [\sigma]$. for $i < 2$. Suppose also $\mathcal{U}_0 \cap \mathcal{L}_{C}$ and $\mathcal{U}_1 \cap \mathcal{L}_{C}$ are largeness classes. By \Cref{hyp-lem-sigma1} it follows that $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{0, \alpha} \cap \mathcal{L}_C$ and $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{1, \alpha} \cap \mathcal{L}_C$ are largeness classes. By \Cref{hyp-lemma-minC} it follows that $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{0, \alpha}$ and $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_{\alpha}} \subseteq \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{1, \alpha}$.
Thus $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{0, \alpha} \cap \bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{1, \alpha} = \bigcap_{\alpha < \omega_1^{ck}} (\mathcal{U}_{0, \alpha} \cap \mathcal{U}_{1, \alpha})$ is a largeness class and thus by \Cref{hyp-lem-sigma1} we have that $\mathcal{U}_0 \cap \mathcal{U}_1$ is a largeness class.
It follow that the intersection $\mathcal{I}$ of every $\Sigma^1_1(C)$ open set $\mathcal{U}$ such that $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class, is a largeness class. Furthermore as $\mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha} \cap \mathcal{L}_C$ is a largeness class for every $\alpha$, the class $\mathcal{I}$ must be included in $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha}$. Also from \Cref{hyp-corpi11accclass} the class $\bigcap_{\alpha < \omega_1^{ck}} \mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha}$ is minimal for $\Pi^1_1$-open sets $\mathcal{U}$ such that $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class. It follows that the class $\mathcal{I} \cap \mathcal{L}_C$ is minimal for $\Sigma^1_1(C)$ and $\Pi^1_1$ open sets.
\end{proof}
We can now detail the class $\mathcal{S}$ involved in the definition of $\mathbb{P}_{\omega_1^{ck}}$ : Let $\mathcal{S}$ be the unique largeness class included in $\mathcal{L}_C$ which is minimal for $\Sigma^1_1(C)$ and $\Pi^1_1$ open sets. Note that $\mathcal{S}$ must be partition regular.
\begin{lemma} \label{hyp-collapse1}
Consider a $\Sigma^0_{\omega_1^{ck}+1}$ class $\mathcal{B} = \bigcup_{n \in \omega} \mathcal{B}_{n}$ with $\Pi^0_{\omega_1^{ck}}$ set $\mathcal{B}_n = \bigcap_{\alpha \in \omega_1^{ck}} \mathcal{B}_{n, \alpha}$. Let $p = (\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$.
Suppose $\sigma \qvdash \mathcal{B}$. Then there is a condition $q \leq p$ together with some $n$ such that $q \Vdash \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$
\end{lemma}
\begin{proof}
Let
$$\mathcal{U} = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n} \}$$
The class $\mathcal{U}$ is a $\Sigma^1_1(C)$-open set and $\mathcal{U} \cap \mathcal{L}_C$ is a largeness class. As $\mathcal{S}$ is minimal for $\Sigma^1_1(C)$-open sets, $\mathcal{S} \subseteq \mathcal{U}$. As $X \in \mathcal{S} \subseteq \mathcal{U}$. Then there is some $\tildeau \subseteq X - \{0, \dots, |\sigma|\}$ and some $n$ such that
$\sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n}$.
Let now
$$\mathcal{V} = \{Y\ :\ \exists \rho \subseteq Y - \{0, \dots, |\sigma \cup \tildeau|\}\ \exists \alpha\ \sigma \cup \tildeau \cup \rho \nqvdash \mathcal{B}_{n, \alpha}\}$$
As $\sigma \cup \tildeau \nqvdash \bigcup_{\alpha \in \omega_1^{ck}} 2^\omega - \mathcal{B}_{n, \alpha}$ then $\mathcal{V} \cap \mathcal{L}_C$ is not a largeness class. Thus there is a cover $Y_0 \cup \dots \cup Y_{k-1} = \omega$ such that $Y_i \notin \mathcal{V} \cap \mathcal{L}_C$ for every $i < k$. As $\mathcal{V} \cap \mathcal{L}_C$ is upward-closed, $X \cap Y_i \notin \mathcal{V} \cap \mathcal{L}_C$ for every $i < k$. As $\mathcal{S} \subseteq \mathcal{L}_C$ is partition regular, there is some $i < k$ such that $X \cap Y_i \in \mathcal{S} \subseteq \mathcal{L}_C$. Therefore we must have $X \cap Y_i \notin \mathcal{V}$ and thus
$$\forall \rho \subseteq X \cap Y_i - \{0, \dots, |\sigma \cup \tildeau|\}\ \forall \alpha\ \sigma \cup \tildeau \cup \rho \qvdash \mathcal{B}_{n, \alpha}$$
Thus $(\sigma \cup \tildeau, X \cap Y_i)$ is an extension of $(\sigma, X)$ such that:
$$(\sigma \cup \tildeau, X \cap Y_i) \Vdash \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$$
\end{proof}
\begin{lemma} \label{hyp-collapse2}
Consider a $\Sigma^0_{\omega_1^{ck}+1}$ class $\mathcal{B} = \bigcup_{n \in \omega} \mathcal{B}_{n}$ with $\Pi^0_{\omega_1^{ck}}$ set $\mathcal{B}_n = \bigcap_{\alpha < \omega_1^{ck}} \mathcal{B}_{n, \alpha}$. Let $p = (\sigma, X) \in \mathbb{P}_{\omega_1^{ck}}$.
Suppose $\sigma \nqvdash \mathcal{B}$. Then there is a condition $q \leq p$ together with some $\beta < \omega_1^{ck}$ such that $q \Vdash \bigcap_{n \in \omega} \bigcup_{\alpha < \beta} 2^\omega - \mathcal{B}_{n, \alpha}$
\end{lemma}
\begin{proof}
Let
$$\mathcal{U} = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \sigma \cup \tildeau \nqvdash 2^\omega - \mathcal{B}_{n} \}$$
The class $\mathcal{U}$ is a $\Sigma^1_1(C)$-open set and $\mathcal{U} \cap \mathcal{L}_C$ is not a largeness class. Let us recall \Cref{def-hyp-mainstuff} together with the notation coming after it: $\mathcal{V}(\mathcal{B}, \sigma)$ is the set
$$\{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \sigma \cup \tildeau \nqvdash \mathcal{B} \}$$
Together with
$$B = \{\tildeau\ :\ \exists n\ \forall \alpha < \omega_1^{ck}\ \mathcal{V}(\mathcal{B}_{n, \alpha}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class} \}$$
with $B = \bigcap_{\alpha < \omega_1^{ck}} B_\alpha$ such that
$$B_\alpha = \{\tildeau\ :\ \exists n\ \forall \beta < \alpha\ \mathcal{V}(\mathcal{B}_{n, \beta}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class} \}$$
and with $\mathcal{U} = \bigcup_{\sigma \in B} [\sigma]$.
Using \Cref{hyp-lem-sigma1}, there is some $\alpha < \omega_1^{ck}$ such that the set
$$\mathcal{U}_\alpha = \{Y\ :\ \exists \tildeau \subseteq Y - \{0, \dots, |\sigma|\}\ \exists n\ \forall \beta < \alpha\ \mathcal{V}(\mathcal{B}_{n, \beta}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is not a largeness class}\}$$
is such that $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is not a largeness class. Thus there is a cover $Y_0 \cup \dots \cup Y_{k-1} \supseteq \omega$ such that $Y_i \notin \mathcal{U}_\alpha \cap \mathcal{L}_C$ for every $i < k$. As $\mathcal{U}_\alpha \cap \mathcal{L}_C$ is upward-closed, then also $X \cap Y_i \notin \mathcal{U}_\alpha \cap \mathcal{L}_C$ for every $i < k$. As $X \in \mathcal{S} \subseteq \mathcal{L}_C$ and as $\mathcal{S}$ is partition regular, there is some $i < k$ such that $X \cap Y_i \in \mathcal{S} \subseteq \mathcal{L}_C$. It follows that $X \cap Y_i \notin \mathcal{U}_\alpha$ and thus that:
$$\forall \tildeau \subseteq X \cap Y_i - \{0, \dots, |\sigma|\}\ \forall n\ \exists \beta < \alpha\ \mathcal{V}(\mathcal{B}_{n, \beta}, \sigma \cup \tildeau) \cap \mathcal{L}_C \tildeext{ is a largeness class}$$
Let $\{\beta_m\}_{m \in \omega}$ be such that $\sup_m \beta_m = \alpha$. Let $\tildeau \subseteq Y - \{0, \dots, |\sigma|\}$ and $n \in \omega$. We have for some $m$ that $\mathcal{V}(\mathcal{B}_{n, \beta_m}, \sigma \cup \tildeau) \cap \mathcal{L}_C$ is a largeness class. Then the set
$$\{Y\ :\ \exists \rho \subseteq Y - \{0, \dots, |\sigma|\}\ \exists m\ \sigma \cup \tildeau \cup \rho \nqvdash \mathcal{B}_{n, \beta_m} \} \cap \mathcal{L}_C$$
is a largeness class and then
$$\{Y\ :\ \exists \rho \subseteq Y - \{0, \dots, |\sigma|\}\ \exists m\ \sigma \cup \tildeau \cup \rho \nqvdash \mathcal{B}_{n, \beta_m}\} \cap \mathcal{U}_{C_\alpha}^{\mathcal{M}_\alpha}$$
is a largeness class and thus that $\sigma \cup \tildeau \qvdash \bigcup_{m} 2^\omega - \mathcal{B}_{n, \beta_m}$. As this is true for every $n$ and every $\tildeau \subseteq Y - \{0, \dots, |\sigma|\}$ it follows that $(\sigma, X \cap Y_i)$ is an extension of $(\sigma, X)$ such that
$$(\sigma, X \cap Y_i) \Vdash \bigcap_{n \in \omega} \bigcup_{\beta < \alpha} 2^\omega - \mathcal{B}_{n, \beta}$$
\end{proof}
We now show that if $\mathcal{F} \subseteq \mathbb{P}_{\omega_1^{ck}}$ is sufficiently generic, then $\omega_1^{G_\mathcal{F}} = \omega_1^{ck}$. We use the following fact : If $\omega_1^G > \omega_1^{ck}$, then in particular some $G$-computable ordinal must code for $\omega_1^{ck}$, that is, there must be a $G$-computable function $\Phi$ such that for every $n$, $\Phi(G, n)$ codes, relative to $G$, for an ordinal smaller than $\omega_1^{ck}$ and with $\sup_n |\Phi(G, n)| = \omega_1^{ck}$. We show that this never happens by forcing that for every functional $\Phi$ either for some $n$, $\Phi(G, n)$ does not code for an ordinal smaller than $\omega_1^{ck}$, or there is an ordinal $\alpha < \omega_1^{ck}$ such that $\Phi(G, n)$ always codes for some ordinal smaller than $\alpha$.
Given $G$ and $\alpha$ let $\mathcal{O}^G_\alpha$ be the set of $G$-codes for ordinals smaller than $\alpha$. For $\alpha < \omega_1^{ck}$, the class $\{G\ :\ n \in \mathcal{O}^G_\alpha\}$ is $\Delta^1_1$ uniformly in $\alpha$ and $n$.
\begin{theorem} \label{th-wck}
Suppose $\mathcal{F} \subseteq \mathbb{P}_{\omega_1^{ck}}$ is sufficiently generic. Then $\omega_1^{G_\mathcal{F}} = \omega_1^{ck}$
\end{theorem}
\begin{proof}
Let $p \in \mathbb{P}_{\omega_1^{ck}}$ be a condition. Given a functional $\Phi : 2^\omega \tildeimes \omega \rightarrow \omega$, let
$$\mathcal{B} = \{X\ :\ \exists n\ \forall \alpha < \omega_1^{ck}\ \Phi(X, n) \notin \mathcal{O}_\alpha^X\}$$
Suppose $p \qvdash \mathcal{B}$. Then from \Cref{hyp-collapse1}, there is an extension $q \leq p$ and some $n$ such that
$$q \Vdash \{X\ :\ \forall \alpha < \omega_1^{ck}\ \Phi(X, n) \notin \mathcal{O}_\alpha^X\}$$
It follows from \Cref{hyp-sideprop-truth} that if $\mathcal{F}$ is sufficiently generic for every $\alpha < \omega_1^{ck}$, $\Phi(G_{\mathcal{F}}, n) \notin \mathcal{O}_\alpha^{G_{\mathcal{F}}}$.
Suppose now $p \nqvdash \mathcal{B}$. Then from \Cref{hyp-collapse2}, there is an extension $q \leq p$ and some $\alpha < \omega_1^{ck}$ such that
$$q \Vdash \{X\ :\ \forall n\ \Phi(X, n) \in \mathcal{O}_\alpha^X\}$$
It follows from \Cref{th-hyp-forcingimpliestruth} that if $\mathcal{F}$ is sufficiently generic, $\sup_n \Phi(G_{\mathcal{F}}, n) \leq \alpha$.
\end{proof}
We can finally deduce our final theorem
\begin{theoremnonumber}[Main theorem 4 (Theorem \ref{maintheorem4})]
Let $B$ be non hyperarithmetical. Every set $A$ has an infinite subset $H \subseteq A$ or $H \subseteq \overline{A}$ such that $B$ is not hyperarithmetical in $H$, in particular with $\omega_1^H = \omega_1^{ck}$.
\end{theoremnonumber}
\begin{proof}
By combining \Cref{th-wck} together with \Cref{th-alphaconeavoid}
\end{proof}
\end{document} | math |
بادشاہ سَپُد أمِس زنانہِ سخت ناراض تہٕ ووٚننس کَمرٕ منٛزٕ نیرنہٕ باپَتھ | kashmiri |
தமிழகத்தில் இன்று மின்தடை செய்யப்படும் பகுதிகள்.. இதோ மொத்த லிஸ்ட்..!!!! தமிழகத்தில் உள்ள அனைத்து மாவட்டங்களிலும் உள்ள துணை மின் நிலையங்களில் மின் பராமரிப்பு பணிகள் நடைபெற்று வருகிறது. இந்த மின் பராமரிப்பு பணிகளின் போது சாலைகளின் மின் விநியோகத்திற்கு தடையாக உள்ள மரக்கிளைகள் வெட்டப்பட்டு மின்பாதை சரி செய்யப்பட்டு வருகிறது. அதனைத் தொடர்ந்து பராமரிப்பு பணியின் போது மின் ஊழியர்கள் மற்றும் மின் பயனர்கள் பாதுகாப்பிற்காக பணிகள் முடியும் வரை மின் விநியோகம் தடை செய்யப்பட்டு வருகிறது. இது குறித்து அந்தந்த பகுதி செயற்பொறியாளர்கள் மக்களுக்கு முன்னறிவிப்பு செய்து வருகின்றனர். அதன்படி இன்று பெரம்பலூர் மாவட்டத்தில் உள்ள துணை மின் நிலையத்தில் மாதாந்திர பராமரிப்பு பணிகள் நடைபெறுவதால் மின் விநியோக தடை செய்யப்படும் என்று மின்வாரியம் தெரிவித்துள்ளது. இதனால் பேரளி, அசூர், சித்தளிண் பீல்வாடி, ஒதியம், சிறுகுடல், அருமடல், கீழப்புலியூர், கீ.புதூர், குடிகாடு, வாலிகண்டபுரம், கல்பாடி, கே.எறையூர், நெடுவாசல், கவுல்பாளையம், மருத்தூர் மற்றும் குரும்பப்பாளையம் ஆகிய பகுதிகளில் இன்று காலை 9.30 முதல் 5 மணி வரை மின் விநியோகம் தடை செய்யப்படும் வாரியம் தெரிவித்துள்ளது. | tamil |
নয়াদিল্লি রেলওয়ে স্টেশনে এক মহিলাকে গণধর্ষণ, গ্রেপ্তার ৪ রেল কর্মচারী টিডিএন বাংলা ডেস্ক: নয়াদিল্লি রেলওয়ে স্টেশনে ৩০ বছর বয়সী এক মহিলাকে গণধর্ষণ করার ঘটনায় চারজন রেল কর্মচারীকে গ্রেপ্তার করা হয়েছে পুলিশ জানিয়েছে, অভিযুক্ত চার রেল কর্মী রেলওয়ের বৈদ্যুতিক বিভাগের কর্মচারী বৃহস্পতিবার রাতে প্ল্যাটফর্মে নির্মিত ট্রেনের লাইটিং কামরায় তারা ওই মহিলাকে ধর্ষণ করে অভিযোগ, দুই অভিযুক্ত যখন মহিলাকে ধর্ষণ করছিল, তখন অপর দুই অভিযুক্ত পাহারাদার হয়ে কামরার বাইরে দাঁড়িয়েছিল পুলিশ জানিয়েছে, অভিযুক্তদের নাম হল সতীশ কুমার ৩৫, বিনোদ কুমার ৩৮, মঙ্গল চাঁদ ৩৩ এবং জগদীশ চাঁদ ৩৭ রেলওয়ের ডিসিপি হরেন্দ্র সিং জানিয়েছে, আক্রান্ত মহিলাটি তাঁকে ৩:২৭ টায় ফোন করে ঘটনার কথা জানান পুলিশ ঘটনাস্থলে পৌঁছে তাঁকে উদ্ধার করে জানা গিয়েছে, রেলে চাকরি দেওয়ার নাম করে ওই মহিলাকে ধর্ষণ করা হয়েছে আক্রান্ত ওই মহিলা পুলিশকে জানিয়েছেন, তিনি হরিয়ানার ফরিদাবাদের বাসিন্দা দুবছর আগে স্বামীর সঙ্গে বিচ্ছেদ হয় তাঁর এরপর থেকেই চাকরির সন্ধান করছিলেন তিনি সতীশের সাথে তাঁর দেখা হয়েছিল এক বন্ধুর মাধ্যমে যে তাঁকে রেলওয়েতে চাকরি দেওয়ার প্রতিশ্রুতি দিয়েছিল বৃহস্পতিবার সতীশ তাঁকে তাঁর ছেলের জন্মদিনের অনুষ্ঠানে আমন্ত্রণ জানিয়েছিলেন রাত ১০:৩০ নাগাদ কীর্তি নগর মেট্রো স্টেশনে সতীশের সঙ্গে ওই মহিলার দেখা হয় এর পরে, সতীশ তাঁকে নয়াদিল্লি রেলওয়ে স্টেশনে নিয়ে যায় সেখানে অন্য তিন অভিযুক্তও আসে এরপর চারজনই ট্রেনের লাইটিং কামরায় ওই মহিলাকে ধর্ষণ করে মহিলা নিজের অভিযোগ জানানোর দুঘণ্টার মধ্যে চারজন অভিযুক্তকে গ্রেপ্তার করে তাদের বিরুদ্ধে মামলা দায়ের করে পুলিশ | bengali |
वृहस्पतिवार के दिन इन 4 राशियों की चमकेगी किस्मत, मिलेगा ढेर सारा प्यार और धन लाइव हिंदी खबर : मेष : आज आपको सलाह दी जाती है कि अपने प्रतिस्पर्धियों के साथ बहस न करें। धन और संपत्ति के मामलों में निर्णय लेते समय सावधानी बरतने की जरूरत है। विपणन के मूल निवासी अपने लक्ष्य को पूरा करेंगे। आज का दिन आपके लिए शानदार रहने वाला है। किस्मत दिन भर आपका साथ देगी। सांस की बीमारी से प्रभावित लोगों को सतर्क रहना चाहिए। आप समाज में अपनी अलग पहचान स्थापित करने में सफल होंगे। वृषभ राशि : आज आप आर्थिक योजना आसानी से बना पाएंगे। आज आपके प्रिय का मूड उखड़ सकता है। आपको दोस्तों के साथ अच्छा समय बिताने का मौका मिलेगा। धन खर्च होगा। दांपत्य जीवन में आत्मीयता आएगी। अपने तेज़तर्रार रवैये पर थोड़ा जाँच करें, अन्यथा रिश्तों में दोस्ती टूट सकती है। नौकरियों और शिक्षा के क्षेत्र से जुड़े लोगों के लिए सफल रास्ते खुलेंगे। विरोधी परास्त होंगे। मिथुन राशि : आज का दिन आपके लिए अनुकूल है। हालाँकि, आप कुछ बातों को लेकर संशय में रहेंगे। आज आप शारीरिक रूप से अस्वस्थ हो सकते हैं लेकिन फिर भी मानसिक रूप से प्रसन्न रहेंगे। आप जो चाहते हैं उसे चोट पहुंचाने से बचें। व्यक्तिगत संबंध संवेदनशील और नाजुक होंगे। आंख में कोई समस्या होने की संभावना है। अपने करियर के बारे में गंभीरता से सोचें और अपने लक्ष्य निर्धारित करें। परिवार के सदस्य कई चीजों की मांग कर सकते हैं। राजनेताओं और प्रशासनिक अधिकारियों के लिए आज का दिन शुभ रहेगा। व्यापार में साझेदारों के साथ विवाद हो सकता है। इससे बचने की कोशिश करें। विपरीत लिंग के लोगों से मिलना विशेष रूप से सुखद रहेगा। धन प्राप्ति की संभावना रहेगी। जो लोग आपको कम आंकते हैं वे आपको सम्मानित करना शुरू कर देंगे। व्यवसाय में कुछ चुनौतियों का सामना करना पड़ेगा। सिंह राशि : आज का अधिकांश समय मित्रों के साथ व्यतीत होगा। आज आप अपनी रोजमर्रा की दिनचर्या से अलग कार्यों में व्यस्त रहेंगे। आप दोस्तों और परिवार के साथ घूमने की योजना बना सकते हैं। आप अपने आप को व्यक्त करने पर जोर देते हैं, तो सफलता आपके कदमों को चूम होगा। अगर आप आज शॉपिंग के लिए बाहर जाते हैं, तो आप एक अच्छी ड्रेस पा सकते हैं। यह दिन आपको आपके जीवनसाथी का सबसे अच्छा पहलू दिखाने वाला है। कन्या राशिफल : नए कार्य शुरू करने के लिए समय शुभ है। मीडिया और बैंकिंग के लोग आज अपने उच्च अधिकारियों को अपने काम से खुश रखेंगे। क्षेत्र में अच्छे काम के लिए लोग आपको पहचानेंगे। विरोधी पार्टियां आज आपके मन को काम से हटाने की कोशिश करेंगी, लेकिन विवेक आज आपको इन लोगों से दूर रखेगा। माली के सुधार के कारण आवश्यक खरीदारी करना आसान हो जाएगा। वैवाहिक जीवन आनंदमय रहेगा। तुला राशि : आज सफलता आपके कदमों को चूम होगा। लव लाइफ में समय की कमी आज आपको परेशान कर सकती है। दांपत्य जीवन में खुशहाली रहेगी। यदि आप अपनी जीभ को नियंत्रित नहीं करते हैं तो आप आसानी से अपनी प्रतिष्ठा को धूमिल कर सकते हैं। छिपे हुए दुश्मन आपके बारे में अफवाहें फैलाने के लिए अधीर होंगे। यदि आप नौकरी पेशा हैं, तो पदोन्नति के योग बन रहे हैं। कार्यक्षेत्र में थोड़ी निराशा का सामना करना पड़ सकता है। वृश्चिक राशि : आज आपको व्यापार में नए अवसर मिल सकते हैं। नौकरी में पदोन्नति के मार्ग की बाधाएँ समाप्त होंगी। आप पाएंगे कि हालात सुधर रहे हैं। अतिरिक्त धन को रियल एस्टेट में निवेश किया जा सकता है। कोई पुराना दोस्त घर आ सकता है। आत्मविश्वास के साथ किसी मामले को सुलझाने की आपकी क्षमता आपको शांति के साथ कठिन स्थिति का सामना करने में मदद करेगी। लव लाइफ अच्छी रहेगी। धनु : धार्मिक कार्यों में आपकी रुचि बढ़ेगी। आज का दिन पत्रकारों और पत्रकारों के लिए भाग्य का दिन है। स्थानान्तरण भी दिखाई दे रहे हैं, लेकिन चिंता न करें, यह आपके लिए अच्छे परिणाम लाएगा। बहादुरी के कदम और फैसले आपको अनुकूल पुरस्कार देंगे। आप उन लोगों से अपने हाथ का वादा करेंगे, जो आपकी मदद के लिए कहेंगे। कला क्षेत्र से जुड़े लोगों के लिए दिन अच्छा रहने वाला है। मकर : आज किसी धार्मिक स्थान पर जाने की संभावना है। करियर के लिहाज से यह समय अनुकूल रहेगा। धन के आगमन की संभावना रहेगी। विद्यार्थियों के लिए आज का दिन अच्छा है। दबाने की समस्याएं फिर से उभर सकती हैं और आपको मानसिक तनाव दे सकती हैं। आज यदि आप दूसरों का अनुसरण करके निवेश करते हैं, तो आर्थिक नुकसान लगभग निश्चित है। प्यार भरी शादीशुदा ज़िंदगी आपको खुश रखेगी। आपकी वित्तीय स्थिति पहले से अधिक मजबूत होगी। कुंभ : परिश्रम में सफलता प्राप्त करेंगे। व्यवसाय में लाभ से आपका मन प्रसन्न रहेगा। आज धन खर्च हो सकता है। लव लाइफ में तनाव हो सकता है। सामाजिक मेलजोल पर स्वास्थ्य को प्राथमिकता दी जानी चाहिए। बेहतर होगा कि आप उस व्यक्ति की उपेक्षा करें जो आपसे उधार लेने के लिए आया हो। शेयर बाजार में निवेश करने से आपको भारी धन लाभ होगा। दांपत्य जीवन खुशियों से भरा रहेगा। कोई रुका हुआ काम पूरा होगा। मीन राशि : छात्रों के लिए आज का दिन मुश्किल भरा हो सकता है। किसी काम के पूरा होने से आप खुश रहेंगे। राजनीति में सफलता और उन्नति की संभावना रहेगी। वैवाहिक जीवन में तनाव आ सकता है। आज आपकी कलात्मक और रचनात्मक क्षमताओं की बहुत प्रशंसा होगी और अचानक लाभ भी होने की संभावना है। अपने क्षेत्र में अच्छा करेंगे और सफलता मिलने के योग हैं। निराश होने की जरूरत नहीं है। | hindi |
As the PS3 powers itself on whenever a Bluetooth command is received but will only power off through a series of on-screen menu operations, adapter engineers have had to deal with the challenge of creating a simple way to offer reliable control of the console for system automation macros – so it’s powered on and off when it should be.
The ps3toothfairy’s solution is a special “Power Track” mode that uses an internal variable to track the console’s power status. When the adapter thinks the system is off, it ignores all button pushes until the [PS] command is sent. Then, when you’re finished with the system, sending [Power Off] transmits an automatic internal macro to turn the PS3 off, finally putting the ps3toothfairy into a sleep mode where it will ignore any other command until [PS] is sent once again.
The macro used to turn the PS3 off can be customized between short and long versions, depending on what the system is primarily used for, and the power tracking features can be disabled entirely if desired. It’s also possible to disable the power off macro, but continue to have the adapter track power status. In the end this feature does add some reliability to system automation macros, but faith-based tracking can become out-of-sync by someone using the console’s physical power button, or powering it off through manual steps. At any rate, hitting the [PS] button should correct any synchronization issue.
Should Sony decide in the future to change the system’s menu layout, the ps3toothfairy offers complete onboard reprogramming of the power off macro, allowing up to 24 steps with both custom hold and delay times on each step.
A feature completely unique to the ps3toothfairy is the ability to send all of the PS3’s special combination keys. The Bluetooth protocol used by the original Sony remote supports a wide range of dual-combination commands, but until now they’ve only used [Start] and [Enter] for remote pairing. Should Sony ever require a different combination of commands, the ps3toothfairy is ready and able to send all possible variations. In fact, the initial pairing procedure is a few button pushes longer than other converters because it’s simply using one of these dual-key combinations for the process.
This dual-key mode can be accessed at any time during normal operation by pressing the [Angle] key followed by the two keys you want to combine. However, since this assignment does prevent [Angle] from performing its normal function until pressed a second time, the function (known as “Attention” in the manual) can be reassigned to any other rarely used function, or a transparent mode can be enabled to turn off dual-key operation except during initial bootup.
Finally, the ps3toothfairy is able to hook up to industry standard IR distribution systems with its rear 3.5mm IR port. The adapter will automatically switch between baseband and modulated signal formats, and one of the configuration options is used to select between the internal infrared eye and the external port. For those who don’t like bright lights on A/V equipment, the front status LED can be set to 4 different brightness intensities, including one that’s nearly off.
The ps3toothfairy is backed by a full website with complete online support, a well written manual, a FAQ section, plus links to forum threads. | english |
/*----------------------------------------------------------------------------
"Point Spread Function Estimation from a Random Target"
Copyright 2010-2011 mauricio delbracio ([email protected])
This program is free software: you can redistribute it and/or modify
it under the terms of the GNU Affero General Public License as
published by the Free Software Foundation, either version 3 of the
License, or (at your option) any later version.
This program is distributed in the hope that it will be useful,
but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
GNU Affero General Public License for more details.
You should have received a copy of the GNU Affero General Public License
along with this program. If not, see <http://www.gnu.org/licenses/>.
----------------------------------------------------------------------------*/
/**
* @file detect_pattern.c
* @brief Module code for detecting the pattern in a digital image
* @author Mauricio Delbracio ([email protected])
*/
#include <stdio.h>
#include <stdlib.h>
#include <math.h>
#include <limits.h>
#include "image.h"
#include "lsd.h"
#include "detect_pattern.h"
/** Number of segments that will be detected in the pattern image */
#define NUM_SEG 13
/** Relative Tolerance to consider a segment is detected */
#define TOL 0.1
/** Big Number */
#define BIG_NUMBER 1000000
/** Maximum LSD scale */
#define MAX_SCALE_LSD 4.0
/** Initial LSD scale */
#define INITIAL_SCALE_LSD 1.85
static void error(char *msg)
{
fprintf(stderr, "Detect_Pattern Error: %s\n", msg);
exit(EXIT_FAILURE);
}
/* homemade round function: float to int*/
int roundfi(float x)
{
if ((x <= INT_MIN-0.5) || (x >= INT_MAX+0.5))
error("roundfi() Float to int conversion out of range");
if (x >= 0)
return (int) (x+0.5);
return (int) (x-0.5);
}
/**
* @brief Coarse detection of the pattern by using segments detected from LSD
* @param in - input image float
* @return Array of floats containing the OPQR 2D point positions of the
* pattern
*/
static float *detect_pattern_coarse(ImageFloat in)
{
image_double in_lsd;
ntuple_list seg;
float min_val, max_val;
double *seg_length, *seg_midpoint_x, *seg_midpoint_y;
char is_in_distance;
double xO, xP, xQ, xR, yO, yQ, yP, yR, l;
int *has_seg, *has_all_seg, point;
/* To keep track of the error from the optimal segment position
* just to choose the closest segment to the optimal location
*/
double seg_error[NUM_SEG];
double xi1, xi2, yi1, yi2, xj1, xj2, yj1, yj2, xjN1, xjN2, yjN1, yjN2;
double xoMp, xpMq, xqMr, xrMo, yoMp, ypMq, yqMr, yrMo;
double xC, yC;
/* Structure of the Pattern Seg0,Seg1,...,Seg10,Seg_Orient0,Seg_Orient1 */
const double x1S[] = { 3, 6, 7, 7, 7, 6, 3, 0, -2, -2, -2, 1, 8 };
const double y1S[] = { 0, 0, -1, -4, -7, -9, -9, -9, -7, -4, -1, 1, 0 };
const double x2S[] = { 3, 6, 8, 8, 8, 6, 3, 0, -1, -1, -1, 1, 7 };
const double y2S[] = { 1, 1, -1, -4, -7, -8, -8, -8, -7, -4, -1, 0, 0 };
int i = 0, j = 0, k = 0, c = 0;
double errorc;
double actual_scale;
char ready;
float *opqr = (float *) malloc(8 * sizeof(float));
/* Execute LSD */
/* Convert between images types and renormalize the image to [0,255]*/
min_val = BIG_NUMBER;
max_val = 0;
for (i=0; i< in->ncol *in->nrow;i++)
{
if(in->val[i] < min_val) min_val = in->val[i];
if(in->val[i] > max_val) max_val = in->val[i];
}
in_lsd = new_image_double((unsigned int) in->ncol,
(unsigned int) in->nrow);
for (i = 0; i < in->ncol * in->nrow; i++)
in_lsd->data[i] = (double) 255/(max_val-min_val)*(in->val[i]-min_val);
/* We do a LOOP from INITIAL_SCALE_LSD to MAX_SCALE_LSD */
actual_scale = INITIAL_SCALE_LSD;
ready = 0;
while (actual_scale < MAX_SCALE_LSD && !ready)
{
printf(" -->LSD scale =%f\n",actual_scale);
seg = lsd_scale(in_lsd, actual_scale);
/* allocate the array or exit with an error */
if ((seg_length = (double *) malloc(seg->size * sizeof(double)))
== NULL
|| (seg_midpoint_x = (double *) malloc(seg->size * sizeof(double)))
== NULL
|| (seg_midpoint_y = (double *) malloc(seg->size * sizeof(double)))
== NULL)
{
error("PSF_ESTIM - Unable to allocate double array space");
exit(EXIT_FAILURE);
}
/*
The i component, of the n-tuple number j, of an n-tuple list 'ntl'
is accessed with:
*/
for (i = 0; i < (int) seg->size; i++)
{
/* segment length */
seg_length[i] = dist_l2(seg->values[i * seg->dim],
seg->values[i * seg->dim + 1],
seg->values[i * seg->dim + 2],
seg->values[i * seg->dim + 3]);
/* segment midpoint */
seg_midpoint_x[i] =
0.5 * (seg->values[i * seg->dim]
+ seg->values[i * seg->dim + 2]);
seg_midpoint_y[i] =
0.5 * (seg->values[i * seg->dim + 1]
+ seg->values[i * seg->dim + 3]);
}
/* Accessing to segment j=0...12 associated to segment i
* has_seg[NUM_SEG*i+j], initialization default to 0
*/
if ((has_seg =
(int *) malloc(NUM_SEG * seg->size * sizeof(int))) == NULL
|| (has_all_seg =
(int *) malloc(seg->size * sizeof(int))) == NULL)
{
error("PSF_ESTIM - Unable to allocate double array space");
exit(EXIT_FAILURE);
}
/* has_seg[], Initialization default to -1 */
for (i = 0; i < NUM_SEG * (int) seg->size; i++)
{
has_seg[i] = -1;
}
/* has_all_seg[], Initialization default to 0 */
/* First pass */
for (i = 0; i < (int) seg->size; i++)
{
xi1 = seg->values[i * seg->dim];
xi2 = seg->values[i * seg->dim + 2];
yi1 = seg->values[i * seg->dim + 1];
yi2 = seg->values[i * seg->dim + 3];
/* Reinitialize the error track */
for (j = 0; j < NUM_SEG; j++)
{
seg_error[j] = BIG_NUMBER;
}
for (j = 0; j < (int) seg->size; j++)
{
xj1 = seg->values[j * seg->dim];
xj2 = seg->values[j * seg->dim + 2];
yj1 = seg->values[j * seg->dim + 1];
yj2 = seg->values[j * seg->dim + 3];
/* Convert the (x,y) coordinates to a new Coordinate System
* (xN, yN) having:
* (xi1,yi1) at (0,0)
* (xi2,yi2) at (0,1)
* The vectors x1s,y1s,x2s,y2s are given within this
* new (xN,yN) coordinate system
*/
l = seg_length[i];
xjN1 =
1 / (l * l) * ((yi2 - yi1) * (xj1 - xi1)
- (xi2 - xi1) * (yj1 - yi1));
yjN1 =
1 / (l * l) * ((xi2 - xi1) * (xj1 - xi1)
+ (yi2 - yi1) * (yj1 - yi1));
xjN2 =
1 / (l * l) * ((yi2 - yi1) * (xj2 - xi1)
- (xi2 - xi1) * (yj2 - yi1));
yjN2 =
1 / (l * l) * ((xi2 - xi1) * (xj2 - xi1)
+ (yi2 - yi1) * (yj2 - yi1));
for (c = 0; c < NUM_SEG; c++)
{
is_in_distance =
(fabs(xjN1 - x1S[c]) < TOL * (2 + fabs(x1S[c])))
&& (fabs(yjN1 - y1S[c]) < TOL * (2 + fabs(y1S[c])))
&& (fabs(xjN2 - x2S[c]) < TOL * (2 + fabs(x2S[c])))
&& (fabs(yjN2 - y2S[c]) < TOL * (2 + fabs(y2S[c])));
if (is_in_distance)
{
/* Need to check that there isn't a previous segment
* closer to the optimal location and already marked
* as good (errorc). I just keep the segment with
* minimum total error l1.
*/
errorc = fabs(xjN1 - x1S[c]) + fabs(yjN1 - y1S[c])
+ fabs(xjN2 - x2S[c]) + fabs(yjN2 - y2S[c]);
if (errorc < seg_error[c])
{
has_seg[i * NUM_SEG + c] = j;
seg_error[c] = errorc;
}
}
}
}
/*has_all_seg[i] will be one if all segments are present */
has_all_seg[i] = 1;
for (j=0;j< NUM_SEG;j++)
{
has_all_seg[i] =
has_all_seg[i] && (has_seg[i * NUM_SEG + j] >= 0);
}
if (has_all_seg[i])
{
point = i;
k++;
}
}
ready = (k==1);
actual_scale *= 1.15;
}
if (k > 1)
{
printf("More than one pattern was detected.");
printf("\nCrop the image surounding the desired pattern and re-run.");
exit(EXIT_SEVERAL_PATTERNS_DETECTED);
}
else if(k<1)
{
printf("No pattern was detected. Use another image");
exit(EXIT_NO_PATTERN_DETECTED);
}
/*Calculate C - center, u = unit_length, theta = angle */
/* 1/3*(DET + 0 + 1) = oMp */
xoMp = 0.33333 * (seg_midpoint_x[point]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 0]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 1]]);
yoMp = 0.33333 * (seg_midpoint_y[point]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 0]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 1]]);
/* 1/3*(2 + 3 + 4) = pMq */
xpMq = 0.33333 * (seg_midpoint_x[has_seg[point * NUM_SEG + 2]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 3]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 4]]);
ypMq = 0.33333 * (seg_midpoint_y[has_seg[point * NUM_SEG + 2]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 3]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 4]]);
/* 1/3*(5 + 6 + 7) = qMr */
xqMr = 0.33333 * (seg_midpoint_x[has_seg[point * NUM_SEG + 5]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 6]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 7]]);
yqMr = 0.33333 * (seg_midpoint_y[has_seg[point * NUM_SEG + 5]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 6]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 7]]);
/* 1/3*(8 + 9 + 10) = rMo */
xrMo = 0.33333 * (seg_midpoint_x[has_seg[point * NUM_SEG + 8]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 9]]
+ seg_midpoint_x[has_seg[point * NUM_SEG + 10]]);
yrMo = 0.33333 * (seg_midpoint_y[has_seg[point * NUM_SEG + 8]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 9]]
+ seg_midpoint_y[has_seg[point * NUM_SEG + 10]]);
/*Center */
xC = 0.25 * (xoMp + xpMq + xqMr + xrMo);
yC = 0.25 * (yoMp + ypMq + yqMr + yrMo);
/*O = C + CoMr + CoMp */
xO = xC + (xrMo - xC) + (xoMp - xC);
yO = yC + (yrMo - yC) + (yoMp - yC);
/*P = C + CpMq + CoMp */
xP = xC + (xpMq - xC) + (xoMp - xC);
yP = yC + (ypMq - yC) + (yoMp - yC);
/*Q = C + CqMr + CpMq */
xQ = xC + (xqMr - xC) + (xpMq - xC);
yQ = yC + (yqMr - yC) + (ypMq - yC);
/*R = C + CrMo + CqMr */
xR = xC + (xrMo - xC) + (xqMr - xC);
yR = yC + (yrMo - yC) + (yqMr - yC);
/*Array of OPQR coordinates*/
opqr[0] = (float) xO;
opqr[1] = (float) yO;
opqr[2] = (float) xP;
opqr[3] = (float) yP;
opqr[4] = (float) xQ;
opqr[5] = (float) yQ;
opqr[6] = (float) xR;
opqr[7] = (float) yR;
/* free memory */
free_image_double(in_lsd);
free_ntuple_list(seg);
free(seg_length);
free(seg_midpoint_y);
free(seg_midpoint_x);
free(has_seg);
free(has_all_seg);
return opqr;
}
/**
* @brief Detection of a X corner in the imput image
* @param in - input image float
* @param ptx - approximate x coord. of the X corner; (output) refined position
* @param pty - approximate y coord. of the X corner; (output) refined position
* @return int - 0 if no error
*/
static int detect_xcorner(ImageFloat in, float *ptx, float *pty)
{
ImageFloat mask, src_buff, gx_buff, gy_buff;
float coeff;
int i, j, k, y, x;
float cx = *ptx;
float cy = *pty;
float c2x, c2y;
int max_iters = 400;
float eps = 0.00001;
int wsize = 3;
float a11, a12, a22, p, q, d;
int iter = 0;
float err;
float py, px;
float tgx, tgy, gxx, gxy, gyy, m;
float *mask1D;
/*mask1D = new_vector(2*wsize+1); */
mask1D = (float *) malloc((2 * wsize + 1) * sizeof(float));
mask = new_imageFloat(2 * wsize + 1, 2 * wsize + 1);
coeff = 1. / (mask->ncol * mask->nrow);
/* calculate mask */
for (i = -wsize, k = 0; i <= wsize; i++, k++)
{
mask1D[k] = exp(-i * i * coeff);
}
for (i = 0; i < (int) mask->nrow; i++)
{
for (j = 0; j < (int) mask->ncol; j++)
{
mask->val[i * mask->nrow + j] = mask1D[j] * mask1D[i];
}
}
do {
src_buff = extract_subpx_window(in, wsize, cx, cy);
gx_buff = gradx(src_buff);
gy_buff = grady(src_buff);
a11 = a12 = a22 = p = q = 0;
/* process gradient */
for (y = -wsize, k = 0; y <= wsize; y++)
{
py = cy + (float) y;
for (x = -wsize; x <= wsize; x++, k++)
{
m = mask->val[k];
tgx = gx_buff->val[k];
tgy = gy_buff->val[k];
gxx = tgx * tgx * m;
gxy = tgx * tgy * m;
gyy = tgy * tgy * m;
px = cx + (float) x;
a11 += gxx;
a12 += gxy;
a22 += gyy;
p += gxx * px + gxy * py;
q += gxy * px + gyy * py;
}
}
d = a11 * a22 - a12 * a12;
c2x = 1 / d * (a22 * p - a12 * q);
c2y = 1 / d * (-a12 * p + a11 * q);
err = dist_l2(cx, cy, c2x, c2y);
cx = c2x;
cy = c2y;
free_imageFloat(src_buff);
free_imageFloat(gx_buff);
free_imageFloat(gy_buff);
} while (++iter < max_iters && err > eps);
*ptx = cx;
*pty = cy;
free_imageFloat(mask);
free((void *) mask1D);
return 0;
}
/**
* @brief Precise detection of the pattern by using the Coarse estimation
* @param in - input image float
* @param opqr - Array of floats containing the OPQR 2D point positions of
* the pattern
* @return Array of 12 points where the X marks are subpixecally located.
*/
static float *detect_pattern_fine(ImageFloat in, float *opqr)
{
/* there should be 12 X-checkerboard corners:
* O O1 O2 P P1 P2 Q Q1 Q2 R R1 R2
*0 1 2 3 4 5 6 7 8 9 10 11
*
*NOTE: O - is the X corner neighbor of two black squares
*/
float *p = (float *) malloc(12 * 2 * sizeof(float));
/* O - initial guess */
p[0] = opqr[0];
p[1] = opqr[1];
/* P - initial guess */
p[6] = opqr[2];
p[7] = opqr[3];
/* Q - initial guess */
p[12] = opqr[4];
p[13] = opqr[5];
/* R - initial guess */
p[18] = opqr[6];
p[19] = opqr[7];
/* first detect opqr at subpixel precision */
detect_xcorner(in, &p[0], &p[1]);
detect_xcorner(in, &p[6], &p[7]);
detect_xcorner(in, &p[12], &p[13]);
detect_xcorner(in, &p[18], &p[19]);
/*linear interpolation as initial position guess
* of the secondary points
*/
/* O1 - initial guess and subpixel detection */
p[2] = 0.6666 * p[0] + 0.3333 * p[6];
p[3] = 0.6666 * p[1] + 0.3333 * p[7];
detect_xcorner(in, &p[2], &p[3]);
/* O2 - initial guess and subpixel detection */
p[4] = 0.3333 * p[0] + 0.6666 * p[6];
p[5] = 0.3333 * p[1] + 0.6666 * p[7];
detect_xcorner(in, &p[4], &p[5]);
/* P1 - initial guess and subpixel detection */
p[8] = 0.6666 * p[6] + 0.3333 * p[12];
p[9] = 0.6666 * p[7] + 0.3333 * p[13];
detect_xcorner(in, &p[8], &p[9]);
/* P2 - initial guess and subpixel detection */
p[10] = 0.3333 * p[6] + 0.6666 * p[12];
p[11] = 0.3333 * p[7] + 0.6666 * p[13];
detect_xcorner(in, &p[10], &p[11]);
/* Q1 - initial guess and subpixel detection */
p[14] = 0.6666 * p[12] + 0.3333 * p[18];
p[15] = 0.6666 * p[13] + 0.3333 * p[19];
detect_xcorner(in, &p[14], &p[15]);
/* Q2 - initial guess and subpixel detection */
p[16] = 0.3333 * p[12] + 0.6666 * p[18];
p[17] = 0.3333 * p[13] + 0.6666 * p[19];
detect_xcorner(in, &p[16], &p[17]);
/* R1 - initial guess and subpixel detection */
p[20] = 0.6666 * p[18] + 0.3333 * p[0];
p[21] = 0.6666 * p[19] + 0.3333 * p[1];
detect_xcorner(in, &p[20], &p[21]);
/* R2 - initial guess and subpixel detection */
p[22] = 0.3333 * p[18] + 0.6666 * p[0];
p[23] = 0.3333 * p[19] + 0.6666 * p[1];
detect_xcorner(in, &p[22], &p[23]);
return p;
}
/**
* @brief Detection of the pattern by using the Coarse & Precise estimations
* There is a fisrt Coarse estimation by using LSD and then a second pass
* in order to refine the position of the X-checkerboard marks presented in
* the pattern
* @param in - input image float
* @return Array of 12 points where the X marks are subpixecally located.
*/
float *detect_pattern(ImageFloat in)
{
float *opqr, *checkpoints;
/*opqr pattern location */
/*X-Checkpoints */
/*O O1 O2 P P1 P2 Q Q1 Q2 R R1 R2 pattern location */
/* coarse detection of the pattern by using LSD */
opqr = detect_pattern_coarse(in);
/* precise detection of the pattern by using the checkerboard marks
and Bouguet-OpenCV X-detector */
checkpoints = detect_pattern_fine(in, opqr);
/*Clean opqr */
free((void *) opqr);
return checkpoints;
}
/**
* @brief Gives the positions of the X marks in the analityc pattern
* @return Array of 12 points where the X marks are subpixecally located.
*/
float *pattern_Xpoints(void)
{
/*This function returns the locations - in the analytic pattern -
* where the X checkerboard corners are.
*/
float *p = (float *) malloc(12 * 2 * sizeof(float));
/* 12 points O..P..Q..R.. coord x (odd) and y (even) */
float up_res = UP_RES;
int up_res_pattern_block_size = up_res * PATTERN_BLOCK_SIZE;
int up_res_pattern_block_size_2 = up_res_pattern_block_size/2;
/*From Point O in counterclockwise order*/
/*O*/
p[0] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[1] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*O1 */
p[2] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[3] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 6;
/*O2 */
p[4] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[5] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 3;
/*P*/
p[6] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[7] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P1 */
p[8] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 6;
p[9] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P2 */
p[10] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 3;
p[11] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*Q*/
p[12] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[13] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*Q1 */
p[14] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[15] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 3;
/*Q2 */
p[16] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[17] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 6;
/*R*/
p[18] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[19] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R1 */
p[20] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 3;
p[21] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R2 */
p[22] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 6;
p[23] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
return p;
}
/**
* @brief Gives the positions of the black squares centers in the analytic
* pattern
* @param p Array of 12 points where the black squares centers are
* subpixecally located on output
*/
void pattern_blackSquares(float *p)
{
/* This function returns the locations - in the analytic pattern -
* where the center of the black squares are placed.
*/
/* 12 points coord x (odd) and y (even) */
float up_res = UP_RES;
int up_res_pattern_block_size = up_res * PATTERN_BLOCK_SIZE;
int up_res_pattern_block_size_2 = up_res_pattern_block_size/2;
/*From Point O in counterclockwise order */
/*O*/
p[0] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[1] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 8;
/*O1 */
p[2] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[3] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 5;
/*O2 */
p[4] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[5] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 2;
/*P*/
p[6] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 8;
p[7] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P1 */
p[8] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 5;
p[9] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P2 */
p[10] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 2;
p[11] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*Q*/
p[12] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[13] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 2;
/*Q1 */
p[14] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[15] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 5;
/*Q2 */
p[16] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[17] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 8;
/*R*/
p[18] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 2;
p[19] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R1 */
p[20] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 5;
p[21] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R2 */
p[22] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 8;
p[23] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
return;
}
/**
* @brief Gives the positions of the black squares centers in the analytic
* pattern
* @param p Array of 12 points where the white squares centers are
* subpixecally located on output
*/
void pattern_whiteSquares(float *p)
{
/*This function returns the locations - in the analytic pattern -
* where the center of the white squares are placed.
*/
/* 12 points coord x (odd) and y (even) */
float up_res = UP_RES;
int up_res_pattern_block_size = up_res * PATTERN_BLOCK_SIZE;
int up_res_pattern_block_size_2 = up_res_pattern_block_size/2;
/*From Point O in counterclockwise order */
/*O*/
p[0] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[1] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 7;
/*O1 */
p[2] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[3] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 4;
/*O2 */
p[4] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
p[5] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 1;
/*P*/
p[6] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 7;
p[7] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P1 */
p[8] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 4;
p[9] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*P2 */
p[10] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 1;
p[11] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
/*Q*/
p[12] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[13] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 1;
/*Q1 */
p[14] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[15] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 4;
/*Q2 */
p[16] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 0;
p[17] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 7;
/*R*/
p[18] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 1;
p[19] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R1 */
p[20] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 4;
p[21] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
/*R2 */
p[22] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 7;
p[23] = up_res_pattern_block_size_2 - 0.5 + up_res_pattern_block_size * 9;
return;
}
/**
* @brief Gives the positions of the the center point inside the noise region
* @param p Array of 1 point where the point is located
*/
void pattern_center(float *p)
{
/*This function returns the location - in the analytic pattern -
* of the pattern center point.
*/
/* 1 points coord x (odd) and y (even) */
float up_res = UP_RES;
p[0] = PATTERN_BLOCK_SIZE * up_res * 5 - 0.5;
p[1] = PATTERN_BLOCK_SIZE * up_res * 5 - 0.5;
return;
}
/**
* @brief Gives the positions of the the topmost point inside the noise region
* part at the horizontal center
* @param p Array of 1 point where the point is located
*/
void pattern_top_center(float *p)
{
/* 1 points coord x (odd) and y (even) */
float up_res = UP_RES;
p[0] = PATTERN_BLOCK_SIZE * up_res * 5 - 0.5;
p[1] = PATTERN_BLOCK_SIZE * up_res * 1 - 0.5;
return;
}
/**
* @brief Draw a X at position 'x','y' of length 'w' on image 'in'
* @param x horizontal coordiante (integer)
* @param y vertical coordiante (integer)
* @param w length of the X-mark (integer)
*/
static void draw_x(int x, int y, int w, float val, ImageFloat in)
{
int i;
for(i=-w;i<w;i++)
{
in->val[x + i + (y+i)*in->ncol] = val;
in->val[x - i -1 + (y+i)*in->ncol] = val;
}
return;
}
/**
* @brief Draw the detected corners with intensity equal to the max value
* in the input image
* @param in input image float
* @param checkerboard arrat containing the 12 locations of the X-checkerboards
* @return float image with X's where the points are located
*/
ImageFloat draw_detected_corners_image_maxval(ImageFloat in,
float *checkerboard)
{
/*Checkpoints */
/*O O1 O2 P P1 P2 Q Q1 Q2 R R1 R2 pattern location */
int length = 4;
int i;
float max_val=0;
ImageFloat in_detected;
in_detected = new_imageFloat(in->ncol,in->nrow);
for(i=0;i<in->nrow*in->ncol;i++)
{
in_detected->val[i] = in->val[i];
if(in->val[i]>max_val) max_val = in->val[i];
}
for(i=0; i<12;i++)
/*0.5 is added to draw the segment in the center of the pixel*/
draw_x(roundfi(checkerboard[2*i]+0.5),
roundfi(checkerboard[2*i+1]+0.5),
length, max_val, in_detected);
return in_detected;
}
/**
* @brief Draw the detected corners with intensity equal to the min value
* in the input image
* @param in input image float
* @param checkerboard arrat containing the 12 locations of the X-checkerboards
* @return float image with X's where the points are located
*/
ImageFloat draw_detected_corners_image_minval(ImageFloat in,
float *checkerboard)
{
/*Checkpoints */
/*O O1 O2 P P1 P2 Q Q1 Q2 R R1 R2 pattern location */
int length = 4;
int i;
float min_val = BIG_NUMBER;
ImageFloat in_detected;
in_detected = new_imageFloat(in->ncol,in->nrow);
for(i=0;i<in->nrow*in->ncol;i++)
{
in_detected->val[i] = in->val[i];
if(in->val[i]<min_val) min_val = in->val[i];
}
for(i=0; i<12;i++)
/*0.5 is added to draw the segment in the center of the pixel*/
draw_x(roundfi(checkerboard[2*i]+0.5),
roundfi(checkerboard[2*i+1]+0.5),
length, min_val, in_detected);
return in_detected;
}
/**
* @brief Convert the input random pattern image to a sharp pattern image
of UP_RES x UP_RES larger size by replacing each pixel by a block of
UP_RES x UP_RES pixels with the same gray value. Also normalize
the sharp pattern image to be a FloatImage in [0,1]
* @param pattern pattern float
* @param pat_nx horizontal size of the pattern input
* @param pat_nx vertical size of the pattern input
* @return ImageFloat with pattern rasterized at UP_RES resolution
*/
ImageFloat pattern_to_pattern_image(float* pattern, int pat_nx, int pat_ny)
{
int maxval, i, j, k, l;
float pixval;
ImageFloat imgP;
maxval = 0;
for(i=0;i<pat_nx*pat_ny;i++)
if(pattern[i]>maxval) maxval = pattern[i];
imgP = new_imageFloat ((int) pat_nx * UP_RES, (int) pat_ny*UP_RES);
for(i=0; i < pat_ny;i++)
for (j=0;j< pat_nx;j++)
{
pixval = pattern[j + i*pat_nx]/maxval;
for(k=0;k< (int) UP_RES;k++)
for(l=0;l< (int) UP_RES;l++)
imgP->val[j* (int)UP_RES
+ l + imgP->ncol*(i* (int)UP_RES + k)]
= pixval;
}
return imgP;
}
| code |
ಲಾಕ್ಡೌನ್ ನಿರ್ಬಂಧ ಸಡಿಲಿಕೆ: ಪ್ರಮುಖ ದೇವಸ್ಥಾನಗಳಲ್ಲಿ ಇಂದಿನಿಂದ ದರ್ಶನ ಬೆಂಗಳೂರು: ರಾಜ್ಯದ ಪ್ರಮುಖ ದೇವಸ್ಥಾನಗಳಲ್ಲಿ ಸೋಮವಾರದಿಂದ ಜು. 5 ಭಕ್ತರಿಗೆ ದರ್ಶನಕ್ಕೆ ಅವಕಾಶ ಕಲ್ಪಿಸಲಾಗಿದೆ. ಆದರೆ, ಉಡುಪಿಯ ಕೃಷ್ಣ ಮಠ ಹಾಗೂ ಸವದತ್ತಿ ಯಲ್ಲಮ್ಮ ದೇವಸ್ಥಾನದ ಪ್ರವೇಶಕ್ಕೆ ನಿರ್ಬಂಧ ಮುಂದುವರಿದಿದೆ. ಶೃಂಗೇರಿ ಶಾರದಾಂಬೆ ದೇಗುಲ, ಹೊರನಾಡಿನ ಅನ್ನಪೂರ್ಣೇಶ್ವರಿ ದೇವಸ್ಥಾನ, ಕೊಲ್ಲೂರು ಮೂಕಾಂಬಿಕಾ ದೇವಸ್ಥಾನ, ಮಂದಾರ್ತಿಯ ದುರ್ಗಾ ಪರಮೇಶ್ವರಿ ದೇವಸ್ಥಾನ, ಧರ್ಮಸ್ಥಳದ ಮಂಜುನಾಥೇಶ್ವರ ದೇವಸ್ಥಾನ, ಕುಕ್ಕೆ ಸುಬ್ರಹ್ಮಣ್ಯ ದೇವಸ್ಥಾನಗಳಲ್ಲಿ ದರ್ಶನಕ್ಕೆ ಅವಕಾಶ ಕಲ್ಪಿಸಲಾಗಿದೆ. ಬಾಗಲಕೋಟೆ ಜಿಲ್ಲೆಯ ಹುನಗುಂದ ತಾಲ್ಲೂಕಿನ ಕೂಡಲಸಂಗಮದ ಬಸವಣ್ಣನ ಐಕ್ಯಮಂಟಪ, ಸಂಗಮೇಶ್ವರ ದೇವಾಲಯ, ಬಾದಾಮಿಯ ಬನಶಂಕರಿ ದೇವಸ್ಥಾನ, ಮೈಸೂರಿನ ಚಾಮುಂಡಿಬೆಟ್ಟದ ಚಾಮುಂಡೇಶ್ವರಿ ದೇಗುಲ, ನಂಜನಗೂಡಿನ ಶ್ರೀಕಂಠೇಶ್ವರ ದೇವಾಲಯ. ಗೋಕರ್ಣ ಮಹಾಬಲೇಶ್ವರ ದೇವಸ್ಥಾನಗಳಲ್ಲೂ ದರ್ಶನಕ್ಕೆ ಅವಕಾಶ ಕಲ್ಪಿಸಲಾಗಿದೆ. ದರ್ಶನಕ್ಕೆ ಬರುವ ಭಕ್ತರು ಕೋವಿಡ್ ನಿಯಮಾವಳಿ ಪಾಲಿಸುವುದು ಕಡ್ಡಾಯ ಎಂದು ಎಲ್ಲ ದೇವಸ್ಥಾನಗಳ ಸಮಿತಿಗಳು ತಿಳಿಸಿವೆ. ಸದ್ಯಕ್ಕೆ ದರ್ಶನವಿಲ್ಲ: ಉಡುಪಿಯ ಕೃಷ್ಣಮಠದಲ್ಲಿ ಸದ್ಯ ಭಕ್ತರಿಗೆ ದರ್ಶನಕ್ಕೆ ಅವಕಾಶ ನೀಡಲಾಗಿಲ್ಲ. ಒಂದು ವಾರ ಪರಿಸ್ಥಿತಿ ನೋಡಿಕೊಂಡು ದೇವಸ್ಥಾನದ ಬಾಗಿಲು ತೆರೆಯಬೇಕೆ, ಬೇಡವೆ ಎಂಬ ನಿರ್ಧಾರ ತೆಗೆದುಕೊಳ್ಳಲಾಗುವುದು ಎಂದು ಪರ್ಯಾಯ ಅದಮಾರು ಮಠದ ಈಶಪ್ರಿಯ ತೀರ್ಥ ಸ್ವಾಮೀಜಿ ತಿಳಿಸಿದ್ದಾರೆ. ಸವದತ್ತಿ ಯಲ್ಲಮ್ಮ ದೇವಸ್ಥಾನ ಸಾರ್ವಜನಿಕ ದರ್ಶನ ನಿಷೇಧ ಮುಂದುವರಿಕೆ: ಬೆಳಗಾವಿ ಜಿಲ್ಲೆಯ ಸವದತ್ತಿ ತಾಲ್ಲೂಕಿನ ರೇಣುಕಾ ಯಲ್ಲಮ್ಮ ದೇವಸ್ಥಾನ ಹಾಗೂ ಜೋಗುಳಭಾವಿ ಸತ್ತೆಮ್ಮದೇವಿ ದೇವಸ್ಥಾನಗಳಲ್ಲಿ ಸಾರ್ವಜನಿಕ ದರ್ಶನವನ್ನು ನಿಷೇಧಿಸಿ ಜಿಲ್ಲಾಧಿಕಾರಿ ಎಂ.ಜಿ. ಹಿರೇಮಠ ಭಾನುವಾರ ಆದೇಶಿಸಿದ್ದಾರೆ. ಯಲ್ಲಮ್ಮ ದೇವಸ್ಥಾನಕ್ಕೆ ನೆರೆಯ ಮಹಾರಾಷ್ಟ್ರದ ಭಕ್ತರು ಹೆಚ್ಚಾಗಿ ಬರುವುದರಿಂದ ಮುಂಜಾಗ್ರತಾ ಕ್ರಮವಾಗಿ ಸಾರ್ವಜನಿಕ ದರ್ಶನ ನಿಷೇಧ ಮುಂದುವರಿಸಲಾಗಿದೆ. ಆ ರಾಜ್ಯದಲ್ಲಿ ಕೋವಿಡ್ ಮತ್ತು ಡೆಲ್ಟಾ ಪ್ಲಸ್ ಪ್ರಕರಣಗಳು ಹೆಚ್ಚಿರುವುದು ಇದಕ್ಕೆ ಕಾರಣ ಎಂದು ಹೇಳಲಾಗಿದೆ. ಮೈಸೂರು ಅರಮನೆ, ಮೃಗಾಲಯ ವೀಕ್ಷಣೆಗೆ ಮುಕ್ತ: ಮೈಸೂರು ಅರಮನೆ ಹಾಗೂ ಚಾಮರಾಜೇಂದ್ರ ಮೃಗಾಲಯವು ಜುಲೈ 5 ರಿಂದ ಸಾರ್ವಜನಿಕರ ವೀಕ್ಷಣೆಗೆ ಮುಕ್ತವಾಗಲಿದೆ. ಪ್ರತಿದಿನ ಬೆಳಿಗ್ಗೆ 10 ರಿಂದ ಸಂಜೆ 5.30ರ ವರೆಗೆ ಅರಮನೆಯನ್ನು ಸಾರ್ವಜನಿಕರ ವೀಕ್ಷಣೆಗೆ ತೆರೆಯಲಾಗುವುದು. ಪ್ರವಾಸಿಗರು ಅರಮನೆಯ ವೆಬ್ಸೈಟ್ನಲ್ಲಿ www.mysorepalace.gov.in ಆನ್ಲೈನ್ ಮೂಲಕ ಟಿಕೆಟ್ ಕಾಯ್ದಿರಿಸಬಹುದು. ಅರಮನೆ ದ್ವಾರದ ಬಳಿಯ ಟಿಕೆಟ್ ಕೌಂಟರ್ನಲ್ಲೂ ಟಿಕೆಟ್ ಖರೀದಿಸಬಹುದು ಎಂದು ಅರಮನೆ ಮಂಡಳಿಯ ಉಪನಿರ್ದೇಶಕ ಟಿ.ಎಸ್.ಸುಬ್ರಮಣ್ಯ ಪ್ರಕಟಣೆಯಲ್ಲಿ ತಿಳಿಸಿದ್ದಾರೆ. ಮೃಗಾಲಯ, ಕಾರಂಜಿ ಕೆರೆ ಮುಕ್ತ: ಮೈಸೂರು ಮೃಗಾಲಯ ಮತ್ತು ಕಾರಂಜಿ ಕೆರೆಯನ್ನು ಸೋಮವಾರದಿಂದ ಸಾರ್ವಜನಿಕರ ವೀಕ್ಷಣೆಗೆ ತೆರೆಯಲಾಗುವುದು ಎಂದು ಮೃಗಾಲಯದ ಕಾರ್ಯನಿರ್ವಾಹಕ ನಿರ್ದೇಶಕರು ಪ್ರಕಟಣೆಯಲ್ಲಿ ತಿಳಿಸಿದ್ದಾರೆ. ಎಚ್.ಡಿ.ಕೋಟೆ ತಾಲ್ಲೂಕಿನ ದಮ್ಮನಕಟ್ಟೆ ಸಫಾರಿ ಕೇಂದ್ರ ಸಹ ಆರಂಭಗೊಳ್ಳಲಿದೆ. | kannad |
கேப்டனாக விராட் கோலியின் சிந்தனை இவரைப் போலவே உள்ளது: சஞ்சய் மஞ்ரேக்கர்! இந்திய கிரிக்கெட் அணியின் கேப்டன் விராட் கோலியின் செயல்பாடுகள் முன்னாள் வெஸ்ட் இண்டீஸ் கேப்டன் விவ் ரிச்சர்ட்ஸ் போல இருப்பதாக சஞ்சய் மஞ்ரேக்கர் தெரிவித்துள்ளார். இந்தியா, இங்கிலாந்து அணிகள் மோதும் முதல் டெஸ்ட் போட்டி சென்னையில் நடக்கிறது. இதன் முதல் இன்னிங்சில் இங்கிலாந்து அணி 578 ரன்கள் குவித்தது. ஆனால் இந்திய அணி 337 ரன்களுக்கு ஆல்அவுட் ஆனது. இதன் மூலம் கோலி தலைமையிலான இந்திய அணி, 241 ரன்கள் என்ற மெகா முன்னிலையை இங்கிலாந்து அணிக்கு அளித்தது. இதனால் போட்டி இந்திய அணியின் கையை விட்டு நழுவிச் சென்றது என்றே பலரும் கருதினர். ஆனால் இரண்டாவது இன்னிங்சில் இங்கிலாந்து அணியை 178 ரன்களுக்கு இந்திய அணி சுருட்டியது ரசிகர்களுக்கு புது நம்பிக்கை அளித்தது. இதன் மூலம் இந்திய அணி தற்போது வெற்றி பெற 420 ரன்கள் என்ற சூழ்நிலை உள்ளது. இதில் துவக்க வீரர் ரோகித் சர்மா மட்டும் வெளியேற இந்திய அணியின் நம்பிக்கை நட்சத்திரமான விராட் கோலி, ரிஷப் பண்ட், சுப்மான் கில் உள்ளிட்ட வீரர்கள் களத்தில் உள்ளனர். நாளை கடைசி நாள் ஆட்டம் மட்டும் எஞ்சியுள்ள நிலையில் இந்திய அணி வெற்றிக்கு 381 ரன்கள் தேவை என்ற நிலை உள்ளது. இங்கிலாந்து அணியைப் பொருத்தவரை 9 விக்கெட்டுகளை கைப்பற்ற வேண்டும் என்ற சூழ்நிலையில் உள்ளது. இதனால் நாளைய கடைசி நாள் ஆட்டத்தில் இந்திய அணி வெற்றி பெறவும் அல்லது இங்கிலாந்து வெற்றி பெறவும் அல்லது போட்டி டிராவை நோக்கிச் செல்லும் என மூன்று விதமான முடிவுகளுக்குச் சாத்தியம் ஏற்பட்டுள்ளது. இந்திய கிரிக்கெட் அணி இரண்டாவது இன்னிங்சில் இங்கிலாந்து அணியின் முதல் பந்திலேயே முதல் விக்கெட்டை கைப்பற்றியது முதல் புது உற்சாகம் அடைந்தது. இதற்கிடையில் சுழற்பந்து வீச்சாளர் ரவிச்சந்திரன் அஸ்வினை நம்பி அவருக்கு முதல் ஓவர் அளித்த கேப்டன் கோலியின் செயல்பாடு முன்னாள் கேப்டனான விவியன் ரிச்சர்ட்ஸை நினைவுபடுத்தும் விதமாக உள்ளதாக சஞ்சய் மஞ்ச்ரேக்கர் தெரிவித்துள்ளார். இதுகுறித்து மஞ்ரேக்கர் கூறுகையில், முதல் பந்திலேயே விக்கெட்டை கைப்பற்றிய உடன் விராட் கோலியின் வெளிப்பாட்டைப் பார்க்கும்போது ஒட்டு மொத்த இந்திய அணியின் மனநிலையைப் புரிந்து கொள்ளும் விதமாக இருந்தது. விராட் கோலியின் கேப்டன் பொறுப்பை பற்றி விமர்சனம் செய்கின்றனர். ஆனால் கோலியின் இந்த அணுகுமுறை முன்னாள் ஜாம்பவான் வெஸ்ட் இண்டீஸ் கேப்டன் விவியன் ரிச்சர்ட்ஸை நினைவுபடுத்தும் விதமாக இருந்தது. அவர் எப்போதும் ஒரு உத்வேகத்துடன் தன்னம்பிக்கையுடனும் களத்தில் செயல்படுவார். இப்படி கேப்டன் அதிக தன்னம்பிக்கையுடன் செயல்படும் பொழுது, அந்த போட்டியின் முடிவையே மாற்றும் சூழல் ஏற்படும் என்பது விவியன் ரிச்சர்ட்ஸ் எண்ணமாக இருந்தது. இதை தற்போது விராட் கோலியிடம் பார்க்க முடிகிறது என்றார். | tamil |
ಗ್ರಾಪಂ 2ನೇ ಹಂತದ ಚುನಾವಣೆ: ಬೆಳ್ತಂಗಡಿ, ಕಡಬದಲ್ಲಿ ಬಿರುಸುಗೊಳ್ಳುತ್ತಿದೆ ಮತದಾನ ಬೆಳ್ತಂಗಡಿ, ಡಿ.27: ತಾಲೂಕಿನಲ್ಲಿ 46 ಗ್ರಾಪಂಗಳ 634 ಸ್ಥಾನಗಳಿಗೆ 292 ಮತಗಟ್ಟೆಗಳಲ್ಲಿ ಬೆಳಗ್ಗೆ 7 ಗಂಟೆಯಿಂದಲೇ ಮತದಾನ ಆರಂಭಗೊಂಡಿದೆ. ಚಳಿ ಮಧ್ಯೆಯೂ ಉತ್ಸಾಹದಿಂದ ಮುಂಜಾನೆಯೇ ಮತದಾನಕ್ಕೆ ಸರತಿ ಸಾಲು ಕಂಡುಬರುತ್ತಿದೆ. ಶಾಸಕ ಹರೀಶ್ ಪೂಂಜಾ, ವಿಧಾನ ಪರಿಷತ್ ಸದಸ್ಯ ಹರೀಶ್ ಕುಮಾರ್ ಹಾಗೂ ಪ್ರತಾಪ ಸಿಂಹ ನಾಯಕ್ ಸೇರಿದಂತೆ ರಾಜಕೀಯ ಮುಖಂಡರುಗಳು ಕುಟುಂಬ ಸಮೇತರಾಗಿ ವಿವಿಧ ಮತಗಟ್ಟೆಗಳಿಗೆ ತೆರಳಿ ಮತ ಚಲಾಯಿಸಿದರು. ಶ್ರೀ ಕ್ಷೇತ್ರದ ಧರ್ಮಸ್ಥಳ ಧರ್ಮಾಧಿಕಾರಿ ಡಾ.ಡಿ.ವೀರೇಂದ್ರ ಹೆಗ್ಗಡೆಯವರು ಧರ್ಮಸ್ಥಳ ಎಸ್.ಡಿ.ಎಂ. ಹಿರಿಯ ಪ್ರಾಧಮಿಕ ಶಾಲೆಯ ಮತಗಟ್ಟೆಯಲ್ಲಿ ತಮ್ಮ ಕುಟುಂಬದ ಸದಸ್ಯರೊಂದಿಗೆ ತೆರಳಿ ಮತ ಚಲಾಯಿಸಿದರು. ಮದುವೆ ಸಂಭ್ರಮದ ಮಧ್ಯೆ ಬಂದು ಮತಹಾಕಿದ ಸಹೋದರಿಯರು! ಮುಂಡಾಜೆ ಗ್ರಾಮದ ಕೂಳೂರು ನಿವಾಸಿ ಇಬ್ರಾಹೀಂ ಕೂಳೂರು ಅವರ ಇಬ್ಬರು ಮಕ್ಕಳಾದ ಝುಹುರಾ ಮತ್ತು ಖೈರುನ್ನಿಸಾ ಸಹೋದರಿಯರು ತಮ್ಮದೇ ಮದುವೆ ಸಂಭ್ರಮದ ಮಧ್ಯೆಯೂ ಮುಂಡಾಜೆ ಗ್ರಾಮದ 103 ಭಾಗ ಸಂಖ್ಯೆಯ ಮತಕೇಂದ್ರಕ್ಕೆ ಆಗಮಿಸಿ ತಮ್ಮ ಹಕ್ಕು ಚಲಾಯಿಸುವ ಮೂಲಕ ಸಾಮಾಜಿಕ ಬದ್ಧತೆ ಮೆರೆದರು. ತಾಲೂಕಿನ 46 ಗ್ರಾಪಂಗಳ ಒಟ್ಟು 631 ಸ್ಥಾನಗಳ ಪೈಕಿ ಒಟ್ಟು 6 ಗ್ರಾಪಂಗಳ 7 ಸ್ಥಾನಗಳಿಗೆ ಅವಿರೋಧವಾಗಿ ಆಯ್ಕೆ ನಡೆದಿದೆ. ಪ್ರಸಕ್ತ 624 ಸ್ಥಾನಗಳಿಗೆ ಮತದಾನ ನಡೆಯುತ್ತಿದೆ. ಬೆಳ್ತಂಗಡಿ ತಾಲೂಕಿನ 2,04,205 ಮತದಾರರು ಇಂದು ಒಟ್ಟು 1,439 ಅಭ್ಯರ್ಥಿಗಳ ಭವಿಷ್ಯ ಬರೆಯಲಿದ್ದಾರೆ. ಕಡಬ ವರದಿ: ಕಡಬ ತಾಲೂಕಿನಲ್ಲಿ ಗ್ರಾಮ ಪಂಚಾಯತ್ ಚುನಾವಣೆಯು ಬಿರುಸಿನಿಂದ ಸಾಗುತ್ತಿದೆ. 21 ಗ್ರಾಮ ಪಂಚಾಯತ್ ಗಳ 91 ವಾರ್ಡ್ ಗಳಿಗೆ ಚುನಾವಣೆ ನಡೆಯುತ್ತಿದ್ದು, ಒಟ್ಟು 285 ಸ್ಥಾನಗಳಿಗೆ 642 ಅಭ್ಯರ್ಥಿಗಳು ಅಂತಿಮ ಕಣದಲ್ಲಿದ್ದಾರೆ. ಕಣದಲ್ಲಿರುವ ಅಭ್ಯರ್ಥಿಗಳು ಮತದಾರರನ್ನು ಆಕರ್ಷಿಸುತ್ತಿರುವುದು ಕಂಡು ಬರುತ್ತಿದೆ. ಗ್ರಾಮೀಣ ಭಾಗದ ಮತದಾರರನ್ನು ಕರೆತರಲು ಅಭ್ಯರ್ಥಿಗಳು ವಾಹನದ ವ್ಯವಸ್ಥೆ ಕಲ್ಪಿಸಿದ್ದಾರೆ. ಕಡಬ ತಾಲೂಕಿನ ಗೋಳಿತೊಟ್ಟು, ನೆಲ್ಯಾಡಿ, ಕೌಕ್ರಾಡಿ, ನೂಜಿಬಾಳ್ತಿಲ, ಶಿರಾಡಿ, ಕೊಂಬಾರು, ಬಿಳಿನೆಲೆ, ಐತ್ತೂರು, ಕೊಣಾಜೆ, ಮರ್ದಾಳ, ಕುಟ್ರುಪ್ಪಾಡಿ, ಪೆರಾಬೆ, ರಾಮಕುಂಜ, ಕೊಯಿಲ, ಆಲಂಕಾರು, ಸವಣೂರು, ಬೆಳಂದೂರು, ಕಾಣಿಯೂರು, ಎಡಮಂಗಲ, ಬಳ್ಪ ಮತ್ತು ಸುಬ್ರಹ್ಮಣ್ಯ ಗ್ರಾಮ ಪಂಚಾಯತ್ ಗಳ ಅಧಿಕಾರದ ಚುಕ್ಕಾಣಿ ಹಿಡಿಯಲು ವಿವಿಧ ಪಕ್ಷಗಳ ಬೆಂಬಲಿತರು ಕಸರತ್ತು ನಡೆಸುತ್ತಿದ್ದಾರೆ. | kannad |
Andy has been a member of IMAPS-UK since 2002 and a member of the IMAPS-UK Committee since 2004, taking on the role of Vice Chair in 2005 and becoming Chair in 2007. During the 6 years in office he was involved with overseeing a transition of the IMAPS organisation requested by USA and Europe, revising the office support and hosting the 40th Anniversary MicroTech. Andy also represented IMAPS-UK as part of the UK Electronics Alliance UKEA (now known as ESCO) and introduced a series of very successful 1 day workshops on Microelectronic Packaging.
Andy’s business life revolves around being CEO of PandA Europe, a technical consultancy in Microelectronics interconnection, packaging and assembly technologies. He is also Co-Chair of the SEMI Advanced Packaging Conference committee, a founder member of ESiPAT, the European packaging, assembly and test special interest group and a member of the EPSRC College. PandA Europe provides secretariat services for IMAPS-UK. | english |
IPL শাসন করবেন, KKRএর বিরুদ্ধে অভিষেকেই ব্যাটে হুঙ্কার বেবি এবি ব্রেভিসের এলেন, দেখলেন এবং জয় করলেন মুম্বইয়ের জার্সিতে দেওয়াল্ড ব্রেভিস আইপিএলে এভাবেই আত্মপ্রকাশ করলেন তাও আবার তারকা খচিত নাইট রাইডার্সের বোলিংয়ের বিরুদ্ধে করলেন ১৯ বলের ২৯ তবে আইপিএল অভিষেকে ছোট্ট ক্যামিও ইনিংসে বুঝিয়ে দিলেন বিশ্বক্রিকেটের কেন তিনি সবথেকে উত্তেজনাময় প্ৰতিভা টুর্নামেন্টের প্ৰথম দুটো ম্যাচে হারের পরে মুম্বই কেকেআরের বিরুদ্ধে মেগা ম্যাচে টিম ডেভিডকে বাইরে রেখে নামিয়ে দিয়েছিল বেবি এবি দেওয়াল্ড ব্রেভিসকে ফুটওয়ার্ক থেকে বাউন্ডারি, ওভার বাউন্ডারির মারার ভঙ্গিমা বারেবারেই একজনকে মনে করিয়ে দিল স্বয়ং এবি ডিভিলিয়ার্স! শুরুতেই পুণের এমসিএ স্টেডিয়াম মাতিয়ে দিলেন ব্রেভিস জোড়া ছক্কা, জোড়া বাউন্ডারিতে উত্তেজনা আমদানি করলেন ক্রিকেট মহলেও অষ্টম ওভারে বরুণ চক্রবর্তীর ওভারে শেষে স্ট্যাম্পড আউট হলেন তবে সেই ওভারেই বরুণকে প্ৰথম বলে স্কোয়ার লেগের ওপর দিয়ে ব্রেভিসের নোলুক সিক্স নিয়ে আলোচনা চলবে দীর্ঘদিন স্বল্পস্থায়ী হয়েও ব্রেভিসের ইনিংসে মুগ্ধ ক্রিকেট মহল মুম্বই একাদশে এদিন জোড়া বদল হয়েছিল ব্রেভিসের সঙ্গেই মুম্বইয়ের জার্সিতে বুধবার নেমে পড়লেন সূর্যকুমার যাদবও তিনি আনমোলপ্রীত সিংয়ের বদলে খেলতে নেমেই সুপারহিট নাইট বোলারদের দাপটে একসময় ত্রাহি ত্রাহি রব উঠেছিল মুম্বই শিবিরে ১১ ওভারে মুম্বই টপ অর্ডারের তিনজনকে হারিয়ে বসেছিল মাত্র ৫৪ রান তোলার ফাঁকেসেখান থেকে ৮৩ রানের পার্টনারশিপ গড়ে মুম্বইকে ট্র্যাকে ফেরান সূর্যকুমার যাদব ৩৬ বলে ৫২ এবং তিলক ভার্মা ২৭ বলে ৩৮ শেষ ওভারে কায়রণ পোলার্ড ঝড় তুলে ৫ বলে ২২করে মুম্বইকে ১৬১তে পৌঁছে দেন প্যাট কামিন্সের ওভারে তিনটে ছক্কা হাঁকান ক্যারিবীয় তারকা | bengali |
@(indicators: List[Indicator], form: Form[Indicator])
@import play.i18n._
@main(Messages.get("list.indicators")) {
<meta charset="UTF-8">
<h1>@Messages.get("indicators")</h1>
<input type="text" id="searchInd" placeholder="@Messages.get("filter")" onkeyup="filter('searchInd', 'crudtable', [0,1]);"/>
<br>
<div class="scrolldiv">
<table class="crudtable">
<tr>
<th>@Messages.get("indicator")</th>
<th>@Messages.get("year")</th>
<th>@Messages.get("remove")</th>
</tr>
@for(ind <- indicators) {
<tr>
<td>@ind.getName()</td>
<td>@ind.getYear()</td>
<td>
@helper.form(routes.Admin.deleteIndicator(ind.getCode())) {
<input type="image" src="@{routes.Assets.at("images/delete.gif")}" alt='@{Messages.get("delete.indicator")}'>
}
</td>
</tr>
}
</table>
</div>
<h2>@{Messages.get("add.indicator")}</h2>
@helper.form(action = routes.Admin.newIndicator()) {
@helper.inputText(form("name"))
<input type="submit" value="@{Messages.get("create")}">
}
}
| code |
ಶಾಸಕರ ಅನರ್ಹತೆ ವಿಚಾರ: ಶಿವಸೇನಾದ ಎರಡೂ ಬಣಗಳಿಗೆ ಸುಪ್ರೀಂ ನೋಟಿಸ್, ಆಗಸ್ಟ್ 1ಕ್ಕೆ ಮುಂದಿನ ವಿಚಾರಣೆ ಇನ್ನೂ ಶಾಸಕರನ್ನು ಅನರ್ಹಗೊಳಿಸಿಲ್ಲ ಎಂದು ಸುಪ್ರೀಂ ಕೋರ್ಟ್ ಹೇಳಿದೆ, ಎರಡೂ ಕಡೆಯವರಿಗೆ ನೋಟಿಸ್ ಜಾರಿ ಮಾಡಿದೆ. ದೆಹಲಿ: ಮಹಾರಾಷ್ಟ್ರದ ಶಿವಸೇನಾದಲ್ಲಿನ Shiv Sena ಉದ್ಧವ್ ಠಾಕ್ರೆ Uddhav Thackeray ಬಣ, ಮುಖ್ಯಮಂತ್ರಿ ಏಕನಾಥ್ ಶಿಂಧೆ Eknath Shinde ನೇತೃತ್ವದ ಬಣಕ್ಕೆ ಸೇರಿದ ಶಾಸಕರನ್ನು ಅನರ್ಹಗೊಳಿಸಬೇಕು ಎಂದು ಕೋರಿ ಸುಪ್ರೀಂಕೋರ್ಟ್ ಮೆಟ್ಟಿಲೇರಿದ್ದು ಈ ಅರ್ಜಿಯ ವಿಚಾರಣೆ ಬುಧವಾರ ನಡೆದಿದೆ. ಶಾಸಕರನ್ನು ಅನರ್ಹಗೊಳಿಸಬೇಕು ಎಂದು ಕೋರಿರುವ ಅರ್ಜಿ ವಿಚಾರಣೆ ನಡೆಸಿದ ಸುಪ್ರೀಂ ಈ ಬಗ್ಗೆ ಅಫಿಡವಿಟ್ ಸಲ್ಲಿಸಲು ಶಿಂಧೆ ಬಣಕ್ಕೆ ಕಾಲಾವಕಾಶ ನೀಡಿದ್ದು ಆಗಸ್ಟ್ 1ಕ್ಕೆ ಮುಂದಿನ ವಿಚಾರಣೆ ನಡೆಸುವುದಾಗಿ ಹೇಳಿದೆ. ಶಿವಸೇನಾದಲ್ಲಿ ಶಿಂಧೆ ಬಂಡಾಯವೆದ್ದ ಕಾರಣ ಮಹಾರಾಷ್ಟ್ರದಲ್ಲಿ ಮಹಾ ವಿಕಾಸ್ ಅಘಾಡಿ ಸರ್ಕಾರ ಪತನವಾಗಿತ್ತು. ಈ ಅರ್ಜಿಯಲ್ಲಿ ಹೇಳಿರುವ ಸಮಸ್ಯೆಗಳನ್ನು ಪರಿಹರಿಸಲು ಐವರು ನ್ಯಾಯಮೂರ್ತಿಗಳ ನ್ಯಾಯಪೀಠದ ಅಗತ್ಯವಿರುತ್ತದೆ ಎಂದು ಹೇಳಿದೆ. ಸ್ಪೀಕರ್ ಅವರು ಈಗ ಯಥಾಸ್ಥಿತಿ ಕಾಯ್ದುಕೊಳ್ಳುತ್ತಾರೆ ಮತ್ತು ಅನರ್ಹಗೊಳಿಸುವ ಅರ್ಜಿಗಳ ಬಗ್ಗೆ ಯಾವುದೇ ನಿರ್ಧಾರ ಪ್ರಕಟಿಸುತ್ತಿಲ್ಲ ಎಂದು ನ್ಯಾಯಾಲಯ ಹೇಳಿದೆ. ಅದೇ ವೇಳೆ ಎಲ್ಲ ದಾಖಲೆಗಳನ್ನು ಸುರಕ್ಷಿತವಾಗಿರಿಸುವಂತೆ ಸುಪ್ರೀಂಕೋರ್ಟ್ ಶಾಸಕಾಂಗ ಸಭೆಯ ಕಾರ್ಯದರ್ಶಿಗೆ ಹೇಳಿದೆ. ನ್ಯಾಯಾಲಯದಲ್ಲಿನ ವಿಚಾರಣೆಯ ಅಪ್ಡೇಟ್ಸ್ ಶಿವಸೇನಾದ ಉದ್ಧವ್ ಠಾಕ್ರೆ ಬಣ ಪರವಾಗಿ ಹಿರಿಯ ವಕೀಲ ಕಪಿಲ್ ಸಿಬಲ್ ವಾದಿಸಿದ್ದು, ಈ ವಿಚಾರ ಸುಪ್ರೀಂಕೋರ್ಟ್ ನಲ್ಲಿರುವಾಗ ರಾಜ್ಯಪಾಲರು ನೂತನ ಸರ್ಕಾರದ ಪ್ರಮಾಣವಚನ ಮಾಡಬಾರದಿತ್ತು ಎಂದಿದ್ದಾರೆ. ಅಸೆಂಬ್ಲಿ ಸ್ಪೀಕರ್ ಅವರ ಆಯ್ಕೆ ಸರಿಯಲ್ಲ ಯಾಕೆಂದರೆ ಅನರ್ಹಗೊಳಿಸಬೇಕೆಂದು ತಮ್ಮ ಕಕ್ಷಿದಾರರು ಕೋರಿರುವ ಶಾಸಕರಿಂದ ಅವರು ಚುನಾಯಿತರಾಗಿದ್ದಾರೆ. ಶಾಸಕರ ಅನರ್ಹತೆ ಅರ್ಜಿ ಇನ್ನೂ ಬಾಕಿ ಇದೆ ಎಂದಿದ್ದಾರೆ ಸಿಬಲ್. ಈ ರೀತಿ ಆದರೆ ದೇಶದಲ್ಲಿರುವ ಯಾವುದೇ ಚುನಾಯಿತ ಸರ್ಕಾರವನ್ನು ಪತನ ಮಾಡಬಹುದು. 10ನೇ ವಿಧಿಯಲ್ಲಿನ ಪ್ರತಿಬಂಧದ ಹೊರತಾಗಿಯೂ ರಾಜ್ಯ ಸರ್ಕಾರಗಳನ್ನು ಪತನ ಮಾಡಲಾಗುತ್ತದೆ ಎಂದಾದರೆ ಪ್ರಜಾತಂತ್ರ ಅಪಾಯದಲ್ಲಿದೆ ಎಂದು ಸಿಬಲ್ ಹೇಳಿರುವುದಾಗಿ ಎಎನ್ಐ ವರದಿ ಮಾಡಿದೆ. ಠಾಕ್ರೆ ಬಣದ ಪರವಾಗಿ ಹಾಜರಾದ ಇನ್ನೊಬ್ಬ ವಕೀಲ ಅಭಿಷೇಕ್ ಮನು ಸಿಂಘ್ವಿ , ಶಾಸಕರ ಅನರ್ಹತೆ ದೂರಿನ ಬಗ್ಗೆ ಸ್ಪೀಕರ್ಗೆ ಗೊತ್ತಿದ್ದರೂ ಅವರು ಯಾವುದೇ ನೋಟಿಸ್ ಕಳುಹಿಸಿಲ್ಲ ಎಂದಿದ್ದಾರೆ. 10ನೇ ಶೆಡ್ಯೂಲ್ ನಲ್ಲಿರುವ ಷರತ್ತು ಎಂದು ನೀವು 23ನ್ನು ಹೊಂದಿರಬೇಕು ಮತ್ತು 23 ಇನ್ನೊಂದು ಪಕ್ಷದೊಂದಿಗೆ ವಿಲೀನವಾಗಿರಬೇಕು. ನನ್ನ ಸ್ನೇಹಿತರು ಇನ್ನೊಂದು ಪಕ್ಷದೊಂದಿಗೆ ವಿಲೀನ ಆಗಿಲ್ಲ. ಅವರು ತಮ್ಮನ್ನು ಬಿಜೆಪಿ ಎಂದು ಕರೆದುಕೊಳ್ಳುತ್ತಿಲ್ಲ ಎಂದಿದ್ದಾರೆ ಸಿಂಘ್ವಿ. ಇದಕ್ಕೆ ಉತ್ತರಿಸಿದ ಏಕನಾಥ್ ಶಿಂಧೆ ಪರ ವಾದಿಸುತ್ತಿರುವ ಹಿರಿಯ ನ್ಯಾಯವಾಗಿ ಹರೀಶ್ ಸಾಳ್ವೆ, ಪಕ್ಷದಲ್ಲಿರುವ ಒಂದು ದೊಡ್ಡ ಗುಂಪು ಬೇರೊಬ್ಬರು ತಮ್ಮ ಪಕ್ಷದ ನೇತೃತ್ವ ವಹಿಸಲಿ ಎಂದು ಬಯಸುವುದರಲ್ಲಿ ತಪ್ಪೇನಿದೆ ಎಂದು ಕೇಳಿದ್ದಾರೆ. ಪಕ್ಷದಲ್ಲೇ ನಿಂತು, ತಮ್ಮ ನಾಯಕರನ್ನು ಪ್ರಶ್ನಿಸಿದರೆ ಏನು? ನಾವು ನಿಮ್ಮನ್ನು ವಿಧಾನಸಭೆಯಲ್ಲಿ ಸೋಲಿಸುತ್ತೇವೆ ಎಂದರೆ ಅದು ಪಕ್ಷಾಂತರ ಅಲ್ಲ. ನೀವು ಪಕ್ಷವನ್ನು ತೊರೆದು ಇನ್ನೊಂದು ಪಕ್ಷಕ್ಕೆ ಸೇರಿದರೆ ಮಾತ್ರ ಪಕ್ಷಾಂತರ. ಅದೇ ಪಕ್ಷದಲ್ಲಿ ಇದ್ದರೆ ಅಲ್ಲ ಎಂದಿದ್ದಾರೆ ಸಾಳ್ವೆ. ಪಕ್ಷದಲ್ಲಿ ಒಡಕು ಇಲ್ಲವೆಂದಾದರೆ , 10ನೇ ಶೆಡ್ಯೂಲ್ ಪ್ರಕಾರ ಯಾವ ಪರಿಣಾಮ ಎದುರಿಸಬೇಕಾಗುತ್ತದೆ ಎಂದು ಮುಖ್ಯನ್ಯಾಯಮೂರ್ತಿ ಕೇಳಿದ್ದಾರೆ. ನಾವು ಫ್ಯಾಂಟಸಿ ಜಗತ್ತಿನಲ್ಲಿ ಇದ್ದೇವೆಯೇ? 20 ಶಾಸಕರ ಬೆಂಬಲ ಸಿಗದೇ ಇರುವವರನ್ನು ಮುಖ್ಯಮಂತ್ರಿ ಮಾಡಬೇಕೇ ಎಂದು ಸಾಳ್ವೆ ಕೇಳಿದ್ದಾರೆ. ಈ ಅರ್ಜಿಯಲ್ಲಿ ಕೆಲವು ಸಮಸ್ಯೆಗಳಿವೆ. ಹಾಗಾಗಿ ಇದಕ್ಕೆ ದೊಡ್ಡ ನ್ಯಾಯಪೀಠದ ಅಗತ್ಯವಿದೆ ಎಂದು ಅನಿಸುತ್ತದೆ ಎಂದಿದ್ದಾರೆ ಸಿಜೆಐ. ಠಾಕ್ರೆ ಬಣ ಸಲ್ಲಿಸಿದ ಅರ್ಜಿಗೆ ಉತ್ತರಿಸಲು ಕಾಲಾವಕಾಶ ಬೇಕು ಹಾಗಾಗಿ ಮುಂದಿನ ವಾರಕ್ಕೆ ವಿಚಾರಣೆ ಮುಂದೂಡಬೇಕು ಎಂದು ಸಾಳ್ವೆ ಮನವಿ ಮಾಡಿದ್ದಾರೆ. ಇದಕ್ಕೆ ಸಮ್ಮತಿಸಿದ ನ್ಯಾಯಾಲಯ ಎರಡೂ ಬಣಗಳಿಗೆ ನೋಟಿಸ್ ಜಾರಿ ಮಾಡಿದ್ದು, ಆಗಸ್ಟ್ 1ಕ್ಕೆ ವಿಚಾರಣೆ ಮುಂದೂಡಿದೆ. | kannad |
రైతులపై కేంద్రం దగా.. ఫొటో : నూతన కమిటీతో సిపిఎం జిల్లా కమిటీ సభ్యులు రైతులపై కేంద్రం దగా.. ప్రజాశక్తినెల్లూరు : విద్యుత్ సవరణ చట్టం పేరుతో వ్యవసాయం పంపుసెట్లకు మీటర్లు బిగించే విధానం తీసుకొచ్చి కేంద్రంలోని బిజెపి ప్రభుత్వం రైతులపై దగాకు పాల్పడుతుందని సిపిఎం నాయకులు పేర్కొన్నారు. ఆదివారం సిపిఎం నెల్లూరు రూరల్ 5వ మండల మహాసభ నెల్లూరు రూరల్ మండలంలోని సౌత్ మోపూరు గ్రామంలో పూడిపర్తి జనార్ధన్, బాలు ఆదిశేషయ్య అధ్యక్షతన నిర్వహించారు. ఈ మహాసభకు ముఖ్య అతిథులుగా సిపిఎం జిల్లా సెక్రటేరియట్ సభ్యులు పొట్టేపాలెం చంద్రమౌళి, మాదాల వెంకటేశ్వర్లు హజరై మాట్లాడారు. కేంద్రంలోని బిజెపి ప్రభుత్వం దేశ ప్రజల శ్రమతో నిర్మించుకున్న ప్రభుత్వ రంగ సంస్థలు, పరిశ్రమలను కారుచౌకగా కార్పొరేట్ శక్తులకు అమ్ముతున్నారని పేర్కొన్నారు. వ్యవసాయాన్ని అదానీ, అంబానీ, ఐటిసి ఇఎఫ్సి కంపెనీల కార్పొరేట్ సంస్థలకు అప్పగించి రైతులను కూలీలుగా మార్చే విధంగా నూతన సాగు చట్టాలను తెచ్చారని విమర్శించారు. విద్యుత్ సవరణ చట్టం పేరుతో వ్యవసాయ పంపుసెట్లకు మీటర్లు బిగించే విధానం తీసుకువస్తున్నారని తెలిపారు. కార్మికుల శ్రమను దోచుకునే విధంగా ఎనిమిది గంటల పని విధానం రద్దు చేసి 4 లేబర్ కోడ్లను తీసుకురావడాన్ని మహాసభ తీవ్రంగా వ్యతిరేకిస్తుందని తెలిపారు. కాబట్టి వెంటనే రైతులు పండించిన పంటలకు మద్దతు గ్యారంటీ చేస్తూ చట్టం తేవాలన్నారు. విశాఖ ఉక్కు ప్రయివేటీకరణను ఆపు చేయాలని ప్రతి ఇంటికి కరోనా సహాయంగా నెలకు రూ.7500 ఇవ్వాలని డిమాండ్ చేశారు. మహాసభ అనంతరం నూతన మండల కార్యదర్శిగా ఆలూరు తిరుపాల్ను ఏకగ్రీవంగా ఎన్నుకున్నారు. 11 మందితో నూతన మండల కమిటీని ఎన్నుకున్నారు. మండల కమిటీ సభ్యులుగా ఎడవల్లి రమణయ్య, అట్లా అనిల్, యిగ్గోలు భాస్కర్, జాన లక్ష్మీ ప్రసాదు, పల్లవరపు చెంచయ్య, పూడిపర్తి జనార్ధన్, గుడి రవీంద్ర, గుంటి.నాగరాజు, శ్రీపతి దయాకర్, గంప. సుధీర్లు ఎన్నికయ్యారు. వీరిలో 8 మందిని జిల్లా మహాసభ ప్రతినిధులుగా ఎన్నుకున్నారు. మహాసభలో బాలు. ఆదిశేషయ్య, ఎస్కె.అబ్దుల్లా, ముత్యాల నాగయ్య, శ్రీపతి వెంకయ్య, మానికళ.చెంచురామయ్య, గడ్డం కోటేశ్వరరావు, తదితరులు పాల్గొని మహాసభను జయప్రదం చేశారు. | telegu |
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