text:
\section{Introduction}
\subsection{Polar Codes}
\IEEEPARstart{P}{olar} codes, introduced by Ar\i kan \cite{Ari09}, are a family of codes
that provably achieve the capacity of binary-input memoryless symmetric (BMS)
channels using low-complexity encoding and decoding algorithms. Since their invention, there has been a large body of work that has analyzed (see e.g., \cite{AT09}--\cite{T10})  and extended
 (see e.g., \cite{KU} --\cite{AT})
these codes.

The construction of polar codes is based on an $\ell \times \ell$ matrix
$G$, with entries in $\{0,1\}$, called the kernel matrix. Besides being invertible, the matrix $G$ should have the
property that  none of its column permutations is upper triangular~\cite{KSU}.  We call a matrix $G$
with such properties a \emph{polarizing} matrix and in the following, whenever we speak of a kernel matrix $G$, we assume that  $G$ is polarizing.

The rows of the generator matrix of a polar code with block-length $N= \ell^n$
are chosen from the rows of the matrix
\begin{equation*}
G^{\otimes n}\triangleq\overbrace{G\otimes G\otimes\cdots\otimes G}^{n},
\end{equation*}
where $\otimes$ denotes the Kronecker product. For the case $\ell=2$ and the
choice $G=\bigl[ \begin{smallmatrix} 1 &0 \\ 1& 1 \end{smallmatrix}
\bigr]$, Reed-Muller (RM) codes also fall into this category.  However,
the crucial difference between polar codes and RM codes lies in the
choice of the rows. For RM codes, the rows of the largest
weights are chosen, whereas for polar codes the choice is dependent on the
channel and is made using a method called channel polarization. We briefly review this method
 and explain how polar codes are constructed from it. We also refer the reader to \cite{Ari09}, \cite{Kor09thesis}  and  \cite{KSU} for
a detailed discussion.

\subsection{Channel Polarization}
Let $W$ be a BMS channel, and let $\mathcal{X}=\{0,1\}$ denote its input alphabet,
$\mathcal{Y}$ the output alphabet, and  $W(y \mid x)$ the transition probabilities.
Let $I(W) \in [0,1]$ denote the mutual information between the input and output of $W$ with uniform distribution on the input. The capacity of a BMS channel $W$ is equal to $I(W)$.
Also,  the Bhattacharyya parameter of $W$, denoted by $Z(W)$,
is defined as
\begin{align*}
& Z(W)= \sum_{y \in \mathcal{Y}} \sqrt{W(y\mid 0)W(y \mid 1)}.
\end{align*}
It provides upper and lower bounds of the error probability $P_e(W)$
in estimating the channel input $x$ on the basis of
the channel output $y$ via the maximum-likelihood (ML)
decoding of $W(y|x)$ as follows~\cite[Chapter 4]{RiU08}, \cite{Kor09thesis}.
\begin{equation}\label{eq:PeZ}
\frac12\left(1-\sqrt{1-Z(W)^2}\right)\le P_e(W)\le \frac12 Z(W).
\end{equation}
It is also related to the capacity $I(W)$ via
\begin{align*}
&Z(W)+I(W)\ge1,\\
&[Z(W)]^2+[I(W)]^2\le1,
\end{align*}
both proved in~\cite{Ari09}.

The method of channel polarization is defined as follows.
Take $N=\ell^n$ copies of a BMS channel $W$.
Combine them by using the kernel matrix $G$ to make a new set of $ \ell^n$
channels $\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$.  The construction of
these channels is done by recursively applying a transform called channel
splitting. Channel splitting is a transform which takes a BMS channel $W$
as input and outputs $\ell$ BMS channels $W ^j $, $0 \leq j \leq \ell-1$.
The channels $W ^j $ are constructed according to the following rule:
Consider a random row vector $U_0^{\ell-1}=(U_0,\,\ldots,\,U_{\ell-1})$ that is uniformly distributed over $\{0, 1\}^{\ell}$.
Let $X_0^{\ell-1}=U_0^{\ell-1} G$, where the arithmetic is in $\mathrm{GF}(2)$.
Also, let $Y_0^{\ell-1}$ be the output of $\ell$ uses of $W$ over the input $X_0^{\ell-1}$. We define the
 channel between $U_0^{\ell-1}$ and $Y_0^{\ell-1}$ by the transition probabilities
\begin{equation}
W_\ell(y_0^{\ell-1} \mid u_0^{\ell-1}) \triangleq \prod_{i=0}^{\ell-1} W(y_i \mid x_i)=\prod_{i=0}^{\ell-1} W(y_i\mid (u_0^{\ell-1} G)_i).
\end{equation}
 The channel $W^j :  \{0, 1\} \to \mathcal{Y}^{\ell} \times \{0, 1\}^{j}$ is defined as the BMS channel
with input $u_j$, output $(y_0^{\ell-1},u_0^{j-1})$ and transition probabilities
\begin{equation}
W^j(y_0^{\ell-1}, u_0^{j-1} \mid u_j)= \frac{1}{2^{\ell-1}}\sum_{u_{j+1}^{\ell-1}} W_\ell(y_0^{\ell-1} \mid u_0^{\ell-1}).
\end{equation}
Here and hereafter, $u_i^j$ denotes the subvector $(u_i,\,\ldots,\,u_j)$.

The construction of the channels $\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$ can be visualized
in the following way~\cite{Ari09}. Consider an infinite $\ell$-ary
tree with the root node placed at the top.
To each vertex of the tree, we assign a channel in a way that the
collection of all the channels that correspond to the vertices at depth
$n$ equals $\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$.  We do
this by a recursive procedure. Assign to the root node the channel $W$
itself. From left to right, assign $W^0$ to $W^{\ell-1}$ to the children
of the root node.  In general, if $Q$ is the channel that is assigned
to vertex $v$, we assign $Q^0$ to $Q^{\ell-1}$, from left to right respectively,
to the children of the node $v$.
There are $ \ell^n$ vertices at level $n$ in this $\ell$-ary tree.
Assume that we label these vertices from left to right from $1$ to $
\ell^n$. Let the channel assigned to the $i$th vertex, $1
\leq i \leq \ell^n$, be $W_{ \ell^n} ^{(i)}$.  Also, let the $\ell$-ary representation
of $i-1$ be $b_1 b_2 \cdots b_n$, where $b_1$ is the most significant
digit. Then we have
\begin{equation*}
 W_{ \ell^n} ^{(i)}=(((W^{b_1})^{b_2})^{\cdots})^{b_n}.
\end{equation*}
As an example, assuming $i=7$, $n=3$ and $\ell=2$ we have $W_{8} ^{(7)} =((W^{1})^{1})^{0}$.

The channels $\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$ have the property that,
 as $n$ grows large, a fraction close to $I(W)$
of the channels have capacity close to $1$ (or Bhattacharyya parameter close to $0$); and a fraction close to $1-
I(W)$ of the channels have capacity close to $0$ (or Bhattacharyya parameter close to $1$).
The basic idea behind polar codes is to use those channels with capacity close to $1$
for information transmission.
Accordingly, given the rate $R<I(W)$ and block-length $N=\ell^n$, the rows
of the generator matrix of a polar code of block-length $N$ correspond to a subset of the rows of the matrix $G^{\otimes n}$ whose indices
are chosen with the following rule:  Choose a subset of size $NR$ of the channels
$\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$  with  the least values for the Bhattacharyya parameter and choose
the rows $G^{\otimes n}$ with the indices corresponding to those of the channels.
For example, if the channel $W_{\ell^n}^{(i)}$ is chosen,
then the $j$th row of $G^{\otimes n}$ is selected,
where the $\ell$-ary representation of $j-1$ is the digit-reversed version of that of $i-1$.
We decode using a successive cancellation (SC) decoder. This algorithm decodes the bits one-by-one in
a pre-chosen order that is closely related to how the row indices of $G^{\otimes n}$ are chosen.

\subsection{Problem Formulation and Relevant Work} \label{related}
Let $\mathcal{I}$ be the set of indices of the $NR$ channels in the set
$\{W_{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$  with the least values for the Bhattacharyya parameter.
Let $\mathbb{P}_e^{\text{SC}}(N,R)$ and $\mathbb{P}_e^{\text{MAP}}(N,R)$ denote
the average block
error probability of the SC and the maximum a-posteriori (MAP) decoders, respectively,
with block-length $N$ and rate $R$.
For the SC decoder we have~\cite{Ari09, Kor09thesis},
\begin{equation}\label{eq:ineqZ}
 \max_{i \in \mathcal{I}} \frac 12 \left(1- \sqrt{1-Z(W_{ \ell^n} ^{(i)})^2}\right)  \leq \mathbb{P}_e^{\text{SC}}(N,R)
\leq \sum_{i \in \mathcal{I}} Z(W_{ \ell^n} ^{(i)}).
\end{equation}
This relation evidently shows that the distribution of the Bhattacharyya parameters of the channels $\{W_
{ \ell^n} ^{(i)}\}_{1 \leq i \leq  \ell^n}$ plays a fundamental role
in the analysis of polar codes. More precisely, for $n \in \ensuremath{\mathbb{N}}\triangleq\{0,\,1,\,2,\,\ldots\}$
and $0 < z < 1$, we are interested in analyzing the behavior of
\begin{equation}\label{equ:1}
F(n, z) = \frac{\# \{ i : Z(W_{ \ell^n} ^{(i)}) \leq z \}}{ \ell^n},
\end{equation}
where $\#A$ denotes the number of elements of the set $A$.
There is an entirely equivalent probabilistic description of
\eqref{equ:1}: Define the ``polarization''  process~\cite{AT09}
of the channel $W$ as a channel-valued stochastic process $\{W_n\}_{n\in\ensuremath{\mathbb{N}}}$ with $W_0=W$ and
\begin{equation}
\label{eq:pol-proc}
W_{n+1}=  W_n^{B_n},
\end{equation}
where $\{B_n\}_{n \in \ensuremath{\mathbb{N}}}$ is a sequence of independent and
identically-distributed (i.i.d.) random variables
with distribution $\mathbb{P}(B_0=j)=\frac{1}{\ell}$ for $j\in\{0,\,1,\,\ldots,\,
\ell-1\}$. In other words, the process begins at the root node of the
infinite $\ell$-ary tree introduced above, and in each step it chooses one of the $\ell$
children of the current node with uniform probability.  So at time $n$,
the process $\{W_n\}_{n\in\ensuremath{\mathbb{N}}}$ outputs one of the $ \ell^n$ channels at level $n$ of
the tree uniformly at random. The Bhattacharyya process $\{Z_n\}_{n\in\ensuremath{\mathbb{N}}}$ of the channel
$W$ is defined from the polarization process as $Z_n \triangleq Z(W_n)$.
In this setting we have
\begin{equation}
 \mathbb{P}(Z_n \leq z) = F(n,z).
\end{equation}
It was shown in \cite{AT09} and \cite{Kor09thesis} that the Bhattacharyya
process $\{Z_n\}_{n\in\ensuremath{\mathbb{N}}}$
 converges almost surely to a $\{0,1\}$-valued random variable
$Z_{\infty}$ with $\mathbb{P}(Z_{\infty}=0)=I(W)$. Our objective is to
investigate the asymptotic behavior of  $\mathbb{P}(Z_n \leq z)$. The analysis of
the process $\{Z_n\}_{n\in\ensuremath{\mathbb{N}}}$ around the point $z=0$ is of particular interest, as
this indicates how the ``good'' channels (i.e., the channels that have
mutual information close to $1$) behave.  The asymptotic analysis of the
process is closely related to the ``partial distances'' of the kernel
matrix $G$:
\begin{definition}[Partial Distances]\label{def:partial-d}
We define the \emph{partial distances} $D_i(G)$, $i=0,\cdots, \ell-1$, of an $\ell
\times \ell$ matrix
$G= \biggl[ \begin{smallmatrix} g_{0} \\  \vdots \\ g_{\ell-1} \end{smallmatrix}
\biggr]$ ($g_i$'s are row vectors)
as
\begin{align*}
D_i(G) & \triangleq \dh(\{g_i\},\langle g_{i+1},\dotsc,g_{\ell-1}\rangle), \qquad i=0,\dotsc,\ell-2, \\
D_{\ell-1}(G) & \triangleq \dh(\{g_{\ell-1}\},\{0\}),
\end{align*}
where $d_H(\cdot, \cdot)$ denotes the Hamming distance between two sets of binary sequences,
and where $\langle g_{i+1},\dotsc,g_{\ell-1}\rangle$ denotes the linear space
spanned by $g_{i+1},\,\dotsc,\,g_{\ell-1}$.
The \textit{exponent} of $G$  is then defined as
\begin{align*}
E(G) = \frac{1}{\ell}\sum_{i=0}^{\ell-1} \log_\ell D_i(G),
\end{align*}
and the \textit{second exponent} of $G$ is defined as
\begin{align*}
V(G) = \frac{1}{\ell}\sum_{i=0}^{\ell-1} (\log_\ell D_i(G) - E(G))^2.
\end{align*}
\end{definition}
In other words, the exponent $E(G)$ and the second exponent $V(G)$
are the mean and the variance of the random variable $\log_\ell D_B(G)$,
where $B$ is a random variable taking a value in $\{0,\,1,\,\ldots,\,\ell-1\}$
with uniform probability.
It should be noted that the invertibility of $G$ implies the partial distances $\{D_i(G)\}$
to be strictly positive, making the exponent $E(G)$ finite.
Note also that the condition for a matrix $G$ to be polarizing,
that none of column permutations of $G$ is upper triangular,
implies $\{D_i(G)\}$ to be strictly greater than 1, yielding $E(G)$ to be
strictly positive.

The following theorem partially characterizes the behavior of the process $\{Z_n\}_{n\in\ensuremath{\mathbb{N}}}$ around $z=0$.
\begin{theorem}[\cite{AT09} and \cite{Kor09thesis}] \label{AT}
Let $W$ be a BMS channel and assume that we are using as the kernel matrix an $\ell\times\ell$ matrix $G$ with exponent $E(G)$. For any fixed $\beta$ with $0 < \beta < E(G)$,
\begin{equation*}
 \lim_{n \to \infty}\mathbb{P}(Z_n \leq 2^{-\ell^{n \beta}}) = I(W).
\end{equation*}
Conversely, if $I(W)<1$, then for any fixed $\beta > E(G)$,
\begin{equation*}
 \lim_{n \to \infty}\mathbb{P}(Z_n \geq 2^{-\ell^{n \beta}}) = 1.
\end{equation*}
\IEEEQED
\end{theorem}
An important consequence of Theorem~\ref{AT} is that, as the behavior of
$\mathbb{P}_e ^{\text{SC}}(N,\,R)$ when using polar codes
with the kernel matrix $G$, of block-length $N= \ell^n$ and rate
$R<I(W)$ under SC decoding is asymptotically the same as that of $\max_{i\in\mathcal{I}}Z(W_N^{(i)})$ from~\eqref{eq:ineqZ},
the probability of error behaves as
$2^{-\ell^{nE(G) + o(n)}}$ as $N$ tends to infinity.
A noteworthy point about this result is that the asymptotic analysis
of the probability of error is rate-independent, provided that
the rate $R$ is less than the capacity $I(W)$.
In this paper,
we provide a refined estimate for $\mathbb{P}(Z_n \leq z)$. Specifically, we derive
the asymptotic relation between  $\mathbb{P}(Z_n \leq z)$ and the rate
of transmission $R$. From this we derive the asymptotic behavior of
$\mathbb{P}_e ^{\text{SC}}(N,\,R)$ and its dependence on the rate of transmission. We further derive
lower bounds on the error probability when we perform MAP decoding instead of SC decoding.

An important point to mention here is that the results of this paper are obtained in the asymptotic
 limit of the block-length for any \emph{fixed} rate value $R$. Considering the
regime where $R$ also varies with the block-length is a problem of different interest,
for which we refer the reader to~\cite{AHKU}.

The outline of the paper
is as follows. In Section~\ref{s_r} we state the main results of the
paper. In Section~\ref{a_p} we first define several auxiliary processes
and provide bounds on their asymptotic behavior. Using these bounds,
we then prove the main results. We discuss the implications of the proofs in selecting the set of channel indices in
Section~\ref{select_sec}.
It should be noted that in the following the logarithms
are in base 2 unless explicitly stated otherwise.

\section{Main Results}\label{s_r}
\begin{theorem}\label{main_result}
Consider an $\ell \times \ell$ polarizing kernel matrix $G= \biggl[ \begin{smallmatrix} g_{0} \\  \vdots \\ g_{\ell-1} \end{smallmatrix}
\biggr]$. For a BMS channel $W$, let $\{Z_n= Z(W_n)\}_{n\in\ensuremath{\mathbb{N}}}$ be the Bhattacharyya process of $W$.
Let $Q(t)\triangleq\int_t^\infty e^{-z^2/2}\,dz/\sqrt{2\pi}$ be the error function
and $Q^{-1}(\cdot)$ be its inverse function.
\begin{enumerate}
\item For $R < I(W)$,
\begin{equation} \nonumber
 \lim _{n \to \infty} \mathbb{P}\left(Z_{n} \leq 2^{ -\ell^{nE(G)+ \sqrt{nV(G)} Q^{-1} \left(\frac{R}{I(W)}\right) + f(n) }}\right) =R.
\end{equation}
\item Let
$H = [g_{\ell-1}^{\rm T},\,\cdots,\,g_{0}^{\rm T}]^{-1}$ ($\cdot^{\rm T}$ denotes the transpose) and assume that $D_{i}(H) \leq {D_{i-1}(H) }$ for $1 \leq i \leq \ell-1$. Then, for $R' < 1- I(W)$ we have,
\begin{equation} \nonumber
 \lim _{n \to \infty} \mathbb{P}\left(Z_{n} \geq 1- 2^{ -\ell^{nE(H)+ \sqrt{nV(H)} Q^{-1} \left(\frac{R'}{1-I(W)}\right) + f(n) }}\right) =R'.
\end{equation}
\end{enumerate}
Here, $f(n)$ is any function satisfying $f(n) = o(\sqrt{n})$.
\IEEEQED
\end{theorem}
{\em Discussion:} Theorem~\ref{main_result} characterizes the asymptotic behavior of $\mathbb{P}(Z_n \leq z)$ and
refines Theorem~\ref{AT} in the following way. According to Theorem~\ref{AT}, if we transmit at rate $R$ below the channel capacity,
then the quantity $ \log_\ell(-\log(\mathbb{P}_{e} ^{\text{SC}}(N=\ell^n,\,R)))$ scales like $nE(G) + o(n)$. The first part of Theorem~\ref{main_result} gives one further
term by stating that $o(n)$ is in fact $\sqrt{nV(G)} Q^{-1}\left(\frac{R}{I(W)}\right)+ o(\sqrt{n})$.
The second part of Theorem~\ref{main_result}, on the other hand, characterizes the asymptotic behavior
of $\mathbb{P}(Z_n\leq z)$ near $z=1$, which is important in applications of polar codes for source coding \cite{KU}.
Put together, Theorem~\ref{main_result} characterizes the scaling of the error probability of polar codes with the SC decoder. Similar results hold
for the case of the MAP decoder.
\begin{theorem} \label{main_result2}
Let $W$ be a BMS channel and let $R < I(W)$ be the rate of transmission. Consider an $\ell \times \ell$ kernel matrix $G$ with $\{w_0(G), \cdots, w_{\ell-1}(G) \}$ the Hamming weights
of its rows and
define
\begin{equation}
 E_w(G)=\frac{1}{\ell} \sum_{i=0}^{\ell-1} \log_\ell w_i(G), \quad V_w(G)= \frac{1}{\ell}\sum_{i=0}^{\ell-1} (\log_\ell w_i(G)-E_w(G))^2.
\end{equation}
If we use polar codes of length $N=\ell^n$ and rate $R$ for transmission, then
the probability of error under MAP decoding,  $\mathbb{P}_e^{\text{MAP}}(N,R)$,  satisfies
\begin{equation}
 \log_\ell (-\log (\mathbb{P}_e^{\text{MAP}}(N,R))) \leq n E_w(G) + \sqrt{n V_w(G)}Q^{-1} \left(\frac{R}{I(W)}\right) + o(\sqrt{n}).
\end{equation}
\IEEEQED
\end{theorem}
{\em Discussion:} Let $G$ be according to Ar{\i}kan's original construction~\cite{Ari09},
i.e., $G= \bigl[ \begin{smallmatrix} 1 & 0 \\ 1 & 1  \end{smallmatrix} \bigr]$,
which is the only polarizing matrix for the case $\ell=2$.
For this $G$, we have $w_i(G) = D_i(G)$ for $i = 0$ and $1$. Hence, the block
error probability for the SC decoder and the MAP block error
probability share the same asymptotic behavior according to Theorems~\ref{main_result} and \ref{main_result2}.
For a general $\ell \times \ell$ matrix $G$, however, one
may have strict inequality $E_w(G) > E(G)$, in which case one
still has an asymptotic  gap between the
error probability with SC decoding and the lower bound of
MAP error probability. Whether or not this gap can be filled
or made narrower is an open problem.

\section{Proof of the Main Result} \label{proofs} \label{a_p}

\subsection{Preliminaries}

Let $\{B_n\}_{n \in \ensuremath{\mathbb{N}}}$ be a sequence of i.i.d. random variables that take their
values in $\{0,1, \cdots, \ell-1\}$ with uniform probability, i.e.,
 $\mathbb{P}(B_0=j) = \frac{1}{\ell}$ for $j \in \{0,\,1,\,\ldots,\, \ell -1\}$.
Let $( \Omega, \mathcal{F}, \mathbb{P})$ denote the probability space generated by
the sequence $\{B_n\}_{n\in\ensuremath{\mathbb{N}}}$ and let  $(\Omega_n, \mathcal{F}_n, \mathbb{P}_n)$ be the probability space generated
by $(B_0, \cdots,B_n)$. We now couple the polarization process $\{W_n\}_{n\in\ensuremath{\mathbb{N}}}$
with the sequence $\{B_n\}_{n \in \ensuremath{\mathbb{N}}}$ via~\eqref{eq:pol-proc}.
Consequently, the Bhattacharyya process $\{Z_n = Z(W_n)\}_{n\in\ensuremath{\mathbb{N}}}$ is coupled with the sequence $\{B_n\}_{n\in\ensuremath{\mathbb{N}}}$.
By using the bounds given in \cite[Chapter 5]{Kor09thesis} we have the
following relationship between the Bhattacharyya  parameters of  $W
^i$ and that of $W$: Recall that $\{D_i(G)\}_{0 \leq i \leq \ell-1}$ are the partial distances of the matrix $G$.
We have~\cite{Kor09thesis}
\begin{align} \label{general_Z(G)}
& Z(W)^{D_{i}(G)} \le Z(W^{i}) \le 2^{\ell-i}Z(W)^{D_{i}(G)}.
\end{align}
Also let $H = [g_{\ell-1}^{\rm T},\,\cdots,\,g_{0}^{\rm T}]^{-1}$. Assuming $D_i(H) \leq D_{i-1} (H)$,
\begin{align} \label{general_Z(H)}
&(1-Z(W))^{D_{i}(H)} \le 1-Z(W^{i}) \le 2^{2i+1}(1-Z(W))^{D_{i}(H)}.
\end{align}
\subsection{Proof of Theorem~\ref{main_result}}
We first provide an intuitive picture behind the result of Theorem~\ref{main_result}. For simplicity, assume $\ell=2$ and let the channel $W$ be a binary erasure channel (BEC) with erasure probability $\epsilon$.
The capacity of this channel is $1-\epsilon$.
For such a channel, the Bhattacharyya process has a simple closed form~\cite{Ari09} as $Z_0= \epsilon$ and
\begin{equation} \label{Z_n}
Z_{n+1} =\left\{ \begin{array}{cc}
Z_n^2,& B_n=0,\\
2Z_n-Z_n^2,& B_n=1.
\end{array}\right.
\end{equation}
We know from Section~\ref{related} that as $n$ grows large,
$Z_n$ tends almost surely to a $\{0,1\}$-valued random variable $Z_{\infty}$ with $\mathbb{P}(Z_{\infty}=0)=1-\epsilon$.
The asymptotic behavior of $\{Z_n\}$ can be explained roughly by considering the behavior of $\{-\log Z_n\}$. In particular, it is clear from \eqref{Z_n} that at time $n+1$, $-\log Z_{n}$ is either doubled (when $B_n=0$), or decreased by at most $1$ (when $B_n=1$).
Also, observe that once $-\log Z_n$ becomes sufficiently large, subtracting $1$ from it has negligible effect compared with the doubling operation. Now assume  that $m$ is a sufficiently large number.  Conditioned on the event that $-\log Z_m$ is a very large value (or equivalently, the value of $Z_m$ is very close to $0$: this happens with probability very close to $1-\epsilon$), for $n>m$  the process $\{-\log Z_n\}$ evolves each time by being doubled if $B_n=0$ or remaining roughly the same if $B_n=1$.
We can then use the central limit theorem to characterize the asymptotic behavior of $\{-\log Z_n\}$ for $n\gg m$.

The proof of  Theorem~\ref{main_result} is done by making the above intuitive steps rigorous for a BMS channel $W$ and a  polarizing $\ell \times \ell$ kernel matrix $G$. In a slightly more general setting, we study the asymptotic properties
of $\mathbb{P}(X_n\le x)$ for any generic process $\{X_n\}_{n\in\ensuremath{\mathbb{N}}}$ satisfying the conditions (c1)--(c4) defined as follows.
\begin{definition}\label{def:S}
Let $S$ be a random variable taking values in $[1,\infty)$.
Assume that the expectation and the variance of $\log S$ exist
and are denoted by $\mathbb{E}[\log S]$ and $\mathbb{V}[\log S]$, respectively.
Assume that $\{S_n\}_{n\in\mathbb{N}}$ are i.i.d.\ samples of $S$.
Let $\{X_n \in (0,1)\}_{n\in\mathbb{N}}$ be a random process satisfying the following conditions:
\begin{itemize}
\item[(c1)] There exists a random variable $X_\infty$ such that $X_n\to X_\infty$ holds almost surely.
\item[(c2)] $X_n^{S_n}\le X_{n+1}$.
\item[(c3)] There exists a constant $c \geq 1$ such that $X_{n+1} \le cX_n^{S_n}$ holds.
\item[(c4)] $S_n$ is independent of $X_m$ for $m\le n$.
\end{itemize}
\end{definition}
The random processes $\{Z_n\}_{n\in\mathbb{N}}$ and $\{1-Z_n\}_{n\in\mathbb{N}}$ satisfy the above four conditions
by letting $S_n = D_{B_n}(G)$ and $S_n=D_{B_n}(H)$, respectively.
The fact that these processes satisfy the condition (c1) has been proved in~\cite[Lemma 5.4]{Kor09thesis},
and the result reads that if $G$ is polarizing, then $Z_\infty$ takes only 0 and 1,
with probabilities $I(W)$ and $1-I(W)$, respectively. Conditions (c2) and (c3) also hold because of \eqref{general_Z(G)} and \eqref{general_Z(H)}.

Our objective now is to prove that for such a process $\{X_n\}_{n\in\mathbb{N}}$, we have
\begin{equation} \label{X}
\lim_{n\to\infty}
\mathbb{P}\left(X_n \le 2^{-2^{n \mathbb{E}[\log S] + t\sqrt{n\mathbb{V}[\log S]} + f(n)}}\right)
 = \mathbb{P}(X_\infty=0)Q(t),
\end{equation}
where $f(n)$ is any function such that $f(n)=o(\sqrt{n})$ holds. The results of Theorem~\ref{main_result} then follow
by noting that $\mathbb{P}(Z_\infty=0)=I(W)$ and $\mathbb{P}(1-Z_\infty=0)=\mathbb{P}(Z_\infty=1)=1-I(W)$ hold,
and by substituting
$t=Q^{-1}(R/I(W))$ and $t=Q^{-1}(R'/(1-I(W)))$, respectively, into \eqref{X}.

We prove \eqref{X} by showing
the two inequalities obtained by replacing the equality in \eqref{X}
by inequality in both directions.
As the first step we have:
\begin{lemma}\label{lem:main_direct}
Let $\{X_n\}_{n\in\mathbb{N}}$ be a random process satisfying (c1), (c3) and (c4).
For any $f(n)=o(\sqrt{n})$,
\begin{equation*}
\liminf_{n\to\infty}
\mathbb{P}\left(X_n \le 2^{-2^{n\mathbb{E}[\log S] + t\sqrt{n \mathbb{V}[\log S]} + f(n)}}\right)
 \ge \mathbb{P}(X_\infty=0)Q(t).
\end{equation*}
\end{lemma}
\begin{IEEEproof}
Without loss of generality, we can assume that $c$ in condition (c3) satisfies $c\ge 2$.
Define the process $\{L_n\}_{n\in\ensuremath{\mathbb{N}}}$ as $L_n \triangleq \log X_n$.
From (c3), we have
\begin{align*}
L_n &\le \log c + S_{n-1} L_{n-1},
\end{align*}
and by applying the above relation recursively, for $m\le n-1$ we obtain
\begin{align} \label{rec}
 L_n &\le \left(\sum_{j=m}^{n-1} \prod_{i=j+1}^{n-1} S_i\right)\log c + \left(\prod_{i=m}^{n-1}S_i\right)L_m \nonumber\\
&\le \left(\prod_{i=m}^{n-1}S_i\right)((n-m)\log c + L_m).
\end{align}
Fix $\beta\in(0,\mathbb{E}[\log S])$ and let
\begin{equation} \label{m}
m \triangleq (\log n+\log\log c)/\beta.
\end{equation}
Conditioned on the event $\mathcal{D}_m(\beta)\triangleq \{ X_m < 2^{-2^{\beta m}}\}$, by using \eqref{rec} we obtain
\begin{equation*}
L_n\le -\left(\prod_{i=m}^{n-1} S_i\right) m\log c.
\end{equation*}
Let the event $\mathcal{H}_m^{n-1}(t)$ be defined as
\begin{align*}
 \mathcal{H}_m^{n-1}(t)\triangleq \biggl\{ & \sum_{i=m}^{n-1}\log S_i\ge (n-m) \mathbb{E}[\log S]
+ t\sqrt{(n-m)\mathbb{V}[\log S]}+f(n-m)\biggr\},
\end{align*}
where $f$ is any function such that $f(k)=o(\sqrt{k})$ holds.
Conditioned on $\mathcal{D}_m(\beta)$ and $\mathcal{H}_m^{n-1}(t)$, we have
\begin{equation*}
\log(-L_n) \ge
\log m + \log\log c + (n-m) \mathbb{E}[\log S]\\
+t\sqrt{(n-m)\mathbb{V}[\log S]}
+f(n-m).
\end{equation*}
Hence,
\begin{multline*}
\mathbb{P}\bigg(
\log(-L_n) \ge
\log m + \log\log c + (n-m)\mathbb{E}[\log S]
+t\sqrt{(n-m)\mathbb{V}[\log S]}
+f(n-m)\bigg)\\
\ge
\mathbb{P}(\mathcal{D}_m(\beta)\cap\mathcal{H}_m^{n-1}(t))
=\mathbb{P}(\mathcal{D}_m(\beta))\mathbb{P}(\mathcal{H}_m^{n-1}(t)).
\end{multline*}
The last equality follows from the independence condition (c4).

Note that taking the limit $n\to\infty$ also implies
$m\to\infty$ and $n-m\to\infty$ via~\eqref{m}.
From Theorem~\ref{thm:FirstEXP} (in Appendix), we  have
$\lim_{n\to\infty}\mathbb{P}(\mathcal{D}_m(\beta))=\mathbb{P}(X_\infty=0)$.
We also have $\lim_{n\to\infty}\mathbb{P}(\mathcal{H}_m^{n-1}(t))=Q(t)$
due to the central limit theorem for $\{\log S_i\}$.
We consequently have
\begin{equation*}
\liminf_{n\to\infty}\mathbb{P}\bigg(
\log(-\log X_n) \ge
n \mathbb{E}[\log S] +t\sqrt{n\mathbb{V}[\log S]}
+f(n)\bigg)\\
\ge \mathbb{P}(X_\infty=0)Q(t)
\end{equation*}
for any $f(n)=o(\sqrt{n})$.
\end{IEEEproof}

The second step of the proof of \eqref{X} is to prove the other direction of the inequality. We have:
\begin{lemma} \label{lem:conv}
Let $\{X_n\}_{n\in\mathbb{N}}$ be a random process satisfying (c1), (c2) and (c4).
For any $f(n)=o(\sqrt{n})$,
\begin{equation*}
\limsup_{n\to\infty}
\mathbb{P}\left(X_n \le 2^{-2^{n\mathbb{E}[\log S] + t\sqrt{n\mathbb{V}[\log S]} + f(n)}}\right)\\
\le \mathbb{P}(X_\infty=0)Q(t).
\end{equation*}
\end{lemma}
\begin{IEEEproof}
Let $L_n\triangleq \log X_n$.
From (c2), for $m\le n-1$ we have
\begin{align*}
L_n &\ge S_{n-1} L_{n-1}\\
&\ge \left(\prod_{i=m}^{n-1}S_i\right)L_m,
\end{align*}
and thus
\begin{equation}
\log (-L_n) \le
\sum_{i=m}^{n-1}\log S_i + \log (-L_m).\label{eq:Lupper}
\end{equation}
Hence, for any fixed $m$ and any $\delta\in(0,1)$,
\begin{align}
\nonumber&\limsup_{n\to\infty}\mathbb{P}\left(\log (-L_n) > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n)\right)\\
\nonumber&\le\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n) > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\; X_m\le \delta\bigg)\\
\label{eq:2terms} &\quad+\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n) > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\; X_m> \delta\bigg).
\end{align}
The first term in the right-hand side of~\eqref{eq:2terms} is upper bounded as
\begin{align*}
&\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n)> n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\; X_m\le \delta\bigg)\\
&\stackrel{(a)}{\le}\limsup_{n\to\infty}\mathbb{P}\bigg(\sum_{i=m}^{n-1}\log S_i + \log (-L_m) > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\; X_m\le \delta\bigg)\\
&\stackrel{(b)}{=}Q(t)\mathbb{P}(X_m\le\delta),
\end{align*}
where (a) follows from~\eqref{eq:Lupper}, and where (b) follows from (c4) and the central limit theorem.
The second term in the right-hand side of~\eqref{eq:2terms} is upper bounded as
\begin{align*}
&\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n)  > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\; X_m> \delta\bigg)\\
&\le\limsup_{n\to\infty}\mathbb{P}\left(X_n\le\frac{\delta}{2},\; X_m>\delta\right)\\
&\stackrel{(a)}{\le} \mathbb{P}\left(X_\infty\le\frac{\delta}{2},\;X_m>\delta\right),
\end{align*}
where (a) follows from (c1).
Applying these bounds to~\eqref{eq:2terms}, for any $\delta\in(0,1)$, we have
\begin{align*}
&\limsup_{n\to\infty}\mathbb{P}\left(\log (-L_n) > n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n)\right)\\
&\le\limsup_{m\to\infty} \left\{Q(t)\mathbb{P}(X_m\le\delta) + \mathbb{P}\left(X_\infty\le\frac{\delta}{2},\;X_m>\delta\right)\right\}\\
&\le Q(t)\mathbb{P}(X_\infty\le\delta) + \mathbb{P}\left(X_\infty\le\frac{\delta}{2},\;X_\infty\ge\delta\right)\\
&=Q(t)\mathbb{P}(X_\infty\le\delta).
\end{align*}
By letting $\delta\to0$, we obtain the result.
\end{IEEEproof}

\subsection{Proof of Theorem~\protect\ref{main_result2}}
\begin{lemma}\label{lem:rep}
The MAP error probability of a linear code $\mathcal{C}$ over a BMS channel $W$ is lower bounded by
$Z(W)^{2d_\text{min}}/4$
where $d_\text{min}$ is the minimum distance of $\mathcal{C}$.
\end{lemma}
\begin{IEEEproof}
Within this proof, the notation $\mathbb{P}(\cdots)$ should be understood as
generically denoting the probability of an event $(\cdots)$.
Since the MAP error probability of a linear code over a BMS channel does not depend on transmitted codeword,
we can assume without loss of generality that transmitted codeword is the all-zero codeword,
which is denoted by $\mathbf{0}$.
Let $\pmb{Y}$ be the random variable corresponding to a received sequence when $\mathbf{0}$ is transmitted
and let $P(y\mid c)$ be the likelihood of a codeword $c$ given a received sequence $y$.
Since MAP and ML are equivalent for equiprobable codewords, the MAP error probability is lower bounded as
\begin{align*}
&\mathbb{P}(\cup_{c'\in\mathcal{C}\setminus\{\mathbf{0}\}} \left\{P(\pmb{Y}\mid c') \ge P(\pmb{Y}\mid \mathbf{0})\right\})\ge
\mathbb{P}(P(\pmb{Y}\mid c) \ge P(\pmb{Y}\mid \mathbf{0}))\\
&=
P_e(W^{\otimes w(c)})\\
& \stackrel{(a)}{\ge}
\frac12 \left(1-\sqrt{1-Z(W^{\otimes w(c)})^2}\right)\\
&=\frac12 \left(1-\sqrt{1-Z(W)^{2w(c)}}\right)\\
&\ge
\frac14 Z(W)^{2w(c)}.
\end{align*}
Here, $c$ is an arbitrary codeword in the set $\mathcal{C}\setminus\{\mathbf{0}\}$
and $w(c)$ denotes its Hamming weight.
Also $W^{\otimes m}$  denotes the $m$-parallel channel of $W$ which has the following rule
\begin{equation}
W^{\otimes m}(y_1^m\mid x)\triangleq\prod_{i=1}^m W(y_i\mid x).
\end{equation}
Step (a) follows from~\eqref{eq:PeZ}.
\end{IEEEproof}
It should be noted that the lower bound $P_e(W^{\otimes w(c)})\ge (1/4) Z(W)^{2w(c)}$ in the proof of Lemma~\ref{lem:rep} is not asymptotically tight
in terms of the conventional exponents.
It is possible to obtain tighter lower bounds via more elaborate arguments as in \cite[Chapter 4] {RiU08}. However, since we are only interested in behavior of double exponents,
the above bound turns out to be sufficient for the purpose of proving Theorem~\ref{main_result2}.

In order to prove Theorem~\ref{main_result2}, from Lemma~\ref{lem:rep} it is sufficient to prove that given any $\epsilon>0$ there exists an integer $M \in \ensuremath{\mathbb{N}}$ such that for $n \geq M$,
\begin{equation*}
 \log_\ell (d(n,R)) \leq n E_w(G) + \sqrt{n V_w(G)}\left(Q^{-1} \left(\frac{R}{I(W)}\right)+\epsilon\right),
\end{equation*}
 where $d(n,R)$ is the minimum distance of a polar code using the kernel matrix $G$, with block-length $N=\ell^n$ and rate $R$.
Since a row weight of the generator matrix is an upper bound of the minimum distance for a linear code,
and since the weight of the $i$th row of $G^{\otimes n}$ is equal to $\prod_{j=1}^n w_{i_j}(G)$,
where $i_j$ is the $j$th digit of the $\ell$-ary representation of $i-1$,
it is therefore sufficient to prove that given any $\epsilon>0$, there exists an integer $M \in \ensuremath{\mathbb{N}}$ such that for a polar code of block-length $N=\ell^n \geq \ell^M$ and rate $R$ and set of chosen indices $\mathcal{I}$, there exists $i\in\mathcal{I}$
for which the inequality
\begin{equation}\label{eq:ineqw}
\sum_{j=1}^n \log_\ell w_{i_j}(G) \leq n E_w(G) + \sqrt{n V_w(G)}\left(Q^{-1} \left(\frac{R}{I(W)}\right)+\epsilon\right)
\end{equation}
holds.
In the proof of Theorem~\ref{main_result}, one can observe that the key idea is to
apply central limit theorem for $\{\log S_n=\log D_{B_n}(G)\}_{n\in\ensuremath{\mathbb{N}}}$.
In the same sense, in order to prove Theorem~\ref{main_result2}
we consider the random process $\{\log w_{B_n}(G)\}_{n\in\ensuremath{\mathbb{N}}}$
in addition to $\{\log D_{B_n}(G)\}_{n\in\ensuremath{\mathbb{N}}}$.
Note that these processes are in general correlated
since they are both coupled to the same process $\{B_n\}_{n\in\ensuremath{\mathbb{N}}}$.
These processes are equal with probability one
in the special case where $D_i(G)=w_i(G)$ holds for all $i\in\{0,\,1,\,\ldots,\,\ell-1\}$.
In the same manner as the proof of Theorem~\ref{main_result},
we move on to a more abstract setting, by introducing a random variable $U$ taking values in $[1,\infty)$,
for which we assume that the expectation and the variance of $\log U$ exist
and are denoted by $\mathbb{E}[\log U]$ and $\mathbb{V}[\log U]$, respectively,
and by letting $\{(S_n,U_n)\}_{n\in\mathbb{N}}$ be i.i.d.\ drawings of $(S,U)$,
where $S$ is defined as in Definition~\ref{def:S}.
Let $\{(X_n, S_n, U_n)\}_{n\in\ensuremath{\mathbb{N}}}$ be a random process such that $\{(X_n, S_n)\}_{n\in\ensuremath{\mathbb{N}}}$
satisfies the conditions (c1) to (c4) together with the additional condition (c5) for $\{U_n\}_{n\in\ensuremath{\mathbb{N}}}$.
\begin{itemize}
\item[(c5)] $U_n$ is independent of $X_m$ for $m\le n$.
\end{itemize}
It is easy to see that the stochastic process of the triplets $\{(Z_n, D_{B_n}(G), w_{B_n}(G))\}_{n\in\ensuremath{\mathbb{N}}}$
satisfies  (c1) to (c5).
We first note from the proof of Theorem~\ref{main_result} that for any generic process $\{(X_n, S_n, U_n)\}_{n\in\ensuremath{\mathbb{N}}}$ satisfying (c1) to (c5),  the relation \eqref{X} holds for any
function $f(n)=o(\sqrt{n})$. We also claim that for real numbers $v,t$ such that $v>t$ and for any function $g(n)=o(\sqrt{n})$ we have
\begin{multline}
\limsup_{n\to\infty}
\mathbb{P}\bigg(X_n \le 2^{-2^{n\mathbb{E}[\log S] + t\sqrt{n\mathbb{V}[\log S]} + f(n)}},\\
\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n)
 \bigg)
< \mathbb{P}(X_\infty=0)Q(t).
\label{eq:map_ineq}
\end{multline}
Using the relations \eqref{X} and \eqref{eq:map_ineq} it is easy to see that for generator matrices of polar codes with rate $R$,
the number of rows satisfying~\eqref{eq:ineqw} is asymptotically proportional to the block-length, and hence
there exists at least a row satisfying~\eqref{eq:ineqw}. We now turn to the proof of \eqref{eq:map_ineq}.
\begin{lemma}\label{lem:2d}
Let $\{(X_n, S_n, U_n)\}_{n\in\mathbb{N}}$ be a random process satisfying (c1) to (c5).
For any $f(n)=o(\sqrt{n})$ and $g(n)=o(\sqrt{n})$,
\begin{multline*}
\lim_{n\to\infty}
\mathbb{P}\bigg(X_n \le 2^{-2^{n\mathbb{E}[\log S] + t\sqrt{n\mathbb{V}[\log S]} + f(n)}},\,
\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n)
 \bigg)\\
= \mathbb{P}(X_\infty=0)\mathbb{P}(A_S \ge t, A_U \ge v),
\end{multline*}
where $(A_S,A_U)$ are Gaussian random variables of mean zero whose covariance matrix is equal to that of
\begin{equation*}
\left(\frac{\log S - \mathbb{E}[\log S]}{\sqrt{\mathbb{V}[\log S]}},\;
\frac{\log U - \mathbb{E}[\log U]}{\sqrt{\mathbb{V}[\log U]}}
\right).
\end{equation*}
\end{lemma}
The proof of this Lemma is the same as the proofs of Lemma~\ref{lem:main_direct} and Lemma~\ref{lem:conv}.
The difference is that the central limit theorem is replaced by the two-dimensional central limit theorem.
From $\mathbb{P}(A_S\ge t, A_U\ge v)\le Q(\max\{t,v\})$, the relation \eqref{eq:map_ineq} is obtained for $v>t$. This completes the proof of Theorem~\ref{main_result2}.

\emph{Remark}:  Let $G=\bigl[ \begin{smallmatrix} 1 &0 \\ 1& 1 \end{smallmatrix}\bigl]$.  For this choice of $G$, we have $w_i(G) = D_i(G)$ for $i = 0$ and $1$. Hence, the random variables $S_n= D_{B_n}(G)$ and $U_n=w_{B_n}(G)$ are equal for $n \in \ensuremath{\mathbb{N}}$. Also note that $S_n$ takes its value in the set $\{1,2\}$ uniformly at random. From the proof of Theorem~\ref{main_result2},
the set of indices of the rows of polar codes with the kernel matrix $G$ and rate $R$ correspond to the event
\begin{equation*}
\left\{X_n \le 2^{-2^{n\mathbb{E}[\log S] + Q^{-1}\left(\frac{R}{I(W)}\right)\sqrt{n\mathbb{V}[\log S]} + f(n)}}\right\}.
\end{equation*}
Also, with the same $G$,  the set of indices of a RM code with rate $R'$ correspond to the event
\begin{equation*}
\left\{\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + Q^{-1}(R')\sqrt{n\mathbb{V}[\log U]} + g(n)\right\}.
\end{equation*}
From Lemma~\ref{lem:2d}, it is easy to conclude that
the fraction of the common chosen row indices of $G^{\otimes n}$ between polar codes of
rate $R$ and RM codes of rate $R'$ tends to  $I(W)\min\{\frac{R}{I(W)},R'\}$ as $n \to \infty$.
\if0
\begin{IEEEproof}
The proof is similar to that of Lemma~\ref{lem:conv}.
Let $L_n\triangleq \log X_n$.
From (c2), for $m\le n-1$,
the relation \eqref{eq:Lupper} holds.
Hence, for any fixed $m$ and any $\delta\in(0,1)$,
\begin{align}
\nonumber&\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n) \ge n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\\
\nonumber&\quad\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n)
\bigg)\\
\nonumber&\le\limsup_{n\to\infty}\\
\nonumber&\quad\mathbb{P}\bigg(\log (-L_n)
\ge n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\\
\nonumber&\quad\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n),\;
 X_m\le \delta\bigg)\\
&\quad+\limsup_{n\to\infty}\mathbb{P}\bigg(\log (-L_n) \ge n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\;
 X_m> \delta\bigg)\label{eq:2terms2}
\end{align}
for any $f(n)=o(\sqrt{n})$ and $g(n)=o(\sqrt{n})$ as $n\to\infty$.
The first term in the right-hand side of~\eqref{eq:2terms2} is upper bounded as
\begin{align}
\nonumber&\limsup_{n\to\infty}  \mathbb{P}\bigg(\log (-L_n)
\ge n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\\
\nonumber & \sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n),\;
 X_m\le \delta\bigg)\\
\nonumber&\le\limsup_{n\to\infty}\mathbb{P}\bigg(\sum_{i=m}^{n-1}\log S_i + \log (-L_m) \ge n\mathbb{E}[\log S]+t\sqrt{n\mathbb{V}[\log S]}+f(n),\\
\nonumber&\quad\sum_{i=0}^{n-1} \log U_i > n\mathbb{E}[\log U] + v\sqrt{n\mathbb{V}[\log U]} + g(n),\;
X_m\le \delta\bigg)\\
&\le Q(\max\{t,v\})\mathbb{P}(X_m\le\delta).
\label{eq:2dclt}
\end{align}
The last inequality is obtained by (c4) and the central limit theorem.
The rest of the proof is the same as the proof of Lemma~\ref{lem:conv}.
\end{IEEEproof}
\emph{Remark:}
In the last inequality of~\eqref{eq:2dclt}, by using two dimensional central limit theorem, the tight result is obtained.
\fi

\section{Selection rule of rows} \label{select_sec}
The proof of Lemma~\ref{lem:main_direct}
suggests a way to help us select the good indices in a more computationally efficient way.
In the proof, $\ell$-ary expansion of row indices of $G^{\otimes n}$ corresponds to realizations of $B_1,\dotsc,B_n$.
The proof of Lemma~\ref{lem:main_direct} implies that
it is sufficient to select rows in $\mathcal{D}_m(\beta)\cap\mathcal{H}_m^{n-1}(t)$
in order to achieve the asymptotically optimum performance.
It should be noted that the event $\mathcal{D}_m(\beta)$ applied to the Bhattacharyya process
$\{Z_n=Z(W_n)\}_{n\in\ensuremath{\mathbb{N}}}$ of $W$ depends on the channel $W$, whereas the event $\mathcal{H}_m^{n-1}(t)$ is channel-independent.
This observation leads to the following selection rule:
The first $m=s(n)\triangleq(\log n + \log\log c)/\beta$ digits of the row indices are determined in the channel-dependent way.
Then, the following $(n-m)$ digits are determined in the RM way, i.e.,
those combinations of digits $(B_m,\,\dotsc,\,B_{n-1})$ giving large values of $\sum_{i=m}^{n-1}\log D_{B_i}(G)$ are selected.
In this rule, only the first $\Theta(\log n)$ digits should be determined depending on the channel.

The above argument can further be extended in a recursive manner.
Let $\mathcal{C}_m^{n-1}(\epsilon) \triangleq \{(n-m)^{-1}\sum_{i=m}^{n-1}\log S_i\ge \mathbb{E}[\log S] -\epsilon\}$.
Then, it is sufficient to select rows in
$\mathcal{D}_{m_0}(\beta)\cap\mathcal{C}_{m_0}^{m_1-1}(\epsilon)\cap\mathcal{H}_{m_1}^{n-1}(t)$
where $m_1=s(n)$ and $m_0=s(m_1)$ since $\mathcal{D}_{m_1}(\beta)$ and $\mathcal{D}_{m_0}(\beta)\cap \mathcal{C}_{m_0}^{m_1-1}(\mathbb{E}[\log S] - \beta)$ are asymptotically equal.
(Use $\mathcal{C}_m^{n-1}(\epsilon)$ instead of $\mathcal{H}_m^{n-1}(t)$ in the proof of Lemma~\ref{lem:main_direct}. A similar argument can be found in~\cite[Section IV-B]{Ari09}.)
From this observation, only $\Theta(\log\log n)$ digits have to be determined depending on the channel.
By iterating this argument, we obtain the selection rule in which only
\begin{equation} \label{select}
\Theta(\overbrace{\log\dotsm\log}^{k} n)
\end{equation}
digits depend on the channel for any $k\in\mathbb{N}$.
From the argument so far, we deduce that even though the behavior of $Z_n=Z(W_n)$ depends
on the channel $W$ as well as the whole sequence $\{B_0,\,B_1,\,\ldots,\,B_{n-1}\}$,
the ``fate'' regarding whether it approaches 0 or 1 when $n$ is large, is mostly determined
by the channel $W$ and a prefix of $\{B_0,\,B_1,\,\ldots,\,B_{n-1}\}$ with a relatively small length. Thus,  to choose the  indices of the
 channels $W_{\ell^n}^{(i)}$ that have the best quality,  the first
sublinear number of significant bits of the $\ell$-ary expansion of $i - 1$ are determined
depending on the channel and the rest are determined in a RM-like fashion.
It should be noted that the above argument is valid in the large-$n$ asymptotics.
It does not mean that one can make the number of digits to be determined
in the channel-dependent manner arbitrarily small.

Although the good indices of the rows of $G^{\otimes n}$  can be selected using density evolution~\cite{MT09},
in practice storage and convolution of probability density functions is exponentially (in block-length $N$)
 costly in terms of memory and computation.
Recently, several authors have considered accurate and efficient implementation of
the density evolution procedure~\cite{TV}, \cite{RHTT}.
The above-mentioned construction rule can be useful in reducing the number of convolutions and the number of levels in the quantization of channels.

\if0
Finally, we mention an interesting consequence of the above observation; For simplicity, we consider the kernel $G=\bigl[ \begin{smallmatrix} 1 &0 \\ 1& 1 \end{smallmatrix}
\bigr]$ and assume
that from $G^{\otimes n}$ we construct a polar code of rate $R$ and a RM code of rate $R'$.
Hence,  the indices
corresponding to the polar code are chosen by the method of channel polarization over $W$ and
the indices of the RM code correspond to the $2^n R$ rows of $G^{\otimes n}$ with highest hamming weight. Let $m=\log n$.  Denote
 by $\mathcal{I}_m$ a subset of size $2^mI(W)$ of the indices of the channels $\{W_{\ell^m}^{(i)}\} _{1 \leq i \leq \ell^m}$ that have the smallest Bhattacharyya value.  Equivalently, we can consider $\mathcal{I}_m$ as a subset of $\{0,1\}^m$  by  mapping a member $x \in \mathcal{I}_m$ to the binary expansion of $x-1$. We now construct the set $A \in \{0,1\}^n $ from $\mathcal{I}_m$ as follows: assume $x \in \mathcal{I}_m$  and denote by   $(b_1,\cdots, b_m)$  the binary expansion of $x-1$. Consider all binary $n$-tuples that their first $m$ bits are exactly equal to $(b_1,\cdots, b_m)$ and choose $2^{(n-m)}\frac{R}{I(W)}$ of them that have the largest hamming weight. We let $A_x$  be the set of all the $2^{(n-m)}\frac{R}{I(W)}$ chosen $n$-tuples. Finally, we let $A= \cup_{x \in \mathcal{I}_n} A_x$.  Form  the selection rule mentioned above, as $n$ grows large, almost all the good polar indices  that correspond to the matrix $G^{\otimes n}$ with rate $R$ fall in the set $A$.  Now, since $\frac{|\mathcal{I}_m|}{2^m} = I(W)$ and also for each member of $A$ the contribution of the first $m$ bits to its hamming weight vanishes as $n$ grows, the  size  of the intersection of the sets $A$ and the se oft RM indices with rate $R'$, when normalized by $2^n$, tends to the value $I(W)\min\{R,R'\}$ as $n \to \infty$.
\fi

redpajama_set_name: RedPajamaArXiv


We've featured hundreds of free Windows applications over the course of 2009 that we hoped might bolster your productivity, workflow, or your PC usefulness in one form or another. Here's a look back at the most popular Windows downloads of the year.
2009 was a big year for Windows, and Windows 7 was the most important ingredient in Windows' solid year. (In fact, you'll notice that several of this year's most popular downloads are related to Windows 7 in one way or another.) Sure it's not exactly an application but rather a full-blown operating system, but it only makes sense that a new version of Windows would top the list of Windows downloads for the year. It started with the Windows 7 beta download in January, which had a ton of hiccups. It was released, then pulled, then released again, then extended because of the trouble Microsoft had handling the demand. (Actually, we just think they underestimated the web.) Later, in May, Microsoft released the Windows 7 release candidate. You even jumped on the chance to try Windows 7's beautiful new themes.
Folks who were still using Vista also flocked to Vista's Service Pack 2 (32-bit; 64-bit) to keep their PCs secure and up to date.
One of our very own readers released his popular desktop configuration as a installable utility that brings a handful of great customization and productivity tools to your desktop. It's called Enigma 2.0. Then Rainmeter, another very popular desktop customization tool, set Enigma as its default desktop. Fancy pants.
Ever wish you could enjoy some of the finer tools available to Linux but stay comfortably in your Windows desktop? Sure you could run a virtual machine, but Portable Ubuntu for Windows runs an entire Linux OS as a Windows application. Better yet, it's portable.
The release of Windows 7 left a lot of XP users wishing they could get in on some of that snazzy new eye candy. Seven Remix XP is a free utility that does its best to bring Windows 7's comely looks to XP.
Another result of the Windows 7 launch: A whole lot of us were rebuilding systems from the ground up, which often means a lot of tedious downloading and installing one app at a time. Ninite makes it easy, streamlining the download and installation processes for tons of the most popular free Windows apps, including most of our 2009 Lifehacker Pack.
Apart from all the new eye candy, Windows 7 really tickled our fancy with tons of incredible new keyboard shortcuts. For folks still chilling out with XP or Vista, we released Windows 7 Shortcuts, a lightweight utility written to bring some of the best new shortcuts of Windows 7 to previous versions of Windows.
By virtue of reading Lifehacker, you're more likely than not the most schooled person among your friends and family when it comes to fixing a bum PC. It's a dubious honor, because it also means you generally are the person who gets called when something goes wrong. Computer Repair Utility Kit puts a handful of useful PC repair utilities in one handy, portable suite. Update: It appears the surge of readers looking to download this one brought down the server; luckily helpful reader Richard offers a mirror of the download on his own server here.
Like to keep items you want to access frequently easily accessible on your desktop but don't want to deal with the added clutter? Fences arranges your cluttered desktop icons into containers so you can clean up the mess into useful groups of shortcuts—or optionally hide them altogether.
We're of the mind that Microsoft's security tools are good enough—including the new and impressive Microsoft Security Essentials antivirus app—but that doesn't mean many of you don't get excited when the AVG 9 Free update is available. It's still the favorite antivirus app of Lifehacker readers (who doesn't love free), though we're sensing a slow but steady sea of change on this one.
Google Chrome is just over a year old, but it's made huge strides among early adopters. Chrome came out with its stable 2.0 release in May, then followed up with a Chrome 3.0 release in September. Early adopters willing to try their luck in the beta and dev channels get more features, which we detailed in our power user's guide to Google Chrome. Whichever version of Chrome interests you most, it's clear that it was a pretty good year for Chrome.
Hulu Video Downloader was a fun little app that lasted about as long as you could say Hu... that is, it doesn't work anymore. But when it did, it grabbed videos from the popular video service for your offline viewing pleasure, and you were eager to try it out.
Safari's 4.0 beta release for Windows came with a lot of bugs and some serious eye candy, but despite the interest at release, we can't imagine many people still stick with Safari on Windows over, say, Firefox, Chrome, Opera, etc.
Google very recently announced a free DNS service they boasted as fast, but rather than take their word for it, we pointed you toward namebench (and several readers also pointed toward the excellent DNS Name Server Benchmark). It tests various popular DNS servers to find what's really going to be the fastest choice for your system.
We're sort of junkies for maps and 3D, so when Google Earth 5 was released, we were pleased as punch. The update featured historical imagery, ocean maps, and improved world touring capabilities. Maybe we just like saving ourselves some dough in these tough economic times with a little Google Earth sightseeing.
The notorious Firefox memory slow-downs may have some of us down on the reliable old 'fox, but that doesn't mean we aren't all still eager to grab the latest and greatest releases and stick with it as our default browser—whether it's the big Firefox 3.5 release or the Firefox 3.6 beta (1, 2, 3, or 4). We're looking forward to more great Firefox'ing in 2010.
If your webcam is sitting around collecting dust, try out Motion Detection, a free, motion-detecting security camera application. It'll snap pics and video when it detects movement, can upload the results via FTP, and more fun at-home security stuffs.
Sure it was two years since Thunderbird's 2.0 release, but at least they didn't disappoint. Thunderbird 3.0 comes with solid new search and filtering tools, better looks, and a great new tabbed interface.
Free, open-source DVD ripping and encoding tool HandBrake released a pretty saucy update last month with a ton of fixes and improvements. It's no coincidence that it's always been our reader's favorite video encoder, and this year's big-ish (but still not 1.0) update should only help keep it there.
Linux users have a killer desktop management tool called Compiz Fusion that puts multiple desktop management on a 3D cube that we've always been jealous of on Windows. Open-source application DeskHedron brings a similar three-dimensional desktop management tool to Windows users.
Got a favorite Windows download from 2009 that you'd add to your list of favorites? Let's hear about it in the comments. If you're craving still more popular Windows downloads, you can also take a look back at the most popular free Windows downloads of 2008.


text:
Šalvěj bělolistá (Salvia leucophylla) je druh rostliny z rozsáhlého rodu šalvěj (Salvia) z čeledi hluchavkovitých (Lamiaceae). Vyskytuje se v jihozápadní Kalifornii a přilehlých oblastech Mexika. Její odrůdy se zde používají také jako okrasné rostliny a k produkci medu. V rámci rodu šalvěj je zařazena do podrodu Audibertia. Tento podrod obsahuje asi 20 druhů původních v Severní Americe, u nichž není vyvinut pákový mechanismus tyčinek; centrum jeho druhové rozmanitosti se nachází v kalifornské provincii a přilehlých pustinách Velké pánve.

Popis

Je to stálezelený, silně rozvětvený, aromatický keř, který dosahuje výšky 1 až 1,5 metru, s poléhavými až vzpřímenými výhony. Nadzemní části rostliny jsou bohatě pokryty bělavě šedými plstnatými chlupy.
Vstřícné, křižmostojně postavené listy se dělí na řapík a listovou čepel. Řapík je poměrně krátký. Jednoduchá listová čepel je podlouhle kopinatá, 2 až 8 cm dlouhá, s víceméně zkrácenou až srdčitou bází, tupým nebo zaobleným horním koncem, jemně vroubkovaným okrajem listu a vrásčitým horním povrchem. Listy jsou na jaře zpočátku zelené a v létě popelavě šedé.

Ve svém přirozeném prostředí rostlina kvete přibližně jeden měsíc, v období od dubna do června. Vzpřímené květenství je přerušovaný lichoklas lichopřeslenů. Květenství má průměr 1,5 až 4 cm a obsahuje mnoho květů. Bíle plstnaté listeny, které jsou na spodní straně často fialové, jsou eliptické nebo podlouhlé a dlouhé 8 až 15 milimetrů. Květ je oboupohlavný, zygomorfní, typicky pyskatého tvaru. Kalich je 8 až 12 milimetrů dlouhý, mírně nafouklý a se sotva patrnými kališními zuby. Pět růžových až levandulově zbarvených korunních lístků je srostlých do 6 až 13 mm dlouhé korunní trubky, která je zakončena dvěma nestejně tvarovanými korunními pysky: poměrně široký horní pysk, dlouhý 6 až 8 mm, je obvykle poněkud zprohýbaný a rozšířený, zatímco spodní pysk, dlouhý 4 až 5 milimetrů, má protáhlý, plochý střední lalok a dobře vyvinuté postranní laloky. Tyčinky a pestík vyčnívají velmi daleko z korunní trubky, blizna je dvojklaná. Plod je tvrdka poltící se do čtyř plůdků, které jsou šedohnědé, tmavě skvrnité, úzce eliptické, 2 až 3 milimetry dlouhé.
Ploidie druhu je 2n = 30.

Ekologie a rozšíření
Tato šalvěj roste endemicky v jihozápadní Kalifornii v oblasti táhnoucí se na jih podél centrálního pobřeží Sanfranciského zálivu přes Kalifornské pobřežní pásmo a Transverse Ranges až na sever mexického státu Baja California. Vyskytuje se na plně osluněných místech na pobřežních útesech a suchých otevřených svazích v nadmořských výškách do 1250 metrů, ve společenstvech nazývaných "Coastal Sage Scrub" či nízký chaparral, často v sousedství blízce příbuzné šalvěje černé (Salvia mellifera) a šalvěje bílé (Salvia apiana). Produkce monoterpenických silic funguje alelopaticky proti vyklíčení konkurenčních rostlin v její blízkosti.

Květy jsou opylovány hmyzem, především medonosnými i samotářskými včelami a čmeláky, ale též kolibříky. Někteří motýli prokousávají korunu květu ze strany, aby se dostali přímo k nektaru.

Využití
V Kalifornii se šalvěj bělolistá používá jako atraktivní a nenáročná okrasná rostlina odolná vůči suchu, jako nektarodárná rostlina pro včely a také jako protierozní rostlina v polopřirozené krajině, například ve výsadbách s jinými druhy kalifornských šalvějí, jako je Salvia clevelandii, se zmarlikou Cercis occidentalis nebo škumpou Rhus integrifolia. Při pěstování nejlépe prospívá v dobře propustných, chudých, štěrkovitých půdách s nízkým obsahem humusu a na plném slunci. Existuje několik kultivarů a hybridů: například 'Amethyst Bluff' (výška až 1,8 m, tmavě růžový květ), 'Point Sal Spreader' (půdopokryvná rostlina), 'Bee's Bliss' (půdopokryvná rostlina, větší, levandulově zbarvený květ, kříženec Salvia leucophylla × Salvia sonomensis), a 'Desperado' (výška až 3 m, větší, hustá květenství, kříženec Salvia leucophylla × Salvia apiana). Je však méně vhodná pro chladné a vlhké klima střední Evropy; navzdory své poměrně dobré zimovzdornosti až do −9 °C (zóna 8b) zde trpí přílišnou vlhkostí a musela by být přezimována v suchu.

Z květů šalvěje bělolisté je získáván druhový med, který je tmavě nachové až téměř černé barvy, slabě krystalizující, jemné květinové vůně a chuti. Indiánskými národy byla rostlina využívána k ceremoniálním a rituálním účelům.

Reference

Externí odkazy

Hluchavkovité
Flóra jihozápadu USA
Flóra Mexika
Medonosné rostliny
Subtropické okrasné rostliny

redpajama_set_name: RedPajamaWikipedia


text:
package com.subterranean_security.crimson.viewer.ui.screen.main;

import java.awt.BorderLayout;
import java.awt.event.MouseAdapter;
import java.awt.event.MouseEvent;

import javax.swing.JMenu;
import javax.swing.JMenuBar;
import javax.swing.JPanel;
import javax.swing.MenuSelectionManager;
import javax.swing.event.MenuEvent;
import javax.swing.event.MenuListener;

import com.subterranean_security.crimson.sv.permissions.Perm;
import com.subterranean_security.crimson.viewer.ViewerState;
import com.subterranean_security.crimson.viewer.store.ViewerProfileStore;
import com.subterranean_security.crimson.viewer.ui.UICommon;
import com.subterranean_security.crimson.viewer.ui.UIStore;
import com.subterranean_security.crimson.viewer.ui.UIUtil;
import com.subterranean_security.crimson.viewer.ui.screen.about.AboutDialog;
import com.subterranean_security.crimson.viewer.ui.screen.files.FMFrame;
import com.subterranean_security.crimson.viewer.ui.screen.files.FMFrame.Type;
import com.subterranean_security.crimson.viewer.ui.screen.generator.GenDialog;
import com.subterranean_security.crimson.viewer.ui.screen.netman.NetMan;
import com.subterranean_security.crimson.viewer.ui.screen.settings.SettingsDialog;
import com.subterranean_security.crimson.viewer.ui.screen.users.UserMan;

public class MainMenu extends JPanel {

	private static final long serialVersionUID = 1L;

	private JMenu mnControls;
	private MenuControls mc;

	public MainMenu() {
		init();
	}

	public void closeControls() {
		MenuSelectionManager.defaultManager().clearSelectedPath();
	}

	public void init() {
		setForeground(UICommon.bg);
		setBackground(UICommon.bg);

		setLayout(new BorderLayout(0, 0));

		JMenuBar menuBar = new JMenuBar();
		add(menuBar, BorderLayout.CENTER);

		mnControls = new JMenu();
		mnControls.setIcon(UIUtil.getIcon("icons16/general/multitool.png"));
		mnControls.setToolTipText("Controls");
		mnControls.addMenuListener(new MenuListener() {

			@Override
			public void menuSelected(MenuEvent e) {
				mc.startStreams();
			}

			@Override
			public void menuDeselected(MenuEvent e) {
				mc.stopStreams();
			}

			@Override
			public void menuCanceled(MenuEvent e) {
				mc.stopStreams();
			}
		});
		menuBar.add(mnControls);

		mc = new MenuControls();

		mnControls.add(mc);

		JMenu mnManagement = new JMenu();
		mnManagement.setIcon(UIUtil.getIcon("icons16/general/box_front.png"));
		mnManagement.setToolTipText("Management");
		menuBar.add(mnManagement);

		final WideMenuItem wmFiles = new WideMenuItem(UIUtil.getIcon("icons16/general/folder.png"), "Files",
				"Manage Filesystems");
		wmFiles.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				if (!ViewerState.isOnline()) {
					MainFrame.main.np.addNote("error", "Offline mode is enabled!");
				} else {
					FMFrame fmf = new FMFrame(Type.SV);
					fmf.setVisible(true);
				}

				wmFiles.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnManagement.add(wmFiles);

		WideMenuItem wmNetwork = new WideMenuItem(UIUtil.getIcon("icons16/general/computer.png"), "Network",
				"Manage Networking");
		wmNetwork.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {

				if (!ViewerState.isOnline()) {
					MainFrame.main.np.addNote("error", "Offline mode is enabled!");
				} else if (UIStore.netMan == null) {
					UIStore.netMan = new NetMan();
					UIStore.netMan.setLocationRelativeTo(null);
					UIStore.netMan.setVisible(true);
				} else {
					UIStore.netMan.toFront();
				}

				wmNetwork.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnManagement.add(wmNetwork);

		final WideMenuItem wmGen = new WideMenuItem(UIUtil.getIcon("icons16/general/linechart.png"), "Generator",
				"Create an Installer");
		wmGen.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				if (!ViewerState.isOnline()) {
					MainFrame.main.np.addNote("error", "Offline mode is enabled!");
				} else if (ViewerProfileStore.getLocalViewer().getPermissions()
						.getFlag(Perm.server.generator.generate_jar)) {
					if (UIStore.genDialog == null) {
						UIStore.genDialog = new GenDialog();
						UIStore.genDialog.setLocationRelativeTo(null);
						UIStore.genDialog.setVisible(true);
					} else {
						UIStore.genDialog.setLocationRelativeTo(null);
						UIStore.genDialog.toFront();
					}

				} else {
					MainFrame.main.np.addNote("error", "Insufficient permissions!");
				}

				wmGen.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnManagement.add(wmGen);

		WideMenuItem wmGlobalControl = new WideMenuItem(UIUtil.getIcon("icons16/general/cog.png"), "Control Panel",
				"Global Control Panel");
		wmGlobalControl.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
			}
		});

		mnManagement.add(wmGlobalControl);

		WideMenuItem wmLogs = new WideMenuItem(UIUtil.getIcon("icons16/general/error_log.png"), "Logs",
				"View Crimson Logs");
		wmLogs.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
			}
		});

		mnManagement.add(wmLogs);

		WideMenuItem wmUsers = new WideMenuItem(UIUtil.getIcon("icons16/general/clients.png"), "Users",
				"Manage Users/Groups");
		wmUsers.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {

				if (!ViewerState.isOnline()) {
					MainFrame.main.np.addNote("error", "Offline mode is enabled!");
				} else if (ViewerProfileStore.getLocalViewer().getPermissions().getFlag(Perm.server.users.view)) {
					if (UIStore.userMan == null) {
						UIStore.userMan = new UserMan();
						UIStore.userMan.setLocationRelativeTo(null);
						UIStore.userMan.setVisible(true);
					} else {
						UIStore.userMan.setLocationRelativeTo(null);
						UIStore.userMan.toFront();
					}

				} else {
					MainFrame.main.np.addNote("error", "Insufficient permissions!");
				}

				wmUsers.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnManagement.add(wmUsers);

		final WideMenuItem wmSettings = new WideMenuItem(UIUtil.getIcon("icons16/general/setting_tools.png"),
				"Settings", "Edit Preferences/Settings");
		wmSettings.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				// TODO DONT allow multiple instances
				SettingsDialog sd = new SettingsDialog();
				sd.setVisible(true);

				wmSettings.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnManagement.add(wmSettings);

		JMenu mnAbout = new JMenu();
		mnAbout.setIcon(UIUtil.getIcon("icons16/general/text_area.png"));
		mnAbout.setToolTipText("About");
		menuBar.add(mnAbout);

		final WideMenuItem wmAbout = new WideMenuItem(UIUtil.getIcon("c-16.png"), "About Crimson", "Info on Crimson");
		wmAbout.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				// TODO DONT allow multiple instances
				AboutDialog ad = new AboutDialog();
				ad.setVisible(true);

				wmAbout.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnAbout.add(wmAbout);

		final WideMenuItem wmUpgrade = new WideMenuItem(UIUtil.getIcon("icons16/general/barcode_2d.png"),
				"Serial Manager", "Add/Remove Keys");
		wmUpgrade.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				wmUpgrade.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnAbout.add(wmUpgrade);

		final WideMenuItem wmHelp = new WideMenuItem(UIUtil.getIcon("icons16/general/health.png"), "Help",
				"Open help center");
		wmHelp.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {
				MainFrame.main.np.addNote("info", "Testing the notification area");
				wmHelp.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnAbout.add(wmHelp);

		final WideMenuItem wmTour = new WideMenuItem(UIUtil.getIcon("icons16/general/steering_wheel.png"), "Tour",
				"Explore the interface");
		wmTour.addMouseListener(new MouseAdapter() {
			@Override
			public void mousePressed(MouseEvent e) {

				wmTour.resetBG();
				MenuSelectionManager.defaultManager().clearSelectedPath();
			}
		});

		mnAbout.add(wmTour);

	}
}

redpajama_set_name: RedPajamaGithub


The candidates running to become New York's next attorney general each sought to assure a tech-friendly audience that they would be defenders of network neutrality and entrepreneurship during a candidate forum on Aug 14.
The Communications Workers of America (CWA) is targeting Republican lawmakers and AT&T in a new advertising campaign aimed at highlighting the telecommunication company's job cuts after receiving a boost from the GOP's tax law.
The Communications Workers of America (CWA) is fighting hard against a Federal Communications Commission proposal they say endangers its members and could send CWA work to unskilled non-union contractors.
Lawmakers in the House passed a bill aimed at bolstering skills training for technical jobs in various industries.
As the tech industry has grown in power and influence, its politics have moved to the left. When people want to understand Silicon Valley's political leanings, they often look to CA's 17th Congressional District.


text:
On the Trail of Picasso
Written by Laurence
Pablo Diego José Francisco de Paula Juan Nepomuceno María de los Remedios Cipriano de la Santísima Trinidad Picasso… better known simply as Pablo Picasso, is perhaps Spain's most famous son. His life in Madrid is well documented, and the city is now home to some of his most famous artworks. ShMadrid tells you more on Picasso and where you can observe his works in Madrid.
Related article: Museum Hack: Seeing Madrid's Art for Free
History Picasso
Reina Sofia Museum
Thyssen-Bornemisza Museum
San Fernando Royal Academy of Arts
Photo by rocor via Visualhunt
Regarded as perhaps the greatest, most influential, and most controversial artist of the 20th century, Picasso's works encompass a breathtaking array of media and styles, roughly divided into his blue period, rose period, African period, and cubism. Showing exceptional artistic talent from an early age, Picasso's father sent him to art school at the age of 16. So it was that Picasso left Barcelona and made his home in Madrid for the first time, where he attended the San Fernando Royal Academy of Arts. Whilst he found the formal tuition stifling, it was the exposure to Madrid's art treasures which most influenced the young Picasso, and set him on a course which was to change the art world forever. He would spend long days at the Prado Museum, admiring particularly the works of Velázquez, Goya, and El Greco; the latter influencing his own style greatly. Whilst Spain's most comprehensive collection of Picasso's works lies in Barcelona at the Museu Picasso, Madrid still exhibits a great many works by the artist.
Perhaps Picassos's most famous and controversial work, Guernica, is a massive mural-sized oil painting, depicting the bombing of the village of Guernica during the Spanish Civil War. The painting chaotically depicts people, animals, and buildings in destruction as a comment on the devastation of war. Whilst it helped bring attention to Spain's plight, it cemented Picasso's position as a powerful political enemy of the state. Picasso was adamant that the painting not be exhibited in his home country until after peace was restored, which meant the painting toured the world before finally finding a permanent home in Madrid in 1981. You can now see Guernica at the Reina Sofia Museum in Madrid, as well as dozens of preliminary sketches and small paintings of the work. (Calle de Santa Isabel, 52, 28012)
Related article: The Most Beautiful Parks in Madrid
Photo by jean louis mazieres via Visualhunt
The Thyssen-Bornemiza Museum contains many Picasso works, including some of his most well-known images. The Frugal Meal, from his early blue period, is a masterpiece depicting poverty stricken figures, based on experiences of his time in Montmartre in Paris. Harlequin with a Mirror is a well-known more classical-style Picasso work. Intended to be a self-portrait, the painting references the Italian commedia dell'arte characters, and at the same time, marks a return to the classical Italian style. Picasso depicted scenes of bullfighting in literally hundreds of his works, and his painting Bullfight from 1934 is a typical Picasso work in which violence and the psyche are at odds, the bull often representing the artist's alter ego throughout his oeuvre. Cubism, in which objects are analyzed, broken up and reassembled in an abstracted form, was central to Picasso's early output. A quintessential work of this style is on display at the Thyssen, Man with a Clarinet. A man with a clarinet is depicted, in a way in which only the most basic cube-like elements can be deciphered. (Paseo del Prado, 8, 28014)
Those on the trail of Picasso and his time in Madrid can also visit the Royal Academy of Arts, of which he is an alumnus. The Academy itself is an impressive building, which now functions as a gallery and museum of artworks spanning 6 centuries. Many of Picasso's minor drawings and paintings are on display here, as are those of alumni Dalí, Goya, and de la Renta. It's a great place to soak up the history of Picasso, who spent some of his formative years as a student in these very halls. (Calle de Alcalá, 13, 28014)
*Main photo by jean louis mazieres via Visualhunt
What is your opinion on Picasso's life and work?
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Laurence is an Australian expat, who has been living in Europe for about 10 years.

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The report is based on the various aspects of taxation laws of Australia. The report provides solutions specific to the income tax provisions of Australia. An income tax assesse has to follow numerous taxation provisions for the providing income tax to the government officials. Therefore, it has to go through the various laws such as ITAA 1997, ITRA 1997 and other provisions. Income tax is calculated as per the slab rate defined by the Australian officials and also, further classified by residents and non-residents criteria. For the calculation of income tax, firstly, assessable income is calculated which is further deducted by the allowable deductions provided to the taxpayers of Australia. After the above calculations, tax liability of a tax payer is calculated and other charges are also calculated such as Medicare levy and Medicare levy surcharge. Further, there is a provision of benefits of withheld of PAYG and benefits of franked credits of dividend receivers and many other benefits (Martin, 2010.). All these deductions, reduces the tax liability of a taxpayer.
Therefore, the tax payable by the resident is nil. As he falls in the category of 0%, he does not need to pay anything.
(b) According to the slab rate of Australian taxation, an individual who is a non-resident of Australia has to pay 32.5 % of taxable income below $ 80,000.
Therefore, the non- resident has to pay an amount of $ 4,875 as the tax liability.
(c) According to the slab rate, corporations are taxed at the rate of 30% of the taxable income.
Therefore, the corporation has to pay $ 4,500 as a tax liability for the financial year 2015-2016.
(d) According to the slab rate, an Australian resident with a taxable income of $ 155,000 has to tax at the rate of 37% above $ 80,001 and has to pay $ 17,547 on the taxable income below $ 80,000.
Therefore, the resident has to pay $ 45,297 as the tax liability for the respective financial year.
Therefore, the non-resident has to pay $ 53,750 as a tax liability to the authorities.
(f) According to the slab rate, corporations are taxed at the rate of 30% of the taxable income. The taxable income of the corporation is $155,000.
Therefore, company has to pay $ 46,500 as a tax liability for the financial year 2015-2016.
Thus, the resident individual has to pay $ 89,797 as a tax liability to the authorities for the financial year 2015- 2016.
Thus, the non- resident has to pay $ 98,250 as tax liability for the respective financial year.
Thus, the company has to pay $ 76,500 as a tax liability for the financial year.
(j) An Australian company which is classified as a small business entity is taxed at the rate of 28.5 %.
Thus, the small entity business has to pay $ 29 as the tax liability.
(a) An individual whose taxable income is equal to or below $21,335 does not need to pay medical levy. In the case, the taxable income is $ 18,000, therefore, it does not need to pay Medicare levy.
Medical levy surcharge- the MLS is exempted up to the base income limit of $ 90,000 for singles and for families the amount is $ 180,000. In the above case, the taxable income is within the threshold limits; therefore, medical levy surcharge is not applicable.
(b) A senior individual whose taxable income is equal to or below $33,738 does not need to pay medical levy. In the case, the taxable income is $ 32,000, therefore, it does not need to pay Medicare levy.
Medical levy surcharge- Since, these are within the prescribed limits, therefore the surcharge would not be applicable.
Thus, the individual has to pay $ 900 as a medical levy charge for the financial year.
Medical levy surcharge- In the above case, the taxable income is $45,000 which is in the prescribed limits, therefore, medical levy surcharge will not imposed over it.
Therefore, the individual has to pay $ 900 as a the medical levy rate to the taxation officers.
Medical levy surcharge- The taxable income is $ 45,000, which is within the base income provided by the taxation laws. Therefore, the medical levy surcharge would not be applicable.
(e) According to the Income tax assessment act 1936, company do not need to pay Medicare levy and Medicare levy surcharge. These charges are mainly for the medical facilities of individuals, therefore these are levied over individuals and not over corporations. Since, the benefits of medical facilities can only be availed by individuals and not by corporations.
The medical levy amount for the individual is $ 2,200 for the financial year 2015-2016.
Medical levy surcharge - Individuals who hold appropriate private insurance do not need to pay medical levy surcharge. Since, this charge is mainly for motivating residents of Australia for taking up private health insurance. This would ultimately, reduce the burden over the public health care sector.
The medical levy charge for the individual is $2,200 for the financial year 2015-2016.
= $110,000* 1.25%= $ 1375.
Note- it is assumed that the individual is under the single category.
Thus, the medical levy amount to be paid by the individual is $ 3000.
Medical levy surcharge – this surcharge is calculated for the number of days it has not taken up the private health insurance. As per the rates prescribed, 1.5 % is applicable on the taxable income.
= $ 150,000 * 1.5% = $ 2250.
Note- it is assumed that the Australian resident falls under the single category.
Thus, the medical levy to be paid by the victor & Jackie are $ 2200 and $ 1500 respectively.
Medical levy surcharge - According to the slab rate for the family, Medical levy surcharge of victor & Jackie is 0 % as the income limit is lower than $ 180,000. This surcharge is calculated of the male spouse only because among the couple the higher taxable income is considered, and the taxable income of the female is lower.
(j) According to the slab rate, if an Australian couple has more than 1 child, than the minimum threshold increased by $ 1500. The minimum tier is $ 180,000 for the family category but it will increased by $4,500 ($1500 *3). Therefore, it will become $ 184,500.
The year ended 30th June, 2015.
The individual has full patient private hospital cover; therefore, no Medicare levy surcharge is levied.
1. Accessible income means all the income receipt for a financial year which is subject to taxed. It includes various items such as profits earned through business operations, commission income, compensation received; income earned outside Australia but is a Australia residents (Woellner et. al., 2011). In the above calculations, assessable income is calculated by adding the gross salary and bank interest received. In the above calculations, salary amounts to $32,000 and bank interest received amounts to $ 150.
2. Allowable deductions are the deductions which are provided by the taxation law. These deductions are claimed by the taxpayers for reducing the tax payable amount. In the above calculations, special work clothing is provided as a deduction.
Tax payable (before deductions) = $ 13,500 * 19% = $ 2565.
$ 31, 700 * 2% = $ 634.
6. PAYG is pay as you go. In this system, the taxpayer has to pay tax in advance. In the above calculations, the taxpayer has already paid $ 2,600 as a PAYG. In the above calculations, the assesse has claimed the deductions of PAYG.
7. The total tax payable by the assesse for the financial year (2014-2015) is $ 599. This has to be paid by the assessee to the government officials within the time limits assigned. Otherwise, it would attract liabilities.
8. All the calculations has been done in view of the Australian taxation provisions and legislatives.
1. As per the provisions of ITAA 1997, under section 4-15 income tax has to be paid every year and income tax is calculated for the income earned in the previous year. According to the section, tax payable income is calculated in three steps. Firstly, all the assessable income are added. Secondly, all the deductions are added. Finally, the deductions are deducted from the assessable income.
2. In the above calculations, assessable income is calculated by adding gross salary, Australian sourced dividend income (including franked dividend, gross franked dividend, unranked dividend) (Woellner et. al., 2011).
3. Since, there are no deductions. Therefore, no deductions are provided in the above calculations.
When calculated over taxable income which is $72,987, the tax amounts to around $ 21,377.
Tax payable = $ 35987* 32.5% = $ 11,696.
Total tax = 11696 + 3,572 = $ 15,268 (before benefits).
5. Medicare levy is a charge over the taxable income. According to the taxation law, this charge is levied over individuals only for health care, and not on any corporation or separate legal entity. Basically, Medicare provides health care to Australian residents. This fund is funded by the taxpayers (Robson et. al., 2011). A nominal rate of 2% is levied over the taxable income of the individual. In the above calculations, 2% is calculated over the taxable income of $ 72, 987, which amounts to 1460.
6. Franking credits are the credits provided to the dividend holder, in lieu of, the tax already paid by the company. This is done with a view to eliminate double taxation. When a company earns net profits, it pays tax over it to the officials and then, the rest portion is dividend among the shareholders (Minney, 2010). Therefore, in order to eliminate the tax payment by the individual's assesse, gross amount of tax credits are again added back in the assessable income of assesse and than taxed again, and then provided at the last as a deduction.
7. PAYG (pay as you go) is a system in which a regular payment as expected income tax liability is paid to the officials (Evans, C., 2012). These are actually advance payment of tax. In the above calculations, $ 15,100 is withheld as PAYG. These are further deducted from the tax payable by the assesse.
8. Finally, the total tax payable amount is calculated. In the above question, the tax payable by Rafael's is $ 541.
According to the taxation laws of Australia, the practical questions are solved. Income tax is a tax levied by the Australian authorities as a measure of source of income and all the tax payers who has a source of income and lies within the income tax limits, pays income tax accordingly. In the report, the first question relates to the assessing of income tax on the basis of resident, non-resident and company. Further, the second question relates to the calculation of medical levy and medical levy surcharge. These surcharges are levied only on individuals and not on corporations. In the third question, there is a calculation of income tax payable by an individual. This practical involves calculation of assessable income and then provision of deductions and further calculation of Medicare levy & Medicare levy surcharge. In the final question, also, there is a calculation of income tax liability, with the aid of income tax provisions and legislatives. The overall project, has provided an insight into the income tax provisions and income tax assessment of Australia.
Ato.gov.in, 2016, "Australian government", Australian Taxation office, available at https://www.ato.gov.au/individuals/medicare-levy/medicare-levy-surcharge/income-thresholds-and-rates-for-the-medicare-levy-surcharge/ accessed on 27.11.2017.
Woellner et. al., 2011. Australian Taxation Law Select: legislation and commentary. CCH Australia.
Evans, 2012, "Tax Governance Issues: Managing System Complexity. Economic Papers: A journal of applied economics and policy, 31(1), pp.30-35.
Woellner et. al., 2011, "Australian Taxation Law Select: legislation and commentary". CCH Australia.
Minney, 2010, "The valuation of franking credits to investors", JASSA, (2), p.29.
Robson, A., Ergas, H. and Paolucci, F., 2011, "The analytics of the Australian private health insurance rebate and the Medicare levy surcharge". Agenda: A Journal of Policy Analysis and Reform, pp.27-47.
Martin, 2010, "Native Title Payments and Their Tax Consequences: Is the Federal Government's Recommendation of a Withholding Tax the Best Approach?".


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We had brunch at Open Farm Community a few Saturdays ago. To be honest, I wanted to visit Open Farm Community because of their playground. I saw it on someone's Instagram account and just had to take the kids there to play. The cow and pig are too adorable for words and the sand is nice and clean.
I love how bright and open the place is and the staggering amount of greenery surrounding it. The food is great and the service is very decent. Adrian is happy to go back again just for the coffee. There's a large garden area where the kids can explore and run around before the food comes. There's even a little shop selling children's clothing, bags, waterbottles, etc.
Even though we really liked the food, it is quite pricey and portions are rather small. I also wish they would open earlier, especially on weekends. But I like that they are trying to promote local farming and cooking with locally-grown produce through events such as community gardening sessions (last Sat of the month) and workshops like Pasta Masterclass and Jam & Sauces Masterclass.
a game of ping pong, perhaps?
This entry was posted in Uncategorized and tagged brunch, food, nature, play, playgrounds, shopping on October 24, 2015 by delphine.
I don't have a green thumb and tend to kill most of my plants so a terrarium that needs minimal watering and care sounds perfect to me! We were invited to a workshop at The Plant Story hosted by actualyse.com and the kids made their own little terrariums with whimsical glass creatures.
Anya and Adam were fascinated to learn that the plants in the terrarium did not have to be watered daily and learnt a bit about how plants transpire and photosynthesize. They were also very taken by the little Murano glass creatures that they could decorate their terrariums with.
The kids were really proud of the terrariums they made and truly enjoyed the hands-on experience. I think they learnt some stuff about plants more effectively than if they had merely read about it or watched in on TV!
actualyse.com believes that children learn in different ways and they are there to provide a platform to support learning for all intelligences and learning styles, needs, and all ages. I believe in Multiple Intelligences, and as a parent, I definitely want to cater to my child's learning style. You can take an MI quiz here and find out more about actualyse.com and their vision 'to inspire and empower everyone to be the best they can be'.
This entry was posted in Uncategorized and tagged nature, reviews on June 21, 2015 by delphine.


Reports released to the public by the New York City Education Department on 18,000 teacher rankings are causing quite an uproar.
Coming into play is the Freedom Of Information Act versus privacy rights, public humiliation, main purpose, accuracy and error margins of the report.
The report ranks teachers on student-gains in math and English exams over a five-year period and up until the 2009-2010 school year.
The reason it has taken this long to release the results is because it's been through a long, costly, legal battle set forth by The United Federation of Teachers in an attempt to keep the results out of the media.
Before your red flags rise, there are some good reasons the teachers and the districts didn't want the information made public.
The assessments are made using a mathematical formula. The past performance of the student is determined and entered into a computer program. The program predicts the future performance of the child and, if the child does better than predicted, the teacher receives a high ranking. If the student does worse than predicted, the teacher gets a low ranking.
A good example of why the study is controversial would be "the ceiling factor:" If a teacher scored "high" in their first year it would be virtually impossible to improve on the ranking for the following years which would show no progress. Another good example is a student that changes schools at midyear or has numerous teachers in the same year. Parental involvement, location of the school, teacher transfers, morale, school reputation, quality of the principal, tutors, illnesses, outside distractions, teacher-retirements, terminations and promotions also leave the results very questionable. Twenty three percent of the teachers ranked no longer work in the school system at all. Other critics of the report say there are just blatant mistakes on it. One teacher even received a ranking while she was on maternity leave. Others are being ranked on math when they're English teachers and vice versa.
A typical New York City teacher ranking would show: 5% low, 20% below average, 50% average, 20% above average and 5% high. 521 teachers were ranked in the bottom 5 percent for two or more years, and 696 were repeatedly in the top 5 percent.
Annual reports were given to the teachers at the end of each year so the teachers themselves have seen the short-term results, but now the parents of the students and the public have free access to all the information including both long and short-term results.
The New York City Education Department acknowledges the problems and questionable aspects of the report. In the same breath they defend and uphold it.
The ranking system is not unique to New York City. For many years it has been used in cities across America and some of the bigger media outlets have developed their own ranking systems. The effort is a way to get a better, unbiased view than the single evaluation of a school Principle.
Is it fair? Can it be used as another way for parents to point their finger at teachers for some of the issues they should be taking responsibility for? With the math-score error margin at as much as 35 points and the English-score error margin at 53 points, it's probably not a good idea to put too much stock in it. Although it could be used as a good base to work out problems, highlight concerns and get the study down to a more scientific and reliable system.
Once all legal options to keep the study from going public were exhausted, it was given to the media via CDs on February 24, 2012 and can be found at SchoolBook.


text:
"With You in Mind" can refer to:

 With You in Mind (Marianne Faithfull song), a 1966 song by Marianne Faithfull
 With You in Mind (Allen Toussaint song), a song by Allen Toussaint included in his 1978 album Motion, popularized by Aaron Neville
 With You in Mind (album), a 1962 album by Alma Cogan
 With You in Mind (Alma Cogan song), a song from the album


Human power is not a given, but a precious commodity. Your body has to take in energy from the environment, process it and distribute it throughout your entire body.
Defense allows you to stay calm, centered and vital even when the vibes around you are difficult or aggressive.
Learn how to use energy to create the foundations for good relationships, troubleshoot difficult ones and build on the ones you value.
The realms of energy and spirit come together in your body. CS provides tools to help you have the energy and structure to achieve a spiritual path.


text:
Guilt free flying? How airlines can achieve net-zero emissions
One of the saddest elements of the Covid-19 pandemic, writes Ashley Norris of Transition Earth, has been the way that it has largely put the brakes on international travel.
Yet even before Covid hit environmentally concerned consumers were already starting to agonise over their use of airlines. Airline travel accounts for 2.5% of global carbon emissions.
So what can the airlines do to make their services more sustainable?
The key trend among them appears to be aiming to become net-zero by 2050. Last year a host of airlines including American Airlines, British Airways, Cathay Pacific, Finnair, Iberia, Japan Airlines, Malaysia Airlines, Qantas, Qatar Airways, Royal Air Moroc, Royal Jordanian, S7 Airlines, Sri Lankan Airlines and Fiji Airways all signed up to an initiative called Oneworld which agreed on a range of measures to aim for zero emissions.
Now Cathay Pacific has been giving more details of how it is aiming for zero emissions by 2050.
In its sustainable development report, which was published last week Cathay Pacific says it "embraces its responsibility to lead the charge towards sustainable aviation and ensuring future generations can experience the joy of travel".
Similar to other airlines Cathay Pacific's plans for delivering net-zero are based around a trio of innovations
improving aircraft efficiency
scaling up sustainable aviation fuel use
offsetting remaining emissions via a carbon credit programme.
Improving efficiency is largely based around upgrading exciting aircraft and investing in new more efficient ones. There is also an aim of reducing engine use on the ground.
Cathay Pacific faces a bigger challenge in the shift to sustainable aviation use. At the current time, it only accounts for 2% of the total amount of fuel it uses. The company estimates that it will purchase 1.1 million tonnes of sustainable aviation fuel within the next ten years. There are limitations on its use though, mainly because of a lack of supply. The company hopes that issues around its distribution will be resolved as time goes by.
One potential magic bullet could be to switch to either electric or hydrogen-based fuels. However, the technologies, while developing, are still immature and not suitable for long-range travelling.
The most controversial element of Cathay Pacific's net-zero plans is carbon-offsetting. The company is working with The Gold Standard to purchase verified credits. "The unprecedented pandemic has shaken the world and showed us that 'business-as-usual' is not an option when dealing with an imminent global risk," Cathay Pacific Group's chief executive Augustus Tang said.
"Climate change, potentially a much more disruptive crisis, calls for ramped-up efforts."
Fancy a trip from Liverpool to Belfast or Barcelona to the Balearic Islands but concerned about the carbon footprint of aeroplane travel? Well a a small Bedford-based company is promising a surprising solution: commercial airships. Hybrid Air Vehicles (HAV), which has developed a new environmentally friendly airship 84 years after the Hindenburg disaster, has named a string of routes it hopes to serve from 2025.
The routes for the 100-passenger Airlander 10 airship include Barcelona to Palma de Mallorca in four and a half hours. The company said the journey by airship would take roughly the same time as aeroplane travel once getting to and from the airport was taken into account, but would generate a much smaller carbon footprint. HAV said the CO2 footprint per passenger on its airship would be about 4.5kg, compared with about 53kg via jet plane.
May 27, 2021 Ashley Norris
Tagged American Airlines British Airways Cathay Pacific environment Finnair flying HAV Iberia Japan Airlines Malaysia Airlines Qantas Qatar Airways Royal Air Moroc Royal Jordanian S7 Airlines Sri Lankan Airlines
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