text:
\section{Introduction}

\section{Introduction}
The issue of estimating camera motion from two views is classical by now, and
methods from projective and algebraic geometry towards its solution
were employed at an early stage (e.g. \cite{Sturm1869}). The beginning of this present century
witnesses the application of sophisticated methods from computational
commutative algebra in order to rephrase the equations into a form from which
solutions can be  obtained with relative ease.
The relationship between the views is established by finding correspondences between
point pairs taken from both images. The fundamental matrix
faithfully encodes the geometric
relationship between
the two images.  For normalised cameras, the fundamental matrix coincides with
the essential matrix.
In general, the two matrices are   related
through  the camera calibration. Hence, if the calibration is known, it is sufficient
to estimate the essential matrix in order to solve the relative pose problem.
From a conceptional as well as a computational point of view, it makes sense to use
only few correspondences of image points in order to estimate
the essential matrix $E$.  And different samples of $n$ correspondences
lead to a set of candidate essential matrices from which an optimal choice can
be made.

Since $E=(e_{ij})$ is a projective $3\times 3$ matrix, i.e.\ only determined up to a scalar factor,
the number of parameters to be found is at most $8$. As each pair of corresponding
image points leads to a linear equation in the $e_{ij}$, it suffices to
take $n=8$, as has been showed in \cite{8pt}. However, this linear method does not
take into account further constraints on $E$. So, a $7$-point algorithm was
developped \cite{7pt-a,7pt-b} (cf.\ also \cite{Zhang1998} for an overview). The
$5$-point algorithm of \cite{Nister5pt} uses
the  minimal number of point
correspondences necessary
for determining $E$. The non-linear constraints lead to
homogeneous equations of degree three in four unknowns which are transformed through an
elimination process into a univariate equation of degree $10$.
Hence the number of complex solutions is not more than $10$.
The elimination process can be simplified \cite{easy5pt},
or replaced by Gr\"obner basis methods \cite{SteweniusDiss}.
There is a lot of existing work towards
optimising the performance of this method in the computational algebraic geometry community.
Stewenius et al.\ compare the performance  of elimination and Gr\"obner basis methods
\cite{SEN2006}.

Correspondences often suffer a geometric perturbation, i.e.\
the correct point $u'$ in image $I'$ corresponding to point $u$ in
image $I$ is mostly found up to a small error vector $\epsilon$
such that $u'+\epsilon$ instead of $u'$ is matched to $u$.
In this article,
a hierarchical method
based on interval subdivision is developped to
handle this problem. A natural way of encoding hierarchies
is provided by the $p$-adic numbers, where $p$ is a fixed prime number.
In our case, we can use $p=2$ and represent image points
by pairs of binary expansions
$$
a=\sum\limits_{n=0}^\infty a_i2^i
$$
with coefficients $a_i$ equal to $0$ or $1$. These expansions can be infinite,
theoretically. Practically, the finiteness of resolution means approximation
through truncation. The framework for this method is $p$-adic geometry
which has been applied in video segmentation and data analysis
\cite{B-PXK2001,Murtagh2004,Brad-Jo
}.

In the context of relative pose problems, we will use the so-called
{\em lifting method}
for solving the equations. This is provided by Hensel's lemma which says that
under certain conditions a solution   of a given equation
modulo $p$ can
be expanded to a $p$-adic solution.
Its proof is constructive. In fact, this is  the $p$-adic analogue
of a Newton iteration. Applied to the equations of the $n$-point problems
for $n=5,7,8$, we obtain the result that for many choices
of point correspondences, a $2$-adic solution can be constructed.
The $2$-adic essential matrix then allows to hierarchically
approximate the rigid motion by truncation. A side effect
of the lifting method is its high robustness to geometric perturbations.
The encoding method  ensures further that the number of Newton iterations
required is proportional to the order of resolution desired.

The article is structured as follows. The next two sections are a brief introduction to
$p$-adic numbers, and to Hensel's lemma
in a multivariate formulation. This is followed by a section
on $2$-adic encoding of image pixels for a $p$-adic camera model.
The last section applies Hensel's lemma to  the $2$-adically defined equations
for the eight-, seven-, and five-point problems.

\section{$p$-adic numbers}

The $p$-adic numbers were first described by Hensel in
\cite{pnum}. They are expansions of the form
\begin{align}
a=\sum a_ip^i \label{pexpansion}
\end{align}
into powers of a fixed prime number $p$ and coefficients $a_i\in\mathset{0,\dots,p-1}$.
If there are only finitely many terms in (\ref{pexpansion}),
this defines a rational number. Any natural number has a finite $p$-adic expansion
(\ref{pexpansion})
with $i\ge 0$. The important observation is that expansions with infinitely
many positive powers of $p$ have a meaning.
Namely, by defining
$$
\absolute{a}_p=p^{-\nu_a},
$$
where $\nu_a\in \mathds{Z}$ is the smallest exponent occurring in expansion (\ref{pexpansion}),
one obtains a metric for which the partial sums
$$
a_N=\sum\limits_{i\le N}a_ip^i
$$
converge to $a$:
$$
\absolute{a-a_N}_p=\frac{1}{p^{N+1}}\longrightarrow 0\quad\text{for}\quad N\longrightarrow \infty.
$$
The domain of all $p$-adic numbers is denoted by $\mathds{Q}_p$
and densly contains the rational numbers $\mathds{Q}$ with respect to this metric.
Those $p$-adic numbers $a$ with $\nu_a\ge 0$ are the $p$-adic integers,
denoted as $\mathds{Z}_p$. This domain contains densely the usual integers $\mathds{Z}$.
An equivalent description of $p$-adic integers is given by
$$
\mathds{Z}_p=\mathset{a\in\mathds{Q}_p\mid\absolute{a}_p\le 1}.
$$

Approximation of $p$-adic integers by their partial sums,  simply termed ``truncation'',
has an algebraic formulation. In its simplest form, a $p$-adic integer $a$
can be given by its coefficient $a_0\in\mathset{0,\dots,p-1}$,
and another  $p$-adic integer $b$ having the same coefficient
$b_0=a_0$ approximates $a$ up to that order of magnitude.
This is the case if and only if $a-b$ is divisible by $p$.
Hence, we arrive at
$$
\mathds{Z}_p/p\mathds{Z}_p\cong\mathds{Z}/p\mathds{Z}\cong\mathds{F}_p,
$$
where the latter is the finite field with $p$ elements.
Similarly, the algebraic formulation of approximation up to terms of higher order
is given by
$$
\mathds{Z}_p/p^{N}\mathds{Z}_p\cong\mathds{Z}/p^N\mathds{Z}=:\mathcal{Z}_{p^N}.
$$
In other words, congruences modulo $p^N$ yield finite approximations
of $p$-adic numbers by their partial sums up to terms of order $N$.
It is this property which makes $p$-adic numbers very suitable for hierarchically organised
data. In later sections, we will see how this algebraic formulation can be used in stereo vision.
A standard reference for $p$-adic numbers is \cite{Gouvea}.

\section{Hensel's lemma}

An important method in $p$-adic analysis is the so-called ``lifting''
of zeros of polynomials from $\mathds{F}_p$ to $\mathds{Z}_p$.
This uses the fact that
$$
\mathds{Z}_p/p\mathds{Z}_p\cong\mathds{Z}/p\mathds{Z}\cong\mathds{F}_p
$$
by reducing a given equation $f(X)=0$ over $\mathds{Z}_p$ modulo $p$
to an equation $f\mod p$ over $\mathds{F}_p$.
 Hensel's lemma then gives a criterion when a solution $x_1$ of $f\mod p$
leads to a solution of $f$. This solution $\xi$, if it exists,
is constructed by an iterated process. Namely, first a solution $x_2$
of $f\mod p^2$ is constructed from $x_1$, and from this a solution $x_3$ of $f\mod p^3$ etc.
Each step of this iteration yields an approximation to the true solution
in $\mathds{Z}_p$ in the  sense
that
$$
f(x_i)\equiv 0\mod p^{i},
$$
which in terms of the $p$-adic norm translates to
$$
\absolute{f(x_i)}<\frac{1}{p^i}.
$$
In other words, the sequence $f(x_i)$ converges $p$-adically to the
value $f(\xi)=0$. The construction process guarantees further that
$x_i$ also converges to the $p$-adic solution $\xi$.

\smallskip
We now state a multivariate Hensel's lemma,
but not in the most general form. For this, $\mathds{Z}[X_1,\dots,X_n]$
denotes the polynomial ring in $n$ variables with integer coefficients.

\begin{thm}[Multivariate Hensel's lemma] \label{hensel}
Let $$
f=(f_1,\dots,f_m)\in\mathds{Z}[X_1,\dots,X_n]^m
$$
with $m\le n$, and let $k\ge 2$.
Suppose that the vector $x=(x_1,\dots,x_n)\in\mathds{Z}^n$ is a solution of the congruences
\begin{align*}
f_1(X_1,\dots,X_n)&\equiv 0 \mod p^{k-1}\\
&\vdots\\
f_m(X_1,\dots,X_n)&\equiv 0 \mod p^{k-1}
\end{align*}
and that the matrix
$$
D_f(x)=\left(\frac{\partial f_i}{\partial X_j}\right)
$$
is  modulo $p$ of rank $m$.
Then there exist $t=(t_1,\dots,t_n)\in \mathset{0,\dots,p-1}^n$ such that
$$
f(x+p^{k-1}t)\equiv 0 \mod p^k.
$$
In particular, the equations $f(X)=0$ have a solution in $\mathds{Z}_p$.
\end{thm}

\begin{proof}
Consider the linear part in the Taylor expansion of $f$ in $x$:
$$
f(X)=f(x)+D_f(x)\cdot(X-x)+\text{terms of higher order},
$$
where $X=(X_1,\dots,X_n)$ denotes the vector of variables.
By assumption, it holds true that
$$
f(x)=p^{k-1}\cdot a
$$
for some vector $a\in \mathds{Z}^m$.
Due to the rank condition, the system of congruences
\begin{align}
a+D_f(x)\cdot t\equiv 0 \mod p \label{crucialcong}
\end{align}
has a solution vector $t\in\mathset{0,\dots,p-1}^n$.
Hence,
\begin{align*}
f(x+p^{k-1}t)&\equiv f(x)+D_f(x)\cdot p^{k-1}t
\\
&\equiv p^{k-1}(a+D_f(x)\cdot t)\equiv 0 \mod p^{k}.
\end{align*}
Iteration proves the last assertion.
\end{proof}

Usually, Hensel's lemma is stated in the case of a single univariate polynomial $f$
having modulo $p^{k-1}$ a zero $x$.
The rank condition translates to $f'(x)\not\equiv 0\mod p$, i.e.\ $x$ is a simple zero of $f$
modulo $p$.
The proof is a $p$-adic analogue of a Newton iteration.
From (\ref{crucialcong}) it follows for $m=n$ that the rank condition
implies uniqueness of the lift.
The univariate case can be found e.g.\ in \cite{Gouvea}.

\section{$p$-adic projective cameras}

\subsection{$p$-adic encoding of image pixels}

Assume that a rectangular 2D image $I$ is given by $M\times N$ pixels and that
$m,n$ are minimal such that
$M\le 2^m$, $N\le 2^n$.
We will use a binary encoding of a given image point with pixel coordinates
$$
(x,y)\in \mathset{0,\dots,M-1}\times\mathset{0,\dots,N-1}.
$$
This means that $x$ and $y$ will be represented by a pair of binary
numbers obtained in a hierarchical manner.
Namley, consider one coordinate at a time, say $x$.
It can be arrived at by a sequence of iterated  subdivisions
of the interval ${0<\dots<2^{m}-1}$ into intervals of equal length.
After $m$ iterations each interval obtained contains precisely one pixel $x$-coordinate.
This means that a given value of $x$ lies in a uniquely determined nested sequence of
intervals produced by this subdivision process.
The intervals form a rooted binary tree whose leaves
correspond uniquely to the $x$-coordinates of pixels.
By assigning for a given interval its left half the value $0$, and $1$ for its
right half, we obtain the binary representation
$$
r(x)= \sum\limits_{\nu=0}^{m-1}\alpha_\nu2^\nu,\quad\alpha_\nu\in\mathset{0,1}
$$
by traversing the path from root down to $x$ and picking up the zeros and ones along the way.

We will interpret the natural number $r(x)$ as a $2$-adic integer: $r(x)\in\mathds{Z}_2$.
Each partial sum
$$
r_\ell=\sum\limits_{\nu=0}^\ell\alpha_\nu2^\nu
$$
is a $2$-adic approximation
of $r(x)$. Its error is bounded by the $2$-adic distance
$$
\absolute{r_\ell-r(x)}_2.
$$
In fact, $r(x)$ itself is a $2$-adic approximation to some imaginary ``pixel'' $\xi$
of infinite precision. The number $m$ is given by the degree of resolution
with which $\xi$ is viewed on the given image.
And an infinitely precise pixel would have a $2$-adic expansion with possibly infinitely
many coefficients equal to $1$.

\smallskip
We proceed similarly for the $y$-coordinate, and obtain an encoding
$$
c_2\colon I\to\mathds{Z}_2\times\mathds{Z}_2,\quad (x,y)\mapsto (r(x),s(y)),
$$
where $s$ is the binary encoding of the $y$-coordinate.

\medskip
Any $p$-adic encoding of a number $x$ bears the problem that its $p$-adic
approximations $r_\ell$ are determined by arithmetic properties of $x$,
ignoring their (euclidean) geometric properties. This means that
although $x_\ell$ could be   $p$-adically very close to $x$, it could be
at large distance for the euclidean norm.
Hence, arbitrary $p$-adic encodings of pixel data would be quite unsuitable
for many applications in computer vision.
However, the binary encoding through interval splitting does not
suffer this disadvantage.
Namely, the map $c_2$ respects the image geometry in the following sense:

\begin{prop}
The map
$$
\iota_2\colon \sum\limits_{\nu=0}^{\infty}\alpha_\nu2^\nu\mapsto\sum\limits_{\nu=0}^{\infty}\alpha_\nu2^{-\nu}
$$
yields an inclusion of $\mathds{Z}_2$
into the real interval $[0,2)$. It has the
property
$$
\absolute{r-s}_2<2^{-\ell}\Rightarrow
\absolute{\iota_2(r)-\iota_2(s)}_\infty< 2^{-\ell}
$$
for all $\ell\in\mathds{N}$.
\end{prop}

\begin{proof}
Any sum of negative powers of $2$ can be majorised by
a (possibly shifted) geometric sum. Hence, such a sum converges for the real metric.
This implies that the map $\iota_2$ is well-defined for arbitrary $2$-adic numbers.

The inclusion property is clear, at least for finite $2$-adic expansions.
However,  the map $\iota_2$ is injective on all of $\mathds{Z}_2$.
Otherwise, assume that
$$
\iota_2(a)=\iota_2(b)
$$
for some $a\neq b$. Then $a=\sum a_i2^i$ and $b=\sum b_i2^i$ must differ in some coefficient:
$$
a_n\neq b_n.
$$
Let $n$ be minimal such that this occurs. Then
it follows that
$$
\absolute{\iota_2(a)-\iota_2(b)}_\infty\ge\frac{1}{2^N},
$$
because the inequality holds true for all partial sums with at least $N$ terms.
This is a contradiction. Hence, $\iota_2$ is injective.

ssume $\absolute{r-s}_2<2^\ell$.
This means that
the $2$-adic expansions of  $r$ and $s$ have the first $\ell+1$ terms in common.
The last assertion follows from this.
\end{proof}

As a consequence, we obtain an embedding of the image $I$ into $[0,2)^2$
via composition of $c_2$ and $\iota_2\times \iota_2$.

\subsection{$p$-adic camera model}

Recall that a projective camera is a map between projective
spaces
$$
\kappa\colon\mathds{P}^3\to\mathds{P}^2
$$
given by a $3\times 4$-matrix of rank $3$.
Usually, cameras are modelled as being defined over the real numbers,
i.e.\ they are given by real $3\times 4$-matrices.
However, when dealing with  data coming from  3D-objects from the real world,
cameras are usually approximated by rational matrices.
In this way, we arrive at rational cameras:
$$
\kappa\colon\mathds{P}^3(\mathds{Q})\to\mathds{P}^2(\mathds{Q})
$$
as maps between the rational points of projective spaces.
For such cameras, the real completion is only one choice of many.
Hence, $p$-adic cameras
$$
\kappa\colon\mathds{P}^3(\mathds{Q}_p)\to\mathds{P}^2(\mathds{Q}_p)
$$
could also be considered in order to use methods from $p$-adic geometry
for camera computations.

In fact, since finite resolution prevents stereoscopic vision at arbitrarily large
distance,
it can become possible to shift coordinate systems in such a way that
cameras are described by matrices whose entries are natural numbers,
at least approximately.
In the following subsection, we will show how this said approximation can
be done for the $2$-adic norm. Precisely the hierarchical method
used in the previous subsection in 2D extends in a natural way to 3D
in order to arrive at
$2$-adic camera model
$$
\kappa\colon\mathds{P}^3(\mathds{Z}_2)\to\mathds{P}^2(\mathds{Z}_2).
$$

\subsection{How to interpret a $2$-adic essential matrix}

Assume that through correspondences between some points $u\in I$, $u' \in I'$
in $2$-adic camera images $I,I\cong\mathds{P}^2(\mathds{Z}_2)$,
 a $2$-adic essential matrix
$E\in\mathds{Z}_2^{3\times 3}$ has been produced. This matrix is the limit of
matrices $E_\nu\in\mathcal{Z}_{2^\nu}^{3\times 3}$.
Through its factorisation
$$
E_\nu=T_\nu R_\nu
$$
into a skew-symmetric matrix $T_\nu$ and a rotation $R_\nu$
it allows to determine a point $U_\nu\in\mathds{P}^3(\mathcal{Z}_{2^\nu})$
of which the two cameras obtain the image points $u_\nu,u'_\nu \mod 2^\nu$.
These are related through $E_\nu$:
$$
u_\nu^T\cdot E_\nu\cdot u'_\nu=0.
$$
The $U_\nu$ converge $2$-adically to a point $U\in\mathds{P}^3(\mathds{Z}_2)$,
and the cameras
$$
\kappa,\kappa'\colon\mathds{P}^3(\mathds{Z}_2)\to\mathds{P}^2(\mathds{Z}_2)
$$
yield $\kappa(U)=u,\kappa'(U)=u'$.
In the same way as $u_\nu$ is an approximation of pixel $u$ at resolution $\nu$,
the 3D-point $U_\nu$ is an approximation of voxel $U$ at resolution $\nu$,
because we can apply the map $\iota_2$ in precisely the same manner
also to three dimensions to obtain the inclusion
$
\mathds{P}^3(\mathds{Z}_2)\to [0,2)^3.
$

\section{Reconstruction from  point correspondences}

We will in the following content ourselves with solving the reconstruction problem
with two calibrated cameras by finding (possible candidates for) essential matrices
in the $2$-adic setting.

\bigskip
Given $n$ correspondences,
we arrive at the system of linear equations
\begin{align}
u_i^TE\,u_i',\qquad i=1,\dots,n. \label{essentialcond}
\end{align}
For $n\le 9$, we can use the Hensel lifting method to
solve these linear equations, written as
\begin{align}
Ax=0 \label{linearequation}
\end{align}
with coefficient matrix $A\in\mathds{Z}_2^{n\times 9}$. Due to finite resolution
and our coding method,
the matrix entries lie in some set $\mathset{0,\dots,2^\nu-1}$.
Hence, the matrix does not change its shape modulo $2^\nu$.

Assume that we are given a basis
$B=(b_1,\dots, b_m)$ in normal form of the solution space of (\ref{linearequation})
modulo $2$. We will interpret $B$ as an ordered set, but
also as a matrix whose columns are the vectors $b_1,\dots, b_m$.

\begin{thm}[Linear Hensel's lemma] \label{nptlift}
If the rank $m$ of $A\mod 2$ equals $\Rank(A)$,
then   $B$ lifts to a set of linearly independent solutions of (\ref{linearequation})
with natural numbers as entries.
\end{thm}

\begin{proof}
The linear equation (\ref{linearequation}) is equivalent to a linear equation
in staircase normal form. This normal form can be viewed as a system of
$\Rank(A)$
equations in the $9$ unknowns. By assumption, $m=\Rank(A)$. Hence, Theorem \ref{hensel} applies,
and $B$ lifts to a set of $m$ solution vectors
$$
\tilde{B}=\mathset{\tilde{b}_1,\dots,\tilde{b}_m}
$$
to (\ref{linearequation}).

\medskip
Let us re-examine the proof of Theorem \ref{hensel} in order to see that  $\tilde{B}$
is linearly independent.
First, notice that $B$ is read off the staircase normal form  of $A \mod 2$.
This means that each $b\in B$ has  in some row $j_b$ entry $1$, and below it all entries are zero.
Further, all other entries of row $j_b$ of matrix $B$ are zero. The sequence $(j_b)_{b\in B}$ is strictly increasing
with respect to the order of occurrence of $b$ in $B$.
Now, a lift of $b$ to $b^{(k)}$ modulo $2^k$ has the property that the entry in row $j_b$ is
odd, and all entries below are even. Likewise, all other entries of
row $j_b$ in a lift of $B$ to $B^{(k)}$ modulo $2^k$ are even.
This description shows that the rows given by the sequence $(j_b)$ form a submatrix
of $B^{(k)}$ having rank $m$.
Since this holds true for all $k>0$, it follows that $\tilde{B}$ is linearly independent.

The lifting process stops after a finite number of steps, because of our initial assumptions.
\end{proof}

\subsection{Reconstruction from $8$ points}

Assume now that the rank of $A\mod 2$ be $n$, and that a basis $B=\mathset{b_1,\dots,b_{9-n}}$
of the solution space be given in normal form. Then, by Theorem \ref{nptlift},
finding a lift to $\mathcal{Z}_{2^\nu}$ of $B$
yields a basis of the solution space of (\ref{linearequation}).

As an application, we obtain a  $2$-adic version of the $8$-point algorithm of
\cite{8pt}, simply by setting $n=8$. However,
in the same way as its original, this ignores the rank constraint
$\det(E)=0$ for the essential matrix. Hence, we obtain the result:

\begin{thm}
Under the assumptions above for
the matrix $A\in\mathds{Z}_2^{8\times 9}$,
the corresponding $8$-point problem has a unique solution $E$,
if additionally  $\Rank(E)=2$. If it can be assumed that $A\in\mathcal{Z}_{2^N}$,
then  $E$ is computed after $N-1$ iterations from $E\mod 2$.
\end{thm}

\begin{proof}
Since $\Rank(A\mod 2)=8$, there is one solution basis vector $b\in\mathcal{Z}^9$.
By Theorem \ref{nptlift}, it lifts to a non-trivial solution $e$ of $Ax=0$.
If the the matrix $E$ corresponding to $e$ is of rank $2$, it is the unique solution
to the $8$-point problem. The last assertion is an immediate consequence of
the
 proof of Theorem \ref{nptlift}.
\end{proof}

\subsection{Reconstruction from $7$ points}

The $7$-point method by \cite{7pt-a,7pt-b} yields $7$ linear constraints
(\ref{essentialcond}) plus the cubic constraint
\begin{align}
\det(E)=0.\label{detcond}
\end{align}
Let us write that system of equations as $f(X)=0$.
If the $7$ points are sufficiently in general position, then
the rank of
$D_f(x)\mod 2$ is $8$ for some solution $x$ modulo $2$:
\begin{align}
f(x)\equiv 0\mod 2, \label{opencond1}
\end{align}
and we can lift to a $2$-adic solution.
The reason is:

\begin{lem}
For $h:=\det(E)$ it holds true that
\begin{align}
D_h(e)\not\equiv 0\mod 2,\label{opencond2}
\end{align}
if $e\in \mathds{Z}_2^{3\times 3}$ is a sufficiently general instance of $E$.
\end{lem}

\begin{proof}
The polynomial $h$ is of degree three in the variables given by the entries of $E$.
Let $x$ be such an entry.
Then $h_x:=\frac{\partial h}{\partial x}$ is a polynomial of degree $2$.
Modulo $2$ only those terms of $h_x$ vanish which are of the form $2xy$ for some other variable $y\neq x$.
Since $h=\det(E)$ is a square-free polynomial, this can never happen.
\end{proof}

In $2$-adic analytic geometry, the conditions (\ref{opencond1}) and (\ref{opencond2})
define an open subset in the space of all $3\times 3$-matrices with entries in $\mathds{Z}_2$.
This can be seen easily by translating (\ref{opencond1}) to the
inequality
$$
\absolute{f(x)}_2<1,
$$
and dealing similarly with (\ref{opencond2}).

We can go further and derive explicit conditions for the existence of $\mathds{Z}_2$-rational
solutions. Namely, write down the $1$-parameter solution of the linear equations as
$$
E=xE_1+(1-x)E_2,
$$
and obtain a polynomial
$$
\det(E)=h(x)=ax^3+bx^2+cx+d
$$
 of degree $3$.
Now consider $h(x)\mod 2$. In case $d\equiv 0\mod 2$, zero is a
simple zero in $\mathds{F}_2$ if and only if
\begin{align}
c\equiv 1\mod 2.
\label{simpzerocond1}
\end{align}
In case $d\equiv 1\mod 2$, one is a simple zero in $\mathds{F}_2$ if and only if
\begin{align}
b\equiv c\equiv 0\mod 2\label{simpzerocond2},
\end{align}
in case $a\equiv 1\mod 2$,
and
\begin{align}
b\equiv0,\quad c\equiv 1\mod 2 \label{simpzerocond3}
\end{align}
otherwise.

\begin{thm} \label{2-adic-7pt}
The $7$-point problem has a $\mathds{Z}_2$-rational solution for many choices of $7$ point correspondences. Concretely, this is the case  if (\ref{simpzerocond1}),
(\ref{simpzerocond2}), or (\ref{simpzerocond3}) hold true
in their respective cases, together with the requirement that the rank of $E$ be
precisely $2$.
\end{thm}

\begin{proof}
Solve the $7$-point problem modulo $2$, and lift whenever  possible
as discussed above.
\end{proof}

\begin{rem}
Due to non-linearity of the constraints, we cannot expect anymore to
be able to lift solutions modulo $2$ to solutions which are defined over the natural numbers.
In other words, generally, the iteration will never stop unless an order of resolution
is specified. Then, a $2$-adically approximate solution will be obtained.
This is not different from the classical situation over the real numbers.
\end{rem}

\subsection{Solving the $5$-point non-linear equations}

Given five corresponding pairs of image points yields
(\ref{essentialcond}) with $n=5$. If the rank of the corresponding matrix is five,
the general solution can be written as
\begin{align}
E=xE_1+yE_2+zE_3+wE_4 \label{lincomb}.
\end{align}
Inserting this into the trace condition \cite{tracecondref}
\begin{align}
2\cdot EE^TE-\Trace(EE^T)\cdot E=0
 \label{tracecond}
\end{align}
yields $9$ cubic equations in four variables. We follow the
easily understandable
method of
elimination via hidden variables used by \cite{easy5pt},
and obtain the linear  equation
$$
C(z)\cdot X=0,
$$
after setting $w=1$.
Here,
$$
X=(x^3, y^3,x^2y,xy^2,x^2,y^2,xy,x,1)
$$
is the vector of monomials,
and the entries of $C(z)$ are polynomials in $z$.
Now,  one seeks $z$ such that
\begin{align}
\det(C(z))=0. \label{deg10poly}
\end{align}
The left hand side turns out to be a polynomial of degree $10$.

\smallskip
We will use the same lax formulation of our theorem as for the $7$-point problem,
but will be more specific in the proof:

\begin{thm}
There exists  a $\mathds{Z}_2$-rational solution to the $5$-point problem
for many choices of $5$ corresponding pairs of points.
\end{thm}

\begin{proof}
Let
$$
g(z):=\det(C(z))=\sum\limits_{i=0}^{10}a_iz^i.
$$
We consider two cases.

First, assume $a_0\equiv 0\mod 2$. In this case,
$g(0)\equiv 0\mod 2$, and $z=0$ is a simple zero modulo 2 if and only if
$a_1\equiv 1\mod 2$.

Secondly, if $a_1\equiv 1\mod 2$, then $f(1)\equiv 0\mod 2$ if and only if
the number of odd coefficients in $g(z)$ is even.
Since
$$
f'(z)\equiv a_9z^8+a_7z^6+a_5z^4+a_3z^2+a_1\mod 2,
$$
$1$ is a simple zero modulo 2 if and only if  in addition
the number of odd coefficients in $f'(z)$ is odd.
\end{proof}

\begin{rem}
Hensel's lemma can be interpreted in this context as a $p$-adic stability
result. Namely, assume that, due to correspondence error, a given choice
of $n$ points yields perturbed equations
$$
f(x)+\epsilon(x)=0,
$$
where $f(x)$ contain the ``true'' coefficients perturbed by some noise
coefficients contained in $\epsilon(x)$
with $2$-adic maximum norm
$$
\norm{\epsilon}_2=\max\mathset{\absolute{\epsilon_i}_2\mid i=1,\dots,m}\le 2^{-N},
$$
where $m$ is the number of noise coefficients $\epsilon_i$.
This means that the noise coefficients satisfy the congruence
$$
\epsilon_i\equiv 0\mod 2^N.
$$
Hence, the first $N$ iterations of Hensel lifting will lead to identical
approximations to the solution of the unperturbed equations
$$
f(x)=0.
$$
As the $2$-adic encoding comes from interval subdivisions,
this observation implies a greater stability
in comparison with the classical   approach over the real numbers.
In particular, the existence of a liftable solution modulo $2$
is not affected by perturbations $\epsilon$
with $\norm{\epsilon}_2\le\frac{1}{2}$.
This is definitively in contrast to the situation over the real numbers,
where small perturbations in coefficients can drastically change the number of
real solutions. That issue is addressed for the 5-point relative pose problem e.g.\
in \cite{alt5pt}.
\end{rem}

\section{Conclusion}

An encoding scheme for image pixels
 through  hierarchical
 interval subdivision is proposed. This allows a $2$-adic encoding of pixels driven by geometry.
As an application to stereo vision, the 8-, 7- and 5-point equations are formulated
with coefficients from the ring $\mathds{Z}_2$ of $2$-adic integers.
These polynomial equations are solved using some multivariate forms of Hensel's
lemma. The essential matrices are obtained in the form of sequences of
matrices modulo powers of $2$, corresponding to $2$-adic approximations to
the exact solutions in $\mathds{Z}_2$, whenever these exist.
Conditions on coefficients of equations modulo $2$ implying the existence
of $\mathds{Z}_2$-rational solutions are derived.
One feature of the hierarchical encoding is that the number
of iterations in solving the linear parts of the equations
is logarithmic in the number of pixels.
Also the precision in the solution
of the non-linear equations is directly related to the desired resolution in 3D.
An immediate consequence of Hensel's lemma is that $p$-adically small perturbations
of the equations do  not affect the first approximations to their solution.
Further, the existence of liftable solutions is not affected by relatively large
perturbations.
This indicates a greater computational benefit  from
 the $2$-adic approach
compared to the classical approach using computational complex  algebraic
geometry before discarding non-real solutions.

\begin{acknowledgements}
Sven Wursthorn is   thanked for the introduction into this fascinating topic,
and Boris Jutzi for multiple fruitful discussions.
\end{acknowledgements}

redpajama_set_name: RedPajamaArXiv


text:
P. G. Wodehouse 1881—1975
Blandings Castle (1)
Psmith (3)
School Stories (7)
Jeeves and Wooster (3)
P.G. Wodehouse was a comic writer who enjoyed enormous popular success during a career of more than seventy years and continues to be widely read. Despite the political and social upheavals that occurred during his life, much of which was spent in France and the United States, Wodehouse's main canvas remained that of pre-war English upper-class society, reflecting his birth, education, and youthful writing career.
An acknowledged master of English prose, Wodehouse has been admired both by contemporaries such as Hilaire Belloc, Evelyn Waugh and Rudyard Kipling and by modern writers such as Douglas Adams, Salman Rushdie, Zadie Smith and Terry Pratchett. Sean O'Casey famously called him "English literature's performing flea", a description that Wodehouse used as the title of a collection of his letters to a friend, Bill Townend.
Best known today for the Jeeves and Blandings Castle novels and short stories, Wodehouse was also a playwright and lyricist who was part author and writer of 15 plays and of 250 lyrics for some 30 musical comedies. He worked with Cole Porter on the musical Anything Goes and frequently collaborated with Jerome Kern and Guy Bolton. He wrote the lyrics for the hit song "Bill" in Kern's Show Boat, wrote the lyrics for the Gershwin - Romberg musical Rosalie, and collaborated with Rudolf Friml on a musical version of The Three Musketeers. (source: Wikipedia)
A Damsel in Distress (1919)
A Damsel in Distress is an early novel from comic genius, P.G. Wodehouse, about the aristocratic Marshmoreton family—a precursor to the Blandings series...read more »
A Prefect's Uncle (1903)
One of Wodehouse's early novels set in an English public school, a school story that revolves around cricket, stolen money, and an embarrassing uncle (w...read more »
Death at the Excelsior (1921)
Death At The Excelsior is a highly recommended introduction into the world of Wodehouse. A sterling collection of early short stories from the master of...read more »
Indiscretions of Archie (1921)
It wasn't Archie's fault really. It's true he went to America and fell in love with Lucille, the daughter of a millionaire hotel proprietor and if he di...read more »
Jill the Reckless (1920)
Jill had money and was engaged to be married to Sir Derek Underhill. But when she suddenly becomes penniless, she finds herself no longer engaged. Refus...read more »
Love Among the Chickens (1906)
After seeing his friend Stanley Featherstonehaugh Ukridge for the first time in years, Jeremy Garnet is dragged along on holiday to Ukridge's new chicke...read more »
Mike (1909)
Michael "Mike" Jackson is the youngest son of a renowned cricketing family. Mike's eldest brother Joe is a successful first-class player, while another ...read more »
My Man Jeeves (1919)
My Man Jeeves, first published in 1919, introduced the world to affable, indolent Bertie Wooster and his precise, capable valet, Jeeves. Some of the fin...read more »
Piccadilly Jim (1917)
In Piccadilly Jim, Jimmy Crocker has a scandalous reputation on both sides of the Atlantic and must do an about-face to win back the woman of his dreams...read more »
Psmith in the City (1910)
Psmith in the City was originally released as a serial in The Captain magazine, between October 1908 and March 1909, under the title The New Fold. It co...read more »
Psmith, Journalist (1915)
Continuing the adventures of the silver-tongued Psmith, one of Wodehouse's best loved characters, and his friend Mike Jackson. The story begins with Psm...read more »
Right Ho, Jeeves (1934)
When Jeeves suggests dreamy, soulful Gussie Fink-Nottle don scarlet tights and false beard to win over soppy Madeline Bassett, Bertie Wooster doubts thi...read more »
Something New (1915)
Young neighbours and fellow-writers Ashe Marson and Joan Valentine, newly met and both in need of a change of direction, find themselves drawn down to B...read more »
Tales of St. Austin's (1903)
St Austin's School is the setting for these twelve delightful early Wodehouse stories. A nostalgic look at English public-school life at the turn of the...read more »
The Adventures of Sally (1921)
When Sally Nicholas became an heiress, she had to cope her brother's wild theatrical ambitions and the defection of her fiance, his replacement being a ...read more »
The Clicking of Cuthbert (1922)
Wodehouse's brilliant but human brand of humor is perfectly suited to these stories of love, rivalry, revenge, and fulfillment on the links. While the O...read more »
The Coming of Bill (1919)
The Coming of Bill is the nearest Wodehouse ever came to a serious novel, although the influence of the musical comedies he was writing at the time is n...read more »
The Gem Collector (1909)
The action begins with playboy bachelor Jimmy Pitt in New York; having fallen in love on a transatlantic liner, he befriends a small-time burglar and br...read more »
The Gold Bat (1904)
When O'Hara and Moriarty, two boys at Wrykyn School, tar and feather the statue of a pompous local MP, O'Hara mislays at the scene of their crime a tiny...read more »
The Head of Kay's (1905)
It is the general view at Eckleton school that there never was such a house of slackers as Kay's. Fenn, head of house and county cricketer, does his bes...read more »
The Intrusion of Jimmy (1910)
The intrusion of Jimmy is a fast-paced farce about love and burglary. Playboy Jimmy Pitt is a betting man, and he reckons that breaking into a house isn...read more »
The Little Nugget (1913)
The Little Nugget is one of the novels in which Wodehouse found his feet, a light comic thriller set in an English prep school for the children of the n...read more »
The Man Upstairs (1914)
Wodehouse's well-known gift for satisfying plots and comic surprises is evident on every page, but there are also signs of his debt to earlier writers i...read more »
The Man With Two Left Feet (1917)
A miscellaneous collection mostly of stories concerning relationships, sports and household pets. It does not feature any of Wodehouse's regular charact...read more »
The Pothunters (1902)
When someone breaks into the cricket pavilion and steals two silver cups, the whole school is agog. Could it possibly be an inside job? Nothing less tha...read more »
The Prince and Betty (1912)
The story tells of how unscrupulous millionaire Benjamin Scobell decides to build a casino on the small Mediterranean island of Mervo, dragging in the u...read more »
The White Feather (1907)
In order to save his reputation and the honour of his house at school after he shames himself by running away from a fight between fellow pupils and tou...read more »
Three Men and a Maid (1921)
Wilhelmina Billie Bennett, red-haired daughter of American millionaire Rufus, loves golf, dogs and Tennyson and is to marry Eustace Hignett, the weak, p...read more »
Uneasy Money (1916)
For William, Lord Dawlish, it seemed the realization of his dreams. He could marry the girl he loved. Of course, things are not quite so simple. The fam...read more »
William Tell Told Again (1904)
Wodehouse's retelling of the William Tell legend in prose, verse and with illustrations. First published on November 11, 1904 by Adam & Charles Blac...read more »

redpajama_set_name: RedPajamaCommonCrawl


If you're unfamiliar with Atlantis Shoes, take the time to get familiar because our brand is changing the game. Innovation is our brand's middle name, combining comfort and sustainability to make a standout shoe.
As a Sydney based footwear brand and known as the "most comfortable shoe on earth," finally, we have our very own Atlantis Shoes store popping up for the springtime!
From today until Nov 24, our amazing team of shoe experts will be at your disposal to find the perfect pair of Atlantis Shoes for you at Greenwood Plaza, 36 Blue St, North Sydney.
As an exclusive offer, all new arrival orders placed at our Spring Pop Up will receive 10% OFF by showing this page.


text:
Heritage Honda Gloucester 01452 429020
Jazz e:HEV
New Honda Civic 5 Door
Caring For your Honda
Honda Civic Type-R Breaks Another Lap Record
<img data-cke-saved-src="https://bluesky-cogmisc.cdn.imgeng.in/images/d439b3f8-6b91-45c9-9e05-c1c3adecea08.jpg?width=1088&scale=both&quality=80" src="https://bluesky-cogmisc.cdn.imgeng.in/images/d439b3f8-6b91-45c9-9e05-c1c3adecea08.jpg?width=1088&scale=both&quality=80" style="border-width: 0px;" width="1088" alt="" hite="" honda="" civic="" type-r="" on="" racetrack"="">
The Honda Civic Type-R is making big waves on the tracks once again. When you think of speed, you probably think Porsche or Ferrari.
However, Honda took the most epic Type-R Limited Edition to the table, and what it offered to set a lap record at the infamous Suzuka Circuit.
The Honda Civic Type-R breaks another lap record with an astonishing 2 minutes 23.993 lap, and for a front-wheel-drive that is quite something.
The Suzuka Circuit is well known for its incredible chicanes and corners that pose a challenge to even the most experienced drivers. The figure 8 layout is the track of a race car driver's dreams. The 1.2km back straight passes right over an earlier part of the circuit too, which is stunning to watch.
So how did the Honda Civic Type-R break another lap record? It has some incredible lightweight components, steering upgrades and suspension upgrades. The stripped-back driver-focused interior means the Honda Civic Type-R pushes boundaries in all the right ways. In fact, it's pushing the boundaries of what is possible for front-wheel-drive cars everywhere, and indeed on one of the most demanding tracks in motorsport.
The Suzuka Circuit has been highly regarded for many years and has a reputation for being a vital testbed for developing those highspeed Honda vehicles since 1962. One of the highlights of Formula 1, Suzuka puts most cars to the test.
In February 2020, the Honda Civic Type-R Limited Edition underwent what would be its final performance evaluations. In the end, the development car has all of the same tech and specs that are representative of the production version.
With zero modifications and not a single performance enhancement, the Type-R Limited Edition is now on sale across the EU.
When people think about Honda, they think about reliability, the family car. Little do they know that the Honda Civic Type-R Limited Edition ripped through the Suzuka track with no regard for what a front-wheel-drive vehicle should be able to achieve, and delivered a stunning 2 minute and 23.993-second lap.
What About The Lap Details?
The Honda Civic Type-R Limited Editions cornering grip was what elevated it, and made it stand out. The standard Michelin Cup 2 tyres were worth their weight in gold, and seconds too. The longer corners on the Suzuka Circuit required some understeer, but that led the Honda Civic Type-R Limited Edition to a stunning lap time.
The 320hp VTEC turbo pushed the Honda Civic Type-R Limited to a massive 225kph (140) into the 130R.
What is the downside to all of this? The downside is that you're unlikely to be able to test-drive your Honda Civic Type-R Limited Edition on the chicanes and figures 8 of the Suzuka Circuit anytime soon.
You can, however, book a test drive for the stunning Honda Civic Type-R at Heritage Honda today.
Heritage Honda is situated in Gloucester and is part of the Heritage Automotive Group. Established in 1996 and is now the fastest-growing automotive dealership group across the south-west of England offering new and approved used Honda, including Motability cars, and also providing servicing & MOT Testing.
Heritage Automotive | 03142712 | Vat number: GB675797954 | 16 Lower Road, Churchfields, Salisbury, Wiltshire, SP2 7QD
Heritage Honda Gloucester, A40 Northern Bypass, Longford, Gloucester, GL2 9DJ


text:
How Salt Lake's infrastructure, climate goals make it an 'attractive' Olympic host city
By Carter Williams, KSL.com | Posted - Jan. 10, 2023 at 10:04 p.m.
Devin Logan, Olympic silver medalist in freestyle skiing, Catherine Raney Norman, a four-time Olympic speed skater and Salt Lake City-Utah Committee for the Games chair, Chris Mazdzer, Olympic silver medalist in men's single luge, and Salt Lake City Mayor Erin Mendenhall participate in a Climate and Sport panel discussion at the Outdoor Retailer Snow Show at the Salt Palace Convention Center in Salt Lake City on Tuesday. (Kristin Murphy, Deseret News)
SALT LAKE CITY — Chris Mazdzer has vivid memories of the four Winter Olympics cities he's competed in.
Whether it was Vancouver, Sochi, PyeongChang or Beijing, he and his fellow luge competitors had to travel to the cities well in advance so they could become comfortable with the track they would eventually ride on. These trips offered him a front-row seat into all the construction that goes into hosting the Games, from new venues to other infrastructure to handle such a large global event.
And there's plenty of construction that's happened in recent history.
"It's got to the point where Beijing built all brand new infrastructure. You have Sochi that built a highway down a river gorge, which was absolutely crazy," he said, recalling his adventures.
That's not the case with Salt Lake City, which is currently vying to host either the 2030 or 2034 Winter Games. Mazdzer notes that the Utah Olympic Oval in Kearns, the Utah Olympic Park in Park City and many of the other major venues used for the 2002 Winter Olympics are very much still in use, which isn't always the case for previous host cities.
Salt Lake City Mayor Erin Mendenhall argues that Utah's Olympic infrastructure is strong enough that it could host the Games tomorrow if needed. That's why she contends Salt Lake's bid is "unbelievably unique."
"We have all the infrastructure that we need in a phenomenal way," she said, sitting around an unlit campfire pit set up for Outdoor Retailer's Community Conversation Tuesday afternoon.
Trade show attendees buzzed by roundtable — or campfire — chat as the participants spoke, some lingering for a few minutes before heading to nearby booths. The discussion centered around the relationship between the changing climate and outdoor sports, a fitting discussion for Outdoor Retailer.
Several brands opted to boycott this week's trade show because Emerald Expositions moved the event back to Utah after spending the last five years in Colorado, citing ongoing complaints with Utah public land policies, particularly its stance on shrinking the size of Bears Ears and Grand Staircase-Escalante national monuments.
It's also an important topic for Salt Lake City as Utah navigates through the impacts of an ongoing drought and when it comes to trying to host a future Winter Olympics. Members of the International Olympic Committee announced last month that they were delaying their decision on which city will host the 2030 Winter Games so they can see how climate change impacts all the cities seeking to host it.
Devin Logan, an Olympic silver medalist in freestyle skiing, right, hugs Catherine Raney Norman, four-time Olympic speed skater and Salt Lake City-Utah Committee for the Games chair, after they participated in a Climate and Sport panel discussion at the Outdoor Retailer Snow Show at the Salt Palace Convention Center in Salt Lake City on Tuesday. (Photo: Kristin Murphy, Deseret News)
Even though Utah is currently enjoying its best snowpack since 2011, Devin Logan, an Olympic slopestyle ski silver medalist and Westminster College student, asserts that it hasn't solved Utah's drought situation yet and doesn't guarantee future success, either.
"Just because we're getting a lot of snow right now doesn't mean that next year won't be bad," she said. "Look what we've seen in the last three years."
The same inconsistency is playing out in different cycles in other parts of the world, which led to the IOC's announcement.
Why the Olympics won't influence Salt Lake's goals
Mendenhall isn't worried at all about the IOC's climate goals for host cities.
She believes Utah's Olympic-ready venues offer the most important item, which is the ability to reuse existing infrastructure. This infrastructure includes TRAX, the light-rail transit system in Salt Lake County. The service has only expanded since it was constructed ahead of the 2002 Olympics, offering a way to move people without additional cars.
But the main reason she isn't concerned about the IOC's criteria is that she believes Salt Lake City was already on the path to reducing carbon emissions just because of the impact it has on air quality, drought, wildfires and other key environmental factors.
For example, the Elektron Solar project, an 80-megawatt solar farm located in Tooele County, could be completed this year, though it likely won't be online by March as it was originally scheduled to be. That's expected to be a major step in helping Salt Lake City reach its goal of 100% net renewable energy by 2030, set well before the IOC voiced concerns about climate.
"The truth is, we're targeting those goals for ourselves," Mendenhall said. "I think it's what makes us such an attractive future host city because we're going to do it anyway."
The mayor added Tuesday that Utahns' ability to conserve billions of gallons of water over the past two years shows that Utahns are willing to make sacrifices in the name of conservation. If anything, hosting a future Games may only speed up project timelines for new infrastructure projects that improve transit service or reduce carbon emissions in the region, she said.
How collaboration factors into the equation
None of these goals can be reached without collaboration, though. It's a potentially difficult task considering Salt Lake City is, in Mendenhall's words, "a blue dot in a red state," but some issues are important to both Republicans and Democrats.
It's been called the 'secret sauce of Utah' that we can come together across party lines or other divisions when we recognize the mutual benefit of coming together to solve (a problem).
–Salt Lake City Mayor Erin Mendenhall
While there are several topics the city and state don't necessarily agree on, Salt Lake City and the State of Utah have been unified in their efforts in hosting the Winter Olympics again. The same can be said about other issues, such as concern over the drying of the Great Salt Lake. Those are topics where the best solutions may come from a variety of viewpoints.
The mayor clarified after Tuesday's event that she doesn't think that Salt Lake City can afford to leave statewide discussions over politics, especially when eventual decisions may negatively impact the city in the long run.
"It's been called the 'secret sauce of Utah' that we can come together across party lines or other divisions when we recognize the mutual benefit of coming together to solve (a problem). ... For me, it means, as this blue dot, we have to stay at the table," she told KSL.com. "We can't afford to get angry and walk away the way the city has so many times in the past."
The importance of the Games
Meanwhile, the Olympians participating in Tuesday's event agree that hosting the Winter Olympics again would be great for Utah. That's because the state is still benefitting from hosting the event before.
Catherine Raney Norman, a four-time Olympic speed skater and Salt Lake City-Utah Committee for the Games chair, speaks during a Climate and Sport panel discussion at the Outdoor Retailer Snow Show at the Salt Palace Convention Center in Salt Lake City on Tuesday. (Photo: Kristin Murphy, Deseret News)
Mazdzer and Logan each moved to Utah to be closer to the Olympic training action and at the venues used 21 years ago. Both have glowing reviews of Utah since moving in.
Children who aspire to be future Olympians have a rare opportunity to train in the same venue as them, oftentimes even learning from the Olympians themselves. It's helped grow the number of Utahns who compete in the Winter Olympics.
"I still live in this community and I want to see it grow," Logan said. "(My passion) happened to be athletics, but passion toward anything can help you grow as a person and grow as a community, in general."
However, having that platform may only exist for as long as there is a Winter Olympics. It's why both Olympians invested as much time as they have in environmental issues.
Logan is an athlete alliance member of a group called Protect Our Winters, while Mazdzer became the IOC's first sustainability ambassador.
"We see what sport can bring to a world, what it can bring to the community," Mazdzer said. "I think sport is such a powerful way that we can tie all of this together — sustainability, climate (and) all these amazing initiatives."
5 new outdoors products that might make a splash in 2023
Outdoor Retailer's on-mountain demo day returns to Utah
Outdoor Retailer plans big 'homecoming' as it returns to Utah
Most recent Utah Olympics bid stories
2030 Winter Olympics in Utah? Why it 'makes a lot of sense' to Nathan Chen
Should the IOC 'grab Salt Lake City while it can' to host the 2030 Winter Games?
2030 Winter Games support sinks to new depths in Sapporo
Utah Olympics bidUtahEnvironmentSalt Lake CountyOutdoors
Carter Williams
Carter Williams is an award-winning reporter who covers general news, outdoors, history and sports for KSL.com. He previously worked for the Deseret News. He is a Utah transplant by the way of Rochester, New York.
Wet winter leads to flooding at Bonneville Salt Flats
What do experts say about tree thinning as a drought solution?
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Why Mitt Romney says US support of Ukraine is good for America


text:
from __future__ import print_function

"""
A Python script for changing the name of an object. Demonstrates the use
of tasks in an asynchronous way.
"""

import atexit
import argparse
import getpass

from pyVim import connect
from pyvmomi_tools.cli import cursor

def get_args():
    parser = argparse.ArgumentParser()

    parser.add_argument('-s', '--host',
                        required=True,
                        action='store',
                        help='Remote host to connect to')

    parser.add_argument('-u', '--user',
                        required=True,
                        action='store',
                        help='User name to use when connecting to host')

    parser.add_argument('-p', '--password',
                        required=False,
                        action='store',
                        help='Password to use when connecting to host')

    parser.add_argument('-o', '--port',
                        required=False,
                        action='store',
                        help="port to use, default 443", default=443)

    parser.add_argument('-n', '--name',
                        required=True,
                        action='store',
                        help='Name of the entity to look for.')

    parser.add_argument('-r', '--new_name',
                        required=False,
                        action='store',
                        help='New name of the entity.')

    _args = parser.parse_args()
    if _args.password is None:
        _args.password = getpass.getpass(
            prompt='Enter password for host %s and user %s: ' %
                   (_args.host, _args.user))

    return _args

args = get_args()

# form a connection...
si = connect.SmartConnect(host=args.host, user=args.user, pwd=args.password,
                          port=args.port)

# doing this means you don't need to remember to disconnect your script/objects
atexit.register(connect.Disconnect, si)

# search the whole inventory tree recursively... a brutish but effective tactic
entity = si.content.rootFolder.find_by_name(args.name)

if entity is None:
    print("A object named {0} could not be found", args.name)
    exit()

if args.new_name:
    new_name = args.new_name
else:
    # just because we want the script to do *something*
    new_name = args.name + "0"

print("\nname        : {0}\n", entity.name)
print("\n    renaming from {0} to {1}\n", args.name, new_name)

# rename creates a task...
task = entity.Rename(new_name)

print("task status: \n")

# demonstrate task state polling
while task.is_alive:
    cursor.spinner('renaming')

print("\n\n\nrename finished\n")

redpajama_set_name: RedPajamaGithub


text:
Q: Is there a better way use these two arrays together to achieve the same result? I have an array of numbers called elements, and another array called selection which is a 3 x 3 array of 0's and 1's. If a 1 is selected it means increases sum by the corresponding element value from the elements array. Once all elements are added together it divides them by the number of selected elements.
How can I improve this code as it seems clunky and I am not sure it is very efficient?
   //Selection Array
   selection = new int [][]{
                    {0,1,0,},
                    {1,1,1,},
                    {0,1,0}
                     };

//Calculate number selected
numberinselection=.....code to add up numbers in selection array above

//Add up selected elements

float sum=0;
sum +=2*elements[x][y][subject]*selection[1][1];
sum +=elements[x][yminus][subject]*selection[1][0];
sum +=elements[x][yplus][subject]*selection[1][2];
sum +=elements[xminus][y][subject]*selection[0][1];
sum +=elements[xplus][y][subject]*selection[2][1];

sum = sum/numberinselection;

A: for(i=0;i<3;i++)
  for(j=0;j<3;j++)
      sum+=elements[i][j]*selection[i][j]

Try these codes.

redpajama_set_name: RedPajamaStackExchange


text:
Current Search: info:fedora/islandora:sp_videoCModel (x) » Teed, Cyrus Reed, 1838-1908 (x)
"Fifty-sixth Anniversary of the Birth of Cyrus".
Koreshan Unity, Koreshan Unity
Music program for concert performed on October 18th, 1895 in celebration of Cyrus Teed's fifty-sixth birthday.
"I command you to her...".
Teed, Cyrus Reed, 1838-1908, Koreshan Unity
This is a typed copy of the original letter that Koresh, otherwise known as Cyrus Teed, sent to the women of Planetary court concerning the mistreatment of Victoria Gratia, otherwise known as Annie Grace Glosson Ordway. At the vey top of the letter is a direct command from Koresh to the women.
"My cause".
A letter from Koresh, originally known as Cyrus Teed, to an unknown recipient. In the letter it does talk about the recipient knowing if they are able to follow Koresh's cause. On the back on the letter is an envelope that may have held the letter.
"My Dear Arlena".
Personal correspondence between someone and a possible Arlena. In the letter it talks about the death of an acquaintance but things must carry on and they most focus on more important issues. The writter also talks abou thimself as being merciful. On the back of the letter is a tribute to Victoria Gratia, otherwise known as Annie Ordway, amd a message to the Koreshan Unity.
"My Dear Arlena" August 14, 1905.
This letter was sent to Arlena from Koresh, originally known as Cyrus Teed, addressing her behavior towards Victoria Gratia, originally known as Annie Grace Glosson Ordway. Koresh ask that she, "...get down off your high horse…" and to let Victoria like her. Koresh also tells her his concerns about his followers' feelings towards him. He wants them to want to be with him no matter where he is.
Arlena is receiving a letter from Koresh, otherwise known as Cyrus Teed. Koresh apoloigizes that she could not be in town for the festivals. He also suggest that they may be able to see each other in washington.
"My Dear Daughters of Zion in Jerusalem".
A typed version of Koresh's, otherwise known as Cyrus Teed, letter sent to the Daughters of Zion concerning their attitudes towards Victoria Gratia, otherwise known as Annie Grace Glosson Ordway. It differs from the other two copies (fgcu_kc_2147 and fgcu_kc_2148) because it includes additional content.
A letter from Cyrus.
A personal letter that was written by Koresh, originally known as Cyrus Teed. Koresh writes that a proper Koreshan is someone that follows their master without question. Koresh also demands that the receiver write him back about how the community is. Koresh signs off "Ever faithfully yours, Cyrus Shepherds of Israel". Each page also has articles attached to the back of them.
A list of important events.
Koreshan Unity
List of events that are not in chronological order. The events do have the month, day, and year written next to the title of the event. On the right hand side it has a corresponding page number, the book thoses pages belong to is unknown. Some of the events are related to the teaching of Koresh, orginally known as Cyrus Teed, with it saying "Purity of Koreshan Patriotim".
A Warning and a Call.
Morgan, Thomas, Koreshan Unity
Dr. Thomas Morgan wrote to remind followers of Koreshanity about Koresh's, originally Cyrus Teed, words on astrological events that are connected to biblical prophecies. He also goes into detail about there being two classes of Koreshanity after Koresh's death and that each class was in a different part of Florida.
Absurdities of Science so Called.
A pamphlet that covers the scientific belief of Koreshanity which was written by Cryus Teed, also known as Cryus Teed. It includes a diagram from the Copernican system that helps support Koresh's theory.
Absurdities of Science so Called (1897).
An excerpt from The Flaming Sword, April 1897 edition, which discusses the scientific belief of Koreshanity. The story is written by Koresh, also known as Cyrus Teed and includes a diagram from the Copernican system that helps support Koresh's scientific theory.
Anniversary Souvenir.
Pamphlet from the 95th annual Solar Festival that took place on October 18, 1934. The Solar Festival was to honor the birth of Koresh, originally Cyrus Teed. The festival is denoted by the printed phrase "A.K. 89" meaning Anno Koresh, the annual calendar system created by the Koreshan's. The pamphlet is a souvenir that explains what Koreshan Universology was.
Annual Festival of the Sun A.K. 61.
Program for a festival on October 18th, 1900 in honor of the birth of Koresh, originally named Cyrus Teed, sixty-one years earlier, as denoted by the printed phrase "A.K. 61" meaning Anno Koresh, the annual calendar system created by the Koreshans. The program lists activities such as songs and readings, along with a play and a closing song.
Annual Lunar Festival.
Festival program for the Annual Lunar festival on April 10th, 1903 in celebration of Victoria Gratia's (Annie Ordway) birthday. This was a theatrical production of the biblical story of Jacob and his colorful coat, and this most likely was an original work. The A.K. 63 marking means "After Koresh 63 years". The "A.K. 63" marks the time in the Koreshan's calendar era they created based on Cyrus Teed's birth year as opposed to the more commonly used year label "A.D."
Annual Solar Festival A.K. 63.
Program for a festival on October 18th, 1902 in honor of the birth of Koresh, originally named Cyrus Teed, sixty-three years earlier, as denoted by the printed phrase "A.K. 63" meaning Anno Koresh, the annual calendar system created by the Koreshans. The program lists activities such as songs and readings, along with a closing song.
Biographical Sketch of Dr. Cyrus Read Teed (Koresh).
Bubbett, Laurence Wilmer, 1884-1960, Koreshan Unity
The biography talks about some of Koresh's, otherwise known as Cyrus Teed, past. However, it mainly focuses on the coming of Koreshanity and the locations that he set up in Chicago before landing in Estero. It also talks about what he looked like and how he is still remembered after his death.
Biographical Sketch of Dr. Cyrus Read Teed (Koresh) by Laurence W. Bubbett.
Cover of the pamphlet Biographical Sketch of Dr. Cyrus Read Teed (Koresh) written by Laurie Bubbett.
Cardinal Points of Koreshanity.
The pamphlet covers the nine cardinal points of Koreshanity. The points cover "the Religious, Ethical, and Economic Principles of the Koreshan system."
Chicago Office.
Koreshan Unity Settlement (48) + -
Koreshan Unity Settlement. (41) + -
Gratia, Victoria, 1844-1923 (18) + -
Ordway, Annie Grace Glosson, 1844-1923 (16) + -
Silverfriend, Henry D., 1864-1949 (16) + -
Guiding Star Publishing House (14) + -
Silverfriend, Etta, 1866-1944 (14) + -
Boomer, Bertha Sterling, 1850-1935 (13) + -
Putnam, Jesse E., 1869-1950 (12) + -
Richards, David J., 1879-1937 (12) + -
Hunt, George W., 1864-1942 (11) + -
Hunt, Charles H., 1862-1943 (10) + -
The Flaming Sword (8) + -
Beth-Ophrah (6) + -
Castle, Eleanor M., 1867-1941 (5) + -
Solar Festival (4) + -
Staton, Lou Hosea, 1871-1950 (4) + -
Woman's Mission (4) + -
Boomer, Bertie May, 1880-1941 (3) + -
Moreland, Harold, 1900-1983 (3) + -
Andrews, A. W. K. (Abie W. K.), 1833-1891 (2) + -
Andrews, Virginia Harmon, 1846-1921 (2) + -
Beth Ophrah (2) + -
Bubbett, Evelyn Trickett, 1854-1935 (2) + -
Bubbett, James Harper, 1844-1924 (2) + -
Bubbett, Laurence Wilmer, 1884-1960 (2) + -
Gilbert, Francis Deane, 1901-1981 (2) + -
Hamilton, Christine E., 1850-1922 (2) + -
Estero (Fla.) (67) + -
Manhattan (New York, N.Y.) (2) + -
1221150 (1) + -
Binghampton (N.Y.) (1) + -
Estero Island (Fla.) (1) + -
Estero, (Fla). (1) + -
Florida. (1) + -
Fort Myers (Fla.) (1) + -
Lee County (Fla.) (1) + -
St. Petersburg, Fla. (1) + -
Personal correspondence (9) + -
Biographies (2) + -
Music Program (2) + -
Personal correspondence. (2) + -
Photographs. (2) + -
Sacred works (2) + -
Songbooks (2) + -
Card (1) + -
Chronology (1) + -
Lecture (1) + -
Monastic Constitutions (1) + -
eng (77) + -
Gratia, Victoria, 1844-1923 (1) + -
Graves, Charles Addison, 1845-1922 (1) + -
Morgan, Thomas (1) + -
Image (JPEG2000) (33) + -
E-book (18) + -
FGCU (102) + -
Free Re-use (30) + -
Limited Re-use (22) + -
No Re-use (5) + -


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Biduanda minara är en fjärilsart som beskrevs av Hans Fruhstorfer 1912. Biduanda minara ingår i släktet Biduanda och familjen juvelvingar. Inga underarter finns listade i Catalogue of Life.

Källor

Juvelvingar
minara

redpajama_set_name: RedPajamaWikipedia


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A compression fracture is a collapse of a vertebra. It may be due to trauma or due to a weakening of the vertebra (compare with burst fracture). This weakening is seen in patients with osteoporosis or osteogenesis imperfecta, lytic lesions from metastatic or primary tumors, or infection. In healthy patients, it is most often seen in individuals suffering extreme vertical shocks, such as ejecting from an ejection seat. Seen in lateral views in plain x-ray films, compression fractures of the spine characteristically appear as wedge deformities, with greater loss of height anteriorly than posteriorly and intact pedicles in the anteroposterior view.

Signs and symptoms
Acute fractures will cause severe back pain. Compression fractures which develop gradually, such as in osteoporosis, may initially not cause any symptoms, but will later often lead to back pain and loss of height.

Diagnosis
Compression fractures are usually diagnosed on spinal radiographs, where a wedge-shaped vertebra may be visible or there may be loss of height of the vertebra. In addition, bone density measurement may be performed to evaluate for osteoporosis. When a tumor is suspected as the underlying cause, or the fracture was caused by severe trauma, CT or MRI scans may be performed.

Treatment

Conservative treatment
 Back brace for support while the bone heals—either a Jewett brace for relatively stable and mild injuries, or a thoracic lumbar sacral orthosis (TLSO) for more severe ones.
 Opioids or non-steroidal anti-inflammatory drugs (NSAIDs) for pain. For osteoporotic patients, calcitonin may be helpful.

Surgical
 Kyphoplasty and vertebroplasty are minimally invasive procedures that inject cement into the bone of the back that is fractured. However, the data examining the effectiveness of these procedures is mixed.

References

Further reading

 Medscape article on lytic lesions
 Emedicine article on spinal metastasis

External links

Bone fractures
Orthopedic problems


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Home → Uncategorized → Seneca Valley Trout Unlimited Becomes A TAME Coalition Supporter
Seneca Valley Trout Unlimited Becomes A TAME Coalition Supporter
The Coalition for Transit Alternatives to Mid-County Highway Extended (TAME) announces that Seneca Valley Trout Unlimited (SVTU) Chapter has signed on with TAME's 50+ organizations and elected officials to stop further planning, design and construction for Alternatives 4, 8 & 9 in the Midcounty Corridor Study of Mid-County Highway Extended (known as M-83).
SVTU is a local voice of Trout Unlimited. TU's vision statement is to conserve, protect and restore North America's coldwater fisheries and their watershed. One way SVTU supports TU's national vision is: to help with maintenance activities on Little Seneca Creek in Montgomery Co., southern Catoctin brook trout streams in Frederick Co., and maintenance of warmwater streams like Great Seneca Creek and areas of the Potomac River.
TAME Coalition supports multiple, viable and sustainable transportation alternatives as solutions to replace the old Mid-County Highway Extended (M-83) plan; and is working with Maryland's state and local elected officials, as well as with MNCPPC Planning Board, to remove M-83 from the Master Plan of Highways and Transitways.
Montgomery County Department of Transportation's (MCDOT) Draft Environmental Effects Report (EER) strongly favors building the 50-year-old-designed highway on alternatives with the highest bio-diverse areas and in the densest sub-urban neighborhoods of any of the original 11 alternative alignments (Alternatives 8 and/or 9). Their Draft EER acknowledges an additional nine intersections will fail county congestion standards after construction. MCDOT lists the cost of the proposed 6.2 mile highway to be $350 million. After mitigation costs are added, the total price tag is expected to double.
MCDOT's proposed M-83 Alternatives 8 & 9 (from Montgomery Village Avenue to Rt. 27) do not align on any existing road, and will negatively impact 16 stream and wetland crossings, including Wildcat Branch (Class III natural trout reproducing stream) near Butlers Orchard, and parts of Dayspring Creek ecosystem adjacent to Dayspring Silent Retreat Center and the 300 acre North Germantown-Greenway Park. These two alignments are made up of 90% forest, wetlands and pristine parkland, which would be irreversibly eliminated and paved over, if M-83 were built. The overall effect from deforestation of nearly 100 accumulative acres of mature ecosystems will drastically increase sediment loads in the Seneca Creek watershed, making it impossible to lower sediment levels mandated by the state.
Transportation solutions (for north-south traffic) supported by TAME Coalition focus on:
Transit Alternatives
Mass transit along existing roads being seriously considered as a viable and sustainable alternative
Dedicated Bus Rapid Transit (BRT) lane along MD 355
Building the Corridor Cities Transitway (CCT) from Shady Grove to Clarksburg
Increasing MARC Train to full service, adding a third track
Finishing Metro's Purple Line from Rockville to Silver Spring (east/west traffic)
Transportation solutions benefiting all citizens in Montgomery County
Traffic Demand Management Solutions
Traffic demand management being seriously considered on existing roads
Encouraging more carpooling and vanpooling
Bus lanes on I-270 during peak periods
Better traffic signal operation
Upgrade intersections
Ramp metering
By TAME on August 20, 2014 / Uncategorized
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Thubana bathrocera is a moth in the family Lecithoceridae. It was described by Chun-Sheng Wu in 1997. It is found in Hunan, China.

References

Moths described in 1997
Thubana


text:
What happens after Republican landslides? They overreach!
By Joel Connelly on November 12, 2014 at 6:04 PM
President Ronald Reagan carried a Republican-controlled U.S. Senate into office in 1981, and its members immediately set out to find examples of waste, fraud and abuse in government.
Incoming Senate Majority Leader Mitch McConnell: Can he, dare he, try to govern with the far right pushing for confrontation?
Newly elected Sen. Paula Hawkins, R-Fla., was eager to begin and scheduled a lunch with colleagues to discuss food stamp fraud. She served New York steak. The lunch caused guffaws across Washington, D.C., and generated outrage among defenders of America's poor.
It has become a law of politics: Republican overreach and step in it after they win an election. They believe that the voters have spoken — for good — and that the future belongs to them. By contrast, Democrats fumble in trying to make their majorities work, fail to explain their reforms, and cower when attacked.
After last week's midterm election, it's the GOP's turn. As in 1980 and 1994, the R's are taking over the Senate. They have padded their majority in the House of Representatives.
The beginnings of overreach were evident Wednesday.
Incoming Senate Majority Leader Mitch McConnell and House Speaker John Boehner went on the war path against a climate change accord, negotiated by President Obama and China's President Xi Jinping. The agreement is that the world's two greatest polluters will cap and curb carbon emissions, and move toward new, non-polluting energy sources.
China, choking on the air in its cities, has finally agreed to cap coal pollution. What's wrong with that? The U.S. is rapidly expanding wind and solar energy, and use of biofuels. What's hurtful about getting out ahead on developing a new technology? It's good business.
On the same day, House Republican leaders were mapping a plan to "repeal and replace" Obamacare. Fine. What are they going to replace it with? Guarantees of coverage for those with preconditions? Allowing young people to remain on their parents' health plans? Expanded Medicaid coverage for lower-income Americans? Are they going to restore caps on the amount of coverage?
Newt Gingrich: While he was speaker, House Republicans tried to impeach President Clinton. They managed to lose seats in a midterm election, and Gingrich lost his job.
As when Clinton was in office, the Republicans' political base has been fed a steady diet of hate by right-wing media. It believes the lies and distortions of Fox News. In a wider sense, the well is poisoned toward any notion of an activist government.
In Congress, such extremists as Texas Sen. Ted Cruz will cry out against any effort to actually govern, to work out society's compromises, to act in such fields as minimum wage and immigration reform. Boehner has repeatedly come back from the White House only to be undercut by his own troops.
Partial government shutdowns. The inability to renew such valuable programs as the Export-Import Bank. Attacks on such goals as clean air once championed by Republican presidents (Nixon, Ford and Bush I). Efforts to emasculate the Antiquities Act, first used by Theodore Roosevelt to designate national monuments in the Grand Canyon and our Olympic Mountains.
An example: In 1995, under the new regime of then-House Speaker Newt Gingrich, the Republican chairman of the House Budget Committee proposed turning responsibilities of the U.S. Geological Survey over to the states.
The USGS maps seismic faults, which cross state lines. The USGS evaluates earthquake dangers. The USGS monitors ash clouds from volcanoes, which blow across political boundaries and threaten to clog passenger jet engines.
U.S. Interior Secretary Bruce Babbitt handled the proposal with loving deftness. To underscore the importance of USGS, he visited the David A. Johnston Volcanic Observatory, named for the young volcanologist who was monitoring Mount St. Helens in 1980 when it blew. ("Vancouver, Vancouver, This is it!")
House Speaker John Boehner and Rep. Cathy McMorris Rodgers, R-Wash. Boehner is undercut by his own Republican caucus when he tries to deal.
The ham-handed attempt to impeach President Clinton caused Republicans to lose House seats in the 1998 midterm elections, two of them in this state. Newt Gingrich lost his job as House speaker.
Ideologues will never learn. Columnist Robert Novak was breathlessly forecasting in '98 that Republicans would achieve a veto-proof 60-seat majority in the Senate. (They ended up with no gain at 54 seats.) King County Republicans welcomed five House managers of the Clinton impeachment. Three would soon be defeated.
The Republicans coming to power in 2015 make the Gingrich-era crew look reasonable. The incoming Senate Environment Committee chairman, Sen. Jim Inhofe, R-Okla., has described human-caused climate change as a "hoax" and a "conspiracy." He has cited the Bible in arguing that climate science is a fraud.
America's voters are equalizers, not ideologues. As well, far more voters turn out — more young people, more minorities, more working women — in presidential election years.
Just look at history.
The Reagan landslide of 1984 was followed by Democrats recapturing the Senate in 1986. The Republican tidal wave of 1994 ebbed two years later as President Clinton was re-elected. The excesses in a GOP-led Congress led to Democratic gains in 1998 and 2000 (including the upset of Sen. Slade Gorton in this state).
President Obama and the Democrats suffered a "shellacking" in 2010, but Obama came back in 2012. He became the first Democratic incumbent since Franklin D. Roosevelt to win a majority of the vote in consecutive presidential elections.
The likelihood, then, is revenge and confrontation rather than effective collaboration.
The likely result: Those who sow the wind will reap the whirlwind.


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EFCC Arraigns Former OAU VC, Bursar for N1.4billion Fraud
The Economic and Financial Crimes Commission, EFCC, July 7, 2017 arraigned a former acting Vice Chancellor, Obafemi Awolowo University , Prof. Anthony Elujoba and the University Bursar, Mrs. Aderonke Akeredolu before Justice A. D. Oladimeji of the Osun State High Court sitting in Ede on a 7-count charge bordering on conspiracy, stealing, abuse of office and using position for gratification.
The defendants, however pleaded not guilty to the charges preferred against them.
In view of their plea, prosecution counsel, Festus Ojo prayed the court for a trial date and to remand the defendants in prison custody. The defense counsel however approached the court with a formal application for bail which the prosecution counsel opposed to on the grounds that he was served the application in court and needed time to formally respond.
One of the charges reads: "That you, Professor Anthony Elujoba and Josephine Aderonke Akeredolu whilst being Public Officers of the Obafemi Awolowo University, Ile-Ife in Osun State between the month of August and September 2016 at Ile-Ife within the jurisdiction of this Honourable Court conspired to distribute the sum of N1,060,725,000.00k (One Billion, Sixty Million and Seven Hundred and Twenty Five Thousand Naira) to your colleagues as productivity bonus.
The defendants were arraigned following a petition alleging that Elujoba, alongside the Bursar, diverted over One Billion Naira from the University's Fixed Assets and Endowment Fund without approval from the Federal Ministry of Education and Governing Council of the University to pay some unapproved allowances from which he received N7m (Seven Million Naira) as furniture allowance.
Justice Oladimeji adjourned the case to 11 July, 2017 for hearing of the bail applications and remands the defendants in the EFCC custody.
Forgiveness Day: Christian, Muslim Scholars Team-Up To Ensure Tolerance
Why I Dumped PDP – Former Governor Elechi
EFCC to Appeal Acquittal of Ex-Adamawa Gov, Ngilari
The Economic and Financial Crimes Commission, EFCC, is to appeal the July 20, 2017 Court of Appeal ruling, which acquitted a former Adamawa State governor, Bala Ngilari, of the five-count charge of corruption, which the […]
EFCC Arraigns Justice Fishim for Corruption
The Economic and Financial Crimes Commission, EFCC, today, Tuesday, July 11, 2017, arraigned a serving judge of the National Industrial Court, Justice Agbadu James Fishim before Justice Raliatu Adebiyi of Lagos State High Court, sitting […]
How Ex-NAMA MD, Abdulsalam, Others Diverted N473.2m
(EFCC Press Release) Babatunde Adenekan, a prosecution witness in the case involving Ibrahim Abdulsalam, a former Managing Director of Nigeria Airspace Management Agency (NAMA) and others on Wednesday, July 5, 2017 told Justice Babs O. Kuewumi […]


Ariana Grande has some feelings about Mac Miller's Grammys loss to Cardi B.
The Thank U, Next singer, who did not attend Sunday's show due to differences with producers, live-tweeted throughout the majority of the telecast — but the best rap album category had Grande particularly frustrated as Miller's fifth and final studio album, Swimming, lost to Cardi's Invasion of Privacy.
Miller, 26, died five months ago from an accidental overdose.
Though Grande, 25, tweeted, "f—" and "trash," she later clarified that she wasn't upset with Cardi who was not the "trash" to which she was referring.
"Nothing to do w her. good for her. i promise. i'm sorry," she later wrote on Twitter.
Though he did not win, Miller was honored during the In Memoriam segment of the show. His parents, Karen Meyer and Mark McCormick, were also in attendance — and its their presence that seemed to most upset Grande.
All of these tweets have since been deleted.
Miller, whose real name was Malcolm McCormick, died from a combination of fentanyl, cocaine and alcohol on Sept. 7, PEOPLE confirmed. He was 26.
Miller and Cardi were nominated in the category alongside Nipsey Hussle's Victory Lap, Pusha T's Daytona and Travis Scott's Astroworld.
Though Miller's album was not celebrated with a win by the Recording Academy, his music has been revered since his death.
To honor his legacy and career, his family organized the "Mac Miller: A Celebration of Life" tribute concert in October with Miller's friends and collaborators like Scott, John Mayer, Chance the Rapper, Anderson.Paak, Miguel, SZA and Ty Dolla $ign performing.
Miller isn't the only artist who has been nominated for the upcoming ceremony posthumously.
The 61st Annual Grammy Awards are broadcasting live from Staples Center in Los Angeles on Sunday, Feb. 10, at 8 p.m. ET/5 p.m. PT on CBS.


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Joint Expert Panel report on USS Statement from TPR
Dec 13th '19
A spokesperson for The Pensions Regulator (TPR) said:
"The issues involved with the Universities Superannuation Scheme (USS) are complex and the constructive engagement of all parties will be crucial to consider the direction the Joint Expert Panel (JEP) has set and ultimately protect savers in the scheme.
"The JEP report's high-level recommendations, including for effective governance and clear communication during the valuation process, align with our views. In particular we agree with the need to ensure the scheme's long-term sustainability. The adoption of a dual discount rate broadly aligns with our proposed approach for the new defined benefit (DB) funding code which we will be consulting on next year. It will be important that these discount rates are set at appropriate levels and that there is a clear explanation of the risks for the scheme.
"USS is the largest DB pension in the private sector and understanding the long-term sustainability of the scheme should be a key focus for this, and all DB schemes. As part of our role is to protect savers and ensure that they receive the benefits they have been promised, USS has now entered our one to one supervision programme and so we will continue to develop our relationship with the scheme, while also working with the employers.
"Work is already underway on the scheme's 2020 valuation. We are working with the trustees to understand the valuation timetable to ensure the 15 month statutory deadline is met, and we look forward to seeing how the trustees and stakeholders will take forward the report's recommendations."
Source: TPR


Translation:I have a pen. I have an apple.
Thanks, now I have that meme in my head while learning.
at the risk of being downvoted into oblivion, it would be nice to be able to learn korean without having k-pop thrust at me for just ten seconds.
...this is from a Japanese song.
If you're learning Korean you should know that it was Japanese.


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