text:
\section{Introduction}

\noindent The aim of this article is to show that the topological stable rank (a notion from
topological $K$-theory, recalled below) of $\mathcal{E}'(\mathbb{R})$ is $2$, where $\mathcal{E}'(\mathbb{R})$ is the classical topological algebra
of compactly supported distributions, with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$,
pointwise vector space operations, and convolution taken as multiplication.

We recall some key notation and facts about $\mathcal{E}'(\mathbb{R})$ in
Section~\ref{section_Eprime} below, including its strong dual topology
$\beta(\mathcal{E}',\mathcal{E})$, and in Section~\ref{section_tsr}, we will
recall the notion of topological stable rank of a topological algebra.

We will prove our main result, stated below, in Sections
\ref{section_geq_2} and \ref{section_leq_2}.

\begin{theorem}
\label{main_thm}$\;$

\noindent
Let $\mathcal{E}'(\mathbb{R})$ be the algebra of all compactly supported
distributions on $\mathbb{R}$, with
\begin{itemize}
\item pointwise addition, and pointwise multiplication by complex scalars,
\item convolution taken as the multiplication in the algebra, and
\item the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$.
\end{itemize}
Then the topological stable rank of $\mathcal{E}'(\mathbb{R})$ is equal to $2$.
\end{theorem}

\section{The topological algebra $\mathcal{E}'(\mathbb{R})$}
\label{section_Eprime}

\noindent
For background on topological vector spaces and distributions, we
refer to \cite{Bar}, \cite{Hor}, \cite{Rud}, \cite{Sha}, \cite{Sch} and \cite{Tre}.

Let $\mathcal{E}(\mathbb{R})=C^\infty(\mathbb{R})$ be the space of functions
$\varphi:\mathbb{R}\rightarrow \mathbb{C}$ that are infinitely many times
differentiable.  We equip $\mathcal{E}(\mathbb{R})$ with the topology of uniform
convergence on compact sets for the function and its derivatives. This is
defined by the following family of seminorms: for a compact
subset $K$ of $\mathbb{R}$, and $M\in \{0,1,2,3\cdots\}=\mathbb{Z}_{ \tiny \small\scriptscriptstyle{\geq 0}}$, we define
$$
p_{\scriptscriptstyle{K,M}}(\varphi)=\sup_{0\leq m\leq M}\sup_{x\in K}|\varphi^{(m)}(x)|\;\;\textrm{ for }\varphi\in \mathcal{E}(\mathbb{R}).
$$
The space $\mathcal{E}(\mathbb{R})$ is
\begin{itemize}
 \item metrizable,
 \item a Fr\'echet space, and
 \item a Montel space;
\end{itemize}
see e.g. \cite[Example 3, p.239]{Hor}.

\smallskip

\noindent
By a topological algebra, we mean the following:

\begin{definition}[Topological algebra]$\;$\label{def_top_alg}

\noindent A complex algebra $\mathcal{A}$ is called a {\em topological algebra}
if it is equipped with a topology $\mathcal{T}$ making the following maps
continuous, with the product topologies on the domains:
\begin{itemize}
\item $\mathcal{A}\times \mathcal{A}\owns \;(a,b)\mapsto a+b\;\in \mathcal{A}$
\item $\mathbb{C}\times \mathcal{A}\owns \;(\lambda, a)\mapsto \lambda \cdot a \;\;\in \mathcal{A}$
\item $\mathcal{A}\times \mathcal{A}\owns\; (a,b)\mapsto ab\;\!\;\;\;\;\;\in\mathcal{A}$
\end{itemize}
\end{definition}

\noindent
We equip the dual space $\mathcal{E}'(\mathbb{R})$ of $\mathcal{E}(\mathbb{R})$ with the strong
dual topology $\beta(\mathcal{E}',\mathcal{E})$, defined by the seminorms
$$
p_{\scriptscriptstyle{B}}(T)=\sup_{\varphi\in B} |\langle T, \varphi\rangle|,
$$
for bounded subsets $B$ of $\mathcal{E}(\mathbb{R})$. Then $\mathcal{E}'(\mathbb{R})$, being the strong dual of the Montel space $\mathcal{E}(\mathbb{R})$, is a
Montel space too \cite[5.9, p.147]{Sha}. This has the consequence that a sequence in $\mathcal{E}'(\mathbb{R})$ is convergent in the
$\beta(\mathcal{E}',\mathcal{E})$ topology if and only if it is convergent in the weak dual/weak-$\ast$ topology $\sigma(\mathcal{E}',\mathcal{E})$
of pointwise convergence on $\mathcal{E}(\mathbb{R})$; see e.g. \cite[Corollary~1, p.358]{Tre}.

As usual, let $\mathcal{D}(\mathbb{R})$ denote the space of all compactly supported functions from
$C^\infty(\mathbb{R})$, and $\mathcal{D}'(\mathbb{R})$ denote the space of all distributions.
The vector space $\mathcal{E}'(\mathbb{R})$ can be identified with the subspace of
$\mathcal{D}'(\mathbb{R})$ consisting of all distributions having compact
support. If $\mathcal{D}'(\mathbb{R})$ is also equipped with its strong dual
topology, then one has a continuous injection
 $
\mathcal{E}'(\mathbb{R})\hookrightarrow \mathcal{D}'(\mathbb{R}).
$
For $T,S\in \mathcal{E}'(\mathbb{R})$, we define their convolution
$T\ast S\in \mathcal{E}'(\mathbb{R})$ by
$$
\langle T\ast S,\varphi\rangle=
\Big\langle T,\;\big[\;x\mapsto  \big\langle S,\varphi (x+\bigcdot)\big\rangle \;\big]\;\Big\rangle,
\quad \varphi \in \mathcal{E}(\mathbb{R}).
$$
The map $\ast:\mathcal{E}'(\mathbb{R})\times \mathcal{E}'(\mathbb{R})\rightarrow \mathcal{E}'(\mathbb{R})$ is
(jointly) continuous; see for instance \cite[Chapter VI, \S3, Theorem IV, p. 157]{Sch}.

Thus $\mathcal{E}'(\mathbb{R})$, endowed with the strong dual topology, forms a topological algebra
with pointwise vector space operations,
and with convolution taken as multiplication.
The multiplicative identity element is $\delta_{\scriptscriptstyle{0}}$, the Dirac delta distribution
supported at $0$. In general, we will denote by $\delta_a$ the Dirac
delta distribution supported at $a\in \mathbb{R}$.

We also recall that the Fourier-Laplace
transform of a compactly supported distribution
$T\in \mathcal{E}'(\mathbb{R})$ is an entire function, given by
$$
\widehat{T}(z)=\Big\langle T, \;\big(x\mapsto e^{-2\pi i x z}\big)\;\Big\rangle\quad (z\in \mathbb{C}),
$$
see e.g. \cite[Proposition~29.1, p.307]{Tre}.

\section{Topological stable rank}
\label{section_tsr}

\noindent
An analogue of the Bass stable rank (useful in algebraic
$K$-theory) for topological rings,  called the topological stable rank, was introduced  in the seminal article
\cite{Rie}.

\goodbreak

\begin{definition}[Unimodular tuple, Topological stable rank]$\;$

\noindent
  Let $\mathcal{A}$ be a commutative unital topological algebra with multiplicative identity element denoted by $1$, endowed with
  a topology $\mathcal T$.

  \noindent We define $\mathcal{A}^n:=\mathcal{A}\times \cdots \times \mathcal{A}$ ($n$ times), endowed with the product topology.
\begin{itemize}
\item (Unimodular $n$-tuple).$\;$

  \noindent Let $n\in \mathbb{N}:=\{1,2,3,\cdots\}$.  We
  call an $n$-tuple $(a_1,\cdots,a_n)\in \mathcal{A}^n$ {\em unimodular} if
  there exists $(b_1,\cdots,b_n)\in \mathcal{A}^n$ such that the B\'ezout
  equation
 $
 a_1 b_1+\cdots+a_nb_n=1
 $
 is satisfied. The set of all unimodular $n$-tuples is denoted by
 $U_n(\mathcal{A})$. Note that $U_1(\mathcal{A})$ is the group of invertible
 elements of $\mathcal{A}$. An element from $U_2(\mathcal{A})$ is referred to as a
 {\em coprime} pair. It can be seen that if $U_n(\mathcal{A})$ is dense in $\mathcal{A}^n$, then
 $U_{n+1}(\mathcal{A})$ is dense in $\mathcal{A}^{n+1}$.
\item (Topological stable rank). $\;$

\noindent
If there exists a least natural number
  $n\in \mathbb{N}$ for which $U_n(\mathcal{A})$ is dense in $\mathcal{A}^n$, then that $n$ is called the
  {\em topological stable rank} of $\mathcal{A}$, denoted by
  ${\rm tsr}\;\! \mathcal{A}$. If no such $n$ exists, then
  ${\textrm{tsr}} \mathcal{A}$ is said to be infinite.
\end{itemize}
\end{definition}

\noindent While the notion of topological stable rank was introduced in the
context of {\em Banach} algebras, the above extends this notion in a natural manner to
topological algebras. The topological stable rank of many concrete Banach
algebras  has been determined previously in several works (e.g. \cite{DF}, \cite{Sua1}, \cite{Sua2}). In this article, we determine the topological stable rank of the classical topological algebra $\mathcal{E}'(\mathbb{R})$
from Schwartz's distribution theory.

\section{$\mathrm{tsr}(\mathcal{E}'(\mathbb{R}))\geq 2$}
\label{section_geq_2}

\noindent The idea is that if tsr$(\mathcal{E}'(\mathbb{R}))$ were 1, then we could approximate any $T$ from $\mathcal{E}'(\mathbb{R})$
by compactly supported distributions whose Fourier transform would be zero-free, and by an application of
Hurwitz Theorem, $\widehat{T}$ would need to be zero-free too, which gives a contradiction, since we can easily choose
$T$ at the outset to not allow this.

\begin{proposition}
\label{prop_geq}
$\textrm{\em tsr}(\mathcal{E}'(\mathbb{R}))\geq 2$.
\end{proposition}
\begin{proof}
Suppose on the contrary that ${\textrm{tsr}}(\mathcal{E}'(\mathbb{R}))=1$. Let
$$
T=\frac{\delta_{\scriptscriptstyle{-1}}-\delta_{\scriptscriptstyle{1}}}{2i}\in \mathcal{E}'(\mathbb{R}).
$$
By our assumption, $U_1(\mathcal{E}'(\mathbb{R}))$ is dense in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$.  But
then the set $U_1(\mathcal{E}'(\mathbb{R}))$ is also sequentially dense: This is a consequence of
the fact that a subset $F$ of $\mathcal{E}'(\mathbb{R})$ is closed in
$\beta(\mathcal{E}',\mathcal{E})$ if and only if it is sequentially closed.  (See
\cite[Satz 3.5, p.231]{Obe}, which says that $E'$, with the $\beta (E',E)$-topology,
is sequential whenever $E$ is Fr\'echet-Montel. A locally convex space $F$ is
{\em sequential} if any subset of $F$ is closed if and only if it is
sequentially closed. If $F$ has this property, then the closure of any
subset equals its sequential closure, and therefore being dense is the
same as being sequentially dense. In our case,
$E=\mathcal{E}(\mathbb{R})$ is Fr\'echet-Montel, and so $\mathcal{E}'(\mathbb{R})$ is
sequential. In fact in the remark following \cite[Satz 3.5]{Obe}, the
case of $\mathcal{E}'(\mathbb{R})$ is mentioned as a corollary.)

Thus there exists
a sequence $(T_n)_{n\in \mathbb{N}}$ in $U_1(\mathcal{E}'(\mathbb{R}))$ such that
$T_n\stackrel{n\rightarrow\infty}{\longrightarrow} T$ in
$\mathcal{E}'(\mathbb{R})$. But since each $T_n$ is invertible in $\mathcal{E}'(\mathbb{R})$,
there exists a sequence $(S_n)_{n\in\mathbb{N}}$ in $ \mathcal{E}'(\mathbb{R})$ such that
$$
T_n\ast S_n=\delta_{\scriptscriptstyle{0}}\;\; \textrm{ for all } n\in \mathbb{N}.
$$
Taking the Fourier-Laplace transform, we obtain
$$
\widehat{T_n}(z) \cdot \widehat{S_n}(z)=1 \;\;\textrm{ for all }z\in
\mathbb{C}\textrm{ and all } n\in \mathbb{N}.
$$
In particular, the entire functions $\widehat{T_n}$ are all zero-free.

But as $T_n\stackrel{n\rightarrow\infty}{\longrightarrow} T$ in $\mathcal{E}'(\mathbb{R})$, we now
show that $(\widehat{T_n})_{n\in \mathbb{N}}$ converges to $\widehat{T}$
uniformly on compact subsets of $\mathbb{C}$ as $n\rightarrow \infty$.  The
pointwise convergence of $(\widehat{T_n})_{n\in \mathbb{N}}$ to $\widehat{T}$
is clear by taking the test function $x\mapsto e^{-2\pi i x z}$:
$$
\widehat{T_n}(z)=\langle T_n,e^{-2\pi iz\;\!\bigcdot }\rangle
\stackrel{n\rightarrow\infty}{\longrightarrow}
\langle T,e^{-2\pi iz\;\!\bigcdot }\rangle=\widehat{T}(z).
$$
Now for any $\varphi \in \mathcal{E}(\mathbb{R})$, we know that the sequence
$(\langle T_n,\varphi\rangle)_{n\in \mathbb{N}}$ converges to
$\langle T,\varphi\rangle$, and in particular, the set
$$
\Gamma(\varphi):=\{\langle T_n,\varphi\rangle:n\in \mathbb{N}\}
$$
is bounded, for every $\varphi\in \mathcal{E}(\mathbb{R})$. By the
Banach-Steinhaus Theorem for Fr\'echet spaces (see for example
\cite[Theorem 2.6, p.45]{Rud}), applied in our case to the Fr\'echet
space $\mathcal{E}(\mathbb{R})$, we conclude that
$$
\Gamma=\{T_n:n\in \mathbb{N}\}
$$
is
equicontinuous. Thus for every $\epsilon>0$, there exists a
neighbourhood $V$ of $0$ in $\mathcal{E}(\mathbb{R})$ such that
$T_n(V)\subset B(0,\epsilon):=\{z\in \mathbb{C}:|z|<\epsilon\}$ for all
$n\in \mathbb{N}$. From here it follows that there exist $M\in \mathbb{Z}_{\scriptscriptstyle{\geq 0}}$,
$R>0$ and $C>0$ such that
$$
|\langle T_n,\varphi\rangle|\leq C\Big(1+\sup_{0\leq m\leq M} \sup_{|x|\leq R} |\varphi^{(m)}(x)|\Big) .
$$
By taking $\varphi=(x\mapsto e^{-2\pi i x z})$ in the above, we obtain
$$
|\widehat{T_n}(z)|\leq C'(1+|z|)^{M'} e^{R' |z|},\quad z\in \mathbb{C}, \;n\in \mathbb{N}.
$$
Also, by the Payley-Wiener-Schwartz Theorem \cite[Theorem~4.12, p.139]{Bar} for $T\in \mathcal{E}'(\mathbb{R})$,
we have
$$
|\widehat{T}(z)|\leq C''(1+|z|)^{M''} e^{R'' |z|},\quad z\in \mathbb{C}, \;n\in \mathbb{N}.
$$
It now follows that for some constants $C_*,M_*,R_*$ that
$$
|\widehat{T_n}(z)-\widehat{T}(z)|\leq C_*(1+|z|)^{M_*} e^{R_* |z|},\quad z\in \mathbb{C}, \;n\in \mathbb{N}.
$$
But this means that the pointwise convergent sequence
$(\widehat{T_n})_{n\in \mathbb{N}}$ of entire functions is uniformly bounded
on compact subsets of $\mathbb{C}$ (that is, the sequence constitutes a
normal family).  Then it follows from Montel's Theorem (see e.g.
\cite[Exercise~9.4, p.157]{Ull}) that $(\widehat{T_n})_{n\in \mathbb{N}}$
converges to $\widehat{T}$ uniformly on compact subsets of $\mathbb{C}$ as
$n\rightarrow \infty$.

But now by Hurwitz Theorem (see e.g. \cite[Exercise~5.6, p.85]{Ull}), and considering, say, the compact set
$K=\{z\in \mathbb{C}:|z|\leq 1\}$, we conclude that $\widehat{T}$ must be
either be identically zero on $K$ or that it must be zero-free in $K$.
But $\widehat{T}$ is neither:
$$
\widehat{T}(z)=\frac{e^{2\pi i z}-e^{-2\pi i z}}{2i}=\sin (2\pi z),
$$
a contradiction. Hence ${\textrm{tsr}} (\mathcal{E}'(\mathbb{R}))\geq 2$.
\end{proof}

\noindent An alternative proof of Proposition~\ref{prop_geq}, suggested by Peter Wagner \cite{Wag}, is as follows.
 The theorem of supports (\cite[Theorem~4.3.3]{Hor})
implies that $U_1(\mathcal{E}'(\mathbb{R}))$ equals the set of nonzero multiples
of $\delta_a$ for arbitrary $a\in \mathbb{R}$, and this set is not dense in $\mathcal{E}'(\mathbb{R})$. We give the
details below. First, one can show the following structure result for $U_1(\mathcal{E}'(\mathbb{R}))$.

\begin{proposition}
\label{prop_PW}
$U_1(\mathcal{E}'(\mathbb{R}))=\{c\delta_a: a\in \mathbb{R}, \;0\neq c\in \mathbb{C}\}$.
\end{proposition}
\begin{proof}
It is clear that $\{c\delta_a:a\in \mathbb{R},\;0\neq c\in \mathbb{C} \}\subset U_1(\mathcal{E}'(\mathbb{R}))$ since
$$
(c\delta_a)\ast (c^{-1} \delta_{-a})=\delta_0.
$$
Now suppose that $T\in U_1(\mathcal{E}'(\mathbb{R}))$. Then there exists an $S\in \mathcal{E}'(\mathbb{R})$ such that
$T\ast S=\delta_0$. By the Theorem on Supports \cite[Theorem~4.3.3, p.107]{Hor}, we have
 $$
\textrm{c.h.supp}(T\ast S)=\textrm{c.h.supp}(T)+\textrm{c.h.supp}(S),
$$
where, for a distribution  $E\in \mathcal{E}'(\mathbb{R})$, the notation $\textrm{c.h.supp}(E)$ is used for the closed convex hull of
$\textrm{supp}(E)$, that is, the intersection of all closed convex sets containing $\textrm{supp}(E)$.
So we obtain
$$
\{0\}=\textrm{c.h.supp}(\delta_0)= \textrm{c.h.supp}(T\ast S)=\textrm{c.h.supp}(T)+\textrm{c.h.supp}(S),
$$
from which it follows that $\textrm{c.h.supp}(T)=\{a\}$ and
$\textrm{c.h.supp}(S)=\{-a\}$ for some $a\in \mathbb{R}$. But then
 also  $\textrm{supp}(T)=\{a\}$ and
$\textrm{supp}(S)=\{-a\}$. As distributions with
support in a point $p$ are linear combinations of the Dirac delta distribution  $\delta_p$
and its derivatives $\delta_p^{(n)}$ \cite[Theorem~24.6, p.266]{Tre}, we conclude that
$S$ and $T$ have the form
\begin{eqnarray*}
 T&=&\sum_{n=0}^N t_n \delta_a^{(n)},\\
 S&=&\sum_{m=0}^M s_m \delta_{-a}^{(m)},
\end{eqnarray*}
for some integers $N,M\geq 0$ and some complex numbers $t_n,s_m$ ($0\leq n\leq N$, $0\leq m\leq M$).
Now $T\ast S=\delta_0$ implies that $N=M=0$ and $t_0s_0=1$, thanks to the
 the linear independence of the set
$$
\{\delta_{-a},\delta_{-a}',\delta_{-a}'',\cdots\}\cup\{\delta_{0},\delta_{0}',\delta_{0}'',\cdots\}\cup
\{\delta_{a},\delta_{a}',\delta_{a}'',\cdots\}
$$
in the complex vector space $\mathcal{E}'(\mathbb{R})$.
In particular $t_0\neq 0$.
Thus
$$
T= t_0 \delta_a\in \{c\delta_p: p\in \mathbb{R}, \;0\neq c\in \mathbb{C}  \}.
$$
Consequently, $U_1(\mathcal{E}'(\mathbb{R}))=\{c\delta_a: a\in \mathbb{R}, \;0\neq c\in \mathbb{C}  \}$.
\end{proof}

\noindent Based on the above, we can now give the following
alternative proof of Proposition~\ref{prop_geq}.

\begin{proof} We show $U_1(\mathcal{E}'(\mathbb{R}))$ is not dense in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$. If it were, then it would
be sequentially dense too, and so for each element $T$ of $\mathcal{E}'(\mathbb{R})$, there would exist a sequence in $U_1(\mathcal{E}'(\mathbb{R}))$  that converges to $T$ in the $\beta(\mathcal{E}',\mathcal{E})$ topology, and hence also in the
$\sigma(\mathcal{E}',\mathcal{E})$ topology. But we now show that
 $\delta_0'\in \mathcal{E}'(\mathbb{R})$ cannot be approximated in the $\sigma(\mathcal{E}',\mathcal{E})$ topology by elements from  $U_1(\mathcal{E}'(\mathbb{R}))=\{c\delta_a: a\in \mathbb{R}, \;0\neq c\in \mathbb{C}  \}$. Suppose, on the contrary, that $(c_n\delta_{a_n})_{n\in \mathbb{N}}$
converges to $\delta_0'$ in $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$.

We first note that $(a_n)_{n\in \mathbb{N}}$ is bounded. For if not, then there exists a subsequence
$(a_{n_k})_{k\in \mathbb{N}}$ of $(a_n)_{n\in \mathbb{N}}$ such that $|a_{n_k}|>2$ for all $k\in \mathbb{N}$. Now choose a $\varphi\in \mathcal{D}(\mathbb{R})$
such that $\varphi'(0)=1$ and $\varphi\equiv 0$ on $\mathbb{R}\setminus(-1,1)$. Then we arrive at the contradiction that
$$
0=c_{n_k}\cdot 0= c_{n_k}\cdot \varphi(a_{n_k})= \langle c_{n_k}\delta_{a_{n_k}},\varphi\rangle  \stackrel{k\rightarrow\infty}{\longrightarrow} \langle \delta_0',\varphi\rangle=
-\varphi'(0)=-1.
$$
So $(a_n)_{n\in \mathbb{N}}$ is bounded.

Next we show that $(c_n)_{n\in \mathbb{N}}$ converges to $0$. Let $R>0$ be such that $|a_n|<R$ for all $n\in \mathbb{N}$.
Let $\psi\in \mathcal{D}(\mathbb{R})$ be such that $\psi\equiv 1$ on $[-R,R]$. Then we have
$$
c_n=c_n\cdot 1= c_n\cdot \psi(a_n)=\langle c_n \delta_{a_n},\psi \rangle
\stackrel{n\rightarrow\infty}{\longrightarrow} \langle \delta'_0,\psi\rangle=-\psi'(0)=0.
$$
Finally, we show that $(c_n\delta_{a_n})_{n\in \mathbb{N}}$
converges to $0$ in $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$.
For any $\chi \in \mathcal{D}(\mathbb{R})$, we have
$$
|\langle c_n\delta_{a_n},\chi\rangle|=|c_n|\cdot |\chi(a_n)|\leq |c_n|\cdot \|\chi \|_\infty\stackrel{n\rightarrow \infty}{\longrightarrow }
0\cdot \|\chi \|_\infty=0.
$$
So $(c_n\delta_{a_n})_{n\in \mathbb{N}}$
converges to $0$ in $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$. But this is a contradiction, since $0\neq \delta'_0$ in $\mathcal{E}'(\mathbb{R})$. Consequently, $U_1(\mathcal{E}'(\mathbb{R}))$ is not dense in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$,
and so ${\textrm{tsr}}(\mathcal{E}'(\mathbb{R}))\geq 2$.
\end{proof}

\section{$\mathrm{tsr}(\mathcal{E}'(\mathbb{R}))\leq 2$}
\label{section_leq_2}

\noindent The idea is to reduce the  determination of tsr$(\mathcal{E}'(\mathbb{R}))$ to tsr($\mathbb{C}[z]$)
of the polynomial ring $\mathbb{C}[z]$ as follows. Given a pair from $\mathcal{E}'(\mathbb{R})$, we use mollification to make
a pair in $\mathcal{D}(\mathbb{R})$, and then approximate
the resulting smooth functions by
a linear combination of Dirac distributions with uniform spacing.
The uniform spacing affords
the identification of the linear combination of Dirac deltas with the ring of polynomials.

\noindent
For $n\in \mathbb{N}$, we define the collection ${\mathbf D}_{n}$ of all `finitely
supported Dirac delta combs' with spacing $1/n$ by
$$
{\mathbf D}_{n}:=\textrm{span}\;\! \{ \delta_{\scriptscriptstyle{k/n}}:k\in \mathbb{Z}\},
$$
where `span' means the set of  all (finite) linear combinations.

\begin{lemma}[Approximating a pair of Dirac combs by a {\em unimodular} pair]
\label{lemma_1}
Let $n\in \mathbb{N}$ and $T,S\in {\mathbf D}_{n}$. Then there exist sequences $(T_k)_{k\in \mathbb{N}}$ and
$(S_k)_{k\in \mathbb{N}}$ in ${\mathbf D}_n$, which converge to $T,S$, respectively, in
$(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$, and hence\footnote{Because $\mathcal{E}'(\mathbb{R})$ is a Montel space; see \cite[Corollary~1, p.358]{Tre}.} also in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$,  and are such that for each $k$,
$(T_k,S_k)\in U_2(\mathcal{E}'(\mathbb{R}))$.
\end{lemma}
\begin{proof}
Write
$\displaystyle
T=\sum_{\ell=-L}^{L}t_{\scriptscriptstyle{\ell}} \delta_{\scriptscriptstyle{\ell/n}}$, and
 $\displaystyle
S=\sum_{\ell=-L}^{L}s_{\scriptscriptstyle{\ell}} \delta_{\scriptscriptstyle{\ell/n}},
$ for some $L\in \mathbb{N}$, $t_{\scriptscriptstyle{\ell}},s_{\scriptscriptstyle{\ell}} \in \mathbb{C}$.

\noindent
Define
\begin{eqnarray*}
 p_{\scriptscriptstyle{T}}&:=&t_{\scriptscriptstyle{-L}}+t_{\scriptscriptstyle{-L+1}} z+\cdots +t_{\scriptscriptstyle{L}} z^{2L},\\
 p_{\scriptscriptstyle{S}}&:=&s_{\scriptscriptstyle{-L}}+s_{\scriptscriptstyle{-L+1}} z+\cdots +s_{\scriptscriptstyle{L}} z^{2L}.
\end{eqnarray*}
For a given $k\in \mathbb{N}$, let $\epsilon=1/(2^k \cdot 2L)>0$. Then we can perturb the coefficients of the
polynomials $p_{\scriptscriptstyle{T}}, p_{\scriptscriptstyle{S}}$ within a distance of $\epsilon$ to make them
have no common zeros, that is after perturbation of coefficients they
are coprime in the ring $\mathbb{C}[z]$. Indeed any polynomial $p_{\scriptscriptstyle{T}}, p_{\scriptscriptstyle{S}}$
can be factorized as
$$
p_{\scriptscriptstyle{T}}=C \prod (z-\alpha_{\scriptscriptstyle{\ell}}), \quad
p_{\scriptscriptstyle{S}}=C'\prod (z-\beta_{\scriptscriptstyle{\ell}}),
$$
and if there is some common zero $\alpha_{\scriptscriptstyle{\ell}}=\beta_{\scriptscriptstyle{\ell'}}$, we simply
replace $\beta_{\scriptscriptstyle{\ell'}}$ by $\beta_{\scriptscriptstyle{\ell'}}+\epsilon'$ with an $\epsilon'$
small enough so that the final coefficients (of this new perturbed polynomial obtained from $p_{\scriptscriptstyle{S}}$), which are polynomial
functions of the zeros, lie within the desired
$\epsilon$ distance of the coefficients of $p_{\scriptscriptstyle{S}}$.
So we can choose
$\widetilde{t}_{\scriptscriptstyle{-L,k}},\cdots ,\widetilde{t}_{\scriptscriptstyle{L,k}}$ and
$\widetilde{s}_{\scriptscriptstyle{-L,k}},\cdots ,\widetilde{s}_{\scriptscriptstyle{L,k}}$ such that for all
$\ell=-L,\cdots, L$, we have
$$
|t_{\scriptscriptstyle{\ell}}-\widetilde{t}_{\scriptscriptstyle{\ell,k}}|
<
\frac{1}{2^k}\cdot \frac{1}{2L} \quad \textrm{ and } \quad
|s_{\scriptscriptstyle{\ell}}-\widetilde{s}_{\scriptscriptstyle{\ell,k}}|
<
\frac{1}{2^k}\cdot \frac{1}{2L} ,
$$
and so that
\begin{eqnarray*}
 \widetilde{p}_{\scriptscriptstyle{T,k}}
 &:=&
 \widetilde{t}_{\scriptscriptstyle{-L,k}}+\widetilde{t}_{\scriptscriptstyle{-L+1,k}} z+
 \cdots +\widetilde{t}_{\scriptscriptstyle{L,k}} z^{2L},\\
 \widetilde{p}_{\scriptscriptstyle{S,k}}
 &:=&
 \widetilde{s}_{\scriptscriptstyle{-L,k}}+\widetilde{s}_{\scriptscriptstyle{-L+1,k}} z+
 \cdots +\widetilde{s}_{\scriptscriptstyle{L,k}} z^{2L}
\end{eqnarray*}
have no common zeros. Thus $\widetilde{p}_{\scriptscriptstyle{T,k}},\widetilde{p}_{\scriptscriptstyle{S,k}}$
are coprime in $\mathbb{C}[z]$, and hence there exist polynomials
$q_{\scriptscriptstyle{T,k}},q_{\scriptscriptstyle{S,k}}\in \mathbb{C}[z]$ (\cite[Corollary~8.5, p.374]{Art}) such
that
$$
\widetilde{p}_{\scriptscriptstyle{T,k}}\cdot q_{\scriptscriptstyle{T,k}}
+\widetilde{p}_{\scriptscriptstyle{S,k}} \cdot q_{\scriptscriptstyle{S,k}}=1.
$$
Set $Q_{\scriptscriptstyle{T,k}}:=z^L q_{\scriptscriptstyle{T,k}}$ and
$Q_{\scriptscriptstyle{S,k}}:=z^L q_{\scriptscriptstyle{S,k}}$, and
\begin{eqnarray*}
 P_{\scriptscriptstyle{T,k}}
 &:=&
 \widetilde{t}_{\scriptscriptstyle{-L,k}}z^{-L}+\widetilde{t}_{\scriptscriptstyle{-L+1,k}} z^{-L+1}
 +\cdots +\widetilde{t}_{\scriptscriptstyle{L,k}} z^{L},\\
 P_{\scriptscriptstyle{S,k}}
 &:=&
 \widetilde{s}_{\scriptscriptstyle{-L,k}}z^{-L}+\widetilde{s}_{\scriptscriptstyle{-L+1,k}} z^{-L+1}
 +\cdots +\widetilde{s}_{\scriptscriptstyle{L,k}} z^{L}.
\end{eqnarray*}
Then in the ring $\mathbb{C}[z,z^{-1}]$ of linear combinations of monomials $z^n$, where $n\in \mathbb{Z}$ (i.e. the Laurent polynomial ring $\mathbb{C}[z,z^{-1}]
=\mathbb{C}[z,w]/\langle zw-1 \rangle$; see for example \cite[p.367]{Art}), we have
\begin{equation}
 \label{Bezout_27_10_2018_11:41}
 P_{\scriptscriptstyle{T,k}}\cdot  Q_{\scriptscriptstyle{T,k}}+ P_{\scriptscriptstyle{S,k}}\cdot Q_{\scriptscriptstyle{S,k}}=1.
\end{equation}
Suppose that $Q_{\scriptscriptstyle{T,k}}$ and $Q_{\scriptscriptstyle{S,k}}$ have the expansions
\begin{eqnarray*}
 Q_{\scriptscriptstyle{T,k}}
 &=&
 \tau_{\scriptscriptstyle{L',k}}z^{L+L'}+\tau_{\scriptscriptstyle{L'-1,k}} z^{L+L'-1}+\cdots
 +\tau_{\scriptscriptstyle{0,k}} z^{L},\\
 Q_{\scriptscriptstyle{S,k}}
 &=&
 \sigma_{\scriptscriptstyle{L',k}}z^{L+L'}+\sigma_{\scriptscriptstyle{L'-1,k}} z^{L+L'-1}+\cdots
 +\sigma_{\scriptscriptstyle{0,k}} z^{L}.
\end{eqnarray*}
Finally, set
$$
T_k:=\sum_{\ell=-L}^{L}\widetilde{t}_{\scriptscriptstyle{\ell,k}} \delta_{\scriptscriptstyle{\ell/n}}, \quad
S_k:=\sum_{\ell=-L}^{L}\widetilde{s}_{\scriptscriptstyle{\ell,k}} \delta_{\scriptscriptstyle{\ell/n}},
$$
and
\begin{eqnarray*}
 U_{k}
 &:=&
 \tau_{\scriptscriptstyle{L',k}}\delta_{\scriptscriptstyle{(L+L')/n}}+\tau_{\scriptscriptstyle{L'-1,k}}
 \delta_{\scriptscriptstyle{(L+L'-1)/n}}+\cdots +\tau_{\scriptscriptstyle{0,k}} \delta_{\scriptscriptstyle{L/n}},\\
 V_{k}
 &:=&
 \sigma_{\scriptscriptstyle{L',k}}\delta_{\scriptscriptstyle{(L+L')/n}}+\sigma_{\scriptscriptstyle{L'-1,k}}
 \delta_{\scriptscriptstyle{(L+L'-1)/n}}+\cdots +\sigma_{\scriptscriptstyle{0,k}} \delta_{\scriptscriptstyle{L/n}}.
\end{eqnarray*}
Then it follows from \eqref{Bezout_27_10_2018_11:41} that
\begin{equation}
 \label{Bezout_5_10_2018_12:46}
T_k\ast U_k +S_k \ast V_k=\delta_{\scriptscriptstyle{0}}.
\end{equation}
To see this, we note that $\Phi:\mathbb{C}[z,z^{-1}]\rightarrow {\mathbf{D}}_n$ given by
\begin{eqnarray*}
\Phi(z)=\delta_{\scriptscriptstyle 1/n}&\textrm{ and }&\Phi(1)=\delta_{\scriptscriptstyle 0}
\end{eqnarray*}
defines a ring homomorphism, and then \eqref{Bezout_5_10_2018_12:46} above follows by
applying $\Phi$ on both sides of \eqref{Bezout_27_10_2018_11:41}.
Hence $(T_k,U_k)\in U_2(\mathcal{E}'(\mathbb{R}))$. Also, for any $\varphi \in \mathcal{E}(\mathbb{R})$, we have
\begin{eqnarray*}
\Big|\big\langle (T-T_k),\varphi\big\rangle\Big|
&=&
\left|\sum_{\ell=-L}^{L}(t_{\scriptscriptstyle{\ell}}-\widetilde{t}_{\scriptscriptstyle{\ell,k}})
\langle \delta_{\scriptscriptstyle{\ell/n}},\varphi\rangle\right|
\\
&=&\frac{1}{2^k} \cdot \frac{1}{2L}\cdot  2L\cdot  \sup_{x\in [-
\frac{L}{n},\frac{L}{n}]}|\varphi(x)|\\
&=&
\frac{1}{2^k}  \cdot \sup_{x\in [-
\frac{L}{n},\frac{L}{n}]}|\varphi(x)|
\stackrel{k\rightarrow \infty}{\longrightarrow }0.
\end{eqnarray*}
Hence $T_k\stackrel{k\rightarrow \infty}{\longrightarrow }T$ in
$(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$ as $k\rightarrow \infty$. But then this convergence also is
valid in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$,  by \cite[Corollary~1, p.358]{Tre}, since $\mathcal{E}'(\mathbb{R})$ is a Montel space.
Similarly, $S_k\stackrel{k\rightarrow \infty}{\longrightarrow }S$ in
$(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$ as $k\rightarrow \infty$, and again, the convergence holds in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$.  This completes the proof.
\end{proof}

\begin{lemma}[Approximation in $\mathcal{E}'(\mathbb{R})$ by Dirac combs]
\label{lemma_2}$\;$

\noindent
Let $T\in \mathcal{E}'(\mathbb{R})$. Then there exists a sequence
$(T_n)_{n\in \mathbb{N}}$ such that
\begin{itemize}
\item for all $n\in \mathbb{N}$, $T_n\in {\mathbf D}_n$, and
\item $T_n\stackrel{n\rightarrow \infty}{\longrightarrow} T$ in
  $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$, and hence also in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$.
 \end{itemize}
\end{lemma}
\begin{proof} Let $k\in \mathbb{N}$ be such that the support of $T$ is
  contained in $(-k,k)$. We first produce a mollified
  approximating sequence for $T$. Let $\varphi:\mathbb{R}\rightarrow [0,\infty)$ be any  test
  function in $\mathcal{D}(\mathbb{R})$ with support in $[-a,a]$ for some
  $a>0$, and such that
  $$
  \int_\mathbb{R} \varphi(x) dx =1.
  $$
  Then we know that if we define
 $
\varphi_m(x):=m\cdot \varphi(mx)$ ($m\in \mathbb{N}$),
then for each $m$,
$$
f_m:=T\ast \varphi_m
$$
is a smooth function having
a compact support, and moreover,
$$
T\ast \varphi_m\stackrel{m\rightarrow \infty}{\longrightarrow} T
$$
in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$; see for example \cite[Theorem 3.3,
p.97]{Bar}. So the convergence is also valid in $(\mathcal{E}'(\mathbb{R}), \sigma(\mathcal{E}',\mathcal{E}))$. Moreover, as the support of $ f_m= T\ast \varphi_m$ is
contained in the sum of the supports of $\varphi_m$ and of $T$,
for all $m$ large enough, say $m\geq M$, we have
\begin{eqnarray*}
\textrm{supp}(T\ast \varphi_m)
&\subset &\textrm{supp}(T)+\textrm{supp}(\varphi_m)\\
&\subset &\textrm{supp}(T)+[-a/m,a/m]\\
&\subset& [-k,k].
\end{eqnarray*}
From now on, we will assume that $m\geq M$, so that
$\textrm{supp}(f_m)\subset [-k,k]$.  Now we will approximate $f_m$ by
Dirac comb elements. To this end, we define
$$
T_{m, n}:= \sum_{\ell=0}^{n-1} \frac{2k}{n} \cdot f_m\Big(-k+\frac{2k}{n} \ell\Big) \cdot
\delta_{\scriptscriptstyle{-k+\frac{2k}{n}\ell}}\;\; \in {\mathbf D}_n.
$$
We will show that
$T_{m,n}\stackrel{n\rightarrow\infty}{\longrightarrow} f_m$ in
$(\mathcal{E}'(\mathbb{R}), \sigma(\mathcal{E}',\mathcal{E}))$.  Let $\psi\in \mathcal{E}(\mathbb{R})$. Then
\begin{eqnarray*}
 \langle T_{m,n},\psi\rangle
 &=&
 \left\langle \sum_{\ell=0}^{n-1} \frac{2k}{n} \cdot f_m\Big(-k+\frac{2k}{n} \ell\Big) \cdot
\delta_{\scriptscriptstyle{-k+\frac{2k}{n}\ell}},\; \psi\right\rangle \\
&=& \sum_{\ell=0}^{n-1} \frac{2k}{n}\cdot  f_m\Big(-k+\frac{2k}{n} \ell\Big)
\left\langle\delta_{\scriptscriptstyle{-k+\frac{2k}{n}\ell}},\;\psi\right\rangle\\
&=&
\sum_{\ell=0}^{n-1} \frac{2k}{n} \cdot f_m\Big(-k+\frac{2k}{n} \ell\Big)\cdot
\psi\Big( -k+\frac{2k}{n}\ell\Big).
\end{eqnarray*}
Thus $ \langle T_{m,n},\psi\rangle $ gives a Riemann sum for the
integral of the continuous function $f_m\psi$ with compact support
contained in $[-k,k]$, giving
\begin{eqnarray*}
&& \big|\langle T_{m,n},\psi\rangle -\langle f_{m},\psi\rangle\big|
\\
&=& \!\!\!
 \left|
\sum_{\ell=0}^{n-1} \frac{2k}{n} \!\cdot \!f_m\Big(\!-\!k\!+\!\frac{2k}{n} \ell\Big)\!\cdot \!
\psi\Big( \!-\!k\!+\!\frac{2k}{n}\ell\Big)
\!-\!
\int_{-k}^k \!f_m(x) \psi (x) dx\right|\stackrel{n\rightarrow\infty}{\longrightarrow} 0.
\end{eqnarray*}
Hence $T_{m,n}\stackrel{n\rightarrow\infty}{\longrightarrow} f_m$ in
$(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$. As $\mathcal{E}'(\mathbb{R})$ is a Montel space, this convergence is also valid in $(\mathcal{E}'(\mathbb{R}),\beta(\mathcal{E}',\mathcal{E}))$, and the proof is completed.
\end{proof}

\begin{proposition}
\label{prop_leq}
 $\textrm{\em tsr}(\mathcal{E}'(\mathbb{R}))\leq 2$.
\end{proposition}
\begin{proof}
  Let $T,S\in \mathcal{E}'(\mathbb{R})$. Throughout this proof, $\mathcal{E}'(\mathbb{R})$ is endowed with the strong dual topology $\beta(\mathcal{E}',\mathcal{E})$, and then $(\mathcal{E}'(\mathbb{R}))^2=\mathcal{E}'(\mathbb{R})\times \mathcal{E}'(\mathbb{R})$ is equipped the product topology.
  Let $V$ be a neighbourhood of $(T,S)$ in
  $(\mathcal{E}'(\mathbb{R}))^2$. By Lemma \ref{lemma_2}, it follows that
  $
  \bigcup\limits_{n\in \mathbb{N}} ({\mathbf D}_n\times {\mathbf D}_n)
  $
  is sequentially dense, and hence dense, in $(\mathcal{E}'(\mathbb{R}))^2$.

  Thus there exists a pair
  $(T_*,S_*)\in V\cap ({\mathbf D}_n\times {\mathbf D}_n)$ for some $n\in \mathbb{N}$.  By Lemma~\ref{lemma_1}, there exists a
  sequence $(T_k,S_k)_{k\in \mathbb{N}}$ in
  $({\mathbf D}_n\times {\mathbf D}_n)\cap U_2(\mathcal{E}'(\mathbb{R}))$ that converges to $(T_*,S_*)$ in $(\mathcal{E}'(\mathbb{R}))^2$.
  Since $V$ is also a neighbourhood of
  $(T_*,S_*)$ in $(\mathcal{E}'(\mathbb{R}))^2$,  there exists an index $K$ large enough so that
  for all $k>K$, $(T_k,S_k)\in V$.

  Consequently, $U_2(\mathcal{E}'(\mathbb{R}))$ is
  dense in $(\mathcal{E}'(\mathbb{R}))^2$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{main_thm}]
It follows from Propositions~\ref{prop_geq} and \ref{prop_leq} that
 the topological stable rank of $(\mathcal{E}'(\mathbb{R}), +,\cdot,\ast, \beta(\mathcal{E}',\mathcal{E}))$ is equal to $2$.
\end{proof}

\begin{remarks}$\;$

\medskip

\noindent (1)  From the proofs, it is clear that we have shown that
$U_1(\mathcal{E}'(\mathbb{R}))$ is not dense in $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$, while
$U_2(\mathcal{E}'(\mathbb{R}))$ is sequentially dense, and hence dense, in
$(\mathcal{E}'(\mathbb{R}))^2$ endowed with the product topology with $\mathcal{E}'(\mathbb{R})$ bearing the $\sigma(\mathcal{E}',\mathcal{E})$ topology.

However, we note that $\ast:\mathcal{E}'(\mathbb{R})\times \mathcal{E}'(\mathbb{R})\rightarrow \mathcal{E}'(\mathbb{R})$ is not continuous if we use the $\sigma(\mathcal{E}',\mathcal{E})$ topology on $\mathcal{E}'(\mathbb{R})$: For example, in $(\mathcal{E}'(\mathbb{R}),\sigma(\mathcal{E}',\mathcal{E}))$, we have that
$
\delta_{\pm n}\stackrel{n\rightarrow\infty }{\longrightarrow} 0,
$
so that in the product topology on $(\mathcal{E}'(\mathbb{R}))^2$, we have
 $
(\delta_n,\delta_{-n})\stackrel{n\rightarrow\infty }{\longrightarrow} (0,0)
$. But $\delta_n\ast \delta_{-n}=\delta_{n-n}=
\delta_0\stackrel{n\rightarrow\infty }{\longrightarrow} \delta_0\neq 0=0\ast 0$. So $(\mathcal{E}'(\mathbb{R}), +,\cdot,\ast,\sigma(\mathcal{E}',\mathcal{E}))$ is
not a topological algebra in the sense of our Definition~\ref{def_top_alg}.

\medskip

\noindent (2) We remark that in higher dimensions, with a similar analysis, it can be shown  that ${\textrm{tsr}}(\mathcal{E}'(\mathbb{R}^d))\leq d+1$.

\medskip

\noindent (3) The Bass stable rank (a notion from algebraic $K$-theory, recalled below)
of $\mathcal{E}'(\mathbb{R})$ is not known.

If $\mathcal{A}$ is a commutative unital algebra, then
 $(a_1,\dots,a_n,b)\in  U_{n+1}(\mathcal{A})$ is  called {\it reducible} if there exists an $n$-tuple
 $(\alpha_1,\cdots,\alpha_n)\in \mathcal{A}^n$ such that we have
  $
 (a_1+\alpha_1 b,\cdots, a_n+\alpha_n b)\in U_n(\mathcal{A}).
 $
 It can be seen that if every element of $U_{n+1}(\mathcal{A})$ is reducible, then every element of
 $U_{n+2}(\mathcal{A})$ is reducible too. The {\em Bass stable rank} of $\mathcal{A}$, denoted by $\textrm{bsr} \;\!\mathcal{A}$,
 is the smallest $n\in \mathbb{N}$ such that every element in $U_{n+1}(\mathcal{A})$ is reducible, and if
  no such $n$ exists, then $\textrm{bsr} \;\!\mathcal{A}:=\infty$. It is known that for commutative unital Banach algebras $\mathcal{A}$,
 $\textrm{bsr}\;\! \mathcal{A}\leq {\textrm{tsr}} \;\!\mathcal{A}$ \cite[Theorem~3]{CorLar}. But the validity of such an inequality in the
 context of topological algebras does not seem to be known.
 We conjecture that  $\textrm{bsr}(\mathcal{E}'(\mathbb{R}))=2$.

 \goodbreak

\noindent (4)
There
are also several other natural convolution algebras of distributions on $\mathbb{R}$, for example
\begin{eqnarray*}
 \mathcal{D}_{\scriptscriptstyle{\geq \bigcdot}}'(\mathbb{R})&:=&\{T\in \mathcal{D}'(\mathbb{R}): \textrm{supp}(T) \textrm{ is bounded on the left}\}, \\
 \mathcal{D}_{\scriptscriptstyle{\geq 0}}'(\mathbb{R})&:=&\{T\in \mathcal{D}'(\mathbb{R}): \textrm{supp}(T)\subset[0,\infty)\},
\end{eqnarray*}
 and we leave the determination of the stable ranks
of these algebras as open questions.

\medskip

\noindent (5) \cite[Corollary~3.1]{MaaSas} gives a `corona-type' pointwise condition for coprimeness in $\mathcal{E}'(\mathbb{R})$, reminiscent of the famous Carleson corona condition\footnote{The Hardy algebra $H^\infty(\mathbb{D})$ is the Banach algebra of all bounded and holomorphic functions on the unit disk $\mathbb{D}:=\{z\in \mathbb{C}:|z|<1\}$. The Carleson Corona Theorem \cite{Car} says that $(f_1,f_2)\in U_2(H^\infty(\mathbb{D}))$ if and only if there exists a $\delta>0$ such that for all $z\in \mathbb{D}$, $|f_1(z)|+|f_2(z)|>\delta$.} of coprimeness in the Banach algebra $H^\infty(\mathbb{D})$:

 \smallskip

 \noindent $T_1, T_2\in U_2(\mathcal{E}'(\mathbb{R}))$ if and only if
 there exist positive $C,N,M$ such that
 $
 \textrm{ for all }z\in \mathbb{C},\;\;
 |\widehat{T_1}(z)|+|\widehat{T_2}(z)|\geq C(1+|z|^2)^{-N} e^{-M|\textrm{Im}(z)|}.
 $
\end{remarks}

\medskip

\noindent {\bf Acknowledgements:} I thank Professor Michael Kunzinger
(University of Vienna) for answering my query on whether density in
$\mathcal{E}'$ implies sequential density, for the reference \cite{Obe}, and for
 useful comments. I also thank Professor Peter Wagner (University of Innsbruck)
 for showing me the alternative approach to establishing Proposition~\ref{prop_geq},
 which we have included in this article (Proposition~\ref{prop_PW}).

\noindent

redpajama_set_name: RedPajamaArXiv


text:
Greg Mattison Leaves Michigan for Ohio State
Posted: Mon 3:49 PM, Jan 07, 2019
A top Michigan assistant is leaving to be co-defensive coordinator at Ohio State.
Ohio State announced Monday that Greg Mattison, who spent the past four years as defensive line coach under Jim Harbaugh, will join the staff of new Buckeyes coach Ryan Day.
A second defensive coordinator, Jeff Hafley, also was hired Monday. That means Greg Schiano, who was Urban Meyer's defensive coordinator for three seasons, is out of a job.
Mattison was Michigan's defensive coordinator from 2011 to 2014 and previously was an assistant with the Baltimore Ravens. He was co-defensive coordinator and defensive line coach at Florida for three seasons under Meyer.
Hafley, who spent the past seven seasons as an NFL assistant, also will coach Ohio State's secondary.
(Copyright 2019 The Associated Press. All rights reserved.)
Sooners tops Big 12 preseason football poll
Big 12 preseason football honors
Mountaineers Back in Pads
Marshall Taking on Notre Dame in 2022
Fields Immediately Eligible
Comments are posted from viewers like you and do not always reflect the views of this station.
Please enable JavaScript to view the powered by Disqus.

redpajama_set_name: RedPajamaCommonCrawl


text:
Public Class Form1

    Private Sub Button1_Click(sender As Object, e As EventArgs) Handles Button1.Click
        'Lets get the sum of all Odd numbers using Mod operator
        Dim Odds As Integer = 0
        Dim AnyNumber = Val(TextBox1.Text)
        Dim AnyOtherNumber = Val(TextBox2.Text)

        For i = AnyNumber To AnyOtherNumber
            If i Mod 2 = 1 Then
                Odds += i
            End If
        Next

        Me.TextBox3.Text = Odds
    End Sub

    Private Sub Button2_Click(sender As Object, e As EventArgs) Handles Button2.Click
        'Lets get the sum of all Even numbers using Mod operator
        Dim Evens As Integer = 0
        Dim AnyNumber = Val(TextBox1.Text)
        Dim AnyOtherNumber = Val(TextBox2.Text)

        For i = AnyNumber To AnyOtherNumber
            If i Mod 2 = 0 Then
                Evens += i

            End If
        Next

        Me.TextBox4.Text = Evens
    End Sub

    Private Sub Button3_Click(sender As Object, e As EventArgs) Handles Button3.Click
        'Lets get the Total of all odd and even numbers
        Dim sum1 As Integer
        Dim sum2 As Integer
        Dim Totalsum As Integer

        sum1 = Val(TextBox4.Text)
        sum2 = Val(TextBox3.Text)

        Totalsum = sum1 + sum2

        Me.TextBox5.Text = Totalsum
    End Sub

    Private Sub Button4_Click(sender As Object, e As EventArgs) Handles Button4.Click
        'Clear all fields and start over
        Me.TextBox1.Text = ""
        Me.TextBox2.Text = ""
        Me.TextBox3.Text = ""
        Me.TextBox4.Text = ""
        Me.TextBox5.Text = ""
    End Sub

    Private Sub Button5_Click(sender As Object, e As EventArgs) Handles Button5.Click
        'Lets close the program for now
        Me.Close()
    End Sub
End Class

redpajama_set_name: RedPajamaGithub


Do your projects keep end-customer value sacrosanct at all stages in their life?
Or does the heavy-lifting of all the decisions, compromises and trade-offs end up with key end-user value being sacrificed (often without thought) in order to achieve a technical outcome (e.g. delivery on time?).
Do you feel like project teams are lose sight of the value that a project is delivering?
It's so easy with technology type projects to lose track of customer value. I think above all there is a mapping or translation process that happens in only one direction.
Assuming that you have a good understanding of the precise problems that a customer is trying to solve (perhaps a big assumption), there is inevitably a translation of those problem or value statements into the world of the technology domain, so that the good members of your team(s) can start defining the technology components that will enable that value.
Once the translation to technology talk happens, it never seems to happen in reverse. We start with "customer speak", continue with "technology speak" and from there we continue with "technology speak", even when reporting back to the customer.
But what if there was a relatively simple way for that concept to be maintained? At least it's simple in concept, if not in practice: it's called "focus".
I am often reminded of themes from an old movie when I see a team start to forget what they are trying to achieve, in the midst of hectic project delivery activities.
This movie (Quest for Fire) is set in Palaeolithic Europe 80,000 years ago, and follows a small Neanderthal tribe who know how to use fire, but are unable to make it. The tribe keeps a small fire burning in a little portable altar (seen in photo at left), and assigns a dedicated "Fire Keeper" to keep feeding and tending the flame. However, at one point the fire source is extinguished in an accident, and the film chronicles three men who are sent out to find a new source of fire from another tribe.
Think of the core value proposition in the project as being like this little flame. Properly maintained, the flame can feed larger fires of enthusiasm and commitment. Ultimately, if delivered to the end customer, the fire will kindle a much larger fire of success in whatever domain the customer operates: business, personal, not-for-profit and so forth.
The little flame of value in the project can easily be diminished or extinguished by the many forces that routinely threaten any project of meaningful size. Typical examples are dissonance of vision amongst executive leadership, compulsory technical practice guidelines, changing organisational imperatives, and personal goals (including job survival).
In any normal project this is difficult, but in a project which has been beset by more than its fair share of problems of any kind, end customer value can be the first casualty: extinguished by the mental and physical fatigue, cognitive overload and just plain stress of trying to get a project over the line.
If we lose this flame, it becomes easier and easier to compromise as the team tries to hit its objectives (typically delivery milestones).
"High Availability? who really needs it at launch?" "Alarms? Well people should be checking to make sure everything's alright!" "Testing? Who's got time for these 'theoretical' combinations and permutations or edge case tests" and so forth.
These examples are obvious, if not extreme. But if less obvious components are removed, how are we ever sure that this does not materially degradation customer value. How can a customer answer this question reliably: "we're going to go live without the xyz widget in the database", is that ok?". At best the response will be: "How the !$#% should I know". Mostly likely the response will be: "If you think that's ok". And in any case, by the time the question gets to the customer, the work has probably already happened.
To avoid this in a project requires discipline and focus. The discipline to maintain a mapping of the technical world of the project to the view of the customer. This requires extra work, or at least the maintenance of a work practice throughout the life of the project. The focus means maintaining that deep understanding of the customer value proposition, so that the mappings have currency and meaning.
How do we avoid this? The answer is "Discipline" and "focus".
The team has to understand the that the concept of end-customer value is not just some warm and fuzzy idea, it is a real and practical environment felt everyday by the customer and their stakeholders. The fundamental problem that the project team is trying to solve for the customer is the entire justification for the project, but teams lose sight of this everyday. It must at all costs be maintained throughout the life of the project.
The discipline to maintain a mapping of the technical world of the project to the view of the customer. This requires extra work, or at least the maintenance of a work practice throughout the life of the project. The focus means maintaining that deep understanding of the customer value proposition, so that the mappings have currency and meaning.
Reports, tools, charts and so forth must always contain a realistic view of both domains: technical and customer problem.
As a bare minimum, one person needs to take on the responsibility to be the "Fire Keeper" for your project, and for the customer. This is at least partially the rationale behind the "Voice of the Customer" empty chair in Amazon, and the "Product Owner" in Scrum.
In reality, the paleo FireKeeper had immense support from the tribe, whereas in modern-day project teams there are many different views and the "FireKeeper" can easily find themselves isolated or accused of slowing things down; of jeopardising the success of the project (how ironic).
The team (the whole team), needs to keep feeding the flame of value, lest you end up with a project successfully delivered (from technical perspective) but which fails to meet its market objectives.
To keep the value flame intact, we have to have it in the first place: we have to have a clear understanding of the project's ultimate customer value and to be able to share this with other team members and stakeholders.
It requires discipline and focus, but it can be done.
PreviousWhat is the Single Most Important Skill for a Project Manager?
NextDeliver projects fast. Faster than a startup. You can do it!


text:
Divisiveness And Intimidation Of The Innocent Is Rearing Its Ugly Head: Kufuor
Former President John Agyekum Kufuor is urging all Ghanaians as they take stock of the outgoing year and plan for the New Year, to "put our country first in all that we do and practice what we preach". In a New Year Message, Mr Kufuor wrote: "We have missed opportunities that would have taken our country up the ladder of progress and development. "We have not matched our words with our deeds. Divisiveness, fear and intimidation of innocent people, which should have been things of the past, are beginning to rear their ugly heads up again." He called on Ghanaians to endeavour at all times to strengthen the country's democratic process and never give in to the rule of tyranny or arbitrariness. "It must be the bounding duty of Ghanaians to strengthen constitutional rule and ensure that in all instances, the tenets of the constitution are applied fairly and equally to all citizens of the land irrespective of their political persuasions," Mr Kufuor said. The former President also urged Ghanaians to work harder in the coming year to ensure the prosperity of our dear country. He noted that the out-going year had undoubtedly been a challenging one and Ghanaians should thank God that He had brought us this far.


text:
Safer, newer blood thinners are under-prescribed for elderly and at-risk patients
Philadelphia, December 20, 2021 – Patients with atrial fibrillation are at risk of forming clots that can migrate to the brain and cause dangerous strokes; these can be prevented by chronic administration of blood-thinning drugs (anticoagulants). Introduced to the market in the past decade, direct oral anticoagulants (DOACs) are a much safer option for patients with atrial fibrillation than traditional warfarin treatment, in addition to being much more convenient because of the tight blood-sample monitoring and regular dose adjustment needed for warfarin treatment.
Although these new DOAC drugs were intended to eliminate some of the difficulties that elderly and frail (mostly elderly and/or unwell) people face associated with warfarin therapy, researchers report that DOACs are actually prescribed less frequently for frail patients than for non-frail patients, and that frail patients who receive anticoagulation therapy are still more likely to be prescribed warfarin. Their findings appear in the Canadian Journal of Cardiology, published by Elsevier.
"We were curious if the introduction of DOACs would help to address the known care gap that frail people with atrial fibrillation are less likely to be prescribed anticoagulant drugs like warfarin, because they have more trouble going to the lab frequently for blood work and are at higher risk for bleeding complications," explained lead investigator Finlay A. McAlister, MD MSc, Canadian VIGOUR Centre, and Department of Medicine, University of Alberta, Edmonton, AB, Canada.
The investigators examined prescribing data for 75,796 adults in Alberta, Canada, who had been discharged from an emergency department or hospital with a new diagnosis of nonvalvular atrial fibrillation between April 1, 2009 and March 31, 2019. The Hospital Frailty Risk Score, which is based on the presence or absence of 109 diagnoses such as ulcers, falls, osteoporosis, and dementia, was used to define frailty. More than 17,000 patients met the criteria for being frail, including 76% of patients who were elderly and 24% of patients under 65 years old.
Most of the patients met Canadian guideline criteria for anticoagulation treatment (92.1% of frail patients and 74.2% of non-frail patients). Anticoagulant prescribing increased more in non-frail patients (from 42.4% to 68.2%) than in frail patients (from 29.0% to 52.2%) over the period studied. Frail patients were still less likely to receive any anticoagulant, and even if anticoagulated, frail patients were 34% less likely to receive a DOAC than warfarin.
The investigators noted that DOACs were approved in Canada only shortly before the study period, and clinicians may have proceeded cautiously in those early years. Patients younger than 65 were more likely to receive a DOAC prescription. DOACs are generally covered by younger patients' work-related drug benefits, while the government-sponsored drug program in Canada for patients older than 65 requires documentation that the patient was unable to have their blood monitored or had abnormal results while on warfarin.
In an accompanying editorial, Lorne J. Gula, MD and Allan C. Skanes, MD, London Heart Rhythm Program, Western University, London, Ontario, Canada, wrote, "We hypothesize that clinicians may be more reluctant to anticoagulate elderly frail patients based on the risk of bleeding. But they will also have a higher risk of stroke. This is a well-designed, clinically important study that is a wake-up call for clinicians on several fronts. It is getting late in the game to consider DOACs as an innovation."
Dr. McAllister observed that the proportion of individuals with nonvalvular atrial fibrillation is increasing over time, and there has been a widespread assumption that the treatment gap in frail patients would have improved since the introduction of DOACs. "Our data demonstrate this is a misconception. We shouldn't assume that new therapies will be taken up by clinicians in the way we expect them to. As frail patients have higher event rates and thus stand to potentially derive greater benefit from anticoagulation than their non-frail counterparts, we encourage clinicians to evaluate absolute risks and benefits when making prescribing decisions."
Canadian Journal of Cardiology
10.1016/j.cjca.2021.09.021
Data/statistical analysis
The Introduction of Direct Oral Anticoagulants Has Not Resolved Treatment Gaps for Frail Patients With Nonvalvular Atrial Fibrillation
Tags: atriskbloodelderlynewerPatientsSaferthinnersunderprescribed


text:
Residential »
Real Estate Regulatory Act
Mumbai beats housing slump with emphasis on affordable homes
A Cushman & Wakefield report on Monday said the top eight cities saw the launch of about 25,800 units in the first quarter of 2017TNN | June 06, 2017, 17:00 IST
MUMBAI: New residential project launches saw a steady quarteronquarter decline for the last four quarters, corresponding with the announcement of the stringent Real Estate Regulatory Act (RERA) in March 2016 and demonetization in November 2016, says a real estate consultancy.
A Cushman & Wakefield report on Monday said the top eight cities saw the launch of about 25,800 units in the first quarter of 2017, registering a 16% decline from the corresponding period last year. Overall, residential launches fell by about 8% from April 2016 to March 2017 compared to the same period in 2015-16 (see box).
However, the Mumbai Metropolitan Region saw a huge 54% jump in new launches bet ween April 2016 and March 2017, mainly due to builders offering affordable homes. In the same period, Delhi-NCR saw a massive 50% drop.
The share of affordable segment in total launches improved to 30% from 25% in 2015-16; the high-end and luxury segments fell to 11% from 13%."While sales have been weak across segments, it has been prominent in the high-end and luxury segments over the last quarters owing to demandsupply mismatches," it said.
In Delhi-NCR, Bengaluru and select locations of Mumbai, the combined impact of the prolonged sales slowdown and the pressure of mounting inventory led to a price decline."Developers have restricted new launches and are reducing the effective cost of their offerings by bundling in incentives and add-ons to clear the inventory backlog,'' said the report.
Builders launched a higher number of subvention schemes, such as pay 5% now, 95% on delivery, and some developers offered assurances of compensationrefund of difference, if prices decline in future. "Developers are focusing on completing their existing under-construction projects, especially the ones at an advanced stage, to avoid contravening RERA 's rules and facing action,'' it said.
"Launches in the residential sector are expected to remain restricted over two-three quarters as developers will be making intrinsic changes to their business structure, operations and marketing strategies to comply with RERA norms,'' said Anshul Jain, MD, India, Cushman & Wakefield.
Top 10 locations in India to buy homes in 2017
Tags : Residential, RERA, real estate, Real Estate Regulatory Act, Mumbai, Cushman & Wakefield
Most Read in Residential
Gurugram: Woodshire residents protest against M3M over high maintenance charges
DDA set to offer penthouses worth Rs 3 crore in Dwarka
Gurugram: Mapsko residents urge Haryana CM to cancel CC issued to project
Over 170 buyers of Bandra society suffer after DB Realty delays rent
Allotment letter adequate proof of investment in flat: ITAT
Over 75,000 homebuyers still await their dream houses
No clarity on parking laws in societies, residents cry foul


I was pleasantly surprised to see it was an option, like it was tailor made for me!
My favorites are probably Rata Novus, Faren's Flier, The Golden Cave, The Olmakhan, Labyrinthine Cliffs, and Taimi's Game. There are others I like, but it fluctuates and I've written so much I forget about a lot of things.
I think this is a very good idea. I'll bring it up with the team.


You may not even use all of them, but you can go though which ones you do like. There is quite a range of them.
PS Brushes - 80 in total. Meant to look like different kinds of pencils. You can download them and use them if you do, credit me and show me what you used them for. I'd love to see.
Can I use it for commercial use when drawing commissions?
Thank you so much, you saved hours of my life.
THANK YOU~! It's so nice!
Why is this getting so many views and faves all of a sudden?
because you are helping by sharing... thank you!
Thank you! It's so good for me!! !!
holy cow, Thank you so much!!
These are so lovely! Thank you for sharing!!
These are wonderful! Thank you thank you thank you!


text:
\section{Introduction}
\label{sec_intro}

Fairness research in Machine Learning (ML) has seen rapid growth in recent years; however, it remains largely rooted in Western concerns and histories: the injustices they focus on (\textit{e.g., \@}{}along race and gender), the datasets they study (\textit{e.g., \@}{}ImageNet), the measurement scales they use (\textit{e.g., \@}{}Fitzpatrick), and the legal tenets they draw from (\textit{e.g., \@}{}equal opportunity). This implicitly Western take is troublingly becoming a \textit{universal} ethical framework for ML, \textit{e.g., \@}{}AI strategies from India \cite{NITI}, Tunisia \cite{TunisiaAI}, and Mexico \cite{MexicoAI} all derive from this work, but fails to account for the several assumptions conventional algorithmic fairness makes about the availability and efficacy of the surrounding institutions and infrastructures. However, these infrastructures, values, and legal systems cannot be naively generalised to various non-Western countries.

Let us consider the example of facial recognition technology, where demonstration of fairness failures resulted in bans and moratoria in the US. Several factors led to this outcome:
decades of empiricism on proxies and metrics that correspond to subgroups in the West \cite{fitzpatrick1988validity}; public datasets, APIs, and laws enabling analysis of model outcomes \cite{MachineB33:online, HowFacia39:online}; an ML research/industry responsive to bias reports from users and civil society \cite{Weareimp72:online, buolamwini2018gender}; the existence of government representatives glued into technology policy \cite{JayapalJ44:online}; and an active media that scrutinizes downstream impacts of AI \cite{HowFacia39:online}. However, due to various cultural, ethnic, and infrastructural differences, these factors are often absent or irrelevant in much else of the world. Could fairness have structurally different meanings or mechanisms in non-Western contexts? How do social, economic, and infrastructural factors influence implementation of meaningful fairness?

In this position paper, we present insights from qualitative interviews with 36 Indian scholars and activists, and a discourse analysis of emerging algorithmic deployments in India. India is home to 1.38 billion people and their multiple languages, religions, cultural systems, and ethnicities. AI deployments are prolific in the public sector, \textit{e.g., \@}{}in predictive policing \cite{PredictiveIndia}, facial recognition \cite{FacialRecIndia}, and agriculture \cite{AgricIndia}. Despite this forward momentum, there is a dire lack of conversations on advancing algorithmic fairness for such a large population. Our paper presents a first step towards filling this important gap. We contend that conventional Fair-ML approach may be inappropriate and inimical in India, if it does not engage with local structures. For a country as deeply plural, complex, and contradictory as India---where the distance between models and oppressed communities is large---optimising model fairness alone can be mere tokenism. We call for action along three critical pathways towards algorithmic fairness in India: \textit{Recontextualising}, \textit{Empowering}, and \textit{Enabling} (see Figure~\ref{fig:Indiaframework}). The considerations we present involve collective responsibility of inter-disciplinary Fair-ML researchers, and push the bounds of what is considered to be fairness research.

\vspace{-.7em}
\section{Methodology}
\label{sec_methodology}

We perform a synthesis of (i) 36 interviews with researchers and activists across various disciplines working with marginalized Indian communities, and (ii) observations of current algorithmic deployments and policies in India. We chose our respondents from diverse disciplines: Computer Science (11), Activism (9), Law and Public Policy (6), Science and Technology Studies (5), Development Economics (2), Sociology (2), and Journalism (1). All respondents had 10-30 years of experience working with on social justice in India, on areas including caste, gender, labour, disability, surveillance, health, and constitutional rights. In conjunction with qualitative interviews, we analyzed various algorithmic deployments and emerging policies in India, starting from 2009. We identified and analysed various Indian news publications, policy documents, grassroots fora, and prior research on automation in India. The systematic process followed for the collection, categorization, and synthesis of data to develop the key insights below are described in detail in Appendix~\ref{app_methodology}.

\vspace{-.7em}
\section{Summary of Findings}
\label{sec_findings}

\begin{wrapfigure}{r}{0.47\textwidth}
\vspace{-.7cm}
  \begin{center}
    \includegraphics[width=\linewidth]{fairnessdistance.png}
  \end{center}
\caption{\label{fig:Indiadistance} \small Challenges to algorithmic fairness in India: A Model-Community distance perspective}
\vspace{-.8cm}
\end{wrapfigure}

Our study finds that the conventional fairness approach may be inappropriate, insufficient, or even inimical in India, if it does not engage with local structures. In India where the distance between ML models and dis-empowered communities whom they aim to serve is large---via technical distance, social distance, ethical distance, temporal distance, and physical distance---a myopic take on localising `fair' model outputs alone can backfire. We identify three high-level themes/clusters of challenges (Figure~\ref{fig:Indiadistance}) that engulf the large distance between models deployed in India and the communities they impact.
We summarize them below.

\textbf{Flawed data and model assumptions}: We found numerous ways data and model assumptions fail in India, primarily: \textit{data distortions} owing to infrastructural challenges and technology usage patterns (\textit{e.g., \@}{}SIM card sharing among women \cite{sambasivan2018privacy}) that makes datasets not faithful representations of people and phenomena; \textit{data incompleteness} due to the models gearing for the `good' data profiles that largely favor middle-class men---in a society where 50\% do not have Internet, especially women and lower caste people; \textit{axes of discrimination} and their proxies and indicators that differ from how they manifest in the West (for a detailed account, see Appendix~\ref{app_subgroups_table}); and \textit{lack of Indic justice approaches} like reservations/quotas that present new challenges to fairness evaluations and interventions.

\textbf{ML makers' distance}: Several respondents described a transactional mindset towards Indians, seeing them
as agency-less data subjects that generated large-scale behavioural traces to improve ML models (also see \cite{Digitalc91:online}), resulting in poor recourse and redress for Indian users, especially for those dis-empowered by caste, religion, or gender. The human infrastructures who played a crucial role in providing recourse to marginalised Indian communities (\textit{e.g., \@}{}street-side bureaucrats, and front line workers) were removed in ML systems. Many respondents also pointed out the lack of diverse representation in Indian tech workforce that the distant ML makers often fail to recognize.

\textbf{AI aspiration}: AI is aspirationally viewed in India, with high reliance for high-stakes domains, accompanied with high trust in automation, limited transparency and the lack of an empowered ecosystem to question algorithmic interventions. For instance, respondents pointed to the lack of a buffer zone of journalists, activists, and researchers to keep ML system builders accountable. Lack of transparency and stakeholder participation, along with the its 'neutral' and 'human-free' associations lent misplaced credence to its algorithmic authority, making it further inscrutable.

\vspace{-.7em}
\section{Towards Algorithmic Fairness in India}

To account for the challenges outlined above, we need to understand and design for \textit{end-to-end} chains of algorithmic power, including how AI systems are conceived, produced, utilised, appropriated, assessed, and contested in India. To this end, we propose a research agenda where we call for action along three critical and contingent pathways towards successful AI fairness in India: \textit{Recontextualising}, \textit{Empowering}, and \textit{Enabling} (see Figure~\ref{fig:Indiaframework}). These pathways present new sociotechnical research challenges and require cross-disciplinary and cross-institutional collaborations.

\vspace{-.7em}
\subsection{Recontextualising data and models}

\textbf{Data} plays a critical role in measurements and mitigation of algorithmic bias. We must be (even more than usual) sceptical of Indian datasets (due to challenges outlined above), and study how fairness research could handle the known distortions in the data as well as how we must account for data voids \cite{golebiewski2019data} for which statistical extrapolations might be invalid. The vibrant role played by human infrastructures in providing, negotiating, collecting, and stewarding data points should encourage us to re-imagine data as a dialogue rather than operations, i.e, products of both the beholder and the beheld. How might data consent work in this context? One approach could be to create transitive informed consent, built upon personal relationships and trust in data workers, with transparency on potential downstream applications. Ideas like collective consent \cite{MozillaF26:online}, data trusts \cite{Nesta}, and data co-ops may enhance community agency in datasets, while simultaneously increasing data reliability. Finally, we must question the categories and constructs we model in datasets, and how we measure them. When categories are appropriate for endemic goals (\textit{e.g., \@}{} caste membership for quotas), what form should their distributions and annotations take? Linguistically and culturally pluralistic communities should be given voices in these negotiations in ways that respect Indian norms of representation.

\textbf{Model} (un)fairness detection and mitigation should incorporate the prominent axes of historical injustices in India (see Appendix~\ref{app_subgroups_table}) and tackle the challenges in operationalising them for testing AI; \textit{e.g., \@}{}representational biases of caste and other sub-groups in NLP models, biases in Indic language NLP including challenges from code-mixing, Indian subgroup biases in computer vision, tackling online misinformation, benchmarking using Indic datasets, and fair allocation models in public welfare. It is important to note that operationalising fairness approaches from the West to these axes is often nontrivial. For instance, personal names act as a signifier for various socio-demographic attributes in India, however there are no large datasets of Indian names (like the US Census data, or the SSA data) that are readily available for fairness evaluations. Another important consideration is how the algorithmic fairness interventions work with the existing infrastructures in India that surrounds decision making processes; \textit{e.g., \@}{}in the context of restorative justice initiatives such as reservations/quotas? The quota system gives rise to the problem of matching under distributional constraints \cite{kamada2015efficient, goto2017designing, ashlagi2020assignment} that has not received much attention within the ML research.

\vspace{-.7em}
\subsection{Empowering communities}

\textbf{Participatory Fair-ML knowledge systems} are critical to effectively fill the distance between ML makers and communities.
Context-free assumptions in fairness research, whether in homegrown or international ML systems, can not just fail, but produce harm inadvertently when applied to different infrastructural and cultural contexts. Systems should be created that enable marginalised communities to participate in the production of knowledge systems about themselves in ML frameworks. Grassroots efforts like Deep Learning Indaba \cite{dlindaba} and Khipu \cite{khipu} are helpful examples in bootstrapping AI research in communities. Initiatives like Design Beku \cite{designbeku} and SEWA \cite{sewa} are good decolonial examples of participatorily co-designing with under-served communities. Diverse values in algorithmic fairness can come from justice systems of indigenous Adivasis \cite{xaxa2011tribes}, \textit{dharma} (social ethic) and \textit{karma} (action) from Hinduism (note that Hindu concepts are sometimes rejected by Dalits for reinforcing hierarchy \cite{narayanan2001water}) or (\textit{sanghas}) (communism) espoused by Dr. B.R. Ambedkar \cite{ambedkar2011buddha}, for example.

\begin{figure*}[t!]
\centering
\includegraphics[width=.86\linewidth]{fairnessframework.png}
\caption{Research pathways for Algorithmic Fairness in India.}
\label{fig:Indiaframework}
\end{figure*}

\textbf{Low-resource considerations} pertaining to India should guide ML researchers to think beyond making model outputs fair, and about India's heterogeneous literacies, economics, and infrastructures that interfere with access to those fair outputs. Half the population of India is not online. Layers of the stack like interfaces, devices, connectivity, languages, and costs are all important in ensuring sustainable access and usage. Tackling resource constraints in computing disciplines like \textsc{ICTD} \cite{toyama2015geek} and \textsc{HCI4D} \cite{dray2012human} can help---where constraints have been embraced as design material, \textit{e.g., \@}{}delay-tolerant connectivity, low cost devices, text-free interfaces, partnership with civil society, and capacity building \cite{brewer2005case, heimerl2013local, kumar2015mobile, medhi2006text, sambasivan2019they}. Data infrastructures to build, share and sustain localised datasets would also enhance access equity to holding AI accountable (\textit{e.g., \@}{}\cite{lacuna}).

\textbf{Care in deployments} should be a primary concern.
Critiques were raised in our study on how neo-liberal ML followed a broader pattern of extraction from the `bottom billion' data subjects and AI labourers. Low costs, large and diverse populations, and policy infirmities have been cited as reasons for following double standards in India, \textit{e.g., \@}{}in non-consensual human trials and waste dumping \cite{macklin2004double} (also see \cite{mohamed2020decolonial}). Past disasters in India, like the fatal Union Carbide gas leak in 1984---one of the world's worst industrial accidents---point to faulty design and low quality standards for the `third world' \cite{Courttol47:online}. Similarly, overgrowth of ML capital, with unequal standards, inadequate safeguards, and dubious applications can lead to catastrophic effects (similar analogies have been made for content moderation \cite{roberts2016digital, sambasivan2019they}). ML researchers should study how to ensure meaningful recourse within the ecosystems they are embedding, and respond and engage with Indian realities and user feedback.

\vspace{-.7em}
\subsection{Enabling fair ML ecosystems}

\textbf{Ecosystems for accountability} requires enabling the civil society, media, industry, judiciary, and the state to meaningfully engage in holding AI interventions responsible. Moving from ivory tower research approaches to solidarity with various stakeholders through project partnerships, involvement in evidence-based policy, and policy maker education can help create a sustainable Fair-ML ecosystem based on sound empirical and ethical basis. Investigative journalism on algorithms is lacking in India, but sustainable partnerships may be created with rigorous media sources. Algorithmic fairness and ethics are not mainstream research topics in Indian academia today, but the academy has a pivotal role in advancing fairness in India. A concerted effort is required to create APIs, documentation, and socio-economic datasets to enable meaningful and equitable accountability in the ecosystem.

\textbf{Radical transparency} is required to counteract the inscrutability and authority created by the enthusiastic AI aspiration. Besides the role played by ecosystem-wide regulations and standards, radical transparency as a design principle should be espoused and enacted by Fair-ML researchers committed to India. Transparency on datasets, processes, and models (\textit{e.g., \@}{}\cite{mitchell2019model,gebru2018datasheets,bender2018data}), openly discussing limitations, failure cases, and intended use-cases can help bolster the practical limits to applying computing to human problems. It is also incumbent upon those in power to discuss current Fair-ML interventions as a flawed and evolving approach. Such approaches can help move from the `magic pill' role of fairness as a checklist for ethical issues in India, to a more pragmatic function.

\vspace{-0.7em}
\section{Conclusion}

We are currently facing a watershed moment for AI, with extravagant hype and pervasive applications, with nations being turned into AI superpowers and rulers \cite{Whoeverl94:online}. We bring to light how efforts in the West to make AI fairer may not transfer readily elsewhere, specifically in the case of India, through an empirical bottom-up analysis. We present a roadmap to go beyond localising data and model fairness alone, and present a holistic research agenda to operationalize Fair-ML in India. While it is important to incorporate Indian concepts of fairness into ML that impacts Indian communities, the broader call is to develop general approaches to fairness that reflect local considerations.

\small
\bibliographystyle{plainnat}

\section{Submission of papers to NeurIPS 2020}

NeurIPS requires electronic submissions.  The electronic submission site is
\begin{center}
  \url{https://cmt3.research.microsoft.com/NeurIPS2020/}
\end{center}

Please read the instructions below carefully and follow them faithfully.

\subsection{Style}

Papers to be submitted to NeurIPS 2020 must be prepared according to the
instructions presented here. Papers may only be up to eight pages long,
including figures. Additional pages \emph{containing only a section on the broader impact, acknowledgments and/or cited references} are allowed. Papers that exceed eight pages of content will not be reviewed, or in any other way considered for
presentation at the conference.

The margins in 2020 are the same as those in 2007, which allow for $\sim$$15\%$
more words in the paper compared to earlier years.

Authors are required to use the NeurIPS \LaTeX{} style files obtainable at the
NeurIPS website as indicated below. Please make sure you use the current files
and not previous versions. Tweaking the style files may be grounds for
rejection.

\subsection{Retrieval of style files}

The style files for NeurIPS and other conference information are available on
the World Wide Web at
\begin{center}
  \url{http://www.neurips.cc/}
\end{center}
The file \verb+neurips_2020.pdf+ contains these instructions and illustrates the
various formatting requirements your NeurIPS paper must satisfy.

The only supported style file for NeurIPS 2020 is \verb+neurips_2020.sty+,
rewritten for \LaTeXe{}.  \textbf{Previous style files for \LaTeX{} 2.09,
  Microsoft Word, and RTF are no longer supported!}

The \LaTeX{} style file contains three optional arguments: \verb+final+, which
creates a camera-ready copy, \verb+preprint+, which creates a preprint for
submission to, e.g., arXiv, and \verb+nonatbib+, which will not load the
\verb+natbib+ package for you in case of package clash.

\paragraph{Preprint option}
If you wish to post a preprint of your work online, e.g., on arXiv, using the
NeurIPS style, please use the \verb+preprint+ option. This will create a
nonanonymized version of your work with the text ``Preprint. Work in progress.''
in the footer. This version may be distributed as you see fit. Please \textbf{do
  not} use the \verb+final+ option, which should \textbf{only} be used for
papers accepted to NeurIPS.

At submission time, please omit the \verb+final+ and \verb+preprint+
options. This will anonymize your submission and add line numbers to aid
review. Please do \emph{not} refer to these line numbers in your paper as they
will be removed during generation of camera-ready copies.

The file \verb+neurips_2020.tex+ may be used as a ``shell'' for writing your
paper. All you have to do is replace the author, title, abstract, and text of
the paper with your own.

The formatting instructions contained in these style files are summarized in
Sections \ref{gen_inst}, \ref{headings}, and \ref{others} below.

\section{General formatting instructions}
\label{gen_inst}

The text must be confined within a rectangle 5.5~inches (33~picas) wide and
9~inches (54~picas) long. The left margin is 1.5~inch (9~picas).  Use 10~point
type with a vertical spacing (leading) of 11~points.  Times New Roman is the
preferred typeface throughout, and will be selected for you by default.
Paragraphs are separated by \nicefrac{1}{2}~line space (5.5 points), with no
indentation.

The paper title should be 17~point, initial caps/lower case, bold, centered
between two horizontal rules. The top rule should be 4~points thick and the
bottom rule should be 1~point thick. Allow \nicefrac{1}{4}~inch space above and
below the title to rules. All pages should start at 1~inch (6~picas) from the
top of the page.

For the final version, authors' names are set in boldface, and each name is
centered above the corresponding address. The lead author's name is to be listed
first (left-most), and the co-authors' names (if different address) are set to
follow. If there is only one co-author, list both author and co-author side by
side.

Please pay special attention to the instructions in Section \ref{others}
regarding figures, tables, acknowledgments, and references.

\section{Headings: first level}
\label{headings}

All headings should be lower case (except for first word and proper nouns),
flush left, and bold.

First-level headings should be in 12-point type.

\subsection{Headings: second level}

Second-level headings should be in 10-point type.

\subsubsection{Headings: third level}

Third-level headings should be in 10-point type.

\paragraph{Paragraphs}

There is also a \verb+\paragraph+ command available, which sets the heading in
bold, flush left, and inline with the text, with the heading followed by 1\,em
of space.

\section{Citations, figures, tables, references}
\label{others}

These instructions apply to everyone.

\subsection{Citations within the text}

The \verb+natbib+ package will be loaded for you by default.  Citations may be
author/year or numeric, as long as you maintain internal consistency.  As to the
format of the references themselves, any style is acceptable as long as it is
used consistently.

The documentation for \verb+natbib+ may be found at
\begin{center}
  \url{http://mirrors.ctan.org/macros/latex/contrib/natbib/natnotes.pdf}
\end{center}
Of note is the command \verb+\citet+, which produces citations appropriate for
use in inline text.  For example,
\begin{verbatim}
   \citet{hasselmo} investigated\dots
\end{verbatim}
produces
\begin{quote}
  Hasselmo, et al.\ (1995) investigated\dots
\end{quote}

If you wish to load the \verb+natbib+ package with options, you may add the
following before loading the \verb+neurips_2020+ package:
\begin{verbatim}
   \PassOptionsToPackage{options}{natbib}
\end{verbatim}

If \verb+natbib+ clashes with another package you load, you can add the optional
argument \verb+nonatbib+ when loading the style file:
\begin{verbatim}
   \usepackage[nonatbib]{neurips_2020}
\end{verbatim}

As submission is double blind, refer to your own published work in the third
person. That is, use ``In the previous work of Jones et al.\ [4],'' not ``In our
previous work [4].'' If you cite your other papers that are not widely available
(e.g., a journal paper under review), use anonymous author names in the
citation, e.g., an author of the form ``A.\ Anonymous.''

\subsection{Footnotes}

Footnotes should be used sparingly.  If you do require a footnote, indicate
footnotes with a number\footnote{Sample of the first footnote.} in the
text. Place the footnotes at the bottom of the page on which they appear.
Precede the footnote with a horizontal rule of 2~inches (12~picas).

Note that footnotes are properly typeset \emph{after} punctuation
marks.\footnote{As in this example.}

\subsection{Figures}

\begin{figure}
  \centering
  \fbox{\rule[-.5cm]{0cm}{4cm} \rule[-.5cm]{4cm}{0cm}}
  \caption{Sample figure caption.}
\end{figure}

All artwork must be neat, clean, and legible. Lines should be dark enough for
purposes of reproduction. The figure number and caption always appear after the
figure. Place one line space before the figure caption and one line space after
the figure. The figure caption should be lower case (except for first word and
proper nouns); figures are numbered consecutively.

You may use color figures.  However, it is best for the figure captions and the
paper body to be legible if the paper is printed in either black/white or in
color.

\subsection{Tables}

All tables must be centered, neat, clean and legible.  The table number and
title always appear before the table.  See Table~\ref{sample-table}.

Place one line space before the table title, one line space after the
table title, and one line space after the table. The table title must
be lower case (except for first word and proper nouns); tables are
numbered consecutively.

Note that publication-quality tables \emph{do not contain vertical rules.} We
strongly suggest the use of the \verb+booktabs+ package, which allows for
typesetting high-quality, professional tables:
\begin{center}
  \url{https://www.ctan.org/pkg/booktabs}
\end{center}
This package was used to typeset Table~\ref{sample-table}.

\begin{table}
  \caption{Sample table title}
  \label{sample-table}
  \centering
  \begin{tabular}{lll}
    \toprule
    \multicolumn{2}{c}{Part}                   \\
    \cmidrule(r){1-2}
    Name     & Description     & Size ($\mu$m) \\
    \midrule
    Dendrite & Input terminal  & $\sim$100     \\
    Axon     & Output terminal & $\sim$10      \\
    Soma     & Cell body       & up to $10^6$  \\
    \bottomrule
  \end{tabular}
\end{table}

\section{Final instructions}

Do not change any aspects of the formatting parameters in the style files.  In
particular, do not modify the width or length of the rectangle the text should
fit into, and do not change font sizes (except perhaps in the
\textbf{References} section; see below). Please note that pages should be
numbered.

\section{Preparing PDF files}

Please prepare submission files with paper size ``US Letter,'' and not, for
example, ``A4.''

Fonts were the main cause of problems in the past years. Your PDF file must only
contain Type 1 or Embedded TrueType fonts. Here are a few instructions to
achieve this.

\begin{itemize}

\item You should directly generate PDF files using \verb+pdflatex+.

\item You can check which fonts a PDF files uses.  In Acrobat Reader, select the
  menu Files$>$Document Properties$>$Fonts and select Show All Fonts. You can
  also use the program \verb+pdffonts+ which comes with \verb+xpdf+ and is
  available out-of-the-box on most Linux machines.

\item The IEEE has recommendations for generating PDF files whose fonts are also
  acceptable for NeurIPS. Please see
  \url{http://www.emfield.org/icuwb2010/downloads/IEEE-PDF-SpecV32.pdf}

\item \verb+xfig+ "patterned" shapes are implemented with bitmap fonts.  Use
  "solid" shapes instead.

\item The \verb+\bbold+ package almost always uses bitmap fonts.  You should use
  the equivalent AMS Fonts:
\begin{verbatim}
   \usepackage{amsfonts}
\end{verbatim}
followed by, e.g., \verb+\mathbb{R}+, \verb+\mathbb{N}+, or \verb+\mathbb{C}+
for $\mathbb{R}$, $\mathbb{N}$ or $\mathbb{C}$.  You can also use the following
workaround for reals, natural and complex:
\begin{verbatim}
   \newcommand{\RR}{I\!\!R}
   \newcommand{\Nat}{I\!\!N}
   \newcommand{\CC}{I\!\!\!\!C}
\end{verbatim}
Note that \verb+amsfonts+ is automatically loaded by the \verb+amssymb+ package.

\end{itemize}

If your file contains type 3 fonts or non embedded TrueType fonts, we will ask
you to fix it.

\subsection{Margins in \LaTeX{}}

Most of the margin problems come from figures positioned by hand using
\verb+\special+ or other commands. We suggest using the command
\verb+\includegraphics+ from the \verb+graphicx+ package. Always specify the
figure width as a multiple of the line width as in the example below:
\begin{verbatim}
   \usepackage[pdftex]{graphicx} ...
   \includegraphics[width=0.8\linewidth]{myfile.pdf}
\end{verbatim}
See Section 4.4 in the graphics bundle documentation
(\url{http://mirrors.ctan.org/macros/latex/required/graphics/grfguide.pdf})

A number of width problems arise when \LaTeX{} cannot properly hyphenate a
line. Please give LaTeX hyphenation hints using the \verb+\-+ command when
necessary.

\section*{Broader Impact}

Authors are required to include a statement of the broader impact of their work, including its ethical aspects and future societal consequences.
Authors should discuss both positive and negative outcomes, if any. For instance, authors should discuss a)
who may benefit from this research, b) who may be put at disadvantage from this research, c) what are the consequences of failure of the system, and d) whether the task/method leverages
biases in the data. If authors believe this is not applicable to them, authors can simply state this.

Use unnumbered first level headings for this section, which should go at the end of the paper. {\bf Note that this section does not count towards the eight pages of content that are allowed.}

\begin{ack}
Use unnumbered first level headings for the acknowledgments. All acknowledgments
go at the end of the paper before the list of references. Moreover, you are required to declare
funding (financial activities supporting the submitted work) and competing interests (related financial activities outside the submitted work).
More information about this disclosure can be found at: \url{https://neurips.cc/Conferences/2020/PaperInformation/FundingDisclosure}.

Do {\bf not} include this section in the anonymized submission, only in the final paper. You can use the \texttt{ack} environment provided in the style file to autmoatically hide this section in the anonymized submission.
\end{ack}

\section*{References}

References follow the acknowledgments. Use unnumbered first-level heading for
the references. Any choice of citation style is acceptable as long as you are
consistent. It is permissible to reduce the font size to \verb+small+ (9 point)
when listing the references.
{\bf Note that the Reference section does not count towards the eight pages of content that are allowed.}
\medskip

\small

[1] Alexander, J.A.\ \& Mozer, M.C.\ (1995) Template-based algorithms for
connectionist rule extraction. In G.\ Tesauro, D.S.\ Touretzky and T.K.\ Leen
(eds.), {\it Advances in Neural Information Processing Systems 7},
pp.\ 609--616. Cambridge, MA: MIT Press.

[2] Bower, J.M.\ \& Beeman, D.\ (1995) {\it The Book of GENESIS: Exploring
  Realistic Neural Models with the GEneral NEural SImulation System.}  New York:
TELOS/Springer--Verlag.

[3] Hasselmo, M.E., Schnell, E.\ \& Barkai, E.\ (1995) Dynamics of learning and
recall at excitatory recurrent synapses and cholinergic modulation in rat
hippocampal region CA3. {\it Journal of Neuroscience} {\bf 15}(7):5249-5262.

\end{document}


text:
Hotels in New South Wales
Accommodation in Port Macquarie
Search for a place to stay
See Port Macquarie hotels on a map
Where to stay in Port Macquarie
Exploring Port Macquarie
The thriving city of Port Macquarie is the fastest-growing of all New South Wales cities and towns, and is set along the coastline, 390 kilometres from Sydney. Divided by the Hastings River, the city draws visitors to its famous Billabong Koala Park as well as to enjoy the sight of koalas sharing their original habitation with incoming humans. Port Macquarie is also a popular destination for retirees looking to spend time on the beaches and waterways from their new homes in the waterside residential suburbs stretching from the north shore to Lighthouse Beach.
There is a range of options for all travelers. The most popular accommodation in Port Macquarie is the Little Palm Cabins, which has been booked 5 times from our Hotels.com verified guests. Our guests also highly recommended SUNNY-DAZE - BONNY HILLS in Port Macquarie as it has been reserved 5 times.
Sightseeing in Port Macquarie
- Douglas Vale Historic Homestead & Vineyard
Classified by the Australian National Trust, Douglas Vale was one of the earliest vineyards in the region and was famous for its wines between 1867 and 1918. Visitors can tour the acres of vines and the remaining areas of formal gardens.
- Billabong Koala Wildlife Park
Everyone's favourite attraction here is the Koala Park, with its huge number of protected koalas which are well-used to admiration and petting by visitors. Kangaroos and spider monkeys are kept in the park as well as reptiles, wombats, wallabies, and many more indigenous species. The park boasts Australia's best-known koala breeding centre and makes for a great day out for families.
Transportation in Port Macquarie
Getting to Port Macquarie is easy by car, long-distance bus, ferry, or by air to the city's small airport. Self-drive from Sydney makes use of a scenic route along the coastline and is the most convenient way to tour the Hunter Valley. Coastal walks link many of the city districts, cycling is encouraged, and shuttle buses link the shopping areas, the beaches, and many of the attractions. Two ferries run to nearby points of beauty along the coast.
City, Beach, or Countryside - Finding the Right Place to Stay in Port Macquarie
Port Macquarie's array of accommodation makes finding the right place to stay a breeze. Whether you're...
Water and Wildlife, Heritage and Culture - What to See and Do in Port Macquarie
Port Macquarie is a classic beach-holiday destination - but it's not only about sun, sand, and surf....
Superb Scenery, Beautiful Beaches, Wonderful Wines - A Guide to Port Macquarie
Known as having one of the friendliest climates in Australia, Port Macquarie is a classic family...
Best Special Hotel Deals in Port Macquarie, New South Wales
Last Minute Hotel Deals in Port Macquarie, New South Wales


text:
Man charged with aggravated assault after hours-long standoff with El Paso SWAT
Times staff report
A domestic violence incident sparked a SWAT standoff Friday morning.
The suspect barricaded himself for several hours before surrendering, according to El Paso Police Department.
Central Command officers responded to the call in the 3900 block of La Luz Avenue, police said in an alert issued shortly before noon Friday.
Officers arrived at the residence and heard people arguing from inside; police identified themselves, the suspect, Devan Lovelace, 24, told the officers he was not coming
out of the residence.
A woman exited the home with a child and told police there was a weapon inside.
She was taken to a hospital with nonlife-threatening injuries, an El Paso Police Department spokesperson said.
"We believe it could be a domestic dispute, but we're not clear on all the details right now," police spokesman Sgt. Robert Gomez said at a news conference.
Central Command officers then notified SWAT and the Police Department's Crisis Management Team, which made an attempt to contact him to try to resolve the situation peacefully.
"We've been talking to him," Gomez said.
Gomez said: "We do think that it's an isolated incident. We don't believe there's any danger to the public."
He urged people to stay out of the area.
"We're gonna do what it takes in order to try to end this situation peacefully," he said, adding: "It's up to the person that's in the house and hopefully we can resolve it very soon."
Streets around the scene were closed and Gomez urged residents to have patience and let the police work. He said streets would open again once that could be done safely.
Gomez said he believes the barricaded man had access to weapons but did not have specifics on the types of weapons.
He said police had concerns about the line of fire from the residence, which is why officers in the area were asking onlookers not to stand in certain areas.
"It's just an unknown that we don't want to take a chance on," he said.
By late Friday afternoon, the incident already had lasted more than four hours, with no surrender.
After several hours later, the suspect exited the residence and was placed under arrest
Lovelace was booked into the El Paso County Jail for aggravated assault with a bond of $250,000.
Second SWAT standoff in two days
Northeast Regional Command officers responded to a domestic disturbance call at 12:26 p.m. Thursday in the 5100 block of Cornell Avenue in Northeast El Paso.
Police said an investigation revealed Benjamin Banda, 45, of Northeast El Paso, had entered a residence that he was not allowed to be in because of a protective order that was issued against him a few days earlier.
Officers tried to talk to Banda, but he was noncompliant and refused to talk with the police and leave the home, according to a news release.
The Police Department's SWAT and Crisis Management Team responded and were able to convince him to surrender. Banda was arrested and booked into the El Paso County Jail on bonds totaling $120,000 on charges of violation of bond/protective order and aggravated assault with a deadly weapon.
Jail records show he still was being held late Friday afternoon.
More:Suspect in stabbing sparks police SWAT standoff in Central El Paso
More:El Paso Police: Man arrested after firing gun in a home during argument, hitting child's bedroom


Christmas Trees (Ёлки; also called in English) is a 2010 Russian comedy, the first in a multi-film series. Six different directors created different parts of the film.
The film consists of nine loosely connected stories that all center on New Year's (which is one of the most important holidays in Russia and more or less the equivalent of the American/British Christmas, down to many of the traditions). It takes place in eleven cities all over Russia, with fourteen different main characters. They're all facing problems, and it's only through a New Year's miracle, plus the theory of six degrees of separation, that they can overcome them. One of the main plot points is that Varya, an orphan living in Kaliningrad, has decided that the president of Russia is her father, and she is waiting for him to acknowledge her. The actual president at the time, Dmitry Medvedev, played himself, though his role only consisted of a few phrases.
The film got pretty good reviews and was well received by audiences. New Year's films and movies are an enormous institution in Russian and this film has spawned three sequels so far, and another is in the works.


We needed to upgrade from EL6 to EL7 at work. After the upgrade, we noticed some of the JSONs returned from the APIs had changed: some numbers were suddenly enclosed in double quotes, while some others lost the quotes, which broke customers' code written in strongly typed languages (e.g. Java).
We used the system default perl to drive our web application, using libraries provided by the vendor. While Perl's version changed from 5.10.1 to 5.16.3, the version of JSON::XS responsible for creation of JSON data jumped from 2.27 to 3.01. Both the changes contributed to the problem.
In EL6, both the values are double-quoted, but in EL7, the second one isn't. The output of Devel::Peek shows $$ wasn't magical in 5.10.1, but was in 5.16.3.
Another difference is the behaviour of utf8::decode. In 5.10.1, it doesn't change numbers into strings, but it does in 5.16.3.
That's what you hear from everyone when you encounter a problem with the system perl. It's good advice, however it wouldn't have helped much in this case: If we had used custom perl (through perlbrew or compiling it ourselves) we'd have just probably noticed earlier when upgrading the Perl interpreter or the JSON library.
We discussed several ways how to fix the issue. One of the simplest ones helped us to fix the most urgent cases: when creating the Perl structure to be encoded to JSON, we forced the types by double-quoting values that were supposed to be strings, calling int on integers, and adding floats to zero.
Nevertheless, as the number of fixes grew, we realised it wasn't the best approach. If an unwary programmer from a distant team had just inspected one of the values, its type might have changed; moreover, the change could have happened in one of the environments only. The type definitions were scattered all over the repositories, and even worse, they were redundant—we already had all the API structures documented and formally defined.
Benchmarks showed only a negligible slowdown. After some time, the changes were accepted to the upstream and Cpanel::JSON::XS::Type is now part of Cpanel::JSON::XS.
We weren't alone; read Did the JSON module change? for a similar story. Furthermore, JSON isn't the only place where you can encounter these problems. Any time you communicate with a system with different arsenal of types, you might need ways to enforce them. For example, read Why does DBI implicitly change integers to strings?.
Perl makes no distinction between numbers and strings, they say. But is it really so?
By the way, introduction of the string bitwise operators nicely shows the Perlish way: the operators decide how to interpret the operands.
Adding a second argument [JSON_TYPE_INT] to encode_json fixes the problem.
Thanks to GoodData for supporting open source, to Pali for working on the solution, and to Reini for accepting it.