text:
\section{Introduction}\label{sec: introduction}
Consider the problem illustrated in Fig.~\ref{fig: Coding for computing with side information.} where the encoder observes $X \in \mathcal{X}$ and decoder observes $Y \in \mathcal{Y}$, and both the encoder and decoder want the decoder to compute $f(X,Y) \in \mathcal{Z}$ with a fidelity criterion for a given function $f : \mathcal{X} \times \mathcal{Y} \mapsto \mathcal{Z}$ known to both encoder and decoder, where $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$ are all finite sets and $\mathcal{Z} \subset \mathbb{R}^d$ for some finite natural number $d$. The objective is to find the minimum number of bits the encoder must send such that the decoder can compute $f(X,Y)$ with a fidelity criterion $\epsilon \in [0,\infty)$, i.e.\ if $\widehat{f(X,Y)}$ is the estimate of $f(X,Y)$ obtained by the decoder, then the following should hold \blfootnote{This work was funded in part by the IBM-Illinois Center for Cognitive Computing Systems Research (C3SR), a research collaboration as part of the IBM AI Horizons Network; and in part by grant number 2018-182794 from the Chan Zuckerberg Initiative DAF, an advised fund of Silicon Valley Community Foundation.}
\begin{align}
\norm{f(X,Y) - \widehat{f(X,Y)}} \leq \epsilon,
\end{align}
where $\norm{z_1 - z_2}$ is the Euclidean distance between $z_1,z_2 \in \mathcal{Z}$.
We assume the function $f$ is to be computed for $N$ independent instances of $(X,Y)$ for large $N$.
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    text width=5em, text centered, minimum height=2 em, node distance = 1.5 cm]
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 \begin{figure}
  \tikzstyle{c} = [rectangle, draw, dashed]
\centering

\begin{tikzpicture}[auto]

    \node [virtual] (message_1) {};
    \node [virtual, below = of message_1] (message_v) {};
    \node [block, right = of message_1] (encoder_1) {Enc 1};
	\node [virtual, right = of message_v] (encoder_v) {};
    \node [block, right = 1.5cm of encoder_1] (decoder) {Decoder};
    \node [virtual, below = of decoder] (message_2) {};
    \node [block_e, right = of decoder] (fidelity) {\vspace{10mm}\\$\norm{f(X,Y) - \widehat{f(X,Y)}} \leq \epsilon$};

	\draw[->] (message_1.east) -- node[] {$X$ }+(1.5cm,0pt);
	\draw[->] (message_2.west) -- node[] {$Y$ }+(0cm,1cm);
	\draw[->] (encoder_1.east) -- node[] {$m$} +(1.5cm,0pt);
	\draw[->] (decoder.east) -- node[] {$\widehat{f(X,Y)}$} +(3cm,0pt);
\end{tikzpicture}

\caption{Coding for computing with side information.}
\label{fig: Coding for computing with side information.}
\end{figure}

Orlitsky and Roche gave a single-letter characterization of the problem for $\epsilon = 0$ \cite{OrlitskyR2001}. Function computation with $\epsilon > 0$ has been considered in \cite{DoshiSMJ2010,FeiziM2014} which provide an efficient graph-based encoding scheme using a generalized version of the characteristic graph in \cite{OrlitskyR2001} called the $D$-characteristic graph. The coding scheme provided there is suboptimal since the construction of the graph is based on only pairwise comparison of function values. Drawing on better geometric methods of comparison, we introduce a novel generalization of the characteristic graph from \cite{OrlitskyR2001} that yields a hypergraph-based encoding scheme that is indeed optimal. Thus, we provide an alternate (but fully equivalent) description of the rate-distortion function, which further inspires practically implementable codes. We refer to the optimal rate for this problem as \emph{functional $\epsilon$-entropy} since this rate reduces to \emph{$\epsilon$-entropy} of $f$ defined in \cite{PosnerR1971} when there is no side information $Y$ and the hyperedges of the constructed characteristic hypergraph partition the support set $\mathcal{X}$.  Unlike traditional rate-distortion problems where the rate-distortion function $R(\epsilon)$ is a continuous function of $\epsilon$ (the constraint on the expected distortion), in this case we have a rate function that is discontinuous function of $\epsilon$ and the points of discontinuity can be determined from the characteristic hypergraph for different values of $\epsilon$.

We also show that some of the assumptions in \cite{OrlitskyR2001} that lead to simple coding schemes for $\epsilon = 0$ might not imply the same when $\epsilon > 0$. Although \cite{DoshiSMJ2010, FeiziM2014} show modular schemes, i.e.\ graph-based quantization followed by source coding, are optimal for the problem with $\epsilon = 0$ under some assumptions, the solution provided is NP-hard and an approximate solution is used for coding. Further, the assumptions in \cite{DoshiSMJ2010, FeiziM2014, OrlitskyR2001} that lead to optimal modular schemes only involve the source distributions; one can weaken assumptions that imply optimal modular schemes by considering both the distribution of the source and the function. Although there does not seem to be a simple and elegant dichotomy of functions and sources analogous to Han and Kobayashi's dichotomy of functions \cite{HanK1987} under which modular schemes are optimal, we provide a simple and general class of function-source pairs for which modular schemes are optimal.

After providing the rate-distortion function and an optimal hypergraph-based encoding scheme, we provide algorithms and conditions under which practical coding schemes using \emph{randomized quantization} and polar codes \cite{Arikan2009}  are optimal.

Sec.~\ref{sec_merged: notations and preliminaries} describes preliminary results and the problem model. Sec.~\ref{sec: graph} gives the main result of this paper, the coding theorem and its equivalence to the conditional entropy of a hypergraph. Sec.~\ref{sec: practical} develops practical coding schemes for the problem. Sec.~\ref{sec: properties} shows that functional $\epsilon$-entropy may be discontinuous in $\epsilon$ and Sec.~\ref{sec: conclusion} concludes the paper.

\section{Preliminaries and problem setting}\label{sec_merged: notations and preliminaries}
In this section, first we provide some definitions and discuss some preliminary results; then we formally define the problem.

\subsection{Preliminaries and notations}\label{sec: notations and preliminaries}
For a set of $k$ points $z^k = \{z_1, \ldots, z_k\}$, $z_i \in \mathcal{Z} \subset \mathbb{R}^d$ for some finite $d$, the smallest circle (sphere) enclosing these $k$ points is called the smallest enclosing circle of $z^k$ \cite{Chrystal1885}. Note that the computational complexity of finding the smallest enclosing circle for a set of points is linear in the number of points \cite{Megiddo1983}.

\begin{definition}
A function $f: \mathcal{X} \mapsto \mathcal{Z}$ is $L$-Lipschitz continuous if for all $x_1,x_2 \in \mathcal{X}$, $\norm{f(x_1) - f(x_2)} \leq L \norm{x_1 - x_2}$ for some $L>0$ where $\norm{\cdot}$ is the Euclidean norm.
\end{definition}
\begin{definition}
A function $g: \mathcal{X} \mapsto \hat{\mathcal{Z}}$, $\hat{\mathcal{Z}} \subset \mathbb{R}^d$ is a $\delta$-approximation to a function $f: \mathcal{X} \mapsto \mathcal{Z}$ if for every $x \in \mathcal{X}$, $\norm{f(x) - g(x)} \leq \delta$ where $\norm{\cdot}$ is the Euclidean norm.
\label{def: delta}
\end{definition}

A hypergraph $G$ is a pair $G = (\mathcal{X},E)$ where $\mathcal{X}$ is the set of vertices of $G$ and $E \subseteq \mathcal{P}(\mathcal{X}) \setminus \emptyset$ is the set of hyperedges of $G$, where $\mathcal{P}(\mathcal{X})$ is the powerset of $\mathcal{X}$ \cite{Bretto2013}.

\begin{definition}
A hyperedge $E$ is called a \emph{maximal hyperedge} if $E$ is not a proper subset of any other hyperedge in the hypergraph $G$.
\end{definition}

Let $(W,X) \sim P_{WX}$ be i.i.d. random variables with $(W,X) \in \mathcal{W}\times \mathcal{X}$ and suppose $\mathcal{W} = \{0,1\}$ for simplicity. Then, source Bhattacharyya parameter \cite{Arikan2010} $Z(W|X)$ for the source $(W,X)$ is defined as $Z(W|X) = 2 \sum_{x} P_{X}(x) \sqrt{P_{W|X}(0|x) P_{W|X}(1|x)}$.

\begin{theorem}[\cite{HondaY2013}]
For any $\beta < \tfrac{1}{2}$, i.i.d.\ random variables $(W,X)$ and $U_1^N = W_1^N G_N$,
\begin{align*}
\lim_{N \to \infty} \frac{1}{N} \left \vert \{  i: Z(U_i|U_1^{i-1},X_1^N) \leq 2^{-N^{\beta}} \text{  and  } Z(U_i|U_1^{i-1}) \geq 1 - 2^{-N^{\beta}} \} \right \vert & = I(W;X),\\
\lim_{N \to \infty} \frac{1}{N} \left \vert \{  i: Z(U_i|U_1^{i-1},X_1^N) \geq 1 - 2^{-N^{\beta}} \text{  or  } Z(U_i|U_1^{i-1}) \leq 2^{-N^{\beta}} \} \right \vert & = 1 - I(W;X),
\end{align*}
where $G_N$ is the generator matrix for polar codes.
\label{thm: polar_codes}
\end{theorem}
We will use polar coding to build practical coding techniques for our problem. Thm.~\ref{thm: polar_codes} can be extended to any finite set $\mathcal{W}$ using results from \cite{SasogluTA2009}.

\subsection{Problem setting}\label{sec: problem_Setup}
Let $(X_i,Y_i) \sim P_{X,Y}$ be $N$ i.i.d.\ random variables, where $X_i \in \mathcal{X}$, $Y_i \in \mathcal{Y}$. The encoder in Fig.~\ref{fig: Coding for computing with side information.} observes $\{X_i\}_{i=1}^N$, the decoder observes $\{Y_i\}_{i=1}^N$, and both the encoder and decoder want the decoder to reconstruct $\{f(X_i,Y_i)\}_{i=1}^{N}$ as $\{\hat{Z}\}_{i=1}^{N}$ such that $\frac{1}{N}\sum_{i=1}^{N}\Pr \left[\norm{\hat{Z}_i - f(X_i,Y_i)} > \epsilon\right] \to 0$ as $N \to \infty$ for some fixed fidelity constraint $\epsilon > 0$.
For any $R>0$, we define a $(2^{NR},N, \epsilon)$ code for any fixed function $f: \mathcal{X}\times \mathcal{Y} \to \mathcal{Z}$ as an encoding function $g_e: \mathcal{X}^N \mapsto \{1,\ldots, 2^{NR}\}$ and a decoding function $g_d: \{1,\ldots, 2^{NR}\} \times \mathcal{Y}^N \mapsto \hat{\mathcal{Z}}^N$ where $\hat{\mathcal{Z}} \subset \mathbb{R}^d$ is the reconstruction set. The probability of error is
\[
P_{\epsilon}^{avg}(\hat{Z}, X, Y) = \frac{1}{N}\sum_{i=1}^{N}\Pr \left[\norm{\hat{Z}_i - f(X_i,Y_i)} > \epsilon\right],
\]
that is, $P_{\epsilon}^{avg}(\hat{Z}, X, Y)$ is the average symbol-error probability. A rate $R$ is achievable if there exists a sequence of $(2^{NR},N, \epsilon)$ codes such that $P_{\epsilon}^{avg}(\hat{Z}, X, Y) \to 0$ as $N \to \infty$. The goal is to find the minimum achievable value of $R$ and design practical codes that attain it.

\section{Coding theorem and hypergraph-based coding scheme}\label{sec: graph}

In this section we first provide the rate-distortion function for the problem described in Sec.~\ref{sec_merged: notations and preliminaries}, and then provide a hypergraph-based coding scheme that achieves the rate-distortion function.
\begin{theorem}
Let $(X,Y) \sim P_{XY}$. The encoder and decoder observe $X$ and $Y$ respectively. The decoder estimates the function $f(X,Y)$ as $\widehat{f(X,Y)}$ such that for some fixed $\epsilon > 0$, $P_{\epsilon}^{avg}(\widehat{f(X,Y)}, X, Y) \to 0$ as $N \to \infty$. Then the minimum rate required by the encoder is
\begin{align}
R(\epsilon) = \min_{U-X-Y } I(U;X|Y) \label{eqn: R_Delta}
\end{align}
such that there exists a function $g$ with $\mathbb{E}d_{\epsilon}(X,Y,g(U,Y)) \leq 0$, where the distortion function $d_{\epsilon}$ is defined as $d_{\epsilon}(x,y,z) = \mathbbm{1} \{ \norm{z - f(x,y)} > \epsilon \}.$
\label{thm: I:coding theorem}
\end{theorem}
The proof to this theorem is direct and follows from \cite[Eq.\ (4)]{OrlitskyR2001} when the distortion is set to zero under the $d_{\epsilon}$ distortion function.

We define the \emph{$\epsilon$-characteristic hypergraph}, $G_{f, \epsilon;X|Y}$, of a random variable $X$ with respect to another possibly correlated random variable $Y$, a function $f$, and a fidelity constraint $\epsilon$.

\begin{definition}
The vertex set of \emph{$\epsilon$-characteristic hypergraph}, $G_{f,\epsilon;X|Y}$, is $\mathcal{X}$. For any non-empty subset $S \subseteq \mathcal{X}$ and $y \in \mathcal{Y}$, let $S_y = \{x:x\in S \hspace{2mm}\text{and} \hspace{2mm} p(x,y) > 0\}$. Then $S$ is a hyperedge in $G_{f,\epsilon;X|Y}$ if and only if the radius of the smallest enclosing circle containing the set of points $\{f(x,y): x \in S_y\}$ is less than or equal to $\epsilon$ for all $y \in \mathcal{Y}$.
\label{def: graph}
\end{definition}
Note that for $\epsilon = 0$ the hypergraph in Def.~\ref{def: graph} reduces to the characteristic graph defined in \cite{OrlitskyR2001, Witsenhausen1975} with hyperedges replaced by independent sets. Now we define the hypergraph entropy of a characteristic hypergraph $G_{f,\epsilon;X|Y}$. Let $\Gamma(G_{f,\epsilon;X|Y})$ be the set of hyperedges of $G_{f,\epsilon;X|Y}$. When it is clear from context, we will denote $G_{f,\epsilon;X|Y}$ by $G_{\epsilon}$, and $G_{0}$ is simply written as $G$. We define the functional $\epsilon$-entropy, which is a generalization of $\epsilon$-entropy proposed by \cite{PosnerR1971}.
\begin{definition}
The \emph{functional $\epsilon$-entropy}, $H_{G_{\epsilon}}(X|Y)$, is defined as
\begin{align}
H_{G_{\epsilon}}(X|Y) = \min_{\substack{W-X-Y \\ X \in W \in \Gamma (G_{\epsilon})}} I(W;X|Y),
\label{eqn: graph_entropy}
\end{align}
where $X$ induces a probability distribution over the vertices of the hypergraph $G_{\epsilon}$. The random variable $W$ is obtained by defining transition probabilities $p(w|x)$ over all hyperedges $w$ that contain $x$, i.e.\ $p(w|x) \geq 0$ for all $x \in w \in \Gamma(G_{\epsilon})$ and  $\sum_{w \ni x}p(w|x) = 1$.
\end{definition}
Note that the minimization over $\Gamma (G_{\epsilon})$ can be restricted to $\Gamma_m (G_{\epsilon})$ by the data processing inequality, where $\Gamma_m (G_{\epsilon})$ is the set of maximal hyperedges. Now we show that the optimal rate $R(\epsilon) = H_{G_{\epsilon}}(X|Y)$.
\begin{theorem}
Let $R({\epsilon})$ and $H_{G_{\epsilon}}(X|Y)$ be as defined in \eqref{eqn: R_Delta} and \eqref{eqn: graph_entropy} respectively for some $\epsilon \geq 0$, then $R({\epsilon}) = H_{G_{\epsilon}}(X|Y). $
\label{thm: 2Delta}
\end{theorem}

\begin{proof}
From the definitions of $R(\epsilon)$ and $H_{G_{\epsilon}}(X)$ we need to prove that
\begin{align*}
\min_{\substack{U-X-Y \\ \exists g: \hspace{0.5em}\mathbb{E}d_{\epsilon}(X,Y,g(U,Y)) \hspace{0.1em} \leq \hspace{0.1em}0}} I(U;X|Y) = \min_{\substack{W-X-Y \\ X \in W \in \Gamma (G_{\epsilon})}} I(W;X|Y).
\end{align*}
First we show the left side is less than or equal to the right side.
If $X \in W \in \Gamma (G_{\epsilon})$, then we can find a (partial) function $g$ over $\Gamma (G_{\epsilon}) \times \mathcal{Y}$ such that $\norm{g(w,y) -  f(x,y)} \leq \epsilon$ whenever $p(w,x,y) > 0$, and thus implying $\mathbb{E}d_{\epsilon}(X,Y,g(W,Y)) = 0$.

Let $w \in \Gamma (G_{\epsilon})$ and $y \in \mathcal{Y}$. If $p(x,y) = 0$ for all $x \in w$, then we can leave $g$ undefined since it will not affect our expected distortion. Otherwise form the set $w_y$ which consists of all $x \in w$ such that $p(x,y) > 0$ and define $g(w,y)$ as the center of the smallest enclosing circle of the set $\{f(x,y): x \in w_y\}$. Then, by Def.~\ref{def: graph}, $\norm{ f(x,y) - g(w,y)} \leq \epsilon$ for all $x \in w$ such that $p(x,y) > 0$. Hence, whenever $p(w,x,y) > 0$, we have $d_{\epsilon}(x,y,g(w,y)) = 0$. This shows that $\mathbb{E}d_{\epsilon}(X,Y,g(W,Y)) = 0$, hence the left side is less than or equal to the right side.

Next we show that the right hand side is less than or equal to the left hand side, completing the proof.
Suppose $U-X-Y$ and there exists a $g$ such that $\mathbb{E}d_{\epsilon}(X,Y,g(U,Y)) \leq 0$. We define $W$ such that $X \in W \in \Gamma (G_{\epsilon})$ and show that $I(W;X|Y) \leq I(U;X|Y)$ for this definition. Let $p(u,x,y)$ be the probability distribution underlying $(U,X,Y)$. Set
\begin{equation}
w(u) = \{x: p(u,x) > 0\}
\label{eqn: w_def}
\end{equation}
and define the Markov chain $W-U-XY$ by
\begin{align*}
p(w|u,x,y) =
\begin{cases}
1, & \text{if $w = w(u)$}\\
0, & \text{otherwise.}
\end{cases}
\end{align*}
We first show that $X \in W \in \Gamma (G_{\epsilon})$. If $p(w,x)>0$, this implies there is a $u$ such that $w = w(u)$ and $p(u,x) > 0$. Then, by \eqref{eqn: w_def} we have $x \in w = w(u)$. Thus, we have $x \in w$ whenever $p(w,x) > 0$. Next we show that whenever $p(w) > 0$, then the radius of the smallest enclosing circle of the set $\{f(x,y): x \in w \hspace{2mm} \text{and} \hspace{2mm} p(x,y) > 0\}$ is less than or equal to $\epsilon$ which further implies $W \in \Gamma (G_{\epsilon})$. From \eqref{eqn: w_def}, $p(w) > 0$ implies there exists a $u$ such that $w = w(u)$. Further, if $x \in w$ then $p(u,x) > 0$, and since $U-X-Y$ forms a Markov chain, it follows that whenever $p(x,y) > 0$ we have $p(u,x,y) > 0$. Note that $\mathbb{E}d_{\epsilon}(X,Y,g(U,Y)) = 0$, hence we must have $\norm{f(x,y) -  g(u,y)} \leq \epsilon$ whenever $p(u,x,y) > 0$. Thus, it follows that the circle centered at $g(u,y)$ of radius $\epsilon$ encloses all the points in the set $\{f(x,y): x \in w \hspace{2mm} \text{and} \hspace{2mm} p(x,y) > 0\}$. Hence, the smallest enclosing circle of the set $\{f(x,y): x \in w \hspace{2mm} \text{and} \hspace{2mm} p(x,y) > 0\}$ has radius less than or equal to $\epsilon$, which implies $w \in \Gamma (G_{\epsilon})$.

It remains to show that $W-X-Y$ forms a Markov chain and that $I(W;X|Y) \leq I(U;X|Y)$. The proof to this follows directly from the proof of \cite[Thm.~2]{OrlitskyR2001}.
\end{proof}

Note that setting $\epsilon = 0$, gives us the rate-distortion function in \cite{OrlitskyR2001}.

\section{Towards practical coding scheme}\label{sec: practical}
In \cite{OrlitskyR2001} it was shown that if $p(x,y)>0$ for all $(x,y) \in \mathcal{X} \times \mathcal{Y}$, then every vertex $x$ in the characteristic hypergraph $G$ belongs to exactly one hyperedge of $G$. This property led to the design of optimal modular schemes in \cite{DoshiSMJ2010,FeiziM2014} under some assumptions introduced therein. However, those assumptions depended solely on the sources and not on the function considered. In this section, we introduce a function/source condition and show that for $\epsilon = 0$, this implies non-overlapping clustering of vertices leading to a modular scheme that can be implemented in $\mathcal{O}(N\log{N})$ time where $N$ is the blocklength. Then, by giving a counterexample we show that for $\epsilon >0$, $p(x,y)>0$ for all $(x,y) \in \mathcal{X} \times \mathcal{Y}$ does not imply non-overlapping clustering  of vertices in $G_{\epsilon}$ in contrast to the case $\epsilon = 0$ where this condition on the source implies non-overlapping clustering.

\begin{condition}
For any $y \in \mathcal{Y}$ and $x,x' \in \mathcal{X}$, if $f(x,y) \neq f(x',y)$, then either $p(x,y) = p(x',y) = 0$ or $p(x,y)>0, p(x',y)>0$.
\label{cond: 1}
\end{condition}
Note that Cond.~\ref{cond: 1} encompasses the cases $p(x,y) > 0$ for all $(x,y) \in \mathcal{X}\times \mathcal{Y}$ and $p(x,y) = p(x) p(y)$ for all $(x,y) \in \mathcal{X}\times \mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are the support set of $X$ and $Y$ respectively. The main idea is that whenever Cond.~\ref{cond: 1} holds, each $x \in \mathcal{X}$ belongs to a unique maximal hyperedge in $\Gamma(G)$. Hence quantization followed by entropy coding attains the optimal rate. Consider the following example that illustrates Cond.~\ref{cond: 1} and each vertex belongs to exactly one maximal hyperedge even though $p(x,y) = 0$ for some $(x,y) \in \mathcal{X}\times \mathcal{Y}$.

\begin{example}
Consider the random variables $(X,Y) \in \mathcal{X} \times \mathcal{Y}$ distributed as $P_{XY}$ in Fig.~\ref{fig: example_1}.\  a and let $f(x,y)$ in Fig.~\ref{fig: example_1}.\  b be the corresponding function, where $\mathcal{X} = \mathcal{Y} = \{1,2,3\}$. The hypergraph formed in this case is shown in Fig.~\ref{fig: example_1}.\ c.
\label{example: assumption}
\end{example}

\begin{figure}
\centering
 \begin{tabular}{lrrr}
	\toprule
	\f{$X\backslash Y$} & \f{1} & \f{2} & \f{3} \\
	\midrule
	\f{1} & \f{ $\tfrac{1}{7}$ } & \f{ $\tfrac{1}{7}$ }  & \f{0}\\
	\f{2} & \f{ $\tfrac{1}{7}$ }  & \f{ $\tfrac{1}{7}$ }  & \f{$\tfrac{1}{7}$}\\
	\f{3} & \f{ $\tfrac{1}{7}$ }  & \f{ $\tfrac{1}{7}$ }  & \f{0} \\
	\bottomrule
\end{tabular} \
\hspace{0.5cm}
\begin{tabular}{lrrr}
	\toprule
	\f{$X\backslash Y$} & \f{1} & \f{2} & \f{3} \\
	\midrule
	\f{1} & \f{1} & \f{1} & \f{1} \\
	\f{2} & \f{1} & \f{0} & \f{1}\\
	\f{3} & \f{1} & \f{0} & \f{1} \\
	\bottomrule
\end{tabular} \
\hspace{0.3cm}
 \includegraphics[width=60mm]{assumption_1} \\ \vspace{1mm}
\hspace{-1.5cm} (a) \hspace{3.5cm} (b) \hspace{4.5cm} (c)
\caption{(a) Probability distribution for $(X,Y)$. (b) Function $f(X,Y)$. (c) Corresponding hypergraph $G_0$.}
\label{fig: example_1}
\end{figure}
\subsection{Modular schemes}\label{I: practical_modular}
We show that whenever Cond.~\ref{cond: 1} holds, any $x \in \mathcal{X}$ belongs to exactly one maximal hyperedge $w$ in $\Gamma(G)$.

\begin{theorem}
If Cond.~\ref{cond: 1} holds, then for any $x \in \mathcal{X}$, if $x \in w_1, w_2$ for $w_1, w_2 \in \Gamma_m(G)$, then $w_1 = w_2$.
\label{thm: unique_clustering}
\end{theorem}
\begin{proof}
Without loss of generality, assume that $|w_1| \leq |w_2|$. We know from Def.~\ref{def: graph} that for any $w \in \Gamma(G)$, $x_1,x_2 \in w$ if and only if for all $y \in \mathcal{Y}$ either $f(x_1,y) = f(x_2,y)$, $p(x_1,y) = 0$, or $p(x_2,y) = 0$ holds. If $w_1$ is a singleton set then we are done since this implies $w_1 \subseteq w_2$ but since $w_1$, $w_2$ are maximal sets, we have $w_1 = w_2$. Now take the case when both $w_1, w_2$ are not singleton sets. Assume that $w_1 \neq w_2$, then there is $x', x'' \in \mathcal{X} \setminus x$ such that $x' \neq x''$, and $x' \in w_1, x' \notin w_2$, and $x'' \notin w_1, x'' \in w_2$.

For any $y \in \mathcal{Y}$, under Cond.~\ref{cond: 1} one of the following cases hold:
\begin{enumerate}
\item $f(x,y) = f(x',y) = f(x'',y)$.
\item If $f(x,y) \neq f(x',y)$ (or $f(x,y) \neq f(x'',y)$), then $p(x,y) = p(x',y) = 0$ (or $p(x,y) = p(x'',y) = 0$) by Cond.~\ref{cond: 1}.
\end{enumerate}

Hence, for all $y \in \mathcal{Y}$, we have $f(x',y) = f(x'',y)$ or $p(x',y) = 0$ or $p(x'',y) = 0$, which implies that $x'$ and $x''$ belong to the same hyperedge in $G$ by Def.~\ref{def: graph}. Thus, $x', x''$ belongs to the same maximal hyperedge which implies $w_1 \subseteq w_2$. But since $w_1$ and $w_2$ are maximal sets, it implies $w_1 = w_2$.
\end{proof}

Thm.~\ref{thm: unique_clustering} implies that whenever Cond.~\ref{cond: 1} holds, each $x \in \mathcal{X}$ belongs to exactly one maximal hyperedge in $\Gamma(G)$. Thus, hypergraph-based coding implies the following quantization$+$entropy coding scheme attains the optimal rate. Given any $x \in \mathcal{X}$, encode it using the unique hyperedge it belongs to and then use Slepian-Wolf coding to achieve the rate-distortion function. This scheme can be implemented in $\mathcal{O}(N\log{N})$ time since quantization can be performed in constant time and Slepian-Wolf coding can be implemented in $\mathcal{O}(N \log{N})$ time using polar codes \cite{Arikan2010} where $N$ is the blocklength.

Next consider the case when $\epsilon > 0$. Unlike for the case $\epsilon = 0$, when $\epsilon > 0$, even when $p(x,y) > 0$ for all $(x,y) \in \mathcal{X} \times \mathcal{Y}$ we might not have non-overlapping clustering, i.e.\ we can have vertices belonging to more than one maximal hyperedge.

\begin{figure}[H]
\begin{center}
\includegraphics{counter_example_1.pdf}
\end{center}
\caption{The hypergraph consisting of vertex set $\mathcal{X} = \{x_1,x_2,x_3\}$ and hyperedges $w_1 = \{x_1, x_2\}$ and $w_2 = \{x_2,x_3\}$.}
\label{fig: counterexample_hypergraph.}
\end{figure}

\begin{example}
Let $X$ and $Y$ be independent uniform random variables defined on the support set $\mathcal{X} = \{1,2,3\}$ and $\mathcal{Y} = \{1,2\}$ respectively. Let $f: \mathcal{X} \times \mathcal{Y} \to \mathcal{Z}$, where $\mathcal{Z} \subset \mathbb{R}^2$, and $f$ be defined as $f(1,y) = (1,y)$, $f(2,y) = (2,1.5 + y)$, $f(3,y) = (3,y)$, and let $\epsilon = \frac{\sqrt{13}}{4}$. Then the characteristic hypergraph $G$ is as shown in Fig.~\ref{fig: counterexample_hypergraph.}. The hypergraph $G$ consists of three vertices $\{x_1,x_2,x_3\}$ and two maximal hyperedges $w_1$ and $w_2$. The smallest enclosing circle of the set of points $\{f(1,y), f(2,y)\}$ is centred at $(1.5,y + 0.75)$ and has a radius of $\epsilon = \frac{\sqrt{13}}{4}$. Hence, $w_1 = \{x_1,x_2\}$ forms a hyperedge of $G$. Similarly, $w_2 = \{x_2,x_3\}$ forms a hyperedge, but the smallest enclosing circle of $\{f(1,1), f(3,1)\}$ has a radius of $1$ which is greater than $\epsilon$ and hence $\{x_1,x_3\}$ does not form a hyperedge. Thus, we see that even though $p(x,y) > 0$ for all $(x,y) \in \mathcal{X}\times \mathcal{Y}$, $x_2$ belongs to two different maximal hyperedges $w_1$ and $w_2$.
\label{example: unique_clustering}
\end{example}
\subsection{Partially known functions}\label{II: unknown_functions}
Suppose there is no side information available at the decoder and the function $f$ is unknown to the encoder but it is known that $f$ is an $L$-Lipschitz continuous function. Then as a corollary of Thm.~\ref{thm: 2Delta}, we have the following result which may be of interest in several applications where the actual function is unknown or requires more computational resources than are available at the encoder. For instance if there is a good linear approximation to a computationally heavy function available to the encoder, the encoder might use the simpler function rather than the actual one.

\begin{corollary}
Let $f: \mathcal{X} \mapsto \mathcal{Z}$ be a $L$-Lipschitz continuous function. Then $R(\epsilon)$ can be upper-bounded as
$
R(\epsilon) \leq H_{G_{\epsilon / L}}(X),
$
where $G_{\epsilon / L}$ is constructed with respect to the random variable $X$ and the identity function and hence the upper-bound is achievable by the encoder even when $f$ is unknown.
\label{cor: L-Lipschitz}
\end{corollary}
\begin{proof}
The proof follows from Thm.~\ref{thm: 2Delta} and the properties of $L$-Lipschitz continuous functions. The main idea is that if a set of $k$ points $\{x_1, x_2, \ldots, x_k\}$ has a smallest enclosing circle of radius $r$, then the set of points $\{f(x_1), f(x_2), \ldots, f(x_k)\}$ will have a smallest enclosing circle of radius $r' \leq rL$.
\end{proof}

Now, consider the case where the encoder cannot compute the exact function $f$ but computes $g$, which is an $\delta$-approximation to $f$ as defined in Def.~\ref{def: delta}.
\begin{corollary}
Let $g$ be a $\delta$-approximation to $f$. If the encoder only has access to $g$, then  for $\epsilon > 2 \delta$, $R(\epsilon)$ can be upper-bounded as
$
R(\epsilon) \leq H_{G_{(\epsilon - 2 \delta)}}(X),
$
where $G_{(\epsilon - 2 \delta)}$ is constructed with respect to the random variable $X$ and $g$. Moreover, this upper bound is achievable.
\label{cor: Delta - 2 epsilon}
\end{corollary}

\begin{proof}
Since $g$ is a $\delta$-approximation to $f$, if a set of points have a smallest enclosing circle of radius $r$ with respect to $g$, then the same set of points must have a smallest enclosing circle of radius less than or equal to $r + 2 \delta$ with respect to $f$. Hence, constructing a graph with fidelity constraint $\epsilon - 2 \delta$ and the function $g$ ensures that the maximal distortion with respect to $f$ is less than or equal to $\epsilon$.
\end{proof}

Although Ex.~\ref{example: unique_clustering} shows that for $\epsilon > 0$, there can be overlapping clustering even when $X$ and $Y$ are independent random variables, in Sec.~\ref{subsec: prob_quant_polar} we will show that in cases where there is overlapping clustering we can still use a \emph{randomized} form of quantization followed by polar coding to attain the optimal rate.

\subsection{Quantization and universal source coding  for lossless coding for computing}\label{subsec: quant_usc}

When $X$ is independent of $Y$, we have $p(x,y) = p(x)p(y) > 0$ for all $(x,y) \in \mathcal{X}\times \mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are the support set of $X$ and $Y$ respectively. Hence it follows from \eqref{eqn: graph_entropy} that $H_{G}(X|Y) = H(q(X))$, where $q(X)$ is the quantized value of $X$ corresponding to the unique maximal hyperedge that $X$ belongs to. Moreover, note that the function $q$ depends only on the function $f$ and not on the probability mass function of $X$, which implies that for a fixed function $f$ and any $p(x,y)>0$ such that $X$ is independent of $Y$, there is a universal source coding scheme that attains the functional $\epsilon$-entropy in \eqref{eqn: graph_entropy} which is the minimum number of bits the encoder needs to send when $\epsilon = 0$. We illustrate this case using an example.

\begin{example}
Let $f(X,Y) = \arg \min_{i \in \{1,2\}} g(i, X, Y)$ where $g(1,x,y) = x^2 + 3y$, $g(2,x,y) = x + 2y^2$ as illustrated in Fig.~\ref{fig: lzw}.a, $\arg \min$ takes the minimum value of $i$ in case of equality, and take $\epsilon = 0$. Let $X$ and $Y$ be independent random variables, $P_X$ be the probability mass function of $X \in \mathcal{X} = \{1,2,3,4\}$, and $P_Y$ be the probability mass function of $Y \in \mathcal{Y} = \{1,2\}$. Since $X$ and $Y$ are independent random variables, Thm.~\ref{thm: unique_clustering} implies the optimal rate can be obtained by using quantization followed by universal source coding. The quantization scheme for the function $f$ can be found to be forming two clusters in the characteristic hypergraph, i.e.\ any $x \in \{1,2\}$ maps to one cluster, while $x \in \{3,4\}$ maps to a different cluster. Each row of Fig.~\ref{fig: lzw}.b shows the probability mass function on $\mathcal{X}$, their corresponding entropy $H(X)$, the optimal rate of functional compression for the function $f$, $H_G(X) = H(q(X))$, where $q$ is the quantization function mapping $x \in \{1,2\}$ to one value and $x \in \{3,4\}$ to another, and the rate observed by using LZW algorithm \cite{ZivL1978} for blocklength $10^5$.
\label{example: indepnedent_sources}
\end{example}
\begin{figure}
\centering
 \begin{tabular}{lrr}
	\toprule
	\f{$X$}$\backslash Y$ & \f 1 & \f 2\\
	\midrule
    \f{1} & \f 2 & \f 2\\
    \f 2 & \f 2 & \f 2\\
    \f 3 & \f 1 & \f 2\\
    \f 4 & \f 1 & \f 2\\
	\bottomrule
\end{tabular} \
\hspace{3cm}
\begin{tabular}{lrrr}
	\toprule
	 $\f P_X$ & $\f H(X)$ & $\f H_G(X)$ & \f LZW rate\\
	\midrule
	$[\frac{1}{15}, \frac{4}{15}, \frac{8}{15}, \frac{2}{15}]$ & \f 1.64 &\f  0.92 &\f 1.06\\
	$[\frac{2}{17}, \frac{1}{17}, \frac{8}{17}, \frac{6}{17}]$ & \f 1.65 &\f  0.67 & \f 0.80\\
	$[\frac{1}{3}, \frac{1}{3}, \frac{1}{3}, \frac{1}{6}]$ & \f 1.95 & \f 0.99 & \f 1.14\\
	$[\frac{1}{6}, \frac{1}{6}, \frac{5}{12}, \frac{1}{4}]$ & \f 1.88 & \f 0.92 & \f 1.06\\
	\bottomrule
\end{tabular} \\ \vspace{1mm}
\hspace{-2.3cm} (a) \hspace{8cm} (b)
\caption{(a) Function $f(X,Y)$. (b) $H(X)$, the functional rate $H_G(X)$, and the observed rate using LZW algorithm for different $P_X$.}
\label{fig: lzw}
\end{figure}

\subsection{Randomized quantization and polar coding for computing}\label{subsec: prob_quant_polar}

In this subsection, we consider the case when there is no side information and $\epsilon > 0$. This can be easily extended to the case when $X$ is independent of $Y$ and the main idea of coding remains the same. Note that even in the absence of side information we might not have non-overlapping clustering in the hypergraph, i.e.\ a vertex might belong to more than one hyperedge, hence, quantization followed by universal source coding might not be optimal. This can be observed from Ex.~\ref{example: unique_clustering} with slight modification as well. In this subsection, we show that even when we do not have unique clustering in the hypergraph, we can have practical coding schemes using \emph{randomized quantization} and polar coding. To that end, we provide the following two-step algorithm for attaining the optimal rate asymptotically for any $\epsilon > 0$.
\begin{itemize}[leftmargin=*]
\item \emph{Randomized quantization:} We refer to the process of finding an auxiliary random variable $W$ as randomized quantization since unlike general rate distortion problems, the support set of $W$ is finite and can be determined directly from the corresponding characteristic hypergraph. Hence, every vertex of the hypergraph quantizes to the hyperedges associated with it in a randomized manner. Note that this process is different from random binning in the sense that random binning involves assigning bins to $N$-length sequences for a coding scheme with blocklength $N$, whereas, in randomized quantization we assign probabilities to single elements in $\mathcal{X}$.
Once we have formed the hypergraph, we need to optimize $I(W;X)$ over all conditional probabilities $p(w|x)$ such that $x$ lies in the hyperedge $w$. This is a convex optimization problem over finite variables and can be solved easily. Once we find a suitable $W$, we know from the proof of Thm.~\ref{thm: 2Delta} that we can find a function $g$ such that $\mathbb{E}[d_{\epsilon}(X,g(W))] = 0$, where $g(w)$ is the center of the smallest enclosing circle of the set of points $\{f(x): x \in w \text{ and } p(x)>0\}$. We will assume that $|\mathcal{W}| = 2$ for simplicity, which can be generalized to arbitrary finite-sized $\mathcal{W}$ using ideas from \cite{SasogluTA2009}.

\item \emph{Polar coding:} Once we have found $W$ corresponding to the optimal rate, the next step is to use polar codes to achieve a rate of $I(W;X)$. Define a distortion function $d(x,w) = d_{\epsilon}(x,g(w))$. Then for the chosen $W$, we have $\mathbb{E}[d(X,W)] = 0$. From Thm.~\ref{thm: polar_codes} there exists a set $\mathcal{I} \subset \{1, \ldots, N\}$ and frozen set $\mathcal{I}^c = \{1,\ldots,N\}\backslash \mathcal{I}$ such that $|\mathcal{I}| = NR > NI(W;X)$ and
\[
Z(U_i|U_1^{i-1},X_1^N) \geq 1 - 2^{-N^{\beta}} \text{  or  } Z(U_i|U_1^{i-1}) \leq 2^{-N^{\beta}},
\]
for $\beta < 1/2$ and $N$ sufficiently large. The coding scheme follows the polar coding scheme for lossy compression \cite{HondaY2013} and is described next.

\emph{Codebook generation:} Let $\mathcal{L}_i$ be the family of functions $\lambda_i : \{0,1\}^{i-1} \to \{0,1\}$ and let $\lambda_{\mathcal{I}^c} \in \prod_{i \in \mathcal{I}^c} \mathcal{L}_i$ be shared between the encoder and the decoder. Later we will show that such set of functions $\lambda_{\mathcal{I}^c}$ exist that give us the desired rate and distortion.\\
\emph{Encoder:} For $i \in \mathcal{I}$, the encoder determines $u_i$ as follows
\[
u_i = \begin{cases}
0 \hspace{2em} \text{with probability} & P_{U_i|U_1^{i-1}}(0|u_1^{i-1},x_1^N)\\
1 \hspace{2em} \text{with probability} & P_{U_i|U_1^{i-1}}(1|u_1^{i-1},x_1^N),\\
\end{cases}
\]
and for $i \in \mathcal{I}^c$, the encoder determines $u_i$ as $u_i = \lambda_i(u_1^{i-1})$. The encoder sends $u_{\mathcal{I}}$ to the decoder. Hence the rate of coding is $|\mathcal{I}|/N$.\\
\emph{Decoder:} The decoder upon receiving $u_{\mathcal{I}}$, determines $u_{\mathcal{I}^c}$ as $u_i = \lambda_i(u_1^{i-1})$ for $i \in \mathcal{I}^c$ and outputs $\hat{w}_1^N = x_1^N G_N$.\\
\emph{Analysis:} We want $\mathbb{E}[d(X,\hat{W})] \to 0$ as $N \to \infty$ for some function $\lambda_{\mathcal{I}^c} \in \prod_{i \in \mathcal{I}^c} \mathcal{L}_i$. For a fixed $N$, the average distortion is given by
$
D_N(\lambda_{\mathcal{I}^c}) = \tfrac{\mathbb{E}[d^N(X^N, \hat{W}^N)]}{N},
$
where $d^N(x^N,w^N) = \sum_{i=1}^N d(x_i, w_i)$. From \cite[Thm.~4]{HondaY2013}, it directly follows that there exists a set of functions $\lambda_{\mathcal{I}^c} = \{\lambda_i \in \mathcal{L}_i\}_{i \in \mathcal{I}^c}$ such that $D_N = \mathcal{O}(2^{-N^{\beta'}})$ for some $\beta' < \beta < 1/2$. Thus, we have $D_N(\lambda_{\mathcal{I}^c}) \to 0$ as $N \to \infty$.
\end{itemize}

\section{Properties of $R(\epsilon)$}\label{sec: properties}

\tikzstyle{block_f_1} = [rectangle, draw,
    text width=8em, text centered, minimum height=2.5 em, node distance = 1.2cm]

\begin{figure}
\centering
\begin{tikzpicture}
\node(plot){\includegraphics[scale = 0.25]{discontinuous_rate-eps-converted-to}};
\hspace{-3cm}
\node[block_f_1, below = 0.15 cm of plot](h_1){\includegraphics[height = 2cm, width = 3cm]{discontinuous_rate_1}}; \hspace{-1cm}
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\node[block_f_1, right = of h_3](h_4){\includegraphics[height = 2cm, width = 3cm]{discontinuous_rate_4}};

\draw[->] (h_1)+(3cm, 1.2cm) -- node[] { }+(4cm,4.1cm);
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\draw[->] (h_3)+(0cm, 1.25cm) -- node[] {} +(-3cm,2.63cm);
\draw[->] (h_4)+(0.15cm, 1.2cm)  -- node[] {} +(-4.05cm,1.8cm);
\end{tikzpicture}

\begin{tabular}{r@{\hskip 2 cm}r@{\hskip 2 cm}r@{\hskip 2 cm}r}
\s{ $0 \leq \epsilon < \frac{\sqrt{13}}{4}$,} & \s{$\frac{\sqrt{13}}{4} \leq \epsilon < 1$,} &\s{$1 \leq \epsilon < \frac{13}{12}$,} & \s{$\frac{13}{12} \leq \epsilon$,}\\
\s{$H_{G_{\epsilon}}(X) = \log_2{3}$.} & \s{$H_{G_{\epsilon}}(X) = \frac{2}{3}$.} &\s{$H_{G_{\epsilon}}(X) = \log_2{3} - 1$.} & \s{$H_{G_{\epsilon}}(X) = 0$.}
\end{tabular}
\caption{$R(\epsilon)$ and corresponding hypergraphs.}
\label{fig: discontinuous_rate}
\end{figure}

Clearly $R(\epsilon)$ is a non-increasing function of $\epsilon$. In this section, we show $R(\epsilon)$ may be a discontinuous function of $\epsilon$ and hence from an operational point of view one must design codes with $\epsilon$ close to zero or the right of point of discontinuity for efficient compression algorithms.\cmmnt{Also, note that } We cannot use time-sharing to remove the discontinuity in $R(\epsilon)$ because we have considered maximal distortion\cmmnt{and not expected distortion}. Further, the discontinuity of $R(\epsilon)$ is not obvious from Thm.~\ref{thm: I:coding theorem} but only from the equivalent definition of $R(\epsilon)$ as $H_{G_{\epsilon}}(X)$ that the discontinuity and points of discontinuity can be observed.
We illustrate this property using the following example.

\begin{example}
Consider the function $f: \mathcal{X} \to \mathcal{Z}$, where $\mathcal{Z} \subset \mathbb{R}^2$, and $f$ is defined as $f(1) = (1,1)$, $f(2) = (2,2.5)$, $f(3) = (3,1)$. Then for different values of $\epsilon$ we have different $G_{\epsilon}$ as illustrated in Fig.~\ref{fig: discontinuous_rate}. $R(\epsilon)$ depends on $G_{\epsilon}$ and hence, on increasing the value of $\epsilon$, the values of $\epsilon$ where $G_{\epsilon}$ changes are the points of discontinuity of $R(\epsilon)$ as illustrated in Fig.~\ref{fig: discontinuous_rate}.
\label{example: discontinuous_rate}
\end{example}
\cmmnt{
\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=0.8\linewidth]{u_discontinuous_rate_1}
  \caption{$0 \leq \epsilon < \frac{\sqrt{13}}{4}$, $H_{G_{\epsilon}}(X) = \log_2{3}$.}
  \label{fig:sub1}
\end{subfigure}%
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=0.8\linewidth]{u_discontinuous_rate_2}
  \caption{$\frac{\sqrt{13}}{4} \leq \epsilon < 1$, $H_{G_{\epsilon}}(X) = \frac{2}{3}$.}
  \label{fig:sub2}
\end{subfigure}

\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=0.8\linewidth]{u_discontinuous_rate_3}
  \caption{$1 \leq \epsilon < \frac{13}{12}$, $H_{G_{\epsilon}}(X) = \log_2{3} - 1$.}
  \label{fig:sub3}
\end{subfigure}
\begin{subfigure}{.45\textwidth}
  \centering
  \includegraphics[width=0.65\linewidth]{u_discontinuous_rate_4}
  \caption{$\frac{13}{12} \leq \epsilon$, $H_{G_{\epsilon}}(X) = 0$.}
  \label{fig:sub4}
\end{subfigure}
\caption{The maximal hyperedges formed and the hypergraph entropy corresponding to the characteristic graph $G_{\epsilon}$ for different values of $\epsilon$ in Ex.~\ref{example: discontinuous_rate}.}
\label{fig:Delta_clustering}
\end{figure}
}

\section{Conclusion}\label{sec: conclusion}
This paper considers the problem of coding for computing with a fidelity constraint. The main insight regarding the solution of the problem is obtained by characterizing the rate as the conditional entropy of a hypergraph, which we call functional $\epsilon$-entropy. It is shown that the rate-distortion function for the problem is discontinuous with respect to the fidelity constraint. We also develop practical coding schemes for the problem and provide achievable bounds when the exact function is unknown to the encoder but an approximate function or a class to which the function belongs is known.
The rate provided in this paper for a maximal distortion $\epsilon$ can be seen as an upper-bound to the rate for the rate distortion problem with expected distortion $\epsilon$.
For future work, we want to provide stronger practically achievable bounds for the problem of coding for computing with expected distortion, since practical codes for this problem are still unknown.
\section*{Acknowledgement}\label{sec: acknowledgement}
We appreciate valuable discussions with Souktik Roy, Harshit Yadav, Aditya Deshmukh, Akshayaa Magesh, and Ishita Jain.
\Section{References}
\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
using System;
using System.Collections.Generic;

namespace IuriiO.Apps.Avidoc.Services.TagServices
{
    /// <summary>
    /// Tag service contract.
    /// </summary>
    public interface ITagService
    {
        /// <summary>
        /// Gets all tags.
        /// </summary>
        /// <returns>All tags.</returns>
        List<String> GetTags();
    }
}

redpajama_set_name: RedPajamaGithub


Arrow's impossibility theorem, Arrow's theorem, or Arrow's paradox is a statement from social choice theory, named after economist Kenneth Arrow, who first described it in 1950: Suppose there is a vote, and voters have at least three different options to choose from. Each voter will then rank the options according to his or her preference. Arrow said, that in such a case, there is no way to convert these rankings into a community-wide complete and transitive ranking while also meeting certain criteria.
Arrow demonstrated the theorem in his doctoral thesis. He popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".
Arrow, Kenneth J. (1950). "A Difficulty in the Concept of Social Welfare". Journal of Political Economy 58 (4): 328–346. doi:10.1086/256963. Archived from the original on 2011-07-20. https://web.archive.org/web/20110720090207/http://gatton.uky.edu/Faculty/hoytw/751/articles/arrow.pdf.
This page was last changed on 25 August 2018, at 10:30.


text:
By Maya Stackhouse
Brought to you by: Global Poverty Project
The 2016 Sewer Games
May your immune system be ever in your favor.
If you haven't heard already, the waters of Rio de Janeiro are polluted. Like, really polluted.
Yet, somehow, Rio's government managed to convince the International Olympic Committee that this city by the water should be the host of the 2016 Summer Games.
On a postcard, Rio is a coastal paradise. It is known for its natural settings--its beaches have inspired songs and nightclubs (Ipanema, Copacabana); every apocalyptic movie you've ever seen features the giant statue of Christ the Redeemer atop Corcovado mountain dramatically and symbolically tumbling to the city below. Rio is, in many ways, iconic.
Next summer, Rio will be an icon of something else: filth.
A recent Associated Press investigation has found that athletes competing in outdoor water sports will be swimming and boating in water chock-full of raw sewage. Let me repeat myself: 1,400 of the Games' 10,000 athletes who have to row, canoe, sail, or swim in Guanabara Bay, off Copacabana Beach, and in Rodrigo de Freitas Lake, will do so in human feces.
Just how contaminated are these waters? Keep your trash bin close. The AP analysis found 1.7 million times the level of disease-causing viruses in Rio's waters than would be considered hazardous on a US Southern California beach.
Flickr: BBC World Service
That means everyone who comes into contact with these polluted waters runs the risk of viral infection. Kristina Mena, a US expert in risk assessment for waterborne viruses, estimated from the AP data that international athletes at all of the Rio Games' water venues will have a 99 percent chance of infection if they ingest just 3 teaspoons of water. The athletes that were consulted during the investigation indicated that they regularly consume much more than that during training.
Whether or not a person falls ill depends on their immunity and prior exposure to the viruses found in Rio's waters. Many teams have already begun training in the Olympic and Paralympic venues. Consequently, competitors have been falling ill with fevers,vomiting and diarrhea.
The world has known about Rio's problems with water pollution and hygiene for a while. It's certainly not news to Cariocas, as residents of Rio are known. Nor is it news to their government, who have promised and failed for more than 2 decades to establish even basic sanitation in Brazil's second most populous city.
You'd think the spotlight of the 2016 Olympic Games would be enough to initiate some sort of progress. Yet, Rio's bureaucrats have had so many conflicting interests through the years that not a single sewage treatment plant in Rio has been constructed to functional capacity. Raw waste flows through open-air ditches to streams and rivers that eventually feed into the Olympic venues. The situation is more than a little ironic for a city named after a river.
Flickr: MAHM
In response to the International Olympic Committee's decision to make Rio the first South American host of the Games, former Brazilian President Luiz Inacio Lula da Silva said, "The world has recognized that the time has come for Brazil."
I tend to agree with Mr. Lula da Silva--the time HAS come for Brazil--the time has come to prioritize sanitation (not to mention the health of Brazilians and visitors). Don't let the attention and money that these Olympic Games bring to the nation become a wasted opportunity. Don't let fame become infamy.
Also in response to the IOC, US President Barack Obama said, "As friends to the Brazilian people, we welcome this extraordinary sign of progress." If hosting the Olympics is a sign of progress, isn't that sign somewhat meaningless if, while hosting, the government of Rio de Janeiro can't figure out how to put a basic sewage system in place?
Flickr: Jairo
The fact is that providing water and sanitation for everyone around the world in need would cost around $45 billion annually. Globally, there is that capacity if governments take action to prioritize funding for sanitation, and each year there would be 170,000 fewer deaths from contaminated water--most of them among children.
At this point, athletes competing in Rio next summer will have to rely on their immune systems and hope for the best. But is that the world that we, as global citizens, want to be in?
One in three people worldwide still live without adequate sanitation, 1 billion people still defecate in the open each day, and 748 million people live without access to an improved drinking water source. With those figures staring us in the face, global citizens can't be passive.
Stand up today and change the future of sanitation for everyone by sending an email to the leaders of Sweden urging them to lead the global community on improving hygiene and sanitation around the world in TAKE ACTION NOW.
TopicsWaterbrazilInternational Olympic CommitteeOlympicsWater and SanitationHygieneSanitationSouth AmericaArticle2016Rio de Janeiro

redpajama_set_name: RedPajamaCommonCrawl


text:
Transformative gender justice: Criminal proceedings for conflict-related sexual violence in Guatemala and Peru
Jelke Boesten
Australian Journal of Human Rights
Accepted/In press
King's Authors
Jelke Boesten (Social Justice, International Development)
Drawing on Andrea Durbach's work around post-conflict transformative gender justice, this paper asks if criminal justice for conflict related sexual violence (CRSV) can bring about transformative gender justice in Latin America. The paper offers a comparative analysis of two judicial cases of conflict related rape: the Sepur Zarco case in Guatemala and the Manta y Vilca case in Peru. The paper argues that domestic courts can have important transformative effects on victim-survivors, their families and on criminal justice practices for CRSV, when international standards for evidentiary practice are adhered to within the specific local context of the case in question, as was the case of Sepur Zarco. If international standards of evidentiary practice are not considered, it is much less likely that such cases are transformative, in fact, the process might do harm, as in the case of Manta y Vilca. Therefore, criminal justice processes are not by default transformative, but good practice can be important to transformative gender justice by providing redress for victim-survivors and affected communities, unsettling hierarchies, and building accountability.
By the same authors
Justicia de género transformadora. Procedimientos penales por violencia sexual relacionados con el conflicto en Guatemala y el Perú
Boesten, J., Dec 2022, In: Debates en Sociología. 55, p. 43-68 25 p.
Military Intimacies. Peruvian veterans and narratives about sex and violence
Boesten, J. & Gavilán, L., 17 Nov 2022, (Accepted/In press) In: LATIN AMERICAN RESEARCH REVIEW.
Sexual violence as a weapon of war in Ukraine
Boesten, J., 10 May 2022, In: BMJ (Clinical research ed.). 377, p. o1172
Gender, Transitional Justice and Memorial Arts: Global Perspectives on Commemoration and Mobilization
Boesten, J. & Scanlon, H. (ed.), 18 May 2021, Routledge. 264 p. (Transitional Justice)
Gender, transitional justice and memorial arts: Global perspectives on commemoration and mobilisation
Boesten, J. & Scanlon, H., 1 Jan 2021, Gender, Transitional Justice and Memorial Arts: Global Perspectives on Commemoration and Mobilization. Taylor and Francis Inc., p. 1-274 274 p.
With some of the same researchers
From the same journal
COVID-19 responses and human rights in selected African countries
Okech, A., Mwambari, D. & Olonisakin, F., 11 Feb 2021, In: Australian Journal of Human Rights. 26, 3, p. 549-555 7 p.
Transitions, State-Building and the 'Residual' Refugee Problem: The East Timor and Cambodian Refugee Experience
McDowell, C. & Eastmond, M., 2002, In: Australian Journal of Human Rights. 8, 1, p. 7 - 27 21 p.


Pharmacists play an important role in the delivery of health care services, and can commonly provide immunizations and medication therapy management services. Their knowledge and expertise are vital in educating patients and helping them to prevent and manage chronic diseases, and in providing basic medical testing services. In addition, pharmacists are the most accessible health care professionals in many underserved areas.
The California Pharmacists Association (CPhA) spearheaded SB 493 (Hernandez) which was signed by Governor Jerry Brown on October 1, 2013. This bill not only recognizes pharmacists as health care providers, but also develops a new classification known as the advanced practice pharmacist (APh).
Administer drugs and biologics when ordered by a prescriber.
Provide consultation, training, and education about drug therapy, disease management and disease prevention.
Participate in multidisciplinary review of patient progress, including appropriate access to medical records.
Furnish self-administered hormonal contraceptives (the pill, the patch, and the ring) pursuant to a statewide protocol. This authority is similar to the existing emergency contraception protocol Furnish travel medications recommended by the CDC not requiring a diagnosis.
Furnish prescription nicotine replacement products for smoking cessation pursuant to a statewide protocol if certain training, certification, recordkeeping, and notification requirements are met.
Independently initiate and administer immunizations to patients three years of age and older if certain training, certification, recordkeeping, and reporting requirements are met. A physician protocol is still required to administer immunizations on children younger than three years of age.
Order and interpret tests for the purpose of monitoring and managing the efficacy and toxicity of drug therapies, in coordination with the patient's primary care provider or diagnosing prescriber.
Order and interpret drug therapy-related tests in coordination with the patient's primary care provider or diagnosing prescriber.
Refer patients to other healthcare providers.
Initiate, adjust, and discontinue drug therapy pursuant to an order by a patient's treating prescriber and in accordance with established protocols.


To get started, let's take a look at the average day for local retailers in The Garden State: New Jersey. On an average day, local retailers in New Jersey process 10 transactions at $187 per purchase, which results in $1,863 in daily revenue. Compared to retailers in other states, New Jersey's average daily revenue is 18.2% higher. They average 23.4% fewer transactions on a daily basis and their customers spend 54.3% more per purchase than consumers nationally.
All told, this makes New Jersey the 8th-ranked state for daily revenue generation, which puts them between Nebraska (7th) and New Hampshire (9th) in our national ranking.
Next, let's look beyond the daily trends to the average sales week for retailers in New Jersey. In 2017, local retailers in New Jersey typically made $13,077 each week, thanks to 70 weekly transactions at the average purchase price of $187.
New Jersey's local retailers' average weekly revenue is 81.8% higher than the national average, and they process 23.1% fewer transactions than retailers nationally each week.
Independent retailers in New Jersey typically make $2,297 on Fridays which is 23.3% better than their average day, and 18.3% higher than retailers nationally. On the flip-side, Sunday is typically the slowest day for retailers in New Jersey, generating only $920.
In New Jersey, local retailers tend to be busiest on Saturdays. With an average of 13.4 daily transactions, Saturday is 34.9% better than their average day. On the other end of the spectrum, we have Sunday, their slowest day of the week, when they only average 7.7 transactions.
In New Jersey, Tuesday is the day when consumers are spending the most per purchase, averaging $223 per purchase.
Let's look at the overall revenue contribution of weekdays and weekends. Over the the course of an average week, local retailers in New Jersey make about $1,996 on weekdays and $1,686 on weekends.
Now, let's look at the average sales month for retailers in New Jersey and how annual revenue is distributed across the months of the year, as well as how much retailers in New Jersey typically made each month.
For starters, local retailers in New Jersey average $56,666 in revenue each month, thanks to 303 monthly transactions at an average purchase price of $187.
Compared to retailers nationally, New Jersey's retailers' average monthly revenue is 18.2% higher, and they process about 23.3% fewer transactions on a daily basis.
In the graph above, we see the highest and lowest revenue-generating months for local retailers in New Jersey. This shows us that the average New Jersey retailer makes 10.4% of their annual revenue in December alone. In 2017, the retailers in New Jersey averaged $70,793 for the month. By contrast, January is typically the slowest month for retailers in New Jersey, as it only generates 7.2% of their annual revenue.
While a few percentage points may not seem that significant, there is a $22,025 difference in total monthly revenue between their best and slowest months.
For example, the best month for local retailer revenues in New Jersey is December, when local retailers typically make $2,284 per day.
Next, we see how the average daily transaction volume varies from month to month, revealing the ebbs and flows of shoppers visiting local retailers in New Jersey. We can see that in December–the month when their transaction volume is highest–local retailers average 14 daily transactions.
Which month do consumers spend the most per-purchase with local retailers in New Jersey? January. Consumers tend to spend $214 per purchase, compared to their usual purchase price of $187.
The graphs above give us a week-by-week look at revenue, transactions, and purchase price trends for local retailers in New Jersey across an entire year. Each flat line represents a full Sunday-to-Saturday week.
All told, New Jersey retailers make about $679,997 in annual revenue over 3,631 total transactions at an average purchase price of $187. Compared to the retailers on the national level, retailers in New Jersey made 18.2% more in 2017 and processed 23.4% fewer transactions.
Unsurprisingly, the closer we get to the end of the year, the higher the weeks rank for sales. The end-of-year holiday rush is, without a doubt, the single biggest money-making period for retailers across the country, as well as for retailers in New Jersey.
Notice the general trends of New Jersey's overall revenue, transactions and purchase price as they draw closer to the end of the year. On a national level, transactions spike the closer they get to Christmas, while the average purchase price drops as consumers take advantage of holidays deals. Take a look at the revenue trends for New Jersey and how they compare to the national trend.
During November and December, local retailers average $2,136 in daily revenue which is 14.7% higher than their average day. Those same retailers process 12.1 transactions on a daily basis (21.8% more than their average day). And the consumers' average purchase price drops 5.9% to $176.
All said, the months of November and December account for 19.2% of New Jersey's annual revenue generation and 20.4% of the year's total transactions.
In New Jersey, local retailers typically average $2,374 in daily revenue between Thanksgiving and Christmas. That's a 27.4% bump from their average day. Those same retailers process 14.4 transactions on a daily basis (a 44.7% lift compared to their average day), and the consumers' average purchase price decreases 12% to $165.
All told, the Thanksgiving-to-Christmas sales period accounts for 11.2% of the revenue and 12.7% of the year's transactions for retailers in New Jersey.
For local retailers in New Jersey, Black Friday was the 2nd-best revenue-generating day of the year. Across the state, local retailers averaged $3,787 in daily revenue, which is 103.2% better than their average day.
Compared to the national average retailer, those in New Jersey made 22.6% more in daily revenue on Black Friday. They averaged 25.8 transactions, which is 0.9% less than the national average. Their average purchase price of $147 is 23.8% higher than other states.
For independent retailers in New Jersey, Small Business Saturday was the 42nd-highest revenue generating day of the year. Across the state, local retailers averaged $2,335 in daily revenue, 18 transactions, and a purchase price of $130. Compared to retailers across the country, Small Business Saturday revenues were 16.6% above the national average in New Jersey.
For local retailers in New Jersey, Cyber Monday was their 37th-best day of the year. Across the state, local retailers averaged $2,377 in daily revenue, 10.1 transactions, and an average purchase price of $235.
Compared to the national average retailer, New Jersey's local retailers made 10.4% more in daily revenue on Cyber Monday. Their transactions were 30.4% below the national average, while their average purchase price was 58.5% higher than other states.
Retailers put a lot of focus and work into holidays and big-sale days. Here's how New Jersey retailers' prominent days rank compared to their average day.


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The 10 Best Fossil Watches of 2019
Garrett Parker 8 months ago
Fossil has always been a trusted brand in the watch industry. The company is fairly young compared to other watchmakers given that it was founded only in 1984, but it has established itself to be makers of excellent quality watches. Fossil also markets itself right in the middle, with lower end pieces and higher end pieces distributed equally. The company has created some incredible watch designs in the past and continues to do so to this day. Their style is mostly reflective of stylish daily wear, and it's true. You could literally wear a different Fossil watch every single day of the week. Here are the 10 best Fossil watches of 2019 to get you started on a proper collection.
10. The Minimalist Two-Hand Watch
Minimalism is all the crave these days, and there's no reason why you can't go minimal on your watch. There are times when all you really need is a simple, classic watch that can tell you the time. This is the perfect watch for those occasions. The Fossil Minimalist Two-Handed Watch is true to its name in every way. It's simple in appearance and in function. According to Gearhungry.com, it's also probably one of the lightest watches you'll ever strap on your wrist.
9. The Q Commuter Watch
If rugged is more your style, the Fossil Q Commuter Watch will be perfect for you. This men's watch has all the looks of a classic Fossil watch. The caramel brown leather straps and the simple face are easy to pair with any outfit. It's the kind of watch you can wear every day. It's got a single subdial on the 4 o'clock mark, but that's it as far as extras on the dial. The three crowns on the side add just a touch of detail for extra ruggedness.
8. Georgia Leather in Sand
This particular Fossil watch is designed for the sophisticated woman. The Georgia Leather can be worn for work during the day, but it can easily be dressed up for a night out on the town. It's extremely classy and elegant. The light gold dial is clean and striking. The sand-colored leather is narrow and light. As described on Bestreviews.com, it also happens to cover the back of the watch, which makes this Fossil completely hypoallergenic.
7. Boyfriend Sport in Navy
Not all females like dainty watches. At the same time, there are those that don't want anything clunky and bulky around their wrists. The perfect solution is the Boyfriend Sport. This watch has a bigger face and prominent crowns on the side that makes it stand out almost immediately. The navy-colored leather and face almost gives it a masculine touch, but the style of the bands balances everything out. The dial has three subdials that add depth to the watch. It's a great watch for the woman that's always on the go.
6. Fossil Sport
These unisex watches are some of the coolest Fossil watches out there. They come in a host of incredible colors, and their functionality is comparable to some of the best smart sports watches out there. The Sport has the usual features: a heart monitor, GPS tracking, and a sturdy, reliable band. The Sport's display, as described in this article, is actually comparable to Apple's iWatch or Samsung's Galaxy Watch—without the price tag, of course.
5. Fossil Machine
If you're a fan of clunky metal watches, the Machine is one for you to check out. This particular watch comes in an all-black color scheme, which makes it as versatile as ever. It's got a bold look to it. Couple that with the chronograph functions, and you've got yourself a truly solid watch from Fossil. The bezel of the watch is engraved, adding a subtle detail that only watch aficionados could appreciate. Given its overall look, you'd be surprised how lightweight this watch actually is. Just don't take it in the water because it's not as water resistant as it may look.
4. Riley by Fossil
According to Justwomanthings.com, this watch might be the only watch you'd need to take on a beach vacation.It's chic. It's classy. You'd never think that it's one of the best waterproof watches for women out there. The design of the Riley is gorgeous. It's got a crystal-embellished bezel, a stunning white dial, and luminous indices that you'll find useful if you do take this underwater. The chronograph subdials add another layer of detail and design that really take this watch to the top. The Riley is also as sturdy as it gets with its stainless steel band. It's got everything you could ever want in a stylish and functional watch.
3. Grant Chronograph Navy Watch in Rose Gold
The rose gold style was one that came and never really left. We're still seeing the color everywhere, and it's popularity will probably never die down because of how nice it actually looks—especially when paired with navy. That's exactly what this watch has. It's definitely a looker, but it's way more than that. The Grant Chronograph offers a quartz movement, stainless steel case, soft calfskin leather, and up to 165 feet of water resistance.
2. Flynn Mechanical Black-Tone Stainless Steel
This watch also has an all-black design, but it also comes with touches of light gold in bits and pieces on the dial. The Flynn Mechanical features skeleton windows that allow you a view into the inner workings of the watch. This watch is also unique because the face is totally skewed in design. It may not be for everyone, but it's certainly different. Other features of the watch include a scratch-resistant bezel, a PVD band, and water resistance of up to 165 feet.
1. Fossil Q Wander Leather Touchscreen Smartwatch
It's easy to say that this watch has it all. It's got style. It's got functionality. And it also happens to be a touchscreen smartwatch. There isn't really much you could ask for in a watch. The Q Wander is one of Fossil's most advanced watches to date. It features an Android system that's displayed on its classy, wide face. The band is made with genuine leather, and the overall style of the watch is modern. According to Skinnyscoop.com, this watch is versatile enough to work with an iPhone up to a certain extent. How many watches could really do that? The Fossil Q Wander will give you value, style, and a ton of capabilities to keep up with your every day hustle. It's one of Fossil's best ever.
Garrett Parker
Garrett by trade is a personal finance freelance writer and journalist. With over 10 years experience he's covered businesses, CEOs, and investments. However he does like to take on other topics involving some of his personal interests like automobiles, future technologies, and anything else that could change the world.
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Immerse yourself in the Submerged wall art collection, a collaboration between Y&B and digital artist Kirin Young. Peaceful pastel colours inspired by the original painting they obscure. Submerged in the paint that painted them. A familiar new proverb "Can't see the painting for the paint" maybe a visual metaphor to the realities of one's life? Would you be so calm? These artworks are printed on stretched canvas and mounted as standard in a white wooden box frame. Bespoke sizes and framing service is also available on request.
These classical characters are submerged in one of the paint colours that painted them. Calm pastel colours and blissfully calm faces that appear undisturbed by the rising paint levels... or are they emerging from a milky pampering? Visually inspired by the 'colour field' paintings of the 1950s and 1960s by artists including Mark Rithko and Barnett Newman, these whimsical works feature large area of a single, carefully chosen colour, but with the twist of a not so abstract or expressive composition.


ICIMOD is looking for an expert with substantive conceptual and theoretical knowledge and understanding of gender issues in natural resource management, climate change, and social systems.
The Third Pole and Internews' Earth Journalism Network, with support from the Skoll Global Threats Fund, are providing fellowships for journalists from countries in Hindu Kush Himalaya Region.
WWF-India invites applications from individuals with an inherent sensitivity towards the conservation of nature to coordinate and manage the wetlands conservation programme of WWF-India.
WWF-India invites applications from individuals with demonstrated experience in a leadership role, to provide technical support and strategic guidance to our Dolphin Conservation work.
The individual will be responsible for coordinating the overall achievement of the Living Himalayas Global Initiative objectives.
The Consultant will be responsible for planning and executing research and consulting projects and assignments in environmental services and public health.
The Forum for Policy Dialogue on Water Conflicts in India (Forum to be brief) is happy to announce its second round of the internship programme.
Applications invited for the post of Coordinator– Water Stewardship, WWF-India, Kanpur, U.P.
WWF-India invites applications from individuals for the position of Coordinator– Water Stewardship to coordinate & manage the existing work of water stewardship in Kanpur.
Anyone who has some involvement with, experience of, or interest in community-led total sanitation can send images for the competition.
R4D has announced a request for applications to become a Country or Regional Partner for a new center focused on scaling innovations in the water, sanitation and hygiene (WASH) sector.
The International Glaciological Society (IGS) will publish a special issue of the Annals of Glaciology with the theme 'Glaciology in High-Mountain Asia'.
The Project Leader for water management projects will serve as a key member of the project management team and reports to Group Leader-Projects.
The candidate will be responsible for strengthening RRAN's advocacy agenda through evolving the communications strategy, developing the annual communications plan, and ensuring its implementation.
Selected candidate will carry out Ph.D research in the Interdisciplinary areas of Environmental Sciences and will also be simultaneously assigned teaching to UG level.


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No Age
Losing Feeling [12 inch] (2009)
Reviewer Brian Shultz
To say No Age is truly losing feeling feels harsh. But there's no denying the slightly less celebratory vibe present on their newest endeavor, the Losing Feeling EP, and the more concentrated reliance on the band's abstract atmospheres.
Where the band's 2008 full-length, Nouns was carried along by a steady mix of noisy pop songs and more caustic, dissonant soundscapes, the focus on Losing Feeling seems to be the latter. The band's generally lo-fi approach is still intact, to be certain, but the songs provide more of an abiding ambience. Drummer/vocalist Dean Spunt sings with a careful, deliberate steadiness that's almost too much so through Feeling's first pair of tracks, the title song and the lazier "Genie." Both are backed by a muffled sparkle of electronics and programming, while the latter flitters along with a slower guitar strum from Randy Randall.
"Aim at the Airport" is an indie-electronic Side B starter that's closer to early Styrofoam than later Sonic Youth, splashed with sounds of natural atmosphere (white noise human conversation at the start and beach waves and walking in the pebbles at the end), but the EP gets a sudden jolt with closer "You're a Target." There's a reason this song's being pushed the most out of the quartet, and that's because it's the closest thing here to a full-on rock song. The volume is fuller than the other three, the tempo is entirely more driven and Spurt's delivery is a tinge more adventurous, trying out subtle falsetto bits in between precociously snotty singing. There's some moving waves of sound that are rather thick but drive the track along well anyway, and a fairly dynamic lead guitar riff that provokes attention.
Nouns was assuredly more thought-out and accomplished, but Losing Feeling provides a pretty okay holdover. At least they didn't just toss off some gimmicky remixes.
Lucero: 1372 Overton Park
Relient K: Forget and Not Slow Down


Yesterday's big announcement that Phil Lord and ChrisMiller, the duo behind The Lego Movie and 21 JumpStreet, are making a movie about a young Han Solo movie is enough to keep Star Wars fans busy for a while, but that's not all we're going to get. With San Diego Comic-Con this week, and Lucasfilm's presence, we knew more was on the way, and now some has leaked out ahead of schedule, namely an image showing the entire upcoming Star Wars slate.
This picture started making the rounds on Imgur. Apparently a worker at the convention center where Comic-Con goes down saw this and couldn't resist snapping a quick photo.
The first three aren't such big news. We already knew about J.J. Abrams' Star Wars: The Force Awakens, Gareth Evans' standalone anthology film Star Wars: Rogue One, and RianJohnson's Star Wars: Episode VIII. From there we're getting an Obi-Wan Kenobi movie, Kenobi: Balance of the Force, which is something that has been rumored for a while, but this is confirmation. Then there's Star Wars: Episode IX, which we know that Johnson is going to write a treatment for and possibly direct. And then it looks like we're getting Lord and Miller's Fett/Solo.
The timing is a little bit in question. When the Lord/Miller announcement came down, it arrived with the release date May 25, 2018. You can't really tell if there are dates on this image—maybe they're really small under the logos, but you can't be sure—and if this is the order, we're getting a lot more Star Wars way faster than we thought. Right now, The Force Awakens opens in December, Rogue One hits December of 2016, and Episode VIII is scheduled for May 26, 2017.
But if this is the order (or at least the logical, left to right order you assume, it's possible they've got some wacky pattern in mind, like left to right on the top row, down one, then right to left, but now I sound like a crazy person) holds, within a year, from May 26, 2017 to May 25, 2018, we'll see four Star Wars movies. That's insane, and the order has to be something different, but it really doesn't matter because however you slice this, we're getting a ton of Star Wars over the next few years. That doesn't even take into account all of the books, shows, and comics, that are all official canon from this point on.
I wonder what Lucasfilm has planned for Comic-Con this week? Maybe they're going to announce Ewan McGregor is coming back as Obi-Wan. That would rule. Guess we'll find out on Friday.


From the author of the long-running # 1 bestseller StrengthsFinder 2.0 comes a landmark study of great leaders, teams, and the reasons why people follow.
Nearly a decade ago, Gallup unveiled the results of a landmark 30-year research project that ignited a global conversation on the topic of strengths. More than 3 million people have since taken Gallup's StrengthsFinder assessment, which forms the core of several books on this topic, including the #1 international bestseller StrengthsFinder 2.0.
In recent years, while continuing to learn more about strengths, Gallup scientists have also been examining decades of data on the topic of leadership. They studied more than 1 million work teams, conducted more than 20,000 in-depth interviews with leaders, and even interviewed more than 10,000 followers around the world to ask exactly why they followed the most important leader in their life.
As you read Strengths Based Leadership, you'll hear firsthand accounts from some of the most successful organizational leaders in recent history, from the founder of Teach For America to the president of The Ritz-Carlton, as they discuss how their unique strengths have driven their success. Filled with novel research and actionable ideas, Strengths Based Leadership will give you a new road map for leading people toward a better future.


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Benelli (motorcycles)
For other uses, see Benelli (disambiguation).
Established in 1911, Benelli is one of the oldest Italian motorcycle manufacturers.[1] It once manufactured shotguns, although this part of the business is now a separate company.
Benelli SpA
Giuseppe Benelli
Giovanni Benelli
Francesco Benelli
Filippo Benelli
Domenico Benelli
Antonio "Tonino" Benelli
Qianjiang Group
https://www.benelli.com
Benelli Monalbero Sport 500 cc 1935
Benelli 500 Turismo 500 cc 1939
Engine Benelli Turismo
Benelli was established in Pesaro, Italy in 1911, which possibly makes it the oldest of Italian motorcycle factories in operation. (Moto Guzzi—the oldest motorcycle factory in non-stop operation—was established in 1921.). Peugeot beeing worlds oldest and still producing motorcyclefactory due to establishing in 1898. After losing her husband, the widow Teresa Benelli invested all of the family capital into the business in the hope that it would offer stable work for her six sons: Giuseppe, Giovanni, Francesco, Filippo, Domenico and Antonio ("Tonino"). She also sent Giuseppe and Giovanni to study Engineering in Switzerland. Initially the business had 6 employees in addition to the 5 brothers working (Tonino didn't work because he was too young.)
In the beginning, it was just the Benelli Garage, which repaired bicycles and motorcycles, but was already able to produce all of the spare parts needed for repairs.[2] During World War I, Benelli worked hard fixing parts for the Italian machines in war and in 1919 the first motorcycle was presented to the public. In 1920 the company built its first complete engine in-house, a single-cylinder two-stroke 75 cc model, immediately adapted to a bicycle frame. A year later in 1921, Benelli built its first motorcycle, using their own engine which had by then become a 98 cc model.
Two years after that, using a version specially designed for competitions, Tonino "the terrible" took to the track. He displayed an extraordinary natural talent as a rider and embarked on a very successful career which confirmed the company's exceptional capacity for development and production. Riding a Benelli 175, Tonino Benelli won four Italian championship titles in five years: in 1927, 1928 and 1930 with the single overhead camshaft (SOHC) version, and in 1931 with the double overhead camshaft (DOHC) version. Unfortunately, a bad crash during a race in 1932 cut short his brilliant career and on 27 September 1937 Tonino died following a "silly" road accident.
As World War II loomed, the Benelli company debuted their four-cylinder supercharged 250cc racing bike. This was intended to compete in the 1940 season, building on Benelli's success in the 1939 Isle of Man TT Lightweight 250 cc race. With the start of the war, the Benelli Four was limited to competition in a handful of Italian domestic races.
Beyond World War IIEdit
Benelli 250 Sport Special MKIII 1971
Benelli Tornado 650 S 1972
Wartime destruction caused Benelli to remain out of production until 1949. Unsold war motorcycles to be used by the Italian army were adapted to become civil motorcycles and generate income to restart operations. When production resumed, designs were still largely based on those of the pre-war period. In 1949, Giuseppe Benelli left the family concern to found the Motobi marque.
By 1951 Benelli was offering a range consisting of 98cc and 125cc lightweights (the Leoncino or lion cub) and 350cc and 500cc singles. The Leoncino was available in both two-stroke and four-stroke forms. The need for cheap transport in post-war Italy meant that these lightweight models became immensely successful . Similar models were offered by Moto Guzzi, Ducati and Laverda, and the same economic conditions led to the growth in popularity of Vespa and Lambretta scooters. Benelli also sold motorcycles to American retailer Montgomery Ward, who rebranded them as Riverside and marketed them through their catalog.
In 1962, when Motobi was not in good financial conditions, it was acquired by Benelli, and the combined company produced around 300 motorcycles a day and had 550 employees. Motobi designs continued in production under the Benelli name.
In 1969, Benelli introduced the Benelli Tornado 650, a vertical twin designed primarily for sale in Great Britain and the US. The Tornado earned a reputation for reliability and high performance, despite its somewhat hefty weight at 480 pounds (220 kg). Benelli claimed at the time of launch that the Tornado was capable of 57 hp at 7,400rpm for a top speed of 117 mph.[3] The Tornado (later the Tornado S) was discontinued after 1974, when Benelli introduced a series of "multis" intended to compete with Japanese triples and fours.
The de Tomaso yearsEdit
Towards the end of the 1960s, the arrival of Japanese manufacturers caused a crisis in the European motorcycle industry. The original Benelli company was heavily involved in the American motorcycle market, selling motorcycles under 350cc through Montgomery Ward. The advent of competition from Japan led to Benelli products (still largely of single cylinder pushrod design) losing popularity as they were perceived as old-fashioned in comparison to Hondas of the era which sported overhead cam engines with electric starters, in much the same way as the British motorcycle manufacturers such as Norton, BSA and Triumph were affected in the larger capacity sector.
In 1973 Benelli was acquired by Argentinian industrialist Alejandro de Tomaso together with its competitor Moto Guzzi which led to the development of new models featuring multi-cylinder engines, such as the "350 GTS" and the "500 Quattro", and the Sei powered by a 747cc (later 906cc) six-cylinder in-line engine, which put the firm briefly ahead of the Japanese competition. Although technically advanced, Benelli motorcycles of the 1980s were plagued by problems, and the production was finally stopped in 1988 when the company was merged into Moto Guzzi to create "Guzzi Benelli Moto S.p.A.", and the production plants in Pesaro were sold.
Benelli motorcycles from the 1970s and 1980s, particularly the Sei, often command higher prices than their contemporaries on the secondhand market.[4]
2003 Benelli Tornado Tre 900
Benelli Tnt Cafè Racer 1130cc 2004
Benelli Tnt 1130cc Sport
After an unsuccessful attempt at revival in 1989 backed by Pesaro-based manufacturer Giancarlo Selci, in 1995 Andrea Merloni took charge.
The Tornado Tre 900 super sport bike was launched in 2002, and the TNT, roadster some years later.
Benelli is now part of motor Group Qianjiang, a corporation in Wenling, southeast China. Benelli Q.J. is located in Pesaro, with the same workforce in the same premises as previous proprietor Benelli S.p.A.
Benelli S.p.A. recently forayed into the motorcycle market of India through a joint venture with DSK Group of India based in Pune where DSK holds 51% and benelli S.P.A holds 49% and sold more than 200 high end bikes within the first 4 months of launch. Benelli has launched six models in India, in a wide range of engine displacement and price. Includes the entry level Benelli TNT 25 with a 250 cc single cylinder engine, Benelli TNT 300 with 300cc twin cylinder engine, TNT 600i and 600GT with 600cc four-cylinder motor, TNT 899 with 898cc three-cylinder mill and the flagship TNT R with 1131cc three-cylinder power plant.[5] DSK-Benelli sold over 3000 units in India over 18 months from the start of sales, one third of them being the Inline-four cylinder engine Inline-four engine 600i, making them one of the fastest premium motorcycle selling brands in India.[6]
Benelli also entered the closely guarded Iranian market in 2016 showcasing their bikes at Iran's IRIB (nikrun motor pasargad) conference center in the capital Tehran.[7]
RacingEdit
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Tonino Benelli was the undisputed protagonist of the early racing successes. With an extraordinary natural talent for riding, he began a rewarding career that confirmed his exceptional technical skill and innovation within the company. In the saddle of a Benelli 175, Tonino won the title of Italian Champion in 1927, 1928 and 1930 riding the SOHC version, and in 1931 with the DOHC version. The numerous victories in the European World Championship were a prelude to one of the most sought after successes of all motorcycle producers: the Isle of Man Tourist Trophy, at that time the most important race in the world.
European championshipsEdit
Carlo Baschieri Benelli 175 bialbero
Yvan Goor Benelli 175 bialbero
In 1939, Ted Mellors was victorious, racing in the saddle of the Benelli 250. In 1950, Dario Ambrosini also won the 1950 Isle of Man TT, repeating Mellors' success and consecrating his title by winning both the Italian and the World Championships.
MotoGP World ChampionshipEdit
250 cc class; :
Dario Ambrosini Benelli 250
Kel Carruthers Benelli 250
MotoGP World Constructors championsEdit
In the 1960s two extraordinary riders competed in the races: Tarquinio Provini, who won the Italian Championship with the 250 in 1965 and Renzo Pasolini who won the Championship with the 250 as well as the 350 in 1968 and 1969. In 1969, the enormous effort invested in competition was rewarded with the 250 world title thanks to Kel Carruthers, who also again won the Tourist Trophy in the same year. It would also mark the last world championship for a four-stroke motorcycle until the advent of the modern MotoGP era in 2003. Many of the most famous riders in the history of motorcycling raced with Benelli bikes, from Mike Hailwood to Jarno Saarinen. Former Australian Superbike champion Peter Goddard signed with Benelli in 2001.[8]
Tourist TrophyEdit
Ted Mellors Class 250 cc
Dario Ambrosini Class 250 cc
Kel Carruthers Class 250 cc
Italy portal
Companies portal
List of Benelli motorcycles
List of Italian companies
List of motorcycle manufacturers
^ Announcement of Benelli's purchase by Qianjiang Group Archived 2007-03-19 at the Wayback Machine
^ "Benelli Motorcycles". Archived from the original on 2005-12-10. Retrieved 2007-05-30.
^ James Adam Bolton (November–December 2009). "1973 Benelli 650S Tornado". Motorcycle Classics. Retrieved 2009-10-08.
^ Bike EXIF. "Benelli Sei". Retrieved 2012-05-10.
^ Ram, Ashwin (22 July 2015). "DSK Benelli Pune Organised Ride To Celebrate Sales Record". Indian Cars Bikes. Archived from the original on 14 March 2018. Retrieved 18 September 2018.
^ "DSK Benelli On a Roll After Selling 3,000 Units In India - NDTV CarAndBike".
^ Asre Khodro. "Benelli in Iran".
^ Peter Goddard. "Career Background". Retrieved 2008-01-23.
Wikimedia Commons has media related to Benelli motorcycles.
Benelli official website
OfficineBenelli-Mototeca Storica Marchigiana
Retrieved from "https://en.wikipedia.org/w/index.php?title=Benelli_(motorcycles)&oldid=900584414"


If you are planning to buy real estate property, you need to consider appointing a solicitor for handling the legal aspect of it. Property solicitors specialising in conveyancing would be able to handle the different steps involved in the buying process. A common question a buyer has in this regard is – when do you need to appoint a solicitor? It is necessary to instruct a solicitor right from the pre-contractual stage, i.e., after you have made an offer.
The pre-contractual stage involves a lot of tasks, of which the drawing up of the contract is the most important. The seller, or their solicitor, handles this task. It is at this stage that you may negotiate the terms of the contract. However, if you have appointed a property solicitor, let him/her handle this task on your behalf. They would also be able to explain the terms of the contract to you.
Adequate research of the property is necessary before you sign the deal. Your property conveyancing solicitors would be of help in this as well. Make sure you find out as much as possible about the property before you sign the contract. It is not the responsibility of the seller to tell you about the problems in the property. However, they do have the responsibility to answer properly to your enquiries.
The checks and searches of the property needs to include checking the 'title', the legal document proving the ownership of the property, enquiring from the local authorities about the development works that may affect the property in the recent future, and asking the solicitor of the seller about the details mentioned in the contract. Other searches may vary according to the property of your choice.


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(-) Adult social care
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