text:
\section{Introduction}

\subsection{Categorical product and coloring}
Let $G=(V,E)$ and $G'=(V',E')$ be two graphs. Their {\em categorical product}, denoted $G\times G'$, is the graph defined by $$\begin{array}{rcl}V(G\times G') & = & V\times V' \\ E(G\times G') & = &\{\{(u,u'),(v,v')\}:\,\{u,v\}\in E,\,\{u',v'\}\in E'\}.\end{array}$$
Hedetniemi's conjecture -- one of the most intriguing conjecture in graph theory -- states that the chromatic number of the categorical product of two graphs is the minimum of their chromatic numbers. Hedetniemi's conjecture has been verified in many cases, but the general case is still open. Tardif~\cite{Ta08} and Zhu~\cite{Zh98} provide extensive surveys of this topic. There exists also a categorical product for hypergraphs, defined by D\"ofler and Waller~\cite{DoWa80} in 1980. Using this definition of the categorical product of hypergraphs, Zhu~\cite{Zh92} conjectured in 1992 that a generalization of Hedetniemi's conjecture holds for hypergraphs as well.

Let $\cal{G}=(V,E)$ and $\cal{G}'=(V',E')$ be two hypergraphs. Their {\em categorical product}, denoted $\cal{G}\times\cal{G}'$, is the hypergraph defined by
$$\begin{array}{rcl}
V(\cal{G}\times\cal{G}') & = & V\times V' \\
E(\cal{G}\times\cal{G}') & = &\left\{\{(u_1,u_1'),\ldots,(u_r,u_r')\}:\,r\in\mathbb{Z}_+,\,\{u_1,\ldots,u_r\}\in E,\,\{u_1',\ldots,u_r'\}\in E'\right\},
\end{array}$$
where the $u_i$'s and the $u_i'$'s do not need to be distinct. In other words, a subset of $V\times V'$ is an edge of $\cal{G}\times\cal{G}'$ if its projection on the first component is an edge of $\cal{G}$ and its projection on the second component is an edge of $\cal{G}'$. Note that this product can be made associative by a natural identification and thus defined for more than two hypergraphs.

We recall that a {\em proper coloring} of a hypergraph is an assignment of colors to the vertices so that there are no monochromatic edges. The {\em chromatic number} of a hypergraph $\cal{G}$, denoted $\chi(\cal{G})$, is the minimum number of colors a proper coloring of $\cal{G}$ may have. If the chromatic number is $k$ or less, the hypergraph is {\em $k$-colorable}.
\begin{conjecture}[\cite{Zh92}]\label{conj:zhu}
Let $\cal{G}$ and $\cal{G}'$ be two hypergraphs. We have
$$\chi(\cal{G}\times\cal{G}')=\min\{\chi(\cal{G}),\chi(\cal{G}')\}.$$
\end{conjecture}
The chromatic number of the product is at most the chromatic number of each of the hypergraphs, since a coloring of any of them provides a coloring of the product. The difficult part in this conjecture is thus the reverse inequality.

When $\cal{G}$ and $\cal{G}'$ are graphs, their categorical product according to the definition for hypergraphs is not a graph anymore, since it will in general contain edges of cardinality four. Anyway, the chromatic number does not depend on the definition we take for the categorical product, as it can be easily checked (further explanations are given in the next paragraph). Thus Conjecture~\ref{conj:zhu} is a true generalization of Hedetniemi's conjecture.

We take this remark as an opportunity to give another point of view on this categorical product for hypergraphs. Let $e\in E$ and $e'\in E'$. Consider the set of all simple bipartite graphs with no isolated vertices and with color classes $e$ and $e'$. The edges of the product $\cal{G}\times\cal{G}'$ obtained from $e$ and $e'$ are in one-to-one relation with these bipartite graphs. When $\cal{G}$ and $\cal{G}'$ are graphs, these bipartite graphs are those with two vertices in each color class and no isolated vertices. Such bipartite graphs all have a perfect matching: in other words, any edge of the product of the graphs seen as hypergraphs contains an edge of the product with the usual definition of the categorical product of graphs. It explains why the chromatic number of the categorical product of two graphs does not depend on the definition we take for the categorical product.

\subsection{Kneser hypergraphs}
In 1976, Erd\H{o}s~\cite{Er76} initiated the study of {\em Kneser hypergraphs} $\operatorname{KG}^r(\cal{H})$ defined for a hypergraph $\cal{H}=(V(\cal{H}),E(\cal{H}))$ and an integer $r\geq 2$ by
$$\begin{array}{rcl}
V(\operatorname{KG}^r(\cal{H})) & = & E(\cal{H}) \\
E(\operatorname{KG}^r(\cal{H})) & = & \{\{e_1,\ldots,e_r\}:e_1,\ldots,e_r\in E(\cal{H}),\,e_i\cap e_j=\emptyset\mbox{ for all $i,j$ with $i\neq j$}\}.
\end{array}$$
These hypergraphs enjoy several properties, which are interesting from both graph theoretical and set theoretical point of views,  especially when $r=2$, i.e. when we are dealing with {\em Kneser graphs}. Among many references dealing with the Kneser hypergraphs, one can cite~\cite{AlDrLu09,LaZi07,Zi02}. For Kneser graphs, there are much more references, see~\cite{Ha10,Lo79,SiTa06,SiTa07,St76,VaVe05} among many of them.

There are also ``usual'' Kneser hypergraphs, which are obtained with $\cal{H}=([n],{{[n]}\choose k})$. They are denoted $\operatorname{KG}^r(n,k)$. In the present paper, we prove that Conjecture~\ref{conj:zhu} is true for the usual Kneser hypergraphs, which provides to the best of our knowledge the first non-trivial and explicit family of hypergraphs not being graphs for which Conjecture~\ref{conj:zhu} is true. We actually prove a more general result involving the {\em colorability defect} of a hypergraph.  The $r$-colorability defect of a hypergraph $\cal{H}$, introduced by Dol'nikov~\cite{Do88} for $r=2$ and by K\v{r}\'i\v{z}~\cite{Kr92,KR00} for any $r$, is denoted $\operatorname{cd}_r(\cal{H})$ and is the minimum number of vertices to be removed from $\cal{H}$ so that the remaining induced subhypergraph is $r$-colorable. The {\em subhypergraph of $\cal{H}$ induced by a set $X$}  is denoted $\cal{H}[X]$ and is the hypergraph with vertex set $X$ and with edge set $\{e\in E(\cal{H}):\,e\subseteq X\}$ (note that this definition of an induced subhypergraph departs from the usual way to define it). We have thus
  $$\operatorname{cd}_r(\cal{H})=\min\{|Y|:\,Y\subseteq V(\cal{H})\mbox{ and } \chi(\cal{H}[V(\cal{H})\setminus Y])\leq r\}.$$
  K\v{r}\'i\v{z}~\cite{Kr92,KR00} proved that $\chi(\operatorname{KG}^r(\cal{H}))\geq\operatorname{cd}_r(\cal{H})$. The $r$-colorability defect has then been used by other authors as a tool for exploring further properties of coloring of graphs and hypergraphs~\cite{Me14,SiTa07,Zi02}.

\subsection{Main results}
In this paper, we prove the following theorem.

\begin{theorem}\label{thm:main}
Let $\cal{H}_1,\ldots,\cal{H}_t$ be hypergraphs and $r\geq 2$ be a positive integer. If none of the $\cal{H}_{\ell}$'s have $\emptyset$ as an edge, then
$$\chi\left(\operatorname{KG}^r(\cal{H}_1)\times\cdots\times\operatorname{KG}^r(\cal{H}_t)\right)\geq\frac{1}{r-1}\min_{\ell=1,\ldots,t}\operatorname{cd}_r(\cal{H}_{\ell}).$$
\end{theorem}

The case $t=1$ is K\v{r}\'i\v{z}'s theorem.

Note that the product in this theorem involves an arbitrary number of Kneser hypergraphs, while in Conjecture~\ref{conj:zhu} only two hypergraphs are involved. The reason is that the categorical product of hypergraphs is associative and if the conjecture holds for two hypergraphs, it can be straightforwardly extended to the product of any number of hypergraphs. On the contrary, it is easy to see that Theorem~\ref{thm:main} for only two Kneser hypergraphs does not imply its correctness for any number of Kneser hypergraphs.

We will actually see that a stronger result involving the {\em $r$-alternation number} instead of the $r$-colorability defect holds. The {\em $r$-alternation number} of a hypergraph has been introduced by Alishahi and Hajiabolhassan~\cite{AlHa13} and it provides better lower bounds for Kneser hypergraphs. The definition is given later in the paper.

The fact that Conjecture~\ref{conj:zhu} is true for the usual Kneser hypergraphs is obtained via the easy equality $$\operatorname{cd}_r\left([n],{{[n]}\choose k}\right)=n-r(k-1)$$ and a theorem by Alon, Frankl, and Lov\'asz~\cite{AlFrLo86} stating that $$\chi(\operatorname{KG}^r(n,k))=\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil.$$

\begin{corollary}\label{cor:main}
Let $r\geq 2$ be an integer and let $n_1\ldots,n_t$ and $k_1,\ldots,k_t$ be positive integers such that $n_{\ell}\geq rk_{\ell}$ for $\ell=1,\ldots,t$. We have $$\chi\left(\operatorname{KG}^r(n_1,k_1)\times\cdots\times\operatorname{KG}^r(n_t,k_t)\right)=\min_{\ell=1,\ldots,t}\chi\left(\operatorname{KG}^r(n_{\ell},k_{\ell})\right).$$
\end{corollary}

If Conjecture~\ref{conj:zhu} is true, then the equality of Corollary~\ref{cor:main} is also true when the Kneser hypergraphs have not the same rank $r$. However, we were not able to find a proof of this more general result. When $r=2$, Corollary~\ref{cor:main} implies the already known fact that Hedetniemi's conjecture is true for the usual Kneser graphs~\cite{Do09,He79,VaVe06}.

\section{Proof of the main result}\label{sec:main}

\subsection{Main steps of the proof}

The proof consists in proving the following two lemmas. Their combination provides a proof of Theorem~\ref{thm:main}. They are respectively proved in Section~\ref{subsec:nonprime} and in Section~\ref{subsec:comb_topo}.

\begin{lemma}\label{lem:nonprime}
Let $r'$ and $r''$ be two positive integers. If Theorem~$\ref{thm:main}$ holds for both $r'$ and $r''$, then it holds also for $r=r'r''$.
\end{lemma}

\begin{lemma}\label{lem:main}
Let $\cal{H}_1,\ldots,\cal{H}_t$ be hypergraphs and $p$ be a prime number. If none of the $\cal{H}_{\ell}$'s have $\emptyset$ as an edge, then
$$\chi\left(\operatorname{KG}^p(\cal{H}_1)\times\cdots\times\operatorname{KG}^p(\cal{H}_t)\right)\geq\frac{1}{p-1}\min_{\ell=1,\ldots,t}\operatorname{cd}_p(\cal{H}_{\ell}).$$
\end{lemma}

\begin{proof}[Proof of Theorem~$\ref{thm:main}$]
The theorem is a direct consequence of Lemmas~\ref{lem:nonprime} and~\ref{lem:main}.
\end{proof}

\subsection{Reduction to the case when $r$ is a prime number}\label{subsec:nonprime}

The proof of Lemma~\ref{lem:nonprime} is based on the following lemma. Given a hypergraph $\cal{H}$ and two positive integers $s$ and $C$, we define a new hypergraph $\mathcal{T}_{\cal{H},C,s}$ by
$$\begin{array}{rcl}
V(\mathcal{T}_{\cal{H},C,s}) & = & V(\cal{H}) \\
E(\mathcal{T}_{\cal{H},C,s}) & = & \{A\subseteq V(\cal{H}):\,\operatorname{cd}_s(\cal{H}[A])>(s-1)C\}.
\end{array}$$
\begin{lemma}\label{lem:th}
The following inequality holds for any positive integer $r$ $$\operatorname{cd}_{rs}(\cal{H})\leq r(s-1)C+\operatorname{cd}_r(\mathcal{T}_{\cal{H},C,s}).$$
\end{lemma}

\begin{proof}
Let $A\subseteq V(\cal{H})$ be such that $|A|=|V(\cal{H})|-\operatorname{cd}_r(\mathcal{T}_{\cal{H},C,s})$ and such that $\mathcal{T}_{\cal{H},C,s}[A]$ is $r$-colorable. The existence of such an $A$ is ensured by the definition of $\operatorname{cd}_r(\mathcal{T}_{\cal{H},C,s})$. Consider any proper coloring of $\mathcal{T}_{\cal{H},C,s}[A]$ with $r$ colors and denote by $X_j\subseteq A$ the set of vertices of color $j$. Since the coloring is proper, $X_j$ is never an edge of $\mathcal{T}_{\cal{H},C,s}$. It implies that we have $\operatorname{cd}_s(\cal{H}[X_j])\leq (s-1)C$ for all $j\in[r]$. We can thus remove at most $(s-1)C$ vertices from each $X_j$ and get $Y_j\subseteq X_j$ with $\chi(\cal{H}[Y_j])\leq s$. There is thus a proper coloring of $\cal{H}[\bigcup_{j=1}^rY_j]$ with at most $rs$ colors and $\cal{H}[\bigcup_{j=1}^rY_j]$ is a subhypergraph of $\cal{H}$ obtained by the removal of at most $r(s-1)C+\operatorname{cd}_r(\mathcal{T}_{\cal{H},C,s})$ vertices.
\end{proof}

\begin{proof}[Proof of Lemma~$\ref{lem:nonprime}$]
Assume for a contradiction that there is proper coloring $c$ of $\operatorname{KG}^{r'r''}(\cal{H}_1)\times\cdots\times\operatorname{KG}^{r'r''}(\cal{H}_t)$ with $C$ colors such that $\operatorname{cd}_{r'r''}(\cal{H}_{\ell})>(r'r''-1)C$ for all $\ell$. Lemma~\ref{lem:th} ensures then that each $\mathcal{T}_{\cal{H}_{\ell},C,r'}$ has at least one edge. Let $(e_1,\ldots,e_t)\in\mathcal{T}_{\cal{H}_1,C,r'}\times\cdots\times\mathcal{T}_{\cal{H}_t,C,r'}$. The inequality $\chi(\operatorname{KG}^{r'}(\cal{H}_1[e_1])\times\cdots\times\operatorname{KG}^{r'}(\cal{H}_t[e_t]))>C$ holds because Theorem~\ref{thm:main} holds for $r'$. There is thus a monochromatic edge in $\operatorname{KG}^{r'}(\cal{H}_1[e_1])\times\cdots\times\operatorname{KG}^{r'}(\cal{H}_t[e_t])$.
We define $h(e_1,\ldots,e_t)$ to be the color of such an edge. Since $c$ is a proper coloring of $\operatorname{KG}^{r'r''}(\cal{H}_1)\times\cdots\times\operatorname{KG}^{r'r''}(\cal{H}_t)$, the map $h$ is a proper coloring of $\operatorname{KG}^{r''}(\mathcal{T}_{\cal{H}_1,C,r'})\times\cdots\times\operatorname{KG}^{r''}(\mathcal{T}_{\cal{H}_t,C,r'})$ with $C$ colors. Since Theorem~\ref{thm:main} holds for $r''$, we have
$$C\geq \frac{1}{r''-1}\min_{\ell=1,\ldots,t}\operatorname{cd}_{r''}(\mathcal{T}_{\cal{H}_{\ell},C,r'}).$$ Lemma~\ref{lem:th} implies via a direct calculation that $(r'r''-1)C\geq\min_{\ell\in[t]}\operatorname{cd}_{r'r''}(\cal{H}_{\ell})$, which is in contradiction with the starting assumption.
\end{proof}

\subsection{Proof of the main result when $r$ is a prime number}\label{subsec:comb_topo}

The proof of Lemma~\ref{lem:main} makes use of a ``$Z_p$-Tucker lemma'' proposed in~\cite{Me11} as a slight generalization of a ``$Z_p$-Tucker lemma'' used by Ziegler~\cite{Zi02} in a combinatorial proof of K\v{r}\'i\v{z}'s theorem (proof inspired by Matou\v{s}ek's proof of the special case of usual Kneser graphs~\cite{Mat04}). Before stating it, we introduce some notations.

We denote by $Z_r$ the cyclic and multiplicative group of the $r$th roots of unity. We denote by $\omega$ one of its generators. Let $\boldsymbol{x}=(x_1,\ldots,x_n)$ and $\boldsymbol{x}'=(x'_1,\ldots,x'_n)$ be two vectors of $(Z_r\cup\{0\})^{n}$. By the notation ``$\boldsymbol{x}\subseteq\boldsymbol{x}'$'', we mean that the following implication holds for all $i\in[n]$ $$x_i\neq 0\quad\Longrightarrow\quad x'_i=x_i.$$ The number of nonzero components of a vector $\boldsymbol{x}$ is denoted $|\boldsymbol{x}|$. We introduce also the notation $\operatorname{supp}_j(\boldsymbol{x})$ for the set of indices $i$ such that $x_i=\omega^j$. Note that we have $|\boldsymbol{x}|=\sum_{j=1}^r|\operatorname{supp}_j(\boldsymbol{x})|$.

For $\boldsymbol{x}\in(Z_r\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$, define $\omega\cdot\boldsymbol{x}=(\omega x_1,\ldots,\omega x_n)$. It defines a free action of $Z_r$ on $(Z_r\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$. For $(\omega^j,k)\in Z_r\times[m]$, where $m$ is a positive integer, we define $\omega\cdot(\omega^j,k)$ to be $(\omega^{j+1},k)$. It defines a free action of $Z_r$ on $Z_r\times[m]$. A map between two sets on which $Z_r$ acts freely is {\em equivariant} if it commutes with the action of $Z_r$.

\begin{lemma}[$Z_p$-Tucker lemma]\label{lem:zptucker} Let $p$ be a prime number, $n,m\geq 1$, $\alpha\leq m$ and let
$$\begin{array}{lccc}\lambda:
& (Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\} &
\longrightarrow & Z_p\times [m] \\
& \boldsymbol{x} & \longmapsto & (s(\boldsymbol{x}),v(\boldsymbol{x}))
\end{array}$$ be a $Z_p$-equivariant map satisfying the following two properties:
\begin{itemize}
\item for all $\boldsymbol{x}^{(1)}\subseteq \boldsymbol{x}^{(2)}\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$, if $v(\boldsymbol{x}^{(1)})=v(\boldsymbol{x}^{(2)})\leq\alpha$, then $s(\boldsymbol{x}^{(1)})=s(\boldsymbol{x}^{(2)})$;
\item
for all
$\boldsymbol{x}^{(1)}\subseteq \boldsymbol{x}^{(2)}\subseteq \cdots \subseteq \boldsymbol{x}^{(p)}\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$, if $v(\boldsymbol{x}^{(1)})=v(\boldsymbol{x}^{(2)})=\cdots=v(\boldsymbol{x}^{(p)})\geq\alpha+1$, then the $s(\boldsymbol{x}^{(i)})$ are not pairwise distinct for $i=1,\ldots,p$.
\end{itemize}

Then $\alpha+(m-\alpha)(p-1)\geq n$.
\end{lemma}

The second ingredient of the proof is a $Z_p$-equivariant map $$\varepsilon:(2^{Z_p}\times\cdots\times 2^{Z_p})\setminus\left(\{\emptyset,Z_p\}\times\cdots\times\{\emptyset,Z_p\}\right)\longrightarrow Z_p,$$ where we assume $p$ to be a prime number.
For $B\subseteq Z_p$, we define $\omega\cdot B$ to be $\cup_{b\in B}\{\omega b\}$. Since $p$ is a prime number, it defines a free action of $Z_p$ on $2^{Z_p}\setminus\{\emptyset,Z_p\}$. We extend the action of $Z_p$ on $2^{Z_p}\times\cdots\times 2^{Z_p}\setminus(\{\emptyset,Z_p\}\times\cdots\times\{\emptyset,Z_p\})$ by defining $\omega\cdot(B_1,\ldots,B_t)$ to be $(\omega\cdot B_1,\ldots,\omega\cdot B_t)$. This action is again free. We can thus define such a $Z_p$-equivariant map $\varepsilon(\cdot)$, which gives a
``sign'' to each $t$-tuple $(B_1,\ldots,B_t)\in 2^{Z_p}\times\cdots\times 2^{Z_p}$ such that at least one of the $B_{\ell}$'s is not in $\{\emptyset,Z_p\}$.

\begin{proof}[Proof of Lemma~$\ref{lem:main}$]
Without loss of generality, we assume that
$$\operatorname{cd}_p(\cal{H}_1)=\min_{\ell=1,\ldots,t}\operatorname{cd}_p(\cal{H}_{\ell}).$$

Denote by $V_{\ell}$ and by $E_{\ell}$ respectively the vertex set and the edge set of $\cal{H}_{\ell}$. The cardinality of $V_{\ell}$ is denoted by $n_{\ell}$ and we arbitrarily identify $V_{\ell}$ and $[n_{\ell}]$. Let $c:E_1\times\cdots\times E_t\rightarrow[C]$ be a proper coloring of $\operatorname{KG}^p(\cal{H}_1)\times\cdots\times\operatorname{KG}^p(\cal{H}_t)$ with $C$ colors.
We endow $E_1\times\cdots\times E_t$ with an arbitrary total order $\preceq$ such that $(S_1,\ldots,S_t)\preceq (T_1,\ldots,T_t)$ if $c(S_1,\ldots,S_t)<c(T_1,\ldots,T_t)$. We shall apply the $Z_p$-Tucker lemma (Lemma~\ref{lem:zptucker}) with $n=\sum_{\ell=1}^tn_{\ell}$, $\alpha=n-\operatorname{cd}_p(\cal{H}_1)+p-1$, and $m=\alpha+C-1$. The lower bound on $C$, and thus on the chromatic number, will be a direct consequence of the inequality $\alpha+(m-\alpha)(p-1)\geq n$. To that purpose, we define a map $\lambda:\boldsymbol{x}\in(Z_p\cup\{0\})^{n}\mapsto(s(\boldsymbol{x}),v(\boldsymbol{x}))\in Z_p\times[m]$.

Let $\boldsymbol{x}\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$. From this $\boldsymbol{x}$, we define $t$ vectors $\boldsymbol{y}_1,\ldots,\boldsymbol{y}_t$ as follows: the vector $\boldsymbol{y}_1$ is the vector made of the first $n_1$ entries of $\boldsymbol{x}$, the vector $\boldsymbol{y}_2$ is the vector made of the following $n_2$ entries of $\boldsymbol{x}$,... and the vector $\boldsymbol{y}_t$ is made of the last $n_t$ entries of $\boldsymbol{x}$. The vector $\boldsymbol{y}_{\ell}$ is thus an element of $(Z_p\cup\{0\})^{n_{\ell}}$. We define also $A_{\ell}$ to be the set of $\omega^j$ such that $\operatorname{supp}_j(\boldsymbol{y}_{\ell})$ contains at least one edge of $E_{\ell}$.
Two cases have to be distinguished. \\

\noindent{\bf First case: $A_{\ell}\neq Z_p$ for at least one $\ell\in[t]$.} We define
$$\begin{array}{rcl} v(\boldsymbol{x}) & = & \displaystyle{\sum_{\ell:\,A_{\ell}\in\{\emptyset,Z_p\}}|\boldsymbol{y}_{\ell}|} \\ \\&&+\displaystyle{\sum_{\ell:\,A_{\ell}\notin\{\emptyset,Z_p\}}\left(|A_{\ell}|+\max\left\{|\widetilde{\boldsymbol{y}}_{\ell}|:\,\widetilde{\boldsymbol{y}}_{\ell}\subseteq\boldsymbol{y}_{\ell}\mbox{ and }E(\cal{H}_{\ell}[\operatorname{supp}_j(\widetilde{\boldsymbol{y}}_{\ell})])=\emptyset\mbox{ for all $j$})\right\}\right)}.\end{array}$$
If $A_{\ell}\in\{\emptyset,Z_p\}$ for all $\ell\in[t]$, we define $s(\boldsymbol{x})$ to be the first nonzero entry of $\boldsymbol{x}$. Otherwise, we define $s(\boldsymbol{x})$ to be $\varepsilon(A_1,\ldots,A_t)$. We always have  in this case $1\leq v(\boldsymbol{x})\leq\alpha$.\\

\noindent {\bf Second case: $A_{\ell}=Z_p$ for all $\ell\in[t]$.} For each $j$, there is at least one $t$-tuple $(S_1(j),\ldots,S_t(j))\in E_1\times\cdots\times E_t$ with $S_{\ell}(j)\subseteq \operatorname{supp}_j(\boldsymbol{y}_{\ell})$ for all $\ell\in[t]$. Among all such $t$-tuples and over all $j$, select the one that is minimal for $\preceq$. We define then $v(\boldsymbol{x})$ to be $\alpha+c(S_1(j),\ldots,S_t(j))$ and $s(\boldsymbol{x})$ to be $\omega^j$ with $j\in[p]$ such that $(S_1(j),\ldots,S_t(j))$ has been chosen. Note that because of the definition of $\preceq$ and because $c(\cdot)$ is a proper coloring, $c(S_1(j),\ldots,S_t(j))\neq C$, and thus $\alpha+1\leq v(\boldsymbol{x})\leq\alpha+C-1$, as required. \\

The fact that such a map $\lambda$ is $Z_p$-equivariant can be easily checked. It remains to check that this map satisfies the two required properties for the application of $Z_p$-Tucker lemma.

Let $\boldsymbol{x}\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$ such that $v(\boldsymbol{x})\leq\alpha$. We are necessarily in the first case. We seek the condition under which making one of its zero components a nonzero one does not modify $v(\boldsymbol{x})$.  Such a transformation cannot concern an entry in a $\boldsymbol{y}_{\ell}$ with $\ell$ such that $A_{\ell}=\emptyset$, nor it can concern an entry in a $\boldsymbol{y}_{\ell}$ with $\ell$ such that $A_{\ell}=Z_p$. It remains to see what happens when the transformation concerns an entry in a $\boldsymbol{y}_{\ell}$ with $\ell$ such that $A_{\ell}\notin\{\emptyset,Z_p\}$. If $A_{\ell}$ increases its cardinality by one because of this transformation, then $v(\boldsymbol{x})$ increases by at least one. Hence, making a zero component of $\boldsymbol{x}$ a nonzero one modifies $v(\boldsymbol{x})$ only if at least one of the $A_{\ell}$ is not in $\{\emptyset,Z_p\}$ and if none of the $A_{\ell}$'s are modified by this transformation. Therefore, making a zero component of $\boldsymbol{x}$ an nonzero one while not modifying $v(\boldsymbol{x})$ does not modify $s(\boldsymbol{x})$ as well. For $\boldsymbol{x}\subseteq \boldsymbol{x}'\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$ such that $v(\boldsymbol{x}')\leq\alpha$, we necessarily have $v(\boldsymbol{x})\leq v(\boldsymbol{x}')$. Thus, according to the discussion we just proposed, if $v(\boldsymbol{x}^{(1)})=v(\boldsymbol{x}^{(2)})\leq\alpha$ with $\boldsymbol{x}^{(1)}\subseteq\boldsymbol{x}^{(2)}$, we have $s(\boldsymbol{x}^{(1)})=s(\boldsymbol{x}^{(2)})$.

Let $\boldsymbol{x}^{(1)}\subseteq \boldsymbol{x}^{(2)}\subseteq \cdots \subseteq \boldsymbol{x}^{(p)}\in(Z_p\cup\{0\})^{n}\setminus\{(0,\ldots,0)\}$ be such that $v(\boldsymbol{x}^{(1)})=v(\boldsymbol{x}^{(2)})=\cdots=v(\boldsymbol{x}^{(p)})\geq\alpha+1$. We are necessarily in the second case. Define $(S_1^{(i)},\ldots,S_t^{(i)})$ to be the $t$-tuple used in the computation of $v(\boldsymbol{x}^{(i)})$. Suppose for a contradiction that the $s(\boldsymbol{x}^{(i)})$'s are pairwise distinct for $i=1,\ldots,p$. Then the $(S_1^{(i)},\ldots,S_t^{(i)})$'s form an edge of $\operatorname{KG}^p(\cal{H}_1)\times\cdots\times\operatorname{KG}^p(\cal{H}_t)$ while getting the same color by $c(\cdot)$ since $v(\boldsymbol{x}^{(1)})=v(\boldsymbol{x}^{(2)})=\cdots=v(\boldsymbol{x}^{(p)})$. It contradicts the fact that $c(\cdot)$ is a proper coloring. Hence, the $s(\boldsymbol{x}^{(i)})$ are not pairwise distinct for $i=1,\ldots,p$.
\end{proof}

\section{Improvements via alternation number}

\subsection{Main theorem involving the alternation number}

An {\it alternating sequence} is a sequence $x_1,x_2,\ldots,x_m\in Z_r$ such that any two consecutive terms are different. For any $\boldsymbol{x}=(x_1,x_2,\ldots,x_n)\in (Z_r\cup\{0\})^{n}$ and any permutation $\pi\in\mathcal{S}_n$, we denote by $\operatorname{alt}_{\pi}(\boldsymbol{x})$ the maximum length of an alternating subsequence of the sequence $x_{\pi(1)},\ldots,x_{\pi(n)}$. Note that by definition this subsequence uses only elements of $Z_r$. In particular, if $\boldsymbol{x}=(0,\ldots,0)$, then $\operatorname{alt}_{\pi}(\boldsymbol{x})=0$ for any permutation $\pi$.

Let $\cal{H}=(V,E)$ be a hypergraph with $n$ vertices. We identify $V$ and $[n]$. The {\em $r$-alternation number} $\operatorname{alt}_r(\cal{H})$ of $\cal{H}$ is defined as
$$\operatorname{alt}_r(\cal{H})=\min_{\pi\in\mathcal{S}_n}\max\{\operatorname{alt}_{\pi}(\boldsymbol{x}):\,\boldsymbol{x}\in(Z_r\cup\{0\})^n\mbox{ with }\,E(\cal{H}[\operatorname{supp}_j(\boldsymbol{x})])=\emptyset\mbox{ for $j=1,\ldots,r$}\}.$$
In other words, for each permutation $\pi$ of $[n]$, we choose $\boldsymbol{x}$ such that $\operatorname{alt}_{\pi}(\boldsymbol{x})$ is maximal while none of the $\operatorname{supp}_j(\boldsymbol{x})$'s contain an edge of $\cal{H}$; then, we take the permutation for which this quantity is minimal.
This number does not depend on the way $V$ and $[n]$ have been identified.

We clearly have $|V|-\operatorname{alt}_r(\cal{H})\geq\operatorname{cd}_r(\cal{H})$. Theorem~\ref{thm:main} can now be improved with the help of the alternation number.

\begin{theorem}\label{thm:main_bis}
Let $\cal{H}_1,\ldots,\cal{H}_t$ be hypergraphs and $r\geq 2$ be a positive integer. If none of the $\cal{H}_{\ell}$'s have $\emptyset$ as an edge, then
$$\chi\left(\operatorname{KG}^r(\cal{H}_1)\times\cdots\times\operatorname{KG}^r(\cal{H}_t)\right)\geq\frac{1}{r-1}\min_{\ell=1,\ldots,t}\left(|V(\cal{H}_{\ell})|-\operatorname{alt}_r(\cal{H}_{\ell})\right).$$
\end{theorem}

This theorem implies that Conjecture~\ref{conj:zhu} holds for Kneser hypergraphs $\operatorname{KG}^r(\cal{H})$ of same rank whose chromatic number equals $\left\lceil\frac{1}{r-1}\left(|V(\cal{H})|-\operatorname{alt}_r(\cal{H})\right)\right\rceil$. The {\em $2$-stable Kneser hypergraphs} are such hypergraphs provided that $r-1$ do not divide $n-k$ (proved in~\cite{AlHa13}). These hypergraphs, denoted $\operatorname{KG}^r(n,k)_{2\mbox{\tiny{-stab}}}$, are the Kneser hypergraphs whose vertices are the $k$-subsets $A$ of $[n]$ no elements of which are cyclically adjacent (if $i\neq i'$ are both in $A$, then $2\leq |i-i'|\leq n-2$). Other examples of such hypergraphs $\cal{H}$ having their chromatic number equaling $\left\lceil\frac{1}{r-1}\left(|V(\cal{H})|-\operatorname{alt}_r(\cal{H})\right)\right\rceil$ are given by some {\em multiple Kneser hypergraphs}, see \cite{AlHa13} for more details.

The proof of Theorem~\ref{thm:main_bis} follows the same lines as the proof of Theorem~\ref{thm:main} given in Section~\ref{sec:main}. We have lemmas similar to Lemmas~\ref{lem:nonprime} and~\ref{lem:main}.

\begin{lemma}\label{lem:nonprime_bis}
Let $r'$ and $r''$ be two positive integers. If Theorem~$\ref{thm:main_bis}$ holds for both $r'$ and $r''$, then it holds also for $r=r'r''$.
\end{lemma}

\begin{lemma}\label{lem:main_bis}
Let $\cal{H}_1,\ldots,\cal{H}_t$ be hypergraphs and $p$ be a prime number. If none of the $\cal{H}_{\ell}$'s have $\emptyset$ as an edge, then
$$\chi\left(\operatorname{KG}^p(\cal{H}_1)\times\cdots\times\operatorname{KG}^p(\cal{H}_t)\right)\geq\frac{1}{p-1}\min_{\ell=1,\ldots,t}\left(|V(\cal{H}_{\ell})|-\operatorname{alt}_p(\cal{H}_{\ell})\right).$$
\end{lemma}

Lemma~\ref{lem:nonprime_bis} can be proved via a technical lemma similar to Lemma~\ref{lem:th}. We define $\widetilde{\mathcal{T}}_{\cal{H},C,s}$ by
$$\begin{array}{rcl}
V(\widetilde{\mathcal{T}}_{\cal{H},C,s}) & = & V(\cal{H}) \\
E(\widetilde{\mathcal{T}}_{\cal{H},C,s}) & = & \{A\subseteq V(\cal{H}):\,|A|-\operatorname{alt}_s(\cal{H}[A])>(s-1)C\}.
\end{array}$$

\begin{lemma}\label{lem:th_bis}
The following inequality holds for any positive integer $r$ $$\operatorname{alt}_r(\widetilde{\mathcal{T}}_{\cal{H},C,s})\leq r(s-1)C+\operatorname{alt}_{rs}(\cal{H}).$$
\end{lemma}

We omit the proofs of Lemmas~\ref{lem:nonprime_bis} and~\ref{lem:th_bis} since they are very similar to the proofs given in Section~\ref{subsec:nonprime}. The proof of Lemma~\ref{lem:main_bis} is also almost identical to the proof of Lemma~\ref{lem:main}: it uses the $Z_p$-Tucker lemma (Lemma~\ref{lem:zptucker}) and the sign $\varepsilon(\cdot)$ of Section~\ref{subsec:comb_topo}. We sketch the main changes.

\begin{proof}[Sketch of proof of Lemma~$\ref{lem:main_bis}$] Let $n_{\ell}$ be the cardinality of $V_{\ell}$. Without loss of generality, we assume that $n_1-\operatorname{alt}_p(\cal{H}_1)=\min_{\ell\in[t]}\left(n_{\ell}-\operatorname{alt}_p(\cal{H}_{\ell})\right)$. Moreover, we identify $V_{\ell}$ and $[n_{\ell}]$ in such a way that the permutation for which the minimum is attained in the definition of $\operatorname{alt}_p(\cal{H}_{\ell})$ is the identity permutation $\operatorname{id}$.

Let $c:E_1\times\cdots\times E_t\rightarrow[C]$ be a proper coloring of $\operatorname{KG}^p(\cal{H}_1)\times\cdots\times\operatorname{KG}^p(\cal{H}_t)$ with $C$ colors. We endow $E_1\times\cdots\times E_t$ with an arbitrary total order $\preceq$ such that $(S_1,\ldots,S_t)\preceq (T_1,\ldots,T_t)$ if $c(S_1,\ldots,S_t)<c(T_1,\ldots,T_t)$.

Let $n=\sum_{\ell=1}^tn_{\ell}$, $\alpha=n-n_1+\operatorname{alt}_p(\cal{H}_1)+p-1$, and $m=\alpha+C-1$. We define a map $\lambda:\boldsymbol{x}\in(Z_p\cup\{0\})^{n}\mapsto(s(\boldsymbol{x}),v(\boldsymbol{x}))\in Z_p\times[m]$. To that purpose, we define for an $\boldsymbol{x}\in(Z_p\cup\{0\})^{n}$ the vectors $\boldsymbol{y}_1,\ldots,\boldsymbol{y}_t$ and the sets $A_1,\ldots,A_t$ as in the proof of Lemma~\ref{lem:main}. \\

\noindent{\bf First case: $A_{\ell}\neq Z_p$ for at least one $\ell\in[t]$.} We define
$$\begin{array}{rcl} v(\boldsymbol{x}) & = & \displaystyle{\sum_{\ell:\,A_{\ell}=\emptyset}\operatorname{alt}_{\operatorname{id}}(\boldsymbol{y}_{\ell}) + \sum_{\ell:\,A_{\ell}=Z_p}|\boldsymbol{y}_{\ell}|} \\ \\
&&+\displaystyle{\sum_{\ell:\,A_{\ell}\notin\{\emptyset,Z_p\}}\left(|A_{\ell}|+\max\left\{\operatorname{alt}_{\operatorname{id}}(\widetilde{\boldsymbol{y}}_{\ell}):\,\widetilde{\boldsymbol{y}}_{\ell}\subseteq\boldsymbol{y}_{\ell}\mbox{ and }E(\cal{H}_{\ell}[\operatorname{supp}_j(\widetilde{\boldsymbol{y}_{\ell}})])=\emptyset\mbox{ for all $j$}\right\}  \right)}.\end{array}$$
If $A_{\ell}\in\{\emptyset,Z_p\}$ for all $\ell\in[t]$, we define $s(\boldsymbol{x})$ to be the first nonzero entry of $\boldsymbol{x}$. Otherwise, we define $s(\boldsymbol{x})$ to be $\varepsilon(A_1,\ldots,A_t)$. We always have in this case $1\leq v(\boldsymbol{x})\leq\alpha$ as required.\\

\noindent {\bf Second case: $A_{\ell}=Z_p$ for all $\ell\in[t]$.} For each $j$, there is at least one $t$-tuple $(S_1(j),\ldots,S_t(j))\in E_1\times\cdots\times E_t$ and $S_{\ell}(j)\subseteq \operatorname{supp}_j(\boldsymbol{y}_{\ell})$ for all $j\in[p]$. Among all such $t$-tuples and over all $j$, select the one that is minimal for $\preceq$. We define then $v(\boldsymbol{x})$ to be $\alpha+c(S_1(j),\ldots,S_t(j))$ and $s(\boldsymbol{x})$ to be $\omega^j$ with $j\in[p]$ such that $(S_1(j),\ldots,S_t(j))$ has been chosen. Note that because of the definition of $\preceq$ and because $c(\cdot)$ is a proper coloring, $c(S_1(j),\ldots,S_t(j))\neq C$, and thus $\alpha+1\leq v(\boldsymbol{x})\leq\alpha+C-1$, as required. \\

The map $\lambda$ satisfies the condition of Lemma~\ref{lem:zptucker}. The lower bound on $C$, and thus on the chromatic number, is a consequence of the inequality $\alpha+(m-\alpha)(p-1)\geq n$.
\end{proof}

\subsection{An application to graphs}

Two graphs $G$ and $G'$ are {\em homomorphically equivalent} when there exists a homomorphism from $G$ to $G'$ and a homomorphism from $G'$ to $G$.
Alishahi and Hajiabolhassan~\cite{AlHa14} have defined the {\em altermatic number} of a graph $G$ as the quantity
$$\zeta(G)=\max_{\cal{H}}(|V(\cal{H})|-\operatorname{alt}_2(\cal{H}))$$ where the maximum is taken over all hypergraphs $\cal{H}$ such that $\operatorname{KG}(\cal{H})$ and $G$ are homomorphically equivalent. This definition makes sense since there always exists at least one such a hypergraph, for which actually a isomorphism holds. This hypergraph is called a {\em Kneser representation} of $G$ (see for instance~\cite{LaZi07} for a discussion on Kneser representations).

Theorem~\ref{thm:main_bis} actually shows that for any two graphs $G$ and $G'$, we also have
$$\chi(G\times G')\geq \min\{\zeta(G), \zeta(G')\}.$$

This inequality allows to prove that Hedetniemi's conjecture is true for new families of graphs. If two graphs have their chromatic number equaling their altermatic number, then they satisfy Hedetniemi's conjecture. Usual Kneser graphs have their chromatic number equaling their altermatic number, but this equality holds for other families. By $\operatorname{KG}(G,H)$, we denote the Kneser graphs whose vertex set is the set of all subgraphs of $G$ isomorphic to $H$ and in which two vertices are adjacent if the corresponding subgraphs are edge-disjoint. The above mentioned equality is satisfied by the following families, which become new families satisfying Hedetniemi's conjecture:

\begin{enumerate}
\item \label{item1} the graphs $\operatorname{KG}(G,H)$ where $G$  is a multigraph such that the multiplicity of each edge is at least $2$ and where $H$ is a simple graph, see \cite{AlHa13bis}.
\item \label{item2} the graphs $\operatorname{KG}(G,rK_2)$, where $rK_2$ is a matching of size $r$, when $G$ is a sufficiently large dense graph, see \cite{AlHa14} for more details.
\item \label{item3} the graphs $\operatorname{KG}^2(\mathcal{B})$, where $\mathcal{B}$ is the basis set of the truncation of any partition matroid, see \cite{AlHa13} where this graph is called a {\em multiple Kneser graphs} (it is a special case of the multiple Kneser hypergraphs mentioned right after Theorem~\ref{thm:main_bis}).
\item \label{item4} the graphs $\operatorname{KG}(G,\mathcal{T}_n)$ (this is a new notation), when $G$ is a dense graphs: The vertices of $\operatorname{KG}(G,\mathcal{T}_n)$ are the spanning trees of $G$ and two vertices are adjacent if the corresponding spanning trees are edge-disjoint. For more details, see~\cite{AlHa14bis}.
\end{enumerate}

Moreover, any pair of graphs obtained by iterating the Mycielski construction on a pair of graphs with the chromatic number equaling the altermatic number satisfies Hedetniemi's conjecture. Indeed, Alishahi and Hajiabolhassan~\cite{AlHa14} proved that if $\chi(G)=\zeta(G)$, then this equality holds for the graph obtained via the Mycielski construction applied on $G$.

\begin{remark}
Partial results were already obtained by Alishahi and Hajiabolhassan in~\cite{AlHa14} for the families (\ref{item1}) and (\ref{item2}) above using the {\em strong altermatic number}. There is a general definition of it for any $r$, but we restrain it here to the case $r=2$. We introduce the following quantity for a hypergraph $\cal{H}=(V,E)$:
$$\operatorname{salt}_2(\cal{H})=\min_{\pi\in\mathcal{S}_n}\max\left\{\operatorname{alt}_{\pi}(\boldsymbol{x}):\,\boldsymbol{x}\in(Z_2\cup\{0\})^n\mbox{ with }\,E(\cal{H}[\boldsymbol{x}^+])=\emptyset\mbox{ or }E(\cal{H}[\boldsymbol{x}^-])=\emptyset\right\},$$ where we identify $Z_2$ and $\{+,-\}$, and where $\boldsymbol{x}^+$ (resp. $\boldsymbol{x}^-$) is the set of indices $i$ such that $x_i=+$ (resp. $x_i=-$). In other words, for each permutation $\pi$ of $[n]$, we choose $\boldsymbol{x}$ such that $\operatorname{alt}_{\pi}(\boldsymbol{x})$ is maximal while at most one of $\boldsymbol{x}^+$ and $\boldsymbol{x}^-$ contains an edge of $\cal{H}$; then, we take the permutation for which this quantity is minimal. The {\em strong altermatic number} of a graph $G$ is then the quantity
$$\zeta_s(G)=1+\max_{\cal{H}}(|V(\cal{H})|-\operatorname{salt}_2(\cal{H}))$$ where the maximum is taken over all hypergraphs $\cal{H}$ such that $\operatorname{KG}(\cal{H})$ and $G$ are homomorphically equivalent. These authors got that for any two graphs $G$ and $G'$, we have
$$\chi(G\times G') \geq\min\{\zeta_s(G), \zeta_s(G')\}.$$
Schrijver graphs are graphs whose chromatic numbers equal their strong altermatic numbers. This inequality provides another proof of the fact that Schrijver graphs satisfy Hedetniemi's conjecture. For some graphs of the families (\ref{item1}) and (\ref{item2}) above, the equality between the chromatic number and the strong altermatic number holds, which allows to derive yet another proof that they satisfy Hedetniemi's conjecture.
\end{remark}

\subsection*{Acknowledgment} The research of Hossein Hajiabolhassan is supported by ERC advanced grant GRACOL. Part of this work was done during a visit of Hossein Hajiabolhassan to the Universit\'e Paris Est. He would like to acknowledge Professor Fr\'ed\'eric Meunier for his support and hospitality.

\bibliographystyle{amsplain}

redpajama_set_name: RedPajamaArXiv


text:
2014-15Last 32Round 3
Virtus Rome 73
Cedevita Zagreb 66
January 21, 2015 CET: 20:45
Local time: 20:45 PALAZZETTO DELLO SPORT
Rome beats Cedevita to grab share of first in Group I
Virtus Rome held off Cedevita Zagreb 73-66 to claim a share of first place in Group I at 2-1 alongside Cedevita. Kyle Gibson scored 19 points on 5-of-7 three-point shooting and Jordan Morgan controlled the paint with 17 points and 12 rebounds to lead the hosts. Morgan had 8 of the team's 17 offensive rebounds. Melvin Ejim added 13 points and 7 boards for Rome. Tomislav Zubcic paced Cedevita with 15 points and 8 rebounds and Miro Bilan scored 12. Rok Stipcevic scored twice in a 10-0 run late in the second quarter that gave Rome a 39-31 halftime lead. Cedevita stormed back with a 0-14 charge of its own, but the hosts' strung together 8 straight points late to guarantee the win.
Cedevita had the lead early after a Marko Arapovic triple, but Rok Stipcevic and Morgan connected from downtown in a 10-0 Rome run for a 14-5 lead. Marko Tomas countered with a three, but the hosts continued to roll and extended the difference to 19-8 through Ejim. Luka babic and Mario Delas lowered the difference to 19-12 after 10 minutes. Threes by Tomas and Nemanja Gordic carried the visitors to within 21-20 early in the second. Delas put Cedevita ahead and Babic increased the difference to 21-24. Rome responded and a Zubcic three tied the game at 27-27. Rome pulled away with a 10-0 run in which Stipcevic scored twice. Roko Ukic's first points made it 39-31 at the break.
Pilepic and Morgan traded jumpers right after the break and a layup by Gibson made it a double-digit game, 43-33. Ukic and Babic fueled a 0-6 run Cedevita run which Zubcic capped with a dunk. Gibson struck from downtown, igniting a 7-0 run which Ejim capped with a jumper for a 50-41 Rome lead. Zubcic nailed a couple of triples and Bilan scored twice around the basket to make Cedevita get a 50-53 lead after 10 minutes. Zubcic capped a 0-14 Cedevita run with a layup but Gibson and Lollo D'Ercole soon tied it at 55-55. Bilan scored twice in the paint but a jumper by Ejim and a three-pointer by Gibson made Rome regain the lead, 60-59. Pilepic and Triche exchanged baskets and a three-pointer by Ukic put Cedevita in charge, 62-64. Morgan and Jones fueled an 8-0 run that gave Rome the lead for good, 70-64, in the next-to-last minute. A big triple by Gibson allowed the hosts to seal their win in style.
Referees: PEREZ, MIGUEL ANGEL; SILVA, SERGIO; BARDERA, REGIS
Virtus Rome 19 20 11 23
Cedevita Zagreb 12 19 22 13
Virtus Rome
3 EJIM, MELVIN 30:52 13 5/8 0/4 3/4 4 3 7 2 1 1 1 1 2 14
5 TRICHE, BRANDON 25:52 6 3/7 0/2 1 2 3 5 5 2 4 15
6 JONES, BOBBY 20:55 4 1/3 0/4 2/2 1 2 3 2 4 2 4 4 5
10 D'ERCOLE, LORENZO 21:25 5 1/1 1/5 3 3 3 1 1 2 5
12 SANDRI, DANIELE 7:04 0/1 1 1 1 -1
14 DE ZEEUW, MAXIME 14:02 0/1 2 1 3 2 1 1 3 1
15 KUSHCHEV, OLEKSANDR DNP - - - - - - - - - - - - - - -
22 GIBSON, KYLE 30:52 19 1/5 5/7 2/2 1 1 1 3 3 12
24 STIPCEVIC, ROK 23:00 9 2/4 1/4 2/2 2 2 2 1 2 9
25 MORGAN, JORDAN 25:58 17 6/9 5/7 8 4 12 3 1 2 8 32
30 PULLAZI, REI DNP - - - - - - - - - - - - - - -
Totals 200:00 73 19/38 7/27 14/17 17 18 35 17 14 11 0 3 16 24 91
50% 25.9% 82.4%
Head coach: DALMONTE, LUCA
Cedevita Zagreb
4 MAZALIN, LOVRO 6:14 1 1 -2
5 ZUBCIC, TOMISLAV 21:21 15 2/3 3/7 2/2 4 4 8 1 1 4 1 15
8 PILEPIC, FRAN 27:40 6 2/3 0/3 2/2 2 2 1 3 4 4
9 BABIC, LUKA 24:12 8 3/4 0/2 2/2 2 2 1 1 2 1 1 7
10 DELAS, ANTE DNP - - - - - - - - - - - - - - -
11 ZGANEC, KARLO 3:02 1 1 1
15 BILAN, MIRO 26:47 12 5/6 2/2 2 4 6 1 4 1 3 4 16
17 DELAS, MARIO 13:53 4 2/2 1 2 3 5 2 1 3 1 9
20 GORDIC, NEMANJA 19:39 3 0/3 1/2 2 2 7 2 4 2 4
32 UKIC, ROKO 25:05 7 2/4 1/3 1 1 2 3 1 2
33 TOMAS, MARKO 19:06 8 1/3 2/4 1 2 1 4
35 ARAPOVIC, MARKO 13:01 3 1/1 4 2 2 -1
Totals 200:00 66 17/28 8/22 8/8 7 22 29 19 2 21 3 0 24 16 65
60.7% 36.4% 100%
Head coach: REPESA, JASMIN
There are no quotes yet.
Round 1 Round 2 Round 3 Round 4 Round 5 Round 6
Unics Kazan 103
January 21 18:15 CET LIVE FINAL
Strasbourg 87
CEZ Basketball Nymburk 80
Dinamo Sassari 86
Group J
Limoges CSP 81
FoxTown Cantu 70
Group K
FC Bayern Munich 90
JDA Dijon 86
Group L
Baloncesto Seville 78
PGE Turow Zgorzelec 99
Telenet Ostend 71
Group M
Paris Levallois 75
Pinar Karsiyaka Izmir 85
Besiktas Integral Forex Istanbul 69
Group N
Lokomotiv Kuban 66
CSU Asesoft Ploiesti 92

redpajama_set_name: RedPajamaCommonCrawl


text:
Home > India News > Sanitation Workers Should Have First Claim To Say Vande Mataram: Modi
Sanitation Workers Should Have First Claim To Say Vande Mataram: Modi
Dainik Hawk | 11 Sep 2017 8:38 AM GMT
New Delhi: Prime Minister Narendra Modi on Monday said those who clean our streets should have the first claim to say 'Vande Mataram'.
Speaking on the occasion of 125th anniversary of Swami Vivekananda's 1893 Parliament of Religions address in Chicago, Modi wondered whether we should have the right to say 'Vande Mataram' after what we do to our streets through indiscriminate littering and spitting.
"If there is anyone who should have the right to say 'Vande Mataram' before anyone else, it is those children of Mother India who perform the job of cleaning our streets," he said.
"Whether we clean our surroundings or not, we do not have the right to litter it," he added.
Modi said that he was proud of women who refuse to get married into families whose houses do not have toilets and called for building of toilets before places of worships.
"Toilets first, temples later," said the Prime Minister.
In his tribute to the sage, he said Vivekananda was a visionary.
"Who would have thought anyone would be interested in celebrating a 125-year-old speech... Just with a few words, a youngster from India won over the world and showed it the power of oneness," Modi said about the prescience of the social reformer.
--IANS
Tags: Prime Minister Narendra Modi Vande Mataram Swami Vivekananda


text:
namespace GoogleVR.Demos
{
    using System;
    using UnityEngine;
    using UnityEngine.UI;

#if UNITY_2017_2_OR_NEWER
    using UnityEngine.XR;
#else // UNITY_2017_2_OR_NEWER
    using XRSettings = UnityEngine.VR.VRSettings;
#endif  // UNITY_2017_2_OR_NEWER

    ///
    /// Manages input controllers in Daydream or triggers in Cardboard in Demo scenes.
    ///
    public class DemoInputManager : MonoBehaviour
    {
        /// The name of the **controller main** property.
        public const string CONTROLLER_MAIN_PROP_NAME = "controllerMain";

        /// The name of the **controller pointer** property.
        public const string CONTROLLER_POINTER_PROP_NAME = "controllerPointers";

        /// The name of the **reticle pointer** property.
        public const string RETICLE_POINTER_PROP_NAME = "reticlePointer";

        /// The controller main.
        [Tooltip("Reference to GvrControllerMain")]
        public GameObject controllerMain;

        /// The controller pointers.
        [Tooltip("Reference to GvrControllerPointers")]
        public GameObject[] controllerPointers;

        /// The reticle pointer.
        [Tooltip("Reference to GvrReticlePointer")]
        public GameObject reticlePointer;

        /// The message canvas.
        public GameObject messageCanvas;

        /// The message text.
        public Text messageText;

#if !RUNNING_ON_ANDROID_DEVICE
        /// The Gvr Platforms.
        public enum EmulatedPlatformType
        {
            Daydream,
            Cardboard
        }

        /// The type of the GVR emulated platform.
        [Tooltip("Emulated GVR Platform")]
        public EmulatedPlatformType gvrEmulatedPlatformType = EmulatedPlatformType.Daydream;

        /// The name of the **emulated platform** property.
        public static string EMULATED_PLATFORM_PROP_NAME = "gvrEmulatedPlatformType";
#endif // !RUNNING_ON_ANDROID_DEVICE

        private const string MESSAGE_CANVAS_NAME = "MessageCanvas";
        private const string MESSAGE_TEXT_NAME = "MessageText";
        private const string LASER_GAMEOBJECT_NAME = "Laser";
        private const string CONTROLLER_CONNECTING_MESSAGE = "Daydream controller connecting...";
        private const string CONTROLLER_DISCONNECTED_MESSAGE = "Daydream controller not connected";
        private const string CONTROLLER_SCANNING_MESSAGE = "Scanning for Daydream controller...";
        private const string NON_GVR_PLATFORM =
            "Please select a supported Google VR platform via "
            + "'Build Settings > Android | iOS > Switch Platform'\n";

        private const string VR_SUPPORT_NOT_CHECKED =
        "Please make sure 'Player Settings > Virtual Reality Supported' is checked\n";

        private const string EMPTY_VR_SDK_WARNING_MESSAGE =
        "Please add 'Daydream' or 'Cardboard' under 'Player Settings > Virtual Reality SDKs'\n";

        // Java class, method, and field constants.
        private const int ANDROID_MIN_DAYDREAM_API = 24;
        private const string FIELD_SDK_INT = "SDK_INT";
        private const string PACKAGE_BUILD_VERSION = "android.os.Build$VERSION";
        private const string PACKAGE_DAYDREAM_API_CLASS = "com.google.vr.ndk.base.DaydreamApi";
        private const string METHOD_IS_DAYDREAM_READY = "isDaydreamReadyPlatform";

        // Buttons that can trigger pointer switching.
        private const GvrControllerButton POINTER_BUTTON_MASK =
                GvrControllerButton.App |
                GvrControllerButton.TouchPadButton |
                GvrControllerButton.Trigger |
                GvrControllerButton.Grip;

        private static readonly GvrControllerHand[] ALL_HANDS =
        {
            GvrControllerHand.Right,
            GvrControllerHand.Left,
        };

#if RUNNING_ON_ANDROID_DEVICE
        // Running on an Android device.
        private GvrSettings.ViewerPlatformType viewerPlatform;
#endif // RUNNING_ON_ANDROID_DEVICE

        private bool isDaydream = false;
        private int activeControllerPointer = 0;

        /// Whether Daydream is enabled in this Unity project's Player Settings.
        /// <returns>
        /// Returns `true` if Daydream is enabled in Unity Player Settings.  Otherwise, returns
        /// `false`.
        /// </returns>
        public static bool playerSettingsHasDaydream()
        {
            string[] playerSettingsVrSdks = XRSettings.supportedDevices;
            return Array.Exists<string>(playerSettingsVrSdks,
                element => element.Equals(GvrSettings.VR_SDK_DAYDREAM));
        }

        /// Whether Cardboard is enabled in this Unity project's Player Settings.
        /// <returns>
        /// Returns `true` if Cardboard is enabled in Unity Player Settings.  Otherwise, returns
        /// `false`.
        /// </returns>
        public static bool playerSettingsHasCardboard()
        {
            string[] playerSettingsVrSdks = XRSettings.supportedDevices;
            return Array.Exists<string>(playerSettingsVrSdks,
                element => element.Equals(GvrSettings.VR_SDK_CARDBOARD));
        }

        /// Determines whether the Platform is currently Daydream.
        /// <returns>
        /// Returns `true` if this Platform is currently daydream.  Otherwise, returns `false`.
        /// </returns>
        public bool IsCurrentlyDaydream()
        {
            return isDaydream;
        }

#if RUNNING_ON_ANDROID_DEVICE
        // Running on an Android device.
        private static bool IsDeviceDaydreamReady()
        {
            // Check API level.
            using (var version = new AndroidJavaClass(PACKAGE_BUILD_VERSION))
            {
                if (version.GetStatic<int>(FIELD_SDK_INT) < ANDROID_MIN_DAYDREAM_API)
                {
                    return false;
                }
            }

            // API level > 24, check whether the device is Daydream-ready.
            AndroidJavaObject androidActivity = null;
            try
            {
                androidActivity = GvrActivityHelper.GetActivity();
            }
            catch (AndroidJavaException e)
            {
                Debug.LogError("Exception while getting the Android Activity: " + e);
                return false;
            }

            AndroidJavaClass daydreamApiClass = new AndroidJavaClass(PACKAGE_DAYDREAM_API_CLASS);
            if (daydreamApiClass == null || androidActivity == null)
            {
                return false;
            }

            return daydreamApiClass.CallStatic<bool>(METHOD_IS_DAYDREAM_READY, androidActivity);
        }
#endif  // RUNNING_ON_ANDROID_DEVICE

        private void Start()
        {
            if (messageCanvas == null)
            {
                messageCanvas = transform.Find(MESSAGE_CANVAS_NAME).gameObject;
                if (messageCanvas != null)
                {
                    messageText =
                        messageCanvas.transform.Find(MESSAGE_TEXT_NAME).GetComponent<Text>();
                }
            }

            // Message canvas will be enabled later when there's a message to display.
            messageCanvas.SetActive(false);
#if !RUNNING_ON_ANDROID_DEVICE
            if (playerSettingsHasDaydream() || playerSettingsHasCardboard())
            {
                // The list is populated with valid VR SDK(s), pick the first one.
                gvrEmulatedPlatformType =
                    (XRSettings.supportedDevices[0] == GvrSettings.VR_SDK_DAYDREAM) ?
                        EmulatedPlatformType.Daydream :
                        EmulatedPlatformType.Cardboard;
            }
            isDaydream = (gvrEmulatedPlatformType == EmulatedPlatformType.Daydream);
#else

            // Running on an Android device.
            viewerPlatform = GvrSettings.ViewerPlatform;

            // First loaded device in Player Settings.
            string vrDeviceName = XRSettings.loadedDeviceName;
            if (vrDeviceName != GvrSettings.VR_SDK_CARDBOARD &&
                vrDeviceName != GvrSettings.VR_SDK_DAYDREAM)
            {
                Debug.LogErrorFormat("Loaded device was '{0}', must be one of '{1}' or '{2}'",
                    vrDeviceName, GvrSettings.VR_SDK_DAYDREAM, GvrSettings.VR_SDK_CARDBOARD);
                return;
            }

            // On a non-Daydream ready phone, fall back to Cardboard if it's present in the list of
            // enabled VR SDKs.
            // On a Daydream-ready phone, go into Cardboard mode if it's the currently-paired
            // viewer.
            if ((!IsDeviceDaydreamReady() && playerSettingsHasCardboard()) ||
                (IsDeviceDaydreamReady() && playerSettingsHasCardboard() &&
                    GvrSettings.ViewerPlatform == GvrSettings.ViewerPlatformType.Cardboard))
            {
                vrDeviceName = GvrSettings.VR_SDK_CARDBOARD;
            }

            isDaydream = vrDeviceName == GvrSettings.VR_SDK_DAYDREAM;
#endif  // !RUNNING_ON_ANDROID_DEVICE
            SetVRInputMechanism();
        }

        // Runtime switching enabled only in-editor.
        private void Update()
        {
            UpdateStatusMessage();

            // Scan all devices' buttons for button down, and switch the singleton pointer
            // to the controller the user last clicked.
            int newPointer = activeControllerPointer;

            if (controllerPointers.Length > 1 && controllerPointers[1] != null)
            {
                GvrTrackedController trackedController1 =
                    controllerPointers[1].GetComponent<GvrTrackedController>();
                foreach (var hand in ALL_HANDS)
                {
                    GvrControllerInputDevice device = GvrControllerInput.GetDevice(hand);
                    if (device.GetButtonDown(POINTER_BUTTON_MASK))
                    {
                        // Match the button to our own controllerPointers list.
                        if (device == trackedController1.ControllerInputDevice)
                        {
                            newPointer = 1;
                        }
                        else
                        {
                            newPointer = 0;
                        }

                        break;
                    }
                }
            }

            if (newPointer != activeControllerPointer)
            {
                activeControllerPointer = newPointer;
                SetVRInputMechanism();
            }

#if !RUNNING_ON_ANDROID_DEVICE
            UpdateEmulatedPlatformIfPlayerSettingsChanged();
            if ((isDaydream && gvrEmulatedPlatformType == EmulatedPlatformType.Daydream) ||
                (!isDaydream && gvrEmulatedPlatformType == EmulatedPlatformType.Cardboard))
            {
                return;
            }

            isDaydream = (gvrEmulatedPlatformType == EmulatedPlatformType.Daydream);
            SetVRInputMechanism();
#else
            // Running on an Android device.
            // Viewer type switched at runtime.
            if (!IsDeviceDaydreamReady() || viewerPlatform == GvrSettings.ViewerPlatform)
            {
                return;
            }

            isDaydream = GvrSettings.ViewerPlatform == GvrSettings.ViewerPlatformType.Daydream;
            viewerPlatform = GvrSettings.ViewerPlatform;
            SetVRInputMechanism();
#endif  // !RUNNING_ON_ANDROID_DEVICE
        }

#if !RUNNING_ON_ANDROID_DEVICE
        private void UpdateEmulatedPlatformIfPlayerSettingsChanged()
        {
            if (!playerSettingsHasDaydream() && !playerSettingsHasCardboard())
            {
                return;
            }

            // Player Settings > VR SDK list may have changed at runtime. The emulated platform
            // may not have been manually updated if that's the case.
            if (gvrEmulatedPlatformType == EmulatedPlatformType.Daydream &&
                !playerSettingsHasDaydream())
            {
                gvrEmulatedPlatformType = EmulatedPlatformType.Cardboard;
            }
            else if (gvrEmulatedPlatformType == EmulatedPlatformType.Cardboard &&
                     !playerSettingsHasCardboard())
            {
                gvrEmulatedPlatformType = EmulatedPlatformType.Daydream;
            }
        }
#endif  // !RUNNING_ON_ANDROID_DEVICE

        private void UpdateStatusMessage()
        {
            if (messageText == null || messageCanvas == null)
            {
                return;
            }

#if !UNITY_ANDROID && !UNITY_IOS
            messageText.text = NON_GVR_PLATFORM;
            messageCanvas.SetActive(true);
            return;
#else
#if UNITY_EDITOR
            if (!UnityEditor.PlayerSettings.virtualRealitySupported)
            {
                messageText.text = VR_SUPPORT_NOT_CHECKED;
                messageCanvas.SetActive(true);
                return;
            }
#endif  // UNITY_EDITOR

            bool isVrSdkListEmpty = !playerSettingsHasCardboard() && !playerSettingsHasDaydream();
            if (!isDaydream)
            {
                if (messageCanvas.activeSelf)
                {
                    messageText.text = EMPTY_VR_SDK_WARNING_MESSAGE;
                    messageCanvas.SetActive(isVrSdkListEmpty);
                }

                return;
            }

            string vrSdkWarningMessage =
                isVrSdkListEmpty ? EMPTY_VR_SDK_WARNING_MESSAGE : "";
            string controllerMessage = "";
            GvrPointerGraphicRaycaster graphicRaycaster =
                messageCanvas.GetComponent<GvrPointerGraphicRaycaster>();
            GvrControllerInputDevice dominantDevice =
                GvrControllerInput.GetDevice(GvrControllerHand.Dominant);
            GvrConnectionState connectionState = dominantDevice.State;

            // This is an example of how to process the controller's state to display a status
            // message.
            switch (connectionState)
            {
                case GvrConnectionState.Connected:
                    break;
                case GvrConnectionState.Disconnected:
                    controllerMessage = CONTROLLER_DISCONNECTED_MESSAGE;
                    messageText.color = Color.white;
                    break;
                case GvrConnectionState.Scanning:
                    controllerMessage = CONTROLLER_SCANNING_MESSAGE;
                    messageText.color = Color.cyan;
                    break;
                case GvrConnectionState.Connecting:
                    controllerMessage = CONTROLLER_CONNECTING_MESSAGE;
                    messageText.color = Color.yellow;
                    break;
                case GvrConnectionState.Error:
                    controllerMessage = "ERROR: " + dominantDevice.ErrorDetails;
                    messageText.color = Color.red;
                    break;
                default:
                    // Shouldn't happen.
                    Debug.LogError("Invalid controller state: " + connectionState);
                    break;
            }

            messageText.text = string.Format("{0}\n{1}", vrSdkWarningMessage, controllerMessage);
            if (graphicRaycaster != null)
            {
                graphicRaycaster.enabled =
                    !isVrSdkListEmpty || connectionState != GvrConnectionState.Connected;
            }

            messageCanvas.SetActive(isVrSdkListEmpty ||
                                    (connectionState != GvrConnectionState.Connected));
#endif  // !UNITY_ANDROID && !UNITY_IOS
    }

        private void SetVRInputMechanism()
        {
            SetGazeInputActive(!isDaydream);
            SetControllerInputActive(isDaydream);
        }

        private void SetGazeInputActive(bool active)
        {
            if (reticlePointer == null)
            {
                return;
            }

            reticlePointer.SetActive(active);

            // Update the pointer type only if this is currently activated.
            if (!active)
            {
                return;
            }

            GvrReticlePointer pointer =
                reticlePointer.GetComponent<GvrReticlePointer>();
            if (pointer != null)
            {
                GvrPointerInputModule.Pointer = pointer;
            }
        }

        private void SetControllerInputActive(bool active)
        {
            if (controllerMain != null)
            {
                controllerMain.SetActive(active);
            }

            if (controllerPointers == null || controllerPointers.Length <= activeControllerPointer)
            {
                return;
            }

            controllerPointers[activeControllerPointer].SetActive(active);

            // Update the pointer type only if this is currently activated.
            if (!active)
            {
                return;
            }

            GvrLaserPointer pointer =
                controllerPointers[activeControllerPointer].GetComponentInChildren<GvrLaserPointer>(
                    true);
            if (pointer != null)
            {
                GvrPointerInputModule.Pointer = pointer;
            }
        }
    }
}

redpajama_set_name: RedPajamaGithub


text:
Home Internet Facebook's Zuckerberg Locks Donald Trump's Page For 2 Weeks
Social media giant, Facebook's CEO Mark Elliot Zuckerberg on Wednesday condemned attacks by Donald Trump supporters in DC, as a "mob violence".
President Donald J. Trump's video and several posts were deleted by Facebook, Twitter and YouTube after Trump continued to give false claims over Presidential elections and provoked riots. Twitter and Facebook banned Trump from posting new updates for 12 hours and 24 hours respectively. Twitter even warned that Trump's account could be suspended forever if he keeps making false claims and incites violence.
Mark Zuckerberg, today has given another statement regarding Trump's account ban. Zuckerberg in the statement says that Trump's Facebook and Instagram account will be banned for atleast next two weeks until peaceful transition of power is complete.
In the statement Zuckerberg writes, " The shocking events of the last 24 hours clearly demonstrate that President Donald Trump intends to use his remaining time in office to undermine the peaceful and lawful transition of power to his elected successor, Joe Biden.
His decision to use his platform to condone rather than condemn the actions of his supporters at the Capitol building has rightly disturbed people in the US and around the world. We removed these statements yesterday because we judged that their effect — and likely their intent — would be to provoke further violence.
Following the certification of the election results by Congress, the priority for the whole country must now be to ensure that the remaining 13 days and the days after inauguration pass peacefully and in accordance with established democratic norms.
Over the last several years, we have allowed President Trump to use our platform consistent with our own rules, at times removing content or labeling his posts when they violate our policies. We did this because we believe that the public has a right to the broadest possible access to political speech, even controversial speech. But the current context is now fundamentally different, involving use of our platform to incite violent insurrection against a democratically elected government.
We believe the risks of allowing the President to continue to use our service during this period are simply too great. Therefore, we are extending the block we have placed on his Facebook and Instagram accounts indefinitely and for at least the next two weeks until the peaceful transition of power is complete."
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text:
Peter Fisher (born Péter Fischer; July 1929 – 18 March 2013) was a Hungarian-born cinematographer, photojournalist. He died on 18 March 2013 at the age of 83. "One of the best-known photographers of the Jewish community." He was the most famous for photographs of the actress Jayne Mansfield, "whose visit to Peter's studio stopped traffic in the local streets as fans craned to see her."

Life

He was "born in Budapest into a traditional, rather than observant, Jewish family."
His father was engineer Pál Fischer (1892–1945), his mother was Lola Bader (1905–1973).

His first wife was the Hungarian actress Flóra Kádár (1928–2003) but they were divorced. Secondly he married Blandine Costi and thirdly Carol Joernet.
His younger brother is the pianist and conductor György Fischer. György Fischer's first wife was opera singer Lucia Popp. After being divorced from Popp he married American violinist Ida Bieler.

His cousins are conductors Ádám Fischer and Iván Fischer.

He studied cinematography at the Academy of Drama and Film in Budapest in the 1950s when he got acquainted with his first wife, Flóra Kádár who took degree in 1953 as actress. Because of the fall of the Hungarian Revolution of 1956, he left Hungary for Great Britain but "he was unable to resume his profession without a union card, so moved instead into stills photography."

References

Sources

External links

1929 births
2013 deaths
Hungarian cinematographers
Hungarian photojournalists
British photojournalists
Photographers from London
Photographers from Budapest
Film people from Budapest

redpajama_set_name: RedPajamaWikipedia


text:
Gov. Haslam visits Wilson County; South Mt. Juliet Road and State Route 109 North discussed as possible future major roadway projects under IMPROVE Act
February 14, 2017 Staff Reports Frontpage Lead Stories, General News 0
Governor Bill Haslam spoke at a town hall meeting in Wilson County last Thursday evening, in which State Representative Susan Lynn organized, regarding the proposed IMPROVE Act. The Governor discussed county tax changes and the proposed IMPROVE Act. Photo by Parker Minor.
Tennessee Governor, Bill Haslam, made a special stop in Wilson County last Thursday evening for a town hall meeting, which was organized by State Representative, Susan Lynn, regarding the proposed IMPROVE Act.
Rep. Lynn began the meeting by informing Wilson County citizens of the current funding and taxes which led into talk of the current surplus budget.
She continued breaking down the IMPROVE Act and what it would do for the state and for Wilson County throughout the next eight years if introduced.
"A total of 962 projects throughout the state would begin, including 10 Wilson County road projects," said Lynn.
Local residents and citizens in attendance of the town hall meeting were concerned of the possible outcome of the tax increase. Lynn stated that the last gas tax increase took place in 1989 in the state of Tennessee. Since that point of time, 30 years have passed by and inflation has taken place causing value and expenses of such projects to drastically change and costs twice as much to repair and maintain such roadways.
The Act could possibly also institute a new open container law into the state of Tennessee," said Lynn. "The proposed Act could possibly take away a large penalty that has to be paid by the state in regards for not having such a law."
The Hall income tax is required to be done away with by the year 2022. The proposed act would quicken up the process of extinguishing the income tax.
Gov. Haslam appeared an hour into the meeting for the meeting at the Wilson County Courthouse after being out of state for other business in which he had flown in from directly.
Haslam explained to Wilson County residents the meaning of the IMPROVE Act.
"The new proposed Act would mean tax cuts and a gas tax increase, while also addressing transportation funding concerns and issues," said Haslam. "We're only trying to keep up with our roads, and it costs twice as much when we only have half as much money."
The results of the IMPROVE Act would be an increase in the road user fee or gas tax by 0.07 cents per gallon of gas. For a gallon of diesel fuel, an increased amount of 0.12 cents would be experienced by fleet drivers.
An increases of $5.00 for car registration fees for the average passenger vehicle would be implemented. This would bring in $278 million dollars in new funds to support backlogged transportation projects.
Balancing out the increase in the other discussed taxes would be major tax cuts on food and manufacturing in the county.
If passed, the IMPROVE Act would most likely include State Route 109, a popular debated roadway with over decades of requests from everyday travelers to widen. The project would begin at the intersection of Highway 70 (Lebanon Road) and State Route 109 and would be projected to end at the Wilson County-line, where the new four-lane bridge overpass into Sumner County begins.
The project would be an estimated 7.5 miles of road work and widening that would cost an estimated 18.5 million to complete.
South Mt. Juliet Road is another main road in Wilson County that state and local government have studied.
"When Interstate-40 or I-24 gets congested or backed up with standstill traffic, drivers automatically bail to Hobson Pike or South Mt. Juliet where most of all the traffic jams up during these situations," explained Rep. Lynn. "The project would cost somewhere around $25.4 million to construct the widening between Central Pike and Providence Way on North Mt. Juliet Road."
Other mentioned Wilson County projects would possibly include two I-40 projects in which would decrease traffic congestion along Interstate 840 to Highway 70. The other project would include the areas between State Route 109 and 840. The estimated total for both projects would be expected somewhere around $94 million.
Another Mt. Juliet congested area on Highway 70 that was mentioned was the intersection of Park Glen Drive and Bender's Ferry Road. Between schools and travelers heading to work during the morning hours and daytime workers making their way home around the same time as dinner, the Mt. Juliet area is known for congested traffic most mornings and afternoons.
The intersection of Central Pike from Old Hickory Boulevard, outside of Mt. Juliet in Hermitage, was also discussed, as well as a new possible interchange at I-40 and Central Pike. The interchange at I-40 and Central Pike is listed under the interstate program and estimated at a total of $14.2 million in construction.


text:
Golden Age of Civilizations
No Eiffel Tower
Trade routes disapearing
Huts don´t disapear
Just a big + for that idea. 1 year 2 days ago
I have a question about 1 year 2 weeks ago
Talked to dev. Its all good 1 year 1 month ago
Soo is it a waste to build 1 year 2 months ago
Same problem here. Mine says 1 year 2 months ago
This has become normal for 1 year 2 months ago
There seems to be a problem 1 year 2 months ago
It is not that simple. Many 1 year 3 months ago
You can occupy a city with a 1 year 4 months ago
There is on our to do list. : 1 year 6 months ago
Home › Overview
6. Buildings
Tue, 05/17/2016 - 10:51 — Gábor Auth
Every building modifies the behaviour of your units and cities.
Several schemes are possible for organizing the variety of buildings available in Golden Age of Civilizations. This table arranges them chronologically: each of the eras in the following table groups units which require similar numbers of advances before they can be built.
FB: Food bonus
PB: Production bonus
LB: Luxury bonus
SB: Science bonus
TB: Tax bonus
RC: Reduce corruption
DB: Defense bonus
PL: Population limit
UT: Affected unit type
Pr: Production cost
Pu: Purchase cost
Su: Support cost
Era Name FB PB LB SB TB RC DB PL UT Pr Pu Su
Barracks 0 0 0 0 0 0 50 0 L 32 20 1
Temple 0 0 50 0 0 0 0 0 32 20 1
City walls 0 0 0 0 0 0 200 0 128 80 0
Palace 0 50 50 50 50 0 0 0 128 80 0
Granary 50 0 0 0 0 0 0 0 32 20 1
Courthouse 0 0 0 0 0 75 0 0 32 20 1
Marketplace 0 0 50 0 50 0 0 0 64 40 0
Library 0 0 0 200 0 0 0 0 128 80 4
Aqueduct 200 0 0 0 0 0 0 10 128 80 0
Colosseum 0 0 200 0 0 0 0 0 128 80 4
Harbor 200 0 0 0 0 0 0 0 256 160 8
Cathedral 0 0 400 0 0 0 0 0 512 320 16
University 0 0 0 800 0 0 0 0 1024 640 32
Bank 0 0 200 0 200 0 0 0 512 320 16
Barracks II 0 0 0 0 0 0 400 0 LAND 1024 640 32
Coastal defense 0 0 0 0 0 0 400 0 SEA 1024 640 32
Sewer system 400 0 0 0 0 0 0 20 1024 640 32
Stock exchange 0 0 400 0 400 0 0 0 2048 1280 64
Factory 0 800 0 0 0 0 0 0 4096 2560 128
Supermarket 400 0 0 0 400 0 0 0 4096 2560 128
Police station 0 0 0 0 0 75 0 0 4096 2560 128
Port facility 0 0 0 0 0 0 800 0 SEA 8192 5120 256
Power plant 0 800 0 0 0 0 0 0 8192 5120 256
Hydro plant 0 800 0 0 0 0 0 0 8192 5120 256
Super highways 0 0 0 0 800 0 0 0 16384 10240 512
Offshore platform 0 1600 0 0 0 0 0 0 32768 20480 1024
Airport 0 0 0 0 0 0 800 0 AIR 16384 10240 512
Barracks III 0 0 0 0 0 0 800 0 LAND 16384 10240 512
Mass tranzit 0 0 0 0 0 0 0 0 16384 10240 512
Recycling center 0 0 0 0 0 0 0 0 16384 10240 512
Research laboratory 0 0 0 1600 0 0 0 0 32768 20480 1024
SAM battery 0 0 0 0 0 0 800 0 16384 10240 512
Nuclear plant 0 800 0 0 0 0 0 0 16384 10240 512
Manufacturing plant 0 1600 0 0 0 0 0 0 32768 20480 1024
SDI defense 0 0 0 0 0 0 800 0 16384 10240 512
Solar plant 0 1600 0 0 0 0 0 0 65536 40960 2048
Space structural 0 0 0 0 0 0 0 0 65536 40960 2048
Space component 0 0 0 0 0 0 0 0 131072 81920 4096
Space module 0 0 0 0 0 0 0 0 262144 163840 8192
(Content is available under the GNU General Public License).
‹ 5. Economy up 7. Wonders ›
Wed, 06/15/2016 - 04:54 — Darth Maul
What pts are given for mass
What pts are given for mass transit and recycle plant
Wed, 06/15/2016 - 15:32 — Gábor Auth
At the moment: nothing, they
At the moment: nothing, they will be decrease the pollution.
Ok ty
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text:
A fine selection of independent media sources
Newsletters/RSS
Can Post Office Be Bank? New Services Test Progressive Priority…
October 6, 2021 1:53 pm by IWB
The U.S. Postal Service has quietly begun offering a handful of new or expanded financial services in four cities, a potential first step toward a return to postal banking, which advocates say could help rescue the agency's finances and assist millions of people who have limited or no access to the banking system.
Tatiana Roy, a spokesperson for the Postal Service, said in an email that the pilot program — a collaboration between the Postal Service and the American Postal Workers Union — began Sept. 13 and that it aligns with the goals set out in the 10-year plan the Postal Service announced in May.
Postal banking was not explicitly called for in the plan, which Roy said would help the agency "achieve financial sustainability and service excellence," but it is a longtime desire of progressive politicians and advocates whose attempts to push it through Congress in recent years have been met with little success. It would require an act of Congress to re-establish postal banking beyond the limited services the Postal Service is beginning to test, but the pilot program could act as a proof of concept.
See also Legal weed is wreaking havoc on return to office: 'all of NY is wake and baking'
www.nbcnews.com/politics/politics-news/return-postal-banking-postal-service-tests-new-financial-services-rcna2502
"You would have a far-Right government in every country in Europe"
Another domino is starting to fall…
US Now Lone Suspect in Nord Stream Pipe Bombing
Fact checkers are quick on damage control for Project Veritas video. He was literally talking about directed evolution of COVID-19. How in the hell is that not clear enough for you?
The Food Supply is Contaminated With SPIKE PROTEINS
The Covid Op Was the Catalyst That Propelled the Entire World Into a Soon Fully Digitized Surveillance State
This is EXACTLY how they will take away the CASH. So easy it's RIDICULOUS.
The Russians are Going to Break Through Bakhmut
Woman "forgives" her mothers killer, visits him in jail. Upon release woman gives him a job at the decease mothers house and is quickly killed too.
See also Office janitor urinated in employee water bottles - 13 women tested positive for STD
Tags office, priority, progressive, services Post navigation
AMAZON Turns Negative for 2021…
Canada: Even those working from home must be vaccinated
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text:
Q: Let $G$ be a finite group of even order. Show that $\exists a\neq e, a=a^{-1}.$
Let $G$ be  a finite group of even order. Show that $\exists a\neq e, a=a^{-1}.$

Suppose $\forall a\neq e,a\neq a^{-1}$ and define an equivalence reation on $G$ by $a\sim b$ if and only if $a=b$ or $a=b^{-1}$. Let $S=\{[a]:a\in G\}$ be the set of all equivalence classes. If $a=e, [e]=\{e\}$, and if $a\neq e, [a]=\{a,a^{-1}\}$ with $a\neq a^{-1}$. Then $G=\{e\}\cup(\bigcup\limits_{[a]\in S,[a]\neq [e]}\{a,a^{-1}\})$ and $|G|=2n+1$, a contradiction.
I am wondering if this proof is correct.

A: Yes, it is correct. But you should say that the union is disjoint because it corresponds to the equivalence classes of your equivalence relation.

A: Your proof is fine.
Here's an alternative:
Since $\lvert G\rvert$ is even, it is divisible by two. Two is prime. Therefore, by Cauchy's theorem, there exists an $H\le G$ such that $\lvert H\rvert=2.$ Let $h\in H$ such that $h\neq e$. Then, by closure, $h^{-1}=h\in H\subseteq G$.

redpajama_set_name: RedPajamaStackExchange


Home > Safety & Environmental Help and Guidance > Noise at Work Assessments > Time Weightings – What are they and why are they used?
Time Weightings – What are they and why are they used?
It is important to understand the Lp is an RMS (Root Mean Squared) quantity and is therefore averaged within the constraints of an averaging time constant. These time constants are helpfully termed Slow, Fast and Impulse and are sometimes referred to as the time weighting.
The Lp therefore is not only affected by the frequency response weighting but by the time weighting of the instrument as well. Hence, by selecting Slow on the sound level meter a sharp rise in noise will be shown as a gradual rise in the Lp, alternatively a sharp fall in noise will be shown as a gradual decrease in the Lp.
By using the Fast weighting on the instrument in both of the above scenarios the Lp will rise and fall on a more rapid scale.
The Impulse response is not as commonly used as the above, it is used in situations of sharp impulsive noises, typically a piling operation may require an impulse time weighting. It has a very fast rise time and a very slow exponential fall time.
Peak responses are different from Slow, Fast and Impulse weightings, in that it is not an RMS quantity. It is simply the crest of the sound pressure level and it is shown as the highest peak achieved within the measuring period. The rise time of the peak is extremely sharp and it is not uncommon for the peak to be much higher than the Lp.


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Around 750,000 people live in this part of North Holland.
Your entire family will have an absolute blast at the Madame Tussauds and the Eye Film Institute. Are the little ones still not worn out? Keep the adventure going with a trip to Zuiderbad and theater Tuschinski.
The little ones will be filled with awe on an excursion to the Artis. They'll get an up-close look at how its animals behave while learning about what it takes to look after them.
Gather your travel buddies and head on over to the Heineken Experience for a round or two. Their tap range is deliciously tempting, so make sure you free up the whole afternoon!
The Holland Casino is the place to go for high-stakes gambling and a sparkling atmosphere. Place a bet or two, have a beer and take in the sights and sounds.
Amsterdam has some gorgeous green spaces where you can find a wide range of flower and plant species. The Hortus Botanicus and the Historische Tuin Aalsmeer Horticultural Museum are where you want to head.
From a friendly ball game to simply relaxing on the grass, there are lots of reasons to head to a green oasis like Begijnhof or Museum Square. There's also lots of fun to be had at Amstelpark and Spaarnwoude.
Whether you're looking for tours or tickets to an attraction, Tourist information Center Monnickendam is the place to go. Its friendly team of local staff will make sure you get the most out of your visit to Amsterdam.
If you're a big believer in the rejuvenating effects that being close to the water can bring, you're in the right place. Het Ij and Prinsengracht are just a few of the peaceful natural waterbodies around here that you should add to your list. Don't forget to go and see the vibrant waterbirds at Singel and Emperor's Canal either.
Travellers that love the outdoors will have plenty to include in their itinerary. A day trip to Volendam Harbour and Sloterpark and Sloterplas is sure to recharge those batteries. Want to have a few options on hand? Western Islands is also worth stopping by.
If you're in need of a few quiet places to collect yourself and maybe discover a thing or two, you can't go past the Anne Frank House and the Rijksmuseum. Ask about walking tours or just study the interesting displays at your own pace. Want more? Also pencil in visits to the Van Gogh Museum and the Stedelijk Museum.
Boats have played a large role in the past of many countries. See for yourself how the region was shaped by these courageous sailors at the National Maritime Museum. For even more interesting stories and artefacts, continue your voyage to the Het Scheepvaartmuseum National Maritime Museum and the Nederlands Scheepvaartmuseum.
Satisfy your inquisitive mind with a visit to the Nemo Science Museum. While there, don't forget to ask about any travelling exhibitions. Believe us, you won't want to miss a thing!
As everybody knows, our world is an unbelievable place. Nowhere is this more apparent than in the jam-packed display cabinets at the Tropenmuseum and the Allard Pierson Museum.
In a place that's brimming with excellent art exhibitions, the Hermitage Amsterdam and the Rembrandt House Museum somehow manage to stand out from the bunch. The Cobra Museum of Modern Art and the Electric Ladyland shouldn't be skipped either.
Calling all history fans! You can explore the rich and captivating past of this area at the Amsterdam Museum and the Dutch Resistance Museum. For an even deeper peek into times gone by, also check out the Jewish Historical Museum and the Zaans Museum.
To properly get a sense of the history of Amsterdam, take your time and experience Zaanse Schans and the University of Amsterdam. Why not also check out Munttoren and the VU University of Amsterdam?
Skinny Bridge and Blauwbrug are enduring crossings that should intrigue visitors that have an interest in civil engineering. Torensluis offers a way to take in the waters of Amsterdam from a third angle.
If you find it difficult keeping up with the Joneses, consider for a second the neighbours of Royal Palace. The striking façade and lavish furnishings of this famed residence would be near impossible to top!
The watchful turrets and impenetrable stone walls of Muiderslot were purpose-built to help guard its inhabitants from enemy invasion. This eye-opening architecture alone makes the trip worthwhile, not to mention the captivating history that lies within.
Whatever your beliefs, you're sure to enjoy the tranquil atmosphere at New Church and The Old Church. In addition, enquire into touring Westerkerk and Zuiderkerk.
Don't grab your passport just yet, there's still lots to do! Have you paid a visit to the Public Library and Kalverstraat yet? You'll also find plenty to keep you busy at P.C. Hooftstraat and Amsterdam Chinatown.
Lovely, friendly, helpful staff. Great bed. Very good room. Right in the middle of town.
Excellent location with unique boutique feel, staff there are always professional, friendly and willing to help. The 24/7 drink and snack on offer at the lobby is a bonus. Although the room size is slightly on the small side, everything else makes up for it. Will highly recommended.


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Virginia Tech, North Carolina Gobbler Country Staff Picks
What does the G.C. crew think about the game in Chapel Hill?
By Roy Hatfield@roybhatfield Oct 7, 2016, 11:12pm EDT
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Roy Hatfield (4-0)
The Hokies have a real test with North Carolina. I think the difference maker will be RB Elijah Hood for the 'Heels. I'm not so sure our defense can stop this great back. I do think we will score some points, but the running game thus far hasn't impressed me. With the weather, passing might be a challenge. It pains me to say this, but UNC 29 Virginia Tech 23.
Jawhar Ali (4-0)
The Hokies' defense will face their biggest test of the season against UNC's offense - slowing down a team that has scored 37 points in back-to-back games is no easy task, even for Bud Foster's defense. But I think the real test might be for the offense. If the elements are what they're projected to be, then this game will most likely come down to whoever can run the ball between the tackles better. The Tar Heels' are underwhelming defensively, but the Hokies haven't shown the ability to run the ball consistently up the middle outside of QB draws. On the other side, Elijah Hood might be the best back in the ACC between the tackles and the UNC interior OL has shown the ability to generate consistent push. I hate to say it, but I think UNC comes out on top in a sloppy football game, 24-23.
Joshua Schneider (4-0)
Look, I have NO IDEA what's going to go on in this game. The weather factor mucks it up (literally and figuratively) that I'm having a hard time trying to figure out the narrative. Elijah Hood is questionable and the UNC Tarheels haven't faced a defense yet anywhere close to the Hokies', but the Hokies might be stuck with their inefficient ground game while being completely stuck in the mud. I don't think that this will be a high scoring contest, and it'll just be a slog to watch. Whoever wins won't do it by much, but I'm going to say UNC 28 Hokies 27.
John Schneider (3-1)
The irony of this football game is that the Carolina defense has to stop the Hokie Offense. We've been talking about the other squads, but the game is going to hinge on Tech's O winning the game. Driving the ball and controlling the clock is key. The best defense against Trubisky and company is keeping them off the field.
31 - 24 Hokies
Mohamed Hassan (3-1)
Slowly, but surely the Virginia Tech team that used to win with great defense and special teams is returning to Blacksburg. The difference is that this time, they have a revamped offense to compliment it. The Hokies will look to make a statement against the Tarheels that they are back and this is their division to win. Hokies win 34-23.
Bryan Manning (3-1)
The Hokies returned to the top 25 again this week and have a big test on their hands with a trip to No. 17 North Carolina. Through four games, the Hokies are clicking on all cylinders. Offense, defense and special teams are all contributing.
UNC runs a fast-paced spread, similar to Tech. Think VT defensive coordinator Bud Foster has taken plenty of notes in practice? Sure he has and that's one reason Foster's defense is as good as it's been in a while.
This game will be close and come down to the final minutes. This time, the Hokies will be on the good end and move to 2-0 in ACC play.
Prediction: Virginia Tech 31, North Carolina 24
Jay Johnson (3-1)
The likelihood of rain gives me pause, because it will likely dampen (haha!) the high octane Fuente offense we have seen showcased over the last two games. The running game seemed to stutter a bit against ECU, but in previous games the running attack has benefited from Williams' and McMillian's skillsets. Not too mention the fireworks that Evans showed he was capable of last week. I expect the offensive plan to be built around the rushing attack of Evans, McMillian, and Williams, while hoping to open up some downfield opportunities. The Hokies defense and special teams will take center stage in Chapel Hill. VPI must slow down Trubisky in order to stay in the game. UNC is a dangerous team and still the lead squad in the Coastal. Virginia Tech has a fantastic opportunity to signal their return to the ACC contenders, and I think they do it in a close slugfest. Hokies over the Tar Heels, 27-24.
Virginia Tech @ North Carolina GameCenter
Virginia Tech 34 North Carolina 3
Scouting the Tar Heels' Offense