text:
\section{Introduction}

Let $ \fld $ be an algebraically closed field, let $ \mathcal{C} $ be a modular tensor category over $ \fld $ together with a complete set of simples $ \mathcal{I} $ and let $ \tid \in \mathcal{I} $ denote the tensor identity. The \emph{modular data} of $ \mathcal{C} $ is composed of two $ \mathcal{I} \times \mathcal{I} $-matrices known as the S-matrix and the T-matrix respectively. These matrices satisfy the following equations,
	%
	\begin{align*}
	%
	(\mathcal{S} \mathcal{T})^3 = \lambda \mathcal{S}^2, \ \mathcal{S}^2 = d(\mathcal{C}) C, \ \mathcal{T} C = C \mathcal{T}
	%
	\end{align*}
	%
where $ \lambda \in \fld $, $ d(\mathcal{C}) \in \fld $ is the \emph{dimension}~\cite[Definition 7.21.3]{Etingof15} of $ \mathcal{C} $ and $ C \vcentcolon= (\delta_{I^\vee,J})_{I,J \in \mathcal{I}} $ is the \emph{charge conjugation matrix}~\cite[Section 8.16]{Etingof15}. As $ C^2 = \id $ these equations imply that the modular data gives a projective representation of $ \SLn{2}(\mathbb{Z} $) a.k.a. the \emph{modular} group.

Part of the motivating force behind the study of modular tensor categories comes from their role in the mathematics of \emph{conformal field theory}. In particular, the category of modules over the vertex operator algebra of chiral symmetries naturally carries the structure of an modular tensor category. Let $ P $ be the partition function of the conformal field theory and let $ Z $ be the non-negative integer matrix describing the irreducible multiplicities of $ P $, i.e.\ $ P = \sum_{IJ} Z_{IJ}\chi_I \chi_J $ where the sum ranges over irreducible modules and $ \chi_I $ denotes the character of $ I $. Within this context the condition of conformal invariance for $ P $ may be rewritten as $ [Z,\mathcal{S}] = [Z,\mathcal{T}] = 0 $. This motivates the following definition.

	\begin{DEF} \label{def:modular_invariant}

	For a modular tensor category with tensor identity $ \tid $ and complete set of simples $ \mathcal{I} $, a \emph{modular invariant matrix} is a non-negative integer $ \mathcal{I} \times \mathcal{I} $-matrix that commutes with the modular data and whose $ (\tid,\tid) $-entry is $ 1 $\footnote{The condition that the $ (\tid,\tid) $-entry be $ 1 $ is also physically motivated, see~\cite[Section 5]{MR2164398}.}.

	\end{DEF}

The tube category, denoted $ \mathcal{TC} $, is a category whose objects coincide with those of $ \mathcal{C} $ and whose morphism are described by diagrams in $ \mathcal{C} $ inscribed upon the surface of a cylinder (see~\cite{hardiman_king} for a precise definition). Although $ \mathcal{TC} $ is not semisimple it may be though of as `semisimple in spirit' in the sense that its idempotent completion is semisimple. Indeed, for $ X $ and $ Y $ in $ \mathcal{C} $ we may consider the morphism $ \epsilon_X^Y \in \End_{\mathcal{TC}}(IJ) $ given by~\ref{eq:idempotents}; in~\cite{hardiman2020graphical} it is shown that $ \epsilon_X^Y $ is a idempotent and that the set $ \{\epsilon_I^J\}_{I,J \in \mathcal{I}} $ forms a complete set of orthogonal, primitive idempotents in $ \mathcal{TC} $. The simple object corresponding to $ \epsilon_I^J $ in the idempotent completion of $ \mathcal{TC} $ is denoted $ \eo{I}{J} $.

Let $ \Vect $ denote the category of finite dimensional vector spaces and let $ \mathcal{RTC} $ denote the category of contravariant functors from $ \mathcal{TC} $ to $ \Vect $. We call an object in $ \mathcal{RTC} $ a \emph{representation} of $ \mathcal{TC} $. As $ \mathcal{TC} $ admits a complete set of orthogonal, primitive idempotents the Yoneda embedding $ \Yon \colon \mathcal{TC} \to \mathcal{RTC} $ gives an alternative\footnote{Alternative, yet equivalent by \cite[Section 5.1.4]{Lurie09}.} idempotent completion of $ \mathcal{TC} $~\cite[Proposition 2.10]{hardiman2020graphical}. We use $ \eo{I}{J}^\sharp $ to denote the irreducible representation of $ \mathcal{TC} $ corresponding to the simple object $ \eo{I}{J} $, and, for $ F $ in $ \mathcal{RTC} $, we use $ F_I^J $ to denote the multiplicity space of $ \eo{I}{J}^\sharp $ in $ F $, i.e.
	%
	\begin{align*}
	%
	F_I^J \vcentcolon= \Hom_{\mathcal{RTC}}(\eo{I}{J}^\sharp,F).
	%
	\end{align*}
	%
Therefore, the isomorphism classes of objects in $ \mathcal{RTC} $ are in bijection with non-negative integer $ \mathcal{I} \times \mathcal{I} $-matrices. For $ F $ in $ \mathcal{RTC} $ we write $ Z(F) $ to denote the non-negative integer matrix corresponding to $ F $'s isomorphism class, i.e.\ $ Z(F)_{IJ} = \dim F_I^J $.

This naturally leads to the question of how to characterize objects in $ \mathcal{RTC} $ whose isomorphism class is a modular invariant matrix. We call such an object a \emph{modular invariant representation}. Furthermore, an object $ F $ in $ \mathcal{RTC} $ is called \emph{T-invariant} if it satisfies $ [Z(F),\mathcal{T}] = 0 $ and is called \emph{S-invariant} if it satisfies $ [Z(F),\mathcal{S}] = 0 $. The main result of this paper provides a characterization of modular invariant representations in terms of the graphical calculus of the tube category:

    \newtheorem*{cor:main_result}{Corollary \ref{cor:main_result}}
    \begin{cor:main_result}
    Let $ F $ be in $ \mathcal{RTC} $. Then $ F $ is modular invariant if and only if
        %
        \begin{align}
        %
         \Tr_{F,X}^{\mathcal{T}} = \Tr_{F,X} = \Tr_{F,X}^{\mathcal{S}} \label{eq:trace_equality}
        %
        \end{align}
        %
    for all $ X $ in $ \mathcal{TC} $.
    \end{cor:main_result}

Here $ \Tr_{F,X},\Tr_{F,X}^{\mathcal{T}},\Tr_{F,X}^{\mathcal{S}} \in \End_{TC}(X)^* $ are given by
$ \Tr_{F,X} \colon \alpha \mapsto \tr F(\alpha) $, \eqref{eq:t_twisted_trace} and \eqref{eq:s_twisted_trace} respectively. Alternatively, the values of $ \Tr_{F,X}^{\mathcal{T}} $ and $ \Tr_{F,X}^{\mathcal{S}} $ on $ \alpha \in \End(X) $ are given by taking $ \Tr_{F,X} $ and pre-composing it with an action of the standard generators of the mapping class group of the torus on the graphical description of $ \alpha $. Consequences of this interpreation are discussed in the concluding remarks found in Section~\ref{sec:conclusion}.

The structure of this paper is as follows: Section~\ref{sec:prilim} briefly introduces certain preliminaries on the graphical calculus of modular tensor categories and their associated tube categories. Section~\ref{sec:graph_char_t_inv} proves Corollary~\ref{cor:t_invariance_reformed} which states that the first equality in \eqref{eq:trace_equality} is equivalent to $ F $ being T-invariant. This section is fairly short as the corollary is a reformulation of a previous result from~\cite{hardiman2019extending}. Section~\ref{sec:graph_char_s_inv} proves Theorem~\ref{thm:s_invariance_reformulated} which states that the second equality in \eqref{eq:trace_equality} is equivalent to $ F $ being S-invariant. Finally, Section~\ref{sec:conclusion} combines these two results into Corollary~\ref{cor:main_result} and provides some concluding remarks.

\section{Preliminaries on Graphical Calculus} \label{sec:prilim}

For an overview of graphical calculus in modular tensor categories and their associated tube categories see~\cite[Sections 3-5]{hardiman2020graphical}. We now recall certain specific facts which will be used throughout this paper.

Firstly, we recall that the modular data of a modular tensor category $ \mathcal{C} $ admits the following graphical description,
   %
   \begin{align} \label{eq:S_and_T_matrices}
   %
   \mathcal{T}_{IJ} \vcentcolon= \delta_{I,J}
       \begin{array}{c}
           \begin{tikzpicture}[scale = 0.25]
           \draw[thick] (-1,1) -- (-1,3);
           \draw[thick] (-1,-1) -- (-1,-3);
           \draw[thick] (1,1) to[out = 90, in = 90] (3,1);
           \draw[thick] (1,-1) to[out = -90, in = -90] (3,-1);
           \draw[thick] (3,1) -- (3,-1);

           \draw[thick] (1,1) to[out=-90,in=90] (-1,-1);
           \draw[line width = 0.3cm, white] (-1, 1) to[out=-90,in=90] (1,-1);
           \draw[thick] (-1, 1) to[out=-90,in=90] (1,-1);

           \node at (-1, 4.3) {$ I $};
           \end{tikzpicture}
       \end{array}
   \quad
   \mathcal{S}_{IJ} \vcentcolon=
       \begin{array}{c}
           \begin{tikzpicture}[scale=0.5]
           \draw [thick](-1.5,0) arc (180:0:1);
           \draw[white,line width = 0.3cm] (0.5,0) circle (1);
           \draw[thick] (0.5,0) circle (1);
           \node at (-2,1) {$ I $};
           \node at (2,1) {$ J $};
           \draw [white,line width = 0.3cm](.5,0) arc (0:-180:1);
           \draw [thick](.5,0) arc (0:-180:1);
           \end{tikzpicture}
       \end{array}
   %
   \end{align}
   %
where we have exploited the fact that $ \End(I) = \fld $ for all $ I \in \mathcal{I} $. Secondly, for $ X $ and $ Y $ in $ \mathcal{C} $ we consider the following endomorphism,

	%
	\begin{align} \label{eq:idempotents}
	%
		\epsilon_X^Y = \frac{1}{d(\mathcal{C})} \bigoplus\limits_S d(S)
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.15,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0,5) {};
				\node (v4) at (0,-5) {};
				\node (v2) at (5,0) {};
				\node (v3) at (-5,0) {};

				\node (v5) at (-2.5,2.5) {};
				\node (v6) at (2.5,-2.5) {};
				\node (v7) at (1.5,3.5) {};
				\node (v11) at (3.5,1.5) {};
				\node (v12) at (-1.5,-3.5) {};
				\node (v10) at (-3.5,-1.5) {};
				\node (v13) at (2.5,-2.5) {};

				\draw [thick] (v7) edge (v10);
				\draw [line width =0.5em,white] (v5) edge (v13);
				\draw [thick] (v5) edge (v13);
				\draw [line width =0.5em,white] (v11) edge (v12);
				\draw [thick] (v11) edge (v12);
				\node at (2.75,4.75) {$ X $};
				\node at (4.75,2.75) {$ Y $};
				\node at (-3.75,3.75) {$ S $};
				\node at (3.75,-3.5) {$ S $};

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array}
		\in \End_{\mathcal{TC}}(XY).
	%
	\end{align}
	%

Then (as mentioned in the introduction) $ \epsilon_X^Y $ is a idempotent and that the set $ \{\epsilon_I^J\}_{I,J \in \mathcal{I}} $ forms a complete set of orthogonal, primitive idempotents in $ \mathcal{TC} $. We use $ \eo{I}{J} $ to denote the corresponding simple object in the idempotent completion of $ \mathcal{TC} $.

\section{Graphical Characterization of $ \mathcal{T} $-invariance} \label{sec:graph_char_t_inv}
As before, let $ \fld $ be an algebraically closed field, let $ \mathcal{C} $ be a modular tensor category over $ \fld $ and let $ F $ be an object in $ \mathcal{RTC} $, i.e.\ $ F \colon \mathcal{C}^{\text{\normalfont op}} \to \Vect $. We recall that $ F $ is called T-invariant if $ [Z(F),\mathcal{T}]=0 $. T-invariance was studied in~\cite[Section 4]{hardiman2019extending} where a graphical characterization was given in terms of the following automorphism in $ \mathcal{TC} $,
	%
	\begin{align} \label{eq:twist}
	%
	t_X \vcentcolon=
		\begin{array}{c}
			\begin{tikzpicture}[scale=0.15,every node/.style={inner sep=0,outer sep=-1}]

		 	\node (v1) at (0.5,4.5) {};
		 	\node (v2) at (4.5,0.5) {};
		 	\node (v3) at (-4.5,-0.5) {};
		 	\node (v4) at (-0.5,-4.5) {};

			\node (v6) at (-2,2) {};
			\node (v7) at (2.5,2.5) {};
			\node (v8) at (-2.5,-2.5) {};
			\node (v9) at (2,-2) {};

		 	\node at (4,4) {$ X $};
		 	\node at (-4,-4) {$ X $};
		 	\node at (3.5,-3.5) {$ X^\vee $};
		 	\node at (-3.5,3.5) {$ X^\vee $};

			\draw[thick] (v6) to[out=-45,in=-135] (v7);
			\draw[thick] (v9) to[out=135,in=45] (v8);

		 	\draw[very thick, red]  (v1) edge (v3);
		 	\draw[very thick, red]  (v2) edge (v4);
		 	\draw[very thick]  (v1) edge (v2);
		 	\draw[very thick]  (v3) edge (v4);
		 	\end{tikzpicture}
		\end{array}.
	%
	\end{align}
	%
	\begin{THM}[Theorem 4.4 in \cite{hardiman2019extending}] \label{thm:t_invariance}

	Let $ \mathcal{C} $ be a modular tensor category and let $ F $ be an object in $ \mathcal{RTC} $. $ F $ is T-invariant if and only if $ F(t_X) = \id_{F(X)} $ for all $ X $ in $ \mathcal{C} $.

	\end{THM}

The goal of this section is to rewrite the statement of this theorem into an equivalent statement. Although this reformulation may initially appear fairly arbitrary it illustrates the analogy with Theorem~\ref{thm:s_invariance}, our main result on S-invariance.

For $ X $ in $ \mathcal{TC} $ and $ F $ in $ \mathcal{RTC} $ we consider two elements in $ \End_{\mathcal{TC}}(X)^* $. One is simply $ \Tr_{F,X} \colon \alpha \mapsto \tr F(\alpha) $ and the other is given by
    %
    \begin{align} \label{eq:t_twisted_trace}
    %
    \Tr_{F,X}^{\mathcal{T}} \colon \alpha \mapsto \tr F(\alpha \circ t_X).
    %
    \end{align}
    %
We note that $ \Tr_{F,X}^{\mathcal{T}} $ does not depend on whether or not we compose $ t_X $ on the right or left by~\cite[Lemma 4.2]{hardiman2019extending}.

	\begin{COR} \label{cor:t_invariance_reformed}

	Let $ \mathcal{C} $ be a modular tensor category and let $ F $ be an object in $ \mathcal{RTC} $. $ F $ is T-invariant if and only if $ \Tr_{F,X}^{\mathcal{T}} = \Tr_{F,X} $ for all $ X $ in $ \mathcal{C} $.

	\proof

 	By~\cite[Proposition 4.3]{hardiman2019extending}, $ F(\epsilon_I^J \circ t_{IJ}) = \frac{\mathcal{T}_{II}}{\mathcal{T}_{JJ}} F(\epsilon_I^J) $. Therefore requiring that $ \tr F(\epsilon_I^J \circ t_{IJ}) = \frac{\mathcal{T}_{II}}{\mathcal{T}_{JJ}} \tr F(\epsilon_I^J) = \frac{\mathcal{T}_{II}}{\mathcal{T}_{JJ}} \dim F_I^J = \dim F_I^J = \tr F (\epsilon_I^J) $ is equivalent to requiring that $ \dim F_I^J \neq 0 $ implies $ \frac{\mathcal{T}_{II}}{\mathcal{T}_{JJ}} = 1 $. As $ \mathcal{T} $ is diagonal, this condition (for all $ I,J \in \mathcal{I} $) is precisely the condition that $ Z(F) $ commutes with $ \mathcal{T} $, i.e.\ that $ F $ is T-invariant.

 	We therefore immediately have that $ \Tr_{F,X}^{\mathcal{T}} = \Tr_{F,X} $ implies T-invariance. We now suppose that $ F $ is T-invariant. Then, by Theorem~\ref{thm:t_invariance}, $ F(t_X) = \id_{F(X)} $ and we have $ \Ttwist{\alpha} = \tr F(\alpha \circ t_X) = \tr F(\alpha) = \Tr_{F,X} (\alpha). $

	\endproof

	\end{COR}

\section{Graphical Characterization of $ \mathcal{S} $-invariance} \label{sec:graph_char_s_inv}

The aim of this section is to provide analogues of Theorem~\ref{thm:t_invariance} and Corollary~\ref{cor:t_invariance_reformed} for S-invariance as opposed to T-invariance. We recall that $ F $ is called S-invariant if $ [Z(F),\mathcal{S}]=0 $. For $ I,J,S \in \mathcal{I} $ we consider the following endomorphism,
    %
    \begin{align*}
    %
	\gamma_{IJ}^S \vcentcolon=
		\begin{array}{c}
			\begin{tikzpicture}[scale=0.15,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

			\node (v1) at (0,5) {};
			\node (v4) at (0,-5) {};
			\node (v2) at (5,0) {};
			\node (v3) at (-5,0) {};

			\node (v5) at (-2.5,2.5) {};
			\node (v6) at (2.5,-2.5) {};
			\node (v7) at (1.5,3.5) {};
			\node (v11) at (3.5,1.5) {};
			\node (v12) at (-1.5,-3.5) {};
			\node (v10) at (-3.5,-1.5) {};
			\node (v13) at (2.5,-2.5) {};

			\draw [thick] (v7) edge (v10);
			\draw [line width =0.5em,white] (v5) edge (v13);
			\draw [thick] (v5) edge (v13);
			\draw [line width =0.5em,white] (v11) edge (v12);
			\draw [thick] (v11) edge (v12);
			\node at (-5.25,-2.5) {$ I^\vee $};
			\node at (-2.75,-5.25) {$ J^\vee $};
			\node at (-4,4) {$ S $};
			\node at (4,-4) {$ S $};

			\draw[very thick]  (v1) edge (v3);
			\draw[very thick]  (v2) edge (v4);
			\draw[very thick, red]  (v1) edge (v2);
			\draw[very thick, red]  (v3) edge (v4);
			\end{tikzpicture}
		\end{array}
	\in \End_{\mathcal{TC}}(S)
    %
    \end{align*}
    %
    \begin{PROP} \label{prop:s_matrix_appears}

    For $ I,J,S \in \mathcal{I} $, we have
        %
        \begin{align*}
        %
        \gamma_{IJ}^S = \sum\limits_{A,B} \frac{\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(A)d(B)} \epsilon(S,A,B)
        %
        \end{align*}
        %
	where
		%
		\begin{align*}
		%
		\epsilon(S,A,B) = \frac{d(A)d(B)}{d(\mathcal{C})}\sum\limits_{T,b} d(T)
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0.5,5.5) {};
				\node (v4) at (-2,-7) {};
				\node (v2) at (5.5,0.5) {};
				\node (v3) at (-7,-2) {};

				\node (v5) at (-4,1) {};
				\node (v6) at (1,-4) {};
				\node (v7) at (0,2) {};
				\node (v11) at (2,0) {};
				\node (v12) at (-2,-4) {};
				\node (v10) at (-4,-2) {};
				\node (v13) at (1,-4) {};

				\draw [thick] (v7) edge (v10);
				\draw [line width =0.5em,white] (v5) edge (v13);
				\draw [thick] (v5) edge (v13);
				\draw [line width =0.5em,white] (v11) edge (v12);
				\draw [thick] (v11) edge (v12);
				\node at (-2.125,1.625) {$ A $};
				\node at (0,-0.5) {$ B $};
				\node at (-5,2) {$ T $};
				\node at (4,4) {$ S $};
				\node at (-5.5,-5.5) {$ S $};

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);

				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v9) at (1.5,1.5) {$ b $};
				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v90) at (-3,-3) {$ b^* $};
				\node (v8) at (3,3) {};
				\draw [thick] (v8) edge (v9);
				\node (v14) at (-4.5,-4.5) {};
				\draw [thick] (v90) edge (v14);
				\end{tikzpicture}
			\end{array}
		%
		\end{align*}
		%
	where $ b $ ranges over a basis of $ \Hom_{C}(S,AB) $ and $ b^* $ is the dual element under the perfect pairing $ b \otimes b^* \mapsto \tr(b^* \circ b) $.
	By \cite[Corollary 5.7]{hardiman_king}, $ \epsilon(S,A,B) $ is an idempotent corresponding (in the idempotent completion of $ \mathcal{TC} $) to the $ \eo{A}{B} $ isotypic component of $ S $.
    \proof As mentioned above, by \cite[Corollary 5.7]{hardiman_king}, we have $ \id_S = \sum_{A,B} \epsilon(S,A,B). $
	Pre-composing this identify with $ \gamma_{IJ}^S $ yields
		%
		\begin{align*}
		%
		\gamma_{IJ}^S &=  \sum_{\substack{A,B,T \\ b}}\frac{d(A)d(B)d(T)}{d(\mathcal{C})}
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0.5,5.5) {};
				\node (v4) at (-4.5,-9.5) {};
				\node (v2) at (5.5,0.5) {};
				\node (v3) at (-9.5,-4.5) {};

				\node (v5) at (-4,1) {};
				\node (v6) at (1,-4) {};
				\node (v7) at (0,2) {};
				\node (v11) at (2,0) {};
				\node (v12) at (-2,-4) {};
				\node (v10) at (-4,-2) {};
				\node (v13) at (1,-4) {};
				\node (v15) at (-3.5,-3.5) {};
				\node (v16) at (-7,-2) {};
				\node (v17) at (-2,-7) {};
				\node (v18) at (-8.5,-3.5) {};
				\node (v19) at (-3.5,-8.5) {};
				\node (v14) at (-7,-7) {};

				\draw [thick] (v16) edge (v17);
				\draw [line width =0.5em,white] (v15) edge (v14);
				\draw [thick] (v15) edge (v14);
				\draw [line width =0.5em,white] (v18) edge (v19);
				\draw [thick] (v18) edge (v19);
				\draw [thick] (v7) edge (v10);
				\draw [line width =0.5em,white] (v5) edge (v13);
				\draw [thick] (v5) edge (v13);
				\draw [line width =0.5em,white] (v11) edge (v12);
				\draw [thick] (v11) edge (v12);
				\node at (-2.125,1.625) {$ A $};
				\node at (0,-0.5) {$ B $};
				\node at (-5,2) {$ T $};
				\node at (-7.5,-0.75) {$ I^\vee $};
				\node at (-9.5,-2.5) {$ J^\vee $};
				\node at (4,4) {$ S $};
				\node at (-8,-8) {$ S $};

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);

				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v9) at (1.5,1.5) {$ b $};
				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v90) at (-3,-3) {$ b^* $};
				\node (v8) at (3,3) {};
				\draw [thick] (v8) edge (v9);
				\end{tikzpicture}
			\end{array}\\
		&= \sum_{\substack{A,B,T \\ b}}\frac{d(A)d(B)d(T)}{d(\mathcal{C})}
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0.5,5.5) {};
				\node (v4) at (-5,-10) {};
				\node (v2) at (5.5,0.5) {};
				\node (v3) at (-10,-5) {};

				\node (v5) at (-3.75,1.25) {};
				\node (v7) at (0,2) {};
				\node (v11) at (2,0) {};
				\node (v12) at (-5,-7) {};
				\node (v10) at (-7,-5) {};
				\node (v13) at (1.25,-3.75) {};
				\node (v15) at (-6.5,-6.5) {};
				\node (v16) at (-7,-2) {};
				\node (v17) at (-2,-7) {};
				\node (v18) at (-8.5,-3.5) {};
				\node (v19) at (-3.5,-8.5) {};
				\node (v14) at (-7.5,-7.5) {};

				\draw [thick] (v15) edge (v14);
				\draw [thick] (v7) edge (v10);
				\draw [line width =0.5em,white] (v5) edge (v13);
				\draw [thick] (v5) edge (v13);
				\draw [line width =0.5em,white] (v11) edge (v12);
				\draw [thick] (v11) edge (v12);
				\draw [line width =0.5em,white] (-4.75,-2.75) ellipse (1.2 and 1.2);
				\draw [line width =0.5em,white] (-2.75,-4.75) ellipse (1.2 and 1.2);
				\draw [thick] (-2.75,-4.75) node (v21) {} ellipse (1.2 and 1.2);
				\draw [thick] (-4.75,-2.75) node (v20) {} ellipse (1.2 and 1.2);
				\draw [line width =0.5em,white] (v20) edge (v10);
				\draw [thick] (v20) edge (v10);
				\draw [line width =0.5em,white] (v21) edge (v11);
				\draw [thick] (v21) edge (v11);
				\node at (-2,1.75) {$ A $};
				\node at (0.25,-0.25) {$ B $};
				\node at (-4.75,2.25) {$ T $};
				\node at (-2.75,-2) {$ I $};
				\node at (-0.5,-4) {$ J $};
				\node at (4,4) {$ S $};
				\node at (-8.5,-8.5) {$ S $};

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);

				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v9) at (1.5,1.5) {$ b $};
				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v90) at (-6,-6) {$ b^* $};
				\node (v8) at (3,3) {};
				\draw [thick] (v8) edge (v9);
				\end{tikzpicture}
			\end{array}\\
		&= \sum_{\substack{A,B,T \\ b}}\frac{d(T)\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(\mathcal{C})}
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0.5,5.5) {};
				\node (v4) at (-2,-7) {};
				\node (v2) at (5.5,0.5) {};
				\node (v3) at (-7,-2) {};

				\node (v5) at (-4,1) {};
				\node (v6) at (1,-4) {};
				\node (v7) at (0,2) {};
				\node (v11) at (2,0) {};
				\node (v12) at (-2,-4) {};
				\node (v10) at (-4,-2) {};
				\node (v13) at (1,-4) {};

				\draw [thick] (v7) edge (v10);
				\draw [line width =0.5em,white] (v5) edge (v13);
				\draw [thick] (v5) edge (v13);
				\draw [line width =0.5em,white] (v11) edge (v12);
				\draw [thick] (v11) edge (v12);
				\node at (-2.125,1.625) {$ A $};
				\node at (0,-0.5) {$ B $};
				\node at (-5,2) {$ T $};
				\node at (4,4) {$ S $};
				\node at (-5.5,-5.5) {$ S $};

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);

				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v9) at (1.5,1.5) {$ b $};
				\node [draw,outer sep=0,inner sep=1,minimum size=10,minimum width = 26,fill=white,rotate=-45] (v90) at (-3,-3) {$ b^* $};
				\node (v8) at (3,3) {};
				\draw [thick] (v8) edge (v9);
				\node (v14) at (-4.5,-4.5) {};
				\draw [thick] (v90) edge (v14);
				\end{tikzpicture}
			\end{array} = \sum\limits_{A,B} \frac{\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(A)d(B)} \epsilon(S,A,B)
		%
		\end{align*}
		%
	where the second equality is due to a so-called `handle-slide move' (for a proof within this context see~\cite[Proposition 7.2]{hardiman2020graphical}) and the penultimate equality is due to~\cite[Lemma 3.1.4]{Bak01}.
 	\endproof

    \end{PROP}

Let $ \textbf{T} $ be given by the direct sum of objects in $ \mathcal{I} $, i.e.\ $ \textbf{T} = \bigoplus_S S $. For $ I,J \in \mathcal{I} $ we consider the following (diagonal) endomorphism of $ \textbf{T} $,
	%
	\begin{align*}
	%
	\nu_I^J = \frac{1}{d(\mathcal{C})} \bigoplus_S d(S) \gamma^S_{IJ}.
	%
	\end{align*}
	%

	\begin{THM} \label{thm:s_invariance}

	Let $ \mathcal{C} $ be a modular tensor category and let $ F $ be an object in $ \mathcal{RTC} $. Then $ F $ is $ \mathcal{S} $-invariant if and only if $ \tr F\left(\nu_I^J\right) = \tr F\left(\epsilon_I^J\right) $ for all $ I,J \in \mathcal{I} $.

	\proof

	By Proposition~\ref{prop:s_matrix_appears}, we have
		%
		\begin{align}\label{eq:apply_prop_s}
		%
		F(\nu_I^J) = \frac{1}{d(\mathcal{C})}\bigoplus_S d(S) \sum_{A,B} \frac{\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(A)d(B)} F(\epsilon(S,A,B)).
		%
		\end{align}
		%
	By the definition of $ \epsilon(S,A,B) $, in the idempotent completion of $ \mathcal{TC} $ we have
		%
		\begin{align} \label{eq:idem_isotipic_identity}
		%
		(S,\epsilon(S,A,B)) = S_A^B \vprod \eo{A}{B}
		%
		\end{align}
		%
	where $ S_A^B $ denotes the multiplicity space of the simple $ \eo{A}{B} $ in $ S $. Furthermore, by~\cite[Proposition 7.4]{hardiman2020graphical} $ S_A^B = \Hom_{\mathcal{C}}(S,AB) $. This implies the following sequence of natural isomorphisms,
		%
		\begin{align} \label{eq:iso_spaces}
		%
		F(S)_{F(\epsilon(S,A,B))} = \Hom_{\mathcal{C}}(S,AB) \otimes F(AB)_{F(\epsilon_A^B)} = \Hom_{\mathcal{C}}(S,AB) \otimes F_A^B
		%
		\end{align}
		%
 	where, for $ V $ a vector space and $ i \in \End(V) $, we write $ V_i $ to denote the corresponding subspace. As this implies $ \tr F(\epsilon(S,A,B)) = \hom_{\mathcal{C}}(S,AB) \dim (F_A^B) $, we may now simply compute the trace,
		%
		\begin{align*}
		%
		\tr F(\nu_I^J) &= \frac{1}{d(\mathcal{C})} \sum_{A,B,S} d(S) \frac{\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(A)d(B)} \hom_{\mathcal{C}}(S,AB) \dim (F_A^B) \\
	 	&= \frac{1}{d(\mathcal{C})} \sum_{A,B}  \left(\sum_S d(S) \hom_{\mathcal{C}}(S,AB)\right) \frac{\mathcal{S}_{I,A}\mathcal{S}_{B,J}}{d(A)d(B)} \dim (F_A^B) \\
		&= \frac{1}{d(\mathcal{C})} \sum_{A,B} \mathcal{S}_{I,A}\mathcal{S}_{B,J} \dim (F_A^B) \\
	 	&= \left(\mathcal{S} Z(F) \mathcal{S}^{-1}\right)_{IJ}.
		%
		\end{align*}
		%
	As by definition $ \tr F(\epsilon_I^J) = \dim F_I^J = Z(F)_{IJ} $, this proves the claim. \endproof

	\end{THM}

As in Section~\ref{sec:graph_char_t_inv}, we shall now rewrite this result into an equivalent formulation. Let $ X $ be in $ \mathcal{TC} $ and let $ F $ be in $ \mathcal{RTC} $. As before we consider two elements in $ \End_{\mathcal{TC}}(X)^* $. One is simply $ \Tr_{F,X} \colon \alpha \mapsto \tr F(\alpha) $ again, the other is given by
	%
	\begin{align}  \label{eq:s_twisted_trace}
	%
	\Tr_{F,X}^\mathcal{S} \colon \alpha \mapsto \sum_S \tr F \left(
	\begin{array}{c}
		\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

		\node (v1) at (0,4) {};
		\node (v4) at (0,-4) {};
		\node (v2) at (4,0) {};
		\node (v3) at (-4,0) {};

		\node (v5) at (-2,2) {};
		\node (v6) at (2,-2) {};
		\node (v7) at (2,2) {};
		\node (v11) at (-2,-2) {};
		\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=22,fill=white,rotate=-90] (v9) at (0,0) {$ \alpha_S $};
		\node at (3,3) {$ S $};
		\node at (-3,-3) {$ S $};
		\node at (-3,3) {$ X^\vee $};
		\node at (3,-3) {$ X^\vee $};
		\draw [thick] (v9) edge (v6);
		\draw [thick] (v9) edge (v7);
		\draw [thick] (v9) edge (v11);
		\draw [thick] (v5) edge (v9);

		\draw[very thick, red]  (v1) edge (v3);
		\draw[very thick, red]  (v2) edge (v4);
		\draw[very thick]  (v1) edge (v2);
		\draw[very thick]  (v3) edge (v4);
		\end{tikzpicture}
	\end{array}\right).
	%
	\end{align}
	%
where $ \alpha_S $ is the $ \Hom_{\mathcal{C}}(SX,XS) $ component of $ \alpha $. The following lemma gives a more intuitive description of $ \Tr_{F,X}^\mathcal{S} $.

	\begin{LEMMA} \label{lem:compute_s_twist}

	Let $ X,G $ be in $ \mathcal{C} $ and let $ f $ be in $ \End_{\mathcal{C}}(GX,XG) $. For
		%
		\begin{align*}
		%
		\alpha =
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1}]

				\node (v1) at (0,4) {};
				\node (v4) at (0,-4) {};
				\node (v2) at (4,0) {};
				\node (v3) at (-4,0) {};

				\node (v5) at (-2,2) {};
				\node (v6) at (2,-2) {};
				\node (v7) at (2,2) {};
				\node (v11) at (-2,-2) {};
				\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=22,fill=white,rotate=0] (v9) at (0,0) {$ f $};
				\node at (3,3) {$ X $};
				\node at (-3,-3) {$ X $};
				\node at (-3,3) {$ G $};
				\node at (3,-3) {$ G $};
				\draw [thick] (v9) edge (v6);
				\draw [thick] (v9) edge (v7);
				\draw [thick] (v9) edge (v11);
				\draw [thick] (v5) edge (v9);

				\draw[very thick, red]  (v1) edge (v3);
				\draw[very thick, red]  (v2) edge (v4);
				\draw[very thick]  (v1) edge (v2);
				\draw[very thick]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array} \quad \text{we have} \quad \Stwist{\alpha} = \tr F \left(
				\begin{array}{c}
					\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

					\node (v1) at (0,4) {};
					\node (v4) at (0,-4) {};
					\node (v2) at (4,0) {};
					\node (v3) at (-4,0) {};

					\node (v5) at (-2,2) {};
					\node (v6) at (2,-2) {};
					\node (v7) at (2,2) {};
					\node (v11) at (-2,-2) {};
					\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=22,fill=white,rotate=-90] (v9) at (0,0) {$ f $};
					\node at (3,3) {$ X^\vee $};
					\node at (-3,-3) {$ X^\vee $};
					\node at (-3,3) {$ G $};
					\node at (3,-3) {$ G $};
					\draw [thick] (v9) edge (v6);
					\draw [thick] (v9) edge (v7);
					\draw [thick] (v9) edge (v11);
					\draw [thick] (v5) edge (v9);

					\draw[very thick]  (v1) edge (v3);
					\draw[very thick]  (v2) edge (v4);
					\draw[very thick, red]  (v1) edge (v2);
					\draw[very thick, red]  (v3) edge (v4);
					\end{tikzpicture}
				\end{array} \right).
		%
		\end{align*}
		%
	\proof

	We compute,
		%
		\begin{align*}
		%
		\tr F \left(
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

				\node (v1) at (0,4) {};
				\node (v4) at (0,-4) {};
				\node (v2) at (4,0) {};
				\node (v3) at (-4,0) {};

				\node (v5) at (-2,2) {};
				\node (v6) at (2,-2) {};
				\node (v7) at (2,2) {};
				\node (v11) at (-2,-2) {};
				\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=22,fill=white,rotate=-90] (v9) at (0,0) {$ f $};
				\node at (3,3) {$ X^\vee $};
				\node at (-3,-3) {$ X^\vee $};
				\node at (-3,3) {$ G $};
				\node at (3,-3) {$ G $};
				\draw [thick] (v9) edge (v6);
				\draw [thick] (v9) edge (v7);
				\draw [thick] (v9) edge (v11);
				\draw [thick] (v5) edge (v9);

				\draw[very thick]  (v1) edge (v3);
				\draw[very thick]  (v2) edge (v4);
				\draw[very thick, red]  (v1) edge (v2);
				\draw[very thick, red]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array} \right) &= \sum_{S,b} \tr F \left(
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

				\node (v1) at (-1.5,5.5) {};
				\node (v4) at (0,-4) {};
				\node (v2) at (4,0) {};
				\node (v3) at (-5.5,1.5) {};

				\node (v5) at (-3.5,3.5) {};
				\node (v6) at (2,-2) {};
				\node (v7) at (3,1) {};
				\node (v11) at (-1,-3) {};
				\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=0,fill=white,rotate=-90] (v9) at (1,-1) {\scriptsize $ f $};
				\node at (4,2) {$ X^\vee $};
				\node at (-2,-4) {$ X^\vee $};
				\node at (-0.25,2.25) {$ S $};
				\node at (-4.5,4.5) {$ G $};
				\node at (3,-3) {$ G $};
				\draw [thick] (v9) edge (v6);
				\draw [thick] (v9) edge (v7);
				\draw [thick] (v9) edge (v11);
				\draw [thick] (v5) edge (v9);
				\node [draw,outer sep=0,inner sep=.5,minimum size=9,fill=white,rotate=-45] (v90) at (-0.5,0.5) {\scriptsize $ b $};
				\node [draw,outer sep=0,inner sep=.5,minimum size=9,fill=white,rotate=-45] (v90) at (-2.5,2.5) {\scriptsize $ b^* $};

				\draw[very thick]  (v1) edge (v3);
				\draw[very thick]  (v2) edge (v4);
				\draw[very thick, red]  (v1) edge (v2);
				\draw[very thick, red]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array} \right) \\
 		&= \sum_{S,b}\tr F \left(
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.2,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

				\node (v1) at (-1.5,5.5) {};
				\node (v4) at (0,-4) {};
				\node (v2) at (4,0) {};
				\node (v3) at (-5.5,1.5) {};

				\node (v5) at (-3.5,3.5) {};
				\node (v6) at (2,-2) {};
				\node (v7) at (1.25,2.75) {};
				\node (v11) at (-2.75,-1.25) {};
				\node [draw,diamond,outer sep=0,inner sep=.5,minimum size=0,fill=white,rotate=-90] (v9) at (-0.75,0.75) {\scriptsize $ f $};
				\node at (2.25,3.75) {$ X^\vee $};
				\node at (-3.75,-2.25) {$ X^\vee $};
				\node at (-4.5,4.5) {$ S $};
				\node at (3,-3) {$ S $};
				\draw [thick] (v9) edge (v6);
				\draw [thick] (v9) edge (v7);
				\draw [thick] (v9) edge (v11);
				\draw [thick] (v5) edge (v9);
				\node [draw,outer sep=0,inner sep=.5,minimum size=9,fill=white,rotate=-45] (v90) at (-2.25,2.25) {\scriptsize $ b $};
				\node [draw,outer sep=0,inner sep=.5,minimum size=9,fill=white,rotate=-45] (v90) at (0.75,-0.75) {\scriptsize $ b^* $};

				\draw[very thick]  (v1) edge (v3);
				\draw[very thick]  (v2) edge (v4);
				\draw[very thick, red]  (v1) edge (v2);
				\draw[very thick, red]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array} \right) = \Stwist{\alpha}
		%
		\end{align*}
		%
	where the second equality is due to the commutativity of $ \tr $. \endproof

	\end{LEMMA}

	\begin{THM} \label{thm:s_invariance_reformulated}

	Let $ \mathcal{C} $ be a modular tensor category and let $ F $ be an object in $ \mathcal{RTC} $. Then $ F $ is $ \mathcal{S} $-invariant if and only if $ \Tr_{F,X}^{\mathcal{S}} = \Tr_{F,X} $ for all $ X $ in $ \mathcal{TC} $.

	\proof 	We start start by supposing that $ \Tr_{F,X}^{\mathcal{S}} = \Tr_{F,X} $. As $ \Tr_{F,X} (\epsilon_I^J) = \tr F(\epsilon_I^J) $ and $ \Stwist{\epsilon_I^J} = \tr F (\nu_I^J) $, Theorem~\ref{thm:s_invariance} allows us to conclude immediately.

	Now we suppose that $ F $ is S-invariant. We take $ X $ in $ \mathcal{TC} $, $ \alpha \in \End_{\mathcal{TC}}(X) $ and a basis $ \mathcal{B} \subset \Hom_{\mathcal{TC}}(\epsilon_I^J,X) $. For $ b,b' \in \mathcal{B} $ and $ I,J \in \mathcal{I} $, we consider the scalar
  		%
  		\begin{align*}
  		%
  		\lambda_{IJ}^{bb'} \vcentcolon= b^* \circ \alpha \circ b' \in \End_{\mathcal{TC}}(\epsilon_I^J) = \fld.
  		%
  		\end{align*}
  		%
	As $ \{\epsilon_I^J\}_{I,J \in \mathcal{I}} $ forms a complete set of primitive idempotents in $ \mathcal{TC} $, we have
 		%
 		\begin{align*}
 		%
		\alpha = \sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'} \ b' \circ b^* = \sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'} \sum_{R,S} \begin{array}{c}
			\begin{tikzpicture}[scale=0.3,every node/.style={inner sep=0,outer sep=-1}]

			\node (v1) at (-1.375,2.875) {};
			\node (v4) at (-3.375,-3.125) {};
			\node (v2) at (0.625,0.875) {};
			\node (v3) at (-5.375,-1.125) {};

			\node (v9) at (-0.375,1.875) {};
			\node (v6) at (-4.375,-2.125) {};

			\node at (0.25,2.5) {$ X $};
			\node at (-5,-2.75) {$ X $};
			\node at (-5,0.75) {$ R $};
			\node at (-3.25,2.5) {$ S $};
			\node at (-1.5,-2.75) {$ R $};
			\node at (0.25,-1) {$ S $};

			\node (v11) at (-3.5,-0.75) {};
			\node (v12) at (-3,-1.25) {};
			\draw [thick] (v11) edge (v12);
			\node (v14) at (-1.75,1) {};
			\node (v13) at (-1.25,0.5) {};
			\draw [thick] (v13) edge (v14);

			\node (v15) at (-4.25,0) {};
			\node (v16) at (-2.5,1.75) {};
			\draw [thick] (v15) to (v11);
			\draw [thick] (v16) to (v14);
			\node (v17) at (-0.5,-0.25) {};
			\node (v18) at (-2.25,-2) {};
			\draw [thick] (v13) to (v17);
			\draw [thick] (v12) to (v18);
			\draw [thick] (v14) edge (v11);
			\draw [thick] (v13) edge (v12);
			\node [rotate=-45,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v50) at (-3.25,-1) {$\scriptstyle b'_R $};
			\node [rotate=-45,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v5) at (-1.5,0.75) {$\scriptstyle b_S^* $};
			\draw [thick] (v50) edge (v6);
			\draw[thick]  (v9) edge (v5);

			\draw[very thick,red]  (v1) edge (v3);
			\draw[very thick,red]  (v2) edge (v4);
			\draw[very thick]  (v1) edge (v2);
			\draw[very thick]  (v3) edge (v4);
			\end{tikzpicture}
		\end{array}.
 		%
 		\end{align*}
 		%
	and
		%
		\begin{align*}
		%
		\Tr_{F,X} (\alpha) = \sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'} \tr F (b' \circ b^*) = \sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'} \tr F (b^* \circ b') = \sum_{\substack{I,J \\ b}} \lambda_{IJ}^{bb} \tr F (\epsilon_I^J).
		%
		\end{align*}
		%
	By Lemma~\ref{lem:compute_s_twist}, we can compute $ \Stwist{\alpha} $ as
		%
		\begin{align*}
		%
		& \hspace{1.4em} \sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'} \sum_{R,S} \tr F \left(
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.3,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

				\node (v1) at (-1.375,2.875) {};
				\node (v4) at (-3.375,-3.125) {};
				\node (v2) at (0.625,0.875) {};
				\node (v3) at (-5.375,-1.125) {};

				\node (v9) at (-0.375,1.875) {};
				\node (v6) at (-4.375,-2.125) {};

				\node at (0.25,2.5) {$ X^\vee $};
				\node at (-5,-2.75) {$ X^\vee $};
				\node at (-5,0.75) {$ R $};
				\node at (-3.25,2.5) {$ S $};
				\node at (-1.5,-2.75) {$ R $};
				\node at (0.25,-1) {$ S $};

				\node (v11) at (-3.5,-0.75) {};
				\node (v12) at (-3,-1.25) {};
				\draw [thick] (v11) edge (v12);
				\node (v14) at (-1.75,1) {};
				\node (v13) at (-1.25,0.5) {};
				\draw [thick] (v13) edge (v14);

				\node (v15) at (-4.25,0) {};
				\node (v16) at (-2.5,1.75) {};
				\draw [thick] (v15) to (v11);
				\draw [thick] (v16) to (v14);
				\node (v17) at (-0.5,-0.25) {};
				\node (v18) at (-2.25,-2) {};
				\draw [thick] (v13) to (v17);
				\draw [thick] (v12) to (v18);
				\draw [thick] (v14) edge (v11);
				\draw [thick] (v13) edge (v12);
				\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v50) at (-3.25,-1) {$\scriptstyle b'_R $};
				\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v5) at (-1.5,0.75) {$\scriptstyle b^*_S$};
				\draw [thick] (v50) edge (v6);
				\draw[thick]  (v9) edge (v5);

				\draw[very thick]  (v1) edge (v3);
				\draw[very thick]  (v2) edge (v4);
				\draw[very thick, red]  (v1) edge (v2);
				\draw[very thick, red]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array} \right)
		=\sum_{\substack{I,J \\ b,b' }} \lambda_{IJ}^{bb'}  \sum_{R,S} \tr F \left(
			\begin{array}{c}
				\begin{tikzpicture}[scale=0.3,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

				\node (v1) at (-2,3.5) {};
				\node (v4) at (-2.75,-3.75) {};
				\node (v2) at (1.25,0.25) {};
				\node (v3) at (-6,-0.5) {};

				\node (v9) at (-0.4375,1.8125) {};
				\node (v6) at (-4.3125,-2.0625) {};

				\node at (-5.25,-2.75) {$ X^\vee $};
				\node at (-4,-4) {$ R^\vee $};
				\node at (-6.5,-1.5) {$ R $};
				\node at (0.25,2.75) {$ X^\vee $};
				\node at (1.5,1.5) {$ R^\vee $};
				\node at (-1,4) {$ R $};
				\node at (-5.625,1.375) {$ S $};
				\node at (-3.875,3.125) {$ R $};
				\node at (-0.875,-3.375) {$ S $};
				\node at (0.875,-1.625) {$ R $};

				\node (v11) at (-3.5,-0.75) {};
				\node (v12) at (-3,-1.25) {};
				\draw [thick] (v11) edge (v12);
				\node (v14) at (-1.75,1) {};
				\node (v13) at (-1.25,0.5) {};
				\draw [thick] (v13) edge (v14);

				\node (v15) at (-5.4375,-0.9375) {};
				\node (v16) at (-4.6875,0.6875) {};
				\draw [thick] (v15) to[out=55,in=135] (v11);
				\draw [thick] (v16) to[out=-45,in=135] (v14);
				\node (v17) at (-1.8125,-2.6875) {};
				\node (v18) at (-3.1875,-3.1875) {};
				\draw [thick] (v13) to[out=-45,in=135] (v17);
				\draw [thick] (v12) to[in=50,out=20] (v18);
				\draw [thick] (v14) edge (v11);
				\draw [thick] (v13) edge (v12);
				\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v50) at (-3.25,-1) {$\scriptstyle b'_R $};
				\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v5) at (-1.5,0.75) {$\scriptstyle b^*_S$};
				\draw [thick] (v50) edge (v6);
				\draw[thick]  (v9) edge (v5);
				\node (v7) at (-1.3125,2.6875) {};
				\node (v8) at (-3.25,2.1667) {};
				\draw [thick] (v7) to[out=-130,in=-45] (v8);
				\node (v10) at (0.4375,0.9375) {};
				\node (v19) at (-0.0625,-0.9375) {};
				\draw [thick] (v10) to[out=-130,in=135] (v19);

				\draw[very thick]  (v1) edge (v3);
				\draw[very thick]  (v2) edge (v4);
				\draw[very thick, red]  (v1) edge (v2);
				\draw[very thick, red]  (v3) edge (v4);
				\end{tikzpicture}
			\end{array}
		\right) \\
		&= \sum_{\substack{I,J \\ b,b' }}\lambda_{IJ}^{bb'} \sum_{R,S} \tr F \left(  \begin{array}{c}
			\begin{tikzpicture}[scale=0.3,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

			\node (v1) at (-1.5,2.75) {};
			\node (v4) at (-3.25,-3) {};
			\node (v2) at (0.5,0.75) {};
			\node (v3) at (-5.25,-1) {};

			\node (v9) at (-0.5,1.75) {};

			\node at (-0.5,3) {$ I^\vee $};
			\node at (0.75,1.75) {$ J^\vee $};
			\node at (-5.5,-2) {$ I^\vee $};
			\node at (-4.25,-3.25) {$ J^\vee $};
			\node at (-5,0.75) {$ S $};
			\node at (-3.25,2.5) {$ R $};
			\node at (-1.5,-2.75) {$ S $};
			\node at (0.25,-1) {$ R $};

			\node (v11) at (-3.5,-0.75) {};
			\node (v12) at (-3,-1.25) {};
			\draw [thick] (v11) edge (v12);
			\node (v14) at (-1.75,1) {};
			\node (v13) at (-1.25,0.5) {};
			\draw [thick] (v13) edge (v14);

			\node (v15) at (-4.25,0) {};
			\node (v16) at (-2.5,1.75) {};
			\draw [thick] (v15) to (v11);
			\draw [thick] (v16) to (v14);
			\node (v17) at (-0.5,-0.25) {};
			\node (v18) at (-2.25,-2) {};
			\draw [thick] (v13) to (v17);
			\draw [thick] (v12) to (v18);
			\node (v8) at (-4,-2.25) {};
			\node (v7) at (-4.5,-1.75) {};
			\node (v6) at (-0.75,2) {};
			\node (v10) at (-0.25,1.5) {};
			\draw [thick] (v11) edge (v7);
			\draw [thick] (v12) edge (v8);
			\draw [thick] (v14) edge (v6);
			\draw [thick] (v13) edge (v10);
			\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v50) at (-3.25,-1) {$\scriptstyle b^*_S $};
			\node [rotate=-135,draw,outer sep=0,inner sep=0,minimum size=10,fill=white] (v5) at (-1.5,0.75) {$\scriptstyle b'_R $};
			\draw [thick] (v50) edge (v5);

			\draw[very thick]  (v1) edge (v3);
			\draw[very thick]  (v2) edge (v4);
			\draw[very thick,red]  (v1) edge (v2);
			\draw[very thick,red]  (v3) edge (v4);
			\end{tikzpicture}
		\end{array} \right) = \sum_{\substack{I,J \\ b }}\lambda_{IJ}^{bb} \sum_{T} \frac{d(T)}{d(\mathcal{C})} \tr F \left( \begin{array}{c}
			\begin{tikzpicture}[scale=0.15,every node/.style={inner sep=0,outer sep=-1},rotate=-90]

			\node (v1) at (0,5) {};
			\node (v4) at (0,-5) {};
			\node (v2) at (5,0) {};
			\node (v3) at (-5,0) {};

			\node (v5) at (-2.5,2.5) {};
			\node (v6) at (2.5,-2.5) {};
			\node (v7) at (1.5,3.5) {};
			\node (v11) at (3.5,1.5) {};
			\node (v12) at (-1.5,-3.5) {};
			\node (v10) at (-3.5,-1.5) {};
			\node (v13) at (2.5,-2.5) {};

			\draw [thick] (v7) edge (v10);
			\draw [line width =0.5em,white] (v5) edge (v13);
			\draw [thick] (v5) edge (v13);
			\draw [line width =0.5em,white] (v11) edge (v12);
			\draw [thick] (v11) edge (v12);
			\node at (-5.25,-2.5) {$ I^\vee $};
			\node at (-2.75,-5.25) {$ J^\vee $};
			\node at (-4,4) {$ T $};
			\node at (4,-4) {$ T $};

			\draw[very thick]  (v1) edge (v3);
			\draw[very thick]  (v2) edge (v4);
			\draw[very thick, red]  (v1) edge (v2);
			\draw[very thick, red]  (v3) edge (v4);
			\end{tikzpicture}
		\end{array} \right)\\
		&= \sum_{\substack{I,J \\ b}} \lambda_{IJ}^{bb} \tr F (\nu_I^J) =  \sum_{\substack{I,J \\ b}} \lambda_{IJ}^{bb} \tr F (\epsilon_I^J) = \Tr_{F,X}(\alpha)
		%
		\end{align*}
		%
	where the first equality is due to the invariance of $ \tr $ under conjugation by an isomorphism, the third is due to the fact that $ b^* \circ b' = \delta_{b,b'} \epsilon_I^J $ and the fifth is due to Theorem~\ref{thm:s_invariance}. \endproof

	\end{THM}

	\section{Conclusion} \label{sec:conclusion}

We are now in position to combine the results of Section~\ref{sec:graph_char_t_inv} and Section~\ref{sec:graph_char_s_inv} into our main result.

	\begin{COR} \label{cor:main_result}

	Let $ F $ be in $ \mathcal{RTC} $. Then $ F $ is modular invariant if and only if
		%
		\begin{align*}
		%
		\Tr_{F,X} = \Tr_{F,X}^{\mathcal{T}} = \Tr_{F,X}^{\mathcal{S}}
		%
		\end{align*}
		%
	for all $ X $ in $ \mathcal{TC} $.

	\proof This follows immediately from Corollary~\ref{cor:t_invariance_reformed} and Theorem~\ref{thm:s_invariance_reformulated}. \endproof

	\end{COR}

Taken together,\! $ \Tr_{F,X}^{\mathcal{T}} $ and $ \Tr_{F,X}^{\mathcal{S}} $ are extremely telling. In particular, their image under $ \alpha \in \End_{TC}(X) $  is given by taking the standard trace $ \Tr_{F,X} $ and pre-composing it with an action of the standard generators of the mapping class group of the torus on the graphical description of $ \alpha $\footnote{This is clear for $ \Tr_{F,X}^{\mathcal{T}} $ and for $ \Tr_{F,X}^{\mathcal{S}} $ it is the content of Lemma~\ref{lem:compute_s_twist}.}. Therefore Corollary~\ref{cor:main_result} may be reformulated into stating that modular invariance of an object in $ \mathcal{RTC} $ amounts to a form of invariance under the mapping class group of the torus, giving us a characterization which doesn't distinguish certain generators.

Furthermore, this picture suggests a deeper relationship with the motivational mathematical physics described in the introduction. We recall that the modular invariance of the partition function encodes the conformal invariance of the field theory. We also recall that the partition function gives the value of the theory on a torus parametrised (in the standard way) by a point in the complex upper half-plane $ \mathbb{H} $. As $ \mathbb{H} $ is the Teichm\"uller space, taking the quotient by the mapping class group gives the moduli space. Therefore conformal invariance with respect to the torus also comes down to a form of invariance under the mapping class group of the torus. Drawing out this correspondance and connecting these two actions of the mapping class group is the subject of ongoing reseach.

\bibliographystyle{alpha}

redpajama_set_name: RedPajamaArXiv


text:
package org.drools.guvnor.client.moduleeditor;

import com.google.gwt.core.client.GWT;
import com.google.gwt.event.shared.EventBus;
import com.google.gwt.uibinder.client.UiBinder;
import com.google.gwt.uibinder.client.UiField;
import com.google.gwt.user.client.ui.Widget;

import org.drools.guvnor.client.asseteditor.GuvnorEditor;
import org.drools.guvnor.client.common.LoadingPopup;
import org.drools.guvnor.client.explorer.ClientFactory;
import org.drools.guvnor.client.rpc.Artifact;
import org.drools.guvnor.client.widgets.MessageWidget;
import org.drools.guvnor.client.widgets.MetaDataWidget;
import org.drools.guvnor.client.widgets.RuleDocumentWidget;

/**
 * The generic editor for all types of artifacts.
 */
public class ArtifactEditor extends GuvnorEditor {
    interface ArtifactEditorBinder
        extends
        UiBinder<Widget, ArtifactEditor> {
    }

    private static ArtifactEditorBinder uiBinder  = GWT.create( ArtifactEditorBinder.class );

    @UiField(provided = true)
    final MetaDataWidget                metaWidget;

    @UiField(provided = true)
    final RuleDocumentWidget            ruleDocumentWidget;

    @UiField
    MessageWidget                       messageWidget;

    protected Artifact                  artifact;
    private long                        lastSaved = System.currentTimeMillis();

    /**
     * @param Artifact
     *            artifact
     */
    public ArtifactEditor(Artifact artifact,
                          ClientFactory clientFactory,
                          EventBus eventBus) {
        this( artifact,
              clientFactory,
              eventBus,
              false );
    }

    /**
     * @param Artifact
     *            artifact
     * @param historicalReadOnly
     *            true if this is a read only view for historical purposes.
     */
    public ArtifactEditor(Artifact artifact,
                          ClientFactory clientFactory,
                          EventBus eventBus,
                          boolean historicalReadOnly) {
        this.artifact = artifact;
        boolean readOnly = historicalReadOnly || artifact.isReadonly();

        ruleDocumentWidget = new RuleDocumentWidget( this.artifact,
                                                     readOnly );

        metaWidget = new MetaDataWidget(
                                         clientFactory,
                                         eventBus,
                                         this.artifact,
                                         readOnly,
                                         this.artifact.getUuid() );

        initWidget( uiBinder.createAndBindUi( this ) );
        setWidth( "100%" );
        LoadingPopup.close();
    }

    @Override
    public boolean isDirty() {
        return (System.currentTimeMillis() - lastSaved) > 3600000;
    }

    protected boolean hasDirty() {
        // not sure how to implement this now.
        return false;
    }

}

redpajama_set_name: RedPajamaGithub


A profile of Cpfair.org was found on the social network Facebook.
The web server used by Cpfair.org is run by CyrusOne LLC and is located in Houston, USA. On this web server 17 other websites are hosted. The language of those websites is mostly english.
A Nginx server hosts the websites of Cpfair.org. For managing the data of the webpages the CMS WordPress in the version 4.9.8 is used. The website uses the latest markup standard HTML 5. The website does not specify details about the inclusion of its content in search engines. For this reason the content will be included by search engines.


The Zuri Zanzibar is a new (2018) resort occupying a wonderful location on a pretty white beach in Northern Zanzibar. Guests will appreciate the resorts top-class accommodation, superb cuisine and a relaxed, laid-back vibe - a great combination for a holiday to remember.
This five-star property lies on the north west coast by Zanzibar's only non tidal beach, making it an ideal location for enjoying the clear waters of the Indian Ocean. There's 300m of sand where guests can relax in privacy, a towel service is available and villas and suites have the benefit of their own private area and bar.
If you're feeling a little more active, head to the Water Sports Centre where you can hire out equipment for snorkelling and non-motorised watersports. Diving lessons can also be arranged plus sunset cruises, boat excursions and fishing trips.
Zuri Zanzibar has a 32m infinity pool, a book and DVD library and the wonderful Spice Garden which is a perfect place for a quiet, meditative stroll.
MAUA Wellness has three massage rooms and a salon offering a range of pampering and relaxing treatments, manicures, pedicures and facials. After your treatment, chill out on the Jacuzzi deck or perhaps take a private yoga lesson.
The hotel has a main restaurant, pool restaurant and beach restaurant where guests can savour perfectly prepared, fresh local dishes or international fare. Private beach dinners and romantic candlelit dinners can be arranged on request.
The bungalows, suites and villas showcase environmentally friendly architecture and feature modern amenities including king or single beds, evening breeze air conditioning, satellite TV, Wi-Fi, tea/coffee making facilities, organic bathroom amenities, tropical outdoor shower and fully furnished terrace. Selected interconnected bungalows are available.
Garden Bungalows have tranquil views of the tropical gardens while Zuri Bungalows have partial ocean views.
Ocean Bungalows have wonderful uninterrupted views over the Indian Ocean and are available with or without a private Jacuzzi.
Spacious suites with a separate bedroom, extended living room area, private Jacuzzi and generously sized private terrace area. Available with partial or full ocean views.
The resort's luxury two and three bedroom villas enjoy an exclusive oceanfront location with your own stretch of semi-private beach and a Jacuzzi. Three bedroom villas benefit from a private outdoor infinity pool. Full details available on request.
Upendo: The hotel's main restaurant featuring live cooking shows and serving delicious international cuisine with an African/Indian Ocean twist.
Maischa: A casual pool restaurant and bar where you can chill out and enjoy the ocean views. Serves Southern European and Indian dishes by day and Middle Eastern and Omani cuisine by night.
Beach Restaurant & Bar: Light bites, finger food and a wide range of smoothies, cocktails, aperitifs and other refreshing beverages.
There's also the Dhow Bar which is a private beach bar for guests staying in deluxe suites and villas.


text:
HP PCB Urges Education Deptt to Educate Children About Air, Noise Pollution Caused by Firecrackers Ahead of Diwali
Shimla-Himachal Pradesh Pollution Control Board, ahead of Diwali festival, has urged the department of education to educate children about the harmful effects of air and noise pollution caused due to bursting of firecrackers and also to encourage them to celebrate pollution-free Diwali.
The board has also urged the people of the State to celebrate this festival of light in an eco-friendly way and say no to firecrackers.
The HP PCB said that to access the impact of the bursting of firecrackers during the forthcoming Diwali festival, it will conduct a special monitoring drive in the state. During this drive, the board will assess the ambient air and noise quality in the State.
HP PCB said that necessary directions have been issued to all the regional officers and laboratory In-Charges of the State Board to conduct extensive ambient air quality monitoring derive with effect from October 20 to November 3, 2019.
Ambient noise monitoring will also be conducted during pre- Diwali October on 21 and on the day of Diwali (October 27).
During this monitoring drive, apart from general parameters like particular Matters (PM10, PM 2.5), oxides of Sulphur and Nitrogen, Carbon Monoxide and Ammonia some additional parameters like Lead, Nickel, arsenic, Barium, Iron and Strontium will also be monitored, the PCB said.
PCB said that all the Deputy Commissioners and Superintendent of Police have been directed to ensure that no firework of firecrackers should be allowed to burst in the 100 meters vicinity of the silence zone such as hospitals, educational institutions, courts etc. as per the provision of law.
Sales and use of firecrackers generating noise level exceeding 125 dBA is also prohibited.
Ironically, the Board has no check on various other activities that cause air pollution throughout the year. For example, garbage burning is rampant in the state. While the practice is regular in rural areas and municipal councils, the Municipal Corporation of capital Shimla is also not far behind.
Moreover, there is no check on vehicular emissions as both HP PCB and the State Police refuse to take responsibility for it. The 'No Honking' campaign to reduce noise pollution in the state has also failed miserably as no steps were taken to create awareness or to deter people from honking unnecessarily. Considering these facts, it sounds odd that the PCB would actually be striving to check air and noise pollution ahead of Diwali.
Related Topics:air pollutionDiwali 2019FestivalHP PCBnoise pollution
HP By-Election 2019 Dates Announced, Model Code of Conduct Comes Into Effect
Shimla Law University Student Protest: Support Pours in From Other Varsities, Students Organizations After Authority Orders Vacating Hostels & Withdraws Basic Facilities
HP Tourism to Cook 1100 kg 'Khichari' at Tattapani to Promote Tourism
"Horn Not Ok" Campaign Launched in Shimla and Manali
India's Air Quality Continues to Worsen While China Takes Aggressive Actions to Improve: Study

redpajama_set_name: RedPajamaCommonCrawl


text:
On July 3, 2016 December 13, 2018 By nightmares06In Uncategorized
nightmares06:
5. Wish
Oh the fate of a sprite who can be summoned. Nixie rues this word some days.
The summons was clear, and Nixie paused in her morning ablutions to cock her head to the side.
Aretha glanced over at her sister, her wings glistening in the morning air. "You will answer?" she asked, her tone as patronizing as it ever was. She gave her wings a flick to get the precipitation from the morning rain in the rainforest off, ready for flight.
Nixie ignored the tone of voice, as she always did. Alone of her sisters, she would never miss a summons. "It is who we are," she reminded Aretha gently. "We must stop ignoring the calls."
Before Aretha could respond, Nixie twisted the air and water around herself, vanishing and following the summons as only a sprite of aeternum could. It was the only time they could travel to the human world without using the portal itself to travel, and it brought her right to her destination.
The human world was always a shock, more because of how it changed each time she was summoned than anything. The size she was prepared for; no sprite stood over four inches in height, the fire sprites were the closest to that. Nixie's wings vibrated once, and she alighted on the edge of the book used for the summons, the glow of the summoning magic fading away from her.
Brown eyes full of tears stared back at her, wide at the sight of the small sprite the summons had brought forth. Nixie waited patiently as the young girl, no more than twenty in her estimation, recovered from the realization that the summoning had worked.
"I am Nixie," she introduced herself, as she always did. "Why have you summoned me?"
Hundreds of summonings in the past left her prepared for anything that might happen. Many times, she was brought forth to heal the dying or restore the poisoned waters of war. It was her reason for creation, along with guarding the flower of gold.
The girl's face was framed by dark hair, and it bobbed as she shook her head in surprise. "B-but you… you're real? Really real?"
Nixie did not deign to answer, merely watching the area around her with a curious eye and taking in the objects used to summon her. The circle could use some work, but the symbol in the center was well-drawn.
"Okay," the girl pulled herself together and straightened. "I want you to make me the prettiest girl in school. That way they stop making fun of me! I'm not a geek!"
Nixie sighed at that. "Child, do you not know who you have summoned?" she asked gently. "I am a sprite. Wishes I do not grant. What you want is beyond my power."
"You should ignore what others say," Nixie said firmly, thinking of Aretha and her constant dislike of what Nixie persisted in doing. "All that matters is who we are to ourselves. Letting their words bother you gives them power over you."
She let her wings vibrate again, and lifted off from the book, whipping the pages and the girl's hair into a mess. The girl opened her mouth. "No, wait!"
With a twinkling, Nixie severed the summoning and vanished back into aethernum.
Nixie by aibyou
adviceaeternum spritearethabrothers apartcreatorofuniversesnixiepromptspritewater spritewish


text:
Wisconsin Cheese Makers, UW-River Falls partnering to strengthen industry education pipeline
MADISON – The Wisconsin Cheese Makers Association (WCMA) Board of Directors gathered at the University of Wisconsin-River Falls June 20 to celebrate the renovation of the school's Dairy Pilot Plant, which is now underway, and to enter into a new partnership to strengthen the industry's educational pipeline.
"WCMA members recognize investments in education as investments in the future success of the U.S. dairy processing industry," said WCMA Executive Director John Umhoefer. "We are proud to partner with UW-River Falls to support training programs that will deliver a truly skilled dairy workforce."
On June 20, Umhoefer joined UW-River Falls College of Agriculture, Food and Environmental Sciences Dean Dale Gallenberg in a pledge to work together to develop, execute, and promote new and existing workshops to meet industry needs.
This action builds upon WCMA's financial support of the UW-River Falls Dairy Pilot Plant renovation project. WCMA has directly donated $200,000, with members making generous individual donations to help UW-River Falls achieve a complete modernization of its dairy processing facilities, with new automation, piping, and equipment.
"Students learn at-the-vat in River Falls, and that experience positions them as great assets when they enter the career field," said Umhoefer. "It's critical that hands-on opportunities to learn are available to the dairy industry's next generation of leaders."
The College of Agriculture, Food and Environmental Sciences at UW-River Falls reinstated its Food Science and Technology program in 2016, and now offers a food processing technology minor. The school's dairy processing plant also hosts continuing education classes focused on pasteurization, cheesemaking, and food safety.
The UW-River Falls Foundation is still accepting contributions to this project, including monetary gifts and in-kind equipment donations.
To learn more, contact UW-River Falls Advancement Officer Julie Stucky at 715-425-4647 or via email at julie.stucky@uwrf.edu.


Because frosting really is the most important part of a cake.
Light and not too sweet, Swiss buttercream is made by cooking egg whites and sugar together over a hot water bath until the sugar has dissolved. The mixture is whipped until stiff peaks form, then butter is beaten in to become a smooth frosting. The result is a super silky, stable buttercream that not only spreads well on cakes but can be used to pipe decorations.
This buttercream also relies on egg whites but it's a bit more complicated than the Swiss version. Instead, a hot sugar syrup is slowly added to the egg whites and sugar while they are being whipped together then the butter is beat in. While almost indistinguishable from Swiss buttercream, it is slightly more rich and dense in flavor.
French buttercream follows the same method as Italian but egg yolks are used instead of egg whites. Because of this, it's richer and it has a slight golden color unlike the Swiss and Italian versions, which are very light in color, if not almost white.
The easiest one to make, many don't actually consider this one "real" buttercream. It's made by whipping softened butter together with powdered sugar, milk, and a bit of vanilla extract. While it's fluffy, it's also the most rich and sweet since it's basically just butter and sugar.


text:
Почётные звания России — почётные звания, существовавшие ранее в Российской империи и в СССР и существующие в настоящее время в Российской Федерации.

При учреждении новых государственных наград (в том числе Почётных званий) России соблюдаются геральдические и исторические традиции государственных наградных систем Российской Федерации, СССР и Российской империи.

Почётные звания Российской империи
 Заслуженный артист Императорских театров
 Мануфактур-советник
 Коммерции советник
 Почётный опекун

Почётные звания СССР

Почётные звания РСФСР

Почётные звания Российской Федерации

Почётные звания субъектов Российской Федерации

Примечания

redpajama_set_name: RedPajamaWikipedia


text:
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text:
Septentrion's Walk est un album de 15 titres du groupe Seyminhol sorti en 2005. Cinquième album du groupe, il s'agit en fait d'un digipack comprenant 2 CD. Il est profondément inspiré par l'Histoire, notamment la guerre entre Charlemagne et les Vikings. Il s'en est vendu 2500 copies en France .

Création de l'album

Après le succès rencontré par Northern Recital, le  « vrai » album du groupe Seyminhol, ses membres ont voulu réaliser une suite encore plus poussée en qualité et en professionnalisme. Kevin Kazek, le chanteur, diplômé en Histoire, s'est remis à l'écriture et a fourni l'ensemble du concept de Septentrion's Walk tant au niveau de l'histoire qu'au niveau des textes et des paroles des chansons. Nicolas Pelissier,  le guitariste et claviste, en a composé une grande partie de la musique.

Septentrion's Walk a été enregistré au studio Kirk Production par Gilles Kaufmann. Lors de son enregistrement, Seyminhol a fait appel à de nombreux invités comme la chorale de la ville de Algrange, berceau du groupe. La pochette et le livret de l'album sont encore l'œuvre de Gengis Kahn Artwork. Ceux-ci comprennent diverses illustrations, les paroles des chansons et l'histoire de l'album en français et en anglais.

Le  CD du digipack comprend l'album Septentrion's Walk proprement dit. Tout comme son prédécesseur, il est composé de 15 titres articulés autour de 3 parties : Thorgis Last Peregrination, When the North is Raging et The Wary Ravens of Hroptatyr. Ses titres et ses parties entament leur numérotation là où se terminaient celles de Northern Recital et sont entrecoupés par des interludes qui accentuent l'immersion dans le concept de l'album par les ambiances qu'elles fournissent et ses passages en langue scandinave (voir Veizla ok Drykkja). L'Histoire est au centre de l'album, tout comme pour son prédécesseur, et on peut ressentir dans la lecture du livret toute la recherche faite par le groupe concernant les faits historiques qui se sont déroulés durant la période narrée dans Septentrion's Walk.
Le  CD du digipack contient un titre chanté en français, La  Croisade, que le groupe avait enregistré à l'occasion de la compilation French Steel, une version « radio edit » de Ode to Eternity, un titre de Northern Recital, et son clip vidéo, plus diverses photos du groupe et autres fonds d'écran.
L'album sera encore une fois bien accueilli et bien noté par la presse spécialisée avec une interview en double page centrale du magazine Rock Hard .

L'album est distribué par Brennus Music.

Titres

Part 4 : Thorgis Last Peregrination

 12 : The Christian World's Death-Throes ?
 13 : A Furore Normannorum
 First Interlude : A Fatal Omen
 14 : Laments of a King
 15 : Return of the Long Snake
 16 : Veizla ok Drykkja

Part 5 : When the North is Raging

 Second Interlude : In the Winding of Thought
 17 : Godfred's Will
 18 : And the North, Now, Had the Ships
 19 : Invaders
 20 : The Bloody Rampart

Part 6 : The Wary Ravens of Hroptatyr

 Third Interlude : Dawn of the Gods
 21 : The Conquest is Marching
 22 : Breath of Fate
 Fourth Interlude : The Kiss of Dusk

Septentrion's Walk : l'histoire continue...

Part 4 : Thorgis Last Peregrination
, le monastère-évêché de Lindisfarne n'est plus. Cet îlot de christianisme, chargé de diffuser la parole divine, vient de subir le joug de Thorgis et de ses hommes. Pillant, brûlant et massacrant tour à tour, le jeune viking, guidé par le souffle violent des dieux du Nord, entend bien poursuivre ses actes de barbarie et réaliser d'autres escarmouches.
Jarrow en Northumbrie, Morganwg en Galles du sud, Kintyre et l'île de Man sont visitées puis abandonnées à la terreur et au chaos.
En 795, c'est au tour du couvent de Iona, fondé par Saint Columba, d'être saccagé. Gardien de la chrétienté en cette partie retirée du monde, le raid qu'il subit marque profondément les esprits, et à maintes reprises il croulera sous les coups assénés par le marteau des païens. L'orage et la foudre s'abattent sur les croyants, les perles de leurs chapelets sont rougis par leur sang et gisent sur un sol désolé, livré à la fureur des impies (The Christian World's Death-Throes ?).
La voix de Thorgis s'élève, il vocifère qu'il est le fléau de Dieu, la bête immonde chargée d'éradiquer le christianisme et ses représentants (A Furore Normannorum). Jusqu'en 799 il va mener une vie de forban, infligeant des blessures profondes et douloureuses à l'église et à ses disciples.
Charlemagne, ébranlé par la violence des premières attaques et par le sac de Lindisfarne, va comprendre très tôt que les nouvelles que lui apportent son conseiller Alcuin seront toujours plus brutales (A Fatal Omen). Ce grand souverain, ceint de la couronne de fer de Lombardie depuis 774, a unifié les terres d'Occident sous le signe de la croix et a déployé sa bannière sur tous les champs de bataille. Ayant massacré les Vascons, dompté les Bavarois, les Avars et déporté un grand nombre de Saxons restés fidèles à leur chef, Widukind, il découvre de nouveaux adversaires bien plus dangereux, prêts à détruire son pouvoir. Iona est tombé et le roi ne peut contenir ses larmes (Laments of a King), il sait désormais que cette lutte sera la plus terrible et la plus difficile qu'il aura à soutenir. Un bien funeste héritage pour ses descendants.
Surgissant de l'écume des flots, les vikings, vengeurs sanguinaires, s'apprêtent à déferler sur ses terres comme des vagues déchainées et dévastatrices. Charlemagne reste le seul rempart susceptible de les briser, il incarne la foi, il est le ministre de Dieu sur Terre. Mais pour combien de temps ?
À l'aube de l'an 800, Thorgis décide de reprendre la mer. Il navigue sans relâche vers son pays natal pour informer son hôte de ses projets. Embarqué sur son long serpent (Return of the Long Snake), il rêve de nouvelles gloires et de conquêtes. Odin a toujours guidé ses gestes et lui a assuré la victoire. Il protège désormais sa cargaison pleine de butin et accompagne son trajet. Sous l'égide du dieu borgne, c'est un retour sans encombre pour l'enfant de l'écume. Son destin (« gaefa ») exceptionnel l'a mené en maints lieux et, comme son équipage, il a mûri avec le vacarme des vagues dont le sel a tanné sa peau. Fidèle à sa promesse, il revient vers les siens et s'apprête à recevoir de Godfried de Danemark tous les honneurs dignes d'un grand guerrier. C'est un homme complet et véritable, fier et courageux, il pourra dire au vieux Thorvald, son père, qu'il est habile aux neuf exercices. Il incarne le nord mythique et héroïque, il peut « fara a vikingu » (aller en expédition viking) car il est né chanceux (hamingja), destiné à accomplir une œuvre d'envergure, dont le renom atteindra l'enclos des Ases, Ásgard.
Après plusieurs semaines de navigation, Thorgis arrive enfin chez Godfried, le nouveau Konungr du Jutland, qui a eu vent des exploits du jeune aventurier. Le souverain est majestueux, et son accueil est grandiose et magnifique. Un immense banquet se prépare en l'honneur de celui qui a blessé la chrétienté (Veizla ok Drykkja). Il se déroulera en pleine période de « vetnaetr ». la bière («bjorr ») a été généreusement brassée, le gibier abonde et les jeux organisés comblent les invités. On boit aux ancêtres, on prête serment, on se félicite, on déclame des poèmes dans une ambiance de liesse irréelle, bercée par le son d'une musique entêtante. Godfried prend alors la parole : « vous devez partager couteaux et quartier de viande ». Thorgis répond : « comme parents mais non ennemi ». Ragnar son compagnon poursuit : « vous devez être hommes vivant en paix, aux beuveries et aux banquets ».
Thorgis et Godfried s'échangent des paroles amicales, ils discutent longuement et apprennent à s'apprécier. Pour le premier, le roi est le seul homme capable de lever une armée gigantesque prête à déferler sur l'occident christianisé. Son charisme et sa prestance inspirent le respect auprès de tous. Une grande opération militaire pourrait stopper toute volonté d'extension chrétienne et d'évangélisation du Septentrion.

Part 5 : When the North is Raging

Lorsque le jour se dévoile, Godfried de Danemark pense à ce que lui a conté Thorgis sur les chrétiens et leurs richesses, sur leur roi dont le bras armé de l'épée du Seigneur veut abattre les croyances du Nord. Il se souvient encore de Siegfried Ier de Danemark qui, lui aussi, avait accueilli un rescapé saxon, dont le peuple avait été anéanti par Charlemagne. Il rumine ses projets en silence (In the Winding of Thought), car il sait que s'il ne réagit pas le premier, un jour il devra faire face à une armée immense et redoutable. Il décide donc d'utiliser les prochaines années de son règne pour renforcer son territoire et ses troupes et mettre sur pied une force militaire impressionnante. Il s'organise, se structure (Godfred's Will') mais doit d'abord gouverner comme un roi sacré et bon, qui agit pour le bien des siens. Il a été élu, choisi par son peuple de qui il tient son pouvoir pour être arsaell et fridsaell. Lorsqu'il combattra, il ne pourra remporter la victoire et devenir sigrsaell que s'il a bien rempli les deux premières fonctions. Il sera alors jugé pour ses actes, pour sa générosité aussi, car il est le « seigneur des anneaux » d'or et d'argent qu'il porte au bras et qu'il doit briser pour les distribuer à ses sujets. Il sait tout cela et sa conduite sera exemplaire car il veut sa place à la table de Odin.
804, le moment est venu de montrer à Charlemagne que le Nord est puissant (And the North, now, had the Ships). Godfried de Danemark décide de faire une démonstration de force par terre et par mer à Slesthorp en Frise (région historique). C'est le plus grand port de commerce de l'époque. Plus de deux cents knerrir sont réunis et des milliers d'hommes à pied et à cheval défilent comme pour défier l'Empereur d'Occident. Celui-ci ne réagit pas. Godfried est confiant, mais préfère attendre avant d'attaquer. Il sait que son armée peut se lever rapidement, comme une tempête sur l'océan, et tout dévaster sur son passage. L'apathie de Charlemagne le sert et lui permet de mûrir sa stratégie.
808, après quatre années d'investissement et de discipline, Godfried envahit l'ancien territoire saxon pris par Charlemagne, sur lequel furent installés des Slaves christianisés, les Obodrites (Invaders). Il brule, pille et détruit leur ville commerçante, Rerik, soumet l'ensemble du peuple, déporte les marchands et s'assure que leur chef Drosuk, a bien été capturé et emmené par sa garde. Les guerres menées par Charlemagne aux confins méridionaux de son empire l'éloigne de ces dangers. C'est donc son fils, Charles le Jeune, qui est envoyé pour résoudre le problème et punir les envahisseurs. Mais il tergiverse et parade plus qu'il ne combat. Godfried a su démontrer ses capacités de réactions et son habilité à diriger ses troupes. Il sait que son adversaire le craint mais préfère pour le moment fortifier ses frontières pour s'assurer une défense efficace en cas de conflit plus général. Il se hâte de faire ériger un puissant rempart, le Danevirke, dont le but est de barrer la route du Nord aux soldats de Charlemagne (The Bloody Rampart). Muni de tours et de plusieurs portes, c'est une défense imposante, symbole de la grandeur viking, le sanctuaire inviolable de l'armée du Nord.
Charlemagne voit en Godfried un ennemi redoutable. Il sait jauger le charisme et la valeur de ses adversaires et ce souverain est un démon prêt à tout. Il lui propose donc de traiter, mais Godfried refuse. Maintenant qu'il possède des troupes aguerries, des bateaux nombreux et rapides et une protection monumentale, il ne craint personne, ni même cet Empereur qu'il dévisage et qu'il rabroue. La volonté de fer du viking est inébranlable, c'est un roi juste et droit qui a su prouver à son peuple sa vaillance et sa détermination. Fort de ses réussites, il se prépare à lancer sur l'Empire Carolingien ses soldats d'élites, ses guerriers-fauves. Plus rien ne semble pouvoir l'arrêter désormais et personne ne le fera changer d'avis.

 Part 6 : The Wary Ravens of Hroptatyr

C'est pourtant Charlemagne qui va réagir le premier. En 810, il choisit d'établir son point d'appui au-delà de la rivière Elbe (fleuve), à Itzehoe. C'est, pense-t-il, depuis cette base avancée qu'il pourra défendre son Empire.
Godfried s'énerve; s'impatiente, il vient d'être devancé par son ennemi mais il sait que Odin préside déjà aux opérations et que ses corbeaux, Hugin et Munin, guideront son armée vers la victoire. C'est l'aube d'une nouvelle époque, les dieux du Nord s'apprêtent à chasser le Christ usurpateur et Charlemagne de leurs terres. Ils insufflent aux guerriers force et courage et à leur chef la fougue nécessaire qui lui permettra de mener à bien son dessein (Dawn of the Gods). Que les Dieux chassent le roi de ce pays, qu'ils lui revaillent le pillage de mes biens, que Odin et les puissances se courroucent contre lui. Freyr et Njörd, faites fuir de ses propriétés l'oppresseur du peuple. Que les Ases du pays se fâchent contre l'ennemi des hommes qui viole le sanctuaire. Avec les Dieux comme protecteurs, Godfried se sent invincible et sa conquête peut débuter (The Conquest is Marching), cette fameuse conquête dont il rêve depuis qu'il a entendu les propos de Thorgis, maintenant disparu. Il souhaite honorer sa parole et le serment qu'il a fait au jeune homme et décide de mener ses troupes au combat. Le Septentrion est en marche.
Deux cents longs navires fendent les flots et apparaissent sur le rivage de la Frise (province). Les guerriers se déversent comme un océan de sang sur le pays. Ils massacrent encore et encore, brandissant d'innombrables haches étincelantes qui s'abattent sur les crânes des malheureux. La réussite est totale. Godfried, une fois installé sur ces terres nouvelles, réclame un tribut, le danegeld, et envoie des hérauts prévenir Charlemagne que son prochain objectif est la destruction de son palais d'Aix-la-Chapelle, qu'il brulera lui-même de sa main en même temps que la cathédrale. La menace est prise au sérieux. Alcuin transmet le funeste message à son maître qui décide d'user de trahison pour arriver à ses fins. Il faut mettre un terme à ce conflit qui commence à prendre une tournure dramatique. L'Empereur ne veut pas se ridiculiser devant ces hommes du Nord. Il incarne la puissance divine et universelle, il est garant de la foi. Mais plutôt que de combattre et de risquer sa vie, il préfère la trahison.
Odin sait déjà quel sort est réservé à Godfried. Il s'exclame : « il faut que tu meurs » car il possède le savoir et connait toutes les destinées (Breath of Fate). La vie du roi s'arrête ici, il n'accomplira plus aucune prouesse, comme mortel en tout cas. Il meurt cette année qui avait vu naître sa conquête, assassiné par la main de son propre garde du corps, acheté par Charlemagne. Il ne put donc terminer la guerre qu'il avait commencé, et ses rêves périrent avec lui. Déjà le chant des valkyries fend l'atmosphère et inonde l'esprit de Godfried. Une voix fragile et suave résonne dans sa tête : « meurent les biens, meurent les parents, et toi tu mourras de même, mais la réputation ne meurt jamais, celle, bonne, que l'on s'est acquise ». Lui ne croit pas à sa mort, mais la voix est plus présente, il la sent pénétrer dans son âme. Elle le conduit, le guide vers le palais de Walhalla, là où les braves vivent à jamais. Godfried aura la joie de gouter l'hydromel de l'oubli. Son corps céleste brillera au firmament et son nom restera gravé pour l'éternité dans toutes les mémoires. Chaque vie terrestre se souviendra de ses actes et nombreux sont ceux qui chantent déjà le crépuscule du héros (The Kiss of Dusk'').
Septentrion's Walk se termine sur la mort de Godfried de Danemark. Son successeur et neveu, Hemming de Danemark, devient roi à son tour. Il va traiter avec Charlemagne et signer un traité de paix avec l'Empereur en 811. Celui-ci confirme et fixe l'établissement de la frontière sud du Danemark sur l'Eider (fleuve).

Composition du groupe

 Kevin Kazek : chant
 Christophe Billon-Laroute : basse
 Nicolas Pélissier : guitare, claviers, piano, chœurs
 Julien Truttmann : batterie

Invités

 Christian Le Menn et Bruno Guénan : Scottish bagpipes, practice pipe, small pipe, Irish flute
 Malik Tounsi : percussions
 Roger Baumann : violons
 Déborah Hofer : chœurs féminins, voix de la Valkyrie
 Chorale « Les Chœurs Classiques » de la ville de Algrange

Notes et références

Album musical sorti en 2005


text:
Policy, Law & Regulation
Gezi Park Protests Push Turkey's Media into the Spotlight (Updated)
By now, you'll have heard – perhaps via international media, or on Twitter or Facebook – about the protests that started in defence of one of Istanbul's last remaining green spaces, and have now, in response to heavy-handed policing and broader worries about democratic rights, spread across many cities in Turkey.
I've been following the torrent of communication and coverage on Twitter mainly via a mixture of local and international academics, NGO people and journalists – ranging from Asli Tunç, Yaman Akdeniz, Zeynep Tufekci (who has also blogged a rapid, excellent analysis), and Burcu Baykurt [UPDATE: Burcu has written a very comprehensive post detailing the main media reform issues emerging from the Gezi Park protest movement] to Aaron Stein, Benjamin Harvey, Hugh Pope and Amberin Zaman – as well as feeds like 140 Journos. (Feel free to tweet me or @mediapolicy with further suggestions.)
One of the most widely discussed (on Twitter) aspects of these protests has been the mainstream Turkish media's perceived failure to cover the protests fairly, adequately, or in some cases at all, leading Bloomberg's Benjamin Harvey to tweet the following:
Turks being confronted with the now-undeniable deficiencies of their media may be one of the most important aspects of these protests.
In report after report after report [UPDATE: 3 June 2013 – see comment below for further resources], those deficiencies – and the reasons for them – have been thoroughly, exhaustively anatomised. Asli Tunç and Vehbi Görgülü's Mapping Digital Media Turkey report (2012), for example, gives a very comprehensive overview of the Turkish media sector and its travails, and is part of a 50-country series that mediapolicy.org readers know well. The Carnegie report (also supported by OSF) by Marc Pierini and Markus Mayr on Press Freedom in Turkey, takes a different approach. Introducing the report in January 2013, Pierini wrote:
I didn't conduct yet another inquiry into press freedom. More modestly, I analyzed all the reports published on the subject by governmental and non-governmental, Turkish and foreign entities during the last two years. Although they had different focuses and methodologies, all these reports convey one single image: Turkey's record is bad because it fares well below the country's democratic credentials and is hurting the nation economically and diplomatically on the international scene.
One has to hope that the Gezi Park crisis will lead in some way to genuine reforms in Turkey's media policy and media sector, freeing journalism to play a stronger role in the country's democracy. As Turkey is one of mediapolicy.org's focus countries, we'll definitely follow developments and build up useful resources over the coming weeks and months. We'd love to hear from academics, researchers, civil society and journalists interested in sharing perspectives on media policy and reform in Turkey – please get in touch on email or Twitter.
Somalia Conference: promises of action on media freedom, safety of journalists
"We agreed on the important role a free and independent media should play in Somalia, and welcomed the Federal Government's commitment to investigate and prosecute those responsible for the killing of journalists, and to promote press freedom."
(Somalia Conference 2013: Official Communiqué at GOV.UK.)
Yesterday's conference communiqué was unambiguous on the need to protect the media in Somalia. Here's a selection of the international groups working on media policy issues in Somalia, and a couple of recent reports about the media environment in the country – let us know via the comments what we're missing, and we'll update the list.
– EU fact sheet from Dec 2012 detailing some of the activities of the Somalia Media Support Group of donors, NGOs, and international organisations, and insights into the future strategy for supporting Somalia's media sector to 2015
– CIMA's compilation of where Somalia sits in various international press freedom rankings
– Somalia sits in 2nd position in the CPJ's Impunity Index for killings of journalists
– Article 19 has been tracking the development of Somalia's media law, and recently held a conference on protection of journalists in Mogadishu
– the BBC's media development arm, BBC Media Action, produced a media environment analysis and a policy briefing about the role of the media in 2011
– the InfoAsAid project we featured a couple of weeks ago includes a pretty comprehensive Somalia media/telecoms landscape report from early 2012 (also here)
– the Center for Law and Democracy published a media law and policy review for Somalia in late 2012 (here's a piece from Albany Associates about the report)
– Albany Associates is also supporting the government and the UN's AMISOM more broadly on communications
– NORAGRIC is a less usual source for media landscape information, but here's their March 2012 report on Somalia
– Danish NGO IMS supports a Somali radio station, Radio Ergo
– Global Voices covers Somalia with reasonable regularity, as has the Guardian's Data Blog
And finally, here are some stats on social media usage in Somalia, courtesy of Social Bakers.
The mobile web and press freedom
Last month, the Global Network Initiative (GNI) – a multi-stakeholder coalition of ICT companies, civil society organisations, investors and academics – signed a cooperation agreement with another body called Industry Dialogue, or, to give it its full name, Telecommunications Industry Dialogue on Freedom of Expression and Privacy. Why should journalism and media policy people care about this? Two reasons…
First, as Rebecca Mackinnon has pointed out on this site before, a free, open internet is crucial for press and media freedom – and that includes the mobile internet:
All news organisations – whether their final news product is distributed online, in print, or broadcast – are increasingly dependent on broadband and mobile networks to gather, transmit, compile, and disseminate their reports and investigations. Whether the internet remains open and globally inter-operable affects the ability of all news organisations to obtain fair access to increasingly global or geographically-dispersed audiences.
Beppe Grillo: Italian population living in a "gigantic 'Truman Show'"
Media, Policy, Law & Regulation
So the long-standing debate about the independence of Italy's public broadcaster, RAI – addressed in Mapping Digital Media: Italy, and by the Open Media Coalition, Italy's media reform movement – has now received the Grillo treatment.
Italian comedian Beppe Grillo last week accelerated debate in Italy about the independence of broadcast media and journalism from political interests, releasing poll results showing that, out of 95,000 responses, 99% of respondents wanted a public broadcast channel free from political meddling, and 52% wanted to see more investigative journalism about domestic issues.
Under the hashtag #raisenzapartite ("RAI (the public broadcaster) without the parties"), Grillo wrote a blog post asserting that:
"a part of the Italian population is living in a gigantic "Truman show", and responsibility for this is entirely due to Italian journalists, with the usual few exceptions and in a country like ours, these exceptions deserve every possible praise. […] RAI has to be reorganised and transformed into a public service following the model of the BBC without any connection to the parties, without advertising, producing quality content that has mainly been produced in-house and not like now, when it's entrusted to external companies with the building up of one set of costs on top of another. In Parliament, the M5S, in accordance with its programme, will propose the establishment of a single RAI channel, without any connection to the parties and without advertising. It proposes the sale of the other channels."
It's sure to be a topic of conversation at the International Journalism Festival in Perugia in two weeks, as Italian journalism already is over at the LSE's POLIS project. In the meantime, take another look at the MDM Report, which proposed a wider range of media reform measures that could restore independence to Italy's media:
Clippings, News, Policy, Law & Regulation
Consultation Paper on Issues relating to Media Ownership in India: Written comments on the consultation paper are invited from the stakeholders by 8th March 2013 and countercomments, if any, by 15th March 2013.
Consultation Paper : Telecom Regulatory Authority of India – have your say by 8th March 2013


What stream content is accepted?
It is expected that you to provide a stream that contains more than just GPS and GLONASS. M-GEX has a particular interest in Galileo, SBAS, QZSS and COMPASS observations. Whatever comes in addition to GPS and GLONASS is appreciated.
Which stream upload format is accepted?
Streams in HP-MSM format are preferred. HP-MSM stands for 'High Precision Multiple Signals Messages'. The format is currently under development in RTCM. If your firmware does not yet support HP-MSM, we would do our best to handle incoming vendor specific raw proprietary stream formats like LB2, RT27, SBF, ATOM etc.
How would I have to upload my stream?
Only streams made available via Ntrip transport protocol are accepted. Contact [igs-ip@bkg.bund.de] to receive account details for uploading a stream to Ntrip broadcaster http://mgex.igs-ip.net/home. Note that all modern geodetic dual frequency receivers nowadays support Ntrip through their firmware.
How can I retrieve streams in HP-MSM format?
Please register for an M-GEX Ntrip user account at http://register.rtcm-ntrip.org. Note that streams in vendor specific proprietary formats are not meant for further dissemination.
What about RINEX v3 files converted from streams?
We convert highrate RINEX version 3 files from all available HP-MSM streams. These files are compiled in 15 min batches with 1 sec sampling rate and saved in our GNSS Data Center at ftp://igs.bkg.bund.de/<project>/<highrate_v3>.


I'll be donating a pice to this – Molly and Paul have been doing great work over there for years and I'm happy to help.
May 14th, the Warehouse is hosting a "Light up the Warehouse" party to raise funds to pay for new lights and sound equipment for the theater.
A preview party honoring the artists will be held May 6th. Artists wishing to donate a work of art should email Molly Ruppert at ruppertm@erols.com.
On May 14, the "Light Up the Warehouse" Fundraising party will take place at Warehouse. 100 Sponsor tickets will be available at $150.00 each, which will include and original work of art. Each $150.00 ticket holder may then buy a companion ticket for $50.00 entitling the companion to the dinner, drinks, music, live and silent actions. There are also general tickets available for $100.00 for dinner, drinks, music, live and silent actions. The special artist price for the same is $50.00.
Artists wishing to donate a work of art, or collectors who'd like to get a ticket, should email Molly Ruppert at ruppertm@erols.com or contact her at 202/257 5989 or 202/783-8263 or 202/783-3933.


Why is seraph and embedded crowd separated?
i read the cook book which say seraph is the authentication framework.
and seraph delegates user management.
but embedded crowd functions authentication includes password matching.
so why is seraph and embedded crowd separated? i think seraph and embedded crowd are authentication framework.


text:
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US Department of Health & Human Services List of State Health Departments
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CDC Resources for Emergency Preparedness
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FEMA Resources for Emergency Management In Your Area
TAMU Disability Tips for First Responders
Helpful Software Links:
Viewers, Plug-ins and Browser Information
Numerous utilities for the PC are available as freeware or shareware and there are some that may assist you. Please note that HHSC provides information via this webpage as a public service. Accessible Emergency Information does not endorse any of the products, vendors, consultants, or documentation referenced. Any mention of vendors, products, or services is for informational purposes only. Accessible Emergency Information is not responsible for the privacy practices or the content of such websites.
http://www.adobe.com/prodindex/acrobat/readstep.html
Adobe's Acrobat Reader allows you to view files in PDF format.
Adobe Acrobat PDF Converter
http://www.adobe.com/products/acrobat/access_onlinetools.html
Adobe's online converter program changes PDF files to HTML or text files "on-the-fly" -- the resulting conversion will be returned to you immediately in your browser application.
Viewers for Microsoft Word files
http://office.microsoft.com/downloads/2000/wd97vwr32.aspx
The Microsoft Word® Viewer 97/2000 enables users without Word to open view and print Word documents. Microsoft® viewers are small, freely distributable programs. Please read the Microsoft Office information to download the correct software for your situation.
Viewers for Microsoft Excel files
http://office.microsoft.com/downloads/2000/xlviewer.aspx
The Microsoft Excel® Viewer 97/2000 enables users to open view and print Excel 97 and Excel 2000 spreadsheet files. Microsoft® viewers are small, freely distributable programs. Please read the Microsoft Office information to download the correct software for your situation.
Viewers for Microsoft PowerPoint files
http://office.microsoft.com/downloads/2002/ppView97.aspx
The Microsoft Powerpoint® Viewer enables users to open and view PowerPoint 95, 97, 2000 and 2002 presentations. Microsoft® viewers are small, freely distributable programs. Please read the Microsoft Office information to download the correct software for your situation.
Viewers for Microsoft Visio files
http://office.microsoft.com/downloads/2002/vViewer.aspx
The Microsoft Visio® Viewer 2002 allows anyone to view Visio drawings and diagrams (if created with Visio 5, 2000, or 2002) inside their Microsoft Internet Explorer version 5.0 or later Web browser. Microsoft® viewers are small, freely distributable programs. Please read the Microsoft Office information to download the correct software for your situation.
http://www.winzip.com/
This is a popular Windows utility for creating and working with .ZIP files. WinZip From Nico Mak Computing, Inc.
Translate to Spanish or other languages / Traduzca a los españoles
http://babelfish.altavista.com/
Copy the website address (URL) and AltaVista can translate a webpage, or specific text, from English to another language. Other language translations are also available.
http://www.microsoft.com/windows/
Microsoft reports that the latest version of Microsoft's browser software incorporates numerous new features to make Web surfing easier and more productive. Start by choosing the correct operating system (version of Windows) then following links to download Internet Explorer.
http://home.netscape.com/download/
Includes Navigator, Messenger, Composer, AOL Instant Messenger 2.0,
RealPlayer G2, PalmPilot Synch tools (Windows only), plus multimedia
plug-ins.
Netscape Plug-ins
http://home.netscape.com/plugins/
Plug-ins are software programs that extend the capabilities of Netscape Navigator in a specific way - giving you, for example, the ability to play audio samples.
http://www.realplayer.com/
RealPlayer is a useful streaming media player.
http://www.apple.com/quicktime/
QuickTime by Apple makes it possible to receive and play a variety of streaming audio and video formats.
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