text:
\subsection*{Organisation}
The organisation of the rest of the paper is as follows: In Section 2, we recall some basic definitions related to topological complexity and LS category. In Section 3 we recall the definition of higher $\mathcal{D}$-topological complexity and prove some general properties. In section 4, we introduce the invariant $\tTC{n}{X}$ and use it to prove Theorem A. In the last section, we introduce higher strongly equivariant complexity, obtain its properties, and prove Theorem B,  Theorem C.

\begin{ack}
The first author was supported by PhD research fellowship of Indian Institute of Technology, Kanpur.
\end{ack}

\end{section}

\begin{section}{Preliminary}
Here we review basic concept of LS-category, topological complexity and higher topological complexity of a space $X$. We also recall some equivariant analogues. For details we refer to \cite{ sva, Col, ColG, far, cor, rud, Mar}.

\begin{mysubsection}{LS-category and topological complexity}
Let $q : E \to B $ be a onto map, then the \emph{sectional category} of $q$ is denoted by \emph{secat$(q)$} and define as the minimal positive integer $k$ such that we have an open cover $\{U_i\}_{i = 1}^k$ for $B$ and on each open subset $U_i$ we have a continuous map $s_i : U_i \to E$ with $q \circ s_i : U_i \to B$ is homotopic to the inclusion $\operatorname{Id}_{U_i} : U_i \hookrightarrow B$. The map $s_i$ is called local section for $q$. If $q : E \to B$ is fibration then $\secat(q) =  \genus(q)$, where $genus(q)$ of the fibration $q$ is the minimal positive integer $k$ such that we have an open cover $\{U_i\}_{i = 1}^k$ for $B$ and on each open subset $U_i$ we have a continuous map $s_i : U_i \to E$ satisfying $q \circ s_i = \operatorname{Id}_{U_i} : U_i \hookrightarrow B$. We denote by $P_0X$ be the space of all paths in $X$ starts from some fix point (say $x_0$) and $PX = X^I$ be free path space of $X$. Consider the fibrations $$p_0 : P_0X \to X, ~~\gamma \to \gamma(1);~~p : PX \to X \times X, ~\alpha \to (\alpha(0), \alpha(1)).$$
 \begin{definition}
The \emph{Lusternik-Schnirelmann category} (\emph{LS-category})  of $X$ is defined as $\operatorname{cat}(X):=\genus(p_0)$. The \emph{topological complexity} of $X$ is $\operatorname{TC}(X) := \genus(p)$.
 \end{definition}

For $n\geq 2,$ let $I_n$ denotes the wedge of $n$ intervals $[0, 1]_j, j = 1, 2, \cdots, n$, where $0_j \in [0, 1]_j$ are identified. Consider the mapping space $X^{I_n}$ and the fibration
\begin{equation}\label{een}
e_n : X^{I_n} \rightarrow X^n ,~~~e_n(\alpha) = (\alpha(1_1), \alpha(1_2),\cdots,\alpha(1_n)).
\end{equation}

\noindent The \emph{n-th topological complexity}  of $X$ is defined to be $\operatorname{TC}_n(X) :=\genus(e_n)$. It can be defined alternatively as  $\operatorname{TC}_n(X) = $ genus$(e'_n)$, where
\begin{equation}\label{eenprime}
e'_n : X^{I} \rightarrow X^n ,  ~~~~e'_n(\alpha) = (\alpha(0), \alpha(\frac{1}{n-1}), \alpha(\frac{2}{n-1}),\cdots,\alpha(1)).
\end{equation}
 This is because $e_n$ and $e_n'$ are both fibrational replacement of the diagonal map $X\to X^n$. Clearly $\operatorname{TC}_2(X)$ is nothing but $\operatorname{TC}(X)$.

Topological complexity is closely related to LS-category, satisfying the relation
\begin{equation}\label{eqineq}
\operatorname{cat}(X^{n-1}) \leq \operatorname{TC}_n (X) \leq \operatorname{cat} (X^n) \leq \operatorname{TC}_{n+1} (X).
\end{equation}
\noindent It is clear from the above inequality that $\{\operatorname{TC}_n(X)\}$ is a non-decreasing sequence. If a space $Y$ is homotopy equivalent to $X$, then $\operatorname{TC}_n (Y) = \operatorname{TC}_n (X)$ for any $n \geq 2$. Consequently, $X$ is contractible  if and only if $ \operatorname{TC}_n(X) = 1$ for any $n \geq 2$.
\end{mysubsection}

\begin{mysubsection}{Equivariant LS-category} Throughout the paper, $G$ will denote a discrete group.  A topological space $X$ with an action of a group $G$ is called a $G$-space.  A continuous map $\phi: X\to Y$ between $G$-spaces is called a $G$-map (or an equivariant map) if $\phi(gx)= g\phi(x)$ for all $g\in G$ and $x\in X$. The set $\mathcal{O}(x) = \{ gx \ ; g \in G \}$ is called the \emph{orbit} of $x\in X$ and $G_x = \{ g \in G ; gx = x \}$ is called the \emph{isotropy group} at $x$.
For a subgroup $H$ of $G$, the $H$-\emph{fixed point set} of $X$ is given by $$X^H = \{x \in X ; hx = x \text{ for all } h \in H \}.$$
We call $X$ is \emph{$G$-connected} if the $H$-fixed point set $X^H$ is path-connected for every subgroup $H$ of $G$.  Let $Y$ be an another $G$-space and $\phi, \psi : X \to Y$ be two $G$-maps. Then $\phi$ is said to be $G$-homotopic to $\psi$, written as $\phi \simeq_G \psi$, if there is a $G$-map $F : X \times I \to Y$ with $F(x, 0) = \phi(x)$ and $F(x ,1) = \psi(x)$, where $G$ acts trivially on $I$ and diagonally on $X \times I$. Two $G$-spaces $X,Y$ are called $G$-homotopy equivalent if there are  $G$-maps $\phi: X\to Y$ and $\phi': Y\to X$ such that $\phi\circ\phi' \simeq_G \operatorname{Id}_Y$ and $\phi'\circ\phi \simeq_G \operatorname{Id}_X.$

A subset $U\subset X$ is called \emph{G-invariant}
if $gU \subseteq U$ for all $g \in G$.  Such a $U\subseteq X$ is called \emph{$G$-categorical} if there exists a $G$-homotopy $F: U\times I \to X$ such that $F(-,0)$ is the inclusion map $U \hookrightarrow X$ and Image$(F(-,1)) \subset \mathcal{O}(x')$ for some $x' \in X$. We say $X$ is \emph{$G$-contractible} if $X$ is $G$-categorical.

\begin{definition}
The \emph{G-equivariant LS\mbox{-}category} of $X$, denoted by $\operatorname{cat}_G(X),$
 is the minimum positive integer $k$ such that $X$ can be covered by $k$ open sets $\{U_1, U_2, \cdots, U_k\}$, each of which is $G$-categorical.
 \end{definition}

\noindent Clearly $X$ is $G$-contractible if and only if $\operatorname{cat}_G(X) = 1$. The $G$-equivariant category $\operatorname{cat}_G(X)$ is a $G$-homotopy invariant. Therefore if $X$ is $G$-homotopy equivalent to a point then $\operatorname{cat}_G(X)=1$, i.e. $X$ is $G$-contractible.
The following lemma gives the converse implication.

\begin{lemma}\label{lemma}
 For a $G$-connected $G$-space $X$ with $X^G \neq \phi$, the $G$-contractibility of $X$ implies $X$ is $G$-homotopy equivalent to a point (in $X^G$).

\end{lemma}

\begin{proof}

Since $X$ is $G$-contractible, there is a $G$-homotopy $F'_t : X \to X$ such that $F'_0 = \operatorname{Id}_X$ and $F'_1(x) \in \mathcal{O}(x')$ for some $x' \in X$. Consider  an element $x_0\in X^G$ and the isotropy group $H = G_{x'}$ at $x'$. Then the both elements $x', x_0 \in X^H$. Since $X$ is $G$-connected so $X^H$ is path-connected. Fix a path $\gamma : I \to X^H$, from $x'$ to $x_0$. Note that $H \subseteq G_{\gamma(t)}$ for all $t \in I$. Define a homotopy $F''_t : \mathcal{O}(x') \to X$ by $F''_t(g.x') = g.\gamma(t)$, where $g \in G$. Then $F''_t$ is well defined and $F''_0 = \operatorname{Id}_{\mathcal{O}(x')}$ , $F''_1(g.x') = x_0$. Define another homotopy $F_t = F'_t * F''_t: X \to X$, where $$F'_t * F''_t(x) = \begin{cases}F'_{2t}(x) &\mbox{if } 0 \leq t \leq \frac{1}{2} \\
F''_{2t-1}(F'_1(x)) &\mbox{if }\frac{1}{2} \leq t \leq 1 , \end{cases}$$ then $F_t$ is $G$-equivariant with $F_0 = \operatorname{Id}_X$ and $F_1(x) = x_0$ for all $x \in X$.

\end{proof}
\begin{example}\label{ecat}
Let $\mathbb{Z}_2$-acts on $S^n, n\geq 2$ by reflection. Then $\operatorname{cat}_{\mathbb{Z}_2}(S^n) = 2.$   Assume the reflection keeps the hyperplane perpendicular to $x_n$ fixed. Take $U: x_0 > -\frac{1}{2}\subset S^n $ and $V: x_0 < \frac{1}{2}\subset S^n$. Then usual contraction of $U,V$ are equivariant. So $U,V$ are $\mathbb{Z}_2$-categorical hence $\operatorname{cat}_{\mathbb{Z}_2}(S^n) \leq 2.$ If $\operatorname{cat}_{\mathbb{Z}_2}(S^n) = 1$, then by Lemma \ref{lemma} $S^n$ is $\mathbb{Z}_2$-homotopy equivalent to a point, and in particular homotopy equivalent to a point which is not true. So $\operatorname{cat}_{\mathbb{Z}_2}(S^n) =2.$
\end{example}

\noindent We will need the following lemma in later section. We refer to  \cite[Theorem 3.16]{ColG}, \cite[Theorem 2.23, Example 6.5]{sarkar1}, \cite[Proposition 2.29]{sarkar} for the proof of the lemma.

\begin{lemma}\label{lcatg}
Suppose $Y_i$ is a $G_i$-space for $i=1,2$. Consider $Y_1\times Y_2$ as $G_1\times G_2$-space with the product action.  If $Y_1^{G_1}\neq \phi$ and $Y_2^{G_2}\neq \phi$ then $$\operatorname{cat}_{G_1\times G_2}(Y_1\times Y_2) \leq \operatorname{cat}_{G_1}(Y_1) + \operatorname{cat}_{G_2}(Y_2) -1.$$

\end{lemma}

\end{mysubsection}

\begin{mysubsection}{Deformable subset and $r$-cover} The topological complexity can be interpreted using deformable subsets. Let $A, U\subseteq X$. We call $U$ is \emph{$A$-deformable} if there is a homotopy $h_t : U \to X$ with $h_0 : U \hookrightarrow X$ is inclusion and $h_1(U) \subset A$.  An open cover $\mathcal{C} = \{U_1, U_2, \cdots, U_{r}\}$ is called \emph{$A$-deformable} if each $U_i$ is $A$-deformable. For a $G$-space $X$, let $A, U$ be invariant subsets. Then $U$ is called \emph{$A$-equivariantly deformable} if the above homotopy is an equivariant homotopy. It is known that the topological complexity $\operatorname{TC}_n(X)$ of a space $X$ is the minimum number $k$ such that there is a $\Delta(X)$-deformable open cover of $X^n$ containing $k$ open sets, where $\Delta(X) = \{(x, x,\cdots, x)\in X^n~; x\in X \}$.

\noindent Now we recall some basic results about open covers which are described in (\cite{alex, kolmo, ostrand}). An open cover $\mathcal{C} = \{U_1, U_2, \cdots, U_{r+r'}\}$ of a space $X$ is called \emph{$r$-cover} if every subcollection of $r$ sets from $\mathcal{C}$ also a cover of $X$. We have the following simple observation.

\begin{lemma}\label{lemmacover}
 Let $\{U_1, U_2, \cdots, U_{r+r'-1}\}$ be an $r$-cover and  $\{V_1, V_2, \cdots, V_{r+r'-1}\}$  be an $r'$-cover of a space $X$, then  $\{W_1, W_2, \cdots, W_{r+r'-1}\}$  covers $X$ where $W_i = U_i \cap V_i$.

\end{lemma}

\begin{proof}  Let $x\in X$.  By \cite[Proposition 2.1]{alex} an open cover $\mathcal{C} = \{U_1, U_2, \cdots, U_{r+r'-1}\}$ is an $r$-cover of a space $X$ if and only if each $x \in X$ is contained in at least $r' $ sets of $\mathcal{C}$.  Hence there is a subcollection $\{U_{i_1}, U_{i_2}, \cdots, U_{i_{r'}}\}$ of $\{U_1, U_2, \cdots, U_{r+r'-1}\}$ each of which contains $x$. Then the set  $\{V_{i_1}, V_{i_2}, \cdots, V_{i_{r'}}\}$ covers $X$. So $x\in V_{i_k}$ for some $k\in \{1, 2, \cdots r'\}$ and hence $x\in U_{i_k} \cap V_{i_k} = W_{i_k}$.

\end{proof}
We will need the following result in last section.
\begin{prop}\cite[Theorem 2.4]{alex} \label{propdeform} Assume $F$ is locally compact metric space and $A\subset F$. If $\{U'_i\}_{i=1}^r$ is a $A$-deformable open cover of $F$, then for any $r'\geq 0$ there is a $A$-deformable open $r$-cover $\{U_j\}_{j=1}^{r+r'}$ of $F$ such that $U_j = U'_i$ for $i=j \leq r$ and for $j > r, \ U_j =\sqcup_{i=1}^{r} V_{i}$ is a disjoint union with $V_i \subset U'_i$.

\noindent If $G$ acts on $F$ and $A$ is $G$-invariant and $\{U'_i\}_{i=1}^r$ is a $A$-equivariantly deformable open cover of $F$ by $G$-invariant sets, then for any $r'\geq 0$ there is a $A$-equivariantly deformable open $r$-cover $\{U_j\}_{j=1}^{r+r'}$ of $F$ by $G$-invariant sets such that $U_j = U'_i$ for $i=j \leq r$ and for $j > r, \ U_j = \sqcup_{i=1}^{r} V_{i}$ is a disjoint union with $G$-invariant subsets $V_i \subset U'_i$.

\end{prop}

\end{mysubsection}

\end{section}

\begin{section}{Higher $\DD$- topological complexity}

In \cite{farberm}, Farber, Grant, Lupton and Oprea introduce $\DD$-topological complexity for a path-connected space and proved that for a finite aspherical cell complex, the topological complexity and $\DD$-topological complexity are same. In \cite{farb}, Farber and Oprea define $n$-th $\DD$-topological complexity $\tcd{n}(X)$ and generalised the result. In \cite{farber}, Farber, Grant, Lupton and Oprea proved some properties of $\DD$-topological complexity. In this Section we generalised the results for $n$-th $\DD$-topological complexity.

\begin{definition}

Let $X$ be a path-connected topological space with fundamental group $\pi = \pi_1(X, x_0)$. The \emph{$n$-th $\DD$-topological complexity}, $\tcd{n}(X)$, is defined as the minimum number $k$ such that $X^n$ can be covered by $k$ open subsets, $X^n = U_1 \cup U_2 \cup \cdots \cup U_k$, with the property that for any $i = 1, 2,\cdots, k$ and for every choice of the base point $u_i \in U_i$, the homomorphism $\pi_1(U_i, u_i) \rightarrow \pi_1(X^n, u_i)$ induced by the inclusion $U_i \rightarrow X^n$ takes values in a subgroup conjugate to the diagonal $\Delta \subset \pi^n$, where $\pi^n = \pi \times \pi \times \cdots \times \pi \ (n$-times).

\end{definition}

\noindent We now interpret the above definition as sectional category of a certain covering map.

\begin{prop}\label{pro}

Let $X$ be a path-connected, locally path-connected and semi-locally simply connected topological space with fundamental group $\pi = \pi_1(X, x_0)$. Let $q :\widehat{X^n} \rightarrow X^n$ be the connected covering space corresponding to the diagonal subgroup $\Delta \subset \pi^n = \pi_1(X^n, X_0)$, where $X_0 = (x_0, x_0, \cdots, x_0)$. Then
\begin{center}
$\tcd{n}(X) = \secat (q).$
\end{center}
\end{prop}

\begin{proof}

Let $\secat (q) = k$, with $\{U_1, U_2, \cdots, U_k\}$ be cover of $X^n$ and for each $i, \ s_i : U_i \rightarrow \widehat{X^n}$ be a section on $U_i$ of $q$. Now by the lifting criterion of covering space $i_*(\pi_1(U_i, u_i)) \subseteq q_*(\pi_1(\widehat{X^n},\widehat{x_0}))$ (where \ $\widehat{x_0} \in \widehat{X^n}$), that is, $i_*(\pi_1(U_i, u_i)) \subseteq q_*(\Delta) $. Hence $\tcd{n}(X) \leq \secat (q)$.

Conversely, let $\tcd{n}(X) = k$. Then $i_*(\pi_1(U_i, u_i)) \subseteq$ some conjugate of $\Delta,$ means $i_*(\pi_1(U_i, u_i)) \subseteq q_*(\pi_1(\widehat{X^n},\widehat{x_0}))$. Again by lifting criterion of covering space, a lift $s_i : U_i \rightarrow \widehat{X^n}$ exist, i.e. section exist on $U_i$.  Hence $\tcd{n}(X) \geq \secat (q)$.

\end{proof}

\begin{example}

For a path-connected space $X$, $\tcd{n}(X) = 1$ if and only if $X$ is simply connected. So we have $\tcd{n}(S^m) = 1$ for all $m,n \geq 2$.

\end{example}

\noindent To get an analogue of Equation (\ref{eqineq}) for $\mathcal{D}$-topological complexity, we recall the definition of Lusternik-Schnirelmann one-category.

\begin{definition}
Let $X$ be a path connected, locally path-connected and semi-locally simply connected space with universal cover $P : \widetilde{X} \to X$. The \emph{Lusternik-Schnirelmann one-category} of $X$ is defined to be $\operatorname{cat}_1(X) := \secat(P)$ of $P$.
\end{definition}

\begin{prop}

If $X$ is a path-connected, locally path-connected  and semi-locally simply connected topological space, then

$$\operatorname{cat}_1(X^{n-1}) \leq \tcd{n}(X) \leq \operatorname{cat}_1(X^n).$$

\end{prop}

\begin{proof}
Consider the commutative diagram,

$$\xymatrix{
\overline{X} \ar[dd]^{q'} \ar[rr]  &&  \widehat{X^n} \ar[dd]^{q}  &&  \widetilde{X}^n \ar[ll] \ar[dd]^{P^n}  \\\\
X^{n-1} \ar[rr]_{f} && X^n && X^n \ar[ll]^=
}$$
where $q : \widehat{X^n} \to X^n$ is the cover corresponds to the diagonal group $\Delta \subset \pi^n = \pi \times \pi \times \cdots \times \pi$. The map $f : X^{n-1} \to X^n$ is the inclusion to the first $n-1$ factor, $f(x_1, x_2,\cdots, x_{n-1}) = (x_1, x_2,\cdots, x_{n-1}, *)$, where $* \in X$ is the base point and $\overline{X}$ is the preimage $q^{-1}(f(X^{n-1}))$. Note that  $q_*(\pi_1(\widehat{X^n}))$ and $f_*(\pi_1(X^{n-1}))$ spans $\pi_1(X^n)$. So using property of pullback covering by inclusion map we can say that $\overline{X}$ is covering space corresponding to the subgroup $ f_*^{-1}(q_*(\pi_1(\widehat{X^n})) \cap f_*(\pi_1(X^{n-1})))$ which is trivial. Thus $q' : \overline{X} \to X^{n-1}$ is the universal cover of $X^{n-1}$.

Given an open subset $U \subset X^n$ with a section $s : U \to \widehat{X^n}$ we may restrict it to $f^{-1}(U) \subset X^{n-1}$ getting a section $s' : f^{-1}(U) \to \overline{X}$. This shows that $\operatorname{cat}_1(X^{n-1}) = \secat(q') \leq \secat(q) = \tcd{n}(X)$, thus proving the first inequality.

Next we consider the right square of the diagram. The map $P^n$ is the universal covering and hence $\secat(P^n) = \operatorname{cat}_1(X^n) \geq \secat(q) = \tcd{n}(X)$.  This is the second inequality.
\end{proof}

\begin{corollary}\label{cineq}
For $X$ as above,   $$\operatorname{cat}_1(X^{n-1}) \leq \tcd{n}(X) \leq \operatorname{cat}_1(X^n) \leq \tcd{n+1}(X).$$ In particular $\tcd{n}(X) \leq \tcd{n+1}(X)$, for all $n \geq 2$.

\end{corollary}

Now we relate higher topological complexity $\operatorname{TC}_n(X)$ with higher $\DD$-topological complexity $\tcd{n}(X)$.

\begin{notation}\label{nota}
Let $P: \widetilde{X} \to X$ be the universal cover of $X$. Let $\pi = \pi_1(X)$ denotes the fundamental group of $X$ and $\tX^n/\pi$ stands for the quotient of ${\widetilde{X}}^n$ with respect to the diagonal action of $\pi$.
\end{notation}

\begin{prop}

For a path-connected, locally path-connected and semi-locally simply connected topological space $X$ one has  $\tcd{n}(X) \leq \operatorname{TC}_n(X)$.

\end{prop}

\begin{proof}

 Consider the projection $q : \tX^n/\pi \to X^n$. Clearly $q$ is a covering map with the property that the image of the induced homomorphism $q_* : \pi_1(\tX^n/\pi) \to \pi_1(X^n)$ is the diagonal. Hence by Proposition \ref{pro}, $\tcd{n}(X) = \secat (q)$.

Now we define
$$p : X^I \to \tX^n/\pi , ~~~~~~~~~~~\gamma \mapsto [\widetilde{\gamma}(0), \widetilde{\gamma}(\frac{1}{n-1}), ...,  \widetilde{\gamma}(\frac{j}{n-1}), \cdots, \widetilde{\gamma}(1)],$$ where $\widetilde{\gamma} : I \to \widetilde{X}$ is any lift of the path $\gamma : I \to X$ and the brackets $[x_0, x_1, \cdots, x_{n-1}]$ denote the orbit of the tuple $(x_0, x_1, \cdots,  x_{n-1}) \in {\widetilde{X}}^n$ with respect to the diagonal action of $\pi$. The map $p$ is well-defined although the lift $\widetilde{\gamma}$ is not unique. We obtain the following commutative diagram.
$$
\xymatrix{
X^I \ar[rrrr]^{p} \ar[drr]_{e'_n}&&& &\tX^n/\pi \ar[dll]^{q}\\
&& X^n & &
}
$$

\noindent Clearly, a partial section $s : U \to X^I$ of $e'_n$ gives a partial section $\widetilde{s} = p\circ s : U \to \tX^n/\pi$ of $q$. So we have $\tcd{n}(X) = \secat(q) \leq \secat(e'_n) = \operatorname{TC}_n(X)$.
\end{proof}

For aspherical spaces, $\DD$-topological complexity is same as topological complexity.
\begin{lemma}\cite[Lemma 4.2 ]{farb}\label{pdtc} Let $X$ be an aspherical CW complex. Then
\begin{center}
$\tcd{n}(X) = \operatorname{TC}_n(X)$.
\end{center}

\end{lemma}

Now we show that the $\tcd{n}{(X)}$ is also homotopy invariant.

\begin{prop}\label{pprop}

Assume that $f : X \rightarrow Y$ is a continuous map between path-connected topological spaces such that the induced map $f_* : \pi_1(X) \rightarrow \pi_1(Y)$ is an isomorphism. Then we have,

\begin{center}
$\tcd{n}(X) \leq \tcd{n}(Y)$.
\end{center}

\end{prop}
\begin{proof}

Let $U \subset Y^n$ be an open subset such that the induced homomorphism $\pi_1(U, u) \rightarrow \pi_1(Y^n, u)$ takes values in a subgroup conjugate to the diagonal. Consider the preimage $V = (f \times f \times \cdots \times f)^{-1}(U) \subset X^n$. The map $\pi_1(V) \to \pi_1(X^n),$ induced by the inclusion, can be factored as the composition $$\pi_1(V) \rightarrow \pi_1(U) \rightarrow \pi_1(Y^n) \xrightarrow{(f_*^{-1})^n} \pi_1(X^n).$$  Since the second map takes values in a subgroup conjugate to the diagonal, hence the map $\pi_1(V) \rightarrow \pi_1(X^n)$ also has the same property. Therefore $\tcd{n}(X) \leq \tcd{n}(Y)$.

\end{proof}

\begin{corollary}

The higher $\DD$-topological complexities are homotopy invariant.

\end{corollary}

\begin{proof}
Assume $f: X\to Y$ is a homotopy equivalence with inverse $g: Y\to X$.  Then applying the above proposition to $f$ and $g$ we get $\tcd{n}{(X)} = \tcd{n}{(Y)}.$

\end{proof}

\noindent Since $\tcd{n}{(X)}$ is homotopy invariant, we can define $\tcd{n}{(\pi)} :=\tcd{n}{(K(\pi,1))}$ for a discrete group $\pi$. Note that $\tcd{n}{(\pi)}  = \operatorname{TC}_n(\pi)$ by Lemma \ref{pdtc}.
\begin{prop}\label{pdasp}

Let $X$ be a path-connected CW complex with fundamental group $\pi = \pi_1(X)$. Then
$$\tcd{n}(X) \leq \tcd{n}(\pi).$$
Moreover, if there exist a positive integer $k \geq 2$ such that the homotopy groups $\pi_j(X) = 0$ for all $j$ satisfying $1 < j < k$ and $\pi$ has cohomological dimension $\leq k$, then  $$\tcd{n}(X) = \tcd{n}(\pi).$$

\end{prop}

\begin{proof}

The Eilenberg-Mac Lane complex $K = K(\pi, 1)$ can be constructed by attaching cells of dimension $\geq 3$ to $X$. Consider the inclusion map $i : X \hookrightarrow K$ which induces isomorphism of fundamental groups. So, using the Proposition \ref{pprop} we can say that $\tcd{n}(X) \leq \tcd{n}(K) = \tcd{n}(\pi)$.

For the second part, the Eilenberg-Mac Lane space $K = K(\pi, 1)$ can be obtained from $X$ by attaching cells of dimension $k+1, k+2, \cdots$. Now convert the inclusion $X \hookrightarrow K$ into a fibration with fiber $F$ satisfying $\pi_i(F) = \pi_{i+1}(K, X)$. Since $\pi_1(X) \simeq \pi_1(K)$, we have $\pi_i(F) = \pi_{i+1}(K, X) = 0$ for $i = 0, 1,\cdots, k-1$. The obstructions to finding a section of $X \rightarrow K$ lie in the groups $H^{i+1}(\pi, \pi_i(F)) = H^{i+1}(K, \pi_i(F))$ and all these groups vanish because our computation with $\pi_i(F)$ and our assumption $\operatorname{cd}(\pi) \leq K$. Finally if we apply Proposition \ref{pprop} to the section, which induces on the fundamental groups, we get $\tcd{n}(X) \geq \tcd{n}(K) = \tcd{n}(\pi)$. So $\tcd{n}(X) = \tcd{n}(\pi)$.

\end{proof}

We now show that if  $X$ has a group structure, then the left side inequality of Corollary \ref{cineq} is an equality.

\begin{prop}

For any connected topological group $H, \  \tcd{n}(H) = \operatorname{cat}_1(H^{n-1})$.

\end{prop}

\begin{proof}

Let $F : H^n \rightarrow H^{n-1}$ be the map given by the formula $$F(a_1, a_2,\cdots, a_n) = (a_1a_n^{-1}, a_2a_n^{-1},\cdots, a_{n-1}a_n^{-1}).$$ Denote $\pi = \pi_1(H, e)$ and consider the induced map on fundamental groups

\begin{center}

$\phi = F_* : \pi^n = \pi_1(H^n, e^n) \longrightarrow \pi^{n-1} = \pi_1(H^{n-1}, e^{n-1})$

\end{center}
where $\pi^{n} = \pi \times \pi \times \cdots \times \pi, \ n$-times and $e^n = (e, e,\cdots, e)$ is $n$-tuple in $H^n$. From the definition of $F$, it is clear that $F_*(\alpha_1, \alpha_2,\cdots, \alpha_n) = (\alpha_1 - \alpha_n, \alpha_1 - \alpha_n, \cdots, \alpha_{n-1} - \alpha_n)$. Note that the kernel of $\phi$ is $\Delta \subset \pi^n$, the diagonal subgroup. This gives a pullback  diagram of covering maps
\begin{center}

$\xymatrix{
\widehat{H^n} \ar[dd]_{q} \ar[rr]^{\widetilde{F}}  &&  {\widetilde{H}}^{n-1} \ar[dd]^{P^{n-1}} \\\\
H^n \ar[rr]_{F} && H^{n-1}
}$

\end{center}
where $P : \widetilde{H} \to H$ is universal covering and $q$ is the covering corresponds to the diagonal subgroup. From the diagram we obtain $\tcd{n}(H) = \secat(q) \leq \secat(P^{n-1}) = \operatorname{cat}_1(H^{n-1})$.

\end{proof}

\end{section}

\begin{section}{The invariant $\tTC{{n}}{X}$ }
In this section we introduce an invaraint $\tTC{n}{X}$ which is higher analogue of $\tTC{}{X}$ as defined in \cite{farber}. We use it to prove Theorem (A). Consider maps $E \xrightarrow {p} \overline{X} \xrightarrow{q} X$, where $p$ is a fibration with fiber $F$, $q$ is a covering map with fiber $F_0$ and the space $\overline{X}$ is connected. The composition is a fibration with fiber $F'$ which is homeomorphic to $F \times F_0$.

\begin{definition}
With notations as above, the number $\widetilde{\secat}(E \xrightarrow {p} \overline{X} \xrightarrow{q} X)$ is the minimal integer $k \geq 1$ such that $X$ admits an open cover $X = U_1 \cup U_2 \cup \cdots \cup U_k$, with the property that for each $1\leq i\leq k,$ the fibration $p$ admits a continuous section over the open set $q^{-1}(U_i) \subset \overline{X}$.
\end{definition}
\noindent It is clear from the definition that $\widetilde{\secat}(E \xrightarrow {p} \overline{X} \xrightarrow{q} X) \geq \secat(p)$ and $\widetilde{\secat}(E \xrightarrow {p} \overline{X} \xrightarrow{q} X) = 1$ if and only if $\secat(p) = 1$. The following result is proved in \cite[Proposition 3.2]{farber}. Note that we are counting from $1$ in the definitions of genus and related things.

\begin{lemma}\label{lem}

With notations as above, we have $$\secat(q \circ p) \leq \secat(q) + \widetilde{\secat}(E \xrightarrow {p} \overline{X} \xrightarrow{q} X) -1.$$

\end{lemma}

\noindent Consider the quotient

\begin{center}
$E = \{(\omega , x_1, x_2,\cdots , x_n);  \ \omega \in {\widetilde{X}}^I, \omega (0)=x_1, \omega(\frac{1}{n-1})=x_2, \cdots, \omega(1)=x_n)\}/\pi$,
\end{center}
where $\pi = \pi_1(X)$. Note that $E$ can be identified with  $X^I$ by a choice of lift of path in $X$ to path in $\widetilde{X}$. The quotient by the fundamental group ensures that this is well defined. Recall that $\tX^n/\pi$ stands for the quotient of ${\widetilde{X}}^n$ with respect to the diagonal action of $\pi$ (see Notation \ref{nota}). We define two maps $p,q$ as follows:
 $$p : E \to { \prod_\pi} \widetilde{X},~~~~p([\omega , x_1, x_2,\cdots , x_n]) \mapsto [x_1, x_2,\cdots , x_n], $$ and $$q :  \prod_\pi \widetilde{X}\ \to X^n,~~~q([x_1, x_2,\cdots , x_n]) \mapsto (Px_1, Px_2,\cdots , Px_n).$$ Here $P : \widetilde{X} \to X$ is the universal cover. Now we have the situation
$$
\xymatrix{
X^I \ar[rrr]^{p} &&& { \prod_\pi} \widetilde{X} \ar[rrr]^{q}&&& X^n.}
$$
For such $p,q$, it is clear that $$\secat(p \circ q) = \operatorname{TC}_n(X),~~~~~~\secat(q) = \tcd{n}(X).$$

\begin{definition}

With notations as above, we define $$\tTC{n}{X} = \widetilde{\secat} (X^I \xrightarrow{p} { \prod_\pi} \widetilde{X} \xrightarrow{q} X^n).$$

\end{definition}

Applying the Lemma \ref{lem} in our this particular case we have

\begin{equation}\label{ineq1}
\operatorname{TC}_n(X) \leq \tcd{n}(X) + \tTC{n}{X} -1.
\end{equation}

\begin{lemma}

For a CW-complex $X$, the following statements are equivalent.
\begin{enumerate}[(i)]
\item  For some $n\geq 2$, $\tTC{n}{X} = 1$.

\item The space $X$ is aspherical.

\item  For all $n\geq 2$, $\tTC{n}{X} = 1$.
\end{enumerate}

\end{lemma}

\begin{proof}

(i) $\Rightarrow$ (ii):

 \noindent Suppose that $\widetilde{TC}_n(X) = 1$, for some $n \geq 2$. Then the fibration $p : X^I \to \prod_\pi \widetilde{X}$ has a continuous section. Now for $r \geq 2$ consider the composition

$$\pi_r(X) = \pi_r(X^I) \xrightarrow{p_*} \pi_r(\prod_\pi \widetilde{X}) \xrightarrow{\simeq} \pi_r(X^n) = \bigoplus \pi_r(X),$$
where $\bigoplus \pi_r(X)$ is the direct sum of $n$-copies of $\pi_r(X)$. Since $p$ has a section so this composition must surjective and it is possible only when $\pi_r(X) = 0$, for all $r \geq 2$. So $X$ is aspherical.

\noindent (ii) $\Rightarrow$ (iii):

 \noindent If $X$ is aspherical then $\widetilde{X}$ is contractible. The fiber of $p : X^I \to \prod_\pi \widetilde{X}$  is the mapping space $\operatorname{map}(\bigvee_{(n-1)\text{ copies} } S^1 \to  \widetilde{X})$, which is also contractible. This implies $p$ has a section and hence $\tTC{n}{X} = 1$ for all $n \geq 2$.

\noindent (iii) $\Rightarrow$ (i) is obvious.
\end{proof}

\begin{prop}

Let $Z = X \times Y$ where $X = K(\pi, 1)$ is aspherical and $Y$ is simply connected. Then $\tcd{n}(Z) = \operatorname{TC}_n(X)$ and $\tTC{n}{Z} = \operatorname{TC}_n(Y)$.

\end{prop}

\begin{proof}

The first equality follows from the Proposition \ref{pprop} applying on $X \to X \times Y \to X$, injection and projection. The proof of second equality is similar as (\cite{farber}, Proposition 3.11).

\end{proof}

We now use the higher $\DD$-topological complexity to give an upper bound of $\operatorname{TC}_n(X)$ using connectivity of the universal covering space $\widetilde{X}$ of X. This is a generalisation of the result \cite[Theorem 4.3]{farber}.

\begin{theorem}\label{theorema}

Let $X$ be a finite dimensional simplicial complex such that its universal cover $\widetilde{X}$ is $r$-connected.  Then we have ,
\begin{equation}\label{eq1}
\operatorname{TC}_n(X) \leq \tcd{n}(X) + \ceil[\Bigg]{\frac{n  \dim X - r}{r+1}} .
\end{equation}
In particular if $\widetilde{X}$ is $ (n-1)$-connected, then

$$\operatorname{TC}_n(X) \leq \tcd{n}(X) +  \dim X .$$

\end{theorem}

\begin{proof}

If we have a covering map $q : \overline{B} \to B$, with $B$ finite dimensional simplicial complex and a fibration $p : E \to \overline{B}$ with $(r-1)$-connected fiber for some $r \geq 0$, then by (\cite[Theroem 4.1]{farber}) we have $$\widetilde{\secat}(E \xrightarrow {p} \overline{B} \xrightarrow{q} B) \leq \ceil{ \frac{\dim B - r}{r+1}} + 1.$$ We apply this result to the defining maps  $X^I \xrightarrow{p} { \prod_\pi} \widetilde{X} \xrightarrow{q} X^n$ of $\tTC{n}{X}.$ The fiber of the map $p$ is the mapping space $F= \operatorname{map} (\bigvee_{(n-1)\text{ copies} }S^1 \to  \widetilde{X})$. Since $\widetilde{X}$ is $r$-connected, we get that $F$ is $(r-1)$-connected. This can be seen by considering the fibration $P_0\widetilde{X} \to \widetilde{X}^{n}$ given by the projections at $0, \frac{1}{n-1} , \frac{2}{n-1},\cdots, \frac{n-1}{n-1}=1$. This also has fiber $F$. Looking at the homotopy long exact sequence gives us the desired connectivity of $F$.
So we get
\begin{equation}\label{eqttc}
\tTC{n}{X}\leq  \ceil[\Bigg]{\frac{n  \dim X - r}{r+1}} + 1.
\end{equation}

Combining with the Equation (\ref{ineq1}) we get the Equation (\ref{eq1}).

If $r = n-1$, then $\ceil{\frac{n  \dim X - r}{r+1}} = \ceil{\dim X - \frac{n-1}{n}} = \dim X,$ so we obtain

$$\operatorname{TC}_n(X) \leq \tcd{n}(X) +  \dim X.$$

\end{proof}
The following corollary is a generalisation of \cite[Theorem 3.3]{alex}.

\begin{corollary}\label{cbound}
For a finite dimensional simplicial complex $X$ with fundamental group $\pi$ we have $$\operatorname{TC}_n(X) \leq \operatorname{TC}_n(\pi) + \ceil[\Bigg]{\frac{n  \dim X - 1}{2}} .$$
\end{corollary}

\begin{proof}
Combining Proposition \ref{pdasp} and Proposition \ref{pdtc} we have
$$\tcd{n}(X) \leq \tcd{n}(\pi) = \operatorname{TC}_{n}(\pi) .$$
Since universal cover $\widetilde{X}$ is $1$-connected (simply connected), putting $r=1$ in the Equation (\ref{eq1}) we get the result.
\end{proof}

\section{Higher strongly equivariant topological complexity}

In (\cite{ColG}) Colman and Grant introduced equivariant topological complexity for a $G$-space $Y$. It is denoted by $\operatorname{TC}_G(Y)$ and is defined as the minimum integer $k\geq 1$ such that there exist $G$-invariant open subsets $U_1, U_2, \cdots, U_k$ covering  $Y \times Y$ under the diagonal action of $G$ on $Y \times Y$ and on each open subset there is $G$-equivariant section of the path fibration map $p : Y^I  \to Y \times Y, \ \gamma \to (\gamma(0), \gamma(1))$. In (\cite{sarkar}) Bayeh and Sarkar generalized equivariant version to higher topological complexity. Dranishnikov (\cite{alex}) introduced strongly equivariant topological complexity $\tcs{G}Y$ of a $G$-space $Y$, in which the covering open subsets are $G \times G$ invariant and the sections $s_i : U_i \to Y^I$ are $G$-equivariant with diagonal action of $G$ on $U_i$. In this section we introduce higher strongly equivariant topological complexity $\tcg {n}{G}Y$ of a $G$-space $Y$ and obtain some properties. After that we relate $\tTC{n}{X}$ and $\tcg{n}{\pi}{\widetilde{X}}$, where  $\widetilde{X}$ is the universal cover of $X$ and $\pi = \pi_1(X)$.

\begin{definition}\label{def}
For a $G$-space $Y$, consider $Y^n$ as a $G^n$-space with product action. Consider the fibration $e_n : Y^{I_n} \rightarrow Y^n $ (cf. Equation \ref{een}) for $Y$. The $G$-action on $Y$ naturally induces a $G$-action on $Y^{I_n}. $ We define the \emph{$n$-th strongly equivariant topological complexity} $\tcg{n}{G}{Y}$, as the the minimal number $k$ such that $Y^n$ can be cover by $G^n$-invariant open sets $\{U_i:~ i=1,\cdots, k\}$ and there is a $G$-equivariant continuous section $s_i : U_i \to Y^{I_n}$ of $e_n$ for $i = 1, \cdots, k$ (considering $G$ as the diagonal subgroup of $G^n$). If no such $k$ exist, then $\tcg{n}{G}Y = \infty$.

\end{definition}

\noindent The following lemma shows that we can take the fibration $e_n' : Y^I \to Y^n$ in the above definition (cf. Equation \ref{eenprime}). Later we will use them interchangeably.

\begin{lemma}

Let $Y$ be a $G$-space and $U$ be a $G^n$-invariant open set of $Y^n$, then admitting $G$-equivariant continuous section on $U$ of the maps $e_n : Y^{I_n} \to Y^n$ and $e'_n : Y^{I} \to Y^n$ are equivalent.

\end{lemma}

\begin{proof}

Let $s : U \to Y^{I_n}$ be a $G$-equivariant continuous section of $e_n$. Define a map $\phi : I \to I_n$ by $[\frac{2j-2}{2(n-1)},  \frac{2j-1}{2(n-1)}]$ goes to linearly on $[0 , 1]_j$ in the reverse direction and $[\frac{2j-1}{2(n-1)}, \frac{2j}{2(n-1)}]$ goes to linearly on $[0 , 1]_{j+1}$ for $j= 1, 2, \cdots, n-1$. This map induces $\phi^{*} : Y^{I_{n}} \rightarrow Y^{I}$ such that the following diagram commutes.

 $$
 \xymatrix{
Y^{I_{n}} \ar[rrrr]^{\phi^{*}}  \ar[ddrr]_{e_{n}}& &&&Y^{I} \ar[ddll]^{e'_{n}}\\
&& U \ar@{^{(}->}[d] \ar[llu]_s \ar@{.>}[rru]^{s'} \\ && Y^n
}
$$

\noindent Consider the composition map $s' = \phi^* \circ s : U \to Y^I$. Let $g \in G$ and $(y_1, y_2, \cdots, y_n) \in U$. Using the fact  $s : U \to Y^{I_n}$ is $G$-equivariant, we have,
\begin{equation*}
\begin{split}
\phi^* \circ s (gy_1, gy_2, \cdots, gy_n)(t)  =~ & \phi^* ( s (gy_1, gy_2, \cdots, gy_n))(t)  \\
=~ &   s(gy_1, gy_2, \cdots, gy_n) \circ \phi(t)  \\
=  ~&  g. (s (y_1, y_2, \cdots, y_n)) \circ \phi(t)
\\ =~ &  g. \phi^* \circ s (y_1, y_2, \cdots, y_n)(t).
\end{split}
\end{equation*}
So the map $s' = \phi^* \circ s$ is a $G$-equivariant section of $e'_n$.

Conversely, let $s' : U \to Y^I$ be a $G$-equivariant continuous section of $e'_n$. Define a map $\psi : I_n \to I$ by sending  $[0, 1]_j$($1 \leq j \leq n$) linearly to:
$$ [\frac{j-1}{n-1}, \frac{1}{2}] \ \text{in the reverse direction}, \ \ \ \ \ \ \ \ ~~\text{if} ~~j \leq \frac{n+1}{2}$$
$$ [\frac{1}{2}, \frac{j-1}{n-1}], \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \  ~~~ \text{if} ~~j > \frac{n+1}{2}.$$

 \noindent So the map $\psi$ induces $\psi^{*} : Y^{I} \rightarrow Y^{I_{n}}$ such that the following diagram commutes.
$$
 \xymatrix{
Y^{I} \ar[rrrr]^{\psi^{*}}  \ar[ddrr]_{e'_{n}}& &&&Y^{I_n} \ar[ddll]^{e_{n}}\\
&& U \ar@{^{(}->}[d] \ar[llu]_{s'} \ar@{.>}[rru]^{s} \\ && Y^n
}
$$
 As in previous case the composition $s = \psi^* \circ s' : U \to Y^{I_n}$ will be a $G$-equivariant section of $e_n$.
\end{proof}

So in Definition \ref{def} we can take the fibration $e'_n$ instead of $e_n$.
As in the other cases of higher topological complexity, the sequence $\{\tcg{n}{G}{Y}\}_{n\geq 2}$ is non-decreasing.

\begin{prop}

 Let $Y$ be a $G$-space, then $\tcg{n+1}{G}{Y} \geq \tcg{n}{G}{Y}$, for any $n \geq 2$.

\end{prop}

\begin{proof}

Take the inclusion $Y^n \to Y^{n+1}$ to the first $n$-factors. Given  an open cover $\{U_i:~ i=1,\cdots, k\}$ of  $Y^{n+1}$ by $G^{n+1}$-invariant subsets, take $V_i = U_i \cap Y^n$. Then $V_i$ are $G^n$ invariant with $G^n \hookrightarrow G^{n+1}$ as $(g_1, \cdots, g_n) \mapsto (g_1, \cdots, g_n, e)$. Restrict the section over $U_i$ to $V_i$. This satisfies the desired properties.

\end{proof}

We now show that $n$-th strongly equivariant topological complexity is $G$-homotopy invariant.
\begin{prop}

Let $X,Y$ be $G$-spaces and there are $G$-maps $\phi : X \to Y$, ~$\psi : Y\to X$ such that $\phi\circ\psi \simeq _G \operatorname{Id}_Y$. Then $\tcg{n}{G}{X} \geq \tcg{n}{G}{Y}$. In particular if $X$ is $G$-homotopy equivalent to $Y$ then $\tcg{n}{G}{X} = \tcg{n}{G}{Y}.$

\end{prop}

\begin{proof}

Take an $G^n$-invariant open set $U \subset X^n$ with an equivariant section $s: U \to X^I$ of $e_n'$. Consider $V= (\psi^n)^{-1} (U)\subset Y^n$. Then $V$ is also $G^n$-invariant.
$$\xymatrix{
& Y^I \ar[dd]_{e_n'}  &&  X^I \ar[dd]^{e_n'} \ar[ll]_{\phi_*}  \\\\
  (\psi^n)^{-1}U=V   \ar@{^{(}->}[r] \ar@{..>}[ruu]_{s'} &Y \times Y \times \cdots \times Y  \ar[rr]_{\psi\times \psi \cdots \times \psi} && X\times X \times \cdots \times X & U  \ar@{_{(}->}[l] \ar[uul]_{s}
}$$
Define $s': V\to Y^I$ as $s'(y_1,\cdots, y_n) = \phi_* \circ s\circ\psi^n(y_1,\cdots, y_n)$ where $\phi_* : X^I \to Y^I$ is the map induced by $\phi$. Then   $e_n' \circ s': V \to Y^n$ is the map $ (y_1,\cdots, y_n) \mapsto (\phi\circ\psi(y_1), \cdots , \phi\circ\psi(y_n))$ which is $G$-homotopic to identity.

\end{proof}

\begin{corollary}
For a $G$-connected space $Y$ with $Y^G \neq \phi$, then $Y$ is $G$-contractible if and only if $\tcg{n}{G}{Y} = 1$, for some $n \geq 2$.
\end{corollary}
\begin{proof}
First assume that $Y$ is $G$-contractible. Since $Y$ is $G$-connected and $Y^G \neq \phi$, using  Lemma \ref{lemma} we can say that $Y$ is $G$-homotopy equivalent to a point. Hence the corollary follows from the above proposition.

Conversely, let $\tcg{n}{G}{Y} = 1$ for some $n \geq 2$. Then there is a $G$-equivariant section $s : Y^n \to Y^I$ of $e'_n$. Fix $y_0 \in Y^G$. Define a homotopy $H : Y \times I \to Y$ by $(y, t) \to s(y, y_0, y_0, \cdots y_0)(t)$. Clearly $H$ is a $G$-homotopy between identity map on $Y$ and $C_{y_0}$(constant map on the orbit $\mathcal{O}(y_0)$). So $Y$ is $G$-contractible.

\end{proof}

\noindent We now give some inequalities relating higher equivariant complexity $\tce{n}{G}{Y}$ of \cite{sarkar} and our $\tcg{n}{G}{Y}.$

\begin{prop}\label{prop}
Let $Y$ be a $G$-space. Then the following holds.
\begin{enumerate}[(a)]
\item  For any $n\geq 2$, we have $\tce{n}{G}{Y} \leq \tcg{n}{G}{Y}$.
\item If $H$ and $K$ are subgroups of $G$ such that $Y^H$ is $K$-invariant, then
$$\tce{n}{K}{Y^H} \leq \tcg{n}{K}{Y^H}  \leq \tcg{n}{G}{Y}.$$
In particular,
$$ \operatorname{TC}_n(Y^H) \leq \tcg{n}{G}{Y}, \, ~~~~\operatorname{TC}_n(Y) \leq \tce{n}{K}{Y} \leq \tcg{n}{G}{Y}.$$

\item If $Y$ is not $G$-connected, then $\tcg{n}{G}{Y} = \infty$ for all $n\geq 2$.

\end{enumerate}
\end{prop}

\begin{proof}
\begin{enumerate}[(a)]

\item This simply follows from the definitions.

\item Let us prove the inequality $\tcg{n}{K}{Y^H}  \leq \tcg{n}{G}{Y}$. Let $U$ be an $G^n$-invariant open set of $Y^n$ and $s : U \to Y^I$ be a $G$-equivariant section for $e'_n$. Define $V = U \cap (Y^H)^n$, is $K^n$-invariant as $U$ and $(Y^H)^n$ both are $K^n$-invariant. If we restrict the map $s$ on $V$, then for any $(y_1, \cdots, y_n) \in V$ and $h\in H$, $$h.s_{|_V}(y_1, \cdots, y_n) = s_{|_V}(hy_1, \cdots, hy_n) = s_{|_V}(y_1, \cdots, y_n),$$ i.e. the path $s_{|_V}(y_1, \cdots, y_n)$ is in $(Y^H)^I$. It is also clear that $s_{|_V}$ is $K$-equivariant. So $ s_{|_V} : V \to (Y^H)^I$ is a $K$-equivariant section for $e'_{n{|_{(Y^H)^I}}} : (Y^H)^I \to (Y^H)^n$.

\noindent The others inequality follows from combining (a) with results of \cite[Proposition 3.14]{sarkar}.

\item If $Y^H$ is not connected then, $\operatorname{TC}_n(Y^H) = \infty.$ So this follows from part (b).

\end{enumerate}
\end{proof}
\begin{remark}
We can visualise the different inequalities of the above Proposition \ref{prop} the following picture, with an arrow goes from bigger to smaller number.

$$\xymatrix{
&& \tcg{n}{G}{Y} \ar[dll] \ar[d]  \ar[drr] \\
\tcg{n}{K}{Y} \ar[dr] &&  \tce{n}{G}{Y} \ar[dl] \ar[dr] && \tcg{n}{K}{Y^H}   \ar[dl] \\  & \tce{n}{K}{Y} \ar[d] && \tce{n}{K}{Y^H}  \ar[d] \\ & \operatorname{TC}_n(Y) && \operatorname{TC}_n(Y^H)
}$$
\end{remark}

\begin{lemma}\label{ldof}
We have $\tcg{n}{G}{Y}\leq k$ if and only if there exist an open cover of $Y^n$ containing $k$ open sets such that each open set is $G^n$-invariant and $G$-equivariantly deformable into $\Delta(Y) \subset Y^n$.
\end{lemma}

\begin{proof}

It is enough to prove that, a $G$-equivariant section $s : U \to  Y^{I_n}$ exist for $e_n : Y^{I_n} \to Y^n$ on some $G^n$-invariant open subset $U\subset Y^n$ if and only if there is a $G$-homotopy $H = (H_1, H_2, \cdots, H_n) : U \times I \to Y^n$ with $H((y_1, y_2, \cdots, y_n), 0) \in \Delta(Y)$ and   $H((y_1, y_2, \cdots, y_n), 1) = (y_1, y_2, \cdots, y_n)$. We set $$H_j((y_1, \cdots, y_j, \cdots, y_n), t_j) = s(y_1, \cdots, y_j, \cdots, y_n)(t_j),$$ where $t_j\in [0,1]_j \simeq I$ and $j = 1, \cdots, n$. Then existence of one of $s$ and $H$ implies the other one. Hence the Lemma follows.

\end{proof}

\begin{prop}
Assume that $Y$ is a $G$-connected space with $Y^G \neq \phi$. Consider $Y^m$ with product $G^m$-action. Then $$\operatorname{cat}_{G^{n-1}}(Y^{n-1}) \leq \tcg{n}{G}{Y} \leq \operatorname{cat}_{G^n}(Y^n)\leq n\operatorname{cat}_{G}(Y) - (n-1), ~n \geq 2.$$
\end{prop}

\begin{proof}

For the first inequality, consider the pull-back square:

$$\xymatrix{
P_0 Y \ar[dd]_{p}  \ar@{^{(}->}[rr]^{\operatorname{inc}}  &&  Y^I \ar[dd]^{e'_{n}} \\\\
Y^{n-1}  \ar@{^{(}->}[rr]_{\operatorname{inc}} && Y^{n}
}$$
Here $P_0 Y$ is the set of all paths in $Y$ starting at a base point $y_0\in Y^G$ and $Y^{n-1} \hookrightarrow Y^n,~~ y\mapsto (y_0,y).$ Then for a section of $e'_n$ over $U \subseteq Y^n$, we get a section over $V = (\operatorname{inc})^{-1} U \subseteq Y^{n-1}.$

For the second inequality, take $U\subseteq Y^n$ be $G^n$-categorical with respect to product action. Then there is a $G^n$-homotopy $H: U\times I \to Y^n$ such that $H(a, 0) =a$ and $H(a,1) = (y_0, \cdots, y_0)$ where $y_0 \in Y^G.$ Then $U$ is $G$-equivariantly deformable to $\Delta(Y).$

For the third inequality,  we note that $\operatorname{cat}_{G^n}(Y^n)\leq n\operatorname{cat}_{G}(Y) - (n-1)$ by Lemma \ref{lcatg}.
\end{proof}

\begin{example}

 Let $\mathbb{Z}_2$ acts on $S^m$ by reflection. If $m=1$, then the fixed point set is disconnected. So in this case $\tcg{n}{\mathbb{Z}_2}{S^1} = \infty$ for all $n\geq 2.$  For $m\geq 2$, the fixed point set is $S^{m-1}$ which is path connected. By Example \ref{ecat} $\operatorname{cat}_{\mathbb{Z}_2}(S^m) =2$. In this case $$\tcg{n}{\mathbb{Z}_2}{S^m} \leq n \operatorname{cat}_{\mathbb{Z}_2}(S^m) - (n-1) = n.2 - (n-1) = n+1.$$  Also from Proposition \ref{prop}, we get $$\tcg{n}{\mathbb{Z}_2}{S^m} \geq \operatorname{TC}_n(  (S^m)^{\mathbb{Z}_2}) =\operatorname{TC}_n(S^{m-1}), ~~\tcg{n}{\mathbb{Z}_2}{S^m} \geq \operatorname{TC}_n(S^m).$$
 We know from \cite[Section 4]{rud} that $ \operatorname{TC}_n( S^{m}) = n$ if $m$ is odd and $ \operatorname{TC}_n( S^{m}) = n+1$ if $m$ is even. Thus $$\tcg{n}{\mathbb{Z}_2}{S^m} = n+1, \mbox{ for all } m,n \geq 2.$$
 Note that $\operatorname{TC}_n(S^m/ \mathbb{Z}_2) = \operatorname{TC}_n(D^m)=1.$

\end{example}

\begin{example}
Let $S^3 = \{(z, w)\in \mathbb{C}^2 |~~ |z|^2 +|w|^2 =1\}. $ Consider the $S^1$-action, defined by $\alpha.(z,w)= (\alpha z,w).$ Then the fixed point sets are
$\{(0, w): ~ w\in S^1\} \cong S^1$.  So $ \tcg{n}{S^1}{S^3} \leq n\operatorname{cat}_{S^1}(S^3) - (n-1).$ But $\operatorname{cat}_{S^1}(S^3) = 2$ (cf. \cite[Example 3.20]{sarkar}).  Therefore $$ n \leq \tcg{n}{S^1}{S^3} \leq  n+1.$$

\end{example}

\noindent We have the following Theorem showing that the strongly equivarint complexity of the universal $\tcg{n}{\pi}{\widetilde{X}}$ is same as $\tTC{n}{X}$ where $\pi$ is the fundamental group of $X$. This is a generalisation of (\cite[Proposition 3.8]{farber}). The proof is similar.

\begin{theorem}\label{ttcg}

For any locally finite CW complex $X$, we have $$\tTC{n}{X} = \tcg{n}{\pi}{\widetilde{X}},$$ where $P : \widetilde{X} \to X$ be the universal covering and $\pi = \pi_1(X)$.

\end{theorem}

\begin{proof}

 We first show $\tTC{n}{X} \leq  \tcg{n}{\pi}{\widetilde{X}}.$   Let $e_n' \colon \widetilde{X}^I \to \widetilde{X}^n$ be the map as above. Assume that $\widetilde{U} \subset \widetilde{X}^n$ be an $\pi^n$ invariant open set such that there is a $\pi$-equivariant section $\tilde{s} : \widetilde{U} \to \widetilde{X}^I$ of $e_n'$.  Consider the open set $V= \widetilde{U}/\pi\subset  \tX^n/\pi $ where $\pi$-acts diagonally. We have the following commutative diagram.

$$
\xymatrix{
\widetilde{X}^I \ar[dd]_{/\pi}  \ar@/^{2.5pc}/[rrrr]\sp{e_n'}   &&  \widetilde{U}  \ar[ll]^{\widetilde{s}}\ar[dd]^{/\pi} \ar@{^{(}->}[rr]^{\operatorname{inc}} &&  \widetilde{X}^n \ar[dd]^{/\pi} \ar[ddrr]^{/\pi^n}  \\\\
X^I   \ar@/_{2.5pc}/[rrrr]\sb{p}  && V   \ar[ll]^{s} \ar@{^{(}->}[rr]_{\operatorname{inc}} &&  \tX^n/\pi \ar[rr]^{q} && X^n
}$$
Since $\tilde{s}$ were $\pi$-equivariant, the section $s$ exists.  Note that $V= q^{-1}(\widetilde{U}/\pi^n).$

To prove the other inequality $\tTC{n}{X} \geq  \tcg{n}{\pi}{\widetilde{X}},$ it is enough to show that given a section $s$ as above, it can be lifted to a section $\widetilde{s}.$  Since the $\pi$-action on the top rows are free, the vertical maps are principle $\pi$-bundles. Consider the classifying maps $\xi: X^I\to B\pi$ and  $\xi':  \tX^n/\pi \to B\pi$.  Then the classifying map for the $\widetilde{U}$-bundle is $\xi'\circ\operatorname{inc}$. The existence of $\widetilde{s}$ follows from the following fact of principal bundles: Let $E\to B$ and $E'\to B'$ be two principle $G$-bundles.  Then a map $f: B'\to B$ can be lifted to a bundle map $\tilde{f}: E'\to E$ if and only if $\xi\circ f \simeq \xi' $ where $\xi : B \to BG$ and $\xi': B'\to BG$ are classifying maps of the respective principle $G$-bundles. We apply it to the $\pi$-bundles of the left square of the above diagram. The existence of the bundle map $e'_n$ covering $p$ implies $\xi \simeq \xi' \circ p$.  Now note that $\xi \circ s \simeq \xi' \circ p\circ s = \xi'$. Therefore, by the above fact, the bundle map $\widetilde{s}$ exists making the diagram commutative.

\end{proof}

We now use the strongly equivariant complexity to give an upper bound for higher complexity of total space of a fiber bundle. The following theorem is a generalisation of \cite[Theorem 3.1]{alex}.

\begin{theorem}\label{tfhtc}
Let $E, B$ be two locally compact metric spaces and $p : E\to B$ be a fiber bundle with fiber $F$ and structure group $G$ acting properly on $F$. Then
$$\operatorname{TC}_n(E)\leq \operatorname{TC}_n(B)+ \tcg{n}{G}{F} -1.$$
\end{theorem}

\begin{proof}

Let $\operatorname{TC}_n(B) = r, \tcg{n}{G}{F} = r'$ and $k = r + r' - 1$ such that $r, r', k \geq 1$. Consider $F^n$ as a $G$-space with diagonal action of $G$. Then there is a $\Delta(B)$-deformable open cover $\{U_1, U_2, \cdots, U_r\}$ of $B^n$ and a $G$-equivariantly deformable into $\Delta(F)$ open cover $\{V_1, V_2, \cdots, V_{r'}\}$ of $F^n$ by $G^n$-invariant sets, by Lemma \ref{ldof}. Using Proposition \ref{propdeform} we can extend the above open covers to an $r$-cover  $\{U_1, U_2, \cdots, U_r, \cdots, U_k\}$ and an $r'$-cover $\{V_1, V_2, \cdots, V_{r'}, \cdots, V_k\}$ for $B^n$ and $F^n$ respectively, with the same property.

Consider the universal $F^n$-bundle $q : F^{n} \times_{G^n} E(G^n) \to B(G^n)$ and classifying map $g : B^n \to B(G^n)$ for the $F^n$-bundle $p^n : E^n \to B^n$. Set $O_i = V_i \times_{G^n} E(G^n), \ i = 1, 2, \cdots, k$ so that $\{O_i\}_{i=1}^k$ is an $r'$-cover of $ {F^n \times_{G^n} E(G^n)}$ and $W_i = O'_i \cap (p^n)^{-1}(U_i)$, where $ O'_i = (g')^{-1}(O_i) $. Using Lemma \ref{lemmacover} we can say that $\{W_i\}_{i=1}^k$ covers $E^n$. Now we show that the each set $W_i$ is deformable to $\Delta(E)$, in two steps.

\noindent \textit{Step-I:} Consider the composition map, $W_i \times I \xrightarrow{p^n \times \operatorname{Id}} U_i \times I \xrightarrow{H} B^n $, where $H$ is a deformation of $U_i$ into $\Delta(B)$. Using the homotopy lifting property of the fibre bundle $f=p^{n}_{|_{O'_i}} : O'_i \to B^n$, we can say that $W_i$ can be deformed in $O'_i$ to the preimage $f^{-1}(\Delta(B))$.

$$\xymatrix{
W_i \times \{0\} \ar@{^{(}->}[dd] \ar@{^{(}->}[rr]  &&  O'_i \ar@{^{(}->}[rr] \ar[dd]^{p^{n}_{|_{O'_i}}=f}  &&  E^n  \ar[dd]^{p^n = g^*(q)}  \ar[rr]^{g'}&&  F^{n} \times_{G^n} E(G^n)  \ar[dd]^{q}  \\\\
W_i \times I \ar@{..>}[rruu]   \ar[rr]_{H \circ (p^n\times \operatorname{Id})} && B^n \ar[rr]_{=} && B^n  \ar[rr]_{g}  &&  B(G^n)}$$

\noindent \textit{Step-II:} Now we show that $f^{-1}(\Delta(B))\subseteq O'_i$ can be deformed into $\Delta(E)$. Let $\phi^i_t : V_i \to F^n$ be an $G$-equivariant deformation of $V_i$ into $\Delta(F)$. It defines a deformation of $V_i \times_G EG$ to $\Delta(F) \times_G EG$ in $F^n \times_G EG$. Observe that the bundle $q : F^{n} \times_{G^n} E(G^n) \to B(G^n)$  restricted over $\Delta(BG) \cong BG$ is $ F^{n} \times_{G} EG \to BG$ with the diagonal action of $G$ on $F^n$. Then the above deformation defines a fiberwise deformation of $O_i$ over $\Delta(BG)$ into $\Delta(F\times_G EG)$. This will induce a fiberwise deformation of $O'_i$ over $\Delta(B)$, i.e.  of $f^{-1}(\Delta(B))$, into $\Delta(E)$.

The concatenation of the above two deformation in Step-I and Step-II defines a deformation of $W_i$ into $\Delta(E)$.
\end{proof}

\begin{remark} Following the arguments of \cite[Theorem 3.3]{alex}, the Corollary \ref{cbound} can also be deduced using the above Theorem \ref{tfhtc}.   Let $\widetilde{X}$ denote the universal cover of $X$. Consider the fiber bundle $\widetilde{X}\times_{\pi} E\pi \to B\pi.$ It has fiber $\widetilde{X}$ and structure group $\pi$. Applying Theorem \ref{tfhtc} to this bundle we get the following inequality
 $$\operatorname{TC}_n(\widetilde{X}\times_{\pi} E\pi) \leq \tcg{n}{\pi}{\widetilde{X}} + \operatorname{TC}_n(B\pi)-1.$$  Now by Theorem \ref{ttcg}   $\tcg{n}{\pi}{\widetilde{X}} = \tTC{n}{X}$.  Also from the Equation \ref{eqttc}, with $k=1$,  we have $\tTC{n}{X}\leq  \ceil[\Bigg]{\frac{n  \dim X - 1}{2}} + 1.$ Putting it in the above inequality

 $$\operatorname{TC}_n(\widetilde{X}\times_{\pi} E\pi)\leq \operatorname{TC}_n(\pi) + \ceil[\Bigg]{\frac{n  \dim X - 1}{2}} .$$
Note that the map induced by covering projection $\widetilde{X}\times_{\pi} E\pi \to X$ is a homotopy equivalent, since it has contractible fiber $E\pi$. So $\operatorname{TC}_n(\widetilde{X}\times_{\pi} E\pi)  = \operatorname{TC}_n(X). $

\end{remark}

\end{section}

redpajama_set_name: RedPajamaArXiv


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It began last month, and regular players have noted most of the changes by now. New players will find new table designs, new avatars, better overall graphics and design, improved functionality, easier multi-tabling, and an easier rebuy process. There are many updates and improvements to be found across the lobby and throughout the site.
Interestingly, it seems the term Poker 8 has been lost along the way. The 888poker post highlighting most of the platform's updates doesn't mention Poker 8 once.
We'll just call it the 888poker upgrade.
New table designs, of which there are dozens, are themed and include graphics that match. For instance, a writer for 888poker Magazine described the scene at a recent PKO God of the Arena tournament that included a sandy arena-style table design, swords and shields around the table, and sound effects to match a slashed red cross as a player exited. He also mentioned someone at another table throwing a cake at that newly-emptied seat.
It sounds like the graphics are as close to virtual reality-style gaming as possible without diving all the way into VR.
Other new features delineate the poker variants, and the tables are lit to give the feel of a televised final table. SNAP! Tables are dressed in neon, and all tables feature illuminated community and hole cards when a hand ends so the winning hand is more obvious than before.
In addition, multi-tabling is easier to view and use, with the necessary tables pushing to the foreground when action is required.
Enhanced graphics and sleek table designs can impress all players but are probably more important for new or recreational players. Those who will be staying at the tables for long periods of time need more substance, and it seems that 888poker took that into consideration.
The poker tables now have features to show hand options and a current chip stack total, as well as the option to use the bet slider or new configurations to bet, raise, or fold. There is also a hand history tab in the upper right of the screen to show previous hands in full. Replays can be fast-forwarded or slowed for easier examination of one's own play or that of other players.
Rebuys are also easier, as a button on the screen has been added to manage and track reentries. In addition, there is now a way to deposit more money into one's online account in order to rebuy, all from the tournament screen.
Technically, it doesn't matter if the upgraded online poker platform is called Poker 8 or not. Most sites simply upgrade and tout the results, as has happened with 888poker in this instance.
It is curious as to why the focus has been dropped from the Poker 8 designation, but what matters in the end is the product.
Which Sites Best Protect Players in UK Market?


text:
E D I T O R I A L
P A G E
weather n spotlight
Awaiting Judgement
A follow-up
Teacher abducted for opposing unfair means; fate unknown
CHANDIGARH: The trial of the five-and-a-half-year old case of abduction of Sushil Kumari, a young lecturer of Hisar, has ground to a halt in the court of CBI's Special Judge at Ambala, following the summoning of the entire record by the Punjab and Haryana High Court almost a year ago.
Tarn Taran Gurdwara
AS announced already, the SGPC had declared that no meetings or lectures should be held in the Parkarma of Darbar Sahib, Tarn Taran, without the previous permission of the Local Gurdwara Committee.
A Tribune Special
And when the case last came up for hearing before the Special Judge Mr R.C. Bansal on January 12 this year, he held: "Further orders from the High Court not received. It is stated that the next date of hearing fixed in the hon'ble High Court is March 15, 1999. Further orders of the High Court be awaited for March 31".
Thus until the High Court returns the record to the trial court, with whatever directions it deems fit, the trial cannot be resumed and the bereaved parents of Sushil Kumari have to cool their heels for the verdict — God knows for how long.
A junior lecturer in Political Science at the Government Senior Secondary School for Girls at Hisar, Sushil Kumari was said to be a firm disciplinarian. Detailed on invigilation duty by the district administration in the 10+2 examination held in March, 1993, she was determined to ensure that no student resorted to copying in the examination hall. It was this principled, uncompromising attitude in the line of academic duty that led to her abduction and, possibly, the ultimate tragedy.
According to Mr Gaze Singh, the octogenarian father of Sushil Kumari, his daughter had earned the wrath of Mr Mukesh Kumar, a gunman attached with Mr Manphool Singh, brother of former Haryana Chief Minister, Bhajan Lal. She had foiled Mukesh's attempt to help his wife copy in the examination. She had also displeased Mr Salender Bishnoi, a relative of Mr Manphool Singh, by refusing to help his friend Raj Kumari, daughter of a former DSP Mr K.K. Sharma, in the examination. Both of them, Mr Gaze Singh alleged, had threatened Sushil Kumari with "dire consequences". He alleged that a conspiracy to abduct his daughter was hatched on April 30, 1993.
A divorcee, Sushil Kumari left her house for the school on May 1, 1993. Before leaving, she promised her mother that she would send material for getting the house white-washed. Once gone, she never returned home. Nor did the promised material ever reach her mother.
Although Sushil Kumari was allegedly abducted on May 1, and her father put in a written complaint to the Civil Lines police station at Hisar the same day, the case was not registered for a whole week thereafter. It was only on May 8, when his son-in-law Mr S.S. Saran approached the DIG, Hisar Range, that an FIR was registered and the investigation started. A reward of Rs 1 lakh for furnishing any clue about the "disappearance" of Sushil Kumari was announced. The police failed to make any headway, however.
The Haryana government then wrote two letters to the Centre on July 27 and August 8, requesting a probe by the CBI. About a year later, July 7, 1994, to be precise, the CBI took over investigation by registering a fresh case (No 28(S)/94-CHG) under Section 364, IPC.
At this stage Mr R.S. Dhull, an advocate, preferred a petition before the apex court praying that the CBI be directed to conduct the investigation expeditiously. On September 19, 1995, the apex court ordered the CBI to complete the probe within two months.
In its 31-page investigation report submitted to the Supreme Court the CBI claimed that it had questioned a large number of persons with a view to unveiling the mystery that surrounded the disappearance of Sushil Kumari. It was revealed that Mukesh Kumar used to come to the school during the board examination to help his wife in copying. On being opposed by her, he threatened Sushil Kumari.
Mukesh Kumar was arrested by the CBI on November 28, 1994. He was subjected to sustained interrogation for 10 days, first from November 28 to December 2 and then from December 9 onwards. While still in custody he "expressed his desire to confess his guilt," says the CBI. He was accordingly produced before the Special Judicial Magistrate, Patiala, under Section 164 of the Cr. PC. on December 12.
In his confessional statement Mukesh Kumar said that he was the gunman of Mr Manphool Singh in 1993. During this period he came to know Salendar Bishnoi and Raju. Salender, he said, used to visit the school for helping Raju copy in the examination, which was objected to by Sushil Kumari. She "insulted" Salender and one Kuldip Godara, a friend of Salender, in the presence of other girls.
The statement adds that he (Mukesh Kumar), Salender Kumar, Kuldip Godara, Joginder Singh alias Tiger, Subhash Bagri and and another unnamed friend hatched a plan on April 30, 1993 to abduct her. In furtherance of this conspiracy, he (Mukesh Kumar) along with Kuldip, Salender, Joginder Singh, Subhash and one more person (friend of Subhash) went to the school on May 1, in a red colour Maruti van bearing No HYE or HYS - 5014 and abducted Sushil Kumari. She was confined at a building called Raj Mahal at Hisar from May 1 to 3, whereafter she was strangulated to death with her "dupatta" and her dead body was thrown in a canal near Badopal village at a distance of about 37-38 kms from Hisar.
Less than two weeks later, Mukesh Kumar retracted his confession in the court of the Special Judicial Magistrate, Ambala, on December 24, 1994.
Following the confession Salender Kumar Bishnoi was arrested on December 12, 1994 and intensively interrogated by the CBI until December 24. He admitted having illicit relations with Raj Kumari, alias Raju, since 1990. Mr Sharma's family lived in a portion of their house (No 120-B, New Grain Market, Hisar). He used to help Raju in copying during the board examination of 10+2 in 1993. He arranged for her to sit in a separate room. On her objecting to this, he had an altercation with Sushil Kumari. He disclosed nothing, however, about her alleged abduction and denied his involvement in the same.
The CBI says that on December 12, 1994, it also raided the house of Joginder Singh. He could not be traced. Subsequently, he allegedly committed suicide by consuming an insecticide, as per the report of Dr Khetarpal, the owner of Khetarpal Hospital, Hisar.
Kuldip Godara, Subhash Bagri, son of Mr Mani Ram Bagri, and Raj Kumari were also questioned.
"No material circumstantial or direct evidence," says the CBI report, "could be collected which could connect" Salender Kumar Bishnoi with the crime, apart from Mukesh Kumar's confession.
Nor had any material evidence come on record, it maintains, to "establish any link" between Mukesh Kumar and Subhash Bagri and Kuldeep Singh Godara.
As for Sushil Kumari's fate, the CBI report says: "Despite the best efforts made by the CBI the corpus delicti of Sushil Kumari has not been recovered nor has it been possible to recover any personal effects belonging to her. In all probability she is dead and no more alive, yet since her corpus delicti could not be recovered, it cannot be conclusively stated that she is dead...."
The report concludes: "It has been decided to prosecute Mukesh Kumar for the offences of abduction and illegal confinement of Sushil Kumari. There is no evidence on record to warrant the prosecution of Salender Kumar Bishnoi, Kuldip Godara and Subhash Bagri. Since Salender Kumar Bishnoi was arrested during the course of investigation, it is proposed to get him discharged under Section 169, Cr.P.C"
After perusing the CBI report a two-Judge Bench of the Supreme Court, comprising Mr Justice Kuldip Singh and Mr Justice Saghir Ahmad, ruled on February 13, 1996: "The CBI has found a prima facie case against constable Mukesh Kumar. Mr R.S. Dhull, counsel assisting us in this case, vehemently contends that there is prima facie evidence against other persons. Mr Tulsi states that the challan against Mukesh Kumar shall be filed within two weeks. Needless to say that it is open to Mr Dhull to argue before the court where challan is filed, that apart from Mukesh Kumar, other persons may also be summoned for trial."
On the challan being filed, the trial court at Ambala framed charges against Mukesh Kumar on October 10, 1996, as proposed by the CBI, and recorded the evidence of four persons — Mr Gaje Singh and Mrs Khazani Devi, parents of Sushil Kumari, Mr Kitab Singh Malik, President of the Haryana Teachers' Union and Mr Samar Singh Saran, the sister's husband of Sushil Kumari.
Advocate R.S. Dhull then moved the court under Section 319, Cr. PC to summon Mr Bhajan Lal, former Chief Minister, his sons Mr Chander Mohan and Mr Kuldip Bishnoi, Mr Anil Dawra, the then SSP of Hisar, Subhash Bagri, Salender Bishnoi, Joginder Singh (who has since committed suicide) and Kuldip Godara, as well.
In his order handed down on October 1, 1997, the Special Judge observed: "It is to be made clear that for any criminal charge no person can be summoned to face trial unless there is some legal evidence brought on record against him or her. A person whether he is a Chief Minister or an ordinary man cannot be summoned for criminal charge to the court on the basis of surmises and conjectures. Mere allegations and proclamations that a Chief Minister got another person murdered, simply would not be enough to summon Mr Bhajan Lal".
The Special Judge Mr A.S. Garg (now elevated to High Court) held, however, "that there is sufficient evidence from the mouth" of Gaje Singh, Khazani Devi, Kitab Singh and Samar Singh Saran that Mukesh, Salender Bishnoi, Subhash Bagri Kuldip Godara and deceased Joginder Tiger "had been going to the school and had been insisting on the deceased" to permit copying. "The threats they had been making are enough to conclude that they had lifted the deceased; they had the motive and their motive was frustrated by the deceased by her upright principles and there is a statement of accused Mukesh under Section 164 of the Cr PC which mentions that the deceased, after being confined as such, was taken to Sirsa road, where she was strangulated" and thrown into the canal.
The Judge declined Mr Dhull's application as regards Mr Bhajan Lal, his sons Chander Mohan and Kuldip Bishnoi, SSP Anil Dawra and Ms Raj Kumari, but accepted his plea for summoning Salender Bishnoi, Subhash Bagri and Kuldip Godara.
Dissatisfied, Mr Dhull challenged the order before the High Court, where the matter is still hanging fire.
AS announced already, the Shiromani Gurdwara Parbandhak Committee had declared that no meetings or lectures should be held in the Parkarma of Darbar Sahib, Tarn Taran, without the previous permission of the Local Gurdwara Committee.
It is a matter of pleasure that on the last Amavas fair all conveners of diwans obeyed the instructions of the Shiromani Gurdwara Parbandhak Committee.
The Namdharis formed the only exception. They did not apply for permission and even on being given a written reminder they persisted in their defiance. In fact, they refused to give a reply to the letter.
Further, they sold a tract in the Parkarma again in defiance of the old rules of the temple.
There was consequently great resentment among the thousands of Sikhs gathered in the temple, and but for the presence of responsible members of the Shiromani Gurdwara Parbhandhak Committee, some untoward incident might have happened.
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redpajama_set_name: RedPajamaCommonCrawl


Registration for the 2019 Gig Harbor Turkey Trot is now OPEN!
"[In 2003], Tim Holmes' son challenged him to run a marathon with him. Holmes, who own[ed] Holmes Chiropractic, accepted the challenge and started looking for training opportunities.
'I wanted to do something new, so I invited some of my friends and my patients to join me on Thanksgiving morning for a run. I thought it would be a good way to work up an appetite,' Holmes said. That first year, about 15 people showed up, he said, and the Turkey Trot was born. Now, he says, it has become something of a family tradition for a lot of people. 'We had 20 the second year, and it's just kept growing ever since.'"
The trot begins at 8:00 am on Thanksgiving morning starting on the main road in front of the Gig Harbor post office (3118 Judson St). Please give yourself plenty of time before the race to drop off your cans of food and prepare for the race. You will be able pick up any race packets or preordered T-shirts. You can also pick these up at packet pickup the day before the race (details below under Race Preparations). Day-of-Race Registration starts at 6:30 AM. But please bear in mind that the registration fee is $30 on the day of the race as we want to enourage participants to pre-register well ahead of time to make for a quick and less-chaotic process on the morning of the race. Additional race T-shirts will also be sold both at packet pickup the day before the race and the morning of the race on a first-come, first-served basis. Sizes may be limited.
The Start of both routes is in front of the Gig Harbor post office on Judson St. There will be volunteers along the way to guide you. For the latter half of the 10k, follow the orange arrows on the map. For the 5k, you will turn around approximately at the point on the map that says "5k turn-around". The end of both routes is at the starting line, in front of the post office.
T-shirt and packet pickup day will be on Wednesday, November 21st from 1:00pm to 5:00pm in the St. Anthony Hospital cafeteria. Please do not bring canned food to the packet pickup as our volunteers do not have the means to transport food to the race. Please only bring food to the designated trucks on the morning of the race. Thank you.
This year, proceeds are going to FISH food bank, a local organization called Food Backpacks 4 Kids, the Bischoff food bank, and local high school scholarships.
Start your turkey day right and join us in beautiful downtown Gig Harbor for a 5K or 10K race! Feel free to run or walk the course on your own or with family and friends.


Lego has just put online the first official images of the new Ghostbusters Firehouse Headquarters. The set that will provide a nice shelter for the iconic Ecto 1 will feature exactly four thousand and six hundred and thirty four pieces in a three floors building.
With nine mini-figures that include Peter Venkman, Raymond Stantz, Egon Spengler, Winston Zeddemore, Janine Melnitz, Dana Barrett, Louis Tully, Zombie Driver and Library Ghost, the building will show an endless number of features that will include opening wall for better playability, a huge laboratory, living quarters, a containment unit and a lot more. The Ghostbusters Firehouse Headquarters will be available next January and should retail around the three hundred and fifty dollars. Can't wait to see more detailed pictures of it!
This entry was posted in Ideas and tagged 75827, ghostbusters by pedrolpais. Bookmark the permalink.


A head-banging new study proves that loud music shakes, rattles and rolls the ecosystem.
It's a rare scientific paper that cites both biologist E.O. Wilson and AC/DC guitarist Angus Young.
In fact, there's only one paper with that distinction: "Testing the AC/DC hypothesis: Rock and roll is noise pollution and weakens a trophic cascade," published this week in the journal Ecology and Evolution.
As you might guess from the title, the study — by ecologist Brandon T. Barton and other researchers from Mississippi State University — takes its cue from the famous AC/DC song "Rock and Roll Ain't Noise Pollution" as an avenue to reveal the actual effects of anthropogenic noise (musical or otherwise) on species and their ecosystems.
Here's the refrain from that song, which AC/DC released on their album Back in Black in 1980.
So what was the effect of this music on natural systems? Not so rockin', as it turns out. The researchers fired up their boom boxes and blared music by AC/DC, Guns N' Roses and other hard-rocking bands (as well as a few less musical urban noises, like jackhammers) at some soybeans and their accompanying aphids (a pest insect) and ladybugs (which normally eat the aphids). During a two-week trial — in which Back in Black was played on a continuous 24-hour loop — the ladybugs became less effective predators and ate fewer aphids. This meant there were 40 times more aphids to consume the soybean plants, resulting in plants that were 25 percent smaller.
In other words, rock 'n roll may survive, but the plants exposed to it were less likely to.
This is just the latest study that shows human-generated noise is causing trouble for ecosystems. A 2016 study found that the noise from natural-gas extraction sites robs owls of their ability to hunt. Another study published last year found that engine sounds from highways diminished the ability for nearby animals to find prey (or, conversely, to avoid predators), even when the animals live in parks and other protected areas. A study published earlier this year found that birds living near natural-gas well are experiencing PTSD-like symptoms.
Ah, science: One more way rock 'n roll will survive (yes it will) and live forever.
Now turn down the music, kids. There are some ladybugs doing important work over here.


An artist who distributed small sculptures containing nails around the subway sparked a bomb scare, reports the BBC.
Tonight dancer and MacArthur Fellow Liz Lerman joins Tikkun editor Michael Lerner, biologist Stephen Palumbi, and Stretcher's own Meredith Tromble in conversation about genetics, art, and society moderated Kenneth Foster, YBCA Executive Director and Performing Arts Curator. Lerman's latest work, Ferocious Beauty, opens at YBCA tomorrow night, was created while she was in residence at the Human Genome Project. Lerman says, "Let's bring the scientists and the artists together and ask some questions about how professions demand purity, about the impact of action on the nature of inquiry and whether it taints our purpose." The panel, in the Forum room at the Yerba Buena Center, is free to the public and starts at 6:30 pm. From more information, visit the YBCA Web site.
American audiences, artists, and students are losing access to international artists due to the unwelcoming bureaucratic rigamarole that greets foreigners attempting to enter the States. Under the guise of protecting us from terrorism, the government is "protecting" us from artists, musicians, and scholars…Sandra Kurtz of the Seattle Post-Intelligencer reports on the issue from a Northwest perspective, but the problem is a national one.
Masquerade at the Museum of Contemporary Art in Sydney, Australia is a powerful fairly contemporary survey of portraiture. Besides the names that we in North America are familiar with, such as Cindy Sherman, Ana Mendieta, Sam Taylor-Wood and others, it was fascinating to see artists perhaps more familiar to those "Down Under" such as Ronnie van Hout, Shigeyuki Kihara, and Tod McMillan tackle this genre. It was also great to see Francesca Woodman included in this show (who died quite young) reveal her poignant photographs and up and comer (at least to me) Kalup Donte Linzy's hilarious video about her place in the art world.
Also at the MCA is Taylor-Wood's Crying Men series and her video of David Beckham sleeping. The crying men seemed a bit fake (they are all Hollywood actors after all) but the one of Robert Downey Jr. is gorgeous (as was the vid of Beckham).
Also of interest are The Sydey Morning Herald's open air Bigger Picture Exhibition in Hyde Park encountered on the way to the Archibald, Wynne, and Sulman Prizes at the Art Gallery of New South Wales, the balmy weather, interesting fashion, and the excellent coffee.
arouses huffiness in Jonathon Jones, as he reports in this Guardian article.
Hou Hanru is coming to the San Francisco Art Institute as Director of Exhibitions and Public Programs and Chair of SFAI's Exhibition and Museum Studies program, effective July 1. Hou, who was the artistic director of the 2nd Guangzhou Triennale, has been living and working in Paris as an independent critic and curator and serving as Professor at the Rijksakademie van Beeldende Kunsten, Amsterdam, Netherlands, and as Visiting Professor at Hoger Instituut voor Shone Kunsten, Antwerp, Belgium. For the Guangzhou Triennale Hou co-curated with Hans Ulrich Obrist and Guo Xiaoyan (Guangzhou, China 2005). Other recent curatorial projects include Go Inside, the 3rd Tirana Biennale (Tirana, Albania, 2005), Out of Sight, organized by the De Appel Foundation (Amsterdam, Netherlands, 2005), Nuit Blanche 2004(Paris, 2004), and A L'Ouest Du Sud De L'Est / A L'Est Du Sud De L'Ouest (Villa Arson, Nice, 2004).


On behalf of Pleasure Riders M/C and all the others who have been slaving away getting the new Thunder Ridge facility up and running, I'd like to thank all the riders who traveled from near and far this weekend for the shake-down event for the new location. There were over 140 entries each day, which has got to be some kind of record for a climb in central IL for a very long time! Last year's average for a D17 event was about 80 entries or so…so this was really a fantastic rider turnout!
Over the course of the weekend, we put down close to 600 rides on the brand-new hill, and will be making some minor tweaks before the 2016 AMA Hillclimb Grand Championship. We already had arrangements for additional speakers to spread farther out across the event area, as well as an additional scoreboard to display rider times. We're going to be pulling out all the stops to make the Grand Championship a fantastic event, and it was great to have the support, feedback, and excitement from that record-setting group of riders this past weekend!
Speaking of the 2016 AMA Hillclimb Grand Championship…there is now only 2 weeks left to get signed up. All entries must be postmarked by July 1st – so if you haven't gotten that form in the mail yet, now is the time! The entry form and information pack is available on the Pleasure Riders M/C website – the Thunder Ridge logo above and the PRMC logo below are both linked to the page on the PRMC website where you can get that form – just click the giant form icon that says "Click Me!" to get started.
If anyone has any questions about the event, there's a contact form on the PRMC website, as well as contact info in that information packet…or post a note here on the KOTH forums.
TWO WEEKS! As usual, there will be no on-site or late entries. Get your entries in the mail no later than July 1st, or you won't be part of history when the AMA Hillclimb Grand Championship returns to Illinois for the first time in 28 years! See you there!


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Q: How to declare a variable for high resolution clock in C++? In the example here: https://en.cppreference.com/w/cpp/chrono/high_resolution_clock/now
They declared the clock time point with auto.
auto start = std::chrono::high_resolution_clock::now();

and the doc says it returns 'A time point representing the current time.'
But I am not sure how do declare in my code below because I am used to declaring the variables at the beginning of the function and I don't know what to declare it as. Code has been simplified here to show what I mean. What do I put for the ????
I already tried auto there but the compiler won't allow it. auto orderRecvedTime; gives me this error:
error: non-static data member declared with placeholder 'auto'

#include <iostream>
#include <iterator>
#include <sstream>
#include <string>
#include <string.h>
//#include "load_symbol.h"
//#include "check_symbol.h"
#include "windows.h"
#include <vector>
#include <chrono>
using namespace std;

class order {
  private:
    string orderID;
    ???    orderRecvedTime;
    char   buysell;
    string symbol;
    double price;
    int    qty;

  public:
    void newOrder(string &_orderID, char &_buysell, string &_symbol, double &_price, int &_qty){
        orderID = _orderID;
        buysell = _buysell;
        symbol = _symbol;
        price = _price;
        qty = _qty;
        orderRecvedTime = std::chrono::high_resolution_clock::now();
    }

};

int main() {
    cout << "!!!Hello once more" << endl; // prints !!!Hello once more

    vector<order> thebook;
    string user_order = "";

    string done = "done trading";
    string orderID;
    string orderaction;
    string orderRecvedTime;
    char buysell;
    string symbol;
    double price;
    int qty;

    while (user_order.compare(done) != 0) {
        cout << "enter order"<< endl;
        getline(cin, user_order);

        stringstream lineStream(user_order);
        lineStream >>orderaction>>orderID>> buysell >> symbol >> price>> qty;

        order user_order;
        if (orderaction.compare("D") == 0) {
            cout << "you are making a new order."<< endl;
            user_order.newOrder(orderID, buysell,symbol,price,qty);
            thebook.push_back(user_order);
        }
    }
}

A: std::chrono::high_resolution_clock::time_point orderRecvedTime;

In practice, high_resolution_clock is a type alias for either system_clock or steady_clock, so my advice is to choose either one of those instead for a portable experience.

*

*system_clock is like a watch.  It can tell you what time it is.

*steady_clock is like a stop watch.  It is really good for timing things, but doesn't really know the time of day.

A: pulled this from the debugger msg but if i declare it like this it compiles.
std::chrono::_V2::system_clock::time_point orderRecvedTime;

redpajama_set_name: RedPajamaStackExchange


text:
Atlanta's Internal Platform Turns Public to Provide Real-Time Commuter Data Following Bridge Collapse
A 12-day build, CommuteATL helps Atlantans get around congestion.
by Theo Douglas / April 24, 2017
Videos of motorists driving through billowing black smoke were the signature images when a bridge collapsed on Interstate 85 in Atlanta recently, but in that crisis, city officials seized opportunity — going live 12 days later with a dynamic website that uses data in new ways to battle gridlock.
No one was hurt in the dramatic chain of events during evening rush hour on Thursday, March 30. But the fire underneath the elevated highway segment, which is alleged to have been illegally set, spread to plastic conduit being stored underneath by the Georgia Department of Transportation (GDOT) — burning so hot that it brought down the roadway above. I-85 is expected to remain closed in both directions north of downtown until at least June 15.
On Monday, April 17, work on a gas line buckled lanes on Interstate 20, closing it southeast of the city center for nearly two days. In a case of good timing, public and private agencies had debuted CommuteATL.com, their new website optimized for mobile use, the previous week.
Metrics recorded the day of the I-20 shutdown showed more than 6,000 users had already visited the site. By Thursday, April 20, an average of 1,200 views per day had swelled that total to 9,500.
CommuteATL is powered by technology from Redlands, Calif.-based GIS software provider Esri and partner Waze, creator of a real-time, crowdsourced navigation app.
But it incorporates key information, including GDOT 511 camera footage at intersections, and four city data layers: the latest from Waze, real-time Metropolitan Atlanta Rapid Transit Authority (MARTA) train arrival times, Relay Bike Share station locations, and the latest from the 2.7-mile Atlanta Streetcar loop.
It lets drivers and city officials see traffic jams, closed roads, accidents and alerts as they happen. City planners can communicate with Waze about the routes it creates for users and share citizen feedback. Members of the public get a dashboard with live updates on traffic data that let them pick the best path through the city — whether by foot, bicycle, train, streetcar or vehicle.
In a tweet April 14, Atlanta Mayor Kasim Reed called the site "another tool in the tool box to help you better navigate traffic during this very difficult time."
The website, a collaboration including the mayor, CIO Samir Saini, Chief Resilience Officer Stephanie Stuckey, Deputy Chief Operating Officer and Public Works Commissioner William Johnson, and GIS employees, began as something quite different. It first launched in early April as an internal-facing way for public works and joint operations center workers "to get an understanding of what was happening on the street," Saini told Government Technology.
"But once we built that out for internal use we started looking at each other [saying], 'Well this could be really valuable,'" he said. Conversations among members of the mayor's task force, formed to deal with traffic congestion, helped identify the four main city data layers.
Stuckey contacted Esri's Disaster Response Program, which offers cities consulting and technical support during disasters. The company's local government account manager, Rob Hathcock, said the process was streamlined by pre-existing contracts Atlanta had with Esri and Waze, which meant the city already had most of the tools it needed to create the platform. One exception, he noted, was amplifying its account with ArcGIS, the company's mapping and analytics platform, to host the services.
"When we first pulled in the disaster response team with the city of Atlanta, we kind of did a little bit of discovery, asking them what they wanted us to provide. They couldn't mitigate the issue with the bridge. What they could do was show how the city responded to this, 'What can we show our constituents after the bridge collapsed?'" Hathcock said.
In 2016, Atlanta was accepted into the 100 Resilient Cities network, a $100 million investment by the Rockefeller Foundation to help cities avoid being buffeted by physical, social and economic challenges. It also joined the Waze Connected Citizen Program, a free data-sharing partnership, last year.
Both partnerships helped fast-track the creation of CommuteATL, believed to be the first time a site has sent real-time information two ways: to Atlanta officials on one end and Waze users on the other, then letting City Hall communicate with Waze.
"If that's true, that's great," Saini said. "You only really feel the value of these kinds of partnerships and this data-sharing in times of crisis, or at least it's empathized a whole lot more. We're proud of it."
"What is coming to us is an official source of two buckets of information," Adam Fried, global partnerships manager at Waze, said, describing the feed from City Hall of planned events like road closures and sporting events, as well as real-time events like weather, disasters and protests that affect transportation.
And there's more to come from the website, which had start-up costs that were "virtually nil," Saini said, consisting mainly of the extra staff time needed to stand it up.
He'd like to see CommuteATL become a real app in addition to an optimized website. And at least until I-85 is reopened, officials plan to continue adding data layers regularly while preserving an easy user experience. In the planning stages are ride-share and car-share information, MARTA bus schedules and more Relay Bike Share locations. Last week, the city announced the program's expansion from 100 bikes at 22 locations to 500 bikes at 65 locations less than a year after its launch.
"We're not going to pull the plug on it once the bridge is fixed," Saini said. "We're probably just going to ask citizens, 'Where do you want us to take this thing?'"
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Managing Director, David Manning, has been asked to present at the 11th Energy Storage World Forum in Berlin, on 17th May. David will be addressing the international attendees on the ways in which modern data centres can benefit from new energy storage technology.
As significant energy consumers, one of the primary benefits of a data centre is the ability to self-generate energy to keep IT and data equipment functioning, should power supplies fail. MIGSOLV's Gatehouse data centre in Norwich uses a rotary uninterrupted power supply, banks of batteries and three industrial generators to instantaneously provide enough power for eight days running at full capacity.
Attended by global energy producers and consumers including Eon, EDF and Carrefour, David's presentation will focus on the ways new energy storage technology can be exploited by data centres to satisfy the growing global need for highly secure and always available data.
Prior to the Berlin event, MIGSOLV will be exhibiting at the ICT Spring, Global Tech Conference in Luxembourg, from 15th to 16th May.
As their chosen UK partner, MIGSOLV have been invited by EBRC (the European Business Reliance Centre) in Luxembourg. One of Europe's largest data-centre providers, EBRC have enlisted MIGSOLV to showcase the world-class facilities available for UK companies using technology to revolutionise finance, insurance and banking (collectively known as FinTech).
Billed as 'The place where Digital, Tech & Science converge', ICT Spring is an international expo of new and innovative technology, featuring over 5,000 key decision makers from more than 70 countries. The annual event provides a showcase for the latest trends and developments and their impact on society and the working world.
Closer to home, May will also see the long-anticipated adoption of new GDPR legislation. MIGSOLV has been working closely with new and existing clients to ensure its ultra-secure facilities help them satisfy regulations and keep their customers' data safe.
Further information on both events can be found at www.energystorageforum.com and www.ictspring.com.


The cabinet file 'st170510.cab' required for this installation is corrupt and cannot be used. I'll split points in a minute ... Thinkpads_User 0 LVL 16 Overall: Level 16 Windows 7 8 Windows OS 6 Message Expert Comment by:joinaunion2012-06-08 Wich program are you trying to install? 0 LVL 32 Overall: Level Close any window that opens automatically after you insert the software CD.
I uninstalled it and tried to install this version and met this error. Solution 5: Extract the file and run the installer again. Schließen Ja, ich möchte sie behalten Rückgängig machen Schließen Dieses Video ist in Deutschland leider nicht verfügbar, da es Musik von UMG enthalten könnte, über deren Verwendung wir uns mit der Click Apply to save the changes, and then click Close to exit the System Configuration Utility.
Thanks! 0 Question by:Daniel Wilson Facebook Twitter LinkedIn Google LVL 32 Best Solution byDaniel Wilson As it turns out, the install set was corrupted during download. Solution 6: (CS4) Reinstall Microsoft Windows Installer 3.1 or 4.x. Method two: Copy the CD files to the hard disk and install the software Follow these steps to copy the CD files to the hard disk and install the software. Step three: Delete all temporary files on the computer again Follow the procedure in Step one: Delete all temporary files on the computer.
Source file not found: x:\filename.cab. This could indicate a network error, an error reading from the CD-ROM, or a problem with this package." is not because of Issue in JDK setup or as such. Source file not found: [drive]:\[product name][product version]\data1.cab. How can I get this program installed?
Join them; it only takes a minute: Sign up Error installing JDK up vote 1 down vote favorite I got following error while installing JDK 1.7 on windows 8. "Error 1335. Find Solutions, ask questions, and share advice with other HP product owners. What should I try on this? I then had to re-install java jre, and then install java jdk.
For instructions, see the documentation for the antivirus software or contact the manufacturer. We should contact Sun and inform them about this conflict between Java jdk and norton anti-virus. CUSTOMER SUBMITTED WORKAROUND : Move unpacked cabs and msi files to directory without spaces/diacritics in it (tried one without both, don't know what exactly of these two is the problem). Moving extracted cabs and msi files from above mentioned directory to another one without spaces and diacritics in path name and running msi installer by hand, installation succeeds.
what is the good approach to make sure advisor goes through all the report? Documentation Database Oracle Fusion Middleware Oracle Enterprise Manager Applications Technology Sun Documentation See All ??? Step four: Install the software from the hard disk Return to the hpdisk folder in Windows Explorer, and then double-click the Setup executable file. Click File, point to New, and then click Folder.


text:
Reviews: Dulles Landing
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June 29, 2016 by salonkhouri
We are going to smash a few walnuts this month …
Our July community outreach partner is the Smashing Walnuts foundation, founded by Ellyn and Mark Miller at the urging of their nine year old daughter, Gabriella. Gabriella wanted to help raise funding and awareness for childhood brain cancers. At 9 years old, Gabriella was diagnosed with DIPG (Diffuse Intrinsic Pontine Glioma), an inoperable brain tumor which is terminal upon diagnosis. Her tumor was the size of a walnut. Wanting to spread awareness and raise money for research, Gabriella quickly became a global advocate for children with cancer, raising nearly $300,000 for children's cancer charities. To say Gabriella was a powerhouse is an understatement! We never had the opportunity to meet Gabriella in person, but were blessed to hear her speak at CureFest in 2013, which you can watch below. We were amazed by her eloquence, courage and ability to captivate an audience better than most celebrities.
Less than a year after her diagnosis, Gabriella passed away surrounded by family and friends who promised to continue her work. An interview filmed just two weeks before her death led to the creation of a Federal Bill, the Gabriella Miller Kids First Research Act, allocating $126 million for childhood disease research and kick started the Smashing Walnuts mission to focus on research.
Raising money for research in childhood cancers is critical. There is simply not enough funding in this area to bring about change. In fact, in the last 30 years, only THREE drugs have been developed to treat cancer in children exclusively. According to CureSearch, the amount of research dollars spent per patient is 5 times more for adults than for children. An article published on Forbes.com indicated that the funding for childhood cancer clinical trials in 2008 was $26.4 million. By comparison, federal funding for AIDS research at the same time was $254 million while funding for breast cancer topped $584 million.
An article published on Forbes.com indicated that the funding for childhood cancer clinical trials in 2008 was $26.4 million. By comparison, federal funding for AIDS research at the same time was $254 million and funding for breast cancer topped $584 million.
We are not saying that other cancers should not be funded. We just feel that funding for childhood cancers is disproportionate compared to adult cancers. For example, Of the National Cancer Institute's budget of $4.9 billion, only 4% is allocated toward childhood cancer research. The Salon Khouri team stands beside Smashing Walnuts and many other local, childhood cancer non-profits to firmly state that 4% is not enough for our kids.
Gabriella smashing a walnut to symbolize smashing out DIPG. The walnut was the same size as the inoperable tumor that took her life within just one year of diagnosis.
This July, Salon Khouri will donate $1 from every haircut from our Dulles and Fairfax locations to Smashing Walnuts. We know it's not millions, but we've been shown in many amazing ways that little can make big. We pray the money will lead to a cure so that no other parent will have to endure the loss of their child to cancer.
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