text:
\section{Introduction}
A great interest was shown lately towards problems concerning optimal partitions related to some spectral quantities (see \cite{BucButHen98,BucBut05,CafLin07,HelHofTer09}). Among them, we distinguish two problems which interest us.
Let $\Omega$ be a bounded and connected domain and $\mathfrak P_{k}(\Omega)$ the set of partitions of $\Omega$ in $k$ disjoint and open subdomains $D_{i}$. We look for partitions $\mathcal D=(D_i)_{1\leq i\leq k}$ of $\Omega$ which minimize
\begin{description}
\item[{\bf\quad Problem 1.}] the largest first eigenvalue of the Dirichlet Laplace operator in $D_{i}$:
\begin{equation}
\mathfrak L_{k}(\Omega)= \min\left\{ \max_{1\leq i\leq k} \lambda_1(D_i),(D_{i})_{1\leq i\leq k}\in\mathfrak P_{k}(\Omega)\right\} .
\end{equation}
\item[{\bf\quad Problem 2.}] the sum of the first eigenvalues of the Dirichlet Laplace operator in $D_{i}$:
\begin{equation}
\mathfrak L_{k,1}(\Omega)= \min\left\{\frac 1k\sum_{1\leq i\leq k}\lambda_1(D_i),(D_{i})_{1\leq i\leq k}\in\mathfrak P_{k}(\Omega)\right\}.
\end{equation}
\end{description}

For simplicity we refer to these two problems in the sequel as minimizing the sum or the max. Theoretical results concerning the existence and regularity of the optimal partitions regarding these problems can be found in \cite{BucButHen98,CafLin07,HelHofTer09} (and references therein). Despite the increasing interest in these problems there are just a few cases where optimal partitions are known explicitly, and exclusively in the case of the max.

\paragraph{Structure of the paper}
In the following section, we apply known results to obtain estimates of the energy of optimal partitions in the case of a disk, a square or an equilateral triangle and recall informations about the structure of an optimal partitions.
Then, we focus on the minimization problem for the max: we give explicit results for $k=2,4$ and in other cases, we present the best candidates we obtained by using several numerical methods.
In Section~\ref{sec.compar}, we recall and apply a criterion which shows when a partition optimal for the max cannot be optimal for the sum.
In cases where the previous criterion shows that candidates for the max are not optimal for the sum, we propose better candidates in Section~\ref{sec.sum}. These candidates are either obtained with iterative methods already used in \cite{BouBucOud09,BBN16} or are constructed explicitly.

\section{Applications of known results}
\subsection{Estimates of the energies}
By monotonocity of the $p$-norm, we easily compare the two optimal energies:
\begin{equation}
\frac 1k \mathfrak L_{k}(\Omega)\leq \mathfrak L_{k,1}(\Omega)\leq \mathfrak L_{k}(\Omega).
\end{equation}
More quantitative bounds can be obtained with the eigenmodes of the Dirichlet-Laplacian on $\Omega$.
For $k\geq1$, we denote by $\lambda_{k}(\Omega)$ the $k$-th eigenvalue of the Dirichlet-Laplacian on $\Omega$ (arranged in increasing order and repeated with multiplicity) and by $L_k(\Omega)$ the smallest eigenvalue (if any) for which there exists an eigenfunction with $k$ nodal domains ({\it i.e.} the components of the nonzero set of the eigenfunction). In that case, the eigenfunction gives us a $k$-partition such that the first eigenvalue of the Dirichlet-Laplacian on each subdomain equals $L_{k}(\Omega)$.  We set $L_k(\Omega) =+\infty$ if there is no eigenfunction with $k$ nodal domains.
It is standard to prove (see \cite{HelHofTer09} for example):
\begin{align}
\lambda_{k}(\Omega) & \leq {\mathfrak L}_{k}(\Omega) \leq L_{k}(\Omega),\label{ineq.HHOT}\\
\frac 1k\sum_{i=1}^k\lambda_{i}(\Omega)& \leq {\mathfrak L}_{k,1}(\Omega) \leq L_{k}(\Omega).\label{ineq.HHOTsum}
\end{align}
Let us consider $k=1,2$. Since the first eigenvalue of the Dirichlet-Laplacian is simple and $\Omega$ is connected, then necessarily any eigenfunction associated with $\lambda_{k}(\Omega)$ has one or two nodal sets whether $k=1$ or $2$. Consequently $L_{k}(\Omega)=\lambda_{k}(\Omega)$ for $k=1,2$ and
\begin{equation}\label{eq.k12}
\mathfrak L_{k}(\Omega) = \lambda_{k}(\Omega)=L_{k}(\Omega), \quad \mbox{ when }k=1,2.
\end{equation}
Furthermore, any nodal partition associated with $\lambda_{k}(\Omega)$ is optimal for the max when $k=1,2$.
We say that an eigenfunction associated with $\lambda$ is {\it Courant-sharp} if it has $k$ nodal domains with $k=\min\{j,\lambda_{j}(\Omega)=\lambda\}$.\\

In this paper, we focus on three geometries: a square $\square$ of sidelength $1$, a disk $\ocircle$ of radius $1$ and an equilateral triangle $\triangle$ of sidelength $1$. The eigenvalues are explicit and given in Table~\ref{tab.vp} where  $j_{m,n}$ is the $n$-th positive zero of the Bessel function of the first kind $J_{m}$ and $\lambda_{k}(\Omega)$ is the $k$-th element of the set $\{\lambda_{m,n}(\Omega)\}$. In Table~\ref{tab.estim}, we explicit the lower and upper bounds in \eqref{ineq.HHOT}--\eqref{ineq.HHOTsum}.

\begin{table}[h!]
\begin{center}\begin{tabular}{c | cl}
$\Omega$ & $\lambda_{m,n}(\Omega)$ & $m,n$\\
\hline
& \\[-8pt]
$\square$ & $\pi^2(m^2+n^2)$& $m,n\geq 1$\\[2pt]
$\triangle$ & $\frac{16}9 \pi^2(m^2+mn+n^2)$& $m,n\geq 1$\\[2pt]
$\ocircle$ & $j_{m,n}^2$ & $m\geq0$, $n\geq 1$ {(multiplicity 2 for $m\geq1$)}\\[2pt]
\end{tabular}\end{center}
\caption{Eigenvalues for the Dirichlet-Laplacian on $\Omega=\square,\ \triangle,\ \ocircle$. \label{tab.vp}}
\end{table}

\begin{table}[h!]
{\small\begin{center}\begin{tabular}{|c||r|r|r||r|r|r||r|r|r|}
\hline
& \multicolumn{3}{c||}{Square} & \multicolumn{3}{c||}{Disk}& \multicolumn{3}{c|}{Equilateral triangle}\\
\hline
 $k$ & \multicolumn{1}{c|}{$\frac1k\displaystyle\sum_{i=1}^k\lambda_{i}$} & \multicolumn{1}{c|}{$\lambda_{k}$} & \multicolumn{1}{c||}{$L_{k}$} & \multicolumn{1}{c|}{$\frac1k\displaystyle\sum_{i=1}^k\lambda_{i}$} & \multicolumn{1}{c|}{$ \lambda_{k}$} & \multicolumn{1}{c||}{$L_{k}$} & \multicolumn{1}{c|}{$\frac1k\displaystyle\sum_{i=1}^k\lambda_{i}$} & \multicolumn{1}{c|}{$\lambda_{k}$} & \multicolumn{1}{c|}{$L_{k}$} \\
 \hline
 1  &  19.74 &  19.74 & 19.74 & 5.78 &  5.78 & 5.78  &  52.64 &  52.64 & 52.64 \\
 2  &  34.54 &  49.35 & 49.35 & 10.23 & 14.68 & 14.68  &  87.73 & 122.82 & 122.82 \\
 3  &  39.48 &  49.35 & 98.70 & 11.72& 14.68& 74.89  &  99.43 & 122.82 & 228.10 \\
 4  &  49.35 &  78.96 & 78.96 & 15.38& 26.37 & 26.37  & 127.21 & 210.55 & 210.55 \\
 5  &  59.22 &  98.70 & 256.61 & 17.58 & 26.37 & 222.93 & 147.39 & 228.10 & 368.46 \\
 6  &  65.80 &  98.70 & 246.74 & 19.73 & 30.47 & 40.71  & 160.84 & 228.10 & 543.92 \\
 7  &  74.73 & 128.30 & 493.48 & 22.72 & 40.71 & 449.93   & 185.49 & 333.37 & 684.29 \\
 8  &  81.42 & 128.30 & 197.39 & 24.97 & 40.71 & 57.58  & 203.97 & 333.37 & 491.29 \\
 9  &  91.02 & 167.78 & 177.65 & 27.67& 49.22 & 755.89  & 222.25 & 368.46 & 473.74 \\
10  &  98.70 & 167.78 & 286.22 & 29.82 & 49.22 & 76.94  & 236.87 & 368.46 & 1000.12 \\
\hline
\end{tabular}\end{center}}
\caption{Bounds \eqref{ineq.HHOT}--\eqref{ineq.HHOTsum} when $\Omega=\square$, $\ocircle$ and $\triangle$.\label{tab.estim}}
\end{table}

\begin{rem}\label{rem.k4}
Looking at Table \ref{tab.sect}, we observe that the lower and upper bounds \eqref{ineq.HHOT} for the max are equal when $k=1,2,4$. Thus, in that case, the optimal energy for the max is $\lambda_{k}(\Omega)$ and any associated nodal partition is optimal. We can wonder if it can happen for larger $k$. We will come back to this question in the following section.
\end{rem}

In the case of the disk, for an odd index $k$, the upper bound $L_k$ given in Table~\ref{tab.estim} corresponds to the $k$-th simple eigenvalue on the disk whose associated nodal partition has $k$ concentric annuli. This eigenvalue grows rapidly with $k$. In these cases, we may observe that considering partitions in $k$ equal sectors gives a better upper bound. If we denote by $\Sigma_{\alpha}$ the angular sector of opening $\alpha$ (see \cite{BonLen14} for analysis on angular sectors), this new upper bound writes
\[{\mathfrak L}_{k}(\Omega) \leq \lambda_{1}\left(\Sigma_{\frac{2\pi}k}\right)=j^2_{\frac k2,1}.\]
This heavily improved upper bound is estimated in Table~\ref{tab.sect}.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$k$ & 3 & 5 & 7 & 9\\
\hline
$\lambda_{1}\left(\Sigma_{\frac{2\pi}k}\right)$ & 20.19 & 33.22 & 48.83 & 66.95\\
\hline
\end{tabular}
\caption{Energy of the partitions with $k$ angular sectors.\label{tab.sect}}
\end{center}
\end{table}
We can notice that if we replace $L_{k}(\ocircle)$ in Table~\ref{tab.sect} by these values, we have monotonicity with respect to $k$ for the upper bound which is more consistent with the monotonicity of $\mathfrak L_{k}(\ocircle)$ and $\mathfrak L_{k,1}(\ocircle)$. \\

Bounds \eqref{ineq.HHOT}--\eqref{ineq.HHOTsum} provide a quantative information about the energy of the optimal partition but no information about the structure of the minimal partition. Let us know discuss this point.

\subsection{About the minimal partitions}
As we have mentioned previously, several works are dealing with the existence of optimal partitions (see \cite{BucButHen98,CafLin07,HelHofTer09}\ldots). These results also give information about their regularity.
\begin{thm} \label{thm.ex}
For any $k\geq1$, there exists a regular optimal $k$-partition for any of the two optimization problems.
Furthermore any optimal partition for the max is an equipartition, {\it i.e.} the first eigenvalues on each subdomain are equal.
\end{thm}
Let us recall that a $k$-partition $\mathcal D$ is called \emph{regular} if its boundary, $N(\mathcal D)= \cup_{1\leq i\leq k}\partial D_{i}$, is locally a regular curve, except at a finite number of singular points, where a finite number of half-curves meet with equal angles. We say that $\mathcal D$ satisfies the \emph{equal angle meeting property}.\\

\section{Candidates for the max}
\label{sec.cmax}

\subsection{Explicit solutions}
The following result, established in \cite{HelHofTer09} gives information about the equality case in \eqref{ineq.HHOT} and permits to solve, in some cases, the optimization problem for the max. In that cases, the minimal partitions are nodal.
\begin{thm}\label{thm.HHOT}
The nodal partition of a Courant-sharp eigenfunction is optimal for the max. Conversely, if a minimal partition for the max is nodal, the associated eigenfunction is Courant-sharp.
\end{thm}
This theorem means that if we have one equality in \eqref{ineq.HHOT}, then we have equality everywhere and any optimal partition is nodal.

As observed in \eqref{eq.k12}, we know that the optimal $k$-partition for the max is given by the $k$-th eigenfunction when $k=1,2$. Remark \ref{rem.k4} notices that it is still the case for $k=4$ for the three considered geometries. The following result established in \cite{HelHofTer09, MR3445517,BH2016LMP} show that for other $k$, the situation is very different.

\begin{prop}\label{prop.k24}
If $\Omega$ is a disk $\ocircle$, a square $\square$ or an equilateral triangle $\triangle$, then
\[\lambda_{k}(\Omega)=\mathfrak L_{k}(\Omega)= L_k(\Omega) \qquad\mbox{if and only if}\qquad k=1,2,4.\]
Thus the minimal $k$-partition for the max is nodal if and only if $k=1,2,4$.
\end{prop}
Figure~\ref{fig.PartNod} gives examples of optimal $k$-partitions for the max. Note that since $\lambda_{2}(\Omega)$ is double, the minimal $2$-partition is not unique whereas for $k=4$ we do have 	uniqueness.

\begin{figure}[h!]
\begin{center}
\begin{minipage}[b]{.6\linewidth}
\begin{center}
\includegraphics[height=1.4cm]{CarreVP2}\
\includegraphics[height=1.4cm]{carreVP2quad}\
\includegraphics[height=1.4cm]{DiskVP2.pdf}
\includegraphics[height=1.4cm]{TriEqVP2.pdf}\
\includegraphics[height=1.4cm]{TriEqVP2bis.pdf}
\subcaption{$k=2$}
\end{center}
\end{minipage}
\hfill\begin{minipage}[b]{.38\linewidth}
\begin{center}
\includegraphics[height=1.4cm]{CarreVP4}\
\includegraphics[height=1.4cm]{DiskVP4}
\includegraphics[height=1.4cm]{TriEqVP4}
\subcaption{$k=4$}
\end{center}
\end{minipage}
\end{center}
\caption{Optimal $k$-partitions for the max, $k=2,4$.\label{fig.PartNod}}
\end{figure}

\subsection{Candidates obtained by numerical simulations}

In cases when explicit solutions are not known, some numerical algorithms are available in order to compute minimal partitions for the max. There are two types of methods that have been used in the literature.
\begin{itemize}
\item The first option is to look for partitions which are nodal for a mixed Dirichlet-Neumann problem on $\Omega$ (see for example \cite{BHV}). In this case we are certain to have an equipartition. On the other hand, we force \emph{a priori} the structure of the partition.
\item Another option is to use an iterative approach based on an adaptation of the algorithm introduced in \cite{BouBucOud09}. In \cite{BBN16}, the authors presented an algorithm allowing the minimization of a $p$-norm of eigenvalues, which approaches the optimization problem for the max as $p$ is large. Another variant of the algorithm was presented, which penalizes the difference between eigenvalues on different cells of the partition. These iterative algorithms have the advantage that there is no constraint on the structure of the partition. On the other hand a relaxation method is used in order to compute the eigenvalues, which makes the method less precise.
\end{itemize}

In \cite{BBN16} the authors studied the square, the disk and the equilateral triangle. They used initially the iterative methods to detect the structure of the partition. Secondly, when some parts of the boundaries of the cells of the partitions were segments or easily parametrizable curves, the mixed Dirichlet-Neumann approach was used. Not surprisingly, the Dirichlet-Neumann method yields best results, {\it i.e.} the smallest maximal eigenvalue, when it can be used. We recall below the best partitions obtained with the two above methods when $k=3,5, 6, 7, 8, 9, 10$.
The smallest energies obtained numerically are summed up in Table \ref{tab.Linfty} and the associated partitions are represented in Figures \ref{fig.triangleinf}, \ref{fig.squareinf} and \ref{fig.disqueinf}. In these figures, the partitions with green lines are obtained with the Dirichlet-Neumann approach whereas the others are the results of the iterative method with penalization.

\begin{table}[h!t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$k$ & 3 & 5 & 6 & 7 & 8 & 9 & 10\\
\hline
$\triangle$ &  $142.88$ & $251.99$ & $275.97$ & $345.91$ & $389.31$ & $428.75$ & $451.73$ \\
\hline
$\ocircle$ & $20.19$ & $33.21$ & $39.02$ & $44.03$ & $50.46$ & $58.25$ & $67.19$ \\
\hline
$\square$ & $66.58$ & $104.29$ & $127.11$ & $146.88$ & $161.28$ & $178.08$ & $204.54$ \\
\hline
\end{tabular}
\caption{Numerical estimates for $\mathfrak L_{k}(\Omega)$, $\Omega=\triangle, \square, \ocircle$.\label{tab.Linfty}}
\end{center}
\end{table}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{ccccccc}
\includegraphics[width=0.138\textwidth]{equi3v}
&\includegraphics[width =0.138\textwidth]{equi5v}
&\includegraphics[width =0.138\textwidth]{equi6v}
&\includegraphics[width =0.138\textwidth]{equim7}
&\includegraphics[width =0.138\textwidth]{equi8v}
&\includegraphics[width =0.138\textwidth]{equim9}
&\includegraphics[width =0.138\textwidth]{equi10v}\\
$k=3$ & $k=5$ & $k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates to be minimal $k$-partition of the equilateral triangle for the max.}
\label{fig.triangleinf}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{ccccccc}
\includegraphics[width = 0.138\textwidth,angle=180,origin=c]{SquarePart3maxv}
&\includegraphics[width= 0.138\textwidth]{SquarePart5maxv}
&\includegraphics[width= 0.138\textwidth]{squarek6p50Partitions}
&\includegraphics[width= 0.138\textwidth]{squarek7v1p50Partitions}
&\includegraphics[width= 0.138\textwidth]{squarek8p50Partitions}
&\includegraphics[width= 0.138\textwidth]{squarek9p50Partitions}
&\includegraphics[width= 0.138\textwidth]{squarek10p50Partitions}\\
$k=3$ & $k=5$ & $k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates to be minimal $k$-partition of the square for the max.}
\label{fig.squareinf}
\end{figure}

\begin{figure}[h!]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{ccccccc}
\includegraphics[width=0.138\textwidth]{DiskPart3maxv}
& \includegraphics[width=0.138\textwidth]{DiskPart5maxv}
& \includegraphics[width=0.138\textwidth]{glisdisq_k5Symv}
& \includegraphics[width=0.138\textwidth]{glisdisq_k6Symv}
& \includegraphics[width=0.138\textwidth]{glisdisq_k7Symv}
& \includegraphics[width=0.138\textwidth]{glisdisq_k8Symv}
&\includegraphics[width =0.138\textwidth]{diskk10p50Partitions}\\
$k=3$ & $k=5$ & $k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates to be minimal $k$-partitions of the disk for the max.}
\label{fig.disqueinf}
\end{figure}

\section{Optimal partitions for the max as candidates for the sum\label{sec.compar}}
The aim of this section is to analyze if the optimal partitions obtained numerically for the max can be optimal for the sum.
Let us recall a criterion established  in \cite{HHO10} which prevents a partition optimal for the max to be optimal for the sum.
\begin{prop}\label{l2norm}
Let ${\mathcal D}=(D_1,D_{2})$ be a minimal $2$-partition for the max and $\varphi_{2}$ be a second eigenfunction  of the Dirichlet-Laplacian on $\Omega$ having $ D_1$ and $ D_2$ as nodal domains.
\begin{equation}\label{eq.criter}
\mbox{Suppose that }
\int_{D_1} |\varphi_{2}|^2 \neq \int_{D_2}|\varphi_{2}|^2,\qquad \mbox{ then }\quad
\mathfrak L_{2,1}(\Omega) < \mathfrak L_{2} (\Omega).\hfill\
\end{equation}
\end{prop}
Since any minimal $2$-partition for the max is nodal, the previous criteria can be generalized by considering two neighbors $D_{1},D_{2}$ of an optimal partition for the max.
Note that this criterion gives no information if we apply it to an optimal partition for the max which is composed of congruent domains.
It seems to be the case when $k=2,4$ for the square, $k=2,3,4,5$ for the disk and $k=3,4$ for the equilateral triangle  (see Figures~\ref{fig.PartNod}--\ref{fig.disqueinf}).

We know that partitions minimal for the max are in fact equipartitions (see Theorem~\ref{thm.ex}). Yet the Dirichlet-Neumann approach produces equipartition whereas with the iterative method, the first eigenvalues on the subdomains of the partition obtained numerically are close but not rigorously equal.
Thus we consider from now in this section the partitions obtained with the Dirichlet-Neumann approach.
In such situation, Proposition \ref{l2norm} can be adapted as follows.
\begin{prop}
Let $\mathcal{D} = (D_i)_{1\leq i \leq k}$ be a minimal $k$-partition for the max. We assume that $\mathcal D$ is the nodal partition of an eigenfunction $\varphi$ of a mixed Dirichlet-Neumann problem. Then this partition is not minimal for the sum if we can find two neighboring cells on which the $L^2$ norms of the eigenfunction $\varphi$ are different.
\label{l2normappl}
\end{prop}
\begin{proof}
It is not difficult to see that the previous result is a simple consequence of Proposition \ref{l2norm}. Indeed, if $D_1,D_2$ are two neighbors of a nodal partition associated to a mixed Dirichlet-Neumann problem, then $(D_1,D_2)$ is a minimal partition for the max on ${\rm Int}(\overline{D_1} \cup \overline{D_2})$. If this is not the case, then $\mathcal{D}$ would not be minimal. Furthermore, the restriction of the eigenfunction $\varphi$ to ${\rm Int}(\overline{D_1} \cup \overline{D_2})$ is an eigenfunction for the second eigenvalue on the same domain ${\rm Int}(\overline{D_1} \cup \overline{D_2})$. Thus, if the $L^2$ norms of $\varphi$ are different on $D_1$ and $D_2$, we may apply Proposition \ref{l2norm} to conclude that $(D_1,D_2)$ is not optimal for the sum on ${\rm Int}(\overline{D_1} \cup \overline{D_2})$. This immediately implies that $\mathcal{D}$ is not optimal for the sum on $\Omega$.
\end{proof}

Let us recall that the mixed Dirichlet-Neumann method consists of expressing the minimal partition for the max as a nodal partition corresponding to a mixed problem. When we look for symmetric partitions, we consider a mixed Dirichlet-Neumann problem on a reduced domain and we search the eigenfunction which realizes the minimal energy in the reduced domain, taking care that the associated nodal partition has the desired structure. In the following we use the results of Proposition \ref{l2normappl} which allows us to deduce if the partition can be a candidate for the sum by looking at the $L^2$ norms of the eigenfunction of the mixed problem on the subdomains. This simplifies the computation, since we are not forced to extract each pair of neighbor domains and compute a new second eigenfunction on these domains. Given the eigenfunction of the mixed problem we can identify each of the subdomains by looking at its sign of restrictions to certain rectangles: $\{\varphi>0 \text{ or }\varphi<0\} \cap \{x \in [a,b]\}\cap \{y \in [c,d]\}$. Once we have computed the $L^2$ norms on subdomains corresponding to the mixed problem, we may multiply the norms corresponding to domains cut by symmetry axes by a corresponding factor, to find the $L^2$ norms on the initial partition.  The computation of the $L^2$ norm is made either in FreeFem++ \cite{freefem} or in Melina \cite{melina}.

Let us now examine the candidates obtained with the Dirichlet-Neumann approach: $k \in \{6,7,8,9\}$ for the disk, $k \in \{3,5\}$ for the square and $k \in \{5,6,8,10\}$ for the equilateral triangle. These configurations correspond in Figures~\ref{fig.triangleinf}, \ref{fig.squareinf} and \ref{fig.disqueinf} with green straight doted lines for which the subdomains are not congruent. We present the $L^2$ norms on each one of the subdomains. The situation is clear in the case of the disk and the square where we have only two types of domains: a single domain $D_{1}$ (the interior domain in the case of the square with $k=5$ or the disk) and a domain $D_{2}$ repeated several times by some symmetry (the exterior domains for the square with $k=5$ or the disk). Table \ref{tab.cardis} shows that the $L^2$ norm of the eigenfunctions on the different subdomains are not equal. Thus, in the case of the disk and $k=5,6,7,8$ or the square and $k=3,5$, if the partitions of Figures~\ref{fig.squareinf} and \ref{fig.disqueinf} are optimal for the max, they are not optimal for the sum.
\begin{table}[h!t]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
$\Omega$ & \multicolumn{2}{c|}{Square} & \multicolumn{4}{c|}{Disk}\\
\hline
$k$ & 3 & 5 & 6 & 7 & 8 & 9\\
\hline
$\int_{D_{1}}|\varphi|^2$ &  0.51 & 1.12 & 1.12 & 0.88 & 0.58 & 0.29\\
$\int_{D_{2}}|\varphi|^2$ & 0.75 & 0.72 & 0.78 & 0.85 & 0.92 & 0.96\\
\hline
\end{tabular}
\end{center}
\caption{$L^2$-norm on the square and the disk.\label{tab.cardis}}
\end{table}

In the case of the equilateral triangle we may have up to $5$ different domains in the partition. We denote these subdomains $D_1,D_2,...$ by starting from a vertex of the triangle and going along the sides in a clockwise rotation sense. In the following we write the values of the $L^2$ norms on each one of these domains for $k=2,5,8,6,10$:\renewcommand\labelitemi{{\boldmath$\cdot$}}
\begin{itemize}[leftmargin=*]
\item $k=2$:  $\int_{D_1} |\varphi|^2 = 0.49,\ \int_{D_2}|\varphi|^2 = 0.51$;
\item $k=5$: $\int_{D_1}|\varphi|^2 = 0.45,\ \int_{D_2} |\varphi|^2 = 0.44,\ \int_{D_3}|\varphi|^2 =0.34$;
\item $k=8$: $\int_{D_1}|\varphi|^2 = 0.31,\ \int_{D_2}|\varphi|^2 = 0.30,\ \int_{D_3} |\varphi|^2=0.22,\ \int_{D_4}|\varphi|^2 = 0.17,\ \int_{D_5}|\varphi|^2 =0.23$;
\item $k=6$: $\int_{D_1}|\varphi|^2  = 0.500,\ \int_{D_2} |\varphi|^2 = 0.500$;
\item $k=10$: $\int_{D_1}|\varphi|^2 = 0.599,\ \int_{D_2}|\varphi|^2 = 0.600,\ \int_{D_3}|\varphi|^2 = 0.604$.
\end{itemize}
In each of these cases $k\in\{2,5,8\}$, we find at least one pair of adjacent domains which have different $L^2$ norm, so these candidates are not optimal for the sum. Note that the case $k=2$ has already been treated in \cite{HHO10}.
In the cases $k\in \{6,10\}$ we do not find adjacent domains with significantly different $L^2$ norms, so the criterion does not apply. Moreover, a parametric study was done in \cite{BBN16}, assuming the borders of the cells are polygonal domains. This study shows that the partitions for the sum and for the max likely coincide in these cases.

\section{Candidates for the sum\label{sec.sum}}

\subsection{Iterative method}
Let us apply the iterative method to exhibit candidates for the sum. In Figures~\ref{fig.equisummax}, \ref{fig.disksummax} and \ref{fig.carresummax}, we compare these candidates for the sum (represented in blue) with the candidates for the max (in red) obtained by the iterative method with penalization.
\begin{figure}[h!t]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{cccc}
\includegraphics[width=0.19\textwidth]{sm_equi2}
& \includegraphics[width=0.19\textwidth]{sm_equi3}
& \includegraphics[width=0.19\textwidth]{sm_equi4}
& \includegraphics[width=0.19\textwidth]{sm_equi5}
\\
$k=2$ & $k=3$ & $k=4$ & $k=5$ \\
\end{tabular}
\begin{tabular}{ccccc}
\includegraphics[width=0.19\textwidth]{sm_equi6}
& \includegraphics[width=0.19\textwidth]{sm_equi7}
& \includegraphics[width=0.19\textwidth]{sm_equi8}
&\includegraphics[width=0.19\textwidth]{sm_equi9}
&\includegraphics[width=0.19\textwidth]{sm_equi10} \\
$k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates for the max (red) and the sum (blue).\label{fig.equisummax}}
\end{figure}
\begin{figure}[h!t]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{ccccccccc}
\includegraphics[width=0.19\textwidth]{Disk2}
& \includegraphics[width=0.19\textwidth]{Disk3}
& \includegraphics[width=0.19\textwidth]{Disk4}
& \includegraphics[width=0.19\textwidth]{Disk5}\\
$k=2$ & $k=3$ & $k=4$ &  $k=5$ \\
\end{tabular}
\begin{tabular}{ccccccccc}
\includegraphics[width=0.19\textwidth]{smpart_disk_6}
&\includegraphics[width=0.19\textwidth]{diskk7p1Partition.pdf}
& \includegraphics[width=0.19\textwidth]{smpart_disk_8}
& \includegraphics[width=0.19\textwidth]{smpart_disk_9}
& \includegraphics[width=0.19\textwidth]{smpart_disk_10}\\
$k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates for the max (red) and the sum (blue).\label{fig.disksummax}}
\end{figure}
\begin{figure}[h!t]
\centering
\setlength{\tabcolsep}{1pt}
\begin{tabular}{cccccccc}
\includegraphics[width=0.19\textwidth]{squarek2}
& \includegraphics[width=0.19\textwidth,angle=270,origin=c]{smpart_squa_3}
& \includegraphics[width=0.19\textwidth]{square4}
& \includegraphics[width=0.19\textwidth]{smpart_squa_5}\\
$k=2$ & $k=3$ & $k=4$ & $k=5$\\
\end{tabular}
\begin{tabular}{ccccccccc}
\includegraphics[width=0.19\textwidth]{smpart_squa_6}
& \includegraphics[width=0.19\textwidth]{smpart_squa_7}
& \includegraphics[width=0.19\textwidth]{smpart_squa_8}
& \includegraphics[width=0.19\textwidth]{smpart_squa_9}
& \includegraphics[width=0.19\textwidth]{smpart_squa_10}\\
$k=6$ & $k=7$ & $k=8$ & $k=9$ & $k=10$
\end{tabular}
\caption{Candidates for the max (red) and the sum (blue).\label{fig.carresummax}}
\end{figure}

In these figures, we observe that the candidates are very close except for two cases:
\begin{itemize}
\item the equilateral triangle and $k=4$ : in that case, the optimal partition for the max is known and composed of four equilateral triangles. The candidate for the sum has less symmetry and has 4 singular points on the boundary and two interior singular points. These points collapse two by two to give 3 singular points on the boundary for the max.
\item the square for $k=7$: in that case, the energies of two candidates are very close and it is very difficult to conclude. These two candidates already appear in \cite{CyBaHo}.
\end{itemize}
In Table~\ref{synthese-sum}, we give the energies of the optimal partition for the sum obtained by the iterative method. We can compare these values with those of Table \ref{tab.Linfty} for the max. This is completely coherent with the previous section: when the criteria shows that the candidate to be optimal for the max can not be optimal for the sum, we obtained new partition with lower energy and which is not an equipartition.
\begin{table}[h!]
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
$k$ & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10\\
\hline
$\triangle$ & 121.65 & 143.38 & 207.75 & 251.69 & 277.06 & 338.01 & 387.72 & 426.25 & 453.87 \\
\hline
$\ocircle$ & 14.68 & 20.19 & 26.37 & 33.21 & 38.95 & 44.02 & 50.44 & 57.69 & 63.67 \\
\hline
$\square$ & 49.40 & 66.14 & 78.956  &  103.37 & 125.68 & 144.34  & 159.93 & 177.68 & 201.85 \\
\hline
\end{tabular}
\end{center}
\caption{Lowest energies for the sum with the iterative algorithm.}
\label{synthese-sum}
\end{table}

\subsection{Partitions with curved polygons}
In this section we use, when available, an explicit representation for the partitions in order to exhibit candidates for the sum which are better than those for the max. We notice that for $k\in\{6,8,9\}$ for the disk and $k\in\{3,5\}$ for the square the partition has a simple symmetric structure. It is possible to approximate each of these partitions by assuming that the boundaries are either segments or arcs of circles. One important aspect used in the construction is the equal angle property which implies that all angles around a singular point are equal.

In the case of the disk, for $k\in \{6,8,9\}$ we obtained partitions which have a rotational symmetry of angle $2\pi/(k-1)$. Furthermore, we have a central domain which is a regular curved polygon with $k-1$ sides and $k-1$ external domains, each obtained by joining the edges of the polygons to the boundary of the disk with straight segments. This suggested us to see what happens if we assume that the rounded polygon's edges are arcs of circle. We note that this is not known theoretically. Furthermore, we want that pairs of consecutive arcs make an angle of $2\pi/3$. The general configurations are represented in Figure \ref{curved_polyk6} for $k=6$ and Figure \ref{curved_polyk8} for $k=7,8$.
We describe now the procedure of constructing such a rounded polygon for $k=6$. Let's start with an angle $\angle BAB'$ of measure $2\pi/5$ with $AB'=AB=\ell$. Note that $\ell \in (0,1)$ will be one of the parameters of the problem since we wish to vary these partitions in function of the size of the inner curved polygon. We wish to construct an arc $\arc{BB'}$ which makes equal angles of $\pi/3$ with $AB$ and $AB'$ (this is so that after symmetrization we have a pentagon with $2\pi/3$ angles. In order to construct this arc $\arc{BB'}$ we need the position of the center $C$ of the corresponding circle and the radius of this circle $CB=CB'=R$. Note that $\angle(CB,\arc{BB'}) = \pi/2$ and $\angle(AB,\arc{BB'})=\pi/3$, which implies that $\angle ABC=\pi/6$. By symmetry, we have $\angle AB'C=\pi/6$. Once we know $\theta = \angle BAB'$ and $\ell$, it is possible to find all other elements of the triangle $ABC$ by using, for example, the sine theorem
\[ \frac{AB}{\sin \hat C} = \frac{BC}{\sin\hat A}=\frac{CA}{\sin \hat B} \Leftrightarrow \frac{\ell}{\sin(\frac{\theta}{2}-\frac{\pi}{6})}=\frac{R}{\sin\frac\theta2}=\frac{AC}{\sin\frac \pi6}. \]
Once $AC$ is known we can determine the position of the center $C$ and then we can trace an arc of a circle of radius $R$ going from $B$ to $B'$. Note that the measure of $\angle BCB'$ is also needed in the implementation and is equal to $\theta-\pi/3$. The situation is similar for $k\in \{8,9\}$ with the difference that now $C$ is on the other side of $BB'$ so that the curved polygon has inward arcs. The case $k=8$ is depicted in Figure \ref{curved_polyk8} and the arguments for finding $R$ and $AC$ are essentially the same.
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{curved_poly5}\\
\caption{Explicit construction of curved regular pentagon with angles equal to $2\pi/3$.}
\label{curved_polyk6}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.9\textwidth]{curved_poly7}
\caption{Explicit construction of curved regular heptagons with angles equal to $2\pi/3$.}
\label{curved_polyk8}
\end{figure}
Note that once the arc $\arc{BB'}$ has been constructed, the exterior domain is drawn by continuing segments $AB,AB'$ until they reach the unit circle. Thus we have a way of constructing admissible partitions with the equal angle property which are very similar to the ones obtained with the iterative method.
Once we fix $k \in \{6,8,9\}$ we can optimize the sum of the eigenvalues of the partition with respect to $\ell$.
Let us remark that if we apply this method for $k=7$, we have necessarily a hexagon with straight lines as observed in Figure \ref{fig.disksummax} and analyzed in \cite{BBN16}.

The same method can be applied in the case of the square for $k=3,5$ (see Figure~\ref{curved_polys}). For $k=3$, the equal angle property applied at a boundary singular point imposes that the center $C$ of the circle is along a side of the square.
As can be seen in Figure \ref{curved_polys} we denote by $B$  the triple point and $AB$ is the segment along the boundary of the partition cells along the symmetry axis. With these considerations, the equal angle property implies that $\angle ABC = \pi/6$. See Figure \ref{curved_polys} for more details.
For $k=5$, we note that we have $4$ axes of symmetry and the central domain resembles a curved polygon with $4$ sides. We apply the previous arguments (see Figures \ref{curved_polyk6}-\ref{curved_polyk8}) to construct a regular polygon with $4$ sides and angles of measure $2\pi/3$. Then we extend this polygon to a partition of the square like in Figure \ref{curved_polys}. In both cases, we optimize the sum of the eigenvalues of these partitions with respect to the length $\ell=AB \in (0,1)$.
\begin{figure}[h!t]
\centering
\includegraphics[height=0.3\textwidth]{curved_polysq3}
\hspace{1cm}\includegraphics[height=0.3\textwidth]{curved_polys4}
\caption{Explicit construction of a $3$- and $5$-partition of the square with curved polygons.\label{curved_polys}}
\end{figure}

We implemented this method in FreeFem++  for the five configurations and Table~\ref{tab.encurv} presents the results.
We note that in every cases except for the square and $k=3$, the average of the eigenvalues is smaller than the energy obtained using the iterative method (see Table~\ref{synthese-sum}). However, the partitions we obtained with the two methods are close, as can be seen in Figure \ref{expl-vs-iter}.
\begin{table}[h!]
\begin{center}\begin{tabular}{|c|c|c| c | c| c|}
\hline
& \multicolumn{3}{c|}{Disk} &\multicolumn{2}{c|}{Square} \\
\hline
$k$ & 6 & 8 & 9 & 3 & 5\\
\hline
$\ell$ &0.411 & 0.3975 & 0.3981 & 0.4781 &0.5093\\
\hline
$\frac 1k\sum_{k}\lambda_{1}(D_{i})$ & 38.85 & 50.29 & 57.51 & 66.16 & 103.34\\
\hline
\end{tabular}\end{center}
\caption{Energies of the partition with constant curvatured boundaries.\label{tab.encurv}}
\end{table}

\begin{figure}[h!t]
\centering
\includegraphics[width=0.19\textwidth]{disk_supc6}
\includegraphics[width=0.19\textwidth]{disk_supc8}
\includegraphics[width=0.19\textwidth]{disk_supc9}
\includegraphics[width=0.19\textwidth,angle=270,origin=c]{sq_supc3}
\includegraphics[width=0.19\textwidth]{sq_supc5}
\caption{Explicit partitions (magenta) vs partitions with iterative algorithm (blue).}
\label{expl-vs-iter}
\end{figure}

\ack This work is supported by a public grant overseen by the French National Research Agency (ANR) as part of the ``Investissements d'Avenir'' program (LabEx {\it Sciences Math\'ematiques de Paris} ANR-10-LABX-0098 and ANR {\it Optiform} ANR-12-BS01-0007-02).

\bibliographystyle{acmurl}

redpajama_set_name: RedPajamaArXiv


Many power utilities are planning to introduce or extend packet-switching capabilities in their telecommunication networks. Before selecting technologies, it is essential to define a consistent architectural evolution strategy for the communication network.
Sentinel-Smallworld association enables phased transformations of network infrastructure and operational processes to deliver utility-grade communication services in the new digital power grid.


Journalist, MG is a survivor of torture and a member of Write to Life – a creative writing and performance group that allows survivors to get their voices heard. She wanted to share this letter with you so that you can see what a difference your support makes to people who have nowhere else to turn.
The first time I arrived in the UK as a student, my life back home was already insecure. My father had been detained after a peaceful protest. By the time the course ended, a year later, my father was out of detention but in hiding.
He had no way to send me the money I needed to continue my studies here. There was nothing for it but to go back and collect the money myself. I knew it was dangerous, that I risked losing the money, or worse.
By the time the authorities let me go, I knew I had to leave the country quickly. It was expensive, and a huge risk; I had no idea what would happen to me next. Only when the plane took off I could breathe.
Coming back to the UK, felt like coming to home and safety. No one knew me – but that meant they didn't know my history, or my family's. I was too scared to ask for help with my damaged physical and mental health; I didn't know what might happen with the Home Office if I did. I tried to put the whole experience in a box and keep it locked forever.
But it was all still there, memory becoming trauma. And when my student visa expired, it all blew up. I seemed to have no options. I couldn't stay; I couldn't go back. For a time, I was thinking that taking my own life would be better than going back to have it taken from me.
For a time, I was thinking that taking my own life would be better than going back to have it taken from me.
Every single person greeted me with a smile. At the reception Chantal, Veronica and Tina have always been kind and helpful. My therapist had endless patience. Before I could start PSTD treatment; she had to make sure my overall state was strong enough. It was quite slow, and sometimes I slipped back rather than progressing.
Meanwhile, the Home Office summoned me for my substantive interview. It was supposed to be postponed until I was strong enough and had amassed the evidence I needed, but they ignored all that. On the appointed day, I waited, alone, for four hours, until the interviewer finally arrived.
She said, if I wanted to postpone it, I'd have to go back, from Croydon to where I was living in Barking, get a letter from my GP and come back by the end of the day. I knew that was impossible, so I told her to go ahead with the interview then and there.
I was in the interview from 11.45 to 18.00. She had arrived without my file, so all we had to go on were the documents I had brought with me. I sat there, answering eighty-four questions, all quite random as she knew nothing of my case.
Every time I tried to explain something, she'd stop me halfway through while she caught up with typing it up.
For the next ten days, I couldn't even get out of bed. And in due course, the Home Office rejected my claim; they were not convinced I was a torture victim or that my life would be in danger if I were returned home.
For month after month I had to go to sign at the Reporting Centre as though I was a criminal. I had one last resort: to appeal against the decision.
Meanwhile, I had been put in touch with Dr. Alison Smith from Freedom from Torture's medico-legal report department. With her great expertise, we managed to prepare a detailed report which was properly appreciated in the Court by the honourable Judge.
He granted my appeal and gave me leave to remain.
Even though sometimes I was not capable of controlling my assertive behaviour, I could face my tearful regret quietly near the fountain in the garden. Every time I came for my therapy session, I noticed a tomato plant in my therapist's room.
That was something I could think about when I was alone and completely depressed about my life. Something with the will to survive. When it became cold, we left the plant safely inside the greenhouse.
I am so grateful to my therapist for showing me different ways to look at life. I became more and more involved in the organisation as there was always something to look forward to there: talking to someone with great sympathy, watching the tomato plant grow and survive the harsh winter in the warmth of the greenhouse.
Then I was invited to the Christmas party, my first Christmas party celebration. Another miracle happened that day. I found out there was a creative writing group at Freedom from Torture. Having been a journalist and writer, I found my passion for life again.
They were not convinced I was a torture victim or that my life would be in danger if I were returned home.
After a long time, I feel confident again; that life is not over yet. Sharing a meal with members of the Write to Life group, writing and thinking about various topics, learning from each other's struggle and pain, getting advice from mentors regarding life, career and writing have all restored the meaning in my life. I always feel the peace and positive vibe of the organization.
And Freedom from Torture arranged an amazing holiday for me last summer. I went to my preferred location, the quiet countryside of Norfolk. I learned a lot from my host. I met wonderful people there.
After a stressful confrontation with the Home Office, today I know I can stay safe under the protection of the UK. I had a long and painful journey through my asylum claim process. However, I am really grateful to have a new life surrounded by good people.
I know life is never going to be the same. Still, every day I am working to become a better person.
The journey is not easy at all, but I am determined to help others in need – as soon as I find my feet.


text:
HomeCrypto NewsBitcoin Btc Price, Chart & News
Bitcoin Btc Price, Chart & News
Posted by Muzzamil Ahmed April 22, 2022
By the end of 2019, Bitcoin's price was back down to $7,200. Bitcoin's high point of the year so far remains in the earliest days of January, when it nearly hit $48,000. In that same month, Bitcoin also hit its six-month low as it dipped below $34,000. Bitcoin has lost 40% of its value since its Nov. 10 all-time high above $68,000. The $40,000 benchmark has emerged as a key level for Bitcoin because how it performs from there could determine whether the market enters another bullish or bearish phase, experts say. Bitcoin started the week on a weak note, with its value falling nearly 2% over the last seven days. "Bitcoin biggest bubble in history, says economist who predicted 2008 crash". "Atlanta-based BitPay hooks up with PayPal to expand bitcoin adoption". "Bitcoin investors are bracing for a key technical event — here's what you need to know".
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Laws and regulations for foreign exchange should be amended if needed to cover crypto alongside traditional assets, the global stability watchdog said. Przemysław Kral is set to discuss cryptocurrencies starting from April 7. Investors can use part of their savings to invest in the world's largest cryptocurrency if their employers allow. Click on the "Group converter" link to convert a group of numbers all at one time. Another way to prevent getting this page in the future is to use Privacy Pass. You may need to download version 2.0 now from the Chrome Web Store. Alternatively, you can store your BTC in an external BTC wallet. There are hot and cold wallet solutions available in the market with different pros and cons, so you can explore the options to see which one suits you best.
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In practice, a transaction can have more than one input and more than one output. Bitcoin cash came out of left field, according to Charles Morris, a chief investment officer of NextBlock Global, an investment firm with digital assets. Until just before the decision, the solution known as Segwit2x, which would double the size of bitcoin blocks to 2 megabytes, seemed to have universal support. A 2015 survey showed bitcoin users tend to be overwhelmingly white and male, but of varying incomes. The people with the most bitcoins are more likely to be using it for illegal purposes, the survey suggested.
Bitcoin transactions are fully transparent and can't be censored. It's a financial system backed by thousands of computers, known as 'nodes', around the world, instead of a single central bank or government, i.e. hence the term 'decentralization'. Overall, Bitcoin is considered a highly speculative and risky asset compared to conventional investments. While there is no guarantee you will get any of your money back, Bitcoin has become the most valuable and commonly held among the thousands of cryptocurrencies that have since been created.
Bitcoin is valued as a useful form of money, and is measured by its growth of users, merchants and accepted locations. Learn all about finances in next to no time with our weekly newsletter. Bitcoin skyrocketed to an all-time high over $64,000 in the first half of 2021, then just as quickly fell back below $30,000 over the summer. Bitcoin hit another all-time high over $68,000 in November 2021, but by January 2022 had dropped back below $35,000. At the start of 2020 when the coronavirus pandemic shut down the economy, Bitcoin's price started to accelerate in its upward climb.
CoinDesk journalists are not allowed to purchase stock outright in DCG. Find out more about the BTC USD Exchange Rate, including a graph of historical pricing. We use mid-market exchange rates for all currency conversions. There are hundreds of cryptocurrency exchanges you can use to buy crypto online, but a few of the more popular ones are Coinbase, Gemini, and Kraken. These exchanges are online platforms where you can buy and sell cryptocurrencies. Bitcoin set the stage for blockchain technology and decentralized finance. The U.S. federal investigation was prompted by concerns of possible manipulation during futures settlement dates. The final settlement price of CME bitcoin futures is determined by prices on four exchanges, Bitstamp, Coinbase, itBit and Kraken. Following the first delivery date in January 2018, the CME requested extensive detailed trading information but several of the exchanges refused to provide it and later provided only limited data. The Commodity Futures Trading Commission then subpoenaed the data from the exchanges.
In 2014 mining pool Ghash.io obtained 51% hashing power which raised significant controversies about the safety of the network. The pool has voluntarily capped their hashing power at 39.99% and requested other pools to act responsibly for the benefit of the whole network. Around the year 2017, over 70% of the hashing power and 90% of transactions were operating from China. All bitcoins in existence have been created through this type of transaction. The bitcoin protocol specifies that the reward for adding a block will be reduced by half every 210,000 blocks . Eventually, the reward will round down to zero, and the limit of 21 million bitcoins will be reached c. 2140; the record keeping will then be rewarded by transaction fees only.
But with more bitcoins in circulation, people also expect transaction fees to rise, possibly making up the difference. It's organized through a network known as a blockchain, which is basically an online ledger that keeps a secure record of each transaction and bitcoin price all in one place. Every time anyone buys or sells bitcoin, the swap gets logged. CoinDesk is an independent operating subsidiary of Digital Currency Group, which invests in cryptocurrencies and blockchain startups. As part of their compensation, certain CoinDesk employees, including editorial employees, may receive exposure to DCG equity in the form of stock appreciation rights, which vest over a multi-year period.
Fees can be based on price volatility, and many are charged per transaction. While fees should definitely be a consideration, experts say you also get what you pay for, especially when sticking to the bigger, more established exchanges like Coinbase. If an exchange has more protections, better security, or other features that are important to you, it may be worth slightly higher fees. You can narrow down your search for the right platform by mostly paying attention to security and fees. If you plan to keep your crypto on your account with an exchange, make sure you choose an exchange that uses offline, cold storage, and has strong protections against theft. Some exchanges also have independent insurance policies to help protect investors from potential hacking. Many exchanges let you to leave your investment within your account, which is easiest for most beginners.
This payment depends on the amount of work an individual miner contributed to help find that block. To be sure, only a minority of bitcoin miners and bitcoin exchanges have said they will support the new currency. The new software has all the history of the old platform; however, bitcoin cash blocks have a capacity 8 megabytes. We are thrilled to announce becoming the first European cryptocurrency exchange that was granted regulatory approval to operate in Canada.
Beaxy Exchange Explains Why it Trusts Bitcoin SV in Virtual Meetup – https://t.co/EBm31zRLu7 #BusinessTips
— Sandra Hinshelwood (@Hinshelwood_S) March 7, 2022
About 20% of all bitcoins are believed to be lost -they would have had a market value of about $20 billion at July 2018 prices. The software validates the entire blockchain, which includes all bitcoin transactions ever. This distributed ledger which has reached more than 235 gigabytes in size as of Jan 2019, must be downloaded or synchronized before full participation of the client may occur. Although the complete blockchain is not needed all at once since it is possible to run in pruning mode. A command line-based daemon with a JSON-RPC interface, bitcoind, is bundled with Bitcoin Core. It also provides access to testnet, a global testing environment that imitates the bitcoin main network using an alternative blockchain where valueless "test bitcoins" are used. Regtest or Regression Test Mode creates a private blockchain which is used as a local testing environment.
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One Week Ago One Month Ago 3 Months Ago $40,878$44,936$36,852So what should crypto investors do in light of this volatility? Given crypto's history of volatility, this increase doesn't guarantee a long-term reversal. Bitcoin's price is just as likely to fall back down as it is to continue climbing. The future of cryptocurrency is sure to include plenty more volatility, and experts say that's something long-term crypto investors will have to continue dealing with. In the short term, these factors have created some noise and extra volatility in the crypto and stock markets, but this is usual during times of uncertainty. Volatility is standard in the cryptocurrency market, so experts predict the ups and downs to continue. Based on those principles, the cryptocurrency market — which now consists of thousands of cryptocurrencies — has grown to a valuation of more than $2 trillion.
Research produced by the University of Cambridge estimated that in 2017, there were 2.9 to 5.8 million unique users using a cryptocurrency wallet, most of them using bitcoin. On 15 July 2017, the controversial Segregated Witness software upgrade was approved ("locked-in"). Segwit was intended to support the Lightning Network as well as improve scalability. SegWit was subsequently activated on the network on 24 August 2017. The bitcoin price rose almost 50% in the week following SegWit's approval. On 21 July 2017, bitcoin was trading at $2,748, up 52% from 14 July 2017's $1,835. Supporters of large blocks who were dissatisfied with the activation of SegWit forked the software on 1 August 2017 to create Bitcoin Cash, becoming one of many forks of bitcoin such as Bitcoin Gold. In March 2013 the blockchain temporarily split into two independent chains with different rules due to a bug in version 0.8 of the bitcoin software.
Bitcoin creation and transfer is based on an open source encryption protocol and is not managed by any central authority. Each bitcoin is subdivided into 100 million smaller units called satoshis, defined by eight decimal places. Conversion rates are based on CoinDesk's Bitcoin Price Index and the price indices of other digital https://www.beaxy.com/ assets. World currency prices are based on rates obtained via Open Exchange Rates. Iran announced pending regulations that would require bitcoin miners in Iran to sell bitcoin to the Central Bank of Iran, and the central bank would use it for imports. Iran, as of October 2020, had issued over 1,000 bitcoin mining licenses.
The network also has no central storage; the bitcoin ledger is distributed. As of April 2022, it takes on average 122 sextillion attempts to generate a block hash smaller than the difficulty target. Computations of this magnitude are extremely expensive and utilize specialized hardware. Simplified chain of ownership as illustrated in the bitcoin whitepaper.
The U.S. Commodity Futures Trading Commission has issued four "Customer Advisories" for bitcoin and related investments. A July 2018 warning emphasized that trading in any cryptocurrency is often speculative, and there is a risk of theft from hacking, and fraud. Securities and Exchange Commission warned that investments involving bitcoin might have high rates of fraud, and that investors might be solicited on social media sites. An earlier "Investor Alert" warned about the use of bitcoin in Ponzi schemes. Because of bitcoin's decentralized nature and its trading on online exchanges located in many countries, regulation of bitcoin has been difficult. However, the use of bitcoin can be criminalized, and shutting down exchanges and the peer-to-peer economy in a given country would constitute a de facto ban.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Visual Studio 2015 resetting web project to use IIS Express I have a web project in VS2015 that I want to run on Local IIS. I set this up in the project properties and it works correctly, but whenever I close and reopen Visual Studio it always resets the project to use IIS Express.
EDIT: this happens when I close and reopen the solution, even if Visual Studio has stayed open.
Why is this happening, and how can I change it? (Note - other users are working on the same project so I don't want to store my Local IIS settings in the project file as I assume this would overwrite their settings when they got the latest version of the code.)

A: NibblyPig pointed out in their comment that:

it appears that if you don't change the URL then it doesn't save the
  setting. So I changed my Project Url, pressed save, then changed it
  back and pressed save again and it updated the file.

I'm just putting this as an answer so it's easier to find, but full credit to NibbyPig for finding this.

A: Delete the file projectName.csproj.user.  User specified preferences in this file overwrite the project settings.

A: I edited the .csproj.user file and this fixed the problem. I changed <UseIISExpress>true</UseIISExpress> to <UseIISExpress>false</UseIISExpress> and left everything else the same.
I still don't know why doing this through the UI didn't save, if anyone else can explain it I'll happily accept their answer.

A: I select project and click "Open Folder in File Explorer".
Then find associated .csproj file and edit with Notepad++ works for me.

A: If you don't save settings how would you expect to persist it and be available when next time you open project? I understand your concern that your settings shouldn't affect other team members who are working on same project. Your concern can be taken care if you follow below steps. Whatever settings you will do will remain local and applicable to you only and other team members will not get impacted.

*

*Go to project properties.

*Go to Web tab on property page.

*Untick "Apply server settings to all users as shown below.

*Click Save button in toolbar.
This way your settings will be stored in xxxx.user file and not in project file.

A: This is what worked for me using VS 2015 Update 3. This is related to what NibblyPig said.  But for me only worked when editing the project file directly.

*

*Unload the project. (If the project is not loading at all just go to #2)

*Edit project file.  The only entry that needs to be changed is (almost at the end of the file) <IISUrl>.  There is no need to mess with <UseIISExpress> configuration, you can leave it as it is.

An entry for IISExpress will look similar to this one:
<IISUrl>http://localhost:123456</IISUrl>
Change it to something like this one:
<IISUrl>http://localhost/your_site_in_iis</IISUrl>
Note: "your_site_in_iis" should be an existing website.

*Save and close project file.

*Close the solution (this one is important).

*Open the solution again and the problem should go away.

redpajama_set_name: RedPajamaStackExchange


"Dirty Sheets" is the latest from Chicago kids The Orwells. Newly signed to Atlantic Records, The Orwells are busier than ever with a headlining tour in the UK and an explosive performance on Later With Jools Holland just last week. The track shows a more cohesive sounding Orwells, but it still stays true to their gritty guitar and honest lyrics. With just two EPs under their belt, a full album is surely to be welcomed with an impatient response by fans, and after listening to "Dirty Sheets", I'm excited to listen to what's coming next.


4G Americas white paper provides detailed information on the status of IMT-Advanced, on 3GPP Rel-10 including HSPA+ enhancements and the introduction of LTE-Advanced, and the planning for Rel-11 and beyond.
Additional information on Rel-9 is included in Appendix B and includes a complete update to Section 6 of the 3G Americas white paper, 3GPP Mobile Broadband Innovation path to 4G: Release 9, Release 10 and Beyond-HSPA+, LTE/SAE and LTE-Advanced, which was published in February 2010 (and therefore did not contain some additional changes to the standards before their publication in March 2010). This paper has been prepared by a working group of 4G Americas' member companies and the material represents the combined efforts of many leading experts from 4G Americas' membership.


Public Utility Commission of Oregon Oregon PUC Rules - How do I Receive Notice of Rulemakings?
Oregon PUC Rules - How do I Receive Notice of Rulemakings?
To be placed on the service list for a particular rulemaking, note the docket number on the notice (it will start with AR) and contact the Administrative Hearings Division at (503) 378-4372 or send an email with your complete contact information to diane.davis@state.or.us. Once you are added to a rulemaking service list, your name and address will appear on this web site as part of the Service List in eDockets.
Please remember to contact the Oregon PUC if your contact information changes!


text:
Q: Substring SQL Query Variable I am trying to query the Database for a sub-string, I am substituting a string variable from a form for the query. It is giving me a syntax error.
This is the error:
LINE 1: SELECT * FROM books WHERE title LIKE '%'sometext'%'
[SQL: SELECT * FROM books WHERE title LIKE '%%''%(form)s''%%']
[parameters: {'form': 'some'}]

This is the query:
books = db.execute("SELECT * FROM books WHERE title LIKE '%:form%'", {"form":form.search.data}).fetchall()

If I use an actual string in the SQL query it works. It is an issue with using a variable. Any assistance would be appreciated.

A: you could try using concat  for avoid quote and windchar issue
books = db.execute("SELECT *
      FROM books
      WHERE title LIKE concat('%',:form, '%')" ,
     ........


text:
Stocks Soar, Risk FX Rally Holds
Daily FX Market Roundup October 7, 2020
The best way to describe how the equity and currency markets have been trading this month is erratic volatility. This may not be unusual for October which is historically the most volatile month for stocks but the market's ability to ignore major game changing developments is impressive. The only rational explanation for the 500 point move in the Dow is that investors are looking past near term political jockeying to the stimulus package that will come eventually. Overnight President Trump went on a tweet tirade where he said he is ready to sign a stand alone stimulus check bill and in a show of strength he returned to work in the Oval office. Regardless of who wins the election, a major stimulus package is on its way and the only question is how many businesses will survive until then. Equity traders are overly optimistic if they think the House will approve stimulus checks without a broader agreement or that the US economy will avoid further economic pain without stimulus before the end of the year. Equity and currency traders are reacting to every conflicting headline from the White House and this trend is likely to continue until November 3rd.
Meanwhile, dovish FOMC minutes kept the dollar from rallying against most of the major currencies. One of the few exceptions was USD/JPY which broke a week long range to trade at its strongest level since September 14th. Yet the rise in Treasury yields suggests that the minutes didn't really matter to investors. By altering their inflation target and in turn their forward guidance, the central bank doubled down on their accommodative policy and made it very clear that interest rates will remain on hold for the next few years. They are open to increasing monetary stimulus but that's unlikely unless stocks crash because of a second US virus wave or sudden deterioration in President Trump's health.
The best performing currency was the Australian dollar which shrugged off weaker PMI. According to the latest report, the contraction in service sector activity in Australia deepened with the PMI index falling to 36.2 from 42.5. We've now seen evidence of manufacturing and service sector weakness in the month of September that could justify a deeper correction in the Australian dollar. The same was true for the Canadian dollar which rallied despite a steep drop in IVEY PMI. The index fell to 54.3 from 67.8 last month. Even the move in the New Zealand was counterintuitive as NZD diverged from AUD with no particular catalyst.
Euro rallied despite an unexpected drop in German industrial production. After the sharp rise in factory orders, economists were looking for a 1.5% increase in IP but instead, it fell -0.2%. Virus cases also hit a record high in France and Spain. France is now on its highest alert level while Spain warned of a very hard few weeks ahead after virus cases topped 800K. New restrictions have been announced in many Eurozone nations including Italy and Spain.
Nothing is more important for sterling than Brexit negotiations. Like US stimulus talks, very little headway has been made and both sides remain too far apart. Most positive headlines have come out of Britain while most EU headlines are laced with skepticism. At the end of the day, while both sides stand to lose from a no deal Brexit the EU has the upper hand and is growing tired of this political dance. An agreement is still possible and the latest rally in sterling is a sign that investors are optimistic. The UK government is close to instituting tighter restrictions which could slow the country's fragile recovery.
Kathy Lien


Paige: Three pretty good stories. I like the Reynders quote. Vision 2020 was well organized. Be sure you're introducing all your speakers; the listener needs to know who they're hearing.
Take a little bit more of a pause between stories. The first story blended in with the second. Is that Shiran? Never says.
Christmas at Morningside: OK, but hit the dates and times again at the end. Good quotes from Watson. And it's free.