text:
\section{Introduction}

\subsection{Physical model}

Schr\"odinger operators on metric graphs have long been studied as simplified models of complex quantum mechanical systems, notably large molecules \cite{Pa36,Ru53}.
More recently, under the name of \emph{quantum graphs}, they have been widely used to describe nano-wires, wave-guides and networks \cite[Chapter 8]{Ex15}.
They have also been investigated by mathematicians as a simplified setting in which to understand complex problems in spectral theory, such as quantum chaos \cite{KoSm99}, trace formulas \cite{Ro84,KoSm99,Gu01,KuNo05,KuNo06} and nodal patterns \cite{Ba12}.
From this point of view, they stand midway between one-dimensional and multi-dimensional Schr\"odinger operators.
Indeed, metric graphs are one-dimensional objects with a non-trivial geometry.
The central theme in studying quantum graphs is therefore the interaction of the spectrum with the geometry of the underlying metric graph.

\subsection{Mathematical problem}

We consider in this work a finite, compact and connected metric graph $\Gamma$. Following \cite{Ku20}, we describe it by a set of $N$ edges $\{e_1,\dots,e_N\}$, with $e_n=[x_{2n-1},x_{2n}]$, and a set of $M$ vertices $\{v_1,\dots,v_M\}$, which is a partition of the set of endpoints $\{x_p\,:\,1\le p\le 2N\}$. The associated quantum graph is an operator in the Hilbert space
\begin{equation*}
    L_2(\Gamma)=\oplus_{n=1}^N L_2(e_n).
\end{equation*}
More precisely, we define the \emph{standard Laplacian} $L_\Gamma$ on $\Gamma$, which is the object of our study, in the following way. Its domain $\mbox{Dom}(L_\Gamma)$ consists of functions $f\in L_2(\Gamma)$ satisfying
\begin{enumerate}
    \item for every $1\le n\le N$, $ f|_{e_n}\in W_2^2(e_n)$;
    \item for every $1\le m\le M$,
    \begin{enumerate}
        \item $f(x_p)=f(x_q)$ whenever the endpoints $x_p$ and $x_q$ belong to the vertex $v_m$ (continuity condition),
        \item $\sum_{x_p\in v_m}\partial f(x_p)=0$ (balance condition).
    \end{enumerate}
\end{enumerate}
In the balance condition, $\partial f(x_p)$ is the derivative along the edge with endpoint $x_p$, taken towards the inside of the edge. The conditions in (ii) are called the standard vertex conditions. The action of $L_\Gamma$ on $f\in \mbox{Dom}(L_\Gamma)$ is then given by the differential expression $L_\Gamma f(x)=-f''(x)$ for all $n\in \{1,\dots,N\}$ and $x$ in the interior of $e_n$.

It is known that the operator $L_\Gamma$ is self-adjoint, non-negative and has compact resolvent.
Its spectrum therefore consists of a sequence of non-negative eigenvalues $(\lambda_j)_{j\ge1}$ of finite multiplicity tending to $+\infty$ (we repeat them according to their multiplicities).
Let us note that $\lambda_1$ is always $0$ and that the hypothesis that $\Gamma$ be connected implies that it is simple, with the associated eigenfunction proportional to the constant function $1$.
For the sake of convenience, we will work in the rest of the paper with the \emph{eigenfrequencies} $k_j=\sqrt{\lambda_j}$.

Our goal is to recover information about the geometry of the graph from the knowledge of some of its eigenfrequencies.
This is a variation on Kac's famous problem ``Can one hear the shape of a drum?" \cite{Ka66}, with some specificities that we will address below. The analogue to Kac's problem in our setting would  be showing that two graphs with the same spectrum are isometric.
There are partial results in that direction. For instance, the eigenfrequencies of a quantum graph satisfy Weyl's law: as $j\to+\infty$, $k_j\sim\pi j/{\mathcal{L}}$, where $\mathcal L$ is the total length of the graph, i.e. the sum of the lengths of all its edges ({see for instance \cite{Be13}}). The total length of the graph can therefore be read off the complete spectrum. Furthermore, the answer to Kac's problem is positive provided all edge-lengths of the graph are rationally independent \cite{Gu01, KuNo05}. Nevertheless, if this last condition is not satisfied, we can find graphs with the same spectrum that are not isometric \cite{Gu01, Ba06, Ba09, Ba10}.

We will restrict ourselves to recovering the main topological invariant of a connected graph: its Euler characteristic $\chi$, defined by $\chi=M-N$. It is related to $\beta_1$, the number of independent cycles in the graph, by
$\beta_1=1-\chi$. { Let us point out that $\beta_1$ can also be recovered from the number of zeros of the eigenfunctions, as shown in \cite{AlBaBe18}. These numbers are however less easily accessible to experiments.}

\subsection{Objectives}

We focus on a situation that can be realized experimentally.
A microwave network can be described to a good approximation by a quantum graph, whose eigenfrequencies correspond to resonances of the system.
In \cite{Mi20} the authors describe the measurement of about 50 resonances of such networks and use this experimental result to recover $\chi$.
Contrary to expectations, they find this relatively small number of eigenfrequencies to be sufficient.
Their computation is based on a trace formula as we will see below in more details.
Extending the work in this paper, we want to check the correctness of the method and modify it in order to improve the efficiency, or namely to reduce the number of eigenfrequencies required.

The total length of the graph is a continuous parameter and an infinite number of eigenfrequencies are needed to compute it using Weyl's law, as described above.
By contrast, $\chi$ is integer valued and we can hope to compute it with only a finite number of them.
However, some assumptions on the graph are needed.
Indeed, by attaching a very small loop to some edge in an existing graph, we decrease $\chi$ by $1$ while altering the first $50$ eigenfrequencies by an arbitrarily small amount.
We come back to this point at the end of Section \ref{se:numerics}.
Therefore, our goal will also be to determine a reasonable set of assumptions that guarantees the success of our methods.

\section{Trace formula}
\label{sec:Trace}

\subsection{Main tool}

{All our methods for computing the Euler characteristic are based on the trace formula in \cite{KuNo05, KuNo06} (building on previous work in \cite{Ro84,KoSm99, Gu01}).}
This formula connects the spectrum of $L_\Gamma$, defined in the previous section, with the set of periodic orbits in $\Gamma$, to be described below. More precisely, the spectrum is described by the distribution
\begin{equation}
\label{eq:def_u}
    u(k):=2\delta(k)+\sum_{k_j>0}\Big(\delta(k-k_j)+\delta(k+k_j)\Big),
\end{equation}
which is a tempered distribution of order $0$.
We denote by $\mathcal P$ the set of closed continuous paths in $\Gamma$ that are allowed to turn back on themselves only at vertices, which we call \emph{periodic orbits}, or \emph{orbits}.
If $p\in \mathcal P$, we denote by $\mbox{prim}(p)$ any shortest orbit such that $p$ can be obtained by repeating $\mbox{prim}(p)$.
Thus any two such orbits have the same length, hence $\ell(\mbox{prim}(p))$ is well defined.
The following quantity will play a crucial role in our methods:
\begin{equation}\label{def:lmin}
    \ell_{\mbox{\tiny min}}:=\min\{\ell(p)\,:\,p\in\mathcal P\}.
\end{equation}{
\begin{observation}
    Notice that $\ell_{\mbox{\tiny min}}$ is given by the minimum between the length of the shortest loop and twice the length of the shortest edge.
\end{observation}}

\begin{proposition}[Theorem 1 from \cite{KuNo05, KuNo06}, also \cite{Ku20}]\label{prp:trace_formula}
    We have
\begin{align}
    \label{eq:u}
    u(k) &= \chi \delta(k) + \frac{\mathcal{L}}{\pi} + \frac{1}{\pi} \sum_{p \in \mathcal{P}} \ell(\mbox{prim}(p))S_V(p) \cos k \ell(p),
 \end{align}
where $S_V(p)$ is the product of all the vertex scattering coefficients along the path $p$. { The convergence of the series occurs in the space $\mathcal S'(\mathbb R)$ of tempered distributions.}
\end{proposition}

{
\begin{remark}
The terms $S_V(p)$ will ultimately drop out of our formulas due to our choice of test functions. Since they will not play a role in our analysis we just refer to \cite{KuNo05} for their definition.
\end{remark}}

\begin{remark}
Taking the Fourier transform on both sides of Equation (\ref{eq:u}), we obtain
\begin{equation}
\label{eq:hat_u}
	 \hat u(\ell)=\chi+2\mathcal L \delta(\ell)+\sum_{p\in\mathcal P}\ell(\mbox{prim}(p))S_V(p)\Big(\delta(\ell-\ell(p))+\delta(\ell+\ell(p))\Big),
\end{equation}
which is the form of the Trace Formula that we will be using in the rest of the paper.
\end{remark}

{In all the paper, we define the Fourier transform so that $\hat f(k)=\int_{-\infty}^{+\infty} e^{ik\ell}f(\ell)\,d\ell$ for any $f\in L_1(\mathbb R)$.}
By definition of the Fourier transform of a tempered distribution, for $\varphi$ in the Schwartz class $\mathcal S(\mathbb{R})$, $\hat u[\varphi]=u[\hat\varphi]$, which means, using Equations (\ref{eq:def_u}) and (\ref{eq:hat_u}),
\begin{align}
 \chi\int_{-\infty}^{+\infty}\varphi(\ell)\,d\ell+2\mathcal L\varphi(0)+\sum_{p\in\mathcal P}\ell(\mbox{prim}(p))S_V(p)\left(\varphi(-\ell(p))+\varphi(\ell(p))\right)\nonumber\\
	=2 \hat {\varphi}(0) +\sum_{k_j>0}\Big(\hat {\varphi}(k_j)+\hat {\varphi}(-k_j)\Big).\label{eq:series}
\end{align}
Equation (\ref{eq:series}) also holds for some functions not in $\mathcal S(\mathbb R)$.
For our purpose, we will consider the set $\mathcal T$ of functions $\varphi$ satisfying the following assumptions:
\begin{enumerate}
    \item $\varphi$ is continuous on $\mathbb R$ and nonnegative;
	\item $\mbox{supp}(\varphi) \subseteq [0,T]$ for some $T > 0$;
	\item $\int_{-\infty}^{+\infty} \varphi(\ell)\,d\ell= 1$;
	\item the function $F_\varphi$ is integrable on $(0,+\infty)$, where $F_\varphi(k):=\sup_{y\ge k}|\Re(\hat \varphi(y))|$.
\end{enumerate}
Let us make some comments on Condition (iv).
The function $F_\varphi$ can be seen as the smallest non-increasing majorant of $|\Re(\hat\varphi)|$ on $(0,+\infty)$.
Its integrability is used both to express $\chi$ as the sum of a convergent series and to prove error estimates.

\begin{proposition}
    If $\varphi\in \mathcal T$,
\begin{equation}
    \label{eq:series2}
 \chi+\sum_{p\in\mathcal P}\ell(\mbox{prim}(p))S_V(p)\Big(\varphi(-\ell(p))+\varphi(\ell(p))\Big)=
	2 \hat {\varphi}(0) +2\sum_{k_j>0}\Re\left(\hat {\varphi}(k_j)\right).
\end{equation}
\end{proposition}
We prove the above proposition in \ref{appTrace}.
Let us note that whenever $\varphi$ is in $\mathcal T$, so is $\varphi_t(l):=t\varphi(t\ell)$, for any $t>0$.

We can now state the result by which we recover the Euler characteristic of $\Gamma$.

\begin{proposition}\label{prp:exact}
    For any $\varphi \in \mathcal{T}$ such that $T=1$ in (ii) then
    \begin{equation}
        \label{eq:chi_series}
        \chi= 2 \hat {\varphi_t}(0) +2\sum_{k_j>0}\Re\left(\hat {\varphi_t}(k_j)\right) \qquad \forall t \geq 1/\ell_{\mbox{\tiny min}},
    \end{equation}
    where $\ell_{\mbox{\tiny min}}$ is defined in (\ref{def:lmin}).
\end{proposition}
\begin{proof}
Since $t \geq 1/\ell_{\mbox{\tiny min}}$, the support of $\varphi_t$ is contained in $[0,\ell_{\mbox{\tiny min}}]$.
Equation (\ref{eq:chi_series}) then follows immediately from Equation (\ref{eq:series2}).
\end{proof}
In the rest of the paper, we assume that $t \ge 1/\ell_{\mbox{\tiny min}}$.
When we will analyse our method, it will accordingly be important to know a lower bound of $\ell_{\mbox{\tiny min}}$ in order to guarantee a successful estimate of $\chi$.
{
On the other hand, from the above proposition we also have the following corollary, cf. \cite{Ku08}.
\begin{corollary}\label{cor:exact}
    For any $\varphi \in \mathcal{T}$ then
    \begin{equation}
        \label{eq:chi_limit}
        \chi=  \lim_{t \to \infty} \left( 2 \hat {\varphi_t}(0) + 2\sum_{k_j>0}\Re\left(\hat {\varphi_t}(k_j)\right) \right).
    \end{equation}
\end{corollary}
    The limit and summation signs in (\ref{eq:chi_limit}) cannot be exchanged since $\lim_{t \to \infty} \Re\left(\hat {\varphi_t}(k)\right) = \hat {\varphi}(0)$ independently of $k$.
    Hence it is clear that (\ref{eq:chi_limit}) cannot be approximated by the limit of a finite sum over the  smallest, say $J$, eigenfrequencies.
    Because of Assumption (iv) on $\varphi$, the summation will be shown to be absolutely converging (see Section \ref{sec:gen}).
    Based on that, our plan is to quantify for each $t$ how many eigenfrequencies $J$ are necessary for the truncated summation to differ from (\ref{eq:chi_limit}) by less than $1/2$, and hence the nearest integer to the sum to coincide with $\chi$.
    Theorem \ref{thm:opt_d} gives an optimal (minimal) number $J^*$  of required eigenfrequencies, based on the choice of $t$.
    As one can expect the worse the estimate of $\ell_{\mbox{\tiny min}}$ (smaller lower bound), hence the larger $t$ is, the larger the number $J^*$ of required eigenfrequencies.
}

\begin{remark}
    {The above trace formula holds for the standard Laplacian, specifically for standard vertex conditions imposed at all the vertices.
    In \cite{Bo09, Ku10} are presented versions of the trace formula for more general vertex conditions  where in particular the constant term in $\hat{u}$ may differ from $\chi$.}
    Hence our method might not be applicable in presence of vertex conditions other than standard.
    It is even possible that $\chi$ cannot be recovered knowing the whole spectrum of the graph: see \cite{Ba06, Ba09, Ba10} for an example of two graphs with mixed vertex conditions which have distinct values of $\chi$ despite being isospectral.
\end{remark}

\subsection{Choice of the test function}

Let us apply Formula (\ref{eq:chi_series}) to a triangular function:
\begin{equation*}
    \psi(\ell) := 4\ell \mathds{1}_{[0,1 / 2]}(\ell) + 4(1-\ell)\mathds{1}_{[1 / 2, 1]}(\ell).
\end{equation*}
Everywhere in the paper, $\mathds{1}_A$ stands for the characteristic function of the set $A$. We find
\begin{align*}
    \label{eq:FT_tr}
    &\hat\psi(k) := \int_0^1\psi(\ell)e^{ik\ell}\,d\ell
	=e^{ik/2} \left(\frac{\sin(k/4)}{k/4}\right)^2.
 \end{align*}
Proposition (\ref{prp:exact}) then gives us the following expression for $\chi$, assuming $t \geq 1 / \ell_{\mbox{\tiny min}}$:
\begin{equation}
        \label{eq:chi_tr}
        \chi= 2+2\sum_{k_j>0}\cos(k_j/2t)\left(\frac{\sin(k_j/4t)}{k_j/4t}\right)^2.
\end{equation}

We can also apply the same formula to a truncated trigonometric function:
\begin{equation*}
    \varphi(\ell) := (1-\cos(2 \pi \ell))\mathds{1}_{[0,1]}(\ell)
\end{equation*}
We find
\begin{align*}
    \label{eq:FT_cos}
    \hat\varphi(k) &:= \int_0^1\left(1-\frac{e^{2i\pi\ell}}{2}-\frac{e^{-2i\pi\ell}}{2}\right)e^{ik\ell}\,d\ell\\
	&=-\frac{8\pi^2}{k(k-2\pi)(k+2\pi)}e^{ik/2}\sin(k/2)\\\
 \end{align*}
We deduce another expression for $\chi$, valid whenever $t \geq 1 / \ell_{\mbox{\tiny min}}$:
\begin{equation}
        \label{eq:chi_cos}
        \chi= 2-8\pi^2\sum_{k_j>0}\frac{\sin(k_j/t)}{k_j/t(k_j/t-2\pi)(k_j/t+2\pi)}.
\end{equation}
This formula is the one employed in \cite{Mi20}.

In order to recover $\chi$ from a finite set of eigenfrequencies, we have to replace the series on the right-hand side of Formulas (\ref{eq:chi_tr}) and (\ref{eq:chi_cos}) by partial sums.
Since the terms in the second series go to zero more quickly than in the first, we can expect that fewer eigenfrequencies are necessary to find a close approximation of $\chi$, which we will then be rounded (up or down) to the nearest integer.
Both numerical trials and the analysis of the experimental results in \cite{Mi20} confirm this.
The more regular $\varphi$ is, the faster $\hat{\varphi}$ decays at infinity.
We can then wonder how much we can decrease the number of eigenfrequencies used to compute $\chi$.
To study this question, we will consider the one-parameter family of test functions
\begin{equation*}
    \varphi_d(\ell)=C_d\left(1-\cos(2\pi\ell)\right)^d\mathds{1}_{[0,1]}(\ell),
\end{equation*}
where $d$ is a positive integer{, which we call \emph{order},} and $C_d$ is chosen so that $\int_{-\infty}^{+\infty}\varphi_d(\ell)\,d\ell=1$.
We easily check that the function $\varphi_d$ is of class $\mathcal{C}^{2d-1}$ on $\mathbb R$. { We show in \ref{appFT} that
\begin{equation*}
    C_d=\frac{2^d(d!)^2}{(2d)!}
\end{equation*}
and that the Fourier transform of $\varphi_d$ is
\begin{equation*}
    \hat \varphi_d(k)=\frac{(-1)^d(d!)^2}{\pi\Pi_{j=-d}^d(k/(2\pi)+j)}e^{ik/2}\sin(k/2).
\end{equation*}
}
Our goal is to give rigorous error estimates and to perform numerical trials for this family of functions.

\section{General estimate}
\label{sec:gen}
\subsection{Calculation with exact spectrum}

For a fixed graph $\Gamma$, we assume that only the smallest $J$ eigenvalues of the standard Laplacian are known. We consider the distribution (\ref{eq:def_u}) evaluated on the Fourier transform of a generic function $\varphi \in \mathcal{T}$ and split the expression in the following way
\begin{equation}\label{eq:SR_def}
        u[\hat{\varphi}_t] = \underbrace{2 \sum_{j \leq J} \Re\left(\hat {\varphi_t}(k_j)\right)}_{\textstyle S_J(t)} + \underbrace{2\sum_{j > J} \Re\left(\hat {\varphi_t}(k_j)\right)}_{\textstyle
         R_J(t)}.
\end{equation}
We call $S_J(t)$ the \emph{truncated sum} and $R_J(t)$ the \emph{remainder} or \emph{tail}.
We have
\begin{equation}\label{eq:triang_ineq}
    |S_J(t) - \chi| \leq |u[\hat{\varphi}_t] - \chi| + |R_J(t)|.
\end{equation}
Let $\varphi \in \mathcal{T}$ such that $T = 1$ in Condition (ii) be fixed. Because of Proposition \ref{prp:exact} when $t \geq 1/\ell_{\mbox{\tiny min}}$ the term $|u[\hat{\varphi}_t] - \chi|$ is zero, hence $|S_J(t) - \chi| \leq |R_J(t)|$.
\begin{align}
        |R_J(t)|    &\leq 2\sum_{j > J} |\Re \hat{\varphi}(k_j / t)| \\
                    &\leq 2\sum_{j > J} \sup_{k \geq k_j } \abs{\Re \hat{\varphi}(k / t)} = 2\sum_{j > J} F_\varphi (k_j / t),
\end{align}
where $F_\varphi$, is defined in Condition (iv) of $\mathcal{T}$.
In particular, $F_\varphi$ is positive decreasing to zero \footnote{This follows immediately from the fact that $\Re \hat{\varphi}$ has only isolated zeros. Indeed, it is a real-analytic function which is not identically zero.}.
At this point we employ the classical eigenvalue estimate $k_j \geq (j - M) \pi  / \mathcal{L}$ (see for instance \cite{Be17}) and we bound the summation over the integers of a monotone function by its integral.
\begin{align}
        |R_J(t)|    &\leq 2\sum_{j > J - M} F_\varphi \left( \frac{\pi}{\mathcal{L}}\frac{j}{t}  \right)\\
                    &\leq 2 \int_{J - M}^\infty F_\varphi \left( \frac{\pi}{\mathcal{L}}\frac{x}{t} \right) \,dx \\
                    &= 2\frac{\mathcal{L} t}{\pi} \int_{\frac{\pi}{\mathcal{L} t}(J - M)}^\infty F_\varphi \left( \mbox{x} \right) \,d\mbox{x} =: \mathcal{E}_\varphi(J - M,\mathcal{L}t).
\end{align}
The function $\mathcal{E}_\varphi:\mathbb{R}^2_+ \to \mathbb{R}_+$ is continuous, positive, and monotone with respect to both arguments.
In particular $\mathcal{E}_\varphi(J-M, \cdot)$ is strictly monotone increasing, while $\mathcal{E}_\varphi(\cdot,{  \mathcal{L}t})$ is strictly monotone decreasing to zero.
Since $\chi$ is an integer number (generally negative) it is enough to ensure that $|R_J(t)| < 1/2$ in order to compute $\chi$.
Therefore we define $J_{\mbox{\tiny min}}(M,\mathcal{L}t) :=\min \{ J \in \mathbb{N} : \mathcal{E}_\varphi(J - M,\mathcal{L}t) < 1/2 \}$, this is the minimum number of eigenvalues needed for computing $\chi$.
This can be further extended to the case where only partial information on $\Gamma$ is known. The result is summarised in
\begin{theorem}\label{thm:estimate_exact}
Let $\Gamma$ be a graph which
\begin{itemize}
    \item has at most $\overline{M}$ vertices, i.e. $M \leq \overline{M}$;
    \item has total length at most $\overline{\mathcal{L}}$, i.e. $\mathcal{L} \leq \overline{\mathcal{L}}$;
    \item has its shortest orbits at least $\underline{\ell_{\mbox{\tiny min}}}$ long, i.e. $\ell_{\mbox{\tiny min}} \geq \underline{\ell_{\mbox{\tiny min}}}$.
\end{itemize}
Let $\varphi \in \mathcal{T}$ with $T=1$ and let $t = 1 / \underline{\ell_{\mbox{\tiny min}}}$.
Then $J \geq J_{\mbox{\tiny min}} = J_{\mbox{\tiny min}}(\overline{M},\overline{\mathcal{L}}t)$ eigenvalues of the standard Laplacian on $\Gamma$ suffice for recovering the Euler characteristic of $\Gamma$ via the formula
\begin{equation} \label{eq:exact}
    \chi(\Gamma) = \mbox{\emph{nint}} \left[ S_{J} \left( t \right) \right],
\end{equation}
where $ \mbox{\emph{nint}}(x)$ denotes the nearest integer function.
\end{theorem}
\begin{proof}
    Because $t = 1 / \underline{\ell_{\mbox{\tiny min}}} \geq 1 / \ell_{\mbox{\tiny min}}$ then Proposition \ref{prp:exact} implies $|\chi - u(\hat{\varphi_t})| = 0$, hence (\ref{eq:triang_ineq}) reads $|\chi - S_J(t)| \leq |R_J(t)| \leq \mathcal{E}_\varphi(J - M,\mathcal{L}t)$.
    Because of the monotonicity of $\mathcal{E}_\varphi$, we have $\mathcal{E}_\varphi(J - M,\mathcal{L}t) \leq \mathcal{E}_\varphi(J - \overline{M},\overline{\mathcal{L}}t)$. Hence for $J \geq J_{\mbox{\tiny min}}$ it follows that $|\chi - S_J(t)| < 1/2$, and thus we obtain (\ref{eq:exact}).
\end{proof}
Notice that the cases $S_J(t) = \chi \pm 1/2$ is excluded, hence we do not need to worry of the definition of the function $\mbox{nint}[\cdot]$ over the half integers.

We notice that the hypothesis of the above theorem cannot be relaxed too much.
In particular, the condition $\ell_{\mbox{\tiny min}} \geq \underline{\ell_{\mbox{\tiny min}}}$ is crucial, because adding small loops to any graph perturbs its smallest eigenvalues only a little, but at the same time it does dramatically affect its topological structure, decreasing $\chi$ by $1$.

We also notice that the quantity $\overline{\mathcal{L}}$ can be obtained from other information on the graph when available.
For instance if the number of edges of the graph is at most $\overline{N}$ then $\displaystyle \mathcal{L} \leq \min_{2 \leq j \leq J} ({j} + \overline{N}) \pi / k_j$.

\begin{remark}\label{rmk:extension_thm_exact}
The above Theorem \ref{thm:estimate_exact} can be reproduced after replacing $F_\varphi$ by any other majorant function monotone decreasing. The $J_{\mbox{\tiny min}}$ computed will be different. We will use this remark in Section 4.
\end{remark}

\subsection{Calculation with approximate spectrum}
\label{ss:approxspectrum}
Since we have in mind to apply our method in an experimental context we need to consider the case where the eigenvalues are known only up to some approximation. We denote by $\{\widetilde{k}_j \}_{j=1}^J$ the approximate spectrum where each eigenfrequency is measured with precision $\delta$, meaning that $\abs{\widetilde{k}_j-k_j} \leq \delta$ for all $1 \leq j \leq J$.
Hence we define $\widetilde{S}_J(t) = 2 \sum_{j \leq J} \Re\hat {\varphi_t}(\tilde{k}_j)$ the truncated sum evaluated at the approximated eigenvalues, hence our new aim is to approximate $\chi$ from $\widetilde{S}_J(t)$.
Therefore to the error contribution $\mathcal{E}_\varphi(J - M,\mathcal{L}t)$ we need to add a new term
\begin{equation}
    \abs{\widetilde{S}_J(t) - \chi}  \leq \mathcal{E}_\varphi(J - M,\mathcal{L}t) + \abs{\widetilde{S}_J(t) - S_J(t)}.
\end{equation}

This second term can be bounded linearly in $\delta$ in the following way
{
\begin{align}
     \abs{\widetilde{S}_J(t) - S_J(t)}
    &= \abs{2\sum_{j \leq J} \Re (\hat {\varphi}(\widetilde{k_j}/t) - \hat {\varphi}(k_j/t))} \label{eq:error:start}\\
    &\leq 2 \abs{\sum_{j \leq J} \int_0^1  \left( \cos{\widetilde{k_j}\ell /t}  -  \cos{k_j \ell /t} \right)\varphi(\ell)\,d\ell} \\
    &\leq 2  \abs{ \sum_{j \leq J} \int_0^1 -2 \sin{\frac{ (\widetilde{k_j} - k_j) \ell}{2t}} \sin{\frac{(\widetilde{k_j} + k_j)\ell}{2t}}\varphi(\ell)  \,d\ell} \\
    &\leq  \frac{2\delta}{t} \sum_{j \leq J} \abs{ \int_0^1 \sin{\frac{(\widetilde{k_j} + k_j)\ell}{2t}}\varphi(\ell)  \,d\ell} \label{eq:error:step}\\
    &\leq  \frac{2 \delta J}{t}.\label{eq:error:end}
\end{align}
}
Therefore we define
\begin{equation}\label{eq:error_approx}
    \widetilde{\mathcal{E}}_\varphi(J - M,\mathcal{L},t) := \mathcal{E}_\varphi(J - M,\mathcal{L}t) + \frac{2\delta J}{t}.
\end{equation}
For this new function there does not necessarily always exists $J$ such that $\widetilde{\mathcal{E}}_\varphi(J - M,\mathcal{L},t) <1/2$.
For simplicity we consider separately the following two inequalities, which together are sufficient:
\begin{itemize}
        \item $\mathcal{E}_\varphi(J - M, \mathcal{L} t) < 1/4$;
        \item $2\delta J / t < 1/4$.
\end{itemize}
We derive from them the following two conditions:
\begin{itemize}
        \item \hfill
        \makebox[0pt][r]{%
            \begin{minipage}[b]{\textwidth}
              \begin{equation}
                 \label{eq:cond1}
            J \geq \widetilde{J}_{\mbox{\tiny min}}(M,\mathcal{L}t) := \min \{ J \in \mathbb{N} : J > M,\, \mathcal{E}_\varphi(J - M,\mathcal{L}t) < 1/4 \};
              \end{equation}
          \end{minipage}}
        \item \hfill
        \makebox[0pt][r]{%
            \begin{minipage}[b]{\textwidth}
              \begin{equation}
                 \label{eq:cond2}
            \delta \leq \widetilde{\delta}_{\mbox{\tiny max}}(t,J) := t / (8J).
              \end{equation}
          \end{minipage}}
\end{itemize}
If the first $J$ eigenvalues are known such that both the above conditions are satisfied then we are able to compute the Euler characteristic via the approximated spectrum.
The above analysis can be summarised in the following

\begin{theorem}
    \label{thm:estimate_approx}
Let $\Gamma$, $\varphi$ and $t$ be as in the hypotheses of Theorem \ref{thm:estimate_exact}, and moreover let the first $J$ eigenvalues of the standard Laplacian on $\Gamma$ be known up to error $\delta$. Assume that $J$ and $\delta$ satisfy both conditions (\ref{eq:cond1}) and (\ref{eq:cond2}) with $M, \mathcal{L}$ replaced by $\overline{M}, \overline{\mathcal{L}}$. Then the Euler characteristic of $\Gamma$ can be recovered via the following formula
\begin{equation} \label{eq:approx}
    \chi(\Gamma) = \mbox{\emph{nint}} \left[ \widetilde{S}_{J} \left( t \right) \right].
\end{equation}
\end{theorem}

\section{Example of a family of test functions}

\subsection{Error estimate and optimal order}

We now apply the analysis in Section \ref{sec:gen} to { the test functions $\varphi_d$ introduced at the end of Section \ref{sec:Trace}}. In the rest of the section, we assume that $t=1/\underline{\ell_{\mbox{\tiny min}}}$. We have
\begin{equation*}
    \Re\left(\hat \varphi_d(k)\right)=\frac{(-1)^d(d!)^2\sin(k)}{2\pi\Pi_{j=-d}^d(k/(2\pi)+j)},
\end{equation*}
so that, when $k>2\pi d$,
\begin{equation*}
    F_\varphi(k)\le\frac{(d!)^2}{2\pi(k/(2\pi)-d)^{2d+1}}.
\end{equation*}
Let us write $R_{d,J}(t)$ for the remainder denoted in Section \ref{sec:gen} by $R_J(t)$, in the case where $\hat\varphi=\hat\varphi_d$. According to Remark \ref{rmk:extension_thm_exact}, we find, for $J>2t\mathcal L d+M$,
\begin{equation*}
   \left|R_{d,J}(t)\right|\le \frac{{2}t\mathcal L}\pi\int_{\frac\pi{t\mathcal L}(J-M)}^{+\infty}\frac{(d!)^2}{2\pi(k/(2\pi)-d)^{2d+1}}\,dk=\frac{(d!)^2(2t\mathcal L)^{2d+1}}{2\pi d(J-M-2t\mathcal L d)^{2d}}.
\end{equation*}
Using the inequality $d!\le \sqrt{2\pi d}d^de^{-d+{1/12}}$ (see for instance Formula (3.9) in \cite{Ar64}), we find that
\begin{equation}\label{eq:eps_pre}
    \left|R_{d,J}(t)\right|\le\varepsilon(M,\rho,d,J),
\end{equation}
with  $\rho:=2t\mathcal L$ and
\begin{equation}\label{eq:eps}
    \varepsilon(\mu,\gamma,\alpha,\beta):=\frac{\alpha^{2\alpha}e^{-2\alpha+1/6}\gamma^{2\alpha+1}}{(\beta-\mu-\gamma\alpha)^{2\alpha}}.
\end{equation}
Let us note that we have replaced the discrete (integer) values $d$ and $J$ by the continuous parameters $\alpha$ and $\beta$.
\begin{theorem}\label{thm:opt_d}
For fixed $M$ and $\rho$ positive and $\overline\varepsilon\in(0,1)$, there exists $J^*$, the smallest among the integers $J$ such that $\varepsilon(M,\rho,d,J)\le\overline\varepsilon$ is satisfied for some $d$. There exists then $d^*$, the smallest positive integer $d$ such that $\varepsilon(M,\rho,d,J^*)\le\overline\varepsilon$. Furthermore, these integers are
given by the following formulas
\begin{equation}
    \label{eq:opt_J}
    J^*=\min\{\lceil\beta(\overline\varepsilon,M,\rho,d)\rceil\,:\,d\in\{\lfloor\alpha^*(\overline\varepsilon,\rho)\rfloor,\lceil\alpha^*(\overline\varepsilon,\rho)\rceil\}\}
\end{equation}
and
\begin{equation}
    \label{eq:opt_d}
    d^*=\min\{d\,:\,d\in\{1,\dots,\lceil\alpha^*(\overline\varepsilon,\rho)\rceil\}\mbox{ and }\lceil\beta(\overline\varepsilon,M,\rho,d)\rceil=J^*\},
\end{equation}
where
\begin{equation*}
        \beta(\overline{\varepsilon},M,\rho,d)=M+\rho d\left(1+\frac1e\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right)^{1/(2d)}\right)
    \end{equation*}
and
\begin{equation*}
        \alpha^*(\overline{\varepsilon},\rho)=\frac1{2(1+W(1))}\ln\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right).
    \end{equation*}
\end{theorem}
\begin{remark}\label{rem:wLambert} We denote by $W(1)$ the unique positive number satisfying $W(1)e^{W(1)}=1$. The notation $W(1)$ is standard and denotes the value of the Lambert W-function at $1$ (see \cite[Eq. 4.13.1]{NIST:DLMF}).
\end{remark}
\begin{remark}\label{rem:opt_d}
Existence of the integers $J^*$ and $d^*$ follows immediately from the fact that $\beta\mapsto\varepsilon(M,\rho,d,\beta)$ decreases to $0$ for any positive integer $M$ and $d$ and any $\rho>0$. Our actual goal is thus to prove Equations (\ref{eq:opt_J}) and (\ref{eq:opt_d}), which we use to compute them.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm:opt_d}] It follows from Equation (\ref{eq:eps}) that the unique $\beta\in(M+\rho\alpha,+\infty)$ which solves the equation $\varepsilon(M,\rho,\alpha,\beta)=\overline\varepsilon$ is
\begin{equation}
\label{eq:beta}
    \beta(\overline{\varepsilon},M,\rho,\alpha)=M+\rho \alpha\left(1+\frac1e\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right)^{1/(2\alpha)}\right).
\end{equation}
This implies that, for any positive integer $d$, the smallest integer $J$ such that $\varepsilon(M,\rho,d,J)\le\overline\varepsilon$ is equal to $\lceil\beta(\overline\varepsilon,M,\rho,d)\rceil$.
Differentiating Equation (\ref{eq:beta}) with respect to $\alpha$, we find
\begin{equation*}
    \frac{d}{d\alpha}\beta(\overline\varepsilon,M,\rho,\alpha)=\rho\left(1+\frac1e\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right)^{1/(2\alpha)}\left(1-\frac1{2\alpha}\ln\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right)\right)\right).
\end{equation*}
Using the substitution
\begin{equation*}
    s:=\frac1{2\alpha}\ln\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right)-1,
\end{equation*}
we see that this derivative vanishes if, and only if, $se^s=1$. This equation has a unique positive solution which is, by definition, $s=W(1)$. Therefore, the derivative vanishes only at $\alpha=\alpha^*(\overline\varepsilon,\rho)$, with
\begin{equation*}
    \alpha^*(\overline\varepsilon,\rho)=\frac1{2(1+W(1))}\ln\left(\frac{e^{1/6}\rho}{\overline{\varepsilon}}\right).
\end{equation*}
We can easily check that the derivative is negative below that value and positive above, so that $\alpha\mapsto\beta(\overline{\varepsilon},M,\rho,\alpha)$ is first decreasing and then increasing. The function $d\mapsto\beta(\overline{\varepsilon},M,\rho,d)$ takes its smallest value among all positive integers for the largest integer below $\alpha^*(\overline\varepsilon,\rho)$ or the smallest integer above. Equations (\ref{eq:opt_J}) and (\ref{eq:opt_d}) then follow easily.

\end{proof}

\section{Numerical examples}
\label{se:numerics}

We have numerically tested our method on some graphs for which we have computed the exact spectrum up to machine error precision and then added an error term uniformly distributed with prescribed variance.
In this section we give an overview of the implemented algorithm and show a few examples of its application.
{We recall the quantities that appear in the numerical analysis:
\begin{itemize}
    \item $t$ is the rescaling parameter of the test function;
    \item $d, (d^*)$ is the order of the test function $\varphi_d$, (respectively the minimal order according to equation (\ref{eq:opt_d}));
    \item $J, (J^*)$ is the number of the first smallest eigenfrequencies used to compute $S_J(t)$, (respectively the minimal number according to Equation (\ref{eq:opt_J}));
    \item $S_J(t)$ is the truncated sum defined in (\ref{eq:SR_def}) computed on the first $J$ eigenfrequencies of the exact spectrum where the test function is rescaled by $t$;
    \item $\widetilde{S}_J(t)$ is the same truncated sum as $S_J(t)$, but computed on a sample of the spectrum affected by a certain error term.
\end{itemize}
In view of the result of Proposition \ref{prp:exact} one might expect to obtain a good approximation of $\chi$ by computing the limit as $t \to \infty$ of the right hand side of (\ref{eq:chi_series}). Unfortunately this works out only if all eigenvalues are taken into account in the computation.
}

\subsection{Code overview}

The code consists of three distinct parts each of them developed to run on the software Mathematica \cite{MA11}.
\begin{enumerate}
    \item The first part is a GUI for building a graph with a drawing, drag-and-drop method and inserting the lengths of the edge.
    \item The second part contains an algorithm which computes the eigenvalues of the standard Laplacian of the graph in input.
    This is based on the von Below formula \cite{VB85} which holds for equilateral metric graphs, cf. \cite{Be13, Ku20}.
    This code shares parts with an other study in \cite{Ho20} on how this method can be extended to approximate eigenvalues of non equilateral metric graphs.
    Several samples of approximate spectra $\{\widetilde{k}_j\}_{j \in \mathbb{N}}$ are realized by summing a pseudo-random number uniformly distributed over a prescribed interval to the exact spectrum: $\widetilde{k}_j = k_j + e_j$,  $e_j \sim \mathcal{U}([-\delta,+\delta])$.
    \item The last part computes the parameters $d^*, J^*$, and Formula (\ref{eq:exact}), plots the maps $t,J \mapsto S_J(t)$ {and several instances of the the maps $t,J \mapsto \widetilde{S}_J(t)$ over different samples of spectra with error terms, and compares the error bound $\varepsilon(M,2t\mathcal{L},d,J) + 2 \delta J / t$ with the numerical errors $|\widetilde{S}_J(t)-\chi|$ over different values of $t$ and $J$}.
\end{enumerate}

\subsection{Comparison of the theoretical bounds with the experimental error}

\begin{figure}[ht]
\centering
    \includegraphics[width=.45\textwidth]{lasso}
    \includegraphics[width=.3\textwidth]{K5}
    \caption{On the left the Lasso graph, $\chi = 0$, the loop has length $\ell_1 = 1$ and the pendant edge has length $\ell_2 = 5$, hence $\ell_{\mbox{\tiny min}} = 1$, $\rho = 12$ and $d^* = 1$; on the right the graph $K_5, \chi = -5$, each edge has assigned length one, hence $\ell_{\mbox{\tiny min}} = 2$, $\rho = 10$ and $d^* = 1$.
    The lengths of the edges are not to scale.}
    \label{fig:lasso_K5}
\end{figure}

Here we present some computations of {$S_J(t)$ and $\widetilde{S}_J(t)$ under different values of $J$ and $t$ compared with $\chi$} performed for the graphs Lasso and $K_5$ (see Figure \ref{fig:lasso_K5}). { We used the test function of optimal order $d^*$, equal to $1$ in both cases.}

\begin{figure}[ht]
\centering
    \includegraphics[width=.45\textwidth]{Lasso_t}
    \includegraphics[width=.45\textwidth]{Lasso_comp_tri-vs-cos_t}\\
    \includegraphics[width=.45\textwidth]{Lasso_errcomp_t}
    \includegraphics[width=.45\textwidth]{Lasso_errcomp_J}
    \caption{{Plots for the Lasso graph:
    In the first three plots $J = J^* = 48$ is fixed and the value $t = 1 / \ell_{\mbox{\tiny min}} = 1$ is shown with the dashed vertical line.
    In the bottom right plot $t = 1 / \ell_{\mbox{\tiny min}} = 1$ is fixed.
    The samples of approximate spectra are computed with $\delta \approx 2.6 \cdot 10^{-3}$.}}
    \label{fig:lasso_plots}
\end{figure}

Let us explain how the numerical results are displayed in Figures \ref{fig:lasso_plots} and \ref{fig:K5_plots}.
\begin{description}
        \item[Top left]: the function $t \mapsto S_J(t)$ (in black) is compared with

        { several instances of $t \mapsto \widetilde{S}_J(t)$, corresponding to different samples of the approximate spectrum.}
        The curves are not visibly distinguishable despite the different error terms.
        \item[Top right]: the function $t \mapsto S_J(t)$ is compared for two different test functions, $\varphi_d$ {(black solid curve)} and the triangular function $\psi$ {(blue dashed curve)}.
        \item[Bottom left]:

        { several instances of the error $t \mapsto |\chi - \widetilde{S}_J(t)|$ (purple solid curves) are compared with the theoretical bounds $t \mapsto \varepsilon(M,\rho,d,J) + 2\delta J / t$ (the red thick dashed curve, with the thin curves representing the two addends)}
        We recall that we have assumed $t \geq 1 / \ell_{\mbox{\tiny min}}$ when we computed the theoretical bounds.
        \item[Bottom right]:

        { several instances of the error  $J \mapsto |\chi - \widetilde{S}_J(t)|$ (purple dots) are compared with the theoretical bound $J \mapsto \varepsilon(M,\rho,d,J) + 2\delta J / t$ (the red thick dashed curve, with the thin curves representing the two addends)}.
\end{description}
\begin{figure}[ht]
\centering
    \includegraphics[width=.45\textwidth]{K5_t}
    \includegraphics[width=.45\textwidth]{K5_comp_tri-vs-cos_t}\\
    \includegraphics[width=.45\textwidth]{K5_errcomp_t}
    \includegraphics[width=.45\textwidth]{K5_errcomp_J}
    \caption{{Plots for the graph $K_5$:
    in the first three plots $J = J^* = 41$ is fixed and the value $t = {1 / \ell_{\mbox{\tiny min}}} = 0.5$ is shown with the dashed vertical line.
    In the bottom right plot $t = {1 / \ell_{\mbox{\tiny min}}} = 0.5$ is fixed.
    The samples of approximate spectra are computed with $\delta \approx 1.5 \cdot 10^{-3}$.}}
    \label{fig:K5_plots}
\end{figure}

\begin{observation}~
We now gather some observations which hold for both Figures \ref{fig:lasso_plots} and \ref{fig:K5_plots}.
\begin{enumerate}
    \item In the two plots at the top right we notice that the black curve approximates $\chi$ with higher precision than the blue curve, i.e. it stays within distance $1/2$ from $\chi$ for a certain interval in $t$.
    This shows that the method with $\varphi_d$ is significantly more efficient compared with $\psi$ in terms of number of eigenfrequencies sufficient in order to compute $\chi$.
    \item The two plots at the bottom show that the theoretical bound exceeds the numerical error by at least a factor $10^2$.
    This phenomenon can be explained by the several cancellations occurring inside the tail $R_{d,J}(t)$ which we did not take into account in (\ref{eq:eps_pre}).
    \item {In the bottom left plots it can be observed that both the theoretical bound and the numerical error grow with respect to the parameter $t$, indicating that it is important for the effectiveness of the method to have a good lower bound of $\ell_{\mbox{\tiny min}}$.}
    \item In the bottom right plot we observe that for $J$ growing the numerical error stabilizes at small positive values different for different samples.
    This suggests that the precision does not improve by taking $J \gg J^*$ and that the error mainly depends on the contribution from the first terms in the summation $\widetilde{S}_J(t)$.
    \item The theoretical bound shows to be even worse with respect to the numerical error for $J \gg J^*$, this can be explained by the decreasing magnitude of the Fourier coefficients of the test function $\varphi$ inside $|\widetilde{S}_J(t) - S_J(t)|$, which we did not take into account in  (\ref{eq:error:start}--\ref{eq:error:end}).
\end{enumerate}
\end{observation}

\subsection{Additional examples}

We compare the numerical computation of $\chi$ for the following three equilateral graphs $K_5$, $K_5$ with one edge made pendant, i.e. disconnected at one end-point, and $K_{3,3}$.
These graphs have Euler characteristic respectively $\chi = -5, -4$ and $-3$, they have all three been constructed equilateral with each edge of length one.
The first two graphs are chosen to have very close spectra. We can numerically observe that only $1$ in $5$ of their eigenfrequencies differ significantly (figure \ref{fig:comp_plots}).

\begin{figure}[ht]
\centering
    \includegraphics[width=.45\textwidth]{group.pdf}
    \includegraphics[width=.45\textwidth]{comparison_K5_K5o_K33.pdf}
    \caption{{For each graph in figure we plot the functions $t \mapsto S_{J}(t)$ with $d=1$ and $J = 30$: $K_{3,3}$ (blue dotted curve), $K_5$ with one endpoint disconnected (red dashed curve) and $K_5$ (green solid curve).}}
    \label{fig:comp_plots}
\end{figure}

\subsection{Influence of the {order} of the test function}

From Theorem \ref{thm:opt_d} we can notice that the values of $d^*$ and $J^*-M = \lceil \beta(\overline{\varepsilon},M,\rho,d) - M \rceil$ depend only on the parameters $\rho$ and $\overline{\varepsilon}$.
Following the idea described in Section \ref{ss:approxspectrum}, we fix $\overline{\varepsilon} = 1/4$ and, for increasing values of $\rho$, we compute the values of $d^*$ and bounds of $J^*-M$. The results are shown in the following table.
{
\begin{center}
\begin{tabular}{r c r | c | r c r}
& $\rho$ & & $d^*$ & & $J^* - M$ & \\
 \hline
 $2$ & $\leq \rho \leq $ & $15.6$    & $1$       & $5$ & $\leq J^* - M \leq$ & $65$  \\
 $15.6$ & $\leq \rho \leq $ & $16.5$    & $1$ or $2$& $65$ & $\leq J^* - M \leq$ & $70$  \\
 $16.5$ & $\leq \rho \leq $ & $421$     & $2$       & $70$ & $\leq J^* - M \leq$ & $2912$\\
 $421$ & $\leq \rho \leq $ & $423$     & $2$ or $3$& $2912$ & $\leq J^* - M \leq$ & $2925$\\
 $423$ & $\leq \rho \leq $ & $10^4$    & $3$       & $2925$ & $\leq J^* - M \leq$ & $10^5$
\end{tabular}
\end{center}}

Cases for which $d^*>1$ occur for $\rho > 16.5$ i.e. $\mathcal{L} >  8.3 \cdot \ell_{\mbox{\tiny min}}$. This depends on two factors: the distribution of the length of the edges and the number of edges.
We can identify two extreme cases.
\begin{itemize}
    \item The distribution of the edge length has large variance, i.e. the length of the shortest edge is very small compared to the total length.
    For instance, a Lasso graph with the loop at least $8$ times shorter than the pendant edge.
    \item The distribution of the edge length has null, or very small variance, but the number of edges is relatively large, at least $17$ if the graph does not have loops.
    For instance, the equilateral $K_7$.
\end{itemize}

redpajama_set_name: RedPajamaArXiv


text:
This UFO-shaped floating condo allows you to live wherever there's water
by Alessandro Renesis | Last updated on Aug 23, 2022 at 3:49PM | Published on Aug 23, 2022 | Travel
This UFO-shaped floating condo will allow you to live in Monaco, Dubai, Miami, or basically just wherever else there's water.
And then when the scenery around you gets boring, you can just move it somewhere else.
Image: Anthénea
READ MORE: These SciFi-looking SeaPods are homes you can actually live in
Dubbed 'Anthénea', the floating condo is essentially an ultra-sophisticated boat and can be moored just like normal vessels.
Once you've reached a destination you like, you simply use the anchor and the sand screw and 'set up camp'.
Speaking of which, the Anthénea does not actually use a traditional anchor.
Instead, it uses an innovative anchoring system that makes it easier to operate, while also minimizing damage to the seabed.
And there's more good news because the pod is fitted with solar panels, which means it is self-sufficient.
So what's inside? The pod has a diameter of 9.4 meters (31 feet) and enough room for a family of four.
Watch: Romanian police seize Andrew Tate's multimillion-dollar supercar collection
Self-proclaimed Tesla fan files trademark for 'Cybercatamaran'
In addition to that, you've got a lounge bar and a rooftop that's large enough to accommodate up to 12 guests.
There are several different models available.
The idea is to use the smaller ones as private condos, and then use the larger ones for other things.
For example, the company that makes them says they want to team up with hotels to turn them into floating spas and restaurants.
Or you can also turn them into an AirBnB.
So the big question is: how much?
The smallest available pod will set you back $365,000.
Meanwhile, the largest, top-of-the-line pod costs $730,000.
At the end of the day, this UFO-shaped floating condo is actually cheaper than buying an apartment – or even a yacht – in Monaco or Miami.

redpajama_set_name: RedPajamaCommonCrawl


Absolutely gorgeous renovated home in a desirable neighborhood in Dorchester. This beautiful single family home features 4 bedrooms, 2.5 bathrooms on 3-levels. The home has been tastefully renovated with high-end finishes including custom white kitchen cabinets, Calacatta Statuario quartz counters, Kitchen Aid stainless steel appliance package, beautiful oak wood flooring, Anderson windows, detailed moldings, carrera marble and subway tile bathrooms, gas fireplace, private deck, and a gorgeous fenced-in yard. Two-zone control for central heating and cooling. The finishes in this home are just amazing! Easy to Show.
Listing courtesy of Michael Sylvia of Compass.
Listing provided courtesy of Michael Sylvia of Compass.


A Police Officer, A member of the armed services, or a Fire Brigade volunteer? Yes, those brave individuals protect us in various ways, and are all definitely heroes. There is another type of hero, the selfless individuals who quietly give their time and energy in various ways to improve others lives. One such person has recently completed his mission to raise funds for Make A Wish, and Spread A Smile, two deserving children's charities. This persons name is Jez Allinson, and he might well be the first Stormtrooper to hit his target.
Jez had already achieved a great deal in his running career, completing several marathons dressed as a Stormtrooper to raise money for charity. In the process, Jez earned himself a spot in the Guiness world records as the fastest Star Wars costumed runner to complete the 2017 London Marathon. Not content to leave it at that, Jez had one last goal, one last Stormtrooper run to complete. Starting at midnight on May 17th, when all sensible people were tucked up in bed, Jez set out to run 100KM in Stormtrooper armour. That's just over 62Miles if you are reading this from 1943.
Jez was supported in this effort by family, friends and the good people at Pinewood studios who provided him a safe flat environment to run in. Pinewood studios and Lucasfilm kindly allowed Jez to run inside the Lucasfilm lot at Pinewood East. After a security briefing for the support crew and photographers about what could and could not be shown, it was all over to The Running Stormtrooper to do the hard bit. Jez completed his 100KM in just under 13 hours, and has blasted past his donation goal of £6200 pounds, with over £7000 in the total and still counting. Jez will continue to run, and raise funds in some way, but he says the Stormtrooper armour wont be coming along any more.
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text:
Q: What Is the meaning of the residue of a complex function? Studying complex analysis I saw that, in many cases, the residue's theorem comes really handful. I learned how to find it and how to use it, but I didn't quite understand what it really means "geometrically", if this makes any sense.
Is there an intuitive way to explain what a residue is, or is it just a mathematical tool?

A: Let's start with the prototypical example of the residue theorem in action, taking the contour integral of $1/z$ over a positively oriented circle containing the origin, say $\gamma : \theta \mapsto e^{i \theta}$. Doing the computation, we obtain
\begin{equation}
\int_\gamma \frac1z dz = \int_0^{2\pi} i d\theta = 2\pi i.
\end{equation}
Compare to an analytic function $f: \Omega \to  \mathbb C$ which has an anti-derivative $F$: by the fundamental theorem of calculus, we see that the contour integral of $f$ vanishes
\begin{equation}
\int_\gamma f dz = \int_0^{2\pi} (F \circ \gamma)' (\theta) d \theta = \left[ F(e^{i \theta}) \right]_0^{2\pi} = F(1) - F(1) = 0.
\end{equation}
What you notice is that the integral above vanishes due periodicity being "quotiented out" nicely, in algebraic terms. This is because we have an anti-derivative which is well-defined on our domain $\Omega$.
The complex logarithm
Recall that $1/z$ also has an anti-derivative, the complex logarithm. The issue with the logarithm is that it is not well-defined on the domain of $\mathbb C \setminus 0$, it's what was classically called a "multi-valued function". Remember we say $i\theta = \log z$ if it satisfies the relation
\begin{equation}
e^{i \theta} = z.
\end{equation}
But for any $\theta$ satisfying the above, $\theta + 2\pi n$ for any $n \in \mathbb Z$ would do the trick as well. To illustrate the point, think about the value of the logarithm as you follow its value on $\gamma$. Naturally you want the logarithm to be continuous, so you say to yourself, let's take the principal logarithm, namely $\theta \in [0, 2\pi)$. So you start at 1, and out pops $\log 1 = 0$. Nothing too fishy yet. You rotate around the circle $\log e^{i \pi /2} = \pi i/2$, $\log e^{i 3\pi/2} = 3\pi i/2$... but suddenly you get
\begin{equation}
2\pi i = \log e^{2\pi i} = \log 1 = 0.
\end{equation}
Contradiction! So the logarithm can't be defined on the complex plane appropriately, you have to make what's called a branch cut in order to make it continuous: define the logarithm on $\mathbb C \setminus [0, \infty)$. But notice that the value of the logarithm is different depending on if you approach from the upper half plane, $\theta \downarrow 0$, which gives you $\log e^{i \theta} = 0$, and the lower half plane, $\theta \uparrow 0$, which gives you $2\pi$. Maybe there's a way to capture this idea mathematically.
Winding around the origin
And there is! We can define the complex logarithm on the associated Riemann surface by gluing together sheets of $\mathbb C \setminus [0, \infty)$ together to capture this idea of approaching the branch cut from different directions. Moreover, this new surface captures the idea of winding number, which is how much a closed curve "winds" around a particular point.
For example $\eta : \theta \mapsto e^{i \theta}$ for $\theta \in [0, 2\pi n]$ is a closed curve which wraps around the origin $n$ times. When you plot this in a graph, it looks exactly the same as our original $\gamma$; it's just a circle. But on our Riemann surface, if you follow the logarithm of $\eta$ and this notion of direction of approach to the branch cut, we find that $\eta$ "lifts" to curve taking you from  the first sheet of $\mathbb C \setminus [0, \infty)$ to the $n$-th sheet. When you compute the integral of $1/z$, you get
\begin{equation}
\int_\eta \frac1z dz = 2\pi i n.
\end{equation}
Out pops the winding number $n$, so integrating $1/z$ captures how much curves wind around the origin. This motivates a general formula for the winding number of a curve $\gamma$ around a point $a \in \mathbb C$.
Poles and Residues
So I went on a few tangents which might not be entirely helpful, so let's get back on topic to the original question. We see that $z \mapsto z^k$ has an anti-derivative defined on $\mathbb C \setminus 0$ for all $k \neq -1$, so we don't have to deal with any funny branch cut business that we did for "$\int 1/z = \log z$". From here it is useful to introduce the concept of a Laurent series: if $f: \mathbb C \to \mathbb C$ is a meromorphic function, then it admits a "power series" expansion
\begin{equation}
f(z) = \sum_{k \in \mathbb Z} a_k z^k = \frac{a_{-1}}{z} + \text{stuff that vanishes under integral}.
\end{equation}
When you integrate along $\gamma$, all the extra stuff vanishes and you're left with
$$\int_\gamma f dz = a_{-1} \int_\gamma \frac1z dz = 2\pi i a_{-1} \operatorname{wind} (\gamma, 0)$$
where recall $a_{-1}$ is the residue. Remember the $k$-th order coefficients tell you about the $k$-th order behaviour of a function, so the residue theorem is basically saying, look at the Laurent series, remember $-1$-th order growth is the only one that doesn't vanish, so write $f \sim a_{-1}/z$, and integrate appropriately.

redpajama_set_name: RedPajamaStackExchange


Indian Motorcycles of Shrewsbury at 730 Boston Turnpike was recently discovered under Shrewsbury Yamaha motorcycle fabrication.
RI Engineering at 2 Shaker Rd, Bldg E001 was recently discovered under Shirley Yamaha moped mechanic.
Motorcycles Of Manchester Inc at 108 Washington was recently discovered under Foxboro Yamaha street bike.
C & E Yamaha at 53 Washington Street was recently discovered under Foxboro Yamaha motorcycle inspection.
BMW Motorcycles Cambridge at 1098 Massachusetts Avenue was recently discovered under Yamaha motorcycle mechanic.


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Tdnews
Tunis Daily News
Home Tech Tech: A high-precision experiment has set its sights on pinpointing the so-far...
Tech: A high-precision experiment has set its sights on pinpointing the so-far hypothetical — (Report)
James M. Patterson
One thing is certain: there's something out there we don't yet know. For years now scientists have been looking for "dark matter" or "dark energy" — with our current inventory of particles and forces in nature we just can't explain major cosmological phenomena, such as why the universe is expanding at an ever faster rate.
New theories for "dark energy" have been suggested. One of the candidates is the so-called "symmetron field," which is said to pervade space much like the Higgs field. At the TU Vienna researchers have developed an experiment capable of measuring extremely small forces with the help of neutrons. The measurements were taken during a 100-day campaign at the Institut Laue-Langevin, on its PF2 ultra-cold neutron source. They could have provided pointers to the mysterious symmetrons — but the particles didn't show up. Although this is not the end of the theory, it does at least exclude the possibility of symmetrons existing across a broad range of parameters — and "dark energy" is going to have to be explained differently.
The Symmetron — the Higgs Boson's Little Brother?
According to Hartmut Abele, the project's lead scientist, the symmetron theory would be a particularly elegant explanation for dark matter. "We already have proof of the Higgs field, and the symmetron field is very closely related." However, as with the Higgs particle whose mass was not known until the existence of the particle was confirmed, the physical properties of symmetrons cannot be accurately predicted.
As Abele explains, "nobody can say what the mass of symmetrons is, nor how strongly they interact with normal matter. That's why it's so hard to prove their existence experimentally — or their non-existence." The existence of symmetrons can only be confirmed or refuted within a certain parameter range — symmetrons, in other words, with mass or coupling constants in a specific value range.
Scientists are therefore progressing with caution, from one experiment to the other, testing different parameter ranges. It was already clear that a number of ranges could be excluded. Symmetrons for example with high mass and low coupling constants cannot exist, as they would already have shown up at atomic level. Investigations into the hydrogen atom would have given different results. Similarly, symmetrons in a certain range with very high coupling constants can also be excluded, as they would already have been detected in other experiments using massive pendula.
Using Neutrons as Force Sensors at the Institut Laue-Langevin Neutron Source
That said, there was still plenty of scope for admitting the existence of symmetrons, and this is what the team has now investigated in the experiment. A stream of extremely slow neutrons was shot between two mirror surfaces. The neutrons can be found in two different quantum physical states. The energies of these states depend on the forces exerted on the neutron, and this is what makes the neutron such a sensitive force detector. If the force acting on the neutron just above the surface of the mirror is different from the force further up, this would be a strong pointer to the existence of a symmetron field. Mario Pitschmann from TU Vienna, Philippe Brax from the CEA near Paris and Guillaume Pignol from LPSC in Grenoble have calculated the influence of a symmetron field on the neutron. This effect, however, cannot be observed, despite the extreme accuracy of the measurement.
The precision of the energy difference measurement is about 2×10^-15 electron-volts (a figure owes to Gunther Cronenberg's dissertation). That is the energy required to lift a single electron in the earth's gravitational field a distance of about 30 micrometres, which is an unimaginably small amount of energy.
The ultra-cold neutrons required for the experiment were generated and delivered by the Institut Laue-Langevin's PF2 instrument. "With its unrivalled flux of ultra-cold neutrons, PF2 is practically the only instrument out there for this type of high-precision measurement at extremely low count rates," says Tobias Jenke. Jenke played an important part in the development of the TU Vienna experiment. He is now, together with Peter Geltenbort, responsible for the Institut Laue-Langevin cold neutron source. Austria is a Scientific Member of the Institute and thus has access to its suite of instruments. The experiment is an excellent example of scientific collaboration between Austrian and French researchers.
The Beginning of the End of the Symmetron Theory
For the moment things are not looking too bright for the symmetron theory, although it's too early to completely exclude their existence. "We have excluded a broad parameter domain: if there were any symmetrons with properties in this domain we would have found them." To close the remaining loopholes however, science needs even better measurements — or a major discovery providing a completely different solution to the mystery of dark energy.
Previous articleResearch: What were those colorful, wavy clouds in Edvard Munch's famous painting? Scientists suggest 'mother-of-pearl' clouds inspired the Norwegian artist —
Next articleResearch: Changing algae could alter ecosystems —
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Google : Gmail and YouTube go down in multiple countries, Report
vivo Expands Business in Europe, Report
Adobe MAX 2020: Enabling "Creativity for All" with Creative Cloud Innovation
Hey ISIS, You Suck: Local Muslims Post Anti-ISIS Billboard
A new billboard on Manchester Road in Missouri reads, "HEY ISIS, YOU SUCK!!! From: #ActualMuslims." A group of Muslim-Americans have put up a blunt billboard...
Man who catcalled flight attendant kicked off Alaska Airlines flight
How to choose men's wedding rings
How to become a responsible borrower
Minneapolis: Helicopter crash in Lino Lakes, killing two
USS Mason Missiles fired at US navy destroyer from rebel-held Yemen
Biology: Eliminating microplastics in wastewater directly at the source
Biology: New research compiles four years of corn loss data from 26 states and...
Dire wolves genes show they weren't really wolves, says new research
Researchers discover the oldest, most distant 'supermassive' black hole
Biology: Simulating evolution to understand a hidden switch
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How technology has changed the way we watch movies
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I want to switch from iphone4s to Z10 but my gf is on iphone and for various reasons, she can't switch with me. I work overseas and so we use imessage and facetime constantly.
Skype will be coming soon.
For chat + video, Skype will probably be the best option (once the BB10 update arrives).
There is also IM+ for BB10, which should give you most of the common IM services (AIM, Google Talk, Yahoo...). WhatsApp is also coming in March.
face flow allows video chat even 4 way.Whats app is coming for messaging.
yeah just about to say thanks.Sorry bro I gots no answers for you.
u can actually side load whatsapp now.. works fine on my playbook... but yea there are tons of options... Good luck choosing..
yep, but doesn't communicate with Contacts on Z10...does it do that on the PB?


PELIspec mirror the highly effective design and shape of the familiar metal equivalent. Each PELIspec vaginal speculum is manufactured with built-in failsafe points. The textured handles provide enhanced grip, crystal clear transparency for maximum visibility. PELIspec can be used for gynaecological procedures including screening, vaginal examinations and IUD fittings.


Shining Star's sister is hypnotized and captured. Knightwoman comes to the rescue! Or...does she???
For your convenience, A ZIP file containing all the images of Mixed Heroines 3! is available when you purchase this gallery.


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Reports can be presented to the States Assembly on a wide range of subjects and may include annual reports, consultations, proposed property sales and purchases and delegations of functions.
Filter by: scrutiny - st...
By Scrutiny - States Assembly
Scrutiny Reports
From Month January February March April May June July August September October November December Year 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990 To Month January February March April May June July August September October November December Year 2023 2022 2021 2020 2019 2018 2017 2016 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1990
/2022 Letter - Public Accounts Committee to Solicitor General re. Disability Compliance - 9 June 2021
Report date: 09/06/2021
Presented by: Public Accounts Committee
/2022 Letter - Public Accounts Committee to Chief Executive re. Executive Response to C&AG Report on Management of the Healthcare Response to the COVID-19 Pandemic - 9 June 2021


inside a movie theater last summer left behind a journal thanking and accusing charleston shooter. >>> a powerball jackpot too big to fit on billboards, tonight's numbers are revealed. >>> chilly one again tonight. there's warmer weather headed our way and with it some rain. we'll go over all the specifics. abc columbia news at 11:00, it starts right now. live from main and gervais, abc columbia news at 11:00. >>> donald trump admits he's angry. good evening. thanks for joining us at 11:00. >> and i'm alicia barnes. tonight, donald trump is responding to governor nikki haley's official response to president barack obama's state of the union address. >> in it, hailey cautioned for responding to the siren call of the angriest voices, code for donald trump. during a stop in pensacola, florida, trump wasted no time from the apparent jab. >> one of the commentators said to me that nikki haley, a very nice woman, she said i'm an angry person. they said to me, they said you were an angry person. i thought, i said, i am. i'm very angry. because i hate what's happening to our country, i am angry.
of former north charleston police officer michael slager accused of shooting and killing walter scott last april.we're told the fire was reported around last night in hanahan. charleston county sheriff officials say a glass cocktail bottle was found at the scene. an arson investigation is underway.slager, was released on bond monday. activists want him back in jail because they say he a danger to the 3 we now know the name of the woman that died during a house fire last night.richland county coroner gary watts has tentatively identified the victim as 76-year-old johnnie mae newton.according to the coroner...she died from carbon monoxide poisoning due to smoke inhalation.the fire happened around 5 friday on the 1100 block of quiet lane. richland county deputies are calling the fire suspicious. they are investigating the incident along with the richland county fire marshall's office. deputies are asking anyone with information about the fire to call crimestoppers at 1-triple 8-crime-sc. 3 we now know the identify of the man killed in an early morning shooting in downtown columbia.


The first show of the season for Project Runway and I already disagree with the judges' choices. I didn't think the loser's dress was that bad, personally. I did like the winner and two others really well. Army of Dad said he'd like to see Michael Kors have to take one of these challenges as he is spouting off his criticisms. That made me laugh. The fact that I can get my big surly husband to watch with me is a testament of his love for me when he'd rather be watching Dirty Jobs.
There is a doll of a blonde girl, Alison Kelley, whom Army of Dad already likes. I am digging Michael Knight. His dress - made out of coffee filters, cleverly enough - was one of the top to me. I really liked it.
This looks like a good season.
I agree that Stacey's dress (the white dress that lost) was NOT the worst dress, it was sheer and pretty and feminine even if she did miss some details. What was with the guy Bradley with the duvet cover outfit? That was a losing design, if ever there was one!
I'm posting sassy weekly recaps of Runway this season - I'd love you to come see my blog!


The School of CBD offers higher quality of dance education and fitness for persons of all ages and skill levels. We have five state of the art studios. All teaching methods are geared towards injury-free training. All of our teachers instruct with an injury free approach, keeping in mind that all dancers are not built alike. If you are new to the school, please call us to schedule a tour and register in person!
It is never too late to start dancing! Contemporary Ballet provides dance lessons for adults and for children. We offer classes for the more serious dancer and the more recreational dancer, as well as expanding our program into kids fitness classes as well. Please let us know if you have any questions - call us!
I enjoyed teaching at Contemporary Ballet Dallas. Great teachers and great classes!


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A Milwaukee man has been charged in connection to the fatal overdose of a West Allis woman
A 26-year-old man is facing a reckless homicide charge after allegedly providing heroin to a West Allis woman that caused her fatal overdose.
A Milwaukee man has been charged in connection to the fatal overdose of a West Allis woman A 26-year-old man is facing a reckless homicide charge after allegedly providing heroin to a West Allis woman that caused her fatal overdose. Check out this story on jsonline.com: https://www.jsonline.com/story/communities/southwest/news/west-allis/2019/12/31/milwaukee-man-charged-reckless-homicide-after-fatal-overdose/2775946001/
Elliot Hughes, Milwaukee Journal Sentinel Published 11:56 a.m. CT Dec. 31, 2019
Javoni K. Smith, of Milwaukee, is also facing felony charges of delivering narcotics and possession with intent to deliver cocaine and other narcotics in connection to the July 26 death of Ashley M. Ledvorowski, 30.
Smith was not in police custody as of Dec. 30. He faces a maximum of 85 years in prison if convicted on all counts.
Ledvorowski was found dead around 9:30 p.m. July 26 in a home along the 800 block of South 77th Street in West Allis. Police found drugs and other paraphernalia nearby and an autopsy showed she died of an overdose of Fentanyl and heroin.
After searching Ledvorowski's phone, police traced the drugs to Smith. Using a confidential informant, officers conducted a controlled purchase of heroin from Smith along the 2600 block of South 9th Place on July 29 and 30.
Smith was arrested after the second purchase and was found with 13 grams of crack cocaine and three grams of Fentanyl in his car.
Contact Elliot Hughes at elliot.hughes@jrn.com or 414-704-8958. Follow him on Twitter @elliothughes12.
Read or Share this story: https://www.jsonline.com/story/communities/southwest/news/west-allis/2019/12/31/milwaukee-man-charged-reckless-homicide-after-fatal-overdose/2775946001/
Woman says she was fired after posting 'MAGA 2020' comment on Facebook
Nearly half of Palmyra-Eagle Area School Board members resign
Longtime Germantown gymnastics coach, gym owner dies after battle with cancer
1 dead in car crash in New Berlin
Complaint: Man ingested his mother's ashes with various substances
Wauwatosa's former powerhouse could become a fitness center, cafe


"EM" stands for "Effective Micro-Organisms" and is the liquid that consists of a variety of multiple cultivated organisms such as lactic acid bacteria and photosynthetic bacteria.
Okinawa-born professor, Teruo Higa, a lecturer at the Agriculture Department of Ryukyu University in Nishihara Town, has been studying and cultivating this super liquid for 30 years. About 18 years ago, EM cultivation skills were established and are still used today to ensure that people enjoy a high-quality, safe and comfortable coexistence between us and the environment..
For those not familiar with EM and for those who already have some background knowledge, a tremendous surprise or two can be found in this invisible micro world at this year's "EM Festa '99," to be held at the Okinawa Convention Center on November 6 and 7.
At the EM Festa, the current status and information on EM is featured. For this year's event, a series of up-to-date case studies from Japan and internationally will be part of a panel discussion with EM specialists, led by Professor Higa.
Rachelle Morgan, Monica Durand and Bill Worthey, among many other domestic and international speakers, will discuss "the effects of EM-X on the chronic ailment", "the introduction of the activities of the EM networks in the United States", and "the wheat and cotton cultivation used by EM".
The main themes for this year's EM presentation is "Promote community developments by using EM," " Solve the dioxin problems", "Improve water purification techniques".
During the two-day event, a variety of health foods that use EM ingredients, such as milk and bread, will be sold at the Festa. The EM accessories handicraft workshop and a display of animals, raised with EM feeds, are also offered.
For those who have questions about EM, an information desk is available, so please feel free to ask questions and satisfy your interests about the EM world.


Watch the election results with your friends in the Peace and Freedom Party.
The Peace and Freedom Party and the Party for Socialism and Liberation are running Gloria La Riva for U.S. President and Dennis Banks, famed co-founder of the American Indian Movement, for Vice President.
Come watch the results of the elections as they come out in real time! We will have some snacks and drinks. The election count will go long into the night join us for as long as your able to.
Click on the image below to download the Adobe Acrobat (pdf) document for printing and distribution. There are two copies of the 5.5 inch by 8.5 inch flyer per sheet.


"World Intellectual Property Organization." International Organisations. , 2018. Web. 25 Apr. 2019. <http://www.whitakersalmanack.com?id=-114805CITANCHOR>.
"World Intellectual Property Organization." International Organisations. , 2018. http://www.whitakersalmanack.com/?id=-114805CITANCHOR.


Introduce themselves to new group members.
Brief the group regarding the ride details.
Have a minimal tool kit comprising of a multi-tool, at least 2 inner tubes, a puncture repair kit and pump.
Ensure guests sign the appropriate paper work (Guest Signing on Sheet).
Ensure all members of the group adhere to the Highway Code and Club Code of Conduct.
Keep the group together. This includes regrouping at the top of climbs.


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package com.github.liyiorg.mbg.support.example;

/**
 *
 * @author LiYi
 *
 */
public interface PaginationAble {

	void setLimitStart(Long limitStart);

	Long getLimitStart();

	void setLimitEnd(Long limitEnd);

	Long getLimitEnd();

	void limit(Long limitStart, Long limitEnd);

	void setOrderByClause(String orderByClause);

	String getOrderByClause();

	String getDatabaseType();

	void setDatabaseType(String databaseType);
}

redpajama_set_name: RedPajamaGithub


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Q: Cannot display text from text box while button click? I have created the button(click on me) using the html. When the button is clicked the textbox along with another button(show) is displayed on the screen. The value written in the textbox is shown when the show button is clicked. The textbox and show button is hidden. Again when i click (click on me ) button the textbox and show button is displayed. But when i write the text in textbox and click the show button this time text wont be displayed. That means first time text will be displayed but not after that. I want to display the text repeatedly. I am new to javascript so please help me. I also want to drag and drop the displayed text anywhere on the screen.
Try.html

<form name="frm">

<input type="button" value="Click on Me" id="button" onClick="display();">
</form>

</script>

A: In the display function you are creating the text box & button every time you invoke the function, I just added a checking if the object is already present it'll just change the visibility.

<form name="frm">

<input type="button" value="Click on Me" id="button" onClick="display();">
</form>


Responsive Web Design: Which Big Brands Have Made the Switch?
Responsive Web Design, often abbreviated to RWD, is a design technique used to ensure that the viewing experience of visitors to a website is of a consistently high quality no matter what types of device are used to access it. In this post we ask: which famous brands are using this technique, and how are they using it?
Responsive web design describes a set of design techniques and principles that aim to produce websites that work just as well on one device as they do on any other. Viewing experience is everything. By using fluid design layouts, proportional grids and advanced media queries (CSS), designers are able to ensure a website looks, feels and operates at the optimal standard. In this case, optimal means whatever layout or technology results in the best or most fitting viewing experience for all users, regardless of whether they access the site on a desktop PC, laptop, tablet or smartphone. RWD reduces the need to scroll, pan or resize a page, making reading text and viewing images a simple, enjoyable and intuitive task.
So, Which Brands are Doing it?
A number of leading brands have embraced responsive web design, including Microsoft, Currys and Starbucks.
Currys also uses a fluid layout, but its developers have gone a step further than Microsoft by making sure some of the design elements change several times in accordance with the screen size or viewing area. At full size (PC or laptop), the site displays the Currys logo on the left, a search field in the centre and account and basket links on the right. When viewed on a smaller screen, or by resizing the browser, the entire header switches to a new design and layout. A smaller logo replaces the full-size version and moves to the top centre. Menu and search links appear on the left and the background colour changes to demarcate the area effectively. When the screen size is reduced even further, all text is hidden and the navigational elements are replaced by symbols (e.g., a magnifying glass for the search function).
Starbucks has paid quite a bit of attention to the benefits of RWD. Unlike Microsoft and Currys, however, the site does not currently replace navigational elements with symbols when the viewing area is reduced to mobile proportions. Instead, the horizontal navigation bar simply disappears, leaving users free to click on links beneath the main corporate message.
As tablets and smartphones have become ever more popular, responsive web design is close to becoming a necessity. However, brands like Microsoft, Starbucks and Currys are currently leading the charge – many big firms still rely on separate mobile domains, or have no support for portable devices at all.
This isn't necessarily all that surprising. Designing, developing and testing a new site is a significant investment; it's not something brands are going to commit to lightly.
Microsoft, Starbucks and Currys all use responsive web design in slightly different ways, but essentially they're all following an established format.
Their use of responsive web design certainly makes for easier user experiences. But the techniques are still young – these brands are the first(ish) wave.
It's going to be fascinating to see how web developers dream up new, exciting, creative ways to utilise responsive design in future.


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Run for It
Stories of Slaves Who Fought for Their Freedom
Salete, Marcelo D'
Run For It ? a stunning graphic novel by internationally acclaimed illustrator Marcelo d'Salete ? is one of the first literary and artistic efforts to face up to Brazil's hidden history of slavery. Originally published in Brazil ? where it was nominated for three of the country's most prestigious comics awards ? Run For It has received rave reviews worldwide, including, in the U.S., The Huffington Post. These intense tales offer a tragic and gripping portrait of one of history's darkest corners. It's hard to look away.
Publisher: [United States] :, Fantagraphics Books,, 2017
Additional Contributors: Rosenberg, Andrea
Valenti, Kristy
Read more reviews of Run for It at iDreamBooks.com
KHCPL_Doug May 24, 2018
This book is a collection of short stories, set within the world of slavery in Brazil (which looks a lot like slavery in the U.S.). I don't really know what I expected, but this was much darker and much more depressing than I expected. I think I expected something more along the lines of the underground railroad, so it would be full of stories of inspiration and the courageous people who escaped from slavery. Instead, it's a very somber and horrifying look at the desperation and the very cruel and horrid expectations of life as a slave. I didn't dislike it at all, and I'm very glad I read it. Just be prepared for the stark and brutal reality it represents.
wildct2003 Jan 31, 2018
Didn't get a whole lot out of the book. Minimal dialogue, plot not too clear for most of the stories.
Slavery — Brazil — Comic Books, Strips, Etc
Fugitive Slaves — Brazil — Comic Books, Strips, Etc
Brazil — Slave Trade — History — Comic Books, Strips, Etc


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Ramadan expected to begin on May 6 in most Islamic nations
Photo used for illustrative purposes. WAM
The International Astronomical Centre has predicted the start of the Holy Month of Ramadan on May 6 in most of the Islamic nations.
The sighting of the crescent moon would be impossible from East and Southeast Asia as well as Southern Europe and most of the Arab nations on Sunday (May 5), said Mohamed Shawkat, the Centre Director, affirming that it would be possible to sight the moon only with a telescope in the these regions on that day.
The sighting of the moon would be difficult in Western and South African countries and most of the US, while it would be relatively easier in Central America, he added.
Ramadan UAE Saudi Arabia
Thursday will be the first day of Eid in UAE and Arab world
The committees said Wednesday would be 30th Ramadan.
Fatima Bint Mubarak Academy joins Zayed Sports Tournament
Zayed Sports tournament will give ladies all over the UAE a chance to enjoy a variety of sports, develop their athletic skills, and win valuable prizes.
Balancing health with fasting during the holy month
Basiligo, a healthy restaurant and meal plan service in the UAE that puts together its meals using only fresh and organic ingredients, offers some guidelines for a wholesome lifestyle during Ramadan.
Workshop helps UAE firms meet requirements of Saudi market
The Federation of UAE Chambers of Commerce and Industry in partnership with the Dubai Chamber of Commerce and Industry and the Saudi Food and Drug Authority recently hosted a workshop for UAE business and stakeholders interested in exporting to Saudi Arabia.
Bomber breaches high security zone, kills over 60 worshippers at Police Line mosque in Pakistan
"It was a suicide bombing," Peshawar Police Chief Ijaz Khan told Reuters. At least 47 people were killed and 176 wounded, he said, many of them critically.
VIDEO: Emirates operates demonstration flight powered with fully Sustainable Aviation Fuel
Emirates' demonstration supports broader efforts to reduce CO2 emissions, refine the playbook for future SAF demonstrations and support future certification of 100% drop-in SAF fuel.
RTA to tow or impound vehicles violating traffic rules
RTA has signed an agreement with Emirates Parkings entitling RTA to tow and impound light and heavy vehicles and trailers found in violation of federal and local rules and regulations.
The MoFAIC affirmed that the UAE expresses its strong condemnation of these criminal acts and its permanent rejection of all forms of violence and terrorism aimed at undermining security and stability in contravention of human values and principles.


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Home News Briefs Primeritus Financial buys Consolidated Asset Recovery Systems
Primeritus Financial buys Consolidated Asset Recovery Systems
Primeritus Financial Services Inc, a portfolio company of Kinderhook Industries LLC, said April 10 that it acquired Consolidated Asset Recovery Systems. Financial terms weren't announced. Greenridge Investment Partners and management were the sellers. Consolidated Asset Recovery Systems, of Raleigh, North Carolina, is a service provider of recovery management, skip tracing, and remarketing services to the auto finance industry.
Nashville, Tennessee – Primeritus Financial Services, Inc. ("Primeritus"), a portfolio company of Kinderhook Industries, LLC ("Kinderhook") and a leading service provider of recovery management, skip tracing, and remarketing services to the auto finance industry, announced the acquisition of Consolidated Asset Recovery Systems ("CARS") from Greenridge Investment Partners and management. CARS represents the 7th add-on acquisition for Primeritus and Kinderhook's 72nd automotive-related transaction. Financial terms of the transaction were not disclosed.
Headquartered in Raleigh, NC, CARS was founded in 2005 by Steve Norwood and Terry Groves, two software supply chain veterans, whose vision will continue to drive CARS' success. From the beginning, Steve and Terry decided to leverage SaaS technology to revolutionize the repossession and remarketing industry and experienced 84% YOY revenue growth since 2013.
"The addition of CARS to the Primeritus Family of Companies demonstrates our ongoing efforts to grow our business while further diversifying our brands. We are very excited to grow in size and scale and we will continue to invest in technology and operational capabilities which add value. As we continue this journey we will always embrace great people, great processes, and great solutions," said Mike Thomas, Primeritus Chief Executive Officer.
Steve Norwood, President & CEO for Consolidated Asset Recovery Systems, added, "As a result of this transaction, the companies will become the market's preferred provider of technology and services for managing both the repossession and remarketing of assets. We will continue to operate as separate businesses leveraging synergies in skip, back office functions and remarketing. We are very excited to be a part of the Primeritus family of companies and together will continue to provide the highest level of service and the best technology in this market."
Paul Cifelli, Managing Director of Kinderhook Industries, stated, "We are excited to combine two best of breed providers of recovery management, skip tracing, and remarketing services to the auto finance industry. The combined company represents the leader in the industry with a substantial opportunity to enhance its performance and operational capabilities while leveraging a unique set of technological capabilities to maximize value for its clients."
Kirkland & Ellis LLP served as legal counsel to Primeritus. Financing for the transaction was provided by TwinBrook Capital Partners.
About Kinderhook Industries
Kinderhook Industries, LLC is a private investment firm that manages over $3.0 billion of committed capital. We have made in excess of 200 investments and follow-on acquisitions since inception. Kinderhook's investment philosophy is predicated on matching unique, growth-oriented investment opportunities with exceptional financial expertise and our proprietary network of operating partners. Our focus is on middle market businesses with defensible niche market positioning in the healthcare services, environmental / business services and automotive / light manufacturing sectors. We have a track record of successfully and consistently building industry leaders.
For more information, please visit: www.kinderhook.com
About Primeritus
Primeritus Financial Services is a national provider of repossession management, remarketing, titled services and skip tracing services to the auto finance industry in the U.S. and Puerto Rico. Primeritus provides clients with value-added, outsourced repossession management, skip tracing and remarketing services, leveraging a national network of certified agents and unique investigative techniques to quickly and reliably secure customers' collateral. Through leadership, service and performance, Primeritus Financial Services offer the trifecta of repossession services: locate, recover and remarket.
For more information, please visit: www.primeritus.com
About CARS:
Consolidated Asset Recovery Systems provides technology and services to finance companies, forwarders and repossession agents that enhances repossession and remarketing performance. Combining comprehensive analytics with real-time performance metrics. status and account updates that can then be used to set parameters in the IBEAM artificial intelligence engine to automate task such as assignment and reassignment to suppliers, auctions or transporters. This combined with supplier optimization methodologies delivers the highest performance results.


I feel the same way you do about Mila. That is one of the reason I loved her 'Jackie' character way more than any of my associates did. I could see the thought process behind her creation of the character and her motivations for making decisions the way she did.


Today, Wild West and the sensualist culture have changed our environment. Plans to celebrate and enjoy life in the full glare of the world are scrambling to bury the mind. Children are first affected by such a lifestyle. In the age of becoming mature and accepting reality, these children are getting attracted to the fairy and fantasy world. This transition period is appalling, which is inevitable. Boredom with school, shirking from work, sabotage, wandering, senseless fashion, charm of dangerous drugs, childhood disorders, etc ruin the childhood and the future of the children. But there's no need to worry and panic. Only you have to be aware always, and you can guide the children at the right time and in the right direction for their bright future. In this book, the reason for the children to go stray as well as the ways and practical measures to prevent that have been detailed.


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Family PracticeNursingPediatricsPulmonologyInternal MedicineCritical CareEmergency MedicinePathologyPharmacyInfectious DiseaseFluPublic HealthVaccines
Flu Vaccine May Not Protect Against Main Flu Strain
However, vaccine still likely to protect against severe illness and death
FRIDAY, Dec. 17, 2021 (HealthDay News) -- This season's flu vaccine may do little to prevent infection with the main circulating virus strain because the strain has changed since the vaccine was developed, a new study suggests. The study was posted on a preprint server online and has not been published in a peer-reviewed journal.
However, the vaccine is still likely to protect against severe illness and death, according to the U.S. researchers. They assessed the vaccine's effectiveness against the H3N2 strain. The vaccine is also meant to protect against H1N1 and two strains of influenza B.
Lab tests suggest a "major mismatch" between the vaccine and the new, mutated version of H3N2, study leader Scott Hensley, Ph.D., a professor of microbiology at the University of Pennsylvania in Philadelphia, told CNN.
That finding may help explain a flu outbreak at the University of Michigan last month that affected more than 700 people. More than 26 percent of those who tested positive had been vaccinated -- the same percentage as those who tested negative. That suggests the vaccine was not effective in preventing infection, CNN reported.
However, Hensley said the changes in the virus probably will not protect it against immune system T cells, so this season's flu vaccine is still likely to prevent severe disease and death. "Studies have clearly shown that seasonal influenza vaccines consistently prevent hospitalizations and deaths, even in years where there are large antigenic mismatches," Hensley and colleagues write in their report.
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NCDC Logs 565 Cases In 14 States, Eight Fatalities
By Sylvia LANO On Aug 24, 2021
Nigeria Centre for Disease Control (NCDC) says 565 new cases of COVID-19 have been reported in 13 states and the federal capital territory (FCT).
The agency confirmed the new cases in its update for August 23, 2021.
The states that recorded positive cases include Lagos (205), Rivers (73), Ondo (63), FCT (53), Oyo (53), Ekiti (37), Delta (24), Edo (20), Benue (12), Bayelsa (6), Kwara (6), Osun (6), Ogun (5), and Adamawa (2).
NCDC said the reported figures include a backlog of cases from Benue state reported on August 17 (7), August 18 (1) and August 19 (4).
The agency said 363 persons were discharged on Monday after recovering from COVID-19, while 8 persons died of the infection.
To date, 187,588 cases of COVID-19 have been confirmed, 168,818 recoveries have been made, and 2,276 deaths have been recorded across the country.
In other news, United States Food and Drug Administration on Monday granted full approval to Pfizer/BioNTech COVID-19 vaccine.
The full approval is the first to be given to any COVID vaccine by the US drug regulator. The vaccine is authorised for people aged 16 and above.
President Joe Biden said the FDA approval is "an important moment in our fight against the pandemic".
He addedf
If you're one of the millions of Americans who said that they will not get the shot until it has full and final approval of the FDA, it has now happened. It's time for you to go get your vaccination. Get it today. There is no time to waste.
coronavirus in nigeriancdcNew coronavirus cases
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