Occupational therapists use carefully selected activities to help children develop the motor, sensory, perceptual and organisational skills they need to be independent in everyday activities of self- care, play and school work.
Occupational therapists will help children master practical skills such as dressing (including fastening buttons, buckles and laces), using tools such as cutlery and scissors effectively, as well as developing handwriting skills.
Some children find busy noisy settings stressful and struggle to concentrate. Occupational Therapists consider how efficiently children process sensory information and consider environmental factors which affect task performance and attention control.
Children's task performance can sometimes be improved by using adapted furniture or simple equipment (such as pencil grips and writing slopes) and these can be prescribed.
Occupational Therapists work with children from birth to 18 years.
Following referral, the occupational therapist will arrange an assessment to establish the nature and extent of difficulty.
Intervention will be recommended as appropriate and discussed with parents.
Individual therapy at our clinic, or in school.
Paired or small group work if appropriate.
Home programme and regular review sessions.
Julia Terteryan is an experienced paediatric occupational therapist with a long career in the NHS and overseas, prior to starting independent practice in 2001.
She believes firmly in early intervention and effective multidisciplinary practice and has worked alongside Wiz Chilton (Physiotherapist) for many years.
Julia has established links with many London Day Schools, and provides therapy services within some of these.
Julia and her occupational therapy colleagues are committed to providing therapy which is practical and efficient, whilst being fun and motivating for the children who attend.
Close collaboration with parents and carers, as well as nursery and school staff, is considered fundamental to ensuring children make the best possible progress.
Once your child has been booked for occupational therapy sessions the time of session, name of therapist and cost of sessions will be confirmed for you by email.
You will be invoiced for sessions on a half termly basis by email and we ask that where possible payment is settled by electronic transfer. Accounts are due to be settled within 14 days from date of issue; late payment is subject to a £25 administration charge. You are responsible for payment of fees, but these may occasionally be covered by medical insurance.
We will provide end of term reports for children and these will be emailed to you.
We will always try to accommodate additional meetings, school visits or provide additional reports if these are needed; charges for these will be quoted on an individual basis.
In order to help us communicate effectively please could you ensure your therapist is aware of any changes to email addresses or telephone numbers for you.


text:
Here Are 20 Celebrities You Didn't Know Were LGBTQ+
The man who took over from the legendary Steve Jobs when he passed as Apple's CEO, Tim Cook came out publicly as being gay in 2014. This was in part due to a social responsibility he felt he had to address despite being a very private person. On The Late Show interview, Tim Cook told Stephen Colbert "It became so clear to me that kids were getting bullied in school, even being disclaimed by their own parents and that I needed to do something," Cook said he wanted everyone to know his truth though many people already knew. Inspired by a Dr. Martin Luther King Jr. quote, "Life's most persistent and urgent question is, 'What are you doing for others?" This quote fueled his decision to speak up and that helping shape a world where kids could feel safe was more important to him than his privacy.
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Les photos les plus incroyables de la Corée du Nord
CELEBRIDADES DA ERA DE OURO DE HOLLYWOOD QUE AINDA ESTÃO VIVAS

redpajama_set_name: RedPajamaCommonCrawl


Absolutely amazing coffee. Like biting into the coffee bean in the best way possible. Absolutely worth the price (which isn't even bad?!). If you're on the fence about purchasing just do it!
As good as you get in any coffee shop. I used to pay $4 for a coffee.so in 12 coffees I have covered the cost of a kilo.


text:
\section{Introduction}
\label{s:intro}
\subsection{Motivation}
\label{ss:motive}
    Hybrid systems find wide applications in modern day Cyber-Physical Systems (CPS). In this paper we deal with a subclass of hybrid systems, where we focus on the discrete dynamics and abstract away the continuous dynamics as switching. Such a system is known as discrete-time \emph{switched system} and contains two components --- a family of systems and a switching logic. The \emph{switching logic} selects an \emph{active subsystem} at every instant of time, i.e., the system from the family whose dynamics is currently being followed \cite[\S 1.1.2]{Liberzon}.

    It is well-known that a switched system does not necessarily inherit qualitative properties of its constituent subsystems. For instance, divergent trajectories may be generated by switching appropriately among stable subsystems while a suitably constrained switching logic may ensure stability of a switched system even if all subsystems are unstable (see e.g., \cite[p.\ 19]{Liberzon} for examples with two subsystems). Due to such interesting features, the problem of designing stabilizing switching logics for switched systems has attracted considerable research attention in the past few decades, see e.g., the survey papers \cite{Liberzon_survey, Heemels_survey, Antsaklis_survey} and the references therein.

    The said design problem typically requires the availability of mathematical models of the subsystems. However, on the one hand, in many real-world scenarios, particularly for large-scale complex systems, accurate mathematical models of the subsystems, such as transfer functions, state-space models or kernel representations, are often difficult to infer. On the other hand, we often have access to traces of state trajectories of the subsystems. Such trajectories may be obtained from a simulation model provided by the system manufacturer to study the system behaviour with respect to various sets of inputs prior to their application to the actual system, or from an experiment conducted during the operations of the subsystems. The above interesting facts motivate our research on data-driven design of switching logics that ensure good qualitative and/or quantitative properties of switched systems. In this paper we present an algorithm to design stabilizing switching logics for discrete-time switched linear systems when explicit knowledge of the state-space models of their subsystems are not available. We restrict our attention to sets of switching logics that obey pre-specified restrictions on admissible switches between the subsystems and admissible dwell times on the subsystems.
\subsection{Literature survey}
\label{ss:lit_survey}
    Stability analysis and control synthesis of switched systems without explicitly involving mathematical models of their subsystems, are dealt with recently in \cite{Kenarian2019,Li2018,def}. The work \cite{Kenarian2019} addresses the problem of deciding stability of a discrete-time switched linear system under arbitrary switching logics from a set of finite traces of state trajectories. Probabilistic stability guarantees are provided as a function of the number of available state observations and a desired level of confidence. In \cite{Li2018} reinforcement learning techniques are employed for optimal control of switched linear systems. A Q-learning based algorithm is proposed to design a discrete switching logic and a continuous control logic such that a certain infinite-horizon cost function is minimized. The convergence guarantee of the proposed algorithm is, however, not available. A randomized polynomial-time algorithm for the design of switching logics under the availability of certain information about the multiple Lyapunov(-like) functions \cite[\S3.1]{Liberzon} corresponding to the individual subsystems in an expected sense, is presented in \cite{def}. The authors show that if it is allowed to switch from any subsystem to a certain number of stable subsystems, then a switching logic obtained from the proposed algorithm is stabilizing with overwhelming probability.
\subsection{Our contributions}
\label{ss:contri}
    We assume that the knowledge of the set of admissible switches between the subsystems and the admissible dwell times on the subsystems is available. In addition, we have access to a set of finite traces of state trajectories of the subsystems that satisfies certain properties (henceforth, to be referred to as \emph{subsystems data}). We combine two ingredients: (a) data-based techniques for stability analysis of discrete-time linear systems and (b) multiple Lyapunov-like functions and graph walks based techniques for the design of stabilizing switching logics for switched systems, towards arriving at an algorithm that designs stabilizing periodic switching logics in the absence of accurate state-space models of the subsystems. Our algorithm involves the following steps:
    \begin{itemize}[label = \(\circ\), leftmargin = *]
        \item First, we determine (in)stability of the subsystems based on the subsystems data, and compute multiple Lyapunov-like functions and a set of corresponding scalars.
        \item Second, a class of cycles on the underlying directed graph that satisfies certain conditions involving the above set of scalars and the admissible dwell times on the subsystems, is identified.
        \item Third, a stabilizing switching logic is designed such that it activates the vertices that appear in the above cycles and dwells on them for appropriate durations. If no favourable cycle is found on the underlying directed graph of the switched system, then the algorithm reports a failure.
    \end{itemize}

     The design of stabilizing switching logics for switched systems by involving multiple Lyapunov-like functions and graph-theoretic tools is standard in the literature, see e.g., \cite{abc,xyz3,xyz4} and the references therein. However, the suitable Lyapunov-like functions employed in this task, are commonly designed under complete knowledge of the state-space models of the subsystems. We assume certain properties of the available dataset, and involve techniques of data-based stability analysis of linear systems proposed in \cite{Park2009} to design multiple Lyapunov-like functions and compute a set of relevant scalars. These scalars together with the admissible switches between the subsystems and admissible dwell times on the subsystems are then employed to design stabilizing switching logics. At this point, it is worth highlighting that we do not opt for the construction of state-space models of the subsystems from the given data, and hence the proposed technique does not involve system identification of the switched system under consideration. To the best of our knowledge, this is the first instance in the literature where the design of stabilizing switching logics for switched systems in the setting of multiple Lyapunov-like functions and graph-theoretic tools is addressed without explicit knowledge of the state-space models of the subsystems.
\subsection{Paper organization}
\label{ss:paper_org}
    The remainder of this paper is organized as follows: In \S\ref{s:prob_stat} we formulate the problem under consideration. A set of preliminaries required for our result is presented in \S\ref{s:prelims}. Our result appears in \S\ref{s:mainres}. We present a numerical example in \S\ref{s:numex}, and conclude in \S\ref{s:concln} with a brief discussion of future research directions.
\subsection{Notation}
\label{ss:notation}
    \(\mathbb{R}\) is the set of real numbers and \(\mathbb{N}\) is the set of natural numbers, \(\mathbb{N}_{0} = \mathbb{N}\cup\{0\}\). We denote by \([k_{1}:k_{2}]\) the set \(\{n\in\mathbb{N}_{0}\::\:k_{1}\leq n\leq k_{2}\}\). \(I_{d}\) denotes the \(d\times d\) identity matrix, \(I_{d}^{k}\) denotes its \(k\)-th column, \(k\in\{1,2,\ldots,d\}\), \(0_{n}\) and \(\overline{0}_{n}\) denote the \(d\times 1\) zero matrix and \(d\times d\) zero matrix, respectively. For a matrix \(B\in\mathbb{R}^{d\times d}\), \(B\succ 0\) (resp., \(B\prec 0\)) denotes that \(B\) is positive definite (resp., negative definite), and \(\lambda_{\max}(B)\) denotes the maximal eigenvalue of \(B\).
\section{Problem statement}
\label{s:prob_stat}
    We consider a family of discrete-time linear systems
    \begin{align}
    \label{e:family}
        x(t+1) = A_{i}x(t),\:\:x(0) = x_{0},\:\:i\in\P,\:\:t\in\mathbb{N}_{0},
    \end{align}
    where \(x(t)\in\mathbb{R}^{d}\) is the vector of states at time \(t\), \(\P = \{1,2,\ldots,N\}\) is an index set, and
    \begin{align}
    \label{e:comp_form}
        A_{i} = \pmat{-a_{i,d-1}&\cdots&-a_{i,1}&-a_{i,0}\\1&\cdots&&0\\&\vdots&&\vdots\\&\cdots&1&0}\in\mathbb{R}^{d\times d},\:\:i\in\P
    \end{align}
    are full-rank constant matrices. Let \(\sigma:\mathbb{N}_{0}\to\P\) be a switching logic. A discrete-time switched linear system generated by the family of systems \eqref{e:family} and a switching logic \(\sigma\) is described as
    \begin{align}
    \label{e:swsys}
        x(t+1) = A_{\sigma(t)}x(t),\:\:x(0) = x_{0},\:\:t\in\mathbb{N}_{0},
    \end{align}
    where we have suppressed the dependence of \(x\) on \(\sigma\) for notational simplicity.

    Let \(\P_{S}\) and \(\P_{U}\) denote the sets of indices of the Schur stable and unstable subsystems, respectively, \(\P = \P_{S}\sqcup\P_{U}\).\footnote{A matrix \(B\in\mathbb{R}^{d\times d}\) is Schur stable if all its eigenvalues are inside the open unit disk. We call \(B\) unstable if it is not Schur stable.} We let \(E(\P)\) be the set of ordered pairs \((i,j)\) such that a switch from subsystem \(i\) to subsystem \(j\) is admissible, \(i,j\in\P\), \(i\neq j\), and \(\delta\in\mathbb{N}\) and \(\Delta\in\mathbb{N}\) be the admissible minimum and maximum dwell times on all subsystems, respectively, \(\delta\leq\Delta\). Let \(0=:\tau_{0}<\tau_{1}<\tau_{2}<\cdots\) be the \emph{switching instants}; these are the points in time where \(\sigma\) jumps. A switching logic \(\sigma\) is called \emph{admissible} if it satisfies the following conditions: \((\sigma(\tau_{k}),\sigma(\tau_{k+1}))\in E(\P)\) and \(\tau_{k+1}-\tau_{k}\in[\delta:\Delta]\), \(k=0,1,2,\ldots\). Let \(\mathcal{S}\) denote the set of all admissible \(\sigma\). We are interested in stability of the switched system \eqref{e:swsys}. Recall that
    \begin{defn}
    The switched system \eqref{e:swsys} is \emph{globally asymptotically stable} (GAS) for a given switching logic $\sigma$ if \eqref{e:swsys} is Lyapunov stable and globally asymptotically convergent, i.e., for all $x(0)$, $\displaystyle{\lim_{t\rightarrow+\infty}\norm{x(t)} = 0}$.
    \end{defn}

    The scalars \(a_{i,j}\), \(j=0,1,\ldots,d-1\), \(i\in\P\), are unknown. The set \(E(\P)\) and the scalars \(\delta\), \(\Delta\) are known. In addition, finite traces of state trajectories \(x(0), x(1),\ldots,x(L)\), \(L\in\mathbb{N}\), are available for all subsystems. We will solve the following problem:
    \begin{prob}
    \label{prob:mainprob}
    \rm{
        Given the set of admissible switches between the subsystems, \(E(\P)\), the admissible minimum and maximum dwell times on the subsystems, \(\delta\) and \(\Delta\), and a subsystems dataset, design a switching logic \(\sigma\) that preserves GAS of the switched system \eqref{e:swsys}.
        }
    \end{prob}

    \begin{remark}
    \label{rem:knowledge}
    \rm{
        We observe that availability of the knowledge of the set of admissible switches between the subsystems and the dwell times on the subsystems is no loss of generality. Indeed, admissible switches between the subsystems are often governed by properties of internal components of a system. A distinction between admissible and inadmissible transitions captures situations where switches between certain subsystems may be prohibited. For instance, in automobile gear switching, certain switching orders (e.g., from first gear to second gear, etc.) are followed \cite[\S III]{Antsaklis_survey}. A restriction on minimum dwell time on subsystems arises in situations where actuator saturations may prevent switching frequency beyond a certain limit. Also, in order to switch from one component to another, a system may undergo certain operations of non-negligible durations \cite[\S I]{Heydari_ACC16} resulting in a minimum dwell time restriction on subsystems. Restricted maximum dwell time is natural to systems whose components need regular maintenance or replacements, e.g., aircraft carriers, MEMS systems, etc. Moreover, systems dependent on diurnal or seasonal changes, e.g., components of an electricity grid, have inherent restrictions on admissible dwell time. As a result, the set \(E(\P)\) and the scalars \(\delta\) and \(\Delta\) are naturally pre-specified and not design parameters.
        }
    \end{remark}

        Notice that even though we are allowing stable subsystems, Problem \ref{prob:mainprob} does not admit a trivial solution. Indeed, under the maximum dwell time restriction, finding a stable subsystem \(i\in\P_S\) by employing the available subsystems data and setting \(\sigma(t)=i\) for all \(t\in\mathbb{N}_{0}\) is not allowed. Prior to presenting our solution to Problem \ref{prob:mainprob}, we catalog a set of preliminaries.
\section{Preliminaries}
\label{s:prelims}
\subsection{Multiple Lyapunov-like functions and graph walks based design of stabilizing switching logics}
\label{ss:swsig_design}
    The following fact is well-known:
    \begin{fact}{\cite[Fact 1]{abc}}
    \label{fact:key1}
    \rm{
        For each \(i\in\P\), there exists a pair \((P_{i},\lambda_{i})\in\mathbb{R}^{d\times d}\times\mathbb{R}\), where \(P_{i}\) is a symmetric and positive definite matrix, and \(0 < \lambda_{i} < 1\), if \(i\in\P_{S}\) and \(\lambda_{i}\geq 1\), if \(i\in\P_{U}\), such that, with
        \begin{align}
        \label{e:Lyap_eq1}
            \mathbb{R}^{d}\ni\xi\mapsto V_{i}(\xi) := \xi^\top P_{i}\xi\in[0,+\infty[,
        \end{align}
        we have
        \begin{align}
        \label{e:Lyap_eq2}
            V_{i}(\gamma_{i}(t+1))\leq\lambda_{i}V_{i}(\gamma_{i}(t)),\:\:t\in\mathbb{N}_{0},
        \end{align}
        and \(\gamma_{i}(\cdot)\) solves the \(i\)-th recursion in \eqref{e:family}.
        }
    \end{fact}
    The functions \(V_{i}\), \(i\in\P\) are Lyapunov-like functions corresponding to the subsystems \(i\in\P\). The scalar \(\lambda_{i}\), \(i\in\P\) gives a quantitative measure of (in)stability of subsystem \(i\). The Lyapunov-like functions corresponding to the individual subsystems are related as follows:
    \begin{fact}{\cite[Fact 2]{abc}}
    \label{fact:key2}
    \rm{
        There exists \(\mathbb{R}\ni\mu_{ij}>0\) such that
        \begin{align}
        \label{e:Lyap_eq3}
            V_{j}(\xi)\leq\mu_{ij}V_{i}(\xi)\:\:\text{for all}\:\xi\in\mathbb{R}^{d},\:(i,j)\in E(\P).
        \end{align}
        }
    \end{fact}
    A \emph{tight} estimate of \(\mu_{ij}\), \((i,j)\in E(\P)\), is provided below.
    \begin{proposition}{\cite[Proposition 1]{abc}}
    \label{prop:mu_estimate}
    \rm{
        The scalars \(\mu_{ij}\), \((i,j)\in E(\P)\) can be computed as follows:
        \begin{align}
        \label{e:mu_estimate}
            \mu_{ij} = \lambda_{\max}(P_{j}P_{i}^{-1}),\:\:(i,j)\in E(\P).
        \end{align}
        }
    \end{proposition}

    We associate a directed graph \(G(\P,E(\P))\) with the family of systems \eqref{e:family} and the set of admissible switches between the subsystems, \(E(\P)\), as follows: the set of vertices of \(G\) is the set of indices of the subsystems, \(\P\), and the set of edges of \(G\) is the set of admissible switches between the subsystems, \(E(\P)\). A \emph{walk} on \(G\) is a finite alternating sequence of vertices and edges, \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1},(v_{\ell-1},v_{\ell}),v_{ell}\), where \(v_{k}\in\P\), \((v_{k},v_{k+1})\in E(\P)\), \(k=0,1,\ldots,\ell-1\). \(W\) is a \emph{cycle} if \(v_{\ell}:= v_{0}\) and \(v_{0},v_{1},\ldots,v_{\ell-1}\) are distinct.

    Let \(\overline{w}:\P\to\mathbb{R}\) and \(\underline{w}:E(\P)\to\mathbb{R}\) be defined as
    \begin{align*}
        \overline{w}(i) =
        \begin{cases}
            -\abs{\ln\lambda_{i}},\:\:&\:\text{if}\:i\in\P_{S},\\
            \abs{\ln\lambda_{i}},\:\:&\:\text{if}\:i\in\P_{U},
        \end{cases}
    \end{align*}
    and
    \begin{align*}
        \underline{w}(i,j) = \ln\mu_{ij},\:\:&\:\:(i,j)\in E(\P),
    \end{align*}
    where the scalars \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\) are as described in Facts \ref{fact:key1} and \ref{fact:key2}, respectively.
    \begin{defn}
    \label{d:contrac}
    \rm{
        A cycle \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1},(v_{\ell-1}\),\(v_{0}),v_{0}\) on \(G\) is called \emph{contractive} if there exist \(D_{v_{k}}\in[\delta:\Delta]\), \(k=0,1,\ldots,\ell-1\), such that the following condition holds:
        \begin{align}
        \label{e:contrac}
            \sum_{k=0}^{\ell-1}\overline{w}(v_{k})D_{v_{k}} + \sum_{\substack{{k=0}\\{v_{\ell}:=v_{0}}}}^{\ell-1}\underline{w}(v_{k},v_{k+1}) < 0.
        \end{align}
        }
    \end{defn}

    Given a contractive cycle \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1}\),\\\((v_{\ell-1},v_{0}),v_{0}\) on \(G\) and the corresponding \(D_{v_{0}}\), \(D_{v_{1}},\ldots,D_{v_{\ell-1}}\), the following algorithm \cite[Algorithm 1]{xyz4} constructs a switching logic \(\sigma\).
    \begin{algorithm}[htbp]
			\caption{Construction of switching logics} \label{algo:swsig_construc}
		\begin{algorithmic}[1]
			\renewcommand{\algorithmicrequire}{\textbf{Input:}}
			\renewcommand{\algorithmicensure}{\textbf{Output:}}

			\REQUIRE A contractive cycle, \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1}\),\((v_{\ell-1},v_{0}),v_{0}\) on \(G\) and the corresponding \(D_{v_{0}},D_{v_{1}},\ldots,D_{v_{\ell-1}}\).
			\ENSURE A switching logic, \(\sigma\).

            \STATE Set \(p=0\) and \(\tau_{0} = 0\).
            \FOR {\(k=p\ell,p\ell+1,\ldots,(p+1)\ell-1\)}
                \STATE Set \(\sigma(\tau_{k}) = v_{k-p\ell}\) and \(\tau_{k+1} = \tau_{k} + D_{v_{k-p\ell}}\).
			 \ENDFOR
            \STATE Set \(p=p+1\) and go to 2.
		\end{algorithmic}
	\end{algorithm}

    By construction, \(\sigma\) is periodic with period \(\displaystyle{\sum_{k=0}^{\ell-1}D_{v_{k}}}\).
    \begin{lemma}{\cite[Theorem 1]{xyz4}}
    \label{lem:stab}
    \rm{
        Consider a switching logic \(\sigma\) obtained from Algorithm \ref{algo:swsig_construc}. The following are true:
        \begin{enumerate}[label = \roman*), leftmargin = *]
            \item \(\sigma\in\mathcal{S}\), and
            \item \eqref{e:swsys} is GAS under \(\sigma\).
        \end{enumerate}
        }
    \end{lemma}

    Algorithm \ref{algo:contrac_cycle_detection} \cite[Proposition 2]{xyz3} is geared towards detecting a contractive cycle on \(G\).
    \begin{algorithm}[htbp]
	\caption{Detection of contractive cycle on \(G\)} \label{algo:contrac_cycle_detection}
    	\begin{algorithmic}[1]
    			\renewcommand{\algorithmicrequire}{\textbf{Input}:}
			\renewcommand{\algorithmicensure}{\textbf{Output}:}

			\REQUIRE The underlying directed graph, \(G(\P,E(\P))\), of the switched system \eqref{e:swsys}, the scalars \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\), the admissible minimum and maximum dwell times on the subsystems, \(\delta\) and \(\Delta\).
			\ENSURE A contractive cycle, \(W\) on \(G\), if exists.

            \STATE Associate edge weights, \(\alpha:E(\P)\to\mathbb{R}\), to \(G\), as follows:
            \begin{align*}
                \alpha(i,j) =
                \begin{cases}
                    \ln\mu_{ij} - \abs{\ln\lambda_{i}}\Delta,\:\:\text{if}\:i\in\P_{S},\\
                    \ln\mu_{ij} + \abs{\ln\lambda_{i}}\delta,\:\:\text{if}\:i\in\P_{U}.
                \end{cases}
            \end{align*}

            \STATE Apply a negative cycle detection algorithm to \(G\) to obtain a cycle \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1},(v_{\ell-1},v_{0}),v_{0}\) that satisfies \(\displaystyle{\sum_{\substack{{k=0}\\{v_{\ell}:=v_{0}}}}^{\ell-1}\alpha(v_{k},v_{k+1})<0}\).
            \end{algorithmic}
   \end{algorithm}

\subsection{Data-based design of multiple Lyapunov-like functions}
\label{ss:mlf_design}
    Notice that the existence of contractive cycles on the underlying directed graph of the switched system \eqref{e:swsys} depends on three factors: (i) the scalars \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\), (ii) the connectivity of \(G\), and (iii) the scalars, \(\delta\) and \(\Delta\). While (ii) and (iii) are pre-specified, we require a mechanism to design (i) by employing the available subsystems data. The remainder of this section caters to the above purpose.

    Let \(x_{i}(T)\) denote the \(T\)-th element of the trace \(x(0),x(1),\ldots,x(L)\) for subsystem \(i\), and \(L+1\in[\delta:\Delta]\). We let \(x_{i}^{(p)}(T)\) be the \(p\)-th element of \(x_{i}(T)\), \(p=1,2,\ldots,d\). We define
    \begin{align*}
        q_{i}(T) = \pmat{x_{i}^{(1)}(T+1)\\x_{i}^{(1)}(T)\\\vdots\\x_{i}^{(d)}(T)},\:\:T=0,1,\ldots,L-1.
    \end{align*}
    Let
    \begin{align*}
        \Psi_{i} = \pmat{q_{i}(T) & q_{i}(T+1) & \cdots & q_{i}(T+d-1)}\in\mathbb{R}^{(d+1)\times d},
    \end{align*}
    \(T\in\{0,1,\ldots,L-1\}\) be such that the column vectors are linearly independent.

    Fix \(i\in\P\). Notice that whether \(\Psi_{i}\) is well-defined or not, depends on the initial value, \(x_{i}(0)\), and the length, \(L+1\), of the available trace \((x_{i}(0),x_{i}(1),\ldots,x_{i}(L))\). Indeed, to design \(\Psi_{i}\), we need \(L\geq d\) and \(x_{i}(0)\) is such that the vectors \(q_{i}(T),q_{i}(T+1),\ldots,q_{i}(T+d-1)\) are defined and linearly independent for some \(T\in\{0,1,\ldots,L-1\}\). From \cite[Lemma 3]{Park2004} it follows that for every \(A_{i}\) in the companion form described in \eqref{e:comp_form}, there exists \(x_{i}(0)\in\mathbb{R}^{d}\) such that \(\Psi_{i}\) is well-defined with \(T=0\) and \(L=d\). We will, therefore, operate under
    \begin{assump}
    \label{assump:max_dwell_time}
        \(\Delta\geq d+1\).
    \end{assump}

    \begin{lemma}
    \label{lem:auxres2}
    \rm{
        A subsystem \(i\in\P_{S}\) if and only if there exists a symmetric and positive definite matrix \(P_{i}\in\mathbb{R}^{d\times d}\) and a scalar \(0<\lambda_{i}<1\) such that
        \begin{align}
        \label{e:key_ineq}
            \Psi_{i}^\top\pmat{I_{n} & 0_{n}\\0_{n} & I_{n}}^\top\pmat{P_{i} & \overline{0}_{n}\\\overline{0}_{n} & -\lambda_{i}P_{i}}\pmat{I_{n} & 0_{n}\\0_{n} & I_{n}}\Psi_{i} \prec 0.
        \end{align}
    }
    \end{lemma}

    {\bf Proof}:
        It follows from \cite[Theorem 2]{Park2009} that condition \eqref{e:Lyap_eq2} with \(i\in\P_S\) is equivalent to \eqref{e:key_ineq}.

    \begin{lemma}
    \label{lem:auxres3}
    \rm{
        A subsystem \(i\in\P_{U}\) if and only if there exists a symmetric and positive definite matrix \(P_{i}\in\mathbb{R}^{d\times d}\) and a scalar \(\lambda_{i}\geq 1\) such that \eqref{e:key_ineq} holds.
    }
    \end{lemma}

    {\bf Proof}:
        Fix \(i\in\P_{U}\). We will show that condition \eqref{e:Lyap_eq2} is equivalent to \eqref{e:key_ineq}. There exists \(0 < \eta_{i} < 1\) such that \(\eta_{i}A_{i}\) is Schur stable.

        Now, condition \eqref{e:Lyap_eq2} with \(\gamma_{i}(t+1) = \eta_{i}A_{i}\gamma_{i}(t)\) and \(\lambda_{i} = \overline{\lambda}_{i}\) can be rewritten as
        \[
            \gamma_{i}(t)^\top \eta_{i}A_{i}^\top P_{i}\eta_{i}A_{i}\gamma_{i}(t)\leq\overline{\lambda}_{i}\gamma_{i}(t)^\top P_{i}\gamma_{i}(t),
        \]
        or equivalently,
        \[
            \gamma_{i}(t)^\top A_{i}^\top P_{i}A_{i}\gamma_{i}(t)\leq\frac{\overline{\lambda}_{i}}{\eta_{i}^{2}}\gamma_{i}(t)^top P_{i}\gamma_{i}(t).
        \]

        In view of Lemma \ref{lem:auxres2}, the satisfaction of the above inequality is equivalent to the existence of \(P_{i}\succ 0\) and \(\displaystyle{\frac{\overline{\lambda}_{i}}{\eta_{i}^{2}}}\in\mathbb{R}\) such that condition \eqref{e:Lyap_eq2} holds with \(\displaystyle{\lambda_{i} = \frac{\overline{\lambda}_{i}}{\eta_{i}^{2}}}\). Since \(0<\overline{\lambda}_{i},\eta_{i}<1\), it follows that \(\lambda_{i}\geq 1\).

    Lemmas \ref{lem:auxres2} and \ref{lem:auxres3} provide a mechanism to design pairs \((\lambda_{i},P_{i})\), \(i\in\P\), defined in Fact \ref{fact:key1}, from the available subsystems data. The scalars \(\mu_{ij}\), \((i,j)\in E(\P)\) can then be computed by employing Proposition \ref{prop:mu_estimate}.

    We are now in a position to present our solution to Problem \ref{prob:mainprob}.
\section{Result}
\label{s:mainres}
    Given the set of admissible switches between the subsystems, \(E(\P)\), the admissible minimum and maximum dwell times on the subsystems, \(\delta\) and \(\Delta\), and a finite traces of state trajectories for each subsystem, Algorithm \ref{algo:main_algo} designs a stabilizing periodic switching logic \(\sigma\in\mathcal{S}\) by employing the following steps:
    \begin{itemize}[label=\(\circ\),leftmargin = *]
        \item First, the underlying directed graph, \(G(\P,E(\P))\), of the switched system \eqref{e:swsys} is constructed from the knowledge of \(\P\) and \(E(\P)\).\footnote{Notice that \(\P\) can be constructed from the knowledge of \(E(\P)\). Indeed, if there is a subsystem \(i\in\P\) such that there is no \(j\in\P\), \(j\neq p\) with \((j,i)\in E(\P)\), then a well-defined switching logic does not activate \(i\) anyway.}
        \item Second, the matrices \(\Psi_{i}\), \(i\in\P\) are constructed by employing the available subsystems data.
        \item Third, the matrices \(\Psi_{i}\), \(i\in\P\) are used to compute the scalars \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\) by employing Lemmas \ref{lem:auxres2}-\ref{lem:auxres3} and Proposition \ref{prop:mu_estimate}, respectively. Notice that the choice of the pairs \((P_{i},\lambda_{i})\), \(i\in\P\) (and hence the corresponding scalars \(\mu_{ij}\), \((i,j)\in E(\P)\)) is not unique. We store all pairs \((P_{i},\lambda_{i})\), \(i\in\P\) that satisfy condition \eqref{e:key_ineq} in \(\Lambda_{i}\), \(i\in\P\), and their corresponding \(\mu_{ij}\), \((i,j)\in E(\P)\) in \(\chi_{ij}\), \((i,j)\in E(\P)\).
        \item Fourth, corresponding to all elements in \(\Lambda_{i}\), \(i\in\P\) and their corresponding elements in \(\chi_{ij}\), \((i,j)\in E(\P)\), Algorithm \ref{algo:contrac_cycle_detection} is applied to \(G\) until a contractive cycle \(W\) is obtained. If \(G\) does not admit any such cycle, then Algorithm \ref{algo:main_algo} reports a failure and terminates. Otherwise, \(W\) is employed to design a stabilizing periodic switching logic that activates the subsystems whose corresponding vertices appear in \(W\) and dwells on each stable and unstable subsystems for \(\Delta\) and \(\delta\) units of time, respectively.
    \end{itemize}
    \begin{algorithm}[htbp]
	\caption{Model-free computation of stabilizing periodic switching logics} \label{algo:main_algo}
    	\begin{algorithmic}[1]
    			\renewcommand{\algorithmicrequire}{\textbf{Input}:}
			\renewcommand{\algorithmicensure}{\textbf{Output}:}

			\REQUIRE The sets of indices of the stable and unstable subsystems, \(\P_{S}\) and \(\P_{U}\), the set of admissible switches between the subsystems, \(E(\P)\), the admissible minimum and maximum dwell times on the subsystems, \(\delta\) and \(\Delta\), and a subsystems dataset.
			\ENSURE A stabilizing periodic switching logic, \(\sigma\), or a failure message.

            \STATE {\bf Step I}: Construct the underlying directed graph, \(G(\P,E(\P))\), of the switched system \eqref{e:swsys}.

            \STATE {\bf Step II}: Employ the available subsystems data and construct \(\Psi_{i}\), \(i\in\P\).
            \FOR {\(i=1,2,\ldots,N\)}
                \STATE Construct a well-defined \(\Psi_{i}\).
            \ENDFOR

            \STATE {\bf Step III}: Apply Algorithm \ref{algo:scalar_compute} to compute \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\) from \(\Psi_{i}\), \(i\in\P\).

            \STATE {\bf Step IV}: Detect a contractive cycle on \(G\).
            \FOR {each \((P_{i},\lambda_{i})\in\Lambda_{i}\) and the corresponding \(\mu_{ij}\), \((i,j)\in E(\P)\)}
            \STATE Apply Algorithm \ref{algo:contrac_cycle_detection}.
            \IF {a contractive cycle \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1},(v_{\ell-1},v_{0}),v_{0}\) on \(G\) is detected}
            \STATE Go to Step V.
            \ELSE
            \STATE Output FAIL and terminate.
            \ENDIF
            \ENDFOR

            \STATE {\bf Step V}: Design a stabilizing periodic switching logic.
            \STATE Apply Algorithm \ref{algo:swsig_construc} with \(W\) obtained in Step VI, and obtain a \(\sigma\) with \(\sigma(\tau_{k}) = \Delta\), if \(\sigma(\tau_{k})\in\P_{S}\) and \(\sigma(\tau_{k}) = \delta\), if \(\sigma(\tau_{k})\in\P_{U}\).
    \end{algorithmic}
    \end{algorithm}

    Observe that solving \eqref{e:key_ineq} with both \(P_{i}\) and \(\lambda_{i}\), \(i\in\P\), unknown is a numerically difficult task. To address this issue, in Algorithm \ref{algo:scalar_compute}, we employ a line search technique \cite{LMI_book} as follows:
   \begin{itemize}[label=\(\circ\),leftmargin = *]
        \item For \(i\in\P_{S}\), a finite set of values of \(\lambda_{i}\) on the interval \(]0,1[\) is fixed, and corresponding to each element of this set, the feasibility problem \eqref{e:feasprob} is solved for symmetric and positive definite matrices, \(P_{i}\), \(i\in\P\).
        \item For \(i\in\P_{U}\), a finite set of values of \(\eta_{i}\) on the interval \(]0,1[\) is fixed, and corresponding to each element of this set, Schur stability of the matrices, \(\eta_{i}A_{i}\), is checked. The favourable values of \(\eta_{i}\) are stored and the feasibility problem \eqref{e:feasprob} is solved for symmetric and positive definite matrices, \(P_{i}\), \(i\in\P\) with \(\lambda_{i} = \frac{1}{\eta_{i}^{2}}\).
    \end{itemize}
    From Lemma \ref{lem:stab}, it follows that a switching logic, \(\sigma\), obtained from Algorithm \ref{algo:main_algo} obeys the given restrictions on the admissible switches between the subsystems and the admissible dwell times on the subsystems, and preserves GAS of the switched system \eqref{e:swsys}.
    \begin{algorithm}[htbp]
	\caption{Computation of \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\) from \(\Psi_{i}\), \(i\in\P\)} \label{algo:scalar_compute}
    	\begin{algorithmic}[1]
    			\renewcommand{\algorithmicrequire}{\textbf{Input}:}
			\renewcommand{\algorithmicensure}{\textbf{Output}:}

			\REQUIRE The matrices \(\Psi_{i}\), \(i\in\P\).
			\ENSURE Sets of scalars \(\lambda_{i}\), \(i\in\P\) and \(\mu_{ij}\), \((i,j)\in E(\P)\).

            \STATE {\bf Step I}: Compute \((P_{i},\lambda_{i})\) from \(\Psi_{i}\), \(i\in\P_{S}\).
            \STATE Fix \(h_{s}>0\) (small enough) and compute \(k_{s}\in\mathbb{N}\) such that \(k_{s}\) is the largest integer satisfying \(k_{s}h_{s}<1\).
            \FOR {all \(i\in\P_{S}\)}
                \STATE Set \(\Lambda_{i} = \emptyset\).
                \FOR {\(\lambda_{i} = h_{s},2h_{s},\ldots,k_{s}h_{s}\)}
                    \STATE Solve the following feasibility problem in \(P_{i}\):
                    \begin{align}
	           		\label{e:feasprob}
		                  \minimize\:\:&\:\:1\nonumber\\
		              	  \sbjto\:\:&\:\:
		                  \begin{cases}
 			                \text{condition}\:\eqref{e:key_ineq},\\
                                        P_{i}^\top = P_{i} \succ 0.\\

		                  			\end{cases}
	             	\end{align}
                    \IF {a solution to \eqref{e:feasprob} exists,}
                        \STATE Set \(\Lambda_{i} = \Lambda_{i}\cup\{(P_{i},\lambda_{i})\}\).
                    \ENDIF
                \ENDFOR
            \ENDFOR

            \STATE {\bf Step II}: Compute \((P_{i},\lambda_{i})\) from \(\Psi_{i}\), \(i\in\P_{U}\).
            \STATE Fix \(h_{u} > 0\) (small enough) and compute \(k_{u}\in\mathbb{N}\) such that \(k_{u}\) is the largest integer satisfying \(k_{u}h_{u} < 1\).
            \FOR {all \(i\in\P_{U}\)}
            \STATE Set \(\Gamma_{i} = \emptyset\) and \(\Lambda_{i} = \emptyset\).
            \FOR {\(\eta_{i} = h_{u},2h_{u},\ldots,k_{u}h_{u}\)}
            \IF {\(\eta_{i}A_{i}\) is Schur stable}
                \STATE Set \(\Gamma_{i} = \Gamma_{i}\cup\{\frac{1}{\eta_{i}^{2}}\}\).
            \ENDIF
            \ENDFOR
            \FOR {all \(\lambda_{i}\in\Gamma_{i}\)}
            \STATE Solve the feasibility problem \eqref{e:feasprob} in \(P_{i}\).
            \IF {a solution to \eqref{e:feasprob} exists}
            \STATE Set \(\Lambda_{i} = \Lambda_{i}\cup\{(P_{i},\lambda_{i})\}\).
            \ENDIF
            \ENDFOR
            \ENDFOR

            \STATE {\bf Step III}: Compute \(\mu_{ij}\), \((i,j)\in E(\P)\) from \(P_{i}\), \(i\in\P\).
            \STATE Set \(\chi_{ij} = \emptyset\).
            \FOR {all \((i,j)\in E(\P)\)}
            \FOR {each pair \((P_{i},\lambda_{i}),(P_{j},\lambda_{j})\in\Lambda_{i}\)}
            \STATE Compute \(\mu_{ij} = \lambda_{\max}(P_{j}P_{i}^{-1})\).
            \STATE Set \(\chi_{ij} = \chi_{ij}\cup\{\mu_{ij}\}\).
            \ENDFOR
            \ENDFOR
    \end{algorithmic}
    \end{algorithm}

    It is worth highlighting that only finitely many possibilities of the pairs \((P_{i},\lambda_{i})\), \(i\in\P\) and their corresponding \(\mu_{ij}\), \((i,j)\in E(\P)\) are explored in Algorithm \ref{algo:main_algo}. Consequently, a failure message obtained from Algorithm \ref{algo:main_algo} does not indicate the non-existence of a set of pairs \((P_{i},\lambda_{i})\), \(i\in\P\), such that \(G\) admits a contractive cycle. In addition, Lemma \ref{lem:stab} provides a sufficient condition for the stability of \eqref{e:swsys}. As a result, a failure message obtained from Algorithm \ref{algo:main_algo} also does not conclude the non-existence of stabilizing elements in \(\mathcal{S}\).
\section{A numerical example}
\label{s:numex}
    Consider a switched system \eqref{e:swsys} with \(E(\P) = \{(1,2),(1,4),(1,5)\),\((2,3)\),\((2,4),(2,5),(3,4),(3,5),(4,5)\),\((5,1),(5,4)\}\), \(\delta = 2\) and \(\Delta = 6\). The numerical values of \(A_{i}\), \(i\in\P = \{1,2,3,4,5\}\), are given in Table \ref{tab:data_set1}; they are generated by choosing elements from the interval \([-1,1]\) uniformly at random.
    \begin{table*}[htbp]
	\centering
    \scalebox{1}{
	\begin{tabular}{|c | c | c|}
		\hline
		\(i\) & \(A_{i}\)\\
		\hline
		\(1\) & \(\pmat{ 0.6800919 & -0.0450156 &  0.8332522 &  0.2474328 &  0.6490691\\
                    1 & 0 & 0 & 0 & 0\\
                    0 & 1 & 0 & 0 & 0\\
                    0 & 0 & 1 & 0 & 0\\
                    0 & 0 & 0 & 1 & 0}\)\\

		\hline
        \(2\) & \(\pmat{-0.5271101  & 0.3123521 & -0.6457525 &  0.8071877 &  0.6592868\\
                    1 & 0 & 0 & 0 & 0\\
                    0 & 1 & 0 & 0 & 0\\
                    0 & 0 & 1 & 0 & 0\\
                    0 & 0 & 0 & 1 & 0}\)\\
        \hline
        \(3\) & \(\pmat{-0.442885  & 0.5338433 &  0.6484834  & 0.8689565 &  0.3665952\\
                    1 & 0 & 0 & 0 & 0\\
                    0 & 1 & 0 & 0 & 0\\
                    0 & 0 & 1 & 0 & 0\\
                    0 & 0 & 0 & 1 & 0}\)\\
        \hline
        \(4\) & \(\pmat{0.6482512 & 0.0272578 & -0.4435161 & 0.1849962 & 0.053028\\
                    1 & 0 & 0 & 0 & 0\\
                    0 & 1 & 0 & 0 & 0\\
                    0 & 0 & 1 & 0 & 0\\
                    0 & 0 & 0 & 1 & 0}\)\\
        \hline
        \(5\) & \(\pmat{0.2486157 &  0.0809103 & -0.0931076 &  0.3252463 & -0.1238403\\
                    1 & 0 & 0 & 0 & 0\\
                    0 & 1 & 0 & 0 & 0\\
                    0 & 0 & 1 & 0 & 0\\
                    0 & 0 & 0 & 1 & 0}\)\\
        \hline
	\end{tabular}}
	\caption{Description of the subsystems.}\label{tab:data_set1}
	\end{table*}

    We first demonstrate that the switched system under consideration is not stable under all \(\sigma\in\mathcal{S}\). Towards this end, we fix \(\tau_{0} = 0\), pick \((\sigma(\tau_{k}),\sigma(\tau_{k+1}))\in E(\P)\), \(\tau_{k+1}-\tau_{k}\in[\delta:\Delta]\), \(k=0,1,2,\ldots\), uniformly at random, and plot \((\norm{x(t)})_{t\in\mathbb{N}_{0}}\) corresponding to \(x(0)\in\mathbb{R}^{5}\) chosen uniformly at random from the interval \([-1,1]^{5}\) and the \(\sigma\) designed above. We observe instability of \eqref{e:swsys}, see Figure \ref{fig:x1_plot}. It is, therefore, of interest to identify stabilizing elements of \(\sigma\).
    \begin{figure}
   \centering
        \includegraphics[scale = 0.4]{unstable_fig}
        \caption{Plot of \((\norm{x(t)})_{t\in\mathbb{N}_{0}}\) under a randomly designed \(\sigma\in\mathcal{S}\).}\label{fig:x1_plot}
   \end{figure}

    The values of \(a_{i,j}\), \(j=0,1,\ldots,d-1\), \(i\in\P\) are not known, but the set \(E(\P)\) and the scalars \(\delta\), \(\Delta\) are known. In addition, we have access to subsystems dataset. We will apply Algorithm \ref{algo:main_algo} to design a stabilizing periodic switching logic \(\sigma\in\mathcal{S}\). It involves the following steps:

    \begin{enumerate}[label = Step \Roman*., leftmargin = *]
        \item The underlying directed graph, \(G(\P,E(\P))\), of the switched system \eqref{e:swsys} is constructed with \(\P\) and \(E(\P)\) as described above.
        \item The matrices, \(\Psi_{i}\), \(i\in\P\), are constructed by employing the available dataset. The numerical values of \(\Psi_{i}\), \(i\in\P\) are given in Table \ref{tab:data_set2}.
        \item Algorithm \ref{algo:scalar_compute} is applied with \(h_{s} = h_{u} = 0.1\) to compute the sets \(\Lambda_{i}\), \(i\in\P\) and \(\chi_{ij}\), \((i,j)\in E(\P)\).\footnote{The feasibility problem \eqref{e:feasprob} is solved using the lmisolver tool in Scilab 6.0.2.}
        \item Algorithm \ref{algo:contrac_cycle_detection} is applied on \(G\). A contractive cycle \(W = 4, (4,5), 5, (5,4), 4\) is obtained with \(\lambda_{4}=\lambda_{5} = 0.7\),
            \begin{align*}
            {{P_{4} = \pmat{{3481.4063} & {-1322.4505} & {-349.63603} &  {805.92625} &  {5.8206481}\\
                        {-1322.4505} &  {2144.6438} & {-369.98414} & {-465.86262} & {186.60695}\\
                        {-349.63603} & {-369.98414} & {999.52474} & {-208.06702} & {-135.54574}\\
                        {805.92625} &  {-465.86262} & {-208.06702} &  {594.47594} & {-20.831212}\\
                        {5.8206481} &  {186.60695} & {-135.54574} & {-20.831212} &  {140.49607}}}},\\
            {{P_{5} = \pmat{{3521.175} &  {-60.946413} & {-304.62761} &  {368.42809} & {-457.01911}\\
                            {-60.946413} &  {1337.4811} & {-130.53998} & {-179.42523} &  {0.623482}\\
                            {-304.62761} & {-130.53998} &  {779.40748} & {-138.61513} & {-35.892171}\\
                            {368.42809} & {-179.42523} & {-138.61513} &  {489.07866} & {-167.19363}\\
                            {-457.01911} &  {0.623482} &  {-35.892171} & {-167.19363} &  {189.56074}}}},
            \end{align*}
            \(\mu_{45} = 3.1883122\) and \(\mu_{54} = 3.60176\). A switching logic \(\sigma\) is constructed by employing Algorithm \ref{algo:swsig_construc}. \(\sigma\) activates subsystems \(4\) and \(5\) alternatively, and during each activation dwell on them for \(6\) units of time.
    \end{enumerate}

    \begin{table*}[htbp]
	\centering
    \scalebox{1}{
	\begin{tabular}{|c | c | c|}
		\hline
		\(i\) & \(\Psi_{i}\)\\
		\hline
		\(1\) &
            \(\pmat{ 1.1109433 &  0.1262875 &  1.1877076  & 1.7062463 &  1.9976929\\
   0.7867279 &  1.1109433 &  0.1262875 &  1.1877076 &  1.7062463\\
  -0.3330575 &  0.7867279 &  1.1109433 &  0.1262875 &  1.1877076\\
   0.8915796 & -0.3330575 &  0.7867279 &  1.1109433 &  0.1262875\\
  -0.8272248 &  0.8915796 & -0.3330575 &  0.7867279 &  1.1109433\\
   0.0349348 & -0.8272248 &  0.8915796 & -0.3330575 &  0.7867279}\)\\
        \hline
        \(2\)  &
                    \(\pmat{0.053192 &  -1.4739238 &  0.5816219 & -0.0585455 &  1.4498912\\
   0.3680257  & 0.053192 &  -1.4739238 &  0.5816219 & -0.0585455\\
   0.6760328  & 0.3680257  & 0.053192 &  -1.4739238 &  0.5816219\\
  -0.7886473 &  0.6760328 &  0.3680257 &  0.053192 &  -1.4739238\\
  -0.7397434 & -0.7886473 &  0.6760328 &  0.3680257  & 0.053192\\
   0.1878737 & -0.7397434 & -0.7886473 &  0.6760328 &  0.3680257}\)\\
        \hline
        \(3\) &
                \(\pmat{ 0.6482342 &  0.459006  &  0.9236082 &  0.5121273 &  1.1249577\\
  -0.0060993 &  0.6482342 &  0.459006  &  0.9236082 &  0.5121273\\
   0.7121564 & -0.0060993 &  0.6482342 &  0.459006  &  0.9236082\\
   0.4527117 &  0.7121564 & -0.0060993 &  0.6482342 &  0.459006\\
  -0.2887465 &  0.4527117 &  0.7121564 & -0.0060993 &  0.6482342\\
   0.6074417 & -0.2887465 &  0.4527117 &  0.7121564 & -0.0060993}\)\\
        \hline
        \(4\)  &
                \(\pmat{-0.4584539 & -0.0066052 &  0.1112462 &  0.1332211 & -0.0226489\\
                -0.5687414 & -0.4584539 & -0.0066052 &  0.1112462 &  0.1332211\\
                -0.6945576 & -0.5687414 & -0.4584539 & -0.0066052 &  0.1112462\\
                0.0805042 & -0.6945576 & -0.5687414 & -0.4584539 & -0.0066052\\
                -0.3177508 &  0.0805042 & -0.6945576 & -0.5687414 & -0.4584539\\
                0.4460485 & -0.3177508 &  0.0805042 & -0.6945576 & -0.5687414}\)\\
        \hline
        \(5\)  &
                \(\pmat{0.0308642 & -0.1437207 & -0.0167672 & -0.0429631 &  0.0304562\\
                    -0.1540222 &  0.0308642 & -0.1437207 & -0.0167672 & -0.0429631\\
                    -0.2083555 & -0.1540222 &  0.0308642 & -0.1437207 & -0.0167672\\
                    -0.56438  &  -0.2083555 & -0.1540222 &  0.0308642 & -0.1437207\\
                    -0.2037382 & -0.56438  &  -0.2083555 & -0.1540222 &  0.0308642\\
                    -0.8053248 & -0.2037382 & -0.56438 &  -0.2083555 & -0.1540222}\)\\
        \hline
	\end{tabular}}
	\caption{Description of \(\Psi_{i}\), \(i\in\P\).}\label{tab:data_set2}
	\end{table*}

    We now pick \(100\) different \(x(0)\in\mathbb{R}^{5}\) from the interval \([-1,1]^{5}\) uniformly at random, and plot \(\norm{x(t)}_{t\in\mathbb{N}_{0}}\) under the switching logic \(\sigma\) obtained from Algorithm \ref{algo:main_algo}, see in Figure \ref{fig:x_plot}. GAS of the switched system \eqref{e:swsys} is demonstrated.
    \begin{figure}
   \centering
        \includegraphics[scale = 0.4]{stable_fig}
        \caption{Plot of \((\norm{x(t)})_{t\in\mathbb{N}_{0}}\) under \(\sigma\) obtained from Algorithm \ref{algo:main_algo}.}\label{fig:x_plot}
   \end{figure}
\section{Conclusion}
\label{s:concln}
    In this paper we presented an algorithm that designs stabilizing periodic switching logics for discrete-time switched linear systems under restricted switching. The proposed design technique can be extended to the setting of stabilizing non-periodic switching logics by employing a cycle \(W = v_{0},(v_{0},v_{1}),v_{1},\ldots,v_{\ell-1},(v_{\ell-1},v_{0}),v_{0}\) that is contractive on \(G\) with multiple choices of \(D_{v_{k}}\in[\delta:\Delta]\), \(k=0,1,\ldots,\ell-1\) and/or multiple contractive cycles \(W_{1},W_{2},\ldots,W_{p}\) on \(G\), see e.g., \cite[Remarks 6,7,10]{xyz3} for discussions on multiple Lyapunov-like functions and graph walks based design of stabilizing non-periodic switching logics. We are currently investigating data-based design of switching logics when the subsystems structures are not restricted to the companion form described in \eqref{e:comp_form} and the available subsystems data are noisy. The findings will be reported elsewhere.

redpajama_set_name: RedPajamaArXiv


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Kay leads workshops on using poetry to help heal and cope during times of loss and grief. In the workshop, each participant has time to read, discuss and write, based on a thoughtfully selected set of poems. Although the workshops are not therapeutic counseling sessions, participants bring their own stories of loss and grief to the work that is done there. In part, that work is to identify how poetry "works." As an example, Kay talked about Joseph Stroud's "Stitching the Woe Shirt," a carefully crafted poem about the death of his wife. On reading that poem, one can sense the power that is wielded by the placement of only one introductory word, "inconsolable," on one line. Each word, each line is so powerful that "sometimes grief just oozes" out of the poem.
Mullen was initiated into the world of loss and grief by the death of her dear husband of thirty-five years. Until that time, nothing had prepared her for all that was to come. She had been a classroom teacher, the mother of two, who eventually became a mental health counselor and school counselor. She had, with encouragement from both her husband (an accomplished writer in his own right) and a colleague, published two, full-length poetry collections: Let Morning Begin (Caritas Communications, 2001) and A Long Remembering: Return to Vietnam (Foothills Publishing, 2006). Her work also appeared in various journals such as Valparaiso Poetry Review, Appalachia, Avocet, Crab Creek Review, as well as anthologies: Tatoos on Cedar, Mute Note Earthward, Pontoon, Northwind. She entered PLU's Rainier Writing Workshop MFA program with her husband at her side for the first summer residency. And then he was gone.
Following her husband's death, Kay devoted her time to his memory. She spent countless hours collecting his writings, putting some together into an autobiography, others in a full-length book, and gathered together his letters, short stories and poems. During the seven years following that, she turned her attention to her own work, her "poems of grief, memories of our first year of marriage in Alaska, poems about his illness, memories of our life together and events after his death." The result was her third book of poems, Even the Stones (Caritas Communications, 2012) She also managed to complete her MFA, win a Stafford award and be nominated for a Pushcart prize.
Those years were a crucible in which the strong forces of intellect, professionalism, spirit, faith and love tempered the sense of the "inconsolable" in trying to "make sense of the world after this major loss in my life." From her experience she is able to help others begin or continue the process of healing with poetry being the vehicle. Her most recent workshop was in January, 2013, at Catherine's place (a center for women in Tacoma). Other workshops have been held at Kings Books, at the Olympia library, at Urban Grace church and even at the correctional center in Shelton.
What if a person is relatively unacquainted with poetry—who would be "good" poets to begin reading? Mullen's reply was, in addition to the two poets already mentioned (Rukeyser and Stroud), that a person might want to begin reading poetry by Tess Gallagher, Jack Gilbert, Lola Haskins, Mary Oliver and Billy Collins as a way of beginning to experience the "soul work" that is poetry. A date has not yet been set for Mullen's next workshop.
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Offering pleasant accommodation, comfortable facilities and detectable service, this stylish hotel is an ideal holiday spot for any kind of travellers.
Enjoy delicious Indian, Chinese, Mughlai, Continental and authentic Nepali cuisine at Paanchali Restaurant. The Trishna Coffee Shop overlooking the swimming pool serves delectable edibles in a pleasant and formal surrounding. You can also sip your favourite drink at Manihkana Bar.
Currency exchange, Gymnasium, Internet access.
The Hotel Vaishali Kathmandu is situated in the most centre part of the city and the main tourist attraction centre. The hotel is 9 kilometres from the Kathmandu International Airport.


Environmentally degradable polyesters offer an interesting perspective for a vast number of applications. However, current front-runners like poly(lactide), poly(glycolide) and poly(ε-caprolactone) are either semi-crystalline excluding applications for which optical transparency is desired, or exhibit low glass transition temperatures (T g) resulting in poor dimensional stability at temperatures exceeding the T g. In the present work, copolymers of lactide and mandelide are explored as a method to obtain amorphous, environmentally degradable polyesters with a glass transition temperature exceeding 50 °C. Mandelide and lactide can be successfully copolymerised and the resulting copolymers revealed rising T g values upon increasing the mandelide content. The obtained molecular weights were superior to the molecular weights previously obtained via polycondensation, but were limited by the epimerisation of the mandelide monomer, which passes through an enolic intermediate that is able to initiate the polymerisation.


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Accentus — французский камерный хор, основанный и возглавленный в 1991 году дирижёром Лоранс Экильбе. Состоит из 32 исполнителей.

В репертуар хора входят как классические сочинения, начиная с эпохи барокко, так и, прежде всего, новейшая музыка. Отношения сотрудничества связывают хор, в частности, с Паскалем Дюсапеном, мировые премьеры произведений которого Granum sinapis и Dona eis прозвучали в исполнении хора в 1998 и 1999 гг. соответственно. Среди других композиторов, чью музыку хор исполнял впервые, — Филипп Манури, Брюно Мантовани, Марко Строппа. Наряду с акапелльным пением хор «Accentus» выступает и вместе с ведущими оркестрами Франции и такими дирижёрами, как Пьер Булез, Кристоф Эшенбах, Джонатан Нотт, Жан Кристоф Спинози. Трижды (2002, 2005, 2008) хор был удостоен премии Виктуар де ля мюзик как лучший вокальный коллектив Франции.

По мнению российского критика, обозревавшего московские гастроли хора в 2002 году,

сильное место в технике французов — вертикаль, особое гармоническое мышление каждого хориста, который согласуется не с соседом по партии, а с остальными тонами созвучия (а также с индивидуальным камертоном, извлекаемым из кармана). Ложка дёгтя: приоритет авангардной музыки породил привычку обособлять каждый звук, и голоса одной партии (особенно сопрано) тембрально, а то и интонационно разобщены, страдает линия. Но гармоническое корреспондирование обусловливает удивительную красоту и глубину единого хорового тембра «Accentus» и делает его поистине уникальным.

Примечания

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Хоры Франции
Музыкальные коллективы, появившиеся в 1991 году
Музыкальные коллективы по алфавиту

redpajama_set_name: RedPajamaWikipedia


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Argentine Peso Plummets To Record Low As Inflation Soars
With inflation soaring (and forecast to accelerate further), the Argentine Peso has plummeted the last few days even as the central bank raised 7-day Leliq rate to 51.862%, according to two people with direct knowledge.
The weakness accelerated into the close...
Taking the peso down top a new record low...
Peso has dropped 11% this year, making it the worst performer against USD.
As Bloomberg notes, Argentina's famous grass-fed beef likely emerged as a top inflation driver in February, analysts at JPMorgan wrote in a note on Wednesday. The bank raised its consumer price forecast for that month to 3.9 percent from 3.2 percent.
"The main culprit," JPMorgan analysts Lucila Barbeito and Diego Pereira wrote "is red meat, with prices that surprised to the upside in the past month."
Specifically, economists polled by Argentina's central bank increased their end-2019 inflation expectations to 31.9% from 29%, according to the institution's February survey.
Paging Christine Lagarde!!!
As Daniel Lacalle warned in January, Argentina should urgently take measures to cut country risk.
Argentina, at the close of this article, maintains a high country risk. In fact, the second highest in Latin America and the third of the emerging countries if we include Turkey.
What is the country risk? It is the spread between the yield of the sovereign bond of a nation compared with the United States Treasury bonds, the lowest risk asset. Investors demand higher-yielding bonds due to the risk of the issuer's economy, to compensate for the currency and solvency risk.
Argentina's country risk increased by 130% in 2018 due to delayed reforms, constant devaluation and loss of foreign exchange reserves. The country risk has fallen significantly from its high of 817 basis points to the current 706. However, it remains the second highest in the region after Venezuela.
The reasons why a rich and high-potential country like Argentina has a higher country risk than allegedly weaker developing countries are easy to explain:
The countries with the highest country risk are also those that have abused most of the financing of public spending by the central bank through the printing of currency.
Argentina has a higher country risk than apparently more fragile economies due to the constant refusal on the part of the successive governments to adopt a prudent monetary policy and to defend the purchasing power of the currency. When inflation rises and the currency is devalued due to this misguided monetary policy, the purchasing power of that currency is destroyed, and the country's investor security erodes.
Risk increases because investors perceive that no relevant structural reforms are being carried out, and this limits potential growth and increases the risk of inflation and default. The default probability increases with the downward spiral of printing money to finance increasing public and extractive spending, triggering inflation and raising taxes, which leads to lower growth, less investment and more displacement of the private sector.
That is the reason why countries with a relatively low debt to GDP ratio but huge monetary imbalances fail before countries with high debt to GDP but a prudent monetary policy that does not directly finance public spending. It is not the same to create a transmission mechanism of monetary policy such as quantitative easing -with all its flaws-, which curbs inflationary risk by generating firewalls in the transmission process because it is done through credit, unlike directly prints money to finance current spending. The second destroys the economy, it is a direct transfer of wealth from the savers and salaries to the government and makes confidence in the currency disappear. Argentine citizens know this well.
Country risk must be contained not only because it leads the state to finance itself at a higher cost and with less demand, but because it drags the financing capacity of the real economy, companies and families.
It is not just bad for public finances. If yield soars in the asset that supposedly has less risk of the nation, the sovereign bond, it leads to a whole chain of negative impacts on the financing of the real economy, the cost of financing of companies and families rises much more and credit dries up.
The only way that Argentina will erase its unnecessarily high country risk is by cutting bloated public spending, lowering taxes, attracting investment and improving ease of doing business. But more importantly, it needs to stop financing public spending by printing pesos for once and for all.
In 2018 Argentina was among the four countries in the world with the highest inflation, below Venezuela, South Sudan, and Iran, according to the International Monetary Fund.
Argentina is also among the countries with the highest tax wedge, in fact, the highest total tax burden in the world, according to the World Economic Forum. The total tax rate absorbs all corporate profits.
In the index of economic freedom published by the Heritage Foundation, Argentina has improved; but it continues to appear in one of the worst positions, number 144 of a total of 180 countries. We must emphasize that the Nordic countries, which are used constantly in television debates in Argentina as an example, are among the countries with the greatest economic freedom, ranking between 12 (Denmark) and 26 (Finland). It is worth remembering this factor when talking about prosperity and redistribution. There is no social policy if economic liberty and prosperity are not the primary objective.
Similarly, in the Doing Business index -of ease to create jobs and companies of the World Bank-, Argentina appears in a slightly improved position with respect to 2017, but far below where it should be. Argentina occupies a very poor position, 117 of 190 countries. Again, the Nordic countries rank between number 3 and 13, at the head of the table.
High inflation, high taxes, low economic freedom and very high burdens on investment. Four factors that worsened significantly in the Cristina Fernández de Kirchner tenure and that the Macri administration has not done enough to improve.
Only eliminating the unbridled printing of pesos from the Central Bank and the useless political spending financed with rising money supply, Argentina would climb at least twenty positions in these indexes. If the tax and bureaucratic burden were reduced to levels similar to those of the surrounding countries and the OECD average, Argentina would be – as it deserves – an international center of capital attraction and global investment, and one of the countries with a more solid growth. Only by cutting the impoverishing inflation, the crazy printing of pesos and reducing taxation and bureaucracy, Argentina would climb to the Top 50 in the aforementioned rankings, quickly approaching Chile and Costa Rica, and shortening the enormous distance with France, the United Kingdom or the Netherlands. This would reduce the country risk to at least region average levels.
A public expenditure that already reaches more than 45% of GDP; and denying the devastating effect of the predatory monetary policy lead Argentina to constant and uninterrupted currency crises, unbridled inflation and to discourage long-term investment and prudent saving. The imprudent use of the national currency as a subterfuge to cover enormous internal imbalances has only led to the fact that, again, it has lost 51% of its value in 2018 and that inflation continues to undermine the country's potential .
Argentina has all the ingredients to be a global leading economy. But the erroneous monetary policy, the unbridled political spending and the extractive bureaucracy and taxation make the country a promise that always seems to be a disappointment because of internal factors.


This is every kid's favorite (including big kids). Simply put a marshmallow on a stick, roast it, and while it's still hot, slap it against a piece of chocolate bar and sandwich it between two graham crackers.
Cut a banana lengthwise. Stud it with miniature marshmallows and chocolate chips. Wrap it in aluminum foil and set it on some coals to melt the chocolate and marshmallows. Eat it while it is still warm.
Split the hot dogs lengthwise and roast on the grill. Turn them over, cover the split side with cheddar cheese, and grill skin side down.
Split English muffins. Spread pizza sauce on them. Sprinkle them with mozzarella cheese, and place pepperoni slices on top. Place them on the grill to cook. Cover them with a can or pan to catch the heat and direct it onto the tops of the pizzas. You can also bake these in a Dutch oven.
When you are ready for lunch, build a small fire. A few one-inch diameter dead pine branches make the perfect fuel. Let the coals burn down.
As the fire is burning down, spread the tomato paste on the crust in the pan, sprinkle with Italian seasonings, and prepare the pizza with the toppings. Cover the pan tightly with aluminum foil.
Scrape the coals away and nestle the pan down into the warm bed. If there is too much heat, the crust will burn. Place a hot stick or two on top of the foil. Let the hot bed warm the pizza through and until the cheese is melted. You may have to recharge the bed with reserved hot coals. When the cheese has melted and is bubbly, the pizza is ready.
Makeshift Pizza only works the first day out unless you have some way to keep the perishables cold. (A good snow bank will do.) And the ingredients are a little heavy to be packing far.
Dennis Weaver is the general manager at www.preparedpantry.com">The Prepared Pantry (www.preparedpantry.com">http://www.preparedpantry.com) with recipes, ideas, and the best selection of mixes and ingredients. Visit the free Bakers' Library for more articles like this, free baking guides, and tested recipes.
Making your own pasta is not difficult. It's actually very easy.


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The White House Is Still Trying To Win Over Skeptical Black Lives Matter Activists
As President Obama's administration enters the home stretch, the White House continues to struggle with what will be one of the key pieces of his legacy: his relationship with the Black Lives Matter movement.
By Darren Sands
Darren Sands BuzzFeed News Reporter
Posted on May 15, 2016, at 10:33 p.m. ET
Saul Loeb / AFP / Getty Images
WASHINGTON — With President Obama's term entering the home stretch, White House officials increasingly concerned about his legacy on race issues are turning their attention to the sometimes difficult relationship with the Black Lives Matter movement, and are hoping to expand the number of activists it engages before he leaves office.
As the nation's first black president, issues of race were inevitably going to be a part of Obama's legacy. And while the administration has aggressively pushed agenda items aimed at addressing systemic racism — most notably criminal justice policy — the resurgence of open, and often violent, racial tensions across the country has taken center stage.
Although the administration has been reaching out to BLM activists since its inception in the wake of massive protests in Ferguson, Missouri, in 2014 (Obama has twice met with activists), many within the movement have treated those overtures with some skepticism. That, in turn has frustrated Obama and his top aides, who feel that activists need to work within the system, even as they use direct action protests to bring pressure on it.
"The federal government is a complex entity," a senior administration official said of the frustration some inside White House had explaining what it would take to work effectively with the administration. "It takes a while to appreciate the difference between what can happen at state and local and federal government. It takes a willingness to engage and understand those limitations and those responsibilities in order to figure out how to actually effect positive change."
But while officials may counsel patience, with just nine months left in Obama's presidency, questions of legacy and unresolved tensions have taken on greater significance. Inside the White House, there is a strain of thought among some aides — the same aides who had once obsessed over the protesters' fervor that captivated the entire country — that the activists aren't serious about effecting change, as much as they want, as one person close to the situation said, "a never-ending crisis."
Those tensions came into focus last weekend during Obama's commencement speech at Howard University. Obama's speech struck many of his traditional notes, lauding the efforts of young activists while urging them to for ways to translate their momentum into concrete accomplishments.
"It's thanks in large part to the activism of young people like many of you, from Black Twitter to Black Lives Matter, that America's eyes have been opened — white, black, Democrat, Republican' to the real problems, for example, in our criminal justice system. But to bring about structural change, lasting change, awareness is not enough. It requires changes in law, changes in custom," Obama said at Howard.
Indeed, White House officials have engaged with the movement with success; at the February meeting, Obama and DeRay Mckesson had a back-and-forth over the role the federal government could play in standardizing police use of force. And yet, Obama's speech at Howard was designed to "encourage" even more activists inside the movement to commit themselves to meaningful action, a White House official familiar with Obama's thinking said.
In many ways, the speech was a clear signal to activists that, despite their chafing at Obama's repeated urging that they work within the system, he isn't going to change course, even after Aislinn Pulley, a cofounder of Black Lives Matter Chicago, publicly protested a meeting with the president in February.
Pulley called work to end police brutality and institutional racism a "false narrative" and said the government officials are "perpetrators and enablers" to whom her appearance would amount to "political cover."
White House officials were "livid," according to a source who attended the meeting. Pulley had originally accepted the invitation to participate in the summit, only to pull out and level harsh criticism at the effort.
"They had worked hard to earn her trust," the senior administration official said of White House staffers who engaged with Pulley. "And initially she has indicated that she would come and then she changed her mind. But that's on her. She gets to make that decision for herself. And we're hopeful at some point she will see it to be in the interest of her movement to engage with us. But there are plenty of people who do show up and do engage and that gives us hope."
After the Pulley episode, many observers wondered how committed Obama was to continuing the dialogue. But senior adviser Valerie Jarrett has in recent weeks implored members of the office of public engagement not to give up on a more radical wing of the movement, a group which has clashed with political groups, shouted down presidential candidates, decried the current political system as oppressive and sharply critiqued Obama himself.
"We believe in engagement and bringing as many diverse voices to the table as possible," a senior administration official said in an email to BuzzFeed News. "And we have seen many activists come to the table and engage with us and we look forward to continuing and broadening our relationship with the activist community."
The notion of engaging with institutions of power has been a hallmark of Obama's advice to the Black Lives Matter movement, including in answer to a question in London in which he warned against simply fighting the system.
"Once you've highlighted an issue and brought it to people's attention and shined a spotlight, and elected officials or people who are in a position to start bringing about change are ready to sit down with you," Obama said then. "Then you can't just keep on yelling at them. You can't refuse to meet because that might compromise the purity of your position."
A senior administration official acknowledged that the White House isn't backing down from its strategy, saying "since day one President Obama and his team have widely opened the door to the White House for the purpose of civil engagement. As the president noted in his Howard commencement speech, he believes that real progress requires us all to work together for the greater good."
Darren Sands is a political reporter for BuzzFeed News and is based in New York.
Contact Darren Sands at darren.sands@buzzfeed.com.


This looks interesting although it seems to lean a bit heavily on the whole fragrance = emotion thing. Maybe I'll check it out.
You can reserve a visit here. If you can't get to NYC you can read about it here or watch the video here.
Avery -- You should totally check it out. I'd love to read what you think about it. And you're already kind of in the neighborhood, right?
I totally did. Stay tuned for a report.


*Clear acrylic handles. *½" Connections- 8" spread, easy to install.
*Fits all standard kitchen sinks- ½" IPS Connections- Clear acrylic handles- 8" spread, installation instructions included- limited 10 year warranty.
8" spread, installation instructions included- limited 10 year warranty.


Ilulissat in Greenland does not have a huge choice of hotels so I opted to stay at the hotel Icefiord which looked to be in a decent location and had the added bonus of overlooking the Ilulissat Icefiord.
Location - The Hotel Icefiord is situated in the western part of the town and is around a 15 minute walk from the World of Greenland tour office (This is where most trips will start from). The airport is not in walking distance but a transfer will be included in your package so that will not be an issue. Generally I found the hotel to be conveniently located and within walking distance of all the main sights including the hiking trails.
Accommodation - My room was fairly basic but spacious and clean, and more than I needed really. There was a comfortable bed, a flat screen TV, plenty of storage and cupboards and free tea and coffee. The bathroom had a good sized shower and ice cold drinkable tap water which was a change from the sulphur tinged water in Iceland. It also had a plentiful supply of towels and toiletries.
Evening Meal - I had several meals at the hotel restaurant and was really pleased with the quality. Although fairly pricy (It was still average for eateries in Greenland) the quality and taste of the food was superb. I enjoyed both beef and Musk steaks which were cooked to perfection and served with some delicious fried potatoes. It is advisable to reserve a table in advance as there are a limited amount of restaurants in Ilulissat, all of which fill up quickly.
Breakfast - Breakfast was a buffet meal with various options. Although not the greatest selection of food, I still enjoyed cereal, cold meats, sausage and bacon and fresh cobs with butter. There was also fruit juices and tea & coffee.
Generally I was more than pleased with the Hotel Icefiord - It wasn't outstanding in any one area but more than adequate all around and I couldn't have asked for more. The staff were polite and spoke fluent English, while the terrace that overlooked the bay was a beautiful place to relax and enjoy a drink. If you are travelling to Ilulissat I would recommend this hotel without hesitation!


The 2018 European LLVM Developers Meeting will be held at the Bristol Marriott Hotel City Centre, Bristol, UK, April 16 and 17, 2018.
The meeting serves as a forum for LLVM, Clang, LLDB and other LLVM project developers and users to get acquainted, learn how LLVM is used, and exchange ideas about LLVM and its (potential) applications. The conference will be two full days that include technical talks, BoFs, hacker's lab, tutorials, a poster session and a reception.
For more information, go to the conference website.
Thanks to the conference organizers for their mention of the 2018 ACCU conference occurring the previous week.


Today, many people are wondering if leasing or purchasing is the best when it comes to new cars. Both leasing and buying have wonderful advantages as well as their own disadvantages.
No matter if you plan to lease or purchase you will be more than likely making monthly payments. Leasing does give you a bit of relief in that your payments will be lower than if you were purchasing, however, you will never own the car. If you purchase, your monthly payment will be higher, but you will own the car after all your payments are complete.
You may hear that leasing is better by some of your family or friends and then you will hear some that buying is better. Before you make your own decision, you should learn the similarities and difference of both options.
When you decide to purchase a car, most people have to apply for a loan. Not too many people have enough cash lying around to pay for a brand new car; therefore, they apply for a loan. The loan consists of two charges, the principal charge and the finance charge. The principal charge is the total value of the car and the finance charge is the interest that is placed on the loan. You should shop around and find the best possible interest rate before applying for a car loan; you may find that you can receive lower monthly payment through your own bank than with the car dealership.
Remember, when you are making monthly payments to purchase a car, the car is yours as long as your make your payment. You can travel any place you would like to go and make any additions that you would like to your car or other customizations. It is your car unless you default on the loan and then the car can be repossessed by the loan company that gave you the loan to purchase the car.
There are a couple of disadvantages to purchasing a new car. Your monthly payment will be higher than leasing, is the number one disadvantage and the other is that you will have to sell the car if you do not want the car any longer.
When leasing a car you will have to make monthly payments that consist of two charges, a depreciation charge and a finance charge. The depreciation charge is the price that the car has gone down in price while you were using the car, this will give the dealership compensation for the depreciation value that you used. The finance charge is of course the interest that you are paying on the term of the lease.
When you lease a car, you will never become owner of the car. You cannot make any additions or customizations to the car. You cannot sell the car or trade it in if you get tired of it. If you decide to take it back and the lease is not up, you will have stiff penalties to pay. You will also not be able to go any place you like. When you are leasing a car, you are given a certain amount of miles that you are allowed to drive during the term of your lease. If you go over the amount of miles specified then you will have to pay for every mile that you go over the original amount.
The short-term advantages of leasing are of course the fact that the monthly lease payment is quite a bit cheaper than monthly payments for purchasing. A matter of fact the payments are between 30 to 60 percent lower than most loan payments.
However when it comes to long-term advantages purchasing wins hands down, if you do not plan on keeping the leased car after the term of the lease. If you plan on returning the car at the end of the term of the lease and leasing another car, of course, you will end up paying more in the long run. You will own your car after you have paid off your loan, however, with a leased car you will have to lease another one and another one compared to driving the same owned car over the same amount of time.
Making this decision is a personal one and if you prefer to drive a new car quite often, then leasing will give you this chance even if you will never own the car. You can always sport around in a brand new car.
About the Author: Dennis is the owner of Mr. Car Quote which has car buying tips and a car forum.


Each junior golfers' path to play collegiate golf is unique. For some, it takes more time to hone their game and prepare for the rigors of being a student-athlete. For those needing more time, Bishops Gate Golf Academy (BGGA) offers a Post Graduate program designed to do just that.
BGGA's program meets the growing demand of specialized training for students who are seeking growth before moving forward with future academic and athletic plans. Many graduating seniors today are not ready physically, technically and/or competitively to excel at their desired level of college sports.
In 2016-17 year, BGGA had 12 Post Grad athletes, all with the same goal of making the most of their gap year. Now that the year has come to an end, all Post Grad students have been placed at universities and many are on college golf teams for the fall semester making for a 100% success rate.
Through the Post Grad program, students received a half day of golf coaching, unlimited golf facility access, golf specific performance training, mental training as well as college planning and placement assistance.
The college placement process is complicated. There are a lot of rules and connections to golf programs are important. That's where BGGA plays a significant role.
When your road to playing college golf doesn't take the turn you want, this may be the program that gets you back on the right path.


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Q: strtod() ambiguity: format error and zero conversion I'm considering to use the strtod() function to convert a string to a double in C, which siganture is:
double strtod (const char* str, char** endptr);

being str the string to be converted and endptr a pointer to be set to to the first character after the number.
Documentation says:

On success, the function returns the converted floating point number as a value of type double.
  If no valid conversion could be performed, the function returns zero (0.0).

Thus, as far as I understand, it is not possible to detect situations in which str has a format error (e.g. "foo", "-3ab" or "3o3") from situations in which the str represents 0 (e.g. "0", "0.000", "-0.0", "0e10").
How strtod() function could be used avoiding that problem in zero conversion?
EDIT: I have seen a similar question here. However, I think I'm not asking the same, as my question is about the ambiguity problem between 0 and wrongly formated string, while that other post is about detecting formating errors in general.

A: That's precisely why you have endptr. If after the call endptr == str, then no number was parsed.

A: Also, if you want to detect other errors, like "3.14xyz", strtod will return 3.14, but endptr will point to the 'x', so, if after strtod, 'endptr' is not pointing to 'str' AND 'endptr' points to only whitespace (or EXACTLY 0 if you want to be strict), then 'str' is indeed a valid float. Also errno is set to ERANGE if the string 'str' would provoke an overflow.
Here is an implementation of it:
bool isDouble(const char* s)
{
  char* rest = (char*) s;

  strtod(s, &rest);

  if ((rest == s) || (errno == ERANGE))
    return false;

  // If all WS after the last char used in the conversion, then the string is considered a 'clean float'
  while ((*rest == ' ') || (*rest == '\t') || (*rest == '\n'))
    ++rest;

  return (*rest == 0);
}

redpajama_set_name: RedPajamaStackExchange


text:
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Green Bay Packers at Minnesota Vikings - 12/23/19 NFL Pick, Odds, and Prediction
Photo by Benny Sieu-USA TODAY Sports
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Randy Chambers December 22, 2019 7:23 pm
Green Bay Packers (11-3) at Minnesota Vikings (10-4)
NFL Football: Monday, December 23, 2019 at 8:15 pm (U.S. Bank Stadium)
The Line: Minnesota Vikings -5 -- Over/Under: 47
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The Green Bay Packers and Minnesota Vikings meet Monday in week 16 NFL action at U.S. Bank Stadium.
The Green Bay Packers look for a statement victory after winning eight of their last nine games. The Green Bay Packers have won four of their last six road games. Aaron Rodgers is completing 63.3 percent of his passes for 3,463 yards, 24 touchdowns and two interceptions. Rodgers has two or more touchdown passes in 13 of his last 24 games. Davante Adams and Marquez Valdes-Scantling have combined for 1,215 receiving yards and six touchdowns while Aaron Jones has 45 receptions. The Green Bay Packers ground game is averaging 106.5 yards per contest, and Jones leads the way with 830 yards and 14 touchdowns. Defensively, Green Bay is allowing 20.2 points and 371.3 yards per game. Blake Martinez leads the Green Bay Packers with 140 tackles, Preston Smith has 11.5 sacks and Kevin King has four interceptions.
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The Minnesota Vikings also look for a big division win after winning four of their last five games. The Minnesota Vikings have won all six home games this season. Kirk Cousins is completing 70.5 percent of his passes for 3,481 yards, 25 touchdowns and five interceptions. Cousins has two or more touchdown passes in 12 of his last 23 games. Stefon Diggs and Dalvin Cook have combined for 1,592 receiving yards and five touchdowns while Adam Thielen has 30 receptions. The Minnesota Vikings ground game is averaging 135.9 yards per contest, and Cook leads the way with 1,135 yards and 13 touchdowns. Defensively, Minnesota is allowing 18.5 points and 338.9 yards per game. Eric Kendricks leads the Minnesota Vikings with 110 tackles, Danielle Hunter has 13.5 sacks and Anthony Harris has five interceptions.
The Packers are 5-2 ATS in their last 7 road games, 2-6 ATS in their last 8 Monday games and 6-1 ATS vs. a team with a winning record. The Vikings are 5-2 ATS in their last 7 vs. NFC, 36-15-1 ATS in their last 52 home games and 2-11 ATS in their last 13 Monday games. The under is 5-1 in Packers last 6 games overall. The over is 5-1 in Vikings last 6 games overall. The Packers are 1-4 ATS in their last 5 meetings in Minnesota. The home team is 6-1 ATS in their last 7 meetings. The under is 5-0 in the last 5 meetings in Minnesota and 8-2 in the last 10 meetings.
The Green Bay Packers won the previous meeting back in September but have lost six of the last eight meetings against the Vikings overall. The Packers are finding ways to win games, but they have some issues with receivers not being able to get open and the offensive line not holding up. There's been many games this season where the Packers have barely escaped the jaws of defeat. The Minnesota Vikings defense seems to be getting better each week and Cousins is taking care of the football, throwing just three picks since September 22. Free points and Aaron Rodgers is tempting, but the Vikings are clearly the better and more balanced team. I'll take the Vikings by a touchdown.
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Randy's Pick Minnesota Vikings -5
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Every year in my neighborhood, the main drag of Clark St becomes a street fair: Midsommarfest. Swedish name, but mostly that means booths selling corn on the cob, hippie dresses, kitschy antiques, plastic cups of cheap beer and good sangria. Plus local bands. Really local, as in the woman from my favorite coffee shop, singing and playing guitar instead of bringing me tea and apple pie.
This morning I realized I'd been to a lot of them.
'05:My first Chicago year, about to become a teacher but thinking of sun and sand, hanging on Matt's arm, downing frozen margaritas, a dizzy dream.
'06: Just finished exhausting teaching year, meet up with Matt; we sit on the curb, he hands me a pin with an Eiffel Tower charm that he found, we share falafel and talk for the first time since we broke up.
'07: The bad year; when all of my joints locked up thanks to antibiotics for a UTI (thanks, Urgent Care Clinic!). I only realized what was happening when the word "lupus" kept surfacing in my brain until I googled it and found that a sulfa drug allergy creates a temporary autoimmune response similar to lupus. Holy crap.
'08: Working at a booth for the theater. After a rainstorm, assembling our tangled canopy with K = short fuses, tent poles in the face. Saved by the metal cover band and Caleb's art show.
'09: Gray day, uncrowded = good for wandering, found chips of metal in my frozen margarita from Simon's, ran into Dan Bloom, so very badly wanted to smuggle out Ginger, with the sad eyes, from the adopt-a-dog booth, smuggled out a cup of sangria instead, for sipping on the back porch.


text:
/ Virtual Reality by Tony Lopez
Is it right?
posted June 21, 2019 at 12:30 am by Tony Lopez
"Perhaps America can share its satellite photos of how the incident of June 9 actually happened."
In Tokyo, last May 31, 2019, addressing the 25th Nikkei Conference on "The Future of Asia," President Rodrigo Duterte asked the existential question: "Is it right for a country to claim the whole ocean?"
He was referring to the vigorous, if violent, attempts by China to claim as a sphere of influence, if not outright sovereignty, over the vast entirety of the South China Sea. In part of that sea, the Philippines has a globally recognized 200-mile exclusive economic zone. Manila refers to its EEZ as the West Philippine Sea.
In his Nikkei speech, Duterte declared:
"We are friends with China. I am the moderator country for China and the ASEAN. And the last thing I heard was that the [code on the] conduct of the sea is just about two years away."
"I would not want to impose my own—it would be my last word—but if I get to get a chance to visit Beijing again, I'll try to talk to President Xi Jinping."
"And the longer it takes for the issue to be there it is always a flashpoint for trouble. And not only that because of the absence of the conduct of the sea, France, Britain, America are testing the waters."
"This is not a testing of waters of temperature, my God. It is really testing who can fire the first shot. And I am sad and bewildered, not angry because I cannot do anything."
"I just hope that China would come up with (the code of) conduct of the sea soon and somebody should reach out to the United States. Because if you leave it to them to talk, nothing will happen. There is so much animosity covered by sweet-talking about how they desire to have an agreement. But nobody is pushing and the intrusions as far as China is concerned it's in their waters."
"I love China. It has helped us a bit. But it behooves upon us to ask: Is it right for a country to claim the whole ocean?"
As you probably know, the South China Sea sprawls over 3.5 million square kilometers. Using its so-called nine-dash line map, Beijing claims up to 90 percent of the South China Sea, a claim debunked by the Philippines winning in 2016 an Arbitral Tribunal ruling which said China has no historical rights based on the nine-dash line. The tribunal, however, refused to rule on the issue of sovereignty, and accordingly, did not delineate boundaries covering contested islands in the South China Sea.
History, however, is written by the victors. Might is right. So per its version of history, China owns most, if not all, of the South China Sea. Hence, the Duterte question: "Is it right for China to own the whole ocean?"
According to International Hydrographic Organization Limits of Oceans and Seas, 3rd edition (1953), the South China Sea is south of China, west of the Philippines, east of Vietnam, and east of the Malay Peninsula and Sumatra, up to the strait of Singapore.
The Philippines exercises EEZ, not sovereignty, up to 2.0 million square kms. EEZ means the Philippines owns the resources (fishery, marine life, minerals, etc) under the sea and above the sea, but does not own the area as a sovereign territory, meaning other countries can fish in its waters,—but not to reclaim islands (as China has done) or attack fishing vessels of other countries in the area (as China has done).
An estimated US$3.37 trillion worth of global trade passes through the South China Sea annually, for a third of the global maritime trade; 80 percent of China´s energy imports and 39.5 percent of China´s total trade passes through the South China Sea, according to Wikipedia.
On near midnight of June 9, Beijing probably gave the answer to Duterte's question: At the Reed Bank or Recto Bank, a heavily reinforced Chinese vessel suddenly appeared in the dark horizon and rammed the hapless wooden boat of 22 Filipino fishermen, all but one of whom were asleep at the time of the incident.
Adding insult to injury, and against the law on conduct of vessels in open sea, the Chinese ship Yuemaobinyu, after making sure it had sunk the Philippine boat, Gem Ver, left in a huff, in classic hit-and-run fashion. The 22 Filipino fishermen waded and gasped for dear life in the middle of the ocean in total darkness and biting cold for about six hours before they were fished out of the water by a Vietnamese fishing boat. The Filipinos were fed rice and noodles, given dry clothes, and made to rest until another Filipino boat came to retrieve them.
Now, Duterte may ask: Is it right for an industrial-grade Chinese boat to ram a smaller wooden boat with 22 sleepy fishermen in the middle of the night, and abandon them? In a place that is claimed by the Philippines as its property?
Only the conscience of Beijing and the patience of Duterte can answer that. In the meantime, maybe, America can share its satellite photos of how the incident of June 9 actually happened.
biznewsasia@gmail.com
Topics: 25th Nikkei Conference , Rodrigo Duterte , South China Sea , ASEAN


Trips to the hospital are not always fun, so it can help a lot when we have a doctor with a great sense of humor. When used correctly, humor in a medical setting can be beneficial for both doctors and patients because it creates trust and comfort between the 2.
We at The Elite Indian have compiled some hilarious photos of medical experts flexing their funny bones in different situations, and showing that a great sense of humor can also go hand in hand with stressful occupations.
1. Rinse your hands with warm water.
2. When the purge uprising happens, he'll be ready.
3. Remember to Fortnite after every meal.
4. That stool looks pretty sturdy.
5. Brace yourselves, Halloween is coming.
6. All hands were on deck to finish this Christmas tree.
7. Here's a dentist's take on the poop emoji.
8. A little bit overkill, but my surgeon is a diva.
9. It's awesome when a dentist cares about making kids comfortable.
10. Let the world in and enjoy the view.
11. Hopefully this doctor has womb to spare.
13. Life is short, smile while you still have teeth.
14. What did you have for lunch?
17. These nuts are non-refundable.
19. If you're not this sassy, what are you doing with your life?
Which photo tickled your funny bone? Do you have your own experience with a witty medical practitioner? Let us know in the comments below.


Sure, you can buy your booze from some big box liquor store—but that just means you're feeding the corporate machine, so why not help out the little guy? It helps when your local liquor store sports an insanely large collection of just about everything you could ever want to drink. Track down those hard-to-find beers for your hipster friends, that perfect scotch for your dad or whatever your poison. It doesn't hurt that the super friendly and knowledgeable staff can help you find the perfect way to lessen the stresses of the day with some liquid courage.


Josh likes his flash like he likes his icebergs. As much as the story on paper (or on screen) is important, what is just as important to him is what is not stated. In other words, the story behind the story. Does what you write make him wonder at what's just below the surface? How big is the story there? In addition to his icebergs he loves a good, unexpected end. To top it off, he loves stories that have fun and play with the genre.
Josh Bertetta holds a Ph.D. in Mythological Studies with a degree emphasis in depth psychology from Pacifica Graduate Institute and currently teaches in the Religious Studies department at a private university in central Texas. He began writing flash fiction in August 2014 after completing his first novel and wanted to keep working on his writing chops. Quickly falling in love with flash and the flash community, he finds flash fiction a fun way to engage the imagination and play with genre, voice, and style. He is a father of three boys and loves to garden.