text:
\section{Introduction}

Computing\footnote{A preliminary version of this work appears in \cite{legat2020set}.} controlled invariant set is paramount in many applications~\pcite{blanchini2015set}.
Indeed, the existence of a controlled invariant set is equivalent to the stabilizability of a control system~\pcite{sontag1983lyapunov} and
a (possibly nonlinear) stabilizable state feedback can be deduced from the controlled invariant set~\pcite{barmish1985necessary}.

The stabilizability of a linear time-invariant (LTI) control system is equivalent
to the stability of its uncontrollable subspace (which is readily accessible
in its Controllability Form) \cite[Section~2.4]{wonham1974linear}.
Indeed, the eigenvalues of its controllable subspace can be fixed to any value
by a proper choice of linear state feedback.
The resulting controlled system is stable hence an invariant ellipsoid can be
determined by solving a system of linear equations~\pcite{Liapounoff1907}.
This set is also controlled invariant for the control system.
When a control system admits an ellipsoidal controlled invariant set,
it is said to be \emph{quadratically stabilizable}.
When there exists a linear state feedback such that the resulting autonomous
system admits an ellipsoidal invariant set, it is said to be
\emph{quadratically stabilizable via linear control}.

While the stabilizability of LTI control systems is equivalent to their
quadratic stabilizability via linear control,
it is no longer the case for \emph{uncertain} or \emph{switched} systems \pcite{petersen1985quadratic}.
Furthermore, it is often desirable for \emph{constrained} systems to find a controlled invariant set of maximal
volume (or which is maximal in some direction~\pcite{ahmadi2018robust}).
For such problem, the method detailed above is not suitable as it does not
take any volume consideration but more importantly, the maximal volume
invariant set may not be an ellipsoid and may not be rendered stable via
a linear control.
For this reason, the Linear Matrix Inequality (LMI) \eqref{eq:controlled_LMI} was devised to encapsulate the
controlled invariance of an ellipsoid via linear control \cite[Section~7.2.2]{boyd1994linear}
and the conservatism of the choice of linear control was analysed \pcite{sontag1983lyapunov}.
As the linearity of the control was found to be conservative for uncertain systems \pcite{petersen1985quadratic},
the LMI \eqref{eq:invLMI} was found to encapsulate controlled invariance of an ellipsoid
via \emph{any} state-feedback \cite{barmish1985necessary}.

These LMIs have had a tremendous impact on control, but unfortunately the approach is limited to ellipsoids due to its algebraic nature.
We reinterpret it in a geometric/behavioural framework,
based on convex analysis,
which allows us to formulate a general condition for the controlled invariance of arbitrary
convex sets via any state-feedback in \theoref{invconvex}.
While this condition reduces to \eqref{eq:invLMI} for the special case of ellipsoids,
it provides a new methods for computing controlled invariant convex sets with convex polynomial and piecewise quadratic support functions.

In \secref{alg}, we review the classical LMI's for the invariance and controlled invariance of ellipsoids and discusses the challenges for its generalization to sublevel sets of polynomials.
In \secref{geometric}, we develop a generic condition of control invariance
for continuous-time systems using our geometric approach.
We particularize it for ellipsoids (resp. sets with polynomial and piecewise quadratic support functions)
in \secref{ell} (resp. \secref{poly} and \secref{pell}).
We illustrate these new results with a numerical example in \secref{exem}.

\paragraph*{Reproducibility}
The code used to obtain the results is
published on codeocean \pcite{legat2021continuousCO}.
The set programs are reformulated
by
SetProg \pcite{legat2019set}
into
a Sum-of-Squares program
which is
reformulated into a semidefinite program
by SumOfSquares \pcite{weisser2019polynomial}
which is
solved by Mosek v8 \pcite{mosek2017mosek81034} through MathOptInterface \pcite{legat2020mathoptinterface}.

\section{Algebraic approach}
\label{sec:alg}

Computing an ellipsoidal invariant set for an autonomous system
$\dot{x} = Ax$ where $A \in \R^{n \times n}$ consists in searching for an ellipsoidal set
\[ \mathcal{E}_P = \{\, x \in \R^n \mid x^\Tr P x \leq 1 \,\} \]
that satisfies the \emph{Nagumo condition} \cite[Theorem~4.7]{blanchini2015set}:
$x^\Tr PAx \leq 0$ for any $x \in \mathbb{R}^n$.
The Nagumo condition for ellipsoids is equivalent to the
LMI:
\begin{equation}
  \label{eq:uncontrolled_LMI}
  A^\Tr P + PA \preceq 0.
\end{equation}
which allows to search for ellipsoidal invariant sets using
semidefinite programming.

Consider the continuous-time control linear system
\begin{equation}
  \label{eq:cs}
  \dot{x} = Ax + Bu
\end{equation}
where $A \in \R^{n_x \times n_x}$ and $B \in \R^{n_x \times n_u}$
with the following definition of invariance.

\begin{mydef}[Controlled invariant set]
  \label{def:cis}
  A set $S$ is \emph{controlled invariant} for system \eqref{eq:cs}
  if for any state $x_0 \in S$, there exists a control $u(t)$ such that
  the trajectory with initial state and control $u$ remains in $S$.
\end{mydef}

With the presence of the control $u$ in the system,
the Nagumo condition becomes:
\begin{equation}
  \label{eq:existential}
  \forall x \in \R^{n_x}, \exists u \in \R^{n_u}, x^\Tr P(Ax + Bu) \leq 0.
\end{equation}
The control term $u$, or more precisely the existential quantifier $\exists$ prevents
us to transform this into an LMI directly.

Fixing the control to a linear state feedback $u(x) = Kx$ for some matrix $K$ allows
to fallback to the case of autonomous system $\dot{x} = (A + BK) x$.
Then, the invariance condition can be formulated
as the \emph{Bilinear Matrix Inequality} (BMI):
\[ A^\Tr P + PA + K^\Tr B^\Tr P + PBK \preceq 0 \]

While the matrix inequality is bilinear in $K$ and $P$, and BMI's are NP-hard to solve in general \pcite{toker1995np},
a clever algebraic manipulation
allows to reformulate it as a
Linear Matrix Inequality (LMI) in $Q := P^{-1}$ and $Y := KQ$,
where the sought controlled invariant ellipsoid is given by $\mathcal{E}_P$,
see e.g. \cite[Section~7.2.1, Section~7.2.2]{boyd1994linear} and \cite[Section~4.4.1]{blanchini2015set} for more details.
The linear matrix inequality is
\begin{equation}
  \label{eq:controlled_LMI}
  QA^\Tr + AQ + Y^\Tr B^\Tr + BY \preceq 0.
\end{equation}

Because the algebraic manipulation which allows to reformulate the BMI into
an LMI is done at the level of matrices, it is not clear how this
approach can be generalized to other families of sets such as the
ones considered in \secref{poly} and \secref{pell}.
Moreover, searching for ellipsoidal controlled invariant sets may be rather
restrictive and the conservativeness is amplified for the class of hybrid systems.

One attempt to generalize it to controlled invariant sublevel sets of polynomials of degree $d$
is developed in \cite{Prajna2004}.
While the method allows to consider systems defined by polynomial equations,
we show below that for linear systems, it
has significant restrictions for $d > 2$
and the case $d = 2$ reduces to the ellipsoidal case given by \eqref{eq:controlled_LMI}.
This suggests that the methods is essentially restricted to compute invariant sublevel sets of polynomials
of a most twice the degree of the polynomial equations.
On the other hand, the condition we find in \cororef{poly} is necessary and sufficient.
The remaining of this section particularize the approach for linear systems,
more details can be found in \cite{Prajna2004}.

Let $x_{[d]}$ denote the vector of all monomials of degree $d$ with the variables $x_i$.
Consider the set $\setbuild{x}{x_{[d]}^\Tr P x_{[d]} \le 1}$ for a symmetric positive definite matrix $P$
and a state feedback of the form
$u(x) = K_1(x)x + K_2(x)x_{[d]}$
where $K_1(x), K_2(x)$ are matrices of the appropriate dimensions whose
entries are polynomials in $x$.
The invariance of the set for the autonomous system
$\dot{x} = Ax + BK_1(x)x + BK_2(x)x_{[d]}$
is equivalent to the nonnegativity of the polynomial
\[
  x_{[d]}^\Tr P M(x) B K_2(x) x_{[d]} + x_{[d]}^\Tr PF(x) x
\]
where $F(x) = M(x)BK_1(x) + M(x)A$
and $M(x)$ is the jacobian of the transformation $x \mapsto x_{[d]}$.
This polynomial can be rewritten in the matrix form:
\begin{equation}
  \label{eq:polyneg}
  \begin{bmatrix}
    x_{[d]}\\
    x
  \end{bmatrix}^\Tr
  \begin{bmatrix}
    P M(x) B K_2(x) & PF(x)\\
    0 & 0
  \end{bmatrix}
  \begin{bmatrix}
    x_{[d]}\\
    x
  \end{bmatrix}.
\end{equation}
The following is therefore a sufficient condition for the invariance:
\begin{equation}
  \label{eq:polybmi}
  \forall x \in \R^n, \quad
  \begin{bmatrix}
    2P M(x) B K_2(x) & PF(x)\\
    F^\Tr(x)P & 0
  \end{bmatrix} \preceq 0.
\end{equation}
After a similarity transformation with the block diagonal matrix
$\BlockDiag(P^{-1}, I_{n_x})$ where $I_{n_x} \in \R^{n_x \times n_x}$ is the identity matrix, the condition is rewritten as:
\begin{equation}
  \label{eq:polyLMI}
  \forall x \in \R^n, \quad
  \begin{bmatrix}
    2M(x) B Y(x) & F(x)\\
    F^\Tr(x) & 0
  \end{bmatrix} \preceq 0
\end{equation}
where $Y(x) = K_2(x)P^{-1}$.

By \propref{block}, for \eqref{eq:polyLMI} to hold we need all entries of $F(x)$
to be zero polynomials which is quite restictive.
The example below discusses the
conservativeness of \eqref{eq:polyneg} compared to \eqref{eq:polybmi}.

\begin{myexem}
  Consider the autonomous system $\dot{x} = -x$ and the invariant set $\setbuild{x}{x^4 \le 1}$.
  The condition \eqref{eq:polyneg} is satisfied:
  \[
    \begin{bmatrix}
      x^2\\
      x
    \end{bmatrix}^\Tr
    \begin{bmatrix}
      0 & -2x\\
      0 & 0
    \end{bmatrix}
    \begin{bmatrix}
      x^2\\
      x
    \end{bmatrix}
    =
    -x^4
  \]
  while the matrix
  \[
    \begin{bmatrix}
      0 & -2x\\
      -2x & 0
    \end{bmatrix}
  \]
  of conditions \eqref{eq:polybmi} and \eqref{eq:polyLMI} is indefinite for any nonzero $x$.
\end{myexem}

\section{Geometric approach}
\label{sec:geometric}

In this section we derive a characterization of
the controlled invariance of a closed convex set
under the form of an inequality for its support function.
We start by showing the equivalence of the notion of invariance with
another class of systems that directly models
the geometric behaviours of the trajectories of
control systems with unconstrained input.

Consider the continuous-time \emph{algebraic} linear systems:
\begin{equation}
  \label{eq:as}
  E\dot{x} = Cx.
\end{equation}
with the following definition of invariance.

\begin{mydef}[Invariant set for an algebraic system]
  \label{def:is}
  A set $S$ is \emph{invariant} for system \eqref{eq:as}
  if for any state $x_0 \in S$, there exists a trajectory of the system that
  remains in $S$.
\end{mydef}

Note the use of ``there exists'' instead of ``for all'' in the definition of invariance
as both versions exists; see \cite[Remark~4]{legat2020sum} for more details.

\begin{myprop}
  \label{prop:projB}
  Given a subset $\mathcal{S} \subseteq \R^n$ and
  matrices $A \in \R^{r \times n}, B \in \R^{r \times m}$, the following holds:
  \[ A \mathcal{S} + B \R^m = \pi_{\Image(B)^\perp}{}^{-1} \pi_{\Image(B)^\perp}{} A \mathcal{S} \]
  where $\pi_{\Image(B)^\perp}{}$ is any projection matrix onto the orthogonal subspace of $\Image(B)$.
  \begin{proof}
    Given $x \in \mathcal{S}$ and $y \in \R^r$, we have $y \in A \{x\} + B \R^m$ if and only if $y - Ax \in \Image(B)$ or equivalently, $\pi_{\Image(B)^\perp} y = \pi_{\Image(B)^\perp} Ax$.
  \end{proof}
\end{myprop}

\begin{myprop}
  \label{prop:csas}
  Consider a control system~\eqref{eq:cs}
  and an arbitrary projection matrix $\pi_{\Image(B)^\perp}{}$ onto the orthogonal subspace of $\Image(B)$.
  A set $S$ is controlled invariant for the control system~\eqref{eq:cs} with $\mathcal{U} = \R^{n_u}$,
  as defined in \defref{cis},
  if and only if
  it is controlled invariant for the algebraic system
  \[ \pi_{\Image(B)^\perp}{}\dot{x} = \pi_{\Image(B)^\perp}{}Ax, \]
  as defined in \defref{is}.
  \begin{proof}
    By \propref{projB}, there exists $u \in \R^{n_u}$ such that $\dot{x} = Ax + Bu$
    if and only if $\pi_{\Image(B)^\perp}{}\dot{x} = \pi_{\Image(B)^\perp}{}Ax$.
    As the input $u$ is unconstrained, the result follows.
  \end{proof}
\end{myprop}

The Nagumo condition
for algebraic systems has the following form.

\begin{myprop
  \label{prop:nagumo}
  A closed set $\mathcal{S}$ is invariant for system~\eqref{eq:as},
  as defined in \defref{is},
  if and only if
  \begin{equation}
    \label{eq:invtang}
    \forall x \in \partial \mathcal{S}, \exists y \in T_S(x), Ey = Ax.
  \end{equation}
\end{myprop}

See \secref{convex} for a brief review of the concepts
of convex geometry needed for the remaining of this section.
The invariance condition \eqref{eq:invtang} can be rewritten in terms
of \emph{exposed faces}.

\begin{mytheo}[Controlled invariance of convex set]
  \label{theo:invconvex}
  A convex set $\Csetvar$ is invariant for system \eqref{eq:as}
  with matrices $C, E \in \mathbb{R}^{r \times n}$
  if and only if
  \begin{equation}
    \label{eq:invface}
    \forall z \in \mathbb{R}^r, \forall x \in F_\Csetvar(E^\Tr z), \la z, Cx \ra \le 0.
  \end{equation}
  \begin{proof}
    As $\Csetvar$ is convex, $T_\Csetvar(x)$ is a convex cone. By definition
    of the polar of a cone, $x \in ET_\Csetvar(x)$ if and only if
    $\langle y, x \rangle \le 0$ for all $y \in \polar{[ET_\Csetvar(x)]}$.
    By \propref{podu}, $\polar{[ET_\Csetvar(x)]} = E^{-\Tr}N_\Csetvar(x)$.
    Therefore, the set $\Csetvar$ is invariant if and only if
    \begin{equation}
      \label{eq:invnorm}
      \forall x \in \partial \Csetvar, \forall z \in E^{-\Tr} N_\Csetvar(x), \langle z, Cx \rangle \le 0.
    \end{equation}
    By \propref{normalsupfun}, we have
    \begin{multline*}
      \{\, (x, z) \in \partial \Csetvar \times \mathbb{R}^r \mid E^\Tr z \in N_\Csetvar(x) \,\} = \\
      \{\, (x, z) \in \partial \Csetvar \times \mathbb{R}^r \mid x \in F_\Csetvar(E^\Tr z) \,\}.
    \end{multline*}

  \end{proof}
\end{mytheo}

As we show in the remaining of this section, \theoref{invconvex} allows to reformulate
the invariance as an inequality in terms of the support function.
This allows to combine the invariance constraint with other set constraints that can be formulated in terms of support functions.
Moreover, for an appropriate family of sets, also called \emph{template}, the set program
can be automatically rewritten into a convex program combining all constraints using \emph{set programming} \cite{legat2019set, legat2020set}.
For this reason, we only focus on the invariance constraint and do not detail how to
formulate the complete convex programs with the objective and all the constraints needed to obtain the results of \secref{exem} as these problems are decoupled.

This allows to formulate the invariance of a convex set in terms of its support function
if it is differentiable.
We generalize this result with a relaxed notion of differentiability in \theoref{pdiff}.

\begin{mytheo}
  \label{theo:invexposed}
  Consider a nonempty closed convex set $\Csetvar$ such that
  $\supfun{\cdot}{\Csetvar}$ is differentiable.
  Then $\Csetvar$ is invariant for system \eqref{eq:as}
  with matrices $C, E \in \mathbb{R}^{r \times n}$
  if and only if
  \begin{equation}
    \label{eq:invgrad}
    \forall z \in \mathbb{R}^r, \la z, C \grad \supfun{E^\Tr z}{\Csetvar} \ra \le 0.
  \end{equation}
  \begin{proof}
    By \propref{exposed}, $F_\Csetvar(E^\Tr z) = \{\grad \supfun{E^\Tr z}{\Csetvar}\}$
    hence \eqref{eq:invface} is equivalent to \eqref{eq:invgrad}.
  \end{proof}
\end{mytheo}

\subsection{Ellipsoidal controlled invariant set}
\label{sec:ell}

In this section, we particularize \theoref{invexposed} to the case of ellipsoids.
Since the support function of an ellipsoid $\mathcal{E}_P$ is $\supfun{y}{\mathcal{E}_P} = \sqrt{y^\Tr P^{-1} y}$,
we have the following corollary of \theoref{invexposed}.
\begin{mycoro}[\cite{barmish1985necessary}]
  Given a positive definite matrix $P$, the ellipsoid $\mathcal{E}_P$ is
  controlled invariant for system~\eqref{eq:as} if and only if
  \begin{equation}
    \label{eq:invLMI}
    CP^{-1}E^\Tr + EP^{-1}C^\Tr \preceq 0.
  \end{equation}
\end{mycoro}

Observe that for the trivial case $\Image(B) = \mathbb{R}^n$ for system~\eqref{eq:cs},
\propref{csas} produces a system~\eqref{eq:as} with $r = 0$ hence the
LMI~\eqref{eq:invLMI} will be trivially satisfied for any $P^{-1}$, which is expected.

In comparison to \eqref{eq:controlled_LMI}, for a system~\eqref{eq:as} with matrices
$C, E \in \mathbb{R}^{r \times n}$, the LMI~\eqref{eq:controlled_LMI} has
size $n \times n$ while the LMI~\eqref{eq:invLMI} has only size $r \times r$.
The characterization of controlled invariance of ellipsoids using \eqref{eq:invLMI}
can also be obtained by applying an elimination procedure to reduce \eqref{eq:controlled_LMI};
see \cite[Equation~(7.11)]{boyd1994linear}.
However, uncertain or switched system may need a nonlinear state feedback to be quadratically
stabilizabilized~\cite{petersen1985quadratic}.
For such systems, \eqref{eq:controlled_LMI} is conservative since it assumes a linear feedback
while \eqref{eq:invLMI} does not assume anything about the feedback.
It was shown in \cite{barmish1985necessary} that if \eqref{eq:invLMI} is satisfied then a stabilizing nonlinear continuous
state feedback can be deduced from the solution $P$.
There is even a closed form for the feedback in case of single input~\cite[Eq.~(15)]{barmish1985necessary}.

\subsection{Polynomial controlled invariant set}
\label{sec:poly}
In this section, we derive the algebraic condition for
the controlled invariance of a set with polynomial support function.
This template is referred to as \emph{polyset}; see \cite[Section~1.5.3]{legat2020set}.
\begin{mycoro}
  \label{coro:poly}
  Given a homogeneous\footnote{A polynomial is \emph{homogeneous} if all its monomials have the same total degree} nonnegative polynomial $p(x)$ of degree $2d$, the set $\Csetvar$
  defined by the support function
  $\supfun{y}{\Csetvar} = p(y)^{\frac{1}{2d}}$
  is invariant for system~\eqref{eq:as}
  with matrices $C, E \in \mathbb{R}^{r \times n}$
  if and only if the polynomial
  \begin{equation}
    \label{eq:invSOS}
    z^\Tr C \grad p(E^\Tr z)
  \end{equation}
  is nonpositive for all $z \in \mathbb{R}^r$.
  \begin{proof}
    We have
    \[
      \grad \supfun{y}{\Csetvar} = \frac{1}{p(y)^{1-\frac{1}{2d}}} \grad p(y).
    \]
    If $p(y)$ is identically zero, this is trivially satisfied.
    Otherwise, $p(y)^{1-\frac{1}{2d}}$ is nonnegative and is zero
    in an algebraic variety of dimension $n - 1$ at most.
    Therefore,

    \eqref{eq:invgrad} is equivalent to \eqref{eq:invSOS}.
  \end{proof}
\end{mycoro}

While verifying the nonnegativity of a polynomial is co-NP-hard,
a sufficient condition can be obtained via the standard Sum-of-Squares programming framework; see ~\cite{blekherman2012semidefinite}.

\subsection{Piecewise semi-ellipsoidal controlled invariant set}
\label{sec:pell}

\cite{johansson1998computation} study the computation of piecewise quadratic Lyapunov functions for continuous-time autonomous piecewise affine systems.
\cite{legat2020piecewise} present a convex programming approach
to compute \emph{piecewise semi-ellipsoidal} controlled invariant sets of discrete-time control systems.
In this section, we show that \theoref{invexposed} provides the corresponding condition for continuous-time.

We recall \cite[Definition~2]{legat2020piecewise} below.
\begin{mydef
  A \emph{polyhedral conic partition} of $\R^n$ is a set of $m$ polyhedral cones $(\mathcal{P}_i)_{i=1}^m$
  with nonempty interior
  such that for all $i \neq j$, $\dim(\mathcal{P}_i \cap \mathcal{P}_j) < n$
  and $\cup_{i=1}^m \mathcal{P}_i = \R^n$.
\end{mydef}

Piecewise semi-ellipsoids have a support function of the form
\begin{equation}
  \supfun{y}{\Csetvar} = \sqrt{y^\Tr Q_i y} \qquad y \in \mathcal{P}_i
\end{equation}
where $(\mathcal{P}_i)_{i=1}^m$ is a polyhedral conic partition.
The support function additionally has to satisfy \cite[(2) and (3)]{legat2020piecewise}
to ensure its continuity and convexity.

\begin{mytheo}
  \label{theo:pdiff}
  Consider a polyhedral conic partition $(\mathcal{P}_i)_{i=1}^m$ and
  a nonempty closed convex set $\Csetvar$
  such that
  \[ \supfun{y}{\Csetvar} = f_i(y) \qquad y \in \mathcal{P}_i \]
  for differentiable functions $f_i : \mathcal{P}_i \to \R$.
  The set $\Csetvar$ is invariant for system~\eqref{eq:as}
  with matrices $C, E \in \mathbb{R}^{r \times n}$
  if and only if, for all $i = 1, \ldots, m$ and $z \in \mathbb{R}^r$

  such that $E^\Tr z \in \mathcal{P}_i$,
  we have
  \begin{equation}
    \label{eq:invpgrad}
    \la z, C \grad f_i(E^\Tr z) \ra \le 0.
  \end{equation}
  \begin{proof}
    Given $z \in \mathbb{R}^r$ such that $E^\Tr z$ is in the intersection of the boundary of $\Csetvar$ and
    the interior of $\mathcal{P}_i$,
    the support function is differentiable at $E^\Tr z$
    hence, by \propref{exposed},
    $F_\Csetvar(E^\Tr z) = \{\grad f_i(E^\Tr z)\}$.
    The condition~\eqref{eq:invface} is therefore reformulated as \eqref{eq:invpgrad}.

    Given a subset $I$ of $\{1, \ldots, m\}$ and
    $z \in \mathbb{R}^r$ such that $E^\Tr z$ is in
    the intersection of the boundary of $\Csetvar$ and $\cap_{i \in I} \mathcal{P}_i$,
    $F_{\Csetvar}(E^\Tr z)$ is the convex hull of
    $\grad \supfun{E^\Tr z}{\Csetvar}$ for each $i \in I$.
    For any convex combination (i.e., nonnegative numbers summing to 1) $(\lambda_i)_{i \in I}$,
    \eqref{eq:invpgrad} implies that
    \[ \la z, C \sum_{i \in I} \lambda_i \grad f_i(E^\Tr z) \ra = \sum_{i \in I} \lambda_i \la z, C \grad f_i(E^\Tr z) \ra \le 0. \]
  \end{proof}
\end{mytheo}

\begin{mycoro}
  A piecewise semi-ellipsoid $\Csetvar$
  is invariant for system~\eqref{eq:as}
  with matrices $C, E \in \mathbb{R}^{r \times n}$
  if and only if the quadratic form
  \begin{equation}
    \label{eq:invpell}
    z^\Tr CP_i^{-1}E^\Tr z + z^\Tr EP_i^{-1}C^\Tr z

  \end{equation}
  is nonpositive for all
  $i = 1, \ldots, m$ and
  $z \in \mathbb{R}^r$ such that $E^\Tr z \in \mathcal{P}_i$.
\end{mycoro}

The condition \eqref{eq:invpell} amounts to verifying the positive semidefiniteness of a quadratic form
when restricted to a polyhedral cone. When this cone is the positive orthant, this is called the
\emph{copositivity} which is co-NP-complete to decide \pcite{murty1987some}.
However, a sufficient LMI is given in \cite[Proposition~2]{legat2020piecewise}
and a necessary and sufficient condition is given by a hierarchy of Sum-of-Squares programs \cite[Chapter~5]{parrilo2000structured}.

\section{Numerical example}
\label{sec:exem}

In this section, we study a simple numerical example to illustrate our new approach.
Suppose we want to compute a controlled invariant set for the control system
\[
  \dot{x} =
  \begin{pmatrix}
    0 & 1\\
    0 & 0\\
  \end{pmatrix}
  x +
  \begin{pmatrix}
    0\\
    1
  \end{pmatrix}
  u
\]
with the state constraint $x \in [-1, 1]^2$ and input constraint $u \in [-1, 1]$.
We represent the state set $[-1, 1]^2$ and its polar in green in \figref{poly} and \figref{pell}.

The union of controlled invariant sets is controlled invariant.
Moreover, by linearity, the convex hull of the unions of controlled
invariant sets is controlled invariant.
Therefore, there exists a \emph{maximal} controlled invariant, i.e., a controlled invariant set in which all controlled invariant sets are included, for any family that is closed under union (resp. convex hull);
it is the union (resp. convex hull) of all controlled invariant sets included in $[-1, 1]^2$.

For this simple planar system, the maximal controlled invariant set
can be obtained by hand, it is
\begin{equation*}
  \setbuild{(x_1, x_2) \in [-1, 1]^2}{x_1x_2 \le 0 \text{ or } |x_1| \le 1 - x_2^2/2}
\end{equation*}
Its polar is given by
\begin{align*}
  & \,\setbuild{x}{x_1x_2 \le 0 \text{ and } |x_1 - x_2| \le 1}\\
  \cup & \,\setbuild{x}{x_1(x_1 - x_2) \le 0 \text{ and } |x_1/2 + x_2| \le 1}\\
  \cup & \,\setbuild{x}{x_2(x_2 - x_1) \le 0 \text{ and } (2x_1 - \sign(x_1))^2 + 2x_2^2 \le 1}
\end{align*}
We represent it in yellow in \figref{poly} and \figref{pell}.

As \propref{csas} requires the input to be unconstrained,
we will consider projections onto the first two dimensions
of controlled invariant sets of the following control system:
\[
  \dot{x} =
  \begin{pmatrix}
    0 & 1 & 0\\
    0 & 0 & 1\\
    0 & 0 & 0
  \end{pmatrix}
  x +
  \begin{pmatrix}
    0\\
    0\\
    1
  \end{pmatrix}
  u.
\]
with the state constraint $x \in [-1, 1]^3$
and no input constraint.

Following \propref{csas}, we consider the algebraic system
\[
  \begin{pmatrix}
    1 & 0 & 0\\
    0 & 1 & 0\\
  \end{pmatrix}
  \dot{x} =
  \begin{pmatrix}
    0 & 1 & 0\\
    0 & 0 & 1\\
  \end{pmatrix}
  x
\]
with the state constraint $x \in [-1, 1]^3$.

While the \emph{maximal} invariant set is well defined,
it is not the case anymore when we restrict the set to belong to the family of ellipsoids,
polysets or piecewise semi-ellipsoids for a fixed polyhedral conic partition
as these families are not invariant under union nor convex hull.
The objective used to determine which invariant set is selected
depends on the particular application.
Let $\mathcal{D}$ be the convex hull of $\{(-1 + \sqrt3, -1 + \sqrt3), (-1, 1), (1 - \sqrt3, 1 - \sqrt3), (1, -1)\}$.
For this example, we maximize $\gamma$ such that $\gamma\mathcal{D}$ is
included in the projection of the invariant set onto the first two dimensions.
We represent $\gamma\mathcal{D}$ in red in \figref{poly} and \figref{pell}.

For the ellipsoidal template considered in \secref{ell}, the optimal solution
is shown in \figref{poly} as ellipsoids corresponds to polysets of degree 2.
The optimal objective value is $\gamma \approx 0.81$.

For the polyset template considered in \secref{poly}, the optimal solution
are represented in \figref{poly}.
The optimal objective value for degree 4 (resp. 6, 10 and 20)
is $\gamma \approx 0.91$.
(resp.
$\gamma \approx 0.93$,
$\gamma \approx 0.96$ and
$\gamma \approx 0.98$).

For the piecewise semi-ellipsoidal template, we consider
polyhedral conic partitions
made of the conic hull of each facet of the polytope with extreme points
\begin{equation}
  \label{eq:sphere}
  (\cos(\alpha)\cos(\beta), \sin(\alpha)\cos(\beta), \sin(\beta))
\end{equation}
where $\alpha = 0, 2\pi/m_1, 4\pi/m_1, \ldots, 2(m_1-1)\pi/m_1$ and
$\beta = -\pi/2, \ldots, -2\pi/(m_2-1), -\pi/(m_2-1), 0, \pi/(m_2-1), 2\pi/(m_2-1), \ldots, \pi/2$.

The optimal objective value for $m = (4, 3)$ (resp. $(8, 5)$) is $\gamma \approx 0.89$ (resp. $\gamma \approx 0.92$).
The corresponding optimal solution is shown in \figref{pell}.

\begin{figure}[!ht]
  \centering
  \polyset{ell}
  \polyset{4}
  \polyset{6}

  \polyset{10}

  \polyset{20}
  \caption
    In blue are the solution for polysets of different degrees.
    The degrees from top to bottom are respectively 2, 4, 6, 10 and 20.
    The green set is the safe set $[-1, 1]^2$, the yellow set is the maximal
    controlled invariant set and the red set is $\gamma \mathcal{D}$.
    The sets are represented in the primal space in left figures
    and in polar space in the right figures.

  }
  \label{fig:poly}
\end{figure}

\begin{figure}[!ht]
  \centering
  \polyset{piece1}
  \polyset{piece2}
  \caption
    In blue are the solution for piecewise semi-ellipsoids for two
    different polyhedral conic partitions.
    The partitions from top to bottom are as described in \eqref{eq:sphere}
    with $m = (4, 3)$ (resp. $(8, 5)$).
    The green set is the safe set $[-1, 1]^2$, the yellow set is the maximal
    controlled invariant set and the red set is $\gamma \mathcal{D}$.
    The sets are represented in the primal space in left figures
    and in polar space in the right figures.

  }
  \label{fig:pell}
\end{figure}

\section{Conclusion}
We proved a condition for continuous-time controlled invariance of a convex set
based on its support function.
We particularized the condition for three templates:
ellipsoids, polysets and piecewise semi-ellipsoids.
In the ellipsoidal case, it reduces to a known LMI,
in the polyset case, it provides a condition significantly less conservative than the existing one\footnote{Indeed, it is equivalent to invariance by \cororef{poly} and we showed in \secref{alg} that the existing approach is quite conservative.}
and in the piecewise semi-ellipsoidal case, it provides the first convex programming approach for continuous-time controlled invariance to the best of our knowledge.

As future work, we aim to apply this framework to other families such as
the \emph{piecewise polysets} defined in \cite{legat2020set}.
Moreover, instead of considering a uniform discretization of the hypersphere
as in \eqref{eq:sphere}, a more adaptive methods could be considered.
The sensitivity information provided by the dual solution of the optimization program could for instance determine
which pieces of the partition should be refined.

Finally, as the discrete-time version of this work developed in \cite{legat2020sum, legat2020piecewise}
also requires the set to be represented by its support function for the
optimization program to be convex,
both these methods and the method of this paper could be combined to compute controlled invariant sets for
hybrid automata using the condition of this paper for the invariance subject to the dynamics of each mode
and the condition of \cite{legat2020sum, legat2020piecewise} for the invariance subject to each reset map.

redpajama_set_name: RedPajamaArXiv


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Deshaun Watson was born on September 14, 1995. Deshaun Watson attended Gainesville High School in Gainesville, Georgia. Deshaun Watson arrived at Gainesville in the Fall of 2010. Gainesville Head Coach Bruce Miller had planned to start a rising junior to quarterback his spread offense, but Watson won the starting spot.
Deshaun Watson was the first freshman quarterback Coach Miller had ever started. During Deshaun Watson career he set numerous state records, including total yards (17,134), total touchdowns (218), career passing yards (13,077), and career passing touchdowns (155).
Deshaun Watson also rushed for 4,057 yards and 63 touchdowns. Watson excelled in his junior year, winning a state championship and earning accolades such as Junior All-American, 2014 Player to Watch, and 2014 Top 100 Recruit. During his sophomore and senior years, he was able to lead the Gainesville High School Football Team to the state semi-finals.
Deshaun Watson received offers from numerous colleges before committing to Clemson. Some of the offers he considered most seriously came from schools such as the University of Alabama, Auburn University, Florida State University, University of Georgia, Georgia Tech, Louisiana State University, Ohio State, University of Oregon, and University of Southern California.
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Hashim Amla vs Reeza Hendricks: Who will open for the Proteas at the World Cup?
The Proteas can only take one of these batsmen for this year's Cricket World Cup. Will they choose experience over potential?
JOHANNESBURG - The much-anticipated announcement of the Proteas 15-man World Cup squad looms larger than ever, with Cricket South Africa set to reveal the men that will represent the country in England on Thursday 18 April.
EWN Sport outlines the batting form of 2 key batsmen in recent times.
Hashim Amla: The Amla has been the most concerning out of all the prospective batsmen that are gunning for a spot in the final World Cup squad. Though he is a veteran of 7,910 runs at an average of 49.74 in ODI cricket for his country, the 36-year-old has battled to replicate the form that saw him revered around the world and broke the shackles of being just a Test grinder that imprisoned him for so long.
His time away from the game to tend to his ailing father hasn't proved to be advantageous for him because his return to cricket for the Cobras in the T20 Challenge has been less than impressive. With scores of 32, 7, 2 and 16 have raised eyebrows on his selection to the squad.
Reeza Hendricks: The Lions top order batsman has blown hot and cold since his magical debut in ODI cricket, scoring a wonderful century. He's had some solid starts in most of the ODI's this past summer but without seizing the opportunity to make a big score and cement his place in the line-up. In the T20 Challenge he has also struggled, recording scores of 0, 0, and 11 suggest he too is under some pressure to make an impression before the World Cup squad is announced.
These two players might prove to be the biggest headaches for convenor of selectors Linda Zondi when he makes a final pick.


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text:
What is a Lutheran
How to meditate on the Passion of Christ
St Paul's African House
St Paul's Lutheran Choir
Concerts in the Chapel
A safe place for all
Lutherans in Africa
Burundi mission
St Pauls Shepparton Lutheran Church
St Paul's Lutheran Church Choir
St Paul's Lutheran Church Choir began around 2008 when former refugees from the Democratic Republic of Congo, Burundi and South Sudan expressed a desire to sing songs of faith in their own languages during regular Sunday worship services.
From these humble beginnings the choir has grown in number and stature. While continuing to see their primary role as serving the congregation each Sunday, the choir's passion for singing and the gospel has seen them performing at the Keith Murdoch Oration in the Sate Library, the Premier's Gala Dinner at Crown Casino and the Melbourne Entertainment Centre, the 'World Famous Spiegeltent' at the Melbourne Arts Centre, the SheppARTon Festival, numerous Emerge festivals, the 2018 Multi-culutral round of Melbourne Victory and more.
At present the choir is made up of people from both D.R. Congo, Burundi and South Sudan and they regularly sing in Kiswahili, Kirundi and English.
If you would like to enquire about the choir's availability to sing at an upcoming event, please email us here

redpajama_set_name: RedPajamaCommonCrawl


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text:
NOWCAST WYFF News 4 at 11pm
Friday Night Hits
MeTV Carolina
Advertise with WYFF4
Greenville County gang member conspired to kill witness in upcoming trial, AG says
Antonio Devon Williamson sentenced for conspiracy to commit murder
Updated: 12:34 PM EDT May 11, 2018
arrest generic.jpg
A member of a Greenville County gang conspired to kill a witness in an upcoming trial, according to U.S. Attorney Beth Drake.Evidence presented at a change of plea hearing showed Antonio Devon Williamson, 30, of Florence, conspired with fellow members of the Elm Street Piru Bloods gang to murder the witness, Drake said. She said the evidence showed the witness had been the victim of an armed robbery, committed by Brian Lewis, the leader of the Elm Street Piru gang for all of South Carolina. Lewis commissioned Williamson and others, as members and leaders in Elm Street Piru, to murder the anticipated witness in order to avoid a lengthy sentence for that crime, Drake said. Law enforcement officials were able to intercept the conspiracy before there any attempt to kill the witness. The case was investigated by the Federal Bureau of Investigation and the Greenville County Sheriff's Office. Assistant U.S. Attorney D. Josev Brewer of the Greenville office handled the case. Williamson was sentenced in federal court in Anderson, for a conspiracy to murder a trial witness and was sentenced to 60 months in prison and three years of supervised release.
GREENVILLE COUNTY, S.C. —
A member of a Greenville County gang conspired to kill a witness in an upcoming trial, according to U.S. Attorney Beth Drake.
Evidence presented at a change of plea hearing showed Antonio Devon Williamson, 30, of Florence, conspired with fellow members of the Elm Street Piru Bloods gang to murder the witness, Drake said.
She said the evidence showed the witness had been the victim of an armed robbery, committed by Brian Lewis, the leader of the Elm Street Piru gang for all of South Carolina.
Lewis commissioned Williamson and others, as members and leaders in Elm Street Piru, to murder the anticipated witness in order to avoid a lengthy sentence for that crime, Drake said.
Law enforcement officials were able to intercept the conspiracy before there any attempt to kill the witness.
The case was investigated by the Federal Bureau of Investigation and the Greenville County Sheriff's Office. Assistant U.S. Attorney D. Josev Brewer of the Greenville office handled the case.
Williamson was sentenced in federal court in Anderson, for a conspiracy to murder a trial witness and was sentenced to 60 months in prison and three years of supervised release.
WYFF Greenville
©2023, Hearst Television Inc. on behalf of WYFF-TV.


Through a string of coincidences, seven authors met in the borderless world of literary e-zines and initiated an international group writing project.
The idea of the project: to develop a future vision of our world caught in a global ecological crisis, a world of flowers turning gray.
Since the start, the story develops steadily in unexpected patterns and shapes, with each author reflecting on his/her own culture and continent while interweaving the different storylines and settings into one novel.
"2028 - a collaborative world novel".


The primary investment objective of the Scheme is to seek to provide long term capital growth by investing predominantly in the JPMorgan Funds – Europe Dynamic Fund an equity fund which invests primarily in an aggressively managed portfolio of European companies. However there can be no assurance that the investment objective of the Scheme will be realised. .


text:
"Yazid was with Jamaah Islamiyah"
A TERRORISM expert told the High Court that the testimony of the allegedly reformed terrorist Hambali pointed to former Internal Security Act detainee Yazid Sufaat being a member of terror group Jemaah Islamiyah.
Dr Rohan Gunaratna of Singapore's Nanyang Technological University said yesterday that this was revealed through the testimonies of Hambali, whose real name is Riduan Isamuddin, and others then detained by United States authorities following the attack on the World Trade Centre in New York on Sept 11, 2001.
"The fact that Yazid joined Jemaah Islamiyah (Malaysia) was established through the testimonies of detainees and debriefings of Hambali in the United States," Rohan said in reply to a question from deputy public prosecutor Nurdeenie Abdul Rashid.
Rohan, head of the International Centre for Political Violence and Terrorism Research of Nanyang Technological University's S. Rajaratnam School of International Studies, told judge Datuk Mohd Azman Husin that the 9/11 Commission Report detailed how Hambali recruited Yazid.
GPO / ICPVTR / Print
A TERRORISM expert told the High Court that the testimony of the allegedly reformed terrorist Hambali pointed to former Internal Security Act detainee Yazid Sufaat bein ...


text:
The Brahma Satya Reiki
by Caroline Todd May 29, 2021
Brahma Satya Reiki is a recent evolution of Reiki which originated from India as a result of descent of Shiva-Shakti as Brahma Satya. This descent was said to occur sometime in August of 1997. It was during this time that Deepak Hardikar was drawn into the waves of this descent in deep meditative states that the brahma sutras, or the universal keys were revealed. Along with this revelation came the existence of the system now known as Brahma Satya Reiki. In further explaining Brahna Satya Reiki, one must be able to learn and understand its name. "Reiki" is a Japanese word that is the result of joining together of two words. One of the words is "Rei" which means Universal and "Ki" which means the Life Force Energy. As for the Indian origin of the name, "Brahma" means Brahmaan or the Universe and "Satya" means the Truth.
Brahma Satya Reiki therefore can be further described as the Universal Truth of the Universal Life Force Energy. Through this new type of Reiki method, it is said that Reiki exists in the form of Shakti, Shiva-Shakti and Shiva. These are considered as the three primal energies which are radiated out of the very source of universal existence.Through this, practitioners believe Reiki as also to refer to the Divine energy. Each of the three primal energies represents the primal vibrations and has their own special and unique characteristics. The first primal energy, Shakti, is believed to provide inner strength essential for the realization of any type of task. Shiva is said to be the primal energy that is responsible for the completion or dissolution of any task. Shiva-Shakti is said to be that primal energy which combines the two other primal energies of Shiva and Shakti.
Why not try: Beauty Reiki


Katsuyuki Motohiro (本広克行, Motohiro Katsuyuki, born July 13, 1965) is a Japanese film director.
^ "本広克行 もとひろ・かつゆき". allcinema.net (in Japanese). Stingray. Retrieved January 8, 2015.
^ "Ajin - Demi Human Manga Gets Live-Action Film With Rurouni Kenshin's Takeru Satoh". Anime News Network. 2016-11-03. Retrieved 2016-11-05.
^ "Live-Action Laughing Under the Clouds Film Casts Shunsuke Daito". Anime News Network. September 3, 2016. Retrieved September 4, 2016.
This page was last edited on 10 June 2018, at 04:31 (UTC).


text:
(100188) 1994 CM1 es un asteroide perteneciente al cinturón de asteroides, descubierto el 9 de febrero de 1994 por el equipo del Observatorio Astronómico de Farra d'Isonzo desde el Observatorio Astronómico de Farra d'Isonzo, Farra d'Isonzo, Italia.

Véase también
 Lista de asteroides del (100101) al (100200)
 Cuerpo menor del sistema solar

Referencias

Asteroides del cinturón principal
Objetos astronómicos descubiertos por Observatorio Astronómico de Farra d'Isonzo
Objetos astronómicos descubiertos desde el Observatorio Astronómico de Farra d'Isonzo
Objetos astronómicos descubiertos en 1994
Wikiproyecto:Asteroides/Artículos de asteroides

redpajama_set_name: RedPajamaWikipedia


text:
Q: laravel migration foreign key nullable I'm building a project with Laravel version 7.25.0. I coded my migration files and they contain some foreign keys. I added nullable() at the end of each foreign key but they don't work. I looked at similar questions but none of them solved my problem. Here is my migration file's up function
    {
        Schema::create('tests', function (Blueprint $table) {
            $table->id();
            $table->string('title')->nullable();
            $table->tinyInteger('level')->nullable();
            $table->tinyInteger('try_count')->nullable();
            $table->char('room_name')->nullable();
            $table->boolean('timeless')->nullable();
            $table->integer('time')->nullable();
            $table->foreignId('school_id')->constrained('schools')->nullable();
            $table->foreignId('year_id')->constrained('years')->nullable();
            $table->foreignId('course_id')->constrained('courses')->nullable();
            $table->string('type')->nullable();
            $table->date('start_date')->nullable();
            $table->date('end_date')->nullable();
            $table->time('start_time')->nullable();
            $table->time('end_time')->nullable();
            $table->text('description')->nullable();
            $table->boolean('timeout')->nullable();
            $table->boolean('question_sorting')->nullable();
            $table->timestamps();
        });
    }

when I migrate it with the other files and not getting any error, this is my phpmyadmin page of that table
phpadmin page table ss

A: Any additional column modifiers must be called before constrained,
So You must put the nullable() before constrained() like this :
$table->foreignId('course_id')->nullable()->constrained('courses');

redpajama_set_name: RedPajamaStackExchange


VADODARA: Two leopards were caged in Dhanpur taluka of late on Thursday after three persons were killed by the wild animals last month. It is believed that one of the leopards was ?man eater? that had earlier gone to neighbouring (MP).
The leopard had killed two girls on November 21 and 22 and an elderly woman on November 26. Several efforts were made earlier to cage it and forest officials from Sasan were also roped in for the operation to nab it.
It was subsequently ascertained that the leopard had gone to MP as its pugmarks were seen on the both the banks of the Panam river. The river is on the border of the two states. In wake of the development, forest officials from the state had alerted their counterparts in MP. But with chances of the leopard returning, the forest department had not removed the cages.
The leopard was caged late on Thursday evening at Kotambi in Dhanpur taluka.
Forest officials said that they were confident that the leopard was one that had killed three persons last month. ?The caged leopard?s pugmarks are almost same as those of the leopard that had attacked people earlier,? said district forest officer J L Jhala.
Jhala said that the leopard?s canine tooth was also damaged and it had an infection in its teeth. This might have made preying difficult for it.
?In the case of one of the persons that it had killed, the leopard had eaten only soft parts of the deceased?s body like ears, nose and lips. This had given us an indication that it might be having problems with its teeth,? said Jhala.
The other leopard was caged at Punakota that is also located near Dhanpur. This leopard is much younger and smaller than the one trapped at Kotambi.
Jhala said that two persons had seen the leopard that was attacking humans from a short distance. ?
We will show both the leopards to them and try to identify the one that was attacking people,? Jhala said.


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Rosebud Perfume Co. Smith's Lip Balm Mocha Rose with vanilla is a blend of essential oils with mocha, vanilla, and chocolate. Rosebud Perfume Co. Smith's Lip Balm Mocha Rose is a perfectly delicious lip balm experience.


text:
It was 'brutal:' Ashland, Lex girls survive wind to move on
Jon Spencer
Mansfield News Journal
FINDLAY - Mandy Ruminski was blown away when she heard her score would likely be the only one to break 80 in Wednesday's Division I sectional golf tournament at Sycamore Springs.
About an hour later it became official.
"Was it the wind?" she said.
That would be putting it mildly on a day that was anything but.
Ruminski shot a 76 for medalist honors and Ashland won its sixth straight sectional title behind the only other three scores that fell below 90.
The results were as frustrating as the conditions were troubling. All but 16 of the 59 scores were 100 or higher.
Lexington was the repeat runner-up with three scores over 100. Even Ashland counted a triple-digit score en route to a 361-379 margin of victory.
Both of those teams move on to next Tuesday's district tournament at Stone Ridge in Bowling Green, along with Ontario sophomore Brooklyn Adkins, now a two-time individual qualifier after carding a 94.
"The wind was crazy," Ruminski said. "It was even moving my putts, which was weird. But, in all honestly, I didn't feel like the wind was my problem today. It definitely affected some shots, but the mistakes were my own. I couldn't really blame it on anything. I just need to fine tune some things.
"I just need to work out a few kinks and I should be good to go."
Every other golfer in the field would have killed to have Ruminski's score, but she wasn't happy with her double-bogey on the last hole or finishing two strokes higher than last year, when she was medalist runner-up to Ashland's Kira Moore.
Moore wasn't thrilled, either. She went for a long walk after finishing with an 84. It was 12 strokes higher than she shot last year in becoming sectional medalist for the second time, but still ended up being the second-best score.
"I'm always tough on myself, but I think that's how you get better, too," Moore said. "I almost lost my balance a couple of times because the wind was gusting really hard."
By the time all the scores were in, Moore had come around to the realization that Wednesday's round was largely an aberration thanks to the forces of nature at work.
She preferred to talk about her swing coach Tyson Deskins, the general manager at Shelby Country Club and an owner of 419 Golf in Ontario, and how he has been a big help to the Arrows.
After that the girls joined first year head coach Tom Marquette for what has become the traditional team celebration photo, with index fingers raised in the air.
"Today wasn't about the scoring," Marquette said. "It was about managing the course. I was most pleased with the mental toughness the kids showed today. That's what we've been looking for and hoping for and really got it today.
"Some girls hung on to their rounds and had decent scores despite the conditions. That wind was brutal."
Other than Ruminski and Moore, the only other golfers breaking 90 were Ashland's No. 3, Sarah Anderson (87), and No. 2, Klaira Paramore (88). The Arrows also counted Emma Packard's 102 with Livia Sponsler's 104 being the drop score.
"We've had a plan since the beginning of the year," Marquette said. "It was a process to get to this point where we're playing good golf at the end."
Lex coach Mike Kathrein was also able to keep things in perspective because of the weather.
"The wind definitely affected things," he said. "The girls don't realize it affects the ball on the putting green as well, especially if you're down wind. It's going to get the ball moving a little more.
"You just know that everybody's playing in the same conditions. It levels out. You're going to get some bad breaks because of the wind. That 76 for Amanda is a heckuva good score in these conditions."
Kathrein was even more impressed with Ruminski's 35 on the front side.
"The front is longer and more open to the wind," he said. "The back has more trees and can shelter some of the shots. The front side is more exposed, so a 35 on that side is quite good."
Lex played a practice round at Sycamore Springs last Saturday and Kathrein's daughter, Lainey, shot an 80, two strokes better than Ruminski. It was a different story Wednesday. Kathrein had an 8 on her first hole and never recovered, shooting a 99.
Trinity Wolfe (100) and Hannah Smith (104) had the other Lex scores that counted. Alissa Wolfe (106) rounded out things for the team.
"If you were hitting the ball even slightly in the direction the wind was blowing ... it just took the ball," Lainey Kathrein said. "Some girls had the wind take their putt right off the green.
"Hopefully, we get it out of our system and we can rally through at districts. If we get out we get out."
The younger Kathrein knows real adversity and the wind isn't it. She's battled illness all season, even spending time in the hospital as doctors try to get to the bottom of what appears to be an auto-immune disease.
"Something in my body is attacking itself," she said. "We aren't quite sure what yet. It just makes me really tired.
"It's been rough. I expected to have a much lower average. I was sick way before the season started, but I didn't know I was sick until I got really sick, which explains my preseason struggles. Now I'm just trying to get back to normal."
In becoming a repeat district qualifier for Ontario, Adkins avoided having to survive a two-hole playoff like last year. The final qualifying score Wednesday was a 99, so she made it with plenty of room to spare.
"I just need to keep looking at the positives," she said. "I need to play my own game, not anyone's else's. (Stone Ridge) is a nice course, but very hard ... very narrow."


text:
Iran official admits most environmental problems happened under the Regime
News / Atrocities Human Rights Violation /
An Iranian regime's environment official has admitted that the vast majority of environmental problems in Iran have occurred in the last 25 years and that the entire country was under threat.
Ahmad Reza Lahijanzadeh, the Marine Environment and Wetlands deputy director of Iran's Department of Environment, explained that more of the damage done to the environment was the result of "development programs" implemented since 1994.
He told a meeting in Hormozagan: "If one investigates the destruction of the marine environment from northern Iran and the Caspian Sea, the gradual and steady disappearance of the coastline is evident in both Mazandaran and Gilan provinces, to the extent that one cannot even find 100 meters off the shore area."
During this meeting, he also said that Iran could soon see the "devastating destruction of the Caspian Hyrcanian mixed forests in Alborz" and that Iran's aquifers might even dry out, especially in areas like Semnan and Fars, because of the reduction of groundwater and that in some areas this has become "irreversible".
Lahijanzadeh described Khuzestan province, as a symbol of the country's unstable development, because, even though it had enough water sources, fertile soil, and oil and gas reserves to sustain the whole country, it has been destroyed by mismanagement and corruption.
Elsewhere, he highlighted that the petrochemical industry has ruined the fishing industry in Mahshahr, citing that it "reduced the city's fishing capacity to about 900 tons", sacrificing a long-term benefit for a short-lived profit.
Lahijanzadeh then spoke about "Iran's most polluted city", Asaluyeh, which is the most polluted area in the region and is seeing the widespread slaughter of aquatic life and reduced environmental diversity.
It is no coincidence that Asaluyeh is thought to contain one of the largest oil, gas, and petrochemical projects in Iran. These plants are built with a lack of attention to environmental standards, something environmentalists have repeatedly warned the government about, and cause great damage to the people and the environment there. More worrying still is the concentration of heavy industrial activities in the surrounding province and lack of wastewater treatment facilities that endanger the Persian Gulf.
In July, the Iran regime's Supreme Council for Environmental Protection held a meeting after warnings about the critical state of the environment and the disregard of this by regime officials, but the government resolved that they were not accountable for serious environmental damages in Iran, which begs the question; who is?
Source » irannewsupdate
Iranian people continue to strike and protest against the Regime
More than 200 protests in Iran during October
Iranian lawmakers want English classes banned in public schools
October 14, 2019 ifmat ifmat 0
Prince Reza Pahlavi says Iranian people cannot take it anymore
November 25, 2019 ifmat ifmat 0
More than 177 Iran protests in August
September 9, 2019 ifmat ifmat 0


The Rose Metal Press Field Guide to Writing Flash Fiction is the best of its kind. For both beginner and advanced practitioners of this hot and wildly popular genre, the Field Guide offers tons of practical advice, hands-on exercises, and intriguing histories of the form. The back matter is chock full of recommended reading lists and suggested anthologies. It's no wonder this fabulous book is already in its second printing and has received a starred review from Publisher's Weekly.
Tara L. Masih, editor of the Field Guide, presents the first comprehensive essay on flash in the introduction. After first reading Tara's intro, I was astounded at the rich history of flash. It's been around since the dawn of myth; and in Shouhua Qi's essay "Old Wine in New Bottles," he gives various examples of Chinese flash dating as far back as 350 BC. Masih describes 14th century Italian flash, the burgeoning of flash in America beginning with Washington Irving and Edgar Allan Poe, and even the feminist flash started by Louisa May Alcott and Kate Chopin. We learn the development of flash and its current style and means of consumption via both print and the internet. The final two sections of the book give you an idea of where flash is going in the future and how you can join the party.
The Field Guide is structured as a series of individual essays, each taking a different approach to flash. Such esteemed writers as Stuart Dybek, Jayne Anne Phillips, Robert Olen Butler, and Steve Almond give their two cents on the form along with advice on inspiration, taking risks, beginning and ending successful flash, and finally how to best tackle the editing process. There is truly something in the Field Guide for everyone. Jayne Anne Phillips, in her essay "'Cheers' (or) How I Taught Myself to Write," introduces the one-page form and how to play with its density to create the most powerful and moving stories in the shortest space. She suggests working with "instruction booklets, tax forms, newspapers, and cookbooks" to get your writing into a direct mode. As with all the other essays in the Field Guide¸ Phillips' essay concludes with a prompt and a story example to illustrate how to use the prompt. The writing advice and exercises contained here offer more than just basic advice on flash fiction. In a way, this is a fun manual for writers of both poetry and prose, no matter what the form. Lex Williford's essay, "Forty Stories in the Desert" gives instructions for how to make your own Rorschach blots and then generate lists of images from them. Williford also gives instructions on how to use the lists to create "15 minute fictions." Practicing one 15-minute fiction a day for five weeks will "encourage the discovery of images, characters, and storylines," says Williford, who has used this exercise successfully for years in the classroom. This same essay is also rife with such wonderful advice as how to create surprising reversals in both poetry and prose.
As a poet, I have used many of these exercises to generate new poems and breathe life into drafts that I thought were stuck. One of my favorite entries, "The Myth-ing Link (Or, Linking up to Myth)" by Pamelyn Casto, explores how to write flash using myth as a formative structure. She gives examples of how to re-create or breathe new life into traditional myths, or de-familiarizing myths by giving them a special and unexpected twist. The writing prompts at the end are fantastic with no less than seven ideas for using a particular myth to create new stories and characters. I've taken some of these ideas, such as putting Pandora on trial and giving her voice using a Q&A format. Steve Almond's contribution shows how you can take bad poems and turn them into successful flash. And Kim Chinquee explores the relationship between prose poem and flash in her essay "Flash Fiction, Prose Poetry, and Men Jumping Out of Windows." The Field Guide is as fine a writing resource as I have yet come across. A highly recommended resource for individual libraries or for use in workshop.
This entry was posted in non-fiction and tagged Rose Metal Press, Tara L Masih, tara mcdaniel. Bookmark the permalink.


On a cold Saturday afternoonin mid-January, Matthew Smith and his dog Jericho are visiting a Walmart north of Baltimore, and everyone's just taking it in stride. That's a sign of progress.
The 42-year-old Smith, injured in a high-speed motorcycle crash in 1994 that left him with a pelvic fracture and nerve damage, gets around with the help of a wheelchair. Jericho, a fawn-colored former shelter dog, has been trained to assist him by Animal Farm Foundation (AFF), a nonprofit in upstate New York that advocates on behalf of pit-bull-type dogs.
With one hand holding a vest wrapped around Jericho's chest, Smith gets pulled through the megastore's aisles, attracting nothing more than a few curious looks from fellow shoppers. The dog remains calm and focused. When Smith waits to pick up an order, Jericho lies patiently at his feet. On the way back to Smith's Jeep, dogs barking in a car fail to distract Jericho from making a beeline through the parking lot.
People are sometimes surprised to see a dog they perceive as a pit bull working as an assistance dog, but Smith says he's "never had a negative response. Everybody absolutely just loves him." One drawback is that everyone wants to pet Jericho; a message printed on his vest asks them to refrain because he's working.
The role of assistance dog has traditionally been reserved for dogs bred for that purpose, many of them purebreds such as Labradors or golden retrievers. The AFF program enables qualified pit-bull-type dogs from shelters around the country to join the club.
Shelter dogs still face the stereotype that they are somehow "broken," and those labeled "pit bulls" are sometimes considered less valuable than other canines in the shelter, says Caitlin Quinn, AFF's director of operations. AFF's mission includes eliminating barriers to adoption for shelter dogs and combating discrimination against pit-bull-type dogs, she adds, so developing a program that trains these dogs to serve people with disabilities was natural for the organization.
Shelters and rescues that think they have suitable candidates for the program may apply through the AFF website. The definition of a pit bull can vary considerably among organizations; dogs labeled as such upon intake are eligible. AFF, which has made six assistance dog matches so far and has two more planned for this summer, provides the training and the dogs at no cost to clients, and covers the cost of transporting the dogs.
AFF transports select candidates to its shelter in New York, where the training process starts with several weeks of playgroup sessions. If all goes well there, AFF trainer Apryl Lea brings the dogs to her home—a converted pajama factory in Kingston, N.Y.—for additional training and socialization. The dogs experience the sights and sounds of an urban environment as well as life with other canines (Lea has seven of her own, and typically has three to five dogs in training). She determines whether the candidates have what it takes to become assistance dogs and how their traits might mesh with the lifestyles of potential clients. Dogs who don't make the grade are put up for adoption by AFF or returned to their home shelters.
"There's a lot of very hands-on matchmaking," Quinn says. Lea, who's accredited by Assistance Dogs International, will visit potential clients' homes to get an idea of their daily lives, including whether they work, how far they travel and their leisure activities.
Once AFF matches a dog to a client, Lea works with the pooch on the particular tasks he'll be performing (which could range from opening refrigerator doors to responding to seizures), and prepares the client to handle an assistance dog. After the dog goes home with the client, Lea continues to work with them through calls, video chats and occasional visits.
Today, they're inseparable. Jericho accompanies Smith to his full-time job as an analyst for Comcast Cable and tags along on fishing trips in Smith's bass boat. He took part when Smith and his wife, Ali, flew to Seattle and took a cruise to Alaska. (Their suite on the ship came equipped with a 4-by-4-foot litter box for Jericho.) Having Jericho along enabled Smith to participate in on-shore excursions that he otherwise would have skipped.
Smith marvels at how even-tempered and affectionate his dog is. The one-time stray, who's around 5 years old, was found on a street in Florida, but Smith suspects Jericho—whom he describes as a "sweetheart" and "the biggest baby"—couldn't have been on the streets for long. When he's not working, Jericho will hop into Smith's lap on command, chase well-chewed Frisbees and do face plants in the snow—and he's made fast friends with the Smiths' three cats.
"They just fit so well with each other. ... They have this really tight partnership that service dogs and clients have," Quinn says.
The timing of Jericho's arrival couldn't have been better, Smith says. Years of using his wheelchair and crutches had left him with arthritis, bursitis and torn rotator cuffs, so rolling himself around had gotten difficult. Jericho literally takes the burden off Smith's shoulders, greatly increasing his independence and self-esteem, he says.
For more information, visit animalfarmfoundation.org/pages/Assistance-Dog-Program.


The Carolinas have been asking. ROVE delivers.
ROVE is opening the doors to their state of the art lab as the team demonstrates their ability to architect, design, install, and demo the latest in equipment and offerings.
Join ROVE for lunch at the event location nearest you as we showcase these products in a live hybrid cloud environment.


text:
Sweet Springs, une ville du comté de Saline, dans le Missouri.
 , une communauté du comté de Monroe en Virginie-Occidentale.


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Written by Jonathan Schofield. Published on May 27th 2014.
TASTE of Portugal was an absolute delight from start to finish.
This is octopus as good steak, filling with tender meat behind the purple of the tentacle case.
On Wednesday my seventeen-year-old had a football match at Irlam o'th' Heights. We drove back to Old Trafford through Salford down Langworthy Road and saw, half way down, Taste of Portugal. It looked inviting.
Saturday we were back, faces dropping as we saw all the tables were reserved. The elegant waiter said, "No problem, the reservations are for later, please this way." The waiter turned out to be the affable and informative proprietor, Fernando.
The interior of Taste of Portugal is like a bar you might find in Porto or Lisbon, bric-a-brac heavy, complete with photographs of the food you might eat. There's even a signed shirt from fellow countryman Cristiano Ronaldo, in this instance from his time at Manchester United.
There are probably a lot of Ronaldo shirts about.
We watched the Champions League Final later that evening - appropriately from Lisbon - and Mr Vanity kept plucking his shirt off and chucking it into the crowd.
I almost felt like doing the same after my clam starter (£11). I wanted to celebrate.
Cooked in a 'special' sauce that contained saffron and feisty amounts of garlic, these were the best clams I've had in ages. The spoon came in handy as I souped up the sauce. A hot starter of prawns (£7) without shells were a zingy joy too.
Atlantic facing Portugal was the first European nation to send sailors around Africa to India. The spices they brought back inform the food to a much greater degree than that of its Iberian neighbour, Spain, and they were very evident in Taste of Portugal's cooking.
The frango asado, or fried chicken (£9.50), and its 'super piri piri sauce', showed the heat off. The chicken was glorious, a flavoured crisp skin with moist flesh beneath. One son braved the super piri piri - home made and supplied in a self-apply vial - and loved it. I tried a drop and swear I scorched my tongue. The only failure in the meal was the poor fries that came with the chicken, the rice that also appeared with the dish was fine.
Octopus is specialThe star course - along with the clams - was the octopus (£13.50) with cabbage and spuds plus caramelised onions. This was superb, as simple, rugged and elemental as the cliffs of Cabo de Roco.
In effect this is octopus as good steak, tender meat under the purple of the tentacle heavy flesh, filling and rewarding. It worked beautifully with the cabbage and the potatoes.
A cod dish (£11) mixed with very thin fried potatoes, onion, bound with egg and adorned with olives was pure peasant food. It was a dish that defined hearty and reeked of authenticity but took some ploughing through.
Rich moussePudds were a homemade chocolate mousse (£3.50) rich in cocoa. The custard tart (£1.15) was a lovely little thing dropped solo onto a plate but tasting lush. The presentation is clearly secondary to flavour in Taste of Portugal.
The food was washed down with a pale, bright lager from the north of Portugal called Super Bock (£3.60 a pint). There's a range of Portuguese wines waiting for our next visit.
Seconds of research revealed this apt fact: 'Super Bock has history with Manchester. The supporters of Manchester United from the early 1990s often had games with teams such as FC Porto. The fondness for 'Bock' was developed into a chant for trips in Portugal – 'Super Bock, superstar, gets you more pissed than Stella Artois'.
That sort of drunken tomfoolery is the wrong mood for Taste of Portugal. As Fernando said, "We're a family restaurant, for relaxing and enjoying the food."
He's right. The place was like a vision of Earth as it is in Star Trek, harmony among multiple linguistic groups. A sweet atmosphere.
If you want the bream or the razor clams, ring Fernando up twenty four hours before your visit and he'll get them in fresh.
That sort of personal service sums up Taste of Portugal.
If you like a homely neighbourhood buzz and really good uncomplicated -occasionally exciting - food, you should try this place.
Langworthy Road looks tough in places, but life should be more than a beauty contest with extra hummus.
But why Langworthy, one of the UK's most deprived wards? I want to get on my soap box and make a point about this.
We reviewed another Portuguese restaurant in Chorlton recently, Lusitano.
Chorlton is a much more predictable location, but on these pages we've frequently expressed the desire that instead of every retail unit in Chorlton, the Didsburys and the Heatons turning into a bijou bar or restaurant it'd be better if the love could be spread a little, especially into east and north Manchester and across into Salford.
People still eat and drink in those districts.
The problem is I know residents of south and central Manchester who are actually scared to go there. It's pathetic - would-be urbanites who are really the suburbanites they mock. These people are mainly Labour voters too, critical of 'cruel' Coalition policies, yet people who never stray from their comfort zone.
Yep, Langworthy Road looks tough but life should be more than a beauty contest with extra hummus. Opening in such districts can be a 'risk' worth taking.
"I knew about Langworthy Road, its reputation, when I opened two years ago," says Fernando. "But this is a family restaurant and it was a good site. There are many good people around here. I'm happy. I think we have even helped the area change a little."
Rejuvenation through food, rejuvenation through the good things in life, seems a sound policy.
Following Taste of Portugal's lead are other businesses such as well-regarded bar restaurant, Monty's. Maybe things are on the up in Langworthy. Maybe it's time for people to stray from their comfort zone.
I'll be re-visiting Taste of Portugal. I'm already pining for those clams and I think they're pining for me.
Leaving the EU will mean we can not longer import continental beer? I assume you're joking with the Burton Road comment?
I've been here a few time and it's never let me down. Well done to ManCon for featuring a smaller, out-of-town place too.
On our way back from Gigantic on Saturday night the taxi driver read out an alert, everyone to avoid Langworthy due to kids running wild with bats and knives. And then on Monday someone got shot in the Ashley Brook pub, according to the MEN it may be a case of mistaken identity. So let's not kid anyone it's the next Didsbury just yet.
Part of the reason Salford is a breath of fresh air is the lack of Chorltonites demanding gluten free vegan flatbreads for their noisy and precocious children.
I live on nearby Chimney Pot Park and have done for 6 years. I have never ever witnessed anything untoward. Indeed, CPP itself is charming, as is nearby Buille Hill Park and the Heights. Everywhere has its dodgy bits. Let's not forget that Chorlton borders Whalley Range which is hardly salubrious. Hmmn.... I wonder if this comment will get deleted by the ManCon team, who are the most defensive lot of "journos" I have ever come across. I thought the comments section was for people to comment in. No?
What's wrong with Whalley Range?
Rinkydink have you actually ever been to Whalley Range? It's a beautiful suburb with a solid community. I think you might be in a timewarp in the early 1990s mate.
Jesus, it's amazing how Chorlton is billed to be this bohemian paradise and yet the entire population seems to spend it's time trolling on the internet. Come to Salford where life is lived. I never cease to be amazed how some people take a positive article about a place and troll just to make themselves feel better and get past their complex of not being able to afford to live in Didsbury.
Jonathan, Really enjoyed your review of 'The Taste of Portugal',as you say it makes a nice change for a good restaurant not to be in Town or the southside.I will definitely pay a visit on my next visit to the UK.Octopus as you know is served beautifully in Spain,but I normally get it 'rubbery' in the UK,good to read about a place that obviously knows what he is doing.
I like the way all the left wing Chorlton Guardian readers come one here and show their true Tory colours slamming a city based on out of date perceptions. Salford is pretty decent these days, it isn't the new Chorlton but nor would we want it to be. I hope the days never come when I bump into a Unicorn militant whilst having a quiet sup in Montys on Langworthy Road.


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The Brainerd Snodeos snowmobile club along with Paradise Resort Bar and Grill on South Long Lake are hosting a Radar Run on Saturday February 24, 2018. Registration starts at 8:00am racing at 10:00am We hope to see you there!
The Brainerd Snodeos and Paradise Shores Bar and Grill are hosting a radar run on upper south long lake Saturday, February 25th 2017. The track will have 1000 ft of shaved ice and 2000+ feet of shutdown with a high tech speed trap.Classes for every sled with "Fun Run" section as well. Cash Prizes for top speed of the day as well as trophies for first and second place in every class except "Fun run". 50-65 classes including kids, ATV, traction, non-traction, and vintage.
Registration opens at 7 am runs start at 9 am. 5 runs for $25 and $10 insurance.
For more information find the event on Facebook, or call Kris (770)-733-5858.


text:
package log;

import org.junit.Test;

import static org.junit.Assert.*;

public class LogTest {

    @Test
    public void addSnapshot() {
    }

    @Test
    public void addSimuLog() {
    }

    @Test
    public void printSnapshot() {
    }

    @Test
    public void getSimuLog() {
    }

    @Test
    public void getSnapshotMap() {
    }
}

redpajama_set_name: RedPajamaGithub