text:
\section{Introduction}
\label{sec:int}
This article deals with the analysis of three particular codimension-two bifurcations in nonsmooth systems. Broadly speaking, nonsmooth systems are continuous- or discrete-time dynamical systems featuring some kind of discontinuity in the right-hand side of their governing equations whenever the system's state reaches a {\it discontinuity boundary}.
More specifically, nonsmooth systems include several classes, e.g., piecewise smooth \citep{Filippov88, Bernardo08}, impacting \citep{Brogliato99}, and hybrid \citep{Branicky98b, Lygeros03} systems, which have been largely used in the last decades as models in various fields of science and engineering (see references above and therein).

While methods of numerical continuation allow to easily detect and trace
bifurcation curves in two-parameter planes, understanding the geometry
of bifurcation curves around codimension-two points is a key to the
construction of complex bifurcation diagrams.
In the domain of smooth dynamical systems, the unfolding of the most common
codimension-two points is well known (see, e.g., \citep{Kuznetsov04}), and
this knowledge is exploited in continuation software for the automatic
switching among bifurcation branches at these points
(see, e.g., \citep{Dhooge03,Meijer09}).
The same cannot be said for nonsmooth systems, where,
though efficient numerical tools for bifurcation analysis are finally
starting to appear \citep{Dercole05c,Thota08b},
results are still mostly limited to codimension-one cases.
A reason for this shortcoming can be found in the fact that nonsmooth systems
exhibit, along with the standard bifurcations of smooth systems, a great
number of completely new bifurcations,
called \textit{discontinuity induced bifurcations},
that involve the interaction of the system's invariant sets with the
discontinuity boundaries. Since the characteristics of these bifurcations
depend critically on both the class of nonsmooth system and the geometry of
the involved boundaries, the number of possible scenarios is huge and,
at the moment, truly general results are scarce.
It goes without saying that codimension-two cases involving simultaneous
smooth and discontinuity induced bifurcations, named ``type II'' in \citep{Kowalczyk06}, are even more numerous, and less understood.

In this article we analyse type II bifurcations of periodic orbits (limit cycles), that is, bifurcations involving a periodic orbit (from now on called the bifurcating cycle) that collides with a discontinuity boundary while being at the same time non-hyperbolic.  Rather than aiming at a complete unfolding with reference to a particular class of nonsmooth systems, we concentrate on finding those geometric features that are common to all classes:
this is accomplished by abstracting our analysis from the nature of the involved boundary. As a consequence, our results are incomplete, because they only focus on the geometry of bifurcation curves around the codimension-two point; on the other hand, they apply more in general --- a feature that should be welcome in a field where peculiarity seems to be the rule.

In particular, we show that three codimension-one bifurcation curves generically emanate from a type II point in a two-parameter plane.  One is the smooth bifurcation curve (fold, flip, or Neimark-Sacker), while the other two are the discontinuity induced bifurcations of the bifurcating cycle and of the secondary invariant set involved in the smooth bifurcation (the other cycle, the double-period cycle, or the torus, respectively).  Then we show that, depending on the bifurcation, one or both of these curves are tangent to the smooth bifurcation curve.  Indeed, in the flip and Neimark-Sacker cases, the bifurcating cycle departs from the image of the nonhyperbolic cycle, left frozen in state space, at a linear rate with respect to the bifurcation parameter, whereas the distance between the period-two cycle or the torus from such an image goes as the square root of the parameter perturbation from the bifurcation.  As a consequence, locally to the codimension-two point, the perturbation required by the secondary invariant set to collide with the discontinuity boundary is quadratic with respect to that required by the bifurcating cycle.  Similarly, in the fold case, the rate at which both cycles approach the image of the nonhyperbolic cycle is proportional to the square root of the parameter perturbation, so that the discontinuity induced bifurcation curves are both quadratically tangent to the fold curve.  These rather intuitive results have been observed in many examples, and proved for some specific classes of discontinuous systems (e.g., in \citep{Dankowicz05, Kowalczyk06, Nordmark06, Thota06, Zhao06, Simpson08b, Simpson09a}).  The aim of this paper is to provide formal support to the above geometric arguments and to prove their validity once and for all under very general conditions.

The ensuing exposition is set into the framework of
{\it grazing} bifurcations in continuous-time, where the discontinuity boundary
is smooth, locally to the point of contact with the bifurcating cycle, and
the contact occurs tangentially.
This allows us to keep the terminology as coherent as possible, especially
in the lack of a uniform terminology across all classes of nonsmooth systems.
Nonetheless, the reader will realise that our exposition is general and applies
to any discontinuity induced bifurcation involving a non-hyperbolic cycle in
continuous time or a non-hyperbolic fixed point in discrete time.
In fact, our analysis is based on the reduction of the nonsmooth flow to a
map which is defined and smooth on one side of a boundary, while we do not
describe the behaviour of the map on the other side. The rest of the analysis
is based on the obtained map, as if the problem was originally set in
discrete time.
Thus, in practise, we do not make any assumption on the class of nonsmooth
systems and on the geometry of the discontinuity boundary.

We begin by stating the problem, introducing the basic notation, and
outlining the steps that we follow in the main proofs
(Sect.~\ref{sec:fa}); then we proceed with the detailed analysis of the three generic grazing bifurcations of non-hyperbolic cycles: the grazing-fold, the grazing-flip,
and the grazing-Neimark-Sacker (Sects.~\ref{sec:bfd}--\ref{sec:bns} and Appendices).
Once casted in discrete time, grazing bifurcations are more appropriately
called {\it border collisions}, and this is the name we use in this part
of the paper. Then we presents three specific applications
(Sect.~\ref{sec:ex}) and conclude with some future directions.

\section{The framework of analysis}
\label{sec:fa}
We consider a nonsmooth autonomous flow
$x(t)=\Phi(x(0),t,\alpha)\in{\bf R}^{n+1}$
depending on parameters $\alpha\in{\bf R}^2$.
Namely, the right-hand side of the system's ODEs
\begin{equation}
\label{eq:ode}
\ds\dot{x}(t)=\left.\Frac{\partial}{\partial\tau}
\Phi(x(t),\tau,\alpha)\right|_{\tau=0}=
\Phi_t(x(t),0,\alpha)
\end{equation}
(here and in the following variables and parameters as subscripts denote
differentiation)
is generically smooth, but characterised by zero- or higher-order
discontinuities across some discontinuity boundaries $\mathcal{D}_i$,
defined as the zero set of suitable smooth functions $D_i(x,\alpha)$.
In particular, we can distinguish three types of discontinuity boundaries
(see Fig.~\ref{fig:cyc}):
\begin{figure}[t!]
\centerline{\includegraphics[width=1\textwidth]
{fig_cyc.eps}}
\caption{A generic (hyperbolic) limit cycle $\gamma$ of the nonsmooth flow $\Phi$.
For some $\alpha$ near $\alpha=0$, the cycle passes close to, but does not
touch, the discontinuity boundary $\mathcal{D}$, so that the resulting
Poincar\'e map on $\mathcal{P}$ is defined, locally to $\bar{z}$, only on
one side of the discontinuity boundary $\mathcal{H}$.
The boundary $\mathcal{H}$ divides $\mathcal{P}$ into two regions,
respectively composed of points $z$ from which the orbit of
$\Phi$ does and does not touch $\mathcal{D}$.}
\label{fig:cyc}
\end{figure}
boundaries across which the right-hand side of (\ref{eq:ode}) is
nonsmooth but continuous, so that orbits always cross the boundary
($\mathcal{D}_C$ in the figure);
boundaries across which the right-hand side of (\ref{eq:ode}) is
discontinuous, so that sliding motions are possible ($\mathcal{D}_S$);
boundaries where the right-hand side of (\ref{eq:ode}) is formally
characterised by impulsive components, which define an instantaneous state
transition (or jump) whenever orbits reach the boundary ($\mathcal{D}_I$).

Forward solutions of system (\ref{eq:ode}) are composed of smooth segments,
each corresponding to a smooth orbit terminating at a discontinuity boundary,
or to a sliding motion. Smooth segments are directly connected at crossing
and sliding boundaries, while they are connected through state jumps at
impacting boundaries.
Let $\gamma$ be a periodic orbit of system (\ref{eq:ode}).
In Fig.~\ref{fig:cyc}, $\gamma$ is composed of four segments, three smooth
(solid) orbits and one sliding motion (thick orbit),
and is characterised by a single state jump (thick dashed connection).

Suppose that, when $\alpha=0$, the cycle $\gamma$ grazes (touches tangentially) a
discontinuity boundary $\mathcal{D}$, and no other degeneracies occur on
$\mathcal{D}_C$, $\mathcal{D}_I$, and $\mathcal{D}_S$.  At the same time, suppose that $\gamma$  is non-hyperbolic at $\alpha=0$ (more precisely, the multipliers are not defined at $\alpha=0$, but the smooth bifurcation curve is a path to $\alpha=0$ on which one real or two complex conjugate simple multipliers lie on the unit circle).
Introduce a Poincar\'e section $\mathcal{P}$ along one of the segments of
$\gamma$, say, e.g., the segment touching $\mathcal{D}$ so that the
flow reaches $\mathcal{D}$ after $\mathcal{P}$ for $\alpha=0$.
Also introduce a coordinate $z\in{\bf R}^n$ on $\mathcal{P}$ such that the
intersection $\bar{z}$ of $\gamma$ with $\mathcal{P}$ lies at $z=0$
for $\alpha=0$.
Then, locally to $(z,\alpha)=(0,0)$, the flow $\Phi$ induces a Poincar\'e map
\begin{equation}
\label{eq:pm}
z\mapsto F(z,\alpha)
\end{equation}
(note that the map may not be invertible, e.g., in the presence of sliding motions).  Since we do not discuss the type of boundary $\mathcal{D}$, we limit the definition of $F$ to the values of $(z,\alpha)$ in a neighbourhood of $(0,0)$ for which the orbit originating at $z$ does not touch $\mathcal{D}$.
This introduces an $(n-1)$-dimensional discontinuity boundary $\mathcal{H}$ on the Poincar\'e section $\mathcal{P}$ such that $F$ is defined and smooth on one side of $\mathcal{H}$. In particular, let
$$
\mathcal{D}=\{x:D(x,\alpha)=0\},\quad \mathcal{H}=\{z:H(z,\alpha)=0\},
$$
and assume, without loss of generality, that the flow $\Phi$ touches
$\mathcal{D}$ tangentially while locally remaining on the side $D(x,\alpha)<0$,
and that $F(z,\alpha)$ is defined for $H(z,\alpha)<0$.
Then, the function $H$ can be constructed as follows
(see again Fig.~\ref{fig:cyc}).
Define the $n$-dimensional smooth manifold $\mathcal{T}$ of the points where
the flow is tangent to the level sets of function $D$:
$$
\mathcal{T}=\{x:T(x,\alpha):=
\left\langle\Phi_t(x,0,\alpha),D_x(x,\alpha)\right\rangle=0\}
$$
(vector $D_x(x,\alpha)\in{\bf R}^{n+1}$ is orthogonal to the level sets of
$\mathcal{D}$ at $(x,\alpha)$ and $\langle\cdot,\cdot\rangle$ is the standard
scalar product in ${\bf R}^{n+1}$).
As shown in Fig.~\ref{fig:cyc},
the $(n-1)$-dimensional intersection between $\mathcal{D}$ and $\mathcal{T}$
is transformed, backward in time by the flow, into the discontinuity boundary
$\mathcal{H}$.
Thus, $H(z,\alpha)$ can be defined as the value $D(x,\alpha)$ at
the point $x$ at which the flow first reaches $\mathcal{T}$ (forward in time)
from the initial condition corresponding to $z$ on $\mathcal{P}$.

We can now abandon the continuous-time framework, and focus on
map (\ref{eq:pm}).
For some $\alpha$ in a neighbourhood of $\alpha=0$, the map is characterised
by a fixed point $\bar{z}$, with $H(\bar{z},\alpha)<0$ and, for $\alpha=0$,
the fixed point is non-hyperbolic and lies at the origin $z=0$ and on the
discontinuity boundary $\mathcal{H}$.
We investigate the bifurcation curves rooted at $\alpha=0$ in the
parameter plane $(\alpha_1,\alpha_2)$,
by considering separately the three generic cases, namely
(I) fold (one simple eigenvalue equal to $1$, Sect.~\ref{sec:bfd}),
(II) flip (one simple eigenvalue equal to $-1$, Sect.~\ref{sec:bfp}), and
(III) Neimark-Sacker
(two simple complex conjugate eigenvalues on the unit circle,
Sect.~\ref{sec:bns}).

In each case, we proceed as follows. Locally to $(z,\alpha)=(0,0)$, we
consider the restriction of map (\ref{eq:pm}) to a parameter-dependent
centre manifold $\mathcal{Z}^c$.
Let $u\in{\bf R}^{n_c}$ represent coordinates on $\mathcal{Z}^c$, $n_c=1$ in the
fold and flip cases, $n_c=2$ in the Neimark-Sacker case, with $u=u(z,\alpha)$
for each $z\in\mathcal{Z}^c$ and $\alpha$ in a neighbourhood of
$(z,\alpha)=(0,0)$, $u(0,0)=0$, and let $z=z(u,\alpha)$ denote the inverse
transformation. Restricted to the centre manifold, map (\ref{eq:pm}) reads
\begin{equation}
\label{eq:cm}
u\mapsto f(u,\alpha):=u(F(z(u,\alpha),\alpha),\alpha).
\end{equation}
and the discontinuity boundary $\mathcal{H}$ is given by the zero-set of the function
\begin{equation}
\label{eq:h_func}
h(u,\alpha):=H(z(u,\alpha),\alpha).
\end{equation}
We assume that the three following conditions hold:
\begin{itemize}
 \item[(i)] Map (\ref{eq:cm}) satisfies, at $\alpha=0$, all genericity conditions of the corresponding smooth bifurcation (see, e.g., \citep{Kuznetsov04}).
 \item[(ii)] At $\alpha=0$, the centre manifold $\mathcal{Z}^c$ transversely intersects the
discontinuity boundary $\mathcal{H}$ at $z=0$ (by continuity the transversality persists near $(z,\alpha)=(0,0)$, see Fig.~\ref{fig:cyc}).  Under this condition, the dynamics of map (\ref{eq:pm}) near $(z,\alpha)=(0,0)$ is captured by that on the centre manifold.  In the coordinate $u$ along the centre manifold the condition becomes $h_u(0,0)\neq 0$.
 \item[(iii)] Changing $\alpha$ along the smooth bifurcation curve, the non-hyperbolic fixed point crosses the discontinuity boundary transversely.  This condition ensures that the smooth bifurcation curve intersects the border collision curves in a generic way.
\end{itemize}
As a first step, we reduce map (\ref{eq:cm}) to a normal form (NF)
(the fold, flip, and Neimark-Sacker normal forms)
through a locally invertible change of variable and parameter, say,
$v=v(u,\alpha)$, $\beta=\beta(\alpha)$, where $v(0,0)=0$, $\beta(0)=0$, and
$u=u(v,\beta)$, $\alpha=\alpha(\beta)$ denote the inverse transformation.
Then, second step, we find the expression of the discontinuity boundary (\ref{eq:h_func})
in the new variables and parameters, i.e.,
\begin{equation}
\label{eq:Sigma}
\{v:h^{\mathrm{NF}}(v,\beta):=h(u(v,\beta),\alpha(\beta))=0\}.
\end{equation}
Finally, third step, we analyse the interaction of the normal form map
$$
v\mapsto f^{\mathrm{NF}}(v,\beta):=
v(f(u(v,\beta),\alpha(\beta)),\alpha(\beta))
$$
with the discontinuity boundary (\ref{eq:Sigma}), and we find local asymptotics for the bifurcation curves emanating from $\alpha=0$ in terms of $(\alpha_1,\alpha_2)$-expansions.

The details of the normal form reduction are reported in appendices A.1, B.1, and C.1, while the technicalities on step two are reported in Appendices A.2, B.2, and C.2.  The specific analytical form taken by condition (iii) in the fold, flip, and NS cases is respectively derived in Appendices A.3, B.3, and C.3 in terms of both the original coordinates $z$ and in the coordinates $u$ in the centre manifold. Finally, some details on step three for the Neimark-Sacker case are relegated to Appendix C.4. For simplicity of notation, in the following the 0 superscript stands for evaluation at $(u,\alpha)=(0,0)$ or $(v,\beta)=(0,0)$.

\section{Case I: Border-fold bifurcation}
\label{sec:bfd}
Let the dynamics in the centre manifold $\mathcal{Z}^c$ be described by the
one-dimensional system
\begin{equation}
\label{eq:bfd_cm}
u \mapsto f(u,\alpha), \quad u\in{\bf R}^1,
\end{equation}
with $f^0=0$ (fixed point condition) and $f_u^0=1$ (fold condition).  Under condition (i), map (\ref{eq:bfd_cm}) can be reduced to normal form (first step, see Appendix A.1) with invertible changes of variable and parameter  $v=v(u,\alpha)$, $\beta=\beta(\alpha)$, becoming
\begin{equation}
\label{eq:bfd_nf}
v\mapsto\beta_1+v+sv^2+O(v^3),
\end{equation}
where $s=\mathrm{sign}(f^0_{uu})$.  In these variables, the fold curve has equation $\beta_1=0$ in the plane $(\beta_1,\beta_2)$, and the corresponding non-hyperbolic fixed point is located at $v=0$.

We now turn our attention to the discontinuity boundary (\ref{eq:Sigma}) (second step, see Appendix A.2).
Condition (ii), ensuring transversal intersection of the centre manifold $\mathcal{Z}^c$ and the discontinuity boundary $\mathcal{H}$, implies local existence and uniqueness of a smooth function
$$
\sigma(\beta)=\sigma_{\beta_1}^0\beta_1+\sigma_{\beta_2}^0\beta_2+
O(\|\beta\|^2),
$$
such that the intersection of $\mathcal{H}$
with $\mathcal{Z}^c$ is located at $v=\sigma(\beta)$.  Then by condition (iii) (see Appendix A.3 for the analytical expression) we know that moving along the fold curve, that is, along the $\beta_2$-axis, the fixed point at $v=0$ crosses $\mathcal{H}$ at $\beta_2=0$.  As a consequence, we have $\sigma_{\beta_2}^0\neq0$.

We are now ready to find the equation of the border collisions in the plane $(\beta_1,\beta_2)$ (third step). The two fixed points of the normal form map (\ref{eq:bfd_nf}) are located at $\bar{v}^{\pm}(\beta)=\pm\sqrt{-s\beta_1}+O(\|\beta\|^2)$ ($\bar{v}^-$ being stable and $\bar{v}^+$ unstable for $s=1$, and viceversa for $s=-1$), and lie on the
discontinuity boundary (\ref{eq:Sigma}) along the curves
\begin{equation}
\label{eq:bfd_1}
\pm\sqrt{-s\beta_1}=\sigma_{\beta_1}^0\beta_1+\sigma_{\beta_2}^0\beta_2+
O(\|\beta\|^2).
\end{equation}
Since $\sigma_{\beta_2}^0\neq 0$,
equation (\ref{eq:bfd_1}) for small $\|\beta\|$ becomes
\begin{equation}
\label{eq:bfd_2}
\pm\sqrt{-s\beta_1}\simeq \sigma_{\beta_2}^0\beta_2,
\end{equation}
and gives the asymptotics, locally to $\beta=0$, of the two
border-collision bifurcation curves involving the fixed points $\bar{v}^{\pm}$.  The invertible parameter change $\beta=\beta(\alpha)$ easily provides the asymptotics in the original $\alpha$ parameters.

Depending upon the sign of $s$ in the normal form map (\ref{eq:bfd_nf}), of $\sigma_{\beta_2}^0$ in (\ref{eq:bfd_2}), and of $h_u^0$ in (ii), there are eight generic cases, two of which are reported in Fig.~\ref{fig:bfd}.
\begin{figure}[t!]
\centerline{\includegraphics[width=1\textwidth]
{fig_bfd.eps}}
\caption{Border-fold bifurcation.
Bifurcation curves: LP, fold (limit point, red);
BC$^{s}$, border collision of the stable fixed point ($\bar{v}^{-}$, left; $\bar{v}^{+}$, right) of map (\ref{eq:bfd_nf}) (green), BC$^{u}$, border collision of the unstable fixed point ($\bar{v}^{+}$, left; $\bar{v}^{-}$, right) of map (\ref{eq:bfd_nf}) (blue).
Region labels: $0$, no fixed point in $V^{-}(\beta):=\{v:h^{\mathrm{NF}}(v,\beta)<0\}$;
$1$, $\bar{v}^{-}$ is the only fixed point in $V^{-}(\beta)$;
$2$, both fixed points $\bar{v}^{\pm}$ lie in $V^{-}(\beta)$.}
\label{fig:bfd}
\end{figure}
The other six can be reduced to these two by suitable parameter changes.  In fact, the four cases with $\sigma_{\beta_2}^0<0$ are symmetric with respect to the $\beta_1$-axis to the corresponding cases with $\sigma_{\beta_2}^0>0$, while the four cases with $h_u^0<0$ can be reduced to cases with $h_u^0>0$ by changing the sign of $s$ and rotating the figure.  Note that only half of the $\beta_2$-axis can be said to belong to the fold curve (LP), since along the other half the two fixed points $\bar{v}^{\pm}$
collide at $v=0$ on the undescribed side of the discontinuity boundary
(\ref{eq:Sigma}), i.e., $h^{\mathrm{NF}}(0,\beta)>0$.

\section{Case II: Border-flip bifurcation}
\label{sec:bfp}
Let the dynamics in the centre manifold $\mathcal{Z}^c$ be described by the
one-dimensional system
\begin{equation}
\label{eq:bfp_cm}
u \mapsto f(u,\alpha), \quad u\in{\bf R}^1,
\end{equation}
with $f^0=0$ (fixed point condition) and $f_u^0=-1$ (flip condition).  Through a parameter-dependent translation, we can ensure that $f(0,\alpha)=0$, i.e., that $u=0$ is a fixed point for all $\alpha$ in a neighbourhood of $\alpha=0$.   Under condition (i), map (\ref{eq:bfp_cm}) can be reduced to normal form (first step, see Appendix B.1) with invertible changes of variable and parameter  $v=v(u,\alpha)$, $\beta=\beta(\alpha)$, becoming
\begin{equation}
\label{eq:bfp_nf}
v\mapsto-(1+\beta_1)v + sv^3+O(v^4),
\end{equation}
with $s=\mathrm{sign}((1/4)(f^0_{uu})^2+(1/6)f^0_{uuu})$.  In these variables, the flip curve has equation $\beta_1=0$ in the plane $(\beta_1,\beta_2)$, and the corresponding non-hyperbolic fixed point is located at $v=0$.  Moreover, parameters can be chosen so that the border collision of the fixed point in the origin has equation $\beta_2=0$.

We now turn our attention to the discontinuity boundary (\ref{eq:Sigma}) (second step, see Appendix B.2).
Condition (ii), ensuring transversal intersection of the centre manifold $\mathcal{Z}^c$ and the discontinuity boundary $\mathcal{H}$, implies local existence and uniqueness of a smooth function
$$
\sigma(\beta)=\sigma_{\beta_1}^0\beta_1+\sigma_{\beta_2}^0\beta_2+
O(\|\beta\|^2),
$$
such that the intersection of $\mathcal{H}$ with $\mathcal{Z}^c$ is located at $v=\sigma(\beta)$.  Moreover, thanks to the parameter choice in (\ref{eq:bfp_nf}), $\sigma_{\beta_1}^0=0$, since the fixed point $v=0$ lies on $\mathcal{H}$ when $\beta_2=0$.  Then by condition (iii) (see Appendix B.3 for the analytical expression) we know that moving along the flip curve, that is, along the $\beta_2$-axis, the fixed point at $v=0$ crosses $\mathcal{H}$ at $\beta_2=0$.  As a consequence, we have $\sigma_{\beta_2}^0\neq0$.

We are now ready to find the equation of the border collisions in the plane $(\beta_1,\beta_2)$ (third step).  Near $(v,\beta_1)=(0,0)$ the normal form map (\ref{eq:bfp_nf}) iterated twice
has one fixed point in $v=0$ (which is also a fixed point of map (\ref{eq:bfp_nf}))
and two others in $\bar{v}^{\pm}(\beta)=\pm\sqrt{s\beta_1}+O(\|\beta\|^2)$
(period-two cycle). In particular, $\bar{v}^{\pm}$ lie on discontinuity boundary (\ref{eq:Sigma}) along the curves
\begin{equation}
\label{eq:bfp_1}
\pm\sqrt{s\beta_1}=\sigma_{\beta_2}^0\beta_2+
O(\|\beta\|^2).
\end{equation}
Since $\sigma_{\beta_2}^0\neq 0$,
equation (\ref{eq:bfp_1}) for small $\|\beta\|$ becomes
\begin{equation}
\label{eq:bfp_2}
\pm\sqrt{s\beta_1}\simeq \sigma_{\beta_2}^0\beta_2,
\end{equation}
and gives the asymptotics, locally to $\beta=0$, of the
border-collision bifurcation curves involving the two points $\bar{v}^{\pm}$
of the period-two cycle.  The invertible parameter change $\beta=\beta(\alpha)$ provides the asymptotics in the original $\alpha$ parameters.

Depending upon the sign of $s$ in the normal form map
(\ref{eq:bfp_nf}), of $\sigma_{\beta_2}^0$ in
(\ref{eq:bfp_2}), and of $h_u^0$ in (ii),
there are eight generic cases. However, again, only two cases are relevant
(see Fig.~\ref{fig:bfp}),
\begin{figure}[t!]
\centerline{\includegraphics[width=1\textwidth]
{fig_bfp.eps}}
\caption{Border-flip bifurcation.
Bifurcation curves: PD, flip (period doubling, red);
BC$_1^{s,u}$, border collision of the fixed point $v=0$ (stable and unstable branches, blue);
BC$_2^{s,u}$, border collision of the stable or unstable period-two cycle.
Region labels: $0$, no fixed point or period-two cycle in $V^{-}(\beta):=\{v:h^{\mathrm{NF}}(v,\beta)<0\}$;
$1$, $v=0$ is a fixed point in $V^{-}(\beta)$ and there is no period-two cycle,
or it does not lie entirely in $V^{-}(\beta)$;
$2$, the fixed point $v=0$ coexists in $V^{-}(\beta)$ with the period-two cycle.}
\label{fig:bfp}
\end{figure}
because all others can be reduced to these two by
suitable parameter changes.  Here, both the four cases with $\sigma_{\beta_2}^0<0$ and those with $h_u^0<0$, are symmetric with respect to the $\beta_1$-axis to the corresponding cases with $\sigma_{\beta_2}^0>0$ or $h_u^0>0$.  Also note that only half of the $\beta_2$-axis can be said to belong to the flip curve (PD), since along the other half the fixed point $v=0$ lies on the
undescribed side of the discontinuity boundary (\ref{eq:Sigma}), i.e., $h^{\mathrm{NF}}(0,\beta)>0$.
Similarly, only one of the two branches in (\ref{eq:bfp_2}) constitutes the border-collision curve involving the period-two cycle (stable, BC$_2^s$; unstable, BC$_2^u$), since along the other branch $h^{\mathrm{NF}}(\bar{v}^\pm,\beta)\geq0$.

\section{Case III: Border-Neimark-Sacker bifurcation}
\label{sec:bns}
Let the dynamics in the centre manifold $\mathcal{Z}^c$ be described by the
two-dimensional system
\begin{equation}
\label{eq:bns_cm}
u \mapsto f(u,\alpha), \quad u\in{\bf R}^2,
\end{equation}
with $f^0=0$ (fixed point condition) and with eigenvalues
$\lambda^0$ and $\bar{\lambda}^0$
(the overbar stands for complex conjugation)
of the $2\times 2$ Jacobian $f_u^0$ given by
$$
\lambda(\alpha)=(1+g(\alpha))e^{i\theta(\alpha)},
$$
with $g^0=0$ (Neimark-Sacker, NS, condition).  As in the flip case, assume that $f(0,\alpha)=0$ for all $\alpha$ in a neighbourhood of $\alpha=0$.  Under condition (i), map (\ref{eq:bns_cm}) can be reduced to normal form in polar coordinates (first step, see Appendix C.1) with invertible changes of variable and parameter  $\rho=\rho(u,\alpha), \varphi=\varphi(u,\alpha)$, $\beta=\beta(\alpha)$, becoming
\begin{subequations}
\label{eq:bns_nf}
\begin{eqnarray}
\label{eq:bns_nf_r}
\rho & \mapsto & \rho(1+\beta_1+a(\beta)\rho^2)+\rho^4 R(\rho,\varphi,\beta),\\
\label{eq:bns_nf_p}
\varphi & \mapsto & \varphi+\theta(\alpha(\beta))+
\rho^2 Q(\rho,\varphi,\beta),
\end{eqnarray}
\end{subequations}
where $a^0\neq 0 $. In these variables, the NS curve has equation $\beta_1=0$ in the plane $(\beta_1,\beta_2)$, and the corresponding non-hyperbolic fixed point is located at $v=0$ (with $v_1=\mathrm{Re}(\rho e^{i\varphi})$ and $v_2=\mathrm{Im}(\rho e^{i\varphi})$).  Moreover, parameters can be chosen so that the border collision of the fixed point in the origin has equation $\beta_2=0$.

We now turn our attention to the discontinuity boundary (\ref{eq:Sigma}) (second step, see Appendix C.2).  Condition (ii), ensuring transversal intersection of the centre manifold $\mathcal{Z}^c$ and the discontinuity boundary $\mathcal{H}$, implies local existence and uniqueness of a smooth function
$$
\sigma(\beta)=\sigma_{\beta_1}^0\beta_1+\sigma_{\beta_2}^0\beta_2+
O(\|\beta\|^2),
$$
measuring the distance between the origin and the boundary, with positive/negative values if $h^{\mathrm{NF}}(0,\beta)$ is negative/positive, in order to make $\sigma(\beta)$ differentiable at $\beta=0$.  Moreover, thanks to the parameter choice in (\ref{eq:bns_nf}), $\sigma_{\beta_1}^0=0$, since the fixed point $v=0$ lies on $\mathcal{H}$ when $\beta_2=0$. Then by condition (iii) (see Appendix C.3 for the analytical expression) we know that moving along the NS curve, that is, along the $\beta_2$-axis, the fixed point at $v=0$ crosses $\mathcal{H}$ transversely at $\beta_2=0$.  As a consequence, we have $\sigma_{\beta_2}^0\neq0$.

We are now ready to find the equation of the border collisions in the plane $(\beta_1,\beta_2)$ (third step). Near $\beta=0$, the normal form map (\ref{eq:bns_nf}) has an fixed point in
$\rho=0$ and a closed invariant curve that is contained in the annular region
\begin{equation}
\label{eq:bns_ar}
\left\{(\rho,\varphi):
\sqrt{-\Frac{\beta_1}{a(\beta)}}(1-\beta_1^{\gamma-1/2})\le\rho\le
\sqrt{-\Frac{\beta_1}{a(\beta)}}(1+\beta_1^{\gamma-1/2}),\,
\varphi\in[0,\,2\pi]\right\},\quad\Frac{1}{2}<\gamma<1
\end{equation}
(see Appendix C.4).  The two circles delimiting the annular region (\ref{eq:bns_ar})
touch the discontinuity boundary along the curves
\begin{equation}
\label{eq:bns_1}
\sqrt{-\Frac{\beta_1}{a(\beta)}}(1\pm\beta_1^{\gamma-1/2})=
\sigma_{\beta_2}^0\beta_2+O(\|\beta\|^2).
\end{equation}
Since $\sigma_{\beta_2}^0\neq 0$,
equation (\ref{eq:bns_1}) for small $\|\beta\|$ becomes
\begin{equation}
\label{eq:bns_2}
\sqrt{-\Frac{\beta_1}{a^0}} \simeq \sigma_{\beta_2}^0\beta_2,
\end{equation}
and gives a unique asymptotic, locally to $\beta=0$, for the grazing
bifurcation curves of both circles. The same asymptotic therefore holds
for the grazing bifurcation involving the invariant curve (the uniqueness of the bifurcation curve is granted by the elliptical shape of the invariant curve near $\beta=0$).
Again, the invertible parameter change $\beta=\beta(\alpha)$ provides the asymptotics in the original $\alpha$ parameters.

Depending upon the sign of $a^0$ in the normal form map (\ref{eq:bns_nf}) and of $\sigma_{\beta_2}^0$ in (\ref{eq:bns_2}), there are four generic cases. However, again, only two cases are relevant (see Fig.~\ref{fig:bns}),
\begin{figure}[t!]
\centerline{\includegraphics[width=1\textwidth]
{fig_bns.eps}}
\caption{Border-NS bifurcation.
Bifurcation curves: NS, Neimark-Sacker (red);
BC$^{s,u}$, border collision of the fixed point $v=0$ (stable and unstable branches, blue);
GR$^{s,u}$, grazing of the stable or unstable torus (green).
Region labels: $0$, no fixed point or invariant curve in
$V^{-}(\beta):=\{v:h^{\mathrm{NF}}(v,\beta)<0\}$;
$1$, $v=0$ is a fixed point in $V^{-}(\beta)$ and there is no invariant curve,
or it does not lie entirely in $V^{-}(\beta)$;
$2$, both the fixed point $v=0$ and the invariant curve lie in $V^{-}(\beta)$.}
\label{fig:bns}
\end{figure}
because those with $\sigma_{\beta_2}^0<0$ are symmetric with respect to the $\beta_1$-axis to the cases with $\sigma_{\beta_2}^0>0$.  Also note that only half of the $\beta_2$-axis can be said to belong to the
NS curve, since along the other half the fixed point $v=0$ lies on the
undescribed side of the discontinuity boundary (\ref{eq:Sigma}), i.e.,
$h^{\mathrm{NF}}(0,\beta)>0$.  Similarly, only half of the parabola in (\ref{eq:bns_2}) constitutes the grazing bifurcation curve involving the invariant curve (stable, GR$^s$; unstable, GR$^u$), since along the other half the invariant curve is composed of points $v$ with $h^{\mathrm{NF}}(v,\beta)\geq0$.

\section{Examples}
\label{sec:ex}
We now present three specific examples, one for each of the three
codimension-two bifurcations analysed in the previous sections.
The three examples deal with different classes of nonsmooth systems
(an impacting, a hybrid, and a piecewise smooth system) and describe interesting applications in different fields of science and engineering
(ecology, social sciences, and mechanics).

\subsection*{An impacting model of forest fires}
\label{sec:ex1}
For an example of border-fold bifurcation, we consider the
forest fire impacting model presented in \citep{Dercole05a,Maggi06}.
The model describes the vegetational growth with the following two
(smooth) ODEs:
\begin{eqnarray*}
\dot{B} & = & \ds r_B B \left(1 - \Frac{B}{K_B}\right) - \alpha B T,\\
\dot{T} & = & \ds r_T T \left(1 - \Frac{T}{K_T}\right),
\end{eqnarray*}
one for the surface layer (bush, $B$) and one for the upper layer (trees, $T$).
Fire episodes are represented by instantaneous events (impacts), that occur
when the biomasses $(B,T)$ of the two layers reach one of three
specified impacting boundaries:
a bush ignition threshold $\rho_BK_B$ triggering bush-only fires that map the bush biomass to $\lambda_B\rho_BK_B$, $0<\lambda_B,\rho_B<1$;
a tree ignition threshold $\rho_TK_T$ triggering trees-only fires that map the trees biomass to $\lambda_T\rho_TK_T$, $0<\lambda_T,\rho_T<1$;
and the segment connecting points $(\sigma_BK_B,\rho_TK_T)$ and
$(\rho_BK_B,\sigma_TK_T)$, $0<\sigma_B<\rho_B$, $0<\sigma_T<\rho_T$,
triggering mixed fires with post-fire conditions suitably assigned as a
function of pre-fire conditions (see \citep{Maggi06} for more details).

For the parameter setting $r_1=0.375$, $r_2=0.0625$, $\alpha=0.43$,
$K_B=K_T=1$, $\rho_B=0.85$, $\rho_T=0.93$, $\lambda_B=0.03$, $\lambda_T=0.01$,
$\sigma_B=0.61$, $\sigma_T=0.3$ (corresponding to Mediterranean forests),
the system is characterised by a globally stable period-one cycle composed
of a growth orbit and a mixed fire.
Numerical continuation (by means of {\sc Auto07p} \citep{Auto07p}) of the
cycle in the parameter plane $(\rho_B,\rho_T)$ identifies two (codimension-one)
bifurcations: a fold (red curve in Fig.~\ref{fig:ex1})
\begin{figure}[t!]
\centerline{\includegraphics[scale=0.9]
{fig_ex1.eps}}
\caption{Example of border-fold bifurcation.
Bifurcation curves: fold (red);
border collision of the period-one stable cycle (blue);
border collision of the period-one unstable cycle (green).
Region labels as in Fig.~\ref{fig:bfd}.}
\label{fig:ex1}
\end{figure}
and a grazing of the
growth orbit with the bush ignition threshold (blue curve).
The two curves merge together at the border-fold bifurcation (black) point
and, as predicted by the analysis carried out in Sect.~\ref{sec:bfd}, the
grazing bifurcation of the unstable cycle involved in the fold (green curve)
emanates tangentially to the fold curve from the codimension-two bifurcation point.

\subsection*{A hybrid model of two-party democracies}
\label{sec:ex2}
For an example of border-flip bifurcation, we consider the hybrid model
presented in \citep{Colombo08a} for describing the dynamics of
two-party democracies.  The model describes the evolution of the size of two lobbies (of sizes $L_D$ and $L_R$), one associated to each party (parties $D$ and $R$, respectively), and assumes that the individuals belonging to the lobby of the party at the government erode the welfare ($W$) at a rate proportional to the size of the lobby; a lobby can grow only as long as its party is at the government, and decays otherwise; a small fraction of the lobbyists not at the government defect and switch to the other lobby; elections are held once every $T$ years, and people vote for the party that has the less damaging lobby at the time of the elections.  Altogether, the dynamics is captured by two sets of ODEs,
namely
\begin{eqnarray*}
\dot{W} & = & r(1-W-a_DL_D)W,\\
\dot{L}_D & = & (e_Da_DW-d_D)L_D + k_RL_R,\\
\dot{L}_R & = & (-d_R-k_R)L_R,
\end{eqnarray*}
when the $D$-party is at the government, and
\begin{eqnarray*}
\dot{W} & = &  r(1-W-a_RL_R)W,\\
\dot{L}_D & = & (-d_D-k_D)L_D,\\
\dot{L}_R & = & (e_Ra_RW-d_R)L_R + k_DL_D,
\end{eqnarray*}
when the $R$-party is at the government.  Here, $r$ is the intrinsic growth rate of the welfare, $a$ represents the aggressiveness of a lobby, $e$ is the recruitment coefficient of a lobby, and $d$ and $k$ are respectively the rate at which individuals abandon the lobbies or defect. In the region of the state space where $a_DL_D < a_RL_R$ ($a_DL_D > a_RL_R$) the $D$-lobby ($R$-lobby) is less damaging and thus wins the elections. The condition $a_DL_D = a_RL_R$ therefore defines the discontinuity boundary (see \citep{Colombo08a} for more details).

In the $(a_D,T)$ plane, with parameters $a_R=1$, $r=0.2$, $e_D=e_R=6$, $d_D=d_R=1.8$, $k_D=k_R=0.06$, the system has a very complex bifurcation diagram (see for example Fig.~1 in \citep{Colombo08a}).  In particular, near $a_D=0.38$, $T=3.2$, a flip (red curve in Fig.~\ref{fig:ex2})
\begin{figure}[t!]
\centerline{\includegraphics[scale=0.9]
{fig_ex2.eps}}
\caption{Example of border-flip bifurcation.
Bifurcation curves: flip (red);
border collision of the period-one cycle (blue);
border collision of the period-two cycle (green).
Region labels as in Fig.~\ref{fig:bfp}.
}
\label{fig:ex2}
\end{figure}
and a border collision (blue curve) of a period-$2T$ cycle meet at the border-flip (black) point and, as predicted by the analysis carried out in Sect.~4, a border collision of the period-4T cycle (green curve) emanates from the codimension-two point tangentially to the flip curve.

\subsection*{A piecewise smooth model of railway wheelset dynamics}
\label{sec:ex3}
For an example of border-NS bifurcation, we consider a two degrees of freedom piecewise smooth model of a suspended railway wheelset with dry friction dampers, subject to a sinusoidal disturbance representing the deformations of the track.  The model is based on that presented in \citep{True03, Knudsen92}, where the track deformation was not taken into account, and its analysis will be published elsewhere.  Since a detailed explanation of the equations and parameters  goes beyond the scope of this paper, here we only report the equations and describe a few key parameters (see \citep{Knudsen92} and \citep{True03} for the details).   The model consists of the following piecewise smooth equations:
$$
\begin{array}{rcl}
 \dot{x}_1 & = & \tilde{x}_2,\\
 \dot{x}_2 & = & \Frac{1}{m}(-2F_x-2K_s \tilde{x}_1-	\mbox{sign}(x_2)\mu),\\
 \dot{x}_3 & = & x_4,\\
 \dot{x}_4 & = & \Frac{1}{I}(-2AF_y),
\end{array}
$$
where
$$
\begin{array}{c}
\tilde{x}_1=x_1+a\sin(\omega t),\quad \tilde{x}_2=x_2+a\omega\cos(\omega t),\\ \mu=(\mu_d(1- {\rm sech} (\alpha \tilde{x}_2)) + \mu_s  {\rm sech} (\alpha \tilde{x}_2)),\\
F_x=\Frac{\xi_x F_r}{\Psi \xi_r}, \quad F_y=\Frac{\xi_y F_r}{\Phi \xi_r},\quad F_r=
\left\{\begin{array}{l}
\xi_r C\left(1-\Frac{C \xi_r}{3 \mu_t} + \Frac{C^2 \xi^2_r}{27 \mu^2_t}\right)\mbox{ if $C\xi_r<3\mu_t$},\\
\mu_t \mbox{ otherwise,}\\
\end{array}\right.\\
\xi_x=\Frac{\tilde{x}_2}{V}-x_3, \quad \xi_y=\Frac{Ax_4}{V}+\Frac{\lambda \tilde{x}_1}{r_0}, \quad \xi_r=\sqrt{\left(\Frac{\xi_x}{\Psi}\right)^2 + \left(\Frac{\xi_y}{\Phi}\right)^2}.
\end{array}
$$
Here $\omega=2\pi V/l$, $a$ and $l$ are the amplitude and wavelength of the sinusoidal disturbance, $V$ is the speed of the wheelset, and $\lambda$ measures the conicity of the wheels. The system's state space is therefore partitioned in four regions, depending on the signs of $x_2$ and of $C\xi_r-3\mu_t$, so that $x_2=0$ and $C\xi_r=3\mu_t$ define two discontinuity boundaries.

The system's dynamics was studied, with TC-HAT \citep{Thota08b}, in the $(V,\lambda)$ plane, with the following values of the parameters: $m = 1022$, $K_s = 1e6$, $I = 678$, $A = 0.75$, $a=0.001$, $\mu_d = 1000$, $\alpha = 50$, $\mu_s = 1200$, $\Psi = 0.54219$, $\Phi = 0.60252$, $C = 6.5630e6$, $\mu_t = 1e5$, $r_0 = 0.4572$, $l=10$.  For large values of $V$, a grazing of a stable cycle with the boundary $x_2=0$ and a NS take place (blue and red in Fig.~\ref{fig:ex3}), and meet at
the border-NS (black) point.
\begin{figure}[t!]
\centerline{\includegraphics[scale=0.9]
{fig_ex3.eps}}
\caption{Example of border-NS bifurcation.
Bifurcation curves: Neimark-Sacker (red);
border collision of the period-one cycle (blue);
border collision of the torus (green).
Region labels as in Fig.~\ref{fig:bns}.}
\label{fig:ex3}
\end{figure}
Then, by systematically evaluating 1000 iterations (after transient) of the Poincar\'e map of the torus on a suitable cross-section, and by continuing the line on which the obtained torus image grazes the discontinuity boundary induced on the cross-section, we were able to trace an approximation of the grazing curve of the torus (green in Fig.~\ref{fig:ex3}).  More rigorous methods, based for example on discretisation of the invariant curve (see, e.g., \citep{Kevrekidis85,Dankowicz06}) could be used to obtain a more precise estimate of the quadratic coefficient.  This lies however beyond of the scope of this paper.  As predicted by the analysis carried out in Sect.~\ref{sec:bns}, the curve emanates from the codimension-two point tangentially to the NS curve.

\section{Concluding remarks}
\label{sec:cr}
We have analysed the geometry of bifurcation curves around three codimension-two bifurcations in nonsmooth systems, namely the border-fold, the border-flip, and the border-Neimark-Sacker. Rather than aiming at the complete unfolding of the dynamics of a particular class of nonsmooth systems (e.g., piecewise smooth, impacting, or hybrid) dealing with a particular geometry of the involved discontinuity boundary (e.g., smooth or corner), we have focused on those results which are general to all scenarios.  Our approach applies to continuous-time as well as discrete-time systems, and basically consists of the analysis of a discrete-time (Poincar\'e) map defined only on one side of a boundary in its state space.  Explicit genericity conditions are listed and explained for each codimension-two case.

Of course, the weakness of this approach is that it cannot provide the complete unfolding of the bifurcation, but its power resides in its generality: as shown in the three examples that we have reported, it applies to a very broad class of nonsmooth systems and it may be relevant in various fields of science and engineering.

The natural sequel of this work would certainly aim at more detailed results, and possibly at the complete unfolding, of the codimension-two bifurcations analysed here, with specific reference to some smaller class of nonsmooth systems.

\section*{Acknowledgements}
The first contributions on the topic of this paper were presented and discussed by Mario di Bernardo, Piotr Kowalczyk, and Yuri A. Kuznetsov in the context of piecewise smooth systems (informal meeting at the Bristol Centre for Applied Nonlinear Mathematics, University of Bristol, UK, summer 2003) and by Arne Nordmark for impacting system (at the meeting ``Piecewise smooth dynamical systems: Analysis, numerics and applications'', University of Bristol, Sept.~13--17, 2004).
The authors are grateful to M.~B., P.~K., Yu.~A.~K., and A.~N. for sharing their preliminary results, and to two anonymous reviewers whose criticisms significantly improved the paper.

\bibliographystyle{siam}

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This Webinar series includes 10 sessions. The Special Topics' series is designed for experienced professionals, covering changes to the 2018 I-Codes, specific details associated with stationary power systems, distilleries, processing and extraction facilities, and hotwork. and other equipment and processes. This registration page is for the Special Topic series.


TAMPA -- Having one of the best goal-scorers on earth playing for your team can't be seen as anything but a blessing for a coach.
Tampa Bay Lightning coach Jon Cooper has the luxury of having Steven Stamkos on his bench every game, but if he spends too much time on that bench, the spotlight turns to Cooper, not in a good way.
When the Lightning needed a goal in the third period of Game 5 of the Stanley Cup Final against the Chicago Blackhawks on Saturday, Stamkos was just another player when you look at his ice time.
The Lightning did not get that goal, lost 2-1, and head back to Chicago down 3-2 in the best-of-7 series with Game 6 on Monday (8 p.m. ET; NBC, CBC, TVA Sports).
"At some point, you just can't keep playing the same guy over and over because then they become infective," Cooper said. "But our top players are getting the minutes. It's the full 60 minutes here we have to dwell on. We can't sit here and say, I know we need that goal in the last minute, but I've learned that you keep putting the same guys out the last four minutes, by the time you got to the last two minutes, they've got nothing left."
Cooper managed a bench missing one of his top players, forward Nikita Kucherov, for almost the entire game after he collided with the goal post and left at 5:22 of the first period. That forced Cooper to mix and match his lines, and Stamkos spent most of the first half on his regular line with Valtteri Filppula and Alex Killorn before switching to take Kucherov's spot on a line with Tyler Johnson and Ondrej Palat.
Over the final 7:41 of the third period, Stamkos, Johnson and Palat played three shifts, or as many as Cedric Paquette and J.T. Brown did. Stamkos' ice time over that 7:41 was 2:47, 10 seconds more than Paquette.
Paquette has scored twice in the Final and Stamkos hasn't scored in seven games, but that could not have been a factor here.
"I mean, I'm not the coach. You'd have to ask him that," Stamkos said when asked about shortening the bench at the end of the game. "But we've relied on our depth all season, and the guys that are thrown out there are going to give it their all, and that's all we can ask for."
At one point, Stamkos spent 2:26 on the bench before he was sent on the ice for his final shift of the game at 18:28 of the third period, shortly before goaltender Ben Bishop was pulled for an extra attacker with a little more than a minute left. Cooper never used his timeout to get his top players some extra rest.
Much like a manager in the World Series bringing in his ace on two days' rest to get out of a late-inning jam, being this close to a championship sometimes calls for measures you normally would not use in the regular season. This seemed like one of those times.
"These guys know. We know when they're ready. They tell me when they're ready to go back out," Cooper said. "It's a team. [Stamkos] is a huge part of this. He's trying to score like everybody else. But we can't sit here and say we're going to put one single player out there for the last four or five minutes and expect results.
"He gave it everything he had. So did everybody else. We're struggling to score right now."
The Lightning have to defeat the Blackhawks twice in a row to win the Stanley Cup, and there's nothing saying that would be any different had Stamkos played an extra shift in the third period.
Though the Lightning are inexperienced when compared to the Blackhawks, this is a scenario they have faced in these Stanley Cup Playoffs. In the first round against the Detroit Red Wings, the Lightning lost Game 5 at home, had to win Game 6 in Detroit, and closed the series with a 2-0 win in Game 7 in Tampa.
Now the Lightning need to do the same thing, on a far bigger stage with far greater scrutiny.
"We've been in this position before just a month and a half ago in the Detroit series," Cooper said. "This team has found a way. That's why we are where we are right now.
"We've been in so many different situations in this playoffs, whether we've been up 3-2 or down 3-2, we've gone through this now. People can't sit here and say, 'the inexperienced Lightning.' I think we're the 'Gamer Lightning.' These guys have gamed out this playoffs, and I expect nothing less in Game 6 in Chicago."


Internal combustion engines, in mostly diesel rather than petrol form, power the vast majority of light commercial vehicles on the UK's roads.
Attendances at auction during the early summer period were generally good with buyers continuing to actively seek out the best stock available.
Well, for an exercise that was meant to establish clarity the outcome is about as clear as mud.
REMARKETING: Are AFVs a viable alternative?
Andy Picton's blog: What happened in February?
How will the leasing and rental market fare in 2017?
What Van? celebrates the best in the business.


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text:
Noël van 't End (born 15 June 1991) is a Dutch judoka.

He competed at the 2016 Summer Olympics in Rio de Janeiro, in the men's 90 kg.

In 2021, he won the gold medal in his event at the 2021 Judo World Masters held in Doha, Qatar.

He holds a black belt.

References

External links

1991 births
Living people
Dutch male judoka
Olympic judoka of the Netherlands
Judoka at the 2016 Summer Olympics
World judo champions
People from Houten
European Games competitors for the Netherlands
Judoka at the 2019 European Games
Judoka at the 2020 Summer Olympics
20th-century Dutch people
21st-century Dutch people
Sportspeople from Utrecht (province)

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Kentucky Senior Games are an "OPEN" Senior Games meaning we welcome competitors from other states.
We have added a 40-49 Masters age division. This division is just for the Kentucky Senior Games. There is no qualifying for this division for the nationals as the National Senior Games are for ages 50+ only.
Also we have an adaptive sports division for each competition as well as special emphasis for entries for our disabled veterans. Non-ambulatory veterans are eligible for free entry in to the Kentucky Senior Games through a joint program between the National Senior Games and the Veterans' Administration.. Email Fmiklavcic@aol.com for information on this program.
Get ready for the 2018 Kentucky Senior Games. This year's Games are the qualifying Games for the 2019 National Senior Games in Albuquerque, New Mexico.
Congratulations to those Kentuckians who competed in the 2017 National Senior Games in Birmingham, Alabama, in June. Kentucky athletes brought home 32 gold, 18 silver and 23 bronze medals. Top eight got on the awards stand and we also had ten 4ths, eight 5ths, thirtenn 6ths, eight 7ths, and eighteen 8th places. Great Job!!!! Thanks for representing.
Kentucky athletes posted 26 golds, 16 silvers, and 7 bronze medals during the National Senior Games in Minneapolis, July 3-16, 2015. Jim Blackwell set an NSGA record in the Men's 75-79 High Jump.
The goal of the Kentucky Senior Games is to maintain or improve the physical, mental, and emotional health of all Kentuckians 50 years of age and over through a year-round program of fitness and health promotion, as well as providing competitive opportunities for those 40-49, even though those athletes do not move on to the National Senior Games. This is accomplished through seasonal competitions at local, regional, and state levels.
The Kentucky Senior Games encourages older adults to stay active and fit, revive past fitness and recreational skills or learn new ones. Social events, ceremonies, and the arts are also an integral part of this successful program.
All men and women 50 years of age and over are eligible to participate in the Kentucky Senior Games as well as those 40-49 in the masters division. We also accept out of state participants.
The Kentucky Senior Games is the only qualifying site in the state of Kentucky for the National Senior Games.


Total synthesis of 1-azapenem (5) is reported. The key intermediate 24 was synthesized by the intramolecular cyclization of 17 and subsequent desilylation. Alkylation of 24 and subsequent hydrogenolysis gave 5.


Hey, guys! This is the second post that I got for you guys which is the fourth post in the Sunday Inspiration series!
I've noticed lately that there's this trend going on around fashion bloggers where they wear 70s-inspired outfits along with the items they're wearing. So of course, that calls for bohemian vibes, fringe, boots, and flared pants! I've actually always loved the idea of flared pants but I think these things, right now, are definitely making their comeback. There's just something exclusively classic and wonderful about them, and you can style them with pretty much anything! It carries a huge fashion statement and it's easy to make it look nice!
I think flared pants mostly come in forms of flared jeans. You can find flared jeans in pretty much any store that sells jeans. I mean we all know that skinny jeans are never going out of style, but it's about time we give flared jeans a little chance!
Do make sure there's a perfect colour and pattern combination on your look. Keep things balanced because flares take up a lot of room, therefore attention. It's easy to go a little too far and make your outfit look like it's a little too much, so keep things simple and let the extravagance go to the bottom of your pants.
Don't pair it with sneakers. Please, please, never sneakers with these. Or platforms, or wedges. If it's flared jeans, then wear some nice heels or maybe like a really cool pair of boots. If it's other than jeans (so maybe it's like chiffon or something similar) then sandals can still work. Sneakers would make your legs look a little sluggish and frankly, you'll look like one of those teenagers in the 70s who never brush their hair.
Don't over-accessorize. In relation to number one, you don't want to go for too much. A simple handbag or clutch will do just fine. A nice hat would be cool also! But try to go light on things like necklaces and bracelets, the noticeable stuff.
Do give it a try and go for a high-waisted look. Maybe buy a nice pair of high-waisted jeans, which you can style with a really nice white shirt or blouse. You know, you might as well emphasize on the pants, so why not? And it'll certainly make your legs look extra longer.
Don't wear it with an oversized poncho or kimono or anything similar. In terms of the silhouettes, a poncho or a kimono will definitely be very eye-catching and it will extremely steal some attention. To go big at the top, and big at the bottom would be too dramatic. If you do want to wear flares but you also need layers, go for a neat blazer or a simple cardigan.
What do you think of flared pants? Do you own a pair? How do you like to wear them?
Leave a comment below, and don't forget to follow me on Bloglovin to stay tuned to my other posts!
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It is! A great comeback too!
I wish I could wear flared pants, but I'm only 5'0", so I just look even shorter! But I love these looks.
I used to have a pair. But I am just too short!
Thank you Naomi! Yeah they're definitely coming back!
I really want a pair, but alas I am too short.
Loving this style post! I need a new pair of flared pants!
Gorgeous! These are all such great flare inspiration!
Great post! I'm still trying to figure out how to wear flare for the coming season! and your post is so helpful!!
Glad this trend is back. Flared pants just give the body a nice sleek look.
Flared pants always come back in the Springtime! Thanks for the inspiration! It kinda makes me wish I hadn't thrown away my flared jeans from way back when!
Oh yes, they are so back.
These flared pants inspirations are great.
she's wearing them when only a few are.
Great post! loving flared pants as well!
I'm a true skinny girl fan but this post has made me really want a nice pair of flared pants!
I felt the same when seeing the trend going around!
I don't think I could ever pull of flared trousers but they look incredible on so many people!
really useful post! I've been wanting a pair for flared pants but wasn't sure how to style them!


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P-values.R illustrates how likely it is that significant P-values are misleading when the magnitude of a between group difference is small relative to within group variation. P-values.pdf was produced by running this code.
credible-intervals.R illustrates how a Bayesian estimate of the difference between the mean of (replicates of) the same populations behaves with respect to coverage (the fraction of replicates in which the 95% credible intervals contain the true difference) and sign (the fraction of time in which the estimate of the difference has the same sign as the true difference). The Bayesian model is implemented in Stan (point.stan).


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This entry was posted on July 1, 2016, 7:14 pm and is filed under About Writing, Marketing, Self-Publishing, The Fiction Writer. You can follow any responses to this entry through RSS 2.0. Both comments and pings are currently closed.


text:
Vlastimil Petržela (* 20. července 1953 v Prostějově) je český fotbalový trenér, bývalý československý fotbalový útočník a reprezentant (účastník MS 1982). Od konce února 2018 byl trenérem FC Fastav Zlín, kde nahradil odvolaného Bohumila Páníka. Po konci sezóny na lavičce po dohodě s majitelem klubu skončil. Nahradil jej Michal Bílek.

Hráčská kariéra
Začínal v Kralicích na Hané. V dorosteneckém věku přestoupil do TJ OP Prostějov, kde se stal v sezoně 1973/74 nejlepším střelcem Jihomoravského krajského přeboru. Odsud si jej vytáhla prvoligová brněnská Zbrojovka, v jejímž dresu debutoval v československé lize. Dále hrál třetí ligu za Železárny Prostějov a ve druhé lize za Sigmu ZTS Olomouc. Následně se do nejvyšší soutěže vrátil v Rudé hvězdě Cheb a pražské Slavii, oblékl též reprezentační dres.

Reprezentace
V československém A-mužstvu debutoval 17. června 1982 ve Valladolidu proti Kuvajtu (nerozhodně 1:1) v rámci Mistrovství světa ve fotbale 1982. Naposled reprezentoval 27. března 1983 v Nikósii proti domácímu Kypru (nerozhodně 1:1, zápas kvalifikace na Mistrovství Evropy ve fotbale 1984).

Prvoligová bilance
V nejvyšší soutěži odehrál 116 utkání a vstřelil v nich 36 branek. V ročnících 1974/75 (2/0), 1975/76 (8/0) a 1976/77 (3/0) hrál za Zbrojovku Brno, na jaře 1981 (13/5) a na podzim 1981 (15/9) hrál za Rudou hvězdu Cheb, poté přestoupil do Slavie Praha, kde působil na jaře 1982 (15/4), 1982/83 (27/10), 1983/84 (27/8) a 1984/85 (6/0).

Trenérská kariéra
Jako trenér vedl Slovan Liberec, Slavii, Spartu, Bohemians 1905, Mladou Boleslav, či Sigmu Olomouc, největších úspěchů ale dosáhl v ruském Zenitu Petrohrad, kde mezi lety 2003 a 2006 vyhrál ruský ligový pohár (2003), získal druhé místo v ruské první lize a postoupil do čtvrtfinále Poháru UEFA.
Později se ale pohádal s majitelem klubu Zenit Petrohrad a byl propuštěn. Pokračoval v gamblerství, později se ale vyléčil.

V sezóně 2009/10 vedl druholigový tým FK Viktoria Žižkov, kde před sezónou podepsal tříletou smlouvu. Smlouvu se Žižkovem ale předčasně ukončil, jelikož čtyři kola před koncem soutěže v květnu 2010 byl odvolán. Vedení klubu to zdůvodňovalo skutečností, že se nepodařilo naplnit postupové ambice.

V červnu 2010 podepsal kontrakt s účastníkem druhé nejvyšší slovenské soutěže (první liga) MFK Zemplín Michalovce. Smlouva zněla na jeden rok plus jeden rok opce. V Michalovcích skončil na jaře 2012 s bilancí 25 výher, 11 remíz a 22 proher a získal 86 ligových bodů.

Od roku 2014 trénoval druholigovou Vlašim. V prosinci 2015 se dohodl s vedením Baníku Ostrava, že od ledna 2016 převezme jeho tým, který byl po podzimní části ligy daleko na posledním místě s pouhými čtyřmi body. Jako cíl deklaroval úsilí vybudovat mužstvo, které se může po sestupu rychle vrátit do první ligy, s tím, že záchrana v probíhajícím ročníku už je nemožná.

Odkazy

Reference

Literatura
 Měsíčník Fotbal 8/2000

Externí odkazy

 Vlastimil Petržela v Impulsech Václava Moravce
 Vlastimil Petržela končí v Michalovcích
 Profil na Playerhistory.com
 Hráčský profil - national-football-teams.com
 Profil na zenit-history.ru

Českoslovenští fotbalisté
Čeští fotbalisté
Fotbalisté TJ OP Prostějov
Fotbalisté FC Zbrojovka Brno
Fotbalisté SK Prostějov
Fotbalisté FC Union Cheb
Fotbalisté SK Sigma Olomouc
Fotbalisté SK Slavia Praha
Českoslovenští fotbaloví trenéři
Čeští fotbaloví trenéři
Čeští držitelé profilicence UEFA
Trenéři SK Slavia Praha
Trenéři FC Slovan Liberec
Trenéři AC Sparta Praha
Trenéři Sigmy Olomouc
Trenéři Zenitu Petrohrad
Trenéři Bohemians Praha 1905
Trenéři FK Mladá Boleslav
Trenéři FK Viktoria Žižkov
Trenéři MFK Zemplín Michalovce
Trenéři Baníku Ostrava
Trenéři FC Tescoma Zlín
Narození 20. července
Narození v roce 1953
Narození v Prostějově
Žijící lidé
Muži
Hráči Mistrovství světa ve fotbale 1982


text:
The Only Thing That Matters Right Now Are the Concentration Camps at the Border
Michelle Ruiz
So, President Trump, Should Your Wife "Go Back" to Her Country, Too?
Kellyanne Conway, Defending Trump, Asked a Reporter: "What's Your Ethnicity?"
The Week in Washington: "So Many Lies—From White Lies to Whoppers"
Lynn Yaeger
One Year After Charlottesville, Protests Shift to Washington
It has been one year since race-fueled violence in Charlottesville, Virginia, saw hundreds of members of various hate groups, including neo-Nazis and Ku Klux Klansmen, march through the University of Virginia campus shouting messages of anti-Semitism and white supremacy. The rally, named "Unite the Right," turned violent when it was met with counterprotesters, which resulted in the death of 32-year-old Heather Heyer, who was killed by a 20-year-old neo-Nazi after he drove his vehicle into the crowd.
Today, as the "Unite the Right 2" rally occurred in Washington on the anniversary of Charlottesville, hundreds of demonstrators also gathered in Charlottesville to honor Heyer and once again denounce racism. Meanwhile, white supremacists shifted the location of their march: A small group of white nationalists, about two dozen, marched in downtown Washington, D.C., making their way to Lafayette Square Park near the White House, where they were met with thousands of counterprotestors that were divided by metal fences and heavy police enforcement. Though the event's permit submitted by Jason Kessler, which was issued by the National Park Service, planned for up to 400 participants, the group of white nationalists proved to be much (much) smaller than anticipated, and indeed much smaller than last year's display.
While tensions rose as counterprotestors dwarfed white nationalists on Sunday, the Trump administration remained characteristically tight-lipped. President Trump did acknowledge the Charlottesville anniversary on Twitter, reversing his statement from last year that claimed there was "blame on both sides," writing yesterday, "The riots in Charlottesville a year ago resulted in senseless death and division. We must come together as a nation. I condemn all types of racism and acts of violence. Peace to ALL Americans!" Daughter Ivanka Trump also took to the platform to speak of last year's protests. "While Americans are blessed to live in a nation that protects liberty, freedom of speech and diversity of opinion," she said on Twitter, "there is no place for white supremacy, racism and neo-nazism in our great country."
Below, influential figures remember Charlottesville. Plus, some coverage of today's rally.
https://twitter.com/realDonaldTrump/status/1028271447632957441
https://twitter.com/CynthiaNixon/status/1028657928348282881
https://twitter.com/JoeBiden/status/1028681867858460674
https://twitter.com/SenSanders/status/1028668707583799298
https://twitter.com/IvankaTrump/status/1028440017180315649
https://twitter.com/TheArtist_MBS/status/1028722792051744768
https://twitter.com/scottheins/status/1028722785336614912
https://twitter.com/girlsreallyrule/status/1028719573724528640
https://twitter.com/KarlaAnnCote/status/1028704748898344960
https://twitter.com/JordanUhl/status/1028726821448245249
https://twitter.com/Journo_Christal/status/1028718197967609856
https://twitter.com/herownbeat/status/1028736769737543681
https://twitter.com/edokeefe/status/1028683770608672768
https://twitter.com/cvillenews_desk/status/1028694386752794624
https://twitter.com/RyanW_Miller/status/1028630277978030080
https://twitter.com/latoshamaria/status/1028723181446680577
Read More Culture Stories:
Three Identical Strangers Revisits a Shadowy Scientific Conspiracy
28-Year-Old Alexandria Ocasio-Cortez Might Just Be the Future of the Democratic Party
Confessions of a Recovering Wannabe Carrie Bradshaw
We Finally Know Who Bit Beyoncé
Is Lindsay Lohan on Her Way to Becoming Our Next Lisa Vanderpump?
Amal Clooney Has Some Choice Words for President Trump
Scenes from the Queer Liberation March, Pride's Anti-Corporate Rally

redpajama_set_name: RedPajamaCommonCrawl


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text:
Home News Policeman run over by drug traffickers finally leaves hospital after 3 months
Policeman run over by drug traffickers finally leaves hospital after 3 months
CREDIT: Twitter
A policeman run over by drug traffickers in Algeciras finally leaves hospital after three months, having spent six weeks in ICU.
INSPECTOR Francisco Javier was part of an operation to intercept a high-end vehicle stolen in Marbella which was carrying a cache of 16 drug bundles collected in Punta Carnero.
The fleeing vehicle rammed his patrol car on September 5, causing it to overturn, leaving the officer with "serious injuries".
After being stabilised at the Punta de Europa Hospital in Algeciras, he was transferred to the Puerta del Mar Hospital in Cadiz.
And on Friday morning, November 27, he was finally discharged. Colleagues and nurses surrounded his bedside as he prepared to return home.
The National Police tweeted: "Congratulations! Our comrade Francisco Javier, inspector who was run over by a drug dealer, left the hospital in #Getafe (#Madrid) today and returned to #Algeciras. Plenty of encouragement."
¡¡¡Enhorabuena!!! 👏👏👏
Nuestro compañero Francisco Javier, inspector arrollado por un narco, ha salido hoy del hospital de #Getafe (#Madrid) de vuelta a #Algeciras
Mucho ánimo💪💪💪, todos estamos contigo#HéroesSinCapa #AlmaAzul pic.twitter.com/j3NsO2O7a4
— Policía Nacional (@policia) November 27, 2020
The driver of the vehicle, who was also seriously injured by the impact, was arrested and charged.
Thank you for taking the time to read this news article "Policeman run over by drug traffickers finally leaves hospital after 3 months". For more UK daily news, Spanish daily news and Global news stories, visit the Euro Weekly News home page.
Previous articleMacron slams attack on black man as 'shameful'
Next articleElton John Accepts British LGBT award
Tara Rippin is a reporter for Spain's largest English-speaking newspaper, Euro Weekly News, and is responsible for the Costa Blanca region.
She has been in journalism for more than 20 years, having worked for local newspapers in the Midlands, UK, before relocating to Spain in 1990.
Since arriving, the mother-of-one has made her home on the Costa Blanca, while spending 18 months at the EWN head office in Fuengirola on the Costa del Sol.
She loves being part of a community that has a wonderful expat and Spanish mix, and strives to bring the latest and most relevant news to EWN's loyal and valued readers.
Share your story with us by emailing newsdesk@euroweeklynews.com, by calling +34 951 38 61 61 or by messaging our Facebook page www.facebook.com/EuroWeeklyNews


Support group open to all adults. Meeting begins with general announcements about upcoming activities, followed by individual introductions and the invitation to share one "golden moment" you've experienced during the previous month. Following that, there will be time for questions and follow-up discussion as well as further opportunity for group support and socialization. Some meetings will have speakers on topics of interest to the autism community in the Baltimore area.
Light refreshments and handouts available along with access to our lending library.
Please be aware that support group meetings are for adults as some topics discussed may not be suitable for children, and child care is not provided.


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On 12 January, IHC launched MAHAA JARRAAFU for the Maldives Transport and Contracting Company (MTCC) at the RMC shipyard in Qidong, China. The Easydredge 3700 is the largest ship in IHC's Easydredge® series and the first trailing suction hopper dredger (TSHD) for MTCC.
With the 3,700m3 MAHAA JARRAAFU, which loosely translates as 'mega dredger', MTCC extends its existing fleet of three IHC Beavers and enters the TSHD market.
The Easydredge® will be used for land reclamation, infrastructure and beach nourishment projects on several islands and resorts in Asia. The activities will contribute to the growth of the tourism industry in the Maldives and give a positive boost to employment.
The acquisition of the MAHAA JARRAAFU was strongly supported by the government in the Maldives and President Abdulla Yameen. The contract for the vessel was awarded in December 2014.
"The contract became effective in August 2015 with a delivery time of 20 months from that point in time," says IHC Project Manager Philip van den Broek. "The keel-laying took place on 25 August 2016, and the ship is expected to be delivered in April 2017, just ahead of schedule. This is due to the excellent communication between our partner yard in Qidong, MTCC and all involved IHC departments in Kinderdijk."
The Easydredge 3700 is, like all ships in the series, equipped with the specially designed Easydredge® dredging equipment, including a state-of-the-art IHC dredge pump and a slender hull for increased speed and low fuel consumption.
In addition, MAHAA JARRAAFU has a 'world dredging' equipment package, which includes an adjustable overflow, bottom doors, jet water system and a shore pumping installation. This makes it a versatile TSHD. The ship is also equipped with an extended suction pipe and an underwater pump that enables the vessel to dredge to a depth of 50 metres.


text:
Author: Chelsey Falb
Jackson Street Plaza Featured in Georgia Trend
The balancing act of enhancing the pedestrian experience while designing for economic vitality is one T&H client, The City of Dublin, Georgia is an expert at. For this reason, the The City of Dublin and Downtown Dublin, GA were featured in a Georgia Trend article...
T&H Project In The News: Stanton Springs North
Grading is the next step in bringing Rivian to life at Stanton Springs North! "Thomas & Hutton will work with Rivian and the state of Georgia to score the bids and then present their recommendation to the JDA within two weeks. Grading is expected to begin...
Laurens County VA and Voter Registration Grand Opening
Infusing new life into a vacant Winn Dixie property, Thomas & Hutton provided construction management services for the demolition of 53,000 square feet to shell and the upfit of 11,800 square feet of the facility into a new Veterans Affairs and Voter Registration facility for...
T&H Project In The News: Tide To Town Urban Trail System
Tide to Town, an ambitious network of 30+ miles of bikeable and walkable trails will connect more than 60 neighborhoods and 30 schools in Savannah and Chatham County, from downtown to the marshes and waterways! Two miles of McQueens Trail, alongside the U.S. 80 causeway...
T&H Project In The News: Seaport 16 Trade Center
Seaport 16 Trade Center will court importers, distributors, and light manufacturing outfits looking to locate along I-16 in Bryan County, 25 miles from the Port of Savannah. Notably, the 797-acre speculative site is a short five miles from the Hyundai Electric Vehicle Plant, Hyundai Motor...
T&H Project In The News – Hyundai Incentive Announcement
T&H is grateful for the opportunity to work with the State and The Savannah Harbor-Interstate 16 Corridor Joint Development Authority (JDA) on this project, representing a $5 billion investment in our region. The vision for this project was laid out 15 years ago, and we...
Atlanta Office Olympics
Just a few miles from Centennial Olympic Park, the T&H Office Olympics Torch was passed to our Atlanta office! Donations collected from the afternoon of spirited competition were collected for Atlanta Angels, a nonprofit that supports youth in Atlanta's foster care system through intentional giving, relationship building,...
Greenville 75th Anniversary Celebration
The 75th Anniversary pop-up celebration continued in Greenville on Thursday, July 21st! Thank you to everyone who came out to celebrate three-quarters of a century at UP On The Roof in downtown Greenville. ...
Charlotte Office Olympics
The Office Olympics torch has been passed to CHARLOTTE! The Olympians faced steep competition in games like plan set javelin, tallest tower, trash can basketball, and office putt-putt. Donations to play and spectate were collected, benefiting the Humane Society of Charlotte! Office Olympics is one...
James Thomas, PE of Thomas & Hutton Named One of Charleston's "Forty Under 40" Young Professionals by Charleston Regional Business Journal
The Charleston Regional Business Journal has selected James Thomas, PE, Principal and Civil Project Manager at Thomas & Hutton for the 2022 Class of Forty Under 40. The Forty Under 40 program honors the top 40 young business professionals across the Charleston region who have...


Welcome to the Modern Blog. These postings are meant to encourage participation. We hope to create discussions and stimulate new ideas. Right now we have a few authors who have been recruited into our blogging staff, so let me introduce everyone.


text:
National Museum of African American Music Receives Major Donation from HCA Healthcare
NASHVILLE, Tenn. (TN Tribune) – HCA Healthcare is donating $650,000 to the National Museum of African American Music (NMAAM). In addition to the $1 million previously committed by the HCA Healthcare Foundation, the health care organization has donated $1,650,000 in support of the museum.
HCA Healthcare's most recent gift will help NMAAM tell the story of America's soundtrack in its One Nation Under a Groove gallery, which will officially be identified as the "One Nation Under a Groove Gallery Sponsored by HCA Healthcare."
"We are thrilled to be a part of welcoming this important cultural institution and helping NMAAM open to the world," said Sherri Neal, chief diversity officer, HCA Healthcare. "It is important to reflect on and celebrate the impact African Americans continue to have in shaping the culture of America, especially through music, and we are proud to help tell that story."
"HCA Healthcare has been a supporter of the museum for nearly a decade, and that support has allowed the museum to put on myriad programming ahead of our opening to the public," said H. Beecher Hicks III, president and CEO of NMAAM. "The museum will expand the Music City brand for Nashville, and it wouldn't be possible without the support of local partners like HCA Healthcare."
Those wishing to visit the museum should visit its website, www.NMAAM.org, to stay updated on when tickets will become available.
Tagged: black music museum, HCA, HCA Healthcare, National Museum of African American Music


Suspect appears to have fired at police officers as they approached on foot.
Milwaukee police on Saturday released portions of body-cam video documenting officers' fatal shooting of a suspect in October.
In the video, shot Oct. 10 outside an apartment building near North 101st Street and West Mill Road, police are shown responding to reports of shots fired in a parking lot.
As officers approached the suspect, later identified as 31-year-old Alonzo L. Smith, the body-cam video shows they repeatedly ordered Smith to "show his hands" and put them on the steering wheel of his car.
He appears to have ignored their commands and, at one point, fires at the officers.
The two officers fired at Smith, striking him in the head and the chest as he fled his car on foot, according to Homicide Division Capt. Joseph McLin, who narrates the video.
Smith was taken to a hospital where he died.
An investigation into the shooting is being led by Wauwatosa police. Additional body-cam video will be available after the investigation is complete, Milwaukee police said.
The video is the second involving a police shooting to be made public this week. The earlier video show police shooting what appears to be an unarmed man at close range on a rooftop last year.
In that footage, Jerry Smith Jr. appears to have spread his arms and fingers as officers approached with guns drawn the afternoon of Aug. 31, 2017, and appears to be getting on the ground as two officers shoot him from close range.
Smith survived but underwent several surgeries and will suffer permanent partial paralysis in his right leg, said Daniel Storm, a private investigator working on Smith's federal civil rights lawsuit against the city. Storm provided the video to the Journal Sentinel.
A cellphone video, taken from another building, also appeared to show Smith was unarmed.
Milwaukee County Chief Deputy District Attorney Kent Lovern concluded in June that Officer Adam Stahl and Officer Melvin Finkley were justified in using deadly force because they believed Smith was armed or was reaching for a gun behind an air conditioning unit on the roof.
No gun was ever found. Smith, who has no criminal record, was not charged with any crime.


text:
Q: How to get closest value from oracle I have a table as shown below and column A datatype is Varchar2, and suppose if i enter 8200.11 as input, i should get 8200.13 ,which is the closest value.How will i get it in oracle ?
A
8200.13
7300.46
8100.98
8200.01

A: In Oracle 12C, you would do:
select a
from t
where a is not null
order by abs(to_number(a) - 8200.11)
fetch first 1 row only;

Although the keep syntax is fine, in earlier versions this would probably be written with a subquery:
select t.*
from (select a
      from t
      where a is not null
      order by abs(to_number(a) - 8200.11)
     ) t
where rownum = 1;

You should not be storing numeric values as strings.

A: Demo setup:
create table demo (a) as
select column_value
from   table(sys.ora_mining_number_nt(8200.13, 7300.46, 8100.98, 8200.01));

Test:
select min(a)
       keep (dense_rank first
             order by abs(a - 8200.11)) as closest
from   demo;

   CLOSEST
----------
   8200.13

1 row selected.

SQL Fiddle
The first function returns the first value of x in order of y. (There is also a corresponding last function.) More details
If the column is a varchar2 (why?) then the implicit conversion to number should work unless the column contains unexpected values. From 12.2 onwards you could avoid conversion errors with:
select min(a)
       keep (dense_rank first
             order by abs(to_number(a default null on conversion error) - 8200.11)) as closest
from   demo;

redpajama_set_name: RedPajamaStackExchange


text:
THE VILLAGERS OF COLLINGHAM AND LINTON WHO SERVED IN WORLD WAR ONE
COLLINGHAMANDDISTRICTWARARCHIVE.INFO
This site commemorates the men and women of Collingham, Linton and Micklethwaite who served during World War 1.
BEFORE THE WAR
JOINING UP
ARMY LIFE
SERVICE OF THOSE LISTED ON THE WAR MEMORIALS
LIST OF SERVICE OF ALL PARISHONERS
RECORDS USED
Named on the War Memorial
Arthur Bootland
Walter Bootland
Sidney Herbert Bray
Alfred Brown
George Elliott Dawson
Robert Benyon Neville Gunter
Arthur Johnson
Raymond Johnson
William Herbert Johnson
George Haddington March
James Herring Parkinson
Arthur Henry Rutherfoord
Reginald Woolley Sanderson
William Wheelhouse Smith
William Lee Stoker
John Stanley Wilson
William Pearson Allen Wilson
George Arthur Wray
Killed in the war but not named on the War Memorial
John William Dawson
Albert Edward Frost
Allix James William Griffith
George Hall
Percy Maude
Frederick Noble
Joseph White Ordish
Henry Morris Renton
Arthur Roberts
George Stanley Saville
John William Swann
Harry Taylor
Neville Inchbald Wilson
Go to full list of those who served
Roberts, Arthur
Rank and Unit at End of World War One
Rank Private
Service Number DM2/196301
Service Army
Battalion 895 MT Company
Regiment Royal Army Service Corps
Died: 15th April 1917
Buried Syra New British Cemetery, Syra, Greece
Pre-war Occupation*/marital status**
Trade or Occupation pre-war: Publican's son
* Taken from attestation papers or 1911 census
** Marital status on enlistment or at start of war
Connection with Collingham, Linton or Micklethwaite and reason for inclusion on this web site
Ex-service man who moved to Collingham, Linton or Micklethwaite after the war
Arthur Roberts is included here as his mother, Mrs Harriet Roberts, claimed a dependent's pension, following Arthur's death in the war, and gave her address as The Windmill Inn, Linton.
Arthur was born in 1886, the eldest son of William and Harriet Roberts of Holbeck, Leeds. In 1901, Arthur was living at 16 Dawson Street, Holbeck with his parents. His father, William, was a publican and innkeeper. By 1911, William had died and the family had moved to 1 Cross Princes Street Holbeck, where Harriet was shown as a publican and the children helped with the business. We believe Harriet moved to Linton after the Great War.
Arthur served in the Royal Army Service Corps with a service number of DM2/196301. He served with 895th Mechanised Transport Company of the Royal Army Service Corps. This unit was part of the troops assigned to the Headquarters of the British Egyptian Expeditionary Force from July 1917 to October 1918. The role of 895th (MT) Company was as one of two companies in the Caterpillar Tractor Supply Column.
En route to Egypt, 895 (MT) Coy. RASC were embarked on the RMS Arcadian (formerly the SS Ortona - the last ship that Pacific Steam built for the London-to-Australia route). We believe that the Arcadian was transporting troops to Salonika, Egypt and Palestine. She was accompanied by two Japanese destroyers as protection against submarine attack. Apart from one or two scares no major incident took place between the UK and Malta where one of the two destroyers remained. Arcadian with the other destroyer left Malta, and after a few days came within sight of the African coast where a first enemy submarine was spotted. The troop convoy took shelter for a few days in the mouth of a small North African river and waited for a few days. A report suggests that on the third day the other transport ventured out but returned soon damaged after an attack. Later Arcadian, with the destroyer in support, left the river mouth and successfully made it to Salonika where those troops destined for that front disembarked. Under cover of darkness, Arcadian with the Egyptian contingent set sail again and made their way through the Aegean Archipelago.
On 15 April 1917, shortly after completing a boat drill, and while 26 miles north east of the Greek island of Milos, the Arcadian, loaded with a company of 1,335 troops and crew was hit by a single torpedo from the German submarine UC-74 and sank within six minutes with the loss of 279 lives. A contemporary newspaper article described how four of Arcadian's overcrowded lifeboats were successfully lowered before she sank. The escorting destroyer had two torpedoes launched at her while she was attempting to rescue men from the water; survivors reported that she had lowered three of her own boats while going "at full speed". One of the 279 lost was Arthur Roberts.
Arcadian sometime between 1910 and 1915 (From Wikipedia) Arcadian sinking after being torpedoed in 1917 (From Wikipedia)
A contemporary report states:
The Mercury (Hobart Tasmania) May 17th 1917
SINKING OF THE ARCADIAN.
STORIES OF SURVIVORS.
LONDON, May 15.
Survivors of the troopship Arcadian, which was torpedoed by the enemy in tho Mediterranean last month, reached England to-day. They state that at tho time of the attack the sea was perfectly smooth. The explosion smashed some of the lifeboats. The vessel turned over to port, but righted herself in a few minutes, enabling four boats to be launched. All were crowded, one of them being loaded with more than 70 men. Then the soldiers and the members of the crew began to throw themselves into the water, clinging to rafts and wreckage. A patrol boat did its best to help the swimmers. The submarine unsuccessfully fired two torpedoes at the patrol boat while it was engaged in rescue work. Tho men of the patrol boat performed a fine feat in launching three lifeboats while going at full speed. The Arcadian had gone almost before it could be realised. There was no time for discipline. A large proportion of the engine room staff and of the cooks perished.
Arthur was initially buried at Ekinosa, but on cemetery aggregation his body was exhumed and buried in Syra New British Cemetery on the island of Syra in the Cyclades. Syra New British Cemetery was made in June 1921, to take the scattered Commonwealth war burials from the islands of the Cyclades. Graves were brought in from Amorghos (7); Antiparos(5); Ekinosa (9) - Arthur would have been one of these; Heraklia (26); Kassos (1); Ano Kouphonisia (1); French Consular Cemetery on Milo (Melos) (20); Naxos (2); Paros (1); Santorini (1); Skarpanto (Kerpe) (9); Stampalia (Astypalaa) (3); Syra British Consular Cemetery (12) and St. Trias Churchyard at Livadi, Zea (Keos) (1). More than half of the graves are those of men drowned when the Transport "Arcadian" was torpedoed and sunk on 15 April 1917 in the middle of this group of islands.
Arthur was posthumously awarded the British War Medal as he had served overseas, but since he had not actually entered a theatre of war he was not eligible for the award of the Victory Medal.
First World War Medal Index Cards. The National Archives (WO372).
First World War Medal Index Rolls. The National Archives (WO329).
Commonwealth War Graves Commission Cemetery and Burial Reports
Pension Record Cards and Ledgers. Case number 4/D/13765
If you have any photographs or further details about this person we would be pleased to hear from you. Please contact us via: alan.berry@collinghamanddistrictwararchive.info
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© Alan Berry 2017 - 2021. All rights reserved.


text:
SheHeroes Episode 9: Dr. Verna Gibbs
Dr. Verna Gibbs is a professor of surgery at the University of California, San Francisco. Dr. Gibbs, is also the director of NoThing Left Behind, a national surgical patient safety project. Every year in the U.S. an estimated 4,000 cases of "retained surgical items"(RSI's, as they are known in the medical world), or surgical tools are accidentally left behind in patients leading to serious injury and long term health problems for patients. Dr. Gibbs' project Nothing Left Behind is on the forefront of researching and promoting better hospital practices to prevent this from happening.
Currently, Dr Gibbs is the Chair of the SFVAMC Surgical Service Quality Improvement Committee, the Physician Utilization Management Advisor and the faculty director of the VA Surgical Quality Improvement Program (VASQIP) at the SFVAMC. She is Chair of the Physicians Relations Committee for Quality Improvement at Anthem Blue Cross of California and is a member of the Wellpoint Physicians Advisory Council. She is also a member of the Board of Overseers of Harvard University.
Download Discussion Questions (PDF)
What do you think it must have been like for the Gibbs family to move into an all-white neighborhood and be fearful for their safety? Why do you think that Dr. Gibbs' father and mother chose to do that? Have you ever experienced discrimination?
Dr. Gibbs said that the role models for girls were more limited during the time that she grew up than for boys. Dr. Gibbs played with her brother's chemistry set and people assumed that she wanted to become a Dr. like her father but he was more interested in her brother's future than hers. What do you think about that? Do you think that there are different expectations for girls than boys?
Do you think that it is more important for someone to believe in you or for you to believe in yourself to succeed? This answer will be different for everyone.
Given what you know about Dr. Gibbs early experiences from the video, why do you think she wanted to become a doctor? What do you want to be or do?
Dr. Gibbs says she takes a lot of pride in her family–Dr. Gibbs daughter is named after her mother. She says, "You have to know where you come from and where you are going." Can you answer that question for yourself?
Dr. Gibbs says that to be successful, children need to know how to play. Play is how you learn and build skill sets to work with others. Do you think that your parents and your school agree with that statement? Do you agree with that statement?
More SheHeroes videos on


text:
A talk with Rosamund Kissi-Debrah
The Pipe FactoryThe Pipe Factory 42 Bain Street Glasgow G40 2LA
07nov1:00 pm2:30 pmA talk with Rosamund Kissi-DebrahRosamund Adoo-Kissi-Debrah is a World Health Organization advocate for health and air quality and co-founder of the Ella Roberta Family Foundation, based in London.The Pipe Factory, The Pipe Factory 42 Bain Street Glasgow G40 2LA1:00 pm - 2:30 pm Event TypeWebinar & TalksThemeClimate Justice
Rosamund Adoo-Kissi-Debrah is a World Health Organization advocate for health and air quality and co-founder of the Ella Roberta Family Foundation, based in London. A former teacher, Rosamund became
Rosamund Adoo-Kissi-Debrah is a World Health Organization advocate for health and air quality and co-founder of the Ella Roberta Family Foundation, based in London.
A former teacher, Rosamund became a clean air advocate after her 9-year-old daughter, Ella, died in 2013 from a rare and severe form of asthma.
Rosamund spent several years campaigning for a second coroner's inquest into Ella's death to determine whether it was linked to air pollution. In a landmark decision in December 2020, the coroner ruled that it was. Ella is now the first person in the UK, and possibly the world, to have air pollution listed as a cause of death on her death certificate. Rosamund intends to continue advocating to secure the right for children around world to breathe clean air.
Various inhalers and steroids was unable to control Ella's life threatening attacks, they even occurred in her sleep. Ella would say "Mummy, we don't sleep any more." after being woken by another severe attack. Living with severe asthma is incredibly stressful and very frightening at times.
The lobby for clean air needs to be heard in order to push through the legislation. In the UK, asthma is the number one childhood illness and the foundation continues to raise awareness of the dangers of this debilitating illness. Currently in the UK three people die from asthma everyday and at least two of those deaths are avoidable. In London the picture is increasingly worrying not only does the capital record high levels of air pollution but 872 schools are in areas which breach the EU recommended levels. Asthma in London continues to increase, in a class of 30 at least 3 children will have a diagnosis of asthma. In the UK a child is admitted to hospital every 20 minutes because of an asthma attack.
The number of asthma deaths every year amongst children (from 1-17 years of age) in London is between 8-12 children.
The talk has been organised and will be chaired by Georgia Elliot Smith, CEO of Element 4.
Register: https://www.eventbrite.co.uk/o/robyn-woolston-37143355113
(Sunday) 1:00 pm - 2:30 pm
The Pipe Factory
The Pipe Factory 42 Bain Street Glasgow G40 2LA


Excel allows 256 columns in a worksheet.
Use "Customise" to change the number of columns in this report.
This was neither an error nor bugs in QuickBooks. This was a limitation in Microsoft Excel, as the warning message stated; Excel allows only 256 columns in a worksheet.
In this case, this user has more than 3000 data columns which was far beyond Excel can take. To resolve it, you may customise your report to reduce the number of Class to be displayed.
This entry was posted in QuickBooks and tagged error, export, QuickBooks, report. Bookmark the permalink.


There are countless DIY solutions out there on the interwebs for unfortunate folks who have dropped their expensive smart phones in water. The suggestions for drying a wet phone (and saving yourself from having to buy a new one) range from submerging it in a bowl of uncooked rice to taking it apart and leaving it in the sun for a few days. If you are afraid of using a DIY method for drying out your wet cell phone, though, the Dry-All might be able to rescue your investment.
The Dry-All appears to be nothing more than a bag lined with drying beads. When you lock your wet phone inside for a couple of days, the beads soak up all of the excess moisture and produce a dry, working cell phone. The commercial above shows how easy it is to use, but be aware that it is easily the most annoying commercial ever made. You can buy one to have on hand for the next time your iPhone takes an unscheduled trip into the toilet: prices range from $29.99 for a cell phone bag to $59.99 for a smart phone bag.
See more in Phones & Mobile or under Gadgets. June, 2011.


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Want to learn more about 48 Bluff Tr (Highway 27& King Road)?
I would like to learn more about 48 Bluff Tr (Highway 27& King Road).


For the past two years, Steph has been working at a monthly pop-up café in Aldershot, serving customers tea, coffee, and plenty of hot chocolate.
At first, Steph found the work challenging. She was shy with the customers and didn't know what to say, or how to deal with enquiries. Ruth, who runs the pop-up café, would stand with her as she served the customers, supporting her to ask the right questions.
But quickly, Steph grew in confidence. As she got more used to catering, she was able to start opening up with the customers, starting conversations and chatting to them. The role became far more than simply serving customers hot chocolate, as Steph quickly became a charismatic hostess and the smiling face of the pop-up café.
To make the most of the opportunity, Steph has been working to complete various certificates to make her more effective in the role. She's already completed her food hygiene certificate, and has been supported through a Health and Safety course. Her next step is to complete a First Aid course to really boost her skills set. She wants to be as ready for work as she can be.
Steph's now starting to engage with more of the mechanics behind how the café runs and makes its money. She's become more business savvy, suggesting customers try a slice of cake with their tea, or a biscuit with their hot chocolate. Her warmth towards customers keeps them coming back each month, and the café has really flourished with her growth in confidence.
There's no doubt that the café has been a game-changer for Steph. But there's also no doubt that Steph has been a game-changer for the café. Everyone benefits.