text:
You are at:Home » Featured » Prolific pottery: House yields many clues to past
Prolific pottery: House yields many clues to past
By N&N Admin on February 17, 2016 Featured
This artist's rendering of what a domestic winter house would have looked like at the Toqua Site in Monroe County represents the Mississippian Dallas period (pre-1550), about a century before Cane Notch's likely date. However, protohistoric winter houses across the Mississippian world were remarkably similar up to about 1650, so this image is probably close to what the Cane Notch winter house looked like.
Courtesy McClung Museum of Natural History and Culture, University of Tennessee, Knoxville
(This is the fifth in a series on the Cane Notch archaeological site and subsequent studies. The site is located in Washington County in 12 acres of bottomland along the Nolichucky River. Its first field school and dig occurred in December 2015 and January 2016. What is anticipated to be years of research is just commencing. This article highlights the master's thesis topic of University of Mississippi student Nathan Shreve.)
By Jeff Keeling
Four years ago, Nathan Shreve discovered a bunch of ceramics sticking out of a bank on the Nolichucky River. The find changed his life, and today Shreve is on the cusp of a significant contribution to what the lives of Native Americans living in this area were like in the decades following European contact.
Shreve, who earned a geography degree at Emory and Henry College, was several years out of school when he and his father saw the ceramics on a kayaking trip. The find led eventually to the Cane Notch project, and led Shreve to the University of Mississippi's graduate archaeology program. The recent field school included excavation of a 7-by-7-meter structure, circa 1600-1650, the findings from which will provide the basis for Shreve's master's thesis.
Nathan Shreve, left, on site at the house during the excavation in January.
Photo by Jeff Keeling
Shreve's immediate objective? Study the ceramics found at and near the house to answer questions about "coalescence" – the coming together of Mississippian groups into new communities in the wake of European arrival, exploration and exploitation.
Along with East Tennessee State University professor Dr. Jay Franklin, who's overseeing the Cane Notch project, Shreve is being guided by Drs. Robbie Ethridge and Maureen Meyers at Ole Miss. Ethridge is an ethnohistorian specializing in the "protohistoric" time period of the 17th century, and Meyers is an archaeologist. The entire quartet – and plenty of others in the archaeological world – are excited about what the evidence gathered at Cane Notch might contribute to our understanding of this period when cultures collided.
As Shreve puts it in an early abstract, "This robust protohistoric middle Nolichucky settlement likely represents one of only a handful of locations to test the hypothesis for coalescence. It has been widely noted in the archaeological literature that time gaps exist in many Cherokee locations, therefore this location represents the unique opportunity to closely examine the ways in which Native Americans adapted to the changing dynamics of the 17th century."
Pottery recovered from the Cane Notch house.
Photo courtesy Jay Franklin
The Mississippian Shatter Zone
Ethridge, the ethnohistorian, is a leading scholar on what is known as the "Shatter Zone." Before the arrival of Europeans, Ethridge said in a phone interview, Mississippian chiefdoms flourished over a large area of the Southeast.
"These chiefdoms collapsed due to the opening up of the Indian slave trade, basically Indians raiding other Indians to sell on the market to Europeans – and also other disturbances of contact (including the gun trade)," Ethridge said.
From that chaos evolved new communities, including, very possibly, the one at Cane Notch.
"If this whole hypothesis holds, this could be a community that formed out of the shatter zone," she said. "When your polity falls apart, people didn't just die. They restructured their lives and the survivors from these raids and all these disturbances came together and formed these new kinds of communities. These coalescences of various people from various polities are coming together and reconstructing and restructuring their lives into new kinds of polities."
Cane Notch may help provide clues into the formation of those new communities.
"Seeing it on the ground is going to be really important, because we don't know the mechanisms by which all this restructuring took place," Ethridge said. "We don't know how these new communities formed. We can probably say why they formed, but how they formed still eludes us. I think archaeology is the only thing that is going to give us that answer."
Nathan Shreve
Pottery par excellence
Meyers, whose primary work takes place at a 14th-century mound site in Lee County, Va., said the volume and quality of the ceramics already excavated at Cane Notch holds out promise for great contributions to knowledge about the scantily recorded protohistoric period.
"From the pictures I've seen, the preservation is pretty astounding, as well as the amount and type of artifacts," Meyers said. "My understanding is there's a fair amount of glass beads coming in. The pottery, which I study, sounds like it's in great condition, there's a lot of it, and the house remains itself is in great condition, so that's amazing to be able to get a site from that period in such great condition."
Robbie Ethridge
Material that points to the economics of the site, including probably the deerskin trade, is also likely to prove fruitful, and far beyond Shreve's thesis, Meyers said. Work in Northeast Tennessee has been a missing link, largely due to lack of funding.
"I can't tell you what a black hole that area's been for people like me working in slightly earlier periods and then for people working in the 16th and 17th centuries. If we could fill in this gap, we might really understand what's happening in that time period between the Natives and Europeans. We might really get a better picture of that. So yeah, this is big."
Shreve is hopeful the evidence will point to a coalescence site at Cane Notch.
Maureen Meyers
"At this point it looks like there was a core group of inhabitants," he said. "Whether they absorbed groups from the periphery of East Tennessee and Southwest Virginia, we hope to discover."
He said the French Broad and Holston river valley sites have scant evidence of protohistoric coalescent communities. For instance, they lack the trade beads found at Cane Notch.
"They had to go somewhere. It may not have been the Nolichucky, but it's certainly a possibility that the Nolichucky absorbed populations, because (the site) was so robust."
(Previous stories in this series can be found at jcnewsandneighbor.com/category/featured).


Too many people place an emphasis these days on retained knowledge. This is especially prevalent within technical fields. Computer programmers are often asked to recite arcane utterings of alphabet soup during job interviews. This isn't important. What is important, however, is simply knowing where to find the right answers.
I am not one who generally looks back and reflects too much on my past mistakes. Well, I reflect upon them, but I don't dwell on them. Admitting one's professional mistakes, and coming to terms with them, is part of the evolutionary process of each of our careers.
My thoughts on the so-called 'five minute jobbing' hiring practice.
In my chosen field of information technology, I am accustomed to working with large, diverse teams. These teams tend to work together over protracted periods of time. In order to assist in building team cohesiveness, I formulated what I call my circle theory of hiring.
The other day I found myself, as usual, plugging away on my laptop at my local Starbuck's. A young lady sat near me interviewing a young man for a job as a salesperson at the EB Games (a video game shop) next door. I thought it might be interesting to capture bits of their conversation, in an effort to provide some critical analysis of it here in my blog (yes, I'm bored).
As I was pouring my first cup this morning, I was thinking about some of the great teams I've worked with in my career. Some were fantastic, some were "ok", and a few were, well, rather deplorable. I began trying to determine what differentiated those great teams from the not-so-great ones.


A floor plan and side view diagram of the 4th century BCE Etruscan Francois Tomb at Vulci, central Italy.
Louis-Garden, . "Francois Tomb Plan." Ancient History Encyclopedia. Last modified January 31, 2017. https://www.ancient.eu/image/6311/.
Louis-Garden, . "Francois Tomb Plan." Ancient History Encyclopedia. Ancient History Encyclopedia, 31 Jan 2017. Web. 20 Apr 2019.


The Group of Seven was a group of Canadian painters that came into their own in the early 1900's. The group consisted of artists who explored a variety of techniques over the years. However, one thing that tied this loose association of artists together was the influence of European impressionism. You can look over the art of anyone who was part of the Group of Seven or was associated with the Group and see this influence. Their genius was bringing impressionistic style to Canadian landscapes, and thereby becoming the most important Canadian artists in the first decades of the 1900's.
One thing that the Group of Seven has received both criticism and praise for is their depiction of well lived-in areas of Canada as untouched landscapes. In doing this they showed a new perspective on a region that was considered unpaintable. This makes us think about California, filled with about 39 million people. There is likely no stone or piece of landscape untouched in this well-loved state, but the natural areas are still filled with beauty that deserves to be painted.
Many members and associates of the Group of Seven were noted outdoorsmen and women, their works thus inspiring others out into the Canadian wilderness. It could be imagined that, if the Group of Seven had the opportunity to paint here in California, they would have forged through our woods, climbed rocks, and explored the deserts just as Erin Hanson has. They would have sought out the many natural landscapes here in California and painted it with bold colors and broad brushstrokes. They would have seen the raw landscapes, and portrayed the beauty in front of them.
So, if the Group of Seven came to California, one could imagine much much of what Hanson has brought to life, vibrant paintings that show the joy natural landscapes can impart. We hope that, when you see the art created by Erin Hanson, you feel the fierce love she has for her home state – and you sense wisps of inspiration from those impressionistic painters of the past.


text:
Ла Провиденсија () насеље је у Мексику у савезној држави Нови Леон у општини Грал. Браво. Насеље се налази на надморској висини од 324 м.

Становништво

Према подацима из 2005. године у насељу је живело 6 становника.

Хронологија

Попис

Види још
 Савезне државе Мексика

Референце

Спољашње везе

 Мексичка насеља

Насеља у општини Грал. Браво (Нови Леон)
Википројект географија/Насеља у Мексику

redpajama_set_name: RedPajamaWikipedia


Famous for its relaxed and easy going atmosphere, its Full Moon parties and the crowds of nature lovers and shoe-string budget travelers, Koh Phangan is an island of exceptional beauty whose nature is still largely wild and unspoilt.
Whereas backpackers still dominate the island's tourist trade, the days when the only accommodation in sight were bamboo huts or reed woven bungalows are long gone. Some stunning and luxurious development projects have found home in Koh Phangan's most secluded bays: fully staffed, private villas nestled on cliffs overlooking the sea and totally integrated in the surrounding nature as well as luxurious resorts boasting 5 stars accommodation and facilities can all be found, even if admittedly less easily than in Koh Samui or Phuket. Still, the remoteness of this type of accommodations adds on charm, especially to travelers looking for a peaceful hideaway.
Those looking for a secluded paradise will discover Phangan has quite a few: remote beaches accessible only by the sea, pristine tropical forest that still covers 90% of the island, waterfall and small mountains, and an incredible abundance of wildlife; all of this has attracted many, not only in recent times but even during the past centuries: King Rama V visited the island not less than 10 times, leaving his seal engraved on a rock in Thaan Sadet Waterfall while the current King Bhumibol followed more recently in his steps.
While Koh Phangan is yet to have yoga retreat, spas, and wellness centers on the scale of Koh Samui, such facilities are plenty on the island, with meditation schools and secret hideaways taking the lead and attracting those looking for peace and a space to learn in harmony with nature.
Koh Phangan abundance of tropical rainforest, wild jungle interior and rich coral reefs make it also the ideal destination for the adventurous and active traveler. The reefs running along the North West coast support a vibrant underwater environment making it an excellent place to snorkel and dive. Indeed, the best diving sites in the Gulf of Thailand are all but 20 km away, scattered in between Koh Phangan and Koh Tao, making Phangan a perfect base to go exploring these marine wonders. Overnight mountain treks can be organized to the top of Khao Ra and coastline walks allow to beach hop around the whole island.
Sailing or Speedboat cruises to the Ang Thong Marine Natural Park can be organized from Koh Phangan as well. Ang Thong is a true natural wonder consisting of 40 islands and a vast expanse of marine environment. Snorkelling here is magnificent while the view from the highest hill in Wua Ta Lap, at 240m, is breath-taking.
And of course, party-goers will find their paradise here: Haad Rin, home to the legendary Full Moon Party, hosts the biggest beach party in the world while outdoor events are organized weekly in jungle and beachfront venues.


other factors such as the price of the goods, whether the goods are inherently dangerous and whether an intermediary will be involved (for example, a pharmacist, explaining to consumers the proper use of a prescription drug, rather than the manufacturer).
As a result, goods can be unsafe even though they are not actually defective. For example, they may not be accompanied by a warning notice or packaged appropriately. On the other hand, they can be safe even though they are not of acceptable quality (see Consumer guarantees) or absolutely free from risk.
A manufacturer of caustic soda was held liable because it did not warn consumers of the danger of mixing the product with hot water. Although the caustic soda was not defective, it was unsafe without such a warning.
A manufacturer of a garage roller door was held liable for injuries sustained by a consumer as a result of the door coming off its mountings and falling on her. Liability arose because the manufacturer's installation instructions were inadequate.


I know you can order some books new on Amazon, but I've never seen any in bookstores in a couple years.
I found a few new books at Bargain Books and Half-Off books. I'm thinking they are selling off the dead stock.
You can get a lot from Amazon for only a penny but the shipping and handling is $3.99 so you're really paying $4.00 for a book.
I've only seen them in secondhand shops, apart from the graphic novels.
I found some of the newer covers in a Barnes and Noble right after the first graphic novel came out.
I saw a Little Sister book when I was at Zellers recently (#104 or something like that), but it's been there for about a year without anyone buying it.
And also, a new Ann M. Martin series that sounds spookily similar to BSC but with different names.


text:
<?php

class Google_Service_Analytics_Upload extends Google_Collection
{
  protected $collection_key = 'errors';
  public $accountId;
  public $customDataSourceId;
  public $errors;
  public $id;
  public $kind;
  public $status;

  public function setAccountId($accountId)
  {
    $this->accountId = $accountId;
  }
  public function getAccountId()
  {
    return $this->accountId;
  }
  public function setCustomDataSourceId($customDataSourceId)
  {
    $this->customDataSourceId = $customDataSourceId;
  }
  public function getCustomDataSourceId()
  {
    return $this->customDataSourceId;
  }
  public function setErrors($errors)
  {
    $this->errors = $errors;
  }
  public function getErrors()
  {
    return $this->errors;
  }
  public function setId($id)
  {
    $this->id = $id;
  }
  public function getId()
  {
    return $this->id;
  }
  public function setKind($kind)
  {
    $this->kind = $kind;
  }
  public function getKind()
  {
    return $this->kind;
  }
  public function setStatus($status)
  {
    $this->status = $status;
  }
  public function getStatus()
  {
    return $this->status;
  }
}

redpajama_set_name: RedPajamaGithub


text:
Q: How to reduce or eliminate inductor sound noise in a buck SMPS? I use this circuit as my device power supply:

Here I have R2=5.6K as my SMPS load but at this load I have about 10KHz switching frequency which leads to an annoying audible noise, probably from L1 (the inductor.)
I tried to increase the output current by reducing R2 to 1.2K and I got more than 20KHz switching frequency which eliminated the noise from the human hearing range,  but in this particular application the current consumption has to be as low as possible.
How can I reduce the noise without increasing the output current consumption?

A: The HT7A6322 has a fixed 60KHz oscillator - it shouldn't change depending on load.
I think the problem is that your load is very light.  It is drawing almost no current, causing the regulator to go into burst mode to keep the output regulated.
It runs at 60KHz for a bit, then switches off until the voltage drops, then runs at 60KHz again.  Repeat constantly.
The bursts occur at a rate that depends on the load.  The inductor likely vibrates in response to the bursts rather than the switching frequency.
To make it quiet, you could try a different inductor.  You could also just live with the higher current consumption needed to keep the regulator out of burst mode. Your 1.2k load resistor only draws 10 mA amperes of current, wasting 0.12 watts of power.
Alternatively, find a regulator made to work with extremely low loads.

Please do remember to keep that circuit in an insulated housing with no user access to metal parts from the outside.
Your circuit is not isolated.  There may be only 5V and 12V between the outputs and the circuit ground, but the output ground can be at line voltage depending on how the outlet and your power plug are wired.
That circuit can kill anyone who touches any conducting part of it.

There is a design for an isolated power supply in the datasheet.  Consider using it instead of your unisolated circuit.

redpajama_set_name: RedPajamaStackExchange


Northumberland Paramedics provides ambulance and paramedic services to Northumberland County. Our team of over 100 paramedics works from six stations strategically located throughout the county.
Learn more about Ambulance and Paramedic Services.
Each municipality in Northumberland County provides Fire Services to their community.


text:
\section{Introduction}
Synchronization of dynamical networks is a fascinating, widely studied, and impactful phenomenon; e.g.,
see \cite{Winfree, Buck, Kuramoto} for early applications in the
applied sciences, and the review
\cite{Dorfler-Bullo} --and the many references there-- for a thorough
account on the topic of oscillators synchronization.
In the most typical and studied circumstance, one has a network of $N$ nodes,
the so-called {\it agents}, that obey $N$ identical nonlinear differential equations with vector field $f$,
coupled through linear anti-symmetric coupling.
The key concern is to find conditions that tie together the strength of the coupling,
the structure of the network, and the agent's dynamics, in such a way that the network synchronizes.
This problem has been extensively studied, under a number of different scenarios, for
networks of smooth systems; e.g., see the work of Pecora and coauthors \cite{BarahonaPecora, Pecora.Carroll} for a
study exploiting Lyapunov exponents ideas, and see \cite{Hale} for a study more
along the lines of the theory of dissipative attractors.

After the cited works of Pecora and coauthors, probably the most widely adopted
and successful tools to infer convergence to, and/or stability of,
a synchronized solution in networks of smooth dynamical systems has been that of the
master stability function (MSF).  However,
as remarked in \cite{CdLdB}, this {\sl ``approach requires some degree
of smoothness in the agents' vector fields ... and extensions
need to be found''} when dealing with piecewise smooth systems.
Our goal in this work is to provide such extension.

In fact, in the present work, we study the synchronization problem for networks where each agent satisfies a nonlinear piecewise smooth
system, that is the underlying dynamical system of each agent is governed by a vector field $f$ which is
piecewise smooth.  We will call these  {\sl piecewise smooth networks}.
That is, we consider the network
\begin{equation}\label{PWSnetwork1}
	\dot {x_i}=f({x_i})+\sigma \sum_{j=1}^N a_{ij}E(x_j-x_i),\quad x_i\in {\mathbb R}^n,\,\,\
	i=1\dots, N\ ,
\end{equation}
where $f$ is only piecewise smooth:
$f(x_i)=\left \{ \begin{matrix}
	f^+(x_i), & h(x_i)>0 \\ f^-(x_i) & h(x_i)<0 \end{matrix} \right .$,
$i=1, \ldots, N$, and $h(x)=0$ describes the discontinuity manifold.
In \eqref{PWSnetwork1},
$A\in {\mathbb R}^{N\times N}$ is the adjacency matrix of the
graph describing the network (assumed to be
undirected, simple and connected, so that $A$ is symmetric), $\sigma \ge 0$ is the the coupling strength,
and $E\in {\R^{n\times n}}$ is the coupling matrix describing which components
of the two agents $x_i$ and $x_j$, $i\ne j$, are connected to one another.
Next, we let $D$ be the diagonal matrix with elements $d_{ii}=\sum_{j=1}^N a_{ij}$, and
let $L=-D+A$, $L \in {\mathbb R}^{N \times N}$, the graph Laplacian. Then,
using Kronecker product notation, we rewrite
 \eqref{PWSnetwork1} as
\begin{equation}\label{PWSnetwork}\begin{split}
\dot {\vec{x}}&=F({\vec{x}})+\sigma M{\vec{x}},\itext{where} \\
{\vec{x}}&=\bmat{x_1 \\ \vdots \\ x_N}\in {\mathbb R}^{nN}, \,
F({\vec{x}})=\bmat{f(x_1)\\ \vdots \\ f(x_N)} \in {\mathbb R}^{nN},
\, M= L \otimes E \in {\mathbb R}^{nN\times nN}\ .
\end{split}\end{equation}
As noted, our present interest is in the case when, taking $\sigma=0$ in \eqref{PWSnetwork},
each agent obeys identical piecewise smooth (PWS) dynamics:
\begin{equation}\label{nonsmooth_eq}
	\dot x_i=f(x_i)=\left \{ \begin{matrix}
		f^+(x_i), & h(x _i)>0 \\ f^-(x_i), & h(x_i)<0 \end{matrix} \right .,
	\qquad i=1, \ldots, N,
\end{equation}
with $f^\pm: {\mathbb R}^n \to {\mathbb R}^n $ smooth vector fields, and $h: {\R^n}\to {\mathbb R}$ is
assumed to be at least $\cont^2$.
For each agent, the manifold of discontinuity is the zero set $\{x\in {\R^n} \ :\,\ h(x)=0\}$,
and we will use the following notation:
\begin{equation}\label{notations_eq}
	\Sigma=\{x \in {\mathbb R}^n \,\ | \,\ h(x) =0\},\,\
	R^\pm=\{x \in {\mathbb R}^n \,\ | \,\ h(x) \gtrless 0\}.
\end{equation}
\begin{rem}
A typical case we have seen in many applications is to have
$h(x)=c^Tx-b$, but this is not necessary in our work.
\end{rem}
As customary, we say that a point $x \in \Sigma$ is a \emph{transversal crossing point} if
\begin{equation}\label{CrossPt}
(\nabla h(x)^Tf^-(x) )(\nabla h(x)^T f^+(x) )>0,
\end{equation}
and it is an \emph{attractive sliding point} if
\begin{equation}\label{SlidePt}
\nabla h(x)^Tf^-(x)>0, \quad \nabla h(x)^T f^+(x)<0.
\end{equation}
On $\Sigma$, sliding will be assumed to take place in the sense of Filippov,
whereby on $\Sigma$ the dynamics of an agent is given by
\begin{equation}\label{SlidingDyn}
	\dot x =f_\Sigma:=(1-\alpha)f^{-}(x)+\alpha f^{+}(x)\ ,\quad
	\alpha=\frac{\nabla h(x)^Tf^-(x)}{\nabla h(x)^T(f^-(x)-f^+(x))}\
\end{equation}
Finally, a point
$\bar x \in \Sigma$ is called \emph{tangential exit point into} $R^-$ if a trajectory $x(t)$
sliding on $\Sigma$ reaches it at some value $\bar t$ and there it holds that
\[\nabla h(\bar x)^Tf^-(\bar x)=0, \quad \nabla h(\bar x)^Tf^+(\bar x)<0, \quad
\left[\frac{d}{dt}\nabla h(x(t))^Tf^-(x(t))\right]_{t=\bar t}<0,  \]
and similarly for a tangential exit point into $R^+$.  The combination of transversal crossings,
transversal entries on $\Sigma$, and tangential exits from $\Sigma$, are called {\sl generic events},
or simply {\sl events}.

Let the single agent \eqref{nonsmooth_eq} have a limit cycle
with a finite number of events,
and not entirely contained in $\Sigma$ (see
\cite{Galvanetto_95}, \cite{Dieci.Elia_2014}, \cite{Coombes.Thul_2016} for examples of self sustained
oscillations in discontinuous systems with partial sliding along the discontinuity manifold).  Let
$x_S(t)$ be the corresponding $T$-periodic solution.  Then, given the structure of $M$,
the function ${\vec{x}_S}(t)=\smat{x_S(t)\\ \svdots\\  x_S(t)}$ is a periodic solution of
\eqref{PWSnetwork} of period $T$; we will call this the
{\sl synchronized manifold} or simply the {\sl synchronous solution}.
However, even if ${ x_S}(t)$ happened to be asymptotically stable for the single agent, there is no
guarantee that $\vec{x}_S$ be stable for the network dynamics for all values of $\sigma$;
further, when $N$ is large, the numerical study
of the stability of $\vec{x}_S(t)$ may be prohibitively expensive. This issue can be overcome by
extending the {\sl Master Stability Function} (MSF) tool of Pecora and Carroll,
see \cite{Pecora.Carroll}, to PWS networks.

The MSF technique relies on exploiting the structure of the fundamental matrix solution of the
network, and for this reason in the present work our goal is two-fold.
When
the network synchronizes on ${\vec{x}_S}(t)$,
first we will give the explicit expression of the fundamental matrix
solution along the synchronized manifold. Then, we will extend the {\sl Master Stability Function} (MSF)
to piecewise smooth networks.
Many authors have considered piecewise smooth networks, and some important
studies have been made to resolve the outstanding concern of how
to infer asymptotic convergence in networks of piecewise-smooth systems.
Notable examples are the recent work of \cite{CdLdB2} where discontinuous
diffusive coupling is adopted, and the works  \cite{Coombes.Thul_2016, Coombes.Thul_2018}, where
the authors use the MSF approach
to study limit cycles in piecewise-linear systems.
However, a rigorous justification of the {\bf use of the MSF} for general, {\bf nonlinear, PWS networks}
appears to be lacking, and it is our purpose to give it in this work.

A plan of the paper is as follows.  In Section \ref{FMS}, we derive the precise form of
the monodromy matrix along the synchronous solution.  In Section \ref{MSF}, we
extend the MSF tool to piecewise smooth networks.  Finally, in Section \ref{GalvaSection}
we give detailed numerical study of a network arising in mechanical vibrations and infer that, for
a range of values of $\sigma$, the synchronous manifold is stable. \\
\noindent{\bf Notation}.\\
$e \in {\mathbb R}^N$ is the vector with all elements equal to $1$, so that
$\vec{x}_S=e \otimes x_S$ is the synchronous solution in ${\mathbb R}^{nN}$.\\
$h_i({\vec{x}})  =  h(x_i)$, $i=1, \ldots, N$.
$\Sigma_i =   \{ {\vec{x}} \in {\mathbb R}^{nN} \,\  | \,\ h_i({\vec{x}})=0 \}$,  $i=1, \ldots ,N$, and
${\BSigma} =  \cap_{i=1}^N\Sigma_i$.

\section{Fundamental matrix solution for synchronous periodic solutions}\label{FMS}

The main difficulties we need to address in this section are the following.
\begin{itemize}
\item[(i)]  The network \eqref{PWSnetwork}  has $N$ discontinuity manifolds and solutions might slide on
the intersection of two or more manifolds (in fact, as we will see, a synchronous periodic solution $\vec{x}_S$
with $x_S$ having a sliding portion, will necessarily slide on the intersection of all $N$ manifolds).
But, in general, the sliding vector field on the intersection of the discontinuity manifolds is not uniquely
defined and we need to address how this impacts the form of the fundamental matrix of the linearized system.
In the specific case we consider here, there is no such ambiguity, see
Lemma \ref{OnSigma_prop} and Theorem \ref{X_thm}.
\item[ii)] The monodromy matrix along a periodic solution of the piecewise system
\eqref{nonsmooth_eq} is not continuous:
it has jumps at the entry points (crossing or sliding) on the discontinuity manifold.
These jumps are taken into account via so called jump or saltation matrices, whose
scope is to transform the vector field at the entry time, say $t^-$,
into the vector field at the exiting time, $t^+$.
The correct expression for such matrices is well known in the literature in the case of a single discontinuity
manifold (see \cite{Aizerman.Gantmacher}, \cite{Mueller}, \cite{Leine}).
However, synchronous sliding solutions have to slide on the intersection of $N$ discontinuity
manifolds and the fundamental matrix solution along a synchronous solution must take into
account jumps at this intersection. In the literature, there are results about these jump
matrices relative to the intersection of two discontinuity manifolds, see \cite{Ivanov} for the
case of crossing and \cite{Dieci.Lopez_2011} for the case of sliding, but no result exist for
the intersection of more than two manifolds. Surely this must be because, in the case of sliding solutions,
there is no uniquely defined Filippov sliding vector field on the intersection of discontinuity manifolds,
as noted in i) above. However, this is not the only issue.  Indeed, in general,
on the intersection  of discontinuity manifolds,
the jump matrix itself is not uniquely defined, even if we are willing to select
a specific sliding vector field (again, see \cite{Ivanov} for the case of
crossing and \cite{Dieci.Lopez_2011} for the case of sliding).  This being the case,
the fundamental matrix solution cannot be defined in a unique way.
Theorems \ref{S14_thm} and \ref{S4Sigma_thm} deal with this aspect in case of the
synchronous periodic solution $\vec{x}_S$ of \eqref{PWSnetwork}.
\end{itemize}

After the expression for the monodromy matrix is arrived at, in Section \ref{MSF} we will
see how to extend the MSF tool to PWS networks.

For the above reasons, hereafter we derive the monodromy matrix along the synchronous solution
$\vec{x}_S$ of \eqref{PWSnetwork}.
The main results are given in Theorem \ref{S14_thm} and \ref{S4Sigma_thm}, where we show that the
saltation matrices can be represented in a unique way.
Recalling that $\vec{x}_S=\smat{x_S\\ \svdots \\ x_S}$, where $x_S$ is the periodic solution of
a single agent \eqref{nonsmooth_eq}, we will assume that $x_S$ has a finite number of generic events.
Because of this, we will make the following convenient assumption on the dynamics of $x_S$.

\begin{ass}\label{xs_ass}
We assume that \eqref{nonsmooth_eq} has a periodic solution $x_S(t)$ that:
\begin{itemize}
\item[0)] At $t=0$, $x_S(t)=s_0$ is in $R^-$;
\item[1)] At $t=t_1$, $x_S$ crosses $\Sigma$ transversally
at the point $s_1=x_S(t_1)$ to enter $R^+$;
\item[2)] At $t=t_2$, $x_S$ reaches transversally the attractive sliding point $s_2=x_S(t_2)\in \Sigma$
and $x_S$  begins sliding on $\Sigma$;
\item[3)] At $t=t_3$, $x_S$ reaches the tangential exit point $s_3=x_S(t_3)$, and it
leaves $\Sigma$ to enter into $R^-$;
\item[4)] At $t=T$, $x_S$ is back at $s_0$: $x_S(T)=s_0$.
\end{itemize}
\end{ass}
A Figure of the above situation is on the left of Figure \ref{po_types}.
\begin{rem}\label{wlog_ass}
Other than the need for a finite number of generic events, the results in this section do not depend on the
particular structure of $x_S(t)$ given in Assumption \ref{xs_ass} and can be immediately
extended to any finite number of generic crossings, sliding segments, and tangential exits,
of the periodic orbit of \eqref{nonsmooth_eq}.
\end{rem}

Now, for $N$ agents,
there are $2^N$ subregions (and corresponding vector fields), and we
can represent them using a tree diagram with $2^N$ branches.
We number the regions, and the vector fields, from $1$ to $2^N$ following the branches of the tree.
\begin{exam}\label{regions_ex}
For $N=3$, we have the following correspondence between region numbering and signs
of $h_1$, $h_2$ and $h_3$:
\begin{equation*}
\begin{matrix}
1& 2& 3& 4& 5& 6 & 7 & 8 \\
(---) & (--+) & (-+-) & (-++) & (+--) & (+-+) & (++-) & (+++)
\end{matrix}
\end{equation*}
In each subregion $R_j$, the vector field in \eqref{PWSnetwork} is
$F(\vec{x})=F_j({\vec{x}})=\bmat{f^\pm(x_1)\\ f^\pm(x_2)\\ f^\pm(x_3)}$, $j=1, \ldots, 8$, where
in $f^\pm$ we select the sign in agreement with the region numbering above.
For example $F_3({\vec{x}})=\bmat{f^-(x_1)\\ f^+(x_2)\\ f^-(x_3)}$.
\end{exam}

\begin{rem}\label{attractive_rem}
It is simple, but important, to observe that
if $x \in \Sigma$ is an attractive sliding point for the single agent, then
${\vec{x}}=e \otimes x \in {\BSigma}$ is an attractive sliding point on ${\BSigma}$ for the full network.
Indeed, it is immediate to verify that $F_j({\vec{x}})$ points toward $\Sigma_i$ for all
$i=1,\ldots ,N$, and $j=1, \ldots, 2^N$, i.e.,
\[ \nabla h_i({\vec{x}})^TF_j({\vec{x}}) > 0, \,\quad
\nabla h_i({\vec{x}})^TF_j({\vec{x}}) < 0 .
 \]
Similarly, if $x \in \Sigma$ is a tangential exit point into $R^-$ (respectively, $R^+$) for the single agent,
then ${\vec{x}}=e \otimes x \in {\BSigma}$ is a tangential exit point into $R_1$ (respectively, $R_{2^N}$)
for the full network.
\end{rem}

Remark \ref{attractive_rem} justifies the following fact.
Let $x_S(t)$ satisfy Assumption \ref{xs_ass} and
let ${\vec{s}}_j=e \otimes s_j$, where $s_j$ is defined in Assumption \ref{xs_ass}, $j=1,\ldots, 4$.
Then, the synchronous solution $\vec{x}_S(t)=(e \otimes x_S(t))$ obeys the following evolution:
\begin{itemize}
\item[0)] At $t=0$, ${\vec{x}_S}(0)={\vec{s}}_0$ is in $R_1$;
\item[1)] At $t=t_1$, $\vec{x}_S$ crosses $R_1$ at ${\vec{s}}_1$ and enters into $R_{2^N}$;
\item[2)] At $t=t_2$, $\vec{x}_S$  reaches the attractive sliding point ${\vec{s}}_2$ and starts sliding\footnote{Although, in general, sliding along ${\BSigma}$ is not unambiguously
	defined, presently this is not a concern, since we are just describing the evolution of
	the specific $\vec{x}_S$.}
along ${\BSigma}$;
\item[3)] At $t=t_3$, $\vec{x}_S$ exits ${\BSigma}$ at the tangential exit point $x={\vec{s}}_3$ and enters
into $R_1$;
\item[4)] At $t=T$, $\vec{x}_S$ reaches ${\vec{s}}_0$.
\end{itemize}
\begin{rem}\label{synch_rem}
Note that the synchronous solution $\vec{x}_S$ satisfies the following :
i) it can only evolve in the regions $R_1$ or $R_{2^N}$;
ii) it can only cross the discontinuity manifolds at points on ${\BSigma}$, and
iii) if it slides, it can only slide on  the intersection of all $N$ discontinuity manifolds, i.e., on ${\BSigma}$.
{\bf However}, the solution of a problem relative to perturbed initial conditions in general will not
satisfy the restricted motion described by points i)-iii) above, and it  may
slide on, or cross, some of the $\Sigma_i$'s and not just ${\BSigma}$.
This fact must be taken into account when deriving the expression of the fundamental matrix
solution of the linearized dynamics, in particular of the saltation matrices.
\end{rem}

Next, we study the case $N=2$ in detail. The generalization to the case $N>2$ is simple, and
appropriate modifications required to describe the case $N>2$ are given below.
Following the tree diagram for $N=2$, we have the following four subregions of  phase space
and corresponding vector fields, for ${\vec{x}}=\bmat{x_1\\vec{x}_2}\in {\mathbb R}^{2n}$:
\[
\begin{matrix}
R_1=\{{\vec{x}}\,\ | \,\
h_1({\vec{x}})<0, \,\ h_2({\vec{x}}) <0 \}, & F_1({\vec{x}})=\bmat{f^-(x_1) \\ f^-(x_2)}, \\
R_2=\{{\vec{x}} \,\ | \,\ h_1({\vec{x}})<0, \,\ h_2({\vec{x}}) >0 \}, & F_2({\vec{x}})=\bmat{f^-(x_1)\\ f^+(x_2)}, \\
R_3=\{{\vec{x}} \,\ | \,\ h_1({\vec{x}})>0, \,\ h_2({\vec{x}})<0 \}, & F_3({\vec{x}})=\bmat{f^+(x_1) \\ f^-(x_2)}, \\
R_4=\{{\vec{x}} \,\ | \,\ h_1({\vec{x}})>0, \,\ h_2({\vec{x}})>0 \}, & F_4({\vec{x}})=\bmat{f^+(x_1) \\ f^+(x_2)},
\end{matrix}
\]
and  $F({\vec{x}})$ in \eqref{PWSnetwork} is equal to $F_i({\vec{x}})$ for ${\vec{x}} \in R_i$.
Moreover we have $\Sigma_{1,2}=\{{\vec{x}} \in {\mathbb R}^{2n} \,\ | \,\ h_{1,2}({\vec{x}})=0\}$ and
we consider also the sets $\Sigma_{1,2}^\pm$, defined as follows:
$\Sigma_1^\pm=\{{\vec{x}} \in {\mathbb R}^{2n} \,\ | \,\ h_{1}({\vec{x}})=0$, and $h_2(\vec{x}) \gtrless 0 \}$,
and similarly for  $\Sigma_2^\pm$.
The synchronous solution is ${\vec{x}_S}(t)=\bmat{x_S(t)\\ x_S(t)}$, and (under
Assumption \ref{xs_ass}) it evolves schematically as on the right of
Figure \ref{po_types}.

\begin{figure}\label{po_types}
\begin{minipage}[c]{0.48\linewidth}
	\centering
	\includegraphics[width=8cm,height=8cm]{po_single.pdf}
\end{minipage}
\begin{minipage}[c]{0.48\linewidth}
	\centering
	\includegraphics[width=8cm,height=8cm]{po_double.pdf}
\end{minipage}
	\caption{Schematic of Assumption \ref{xs_ass}, on the left, and of the periodic orbit
	for	the network, on the right.}
\end{figure}

As already pointed out in Remark \ref{synch_rem}, while $\vec{x}_S(t)$ can only evolve in $R_1$ and $R_4$,
and can only cross/slide-on ${\BSigma}=\Sigma_1 \cap \Sigma_2$,  a perturbed solution might
instead cross just $\Sigma_1$ or $\Sigma_2$, and evolve in $R_2$ or $R_3$
and it might slide along $\Sigma_1$ and/or
$\Sigma_2$. Hence, in order to compute the fundamental matrix solution, we will need the
expressions of the sliding vector fields on $\Sigma_1$, $\Sigma_2$ and ${\BSigma}$.
Let $f_{\Sigma}: {\mathbb R}^n \to {\mathbb R}^n$ denote the sliding vector field of \eqref{nonsmooth_eq} on
$\Sigma$ (see \eqref{SlidingDyn}) and let $F_{{\mathcal S}}: {\mathbb R}^{2n} \to {\mathbb R}^{2n}$ denote
the sliding vector field on
${\mathcal S}$, where ${\mathcal S}$ is any of the following:
${\mathcal S}=\Sigma_1^\pm, \Sigma_2^\pm,{\BSigma}$.

The next result provides  --at a point $\vec{x}$ on $\bf \Sigma$--
the four vector fields on $\Sigma_{1,2}^\pm$: these are the vector fields {\emph {that are felt}} at $\vec{x}$.

\begin{lemma}\label{OnSigma_prop}
Let $\vec{x}\in \bf \Sigma$, $\vec{x}=\bmat{x \\ x}$, and suppose that $\vec{x}$ is an attractive
point with respect to both $\Sigma_1$ and $\Sigma_2$: cfr \eqref{SlidePt}.  That is,
we have
\begin{equation}\label{PartAtt}
 0<\frac{\nabla h(x)^Tf^-(x)}{\nabla h(x)^T(f^-(x)-f^+(x))}<1 \ .
\end{equation}
Then, we have
\begin{equation*}\label{sliding_vf_eq}
F_{\Sigma_1^\pm}({\vec{x}})=\begin{pmatrix} f_{\Sigma}(x) \\ f^{\pm}(x) \end{pmatrix}, \,\
F_{\Sigma_2^\pm}({\vec{x}})=\begin{pmatrix} f^{\pm}(x) \\ f_{\Sigma}(x) \end{pmatrix},\,\
F_{{\BSigma}}({\vec{x}})=\begin{pmatrix} f_{\Sigma}(x) \\ f_{\Sigma}(x) \\ \end{pmatrix}=
e \otimes f_{\Sigma}(x),
\end{equation*}
where with $f_{\Sigma}(x)$ we denote the sliding vector field of \eqref{nonsmooth_eq} on
$\Sigma$, as defined in \eqref{SlidingDyn}.
\end{lemma}
\begin{proof}
We prove the statement for $F_{\Sigma_2^-}$. The proofs for the other sliding vector fields are analogous.

Let $\vec{x}^k$ be a sequence of points in $\Sigma_2^-$, converging to $\vec{x}$:
$\displaystyle{\lim_{k\to \infty}}\vec{x}^k =\vec{x}\in {\BSigma}$.  Since $\vec{x}^k\in \Sigma_2^-$,
we have $\vec{x}^k=\bmat{x_1^k \\ x_2^k}$ and $h(x_2^k)=0$ for all $k$.

Now, letting
$\hat F_1(\vec{x}^k)=F_1(\vec{x}^k)+\sigma M\vec{x}^k=
\bmat{f^-(x_1^k) \\ f^-(x_2^k)}+\sigma M\vec{x}^k$ and
$\hat F_1(\vec{x}^k)=F_2(\vec{x}^k)+\sigma M\vec{x}^k=
\bmat{f^-(x_1^k) \\ f^+(x_2^k)}+\sigma M\vec{x}^k$, a Filippov
sliding vector field on $\Sigma_2^-$ is given by the convex combination
$F_{\Sigma_2^-}=(1-\alpha^k)\hat F_1(\vec{x}^k)+\alpha^k \hat F_2(\vec{x}^k)$, where
$\alpha^k$ must be found from the requirement that $F_{\Sigma_2^-}$ is on
tangent plane, that is
$$(0,\nabla h^T)\left[ (1-\alpha^k)\bmat{f^-(x_1^k) \\ f^-(x_2^k)}+\alpha^k
\bmat{f^-(x_1^k) \\ f^+(x_2^k)}+\sigma M\vec{x}^k\right] \ = \ 0\ .$$
Therefore,
$$\alpha^k=\frac{\nabla h^Tf^-(x_2^k)+\sigma(0,\nabla h^T)M\vec{x}^k}{\nabla h^T(f^-(x_2^k)-f^+(x_2^k))}\ .$$
Now, for any $\epsilon>0$,
there exists $K_\epsilon$ such that if $k>K_\epsilon$, then $\|\vec{x}^k-\vec{x}\|<\epsilon$.  This means that
(for $k$ sufficiently large) all points $\vec{x}^k$ are attractive sliding points relative to $\Sigma_2^-$, since
--because of \eqref{PartAtt}-- $0<\alpha^k<1$ for $k$ sufficiently large.
Thus, the sequence of sliding vector fields on $\Sigma_2^-$ is well defined. In the limit
as $k\to \infty$, $\alpha^k \to \frac{\nabla h^Tf^-(x)}{\nabla h^T(f^--f^+)(x)}$
(since $M\vec{x}=0$ for $\vec{x}\in \bf \Sigma$) and hence (see  \eqref{SlidingDyn}) we obtain
$$F_{\Sigma_2^-}(\bf x) \ =  \ \bmat{f^-(x) \\ f_{\Sigma}(x)}.$$
\end{proof}

\begin{rem}
The result {\bf extends} to the case $N>2$ as follows.
The sliding vector field $F_{\Sigma_i}({\vec{x}})$ along a single
$\Sigma_i$, has $f_{\Sigma}(x)$ in the $i$-th block while the components of the vector field
in the $j$-th block, $j \neq i$,
are equal to $f^+(x)$ or $f^-(x)$ in agreement with the sign of $h_j(x)$.
Moreover, $F_{\BSigma}({\vec{x}})=
\begin{pmatrix}f_{\Sigma}(x) \\ \vdots \\ f_{\Sigma}(x) \end{pmatrix}$.
\end{rem}

Next, in the case of $N=2$, let $X(T,0)$ denote the monodromy matrix along $\vec{x}_S(t)$.
Then, because of Assumption \ref{xs_ass}, $X(T,0)$ can be written as:
\begin{equation}\label{monodromy_eq}
X(T,0)=  X_1(T,t_3)X_{\BSigma}(t_3,t_2)S_{4,{\BSigma}}X_4(t_2,t_1)S_{14}X_1(t_1,0),
\end{equation}
where
\[
\begin{matrix}
\frac{d X_i(t,\tau)}{dt}= & (DF_i({\vec{x}_S}(t))+\sigma M)X_i(t,\tau), & X_i(\tau,\tau)=I_{2n}, & i=1,2, \\
& & & \\
\frac{d X_{\BSigma}(t,\tau)}{dt}= & (DF_{\BSigma}({\vec{x}_S}(t))+\sigma M)X_{\BSigma}(t,\tau),
& X_{\BSigma}(\tau,\tau)=I_{2n}. & \\
\end{matrix}
\]
In \eqref{monodromy_eq}, the matrix $S_{14}$ is the jump matrix from $R_1$ into $R_4$,
and the matrix $S_{4,{\BSigma}}$
is the jump matrix from $R_4$ into ${\BSigma}$. In general, see \cite{Ivanov} and \cite{Dieci.Lopez_2011},
the jump matrix on the  intersection of two discontinuity manifolds is ambiguous.  However,
in Theorems \ref{S14_thm}  and \ref{S4Sigma_thm} we show that for our problem
we can use a unique expression for the jump matrix.

\begin{rem}
The {\bf extension} of \eqref{monodromy_eq} {\bf for $N>2$}, is easily obtained
replacing $S_{14}$ and $S_{4,{\BSigma}}$ with $S_{1,2^N}$ and
$S_{2^N,{\BSigma}}$.
\end{rem}

Now, let $\y(t)=\bmat{y_1(t) \\ y_2(t)}$ be the solution of \eqref{PWSnetwork} with
perturbed initial
conditions $\y(0)={\vec{s}}_0+\Delta {\vec{s}}$, $\|\Delta \vec{s}\| \ll 1$.
Recall that the monodromy matrix expresses, at
first order in $\Delta {\vec{s}}$, the evolution of these perturbed initial conditions after one period.
Below we give details on how to compute $S_{14}$.   To go from $R_1$ to $R_4$,
$\y(t)$ might either cross directly ${\BSigma}$ at time
$t_1+\Delta t$ or it might instead cross $\Sigma_1$ and $\Sigma_2$ at two different times
$\Delta t_1$ and $\Delta t_2$ before entering $R_4$ (see Remark \ref{synch_rem}). In \cite{Ivanov},
the form of $S_{14}$ is given and the author points out that the jump matrix is ambiguous.
In Lemma \ref{S14_prop} and
\ref{S14_2_prop} we give the expression of $S_{14}$ for the two different possibilities when
$\y(t)$ crosses $\Sigma_1$ and $\Sigma_2$ at two different
times, or it crosses directly ${\BSigma}$; but, then, Lemma \ref{S14_2_prop} states that only one expression of $S_{14}$ is
needed to assess perturbations of synchronous solutions and in Theorem \ref{S14_thm} we give
this unique expression of $S_{14}$ as a Kronecher product of the identity matrix $I_2$ with the jump
matrix of the single agent.

\begin{lemma}\label{S14_prop}
Let $\vec{x}_S=\bmat{x_S\\vec{x}_S}$ where $x_S$ satisfies Assumption \ref{xs_ass}.
Let	$\y(t)=\bmat{y_1(t)\\ y_2(t)}$
be the solution of \eqref{PWSnetwork} with initial
conditions $\y(0)={\vec{s}}_0+\Delta {\vec{s}}$, $\vec{s}_0=\bmat{s_0\\ s_0}$ and
$\|\Delta {\vec{s}}\|\ll 1$.
Let $(t_1+\Delta t_1)$ and $(t_1+\Delta t_2)$ be the times at which $\y(t)$ crosses respectively
$\Sigma_1$ and $\Sigma_2$ before entering $R_4$.  Assume $\Delta t_1 \neq \Delta t_2$.
Then, the jump matrix $S_{14}$ is given by
\[S_{14}=I_2 \otimes I_n +I_2 \otimes \frac{(f^+(s_1)-f^-(s_1) )\nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)}=I_2 \otimes S_{-,+},\]
with $S_{-,+}=I_n+\frac{(f^+(s_1)-f^-(s_1) )\nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)}$ being the jump matrix for the single agent in \eqref{nonsmooth_eq} from $R^-$ to $R^+$.
\end{lemma}
\begin{proof}
We examine below the case $\Delta t_1<\Delta t_2$. The proof for the case $\Delta t_2>\Delta t_1$ is similar.
Following \cite{Ivanov} and \cite{Dieci.Lopez_2011}, we can rewrite $S_{14}$ as the composition of two
jump matrices: $S_{14}=S_{34}S_{13}$,  with $S_{ij}$ being the jump
matrix from $R_i$ to $R_j$.  Using standard
results on jump matrices for one discontinuity manifold, we have that
(note that $\sigma M\vec{x}=0$ along a synchronous solution)
\[S_{13}= \begin{pmatrix} S_{-,+} & 0  \\ 0 & I_n \end{pmatrix} ,\,\
S_{34}= \begin{pmatrix} I_n & 0 \\ 0 & S_{-,+} \end{pmatrix},\]
with $S_{-,+}$ defined in the statement. Then $S_{34}S_{13}=I_2 \otimes S_{-,+}.$
The case $\Delta t_1>\Delta t_2$, in which the perturbed solution first crosses $\Sigma_2$ to enter into
$R_2$ and then crosses $\Sigma_1$ to enter into $R_4$, gives $S_{14}$ as the product $S_{24}S_{12}$.
It is easy to verify that
$S_{24}S_{12}=S_{34}S_{13}$.
\end{proof}

In the Lemma below we consider the case when the perturbed solution $\y(t)$ crosses $\Sigma_1$ and $\Sigma_2$ at the same time, and
give a unique expression for the jump matrix $S_{14}$ in this case.

\begin{lemma}\label{S14_2_prop}
As in Lemma \ref{S14_prop}, let $\vec{x}_S=\bmat{x_S\\vec{x}_S}$ where $x_S$ satisfies Assumption \ref{xs_ass}.
Let	$\y(t)=\bmat{y_1(t)\\ y_2(t)}$
be the solution of \eqref{PWSnetwork} with initial
conditions $\y(0)={\vec{s}}_0+\Delta {\vec{s}}$, $\vec{s}_0=\bmat{s_0\\ s_0}$ and
$\|\Delta {\vec{s}}\|\ll 1$.
Let $(t_1+\Delta t_1)$ and $(t_1+\Delta t_2)$ be the times at which $\y(t)$ crosses respectively
$\Sigma_1$ and $\Sigma_2$ before entering $R_4$.
Assume $\Delta t_1=\Delta t_2=\Delta t$, so that
necessarily ${\y}(t)$ from $R_1$ crosses $\BSigma$ to enter into $R_4$.
Then, without loss of generality,
\[S_{14}=\begin{pmatrix} I_n+\frac{(f^+(s_1)-f^-(s_1)) \nabla h^T(s_1)}{\nabla h^T(s_1)^Tf^-(s_1)} & {\bf 0} \\
	\frac{(f^+(s_1)-f^-(s_1)) \nabla h^T(s_1)}{\nabla h^T(s_1)f^-(s_1)} & I_n
\end{pmatrix}=I_2 \otimes I_n+\begin{pmatrix} 1 &0 \\ 1 & 0 \end{pmatrix} \otimes
\frac{(f^+(s_1)-f^-(s_1))\nabla h^T(s_1)}{\nabla h^T(s_1)f^-(s_1)}.\]
\end{lemma}
\begin{proof}
Following  \cite{Ivanov} and \cite{Dieci.Lopez_2011},
there are two possible expressions for $S_{14}$ when $\Delta t_1=\Delta t_2$:
\begin{eqnarray*}
	S_{14}^{(1)}=I_{2n}+\left (F_4({\bf s}_1)-F_1({\bf s}_1)\right )
	\frac{\nabla h_1^T(s_1)}{\nabla h_1(s_1)^TF_1({\bf s}_1)}=
	\begin{pmatrix} I_n+\frac{(f^+-f^-) \nabla h^T(s1)}{\nabla h^Tf^-(s_1)} & {\bf 0} \\
		\frac{(f^+-f^-)  \nabla h^T(s1)}{\nabla h^Tf^-(s_1)} & I_n
	\end{pmatrix} ,\\
	S_{14}^{(2)}=I_{2n}+\left (F_4({\vec{s}}_1)-F_1({\vec{s}}_1) \right )
	\frac{\nabla h_2^T(s_1)}{\nabla h_2(s_1)^TF_1({\bf s}_1)}=
	\begin{pmatrix} I_n & \frac{(f^+-f^-) \nabla h(s_1)2^T(s1)}{\nabla h(s_1)^Tf^-(s_1)} \\
		{\bf 0 } & I_n+\frac{(f^+-f^-) \nabla h^T(s1)}{\nabla h^T(s_1)f^-(s_1)}
	\end{pmatrix}.
\end{eqnarray*}
Below we show that the two expressions are equivalent, in the sense that their
action on the input vector $\vec{x}_S(t_1)-\y(t_1)$ is identical.

Explicit computation of $S^{(1)}_{14} ({\vec{x}_S}(t_1)-{\y}(t_1))$ and
$S^{(2)}_{14}({\vec{x}_S}(t_1)-{\y}(t_1))$ gives
\begin{align*}
S_{14}^{(1)} ({\vec{x}_S}(t_1)-{\y}(t_1))= &
\begin{pmatrix}
	(s_1-y_1(t_1))+(f^+(s_1)-f^-(s_1))
	\frac{ \nabla h(s_1)^T(s_1-y_1(t_1))}{\nabla h(s_1)^Tf^-(s_1)} \\
	(f^+(s_1)-f^-(s_1))\frac{ \nabla h(s_1)^T(s_1-y_1(t_1))}{\nabla h(s_1)^Tf^-(s_1)}+(s_1-y_2(t_1))
\end{pmatrix},\\
S_{14}^{(2)} ({\vec{x}_S}(t_1)-{\y}(t_1))= &
\begin{pmatrix}
	(s_1-y_1(t_1))+(f^+(s_1)-f^-(s_1))\frac{\nabla h(s_1)^T(s_1-y_2(t_1))}{\nabla h(s_1)^Tf^-(s_1)} \\
	(f^+(s_1)-f^-(s_1))\frac{ \nabla h(s_1)^T(s_1-y_2(t_1))}{\nabla h(s_1)^Tf^-(s_1)}+(s_1-y_2(t_1))
\end{pmatrix}.
\end{align*}
The statement of the theorem then follows if we show
\begin{equation}\label{S14(1)(2)_eq}
\nabla h(s_1)^T(s_1-y_1(t_1))=\nabla h(s_1)^T(s_1-y_2(t_1))+\text{h.o.t.}.
\end{equation}
We use the following Taylor expansions
\begin{align*}
	y_{1,2}(t_1+\Delta t)& =y_{1,2}(t_1)+f^-(y_{1,2}(t_1))\Delta t_1+O(\Delta t_1^2), \\
	h(y_{1,2}(t_1))& =h(s_1)+\nabla h^T(s_1)(y_{1,2}(t_1)-s_1)+O(\|y_{1,2}(t_1)-s_1\|^2), \\
	0=h(y_{1,2}(t_1+\Delta t))&=h(y_{1,2}(t_1))+
	\nabla h(y_{1,2}(t_1))^Tf^-(y_{1,2}(t_1))\Delta t+\text{h.o.t.}=\\
	& =h(s_1)+\nabla h(s_1)^T(y_{1,2}(t_1)-s_1)+ \nabla h(s_1)^Tf^-(s1)\Delta t+\text{h.o.t.}.
\end{align*}
From the last equality \eqref{S14(1)(2)_eq} follows.
\end{proof}

We now have the following theorem, that gives a unique expression for $S_{14}$.
\begin{theorem}\label{S14_thm}
With same notation as in Lemma \ref{S14_2_prop},
without loss of generality we can use the following expression for the jump matrix $S_{14}$:
\[S_{14}=I_2 \otimes S_{-,+},
\]
where $S_{-,+}$ is the jump matrix of the single agent  \eqref{nonsmooth_eq} from $R^-$ into $R^+$.
\end{theorem}
\begin{proof}
As a consequence of Lemmas \ref{S14_prop} and \ref{S14_2_prop}, the jump matrix
$S_{14}$ can be taken as follows
\[S_{14}= \left \{ \begin{matrix} I_2 \otimes I_n +I_2 \otimes \frac{(f^+(s_1)-f^-(s_1) )\nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)}, & \Delta t_1 \neq \Delta t_2, \\
I_2 \otimes I_n+\begin{pmatrix} 1 & 0 \\ 1 & 0\end{pmatrix} \otimes
\frac{(f^+(s_1)-f^-(s_1)) \nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)} ,
 & \Delta t_1 = \Delta t_2. \end{matrix} \right .\]
Moreover, if $\Delta t_1=\Delta t_2$, then \eqref{S14(1)(2)_eq} implies that
at first order $\nabla h(s_1)^Ty_1(t_1)=\nabla h(s_1)^Ty_2(t_1)$ and similarly to the proof of Lemma \ref{S14_2_prop}, easy computations imply that
 \[\left ( I_2 \otimes \frac{(f^+(s_1)-f^-(s_1)) \nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)} \right ) \y(t_1)=
 \left ( \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} \otimes
 \frac{(f^+(s_1)-f^-(s_1)) \nabla h(s_1)^T}{\nabla h(s_1)^Tf^-(s_1)} \right ) \y(t_1).\] Then, at first order,
 \[\y(t_1^+)-{\vec{x}}(t_1^+) =(I_2 \otimes S_{-,+})(\y(t_1^-)-{\vec{x}}(t_1^-)), \]
 for any $\y(t)$ perturbed solution of ${\vec{x}_S}(t)$.
\end{proof}

\begin{rem}
Lemma \ref{S14_prop} and \ref{S14_2_prop} and Theorem \ref{S14_thm} {\bf extend}
in a straightforward
way to the case $N>2$, via the replacement of $S_{14}$ with
$S_{1,2^N}=I_N \otimes S_{-,+}$,
giving $\y(t_1^+)-{\vec{x}}(t_1^+)=(I_N \otimes S_{-,+})(\y(t_1^-)-{\vec{x}}(t_1^-))$.
\end{rem}

Going back to \eqref{monodromy_eq}, we next need to analyze
$S_{4,{\BSigma}}$, that is the jump matrix from $R_4$ to $\bf \Sigma$.   In general, the
jump matrix from the region $R_4$ to $\bf \Sigma$ is not uniquely defined, which makes it
not possible to give a unique expression for the mondromy matrix.   However,
the jump matrix for ${\vec{x}_S}(t)$ is an exception, as stated in Theorem \ref{S4Sigma_thm},
in the sense that (at first order) we can give a unique expression for the action of
$S_{4,{\BSigma}}$ on $\y(t_2^-)-\vec{x}_S(t_2^-)$. The proof in Theorem \ref{S4Sigma_thm} sums up
results analogous to the ones given in Lemmas \ref{S14_prop} and \ref{S14_2_prop} and in Theorem \ref{S14_thm} for $S_{14}$.

\begin{theorem}\label{S4Sigma_thm}
The jump matrix $S_{4,{\BSigma}}$ can be taken to be
\[S_{4,{\BSigma}}=I_2 \otimes (I_n+\frac{(f_{\Sigma}(s_2)-f^+(s_2)) \nabla h(s_2)^T}{\nabla h(s_2)^Tf^+(s_2)})=I_2 \otimes S_{+,\Sigma},\]
where $S_{+,\Sigma}$ is the jump matrix of the single agent \eqref{nonsmooth_eq} from $R^+$ into
$\Sigma$, and it is explicitly given by
$S_{+,\Sigma}=\displaystyle{I+\frac{(f_\Sigma(s_2)-f^+(s_2))\nabla h(s_2)^T}{\nabla h(s_2)^Tf^+(s_2)}}$, with
$f_{\Sigma}$ given in \eqref{SlidingDyn}.
\end{theorem}
\begin{proof}
Following \cite{Dieci.Lopez_2011}, a perturbed solution of ${\vec{x}_S}(t)$
might either reach directly ${\bf \Sigma}$ in a neighborhood of ${\vec{s}}_2$  or it might first slide along
$\Sigma_1$ or $\Sigma_2$ before reaching ${\bf \Sigma}$.  Let
$\y(t)=\bmat{y_1(t) \\ y_2(t)}$ be the perturbed solution, with $\y(0)={\vec{s}}_0+ \Delta {\vec{s}}$ and
$\|\Delta {\vec{s}}\| \|\ll1$.
\begin{itemize}
\item[i)] We first consider the case in which the perturbed solution reaches $\Sigma_1^+$, slides along
it and then it reaches ${\bf \Sigma}$. Then $S_{4, {\bf \Sigma}}=S_{{\bf \Sigma},\Sigma_1^+}S_{\Sigma_1^+,4}$ with
\begin{align*}
S_{4,\Sigma_1^+} & =I_{2n}+\frac{(F_{\Sigma_1^+}({\vec{s}}_2)-F_4({\vec{s}}_2))\nabla h_1({\vec{s}}_2)^T}{\nabla h_1({\vec{s}}_2)^TF_1({\vec{s}}_2)}=
\begin{pmatrix} I_n+\frac{(f_{\Sigma}({s}_2) -f^+({s}_2))\nabla h(s_2)^T}{\nabla h(s_2)^Tf^+({s}_2)} & 0 \\
	0 & I_n  \end{pmatrix},  \\
S_{\Sigma_1^+,{\bf \Sigma}} & =I_{2n}+\frac{(F_{\Sigma}({\vec{s}}_2)-F_{\Sigma_1^+}({\vec{s}}_2))\nabla h_2({\vec{s}}_2)^T}{\nabla h_2({\vec{s}}_2)^TF_1({\vec{s}}_2)}=
\begin{pmatrix} I_n & 0 \\
	0 & I_n+\frac{(f_{\Sigma}({s}_2) -f^+(x))\nabla h(s_2)^T}{ \nabla h(s_2)^Tf^+({s}_2)} \end{pmatrix},
\end{align*}
where the vector fields $F_{{\bf \Sigma}}$ and $F_{\Sigma_1^+}$ are as in Lemma
\ref{OnSigma_prop}. Then the statement follows in this case. The case in which $\y(t)$
reaches $\Sigma_2^+$ before reaching $\bf \Sigma$ is analogous and gives the same expression for the jump matrix.

\item[ii)] We next consider the case in which $\y(t)$ reaches $\bf \Sigma$ directly at time
$t_2+\Delta t$. As in \cite{Dieci.Lopez_2011}, in this case
$S_{4,{\BSigma}}$ has two expressions, namely
\begin{align*}
S_{4,{\BSigma}}^{(1)} & =I_{2n}+
\frac{(F_{{\BSigma}}-F_4)({\vec{s}}_2)\nabla h_1({\vec{s}}_2)^T}{\nabla h_1({\vec{s}}_2)^TF_4({\vec{s}})}, \\
S_{4,{\BSigma}}^{(2)} & =I_{2n}+
\frac{(F_{{\BSigma}}-F_4)({\vec{s}}_2)\nabla h_2({\vec{s}}_2)^T}{\nabla h_2({\vec{s}}_2)^TF_4({\vec{s}})}.
\end{align*}
Using the same reasonings as in Lemma \ref{S14_2_prop}, at first order
the perturbed solution must
satisfy $\nabla h_1({\vec{s}}_2)^T\y(t_2)=\nabla h_2({\vec{s}}_2)^T\y(t_2)$, i.e.,
$\nabla h(s_2)^Ty_1(t_2)=\nabla h(s_2)^Ty_2(t_2)+\text{h.o.t.}$.
In this case we can use a unique expression for $S_{4,\BSigma}$:
\[S_{4,{\BSigma}}=I_2 \otimes I_n+\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}
\frac{(f_{\Sigma}-f^+)(s_2)\nabla h^T}{\nabla h^Tf^+(s_2)}.\]
\item[iii)] The final argument is analogous to the one in the proof of Theorem \ref{S14_thm}.
We have two possible expressions for the jump matrix:  in  i), when $\y$ does not reach $\bf \Sigma$
directly and in ii) when $\y$ reaches $\bf \Sigma$ directly from $R_4$. In this last case
however, $\nabla h^T(s_2)y_1(t_2)=\nabla h^T(s_2)y_2(t_2)+\text{h.o.t.}$ and this implies that at first order
\[\y(t_2^+)-\vec{x}(t_2^+)=(I_2 \otimes S_{+,\Sigma})(\y(t_2)-\vec{x}(t_2) ). \]
\end{itemize}
The above points i), ii), iii) imply the statement of the theorem.
\end{proof}

\begin{rem}
The {\bf extension} of Theorem \ref{S4Sigma_thm}
to the case of $N>2$ is immediately achieved by taking
$S_{2^N,{\BSigma}}=I_N \otimes S_{+,\Sigma}$.
\end{rem}

Finally, the following theorem gives the complete
expression for the monodromy matrix of \eqref{PWSnetwork} linearized about the synchronous
solution $\vec{x}_S(t)$.  The proof puts together all the results previously derived in this section
and is therefore omitted.
\begin{theorem}\label{X_thm}
Let $x_s(t)$ be limit cycle of \eqref{nonsmooth_eq}, and let Assumption \ref{xs_ass} be satisfied.
Let ${\vec{x}_S}(t)=e \otimes x_S(t)$ be the corresponding synchronous solution of \eqref{PWSnetwork}.
Then, the monodromy matrix of \eqref{PWSnetwork} along $\vec{x}_S$ can be taken as
\[X(T,0)=X_1(T,t_3)X_\Sigma(t_3,t_2)(I_N \otimes S_{+,\Sigma})X_{2^N}(t_2,t_1)(I_N \otimes
S_{-,+})X_1(t_1,0),
\]
with
\[
\begin{matrix}
\frac{d X_i(t,\tau)}{dt}= & (DF_i({\vec{x}_S}(t))+\sigma M)X_i(t,\tau), & X_i(\tau,\tau)=I_{Nn}, & i=1,2^N \\
& & & \\
\frac{d X_{{\BSigma}}(t,\tau)}{dt}= & (DF_{{\BSigma}}({\vec{x}_S}(t))+\sigma M)X_{{\BSigma}}(t,\tau), &
X_{{\BSigma}}(\tau,\tau)=I_{Nn}, & \\
\end{matrix}
\]
where $DF_i$  and $DF_{{\BSigma}}$ are the Jacobian matrices of the vector fields
$F_i$, $i=1,2^N$ and $F_{{\BSigma}}$ respectively.
\hfill\qed
\end{theorem}

The key implication of Theorem \ref{X_thm} is that
the saltation matrices appearing in the expression of $X(T,0)$ can be obtained from the saltation
matrices of a single agent in the network, which is a great simplification.  However,  computation of the
matrices $X_i$ and $X_{{\BSigma}}$ involves all $N$ agents.
In case of large networks, these computations are too expensive,
and an extension of the MSF theory to \eqref{PWSnetwork} is needed.    This is the
purpose of the next section.

\section{Master Stability function}\label{MSF}
The MSF is a very nice technique which allows to study linearized stability of the synchronous solution
of a smooth network of $N$ agents of size $n$ each, by working with $N$ linearized systems
of size $n$, rather than one linearized system of size $nN$, a substantial saving!
The key idea in the MSF technique is to consider the variational equation along the synchronous
solution and to perform a change of coordinates induced by the matrix of eigenvectors of the
Laplacian $L$.   For smooth systems,
this change of coordinates brings the whole network into a block diagonal structure with sub-blocks
of size $n \times n$  and this in turn allows one to study the stability of $N$ systems of dimension
$n$ instead of the stability of one system of dimension $nN$ (see \cite{Pecora.Carroll}).

In the case of piecewise smooth vector fields for the agents, there are
at least two new concerns.  First, the same change of coordinates, while still
bringing the  variational equation along the synchronous solution into block diagonal form, will also change
the equations of the discontinuity manifolds that will now in general involve more than one agent,
and possibly all of them.  Then, we should not expect the saltation matrices to be block
diagonal. However,  we will see
that the transformation preserves the block structure of the
saltation matrices of Theorem \ref{X_thm}.  In other words, the Kronecker products involved in the expression of the
saltation matrices along a synchronous solution
are left unchanged by the coordinate change induced by the eigenvectors
of $L$ (see equation \eqref{jump_unchanged_eq}).
The second concern is related to the portion of the fundamental matrix
solution on $\bf \Sigma$, in particular to the Jacobian of $F_{\BSigma}$.
The general lack of uniqueness in expressing the sliding vector
field on the intersection of two or more discontinuity manifolds is not a concern in this setting,
since, by Lemma \ref{OnSigma_prop},  on
$\bf \Sigma$ we have a unique Flilippov sliding vector field.  However a difficulty is
related to expressing the Jacobian itself, since again the coordinate change
seemingly will destroy the sought block structure.  We deal with this difficulty in
Lemma \ref{Kron_slide_prop} below.
As a final result, in Theorem  \ref{main_thm} we will see that we can use the MSF technique also for PWS networks, and in particular to study stability of the synchronous solution $\vec{x}_S$ by linearized analysis on $N$ systems of size $n$.

In what follows, let $L$ be the graph Laplacian matrix ($L=L^T$),
and $W$ be the matrix of the orthonormal eigenvectors of $L$:
$W^TLW=\Lambda$, with $\Lambda$ diagonal.
Then $M=L \otimes E$ has eigenvalues $ \lambda_i\mu_j$, $i=1, \ldots N$, $j=1, \ldots, n$,
with $\lambda_{i}$'s the eigenvalues of $L$, and $\mu_j$'s the eigenvalues of $E$.
Moreover, with $V=W \otimes I$,
we get $V^{-1}MV=(W^T \otimes I_n)(L \otimes E)(W \otimes I_n)= \Lambda \otimes E$.
In \cite{Pecora.Carroll}, the change of variables $\y=V^{-1}{\vec{x}}$
reduces the variational equation along a synchronous solution
$\vec{x}_S(t)$ into the following block diagonal form
\[\dot z_i=(Df(x_S)+\sigma \lambda_{i}E) z_i, \qquad i=1, \ldots, N,\]
that is we have $N$ systems of size $n$.
The issue with nonsmooth agents is that the change of variables above in general changes the equations of
the discontinuity manifolds as well, and this makes it impossible to study $N$ systems independently.
In particular, the new equations of the discontinuity manifold(s) might involve all the agents.
\begin{exam}
To illustrate the last statement, take $L=\begin{pmatrix} -1 & 1 \\ 1 & -1 \end{pmatrix}$ and
$M=L \otimes I_2$. Then, using the same notations as before,
$W=\frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ and
$V=W \otimes I_2$.  If the discontinuity surface for the single agent is the plane
$\{ x \in {\mathbb R}^n | h(x)=e_1^Tx=0 \}$ , then $\Sigma_1$ and $\Sigma_2$ in the
$\y$ coordinates become respectively
$e_1^T\y=-e_3^T\y$ and $e_1^T\y=e_3^T\y$ .
It follows that the agents cannot be studied independently even though the variational
equations for the fundamental matrix solution are in block diagonal form.
\end{exam}

Nevertheless, we will now see that the special structure of the fundamental matrix solution for the case of
a synchronous periodic solution of \eqref{PWSnetwork}, allows to study the stability
of $\vec{x}_S$ via $N$ systems of dimension $n$.

With $V=W \otimes I$, consider the monodromy matrix for the linearization along the synchronous
periodic solution.  Let $\y_S(t)=V^{-1}\vec{x}_S(t)$ and
$Y(T,0)=V^{-1}X(T,0)V$. Then
\begin{align*}
Y(T,0) =Y_1(T,t_3)Y_{\BSigma}(t_3,t_2)(V^{-1}S_{2^N,{\BSigma}}V)
Y_{2^N}(t_2,t_1)(V^{-1}S_{1,2^N}V)Y_1(t_1,0),
\end{align*}
with $Y_i(t,\tau)=V^{-1}X_i(t,\tau)V$, $i=1,2^N$, and
$Y_{{\BSigma}}(t,\tau)=V^{-1}X_{{\BSigma}}(t,\tau)V$.

First notice that the particular structure of the jump matrices is such that
\begin{eqnarray}\label{jump_unchanged_eq}
(V^{-1}S_{1,2^N}V)= & (W^T \otimes I_n) (I_N \otimes  S_{-,+})(W \otimes I_n)= \\
&  (W^T I_N W) \otimes ( I_n S_{-,+} I_n) =I_N \otimes S_{-,+},
\end{eqnarray}
and similarly for $S_{2^N,{\bf  \Sigma}}$.

Secondly, we show that the $Y_i$'s can be obtained solving block diagonal
systems of ODEs.
Note that
\begin{align*}
V^{-1} DF_i(V{\vec{x}_S}(t))V & =(W^{T} \otimes I_n)(I_N \otimes Df_*(x_S(t))(W \otimes I_n) \\
& =I_N \otimes Df_*(x_S(t)), \,\
\end{align*}
with $Df_*=Df^-$ for $i=1$, and $Df_*=Df^+$ for $i=2^N$, and hence
\begin{equation}\label{Yi_eq1}
\frac{d Y_i(t,\tau)}{dt}= \left[I_N \otimes Df_*(x_S(t))+\sigma \Lambda  \otimes E\right]Y_i(t,\tau),   \quad Y_i(\tau,\tau)=I_{Nn},    \quad  i=1,2^N.
\end{equation}
The following Lemma shows how to rewrite the sliding vector field
$V^{-1}DF_{\BSigma}(\vec{x}_S)V$ as a Kronecker product as well.
\begin{lemma}\label{Kron_slide_prop}
	Let $\vec{x}_S(t)$ be a synchronous periodic solution of \eqref{PWSnetwork} and let
	$F_{\BSigma}(\vec{x}_S(t))$ be the sliding vector field defined as in Lemma
	\ref{OnSigma_prop}. Then
\begin{equation}\label{DerSlideFMS}\begin{split}
V^{-1} DF_{\BSigma}(\vec{x}_S(t))V & =I_N \otimes Df_{\Sigma}(x_S(t))+  \\
&	\{\frac\sigma{\nabla h^T(f^--f^+)} \Lambda \otimes [(f^+-f^-)\nabla h^TE]\}_{x_S(t)}
+\sigma M\ .
\end{split}\end{equation}
\end{lemma}
\begin{proof}
For simplicity, we will show the statement for $N=2$, the generalization for $N>2$ is immediate.
	The sliding vector field on $\Sigma$ for \eqref{PWSnetwork} is defined in
	Lemma \ref{OnSigma_prop}.	That is, $F_{\BSigma}$ is
\begin{equation*}
		F_{\BSigma}(\vec{x}) = \left [ \begin{matrix}
			(1-\alpha_1(\vec{x})))f^-(x_1)+\alpha_1(\vec{x})f^+(x_1) \\
			(1-\alpha_2(\vec{x})))f^-(x_2)+\alpha_2(\vec{x})f^+(x_2)
		\end{matrix}\right ] +\sigma M \vec{x}_S,
\end{equation*}
where we have kept the term $M\vec{x}_S$ even though it is $0$ (since $\vec{x}_S$
is synchronous), to clarify the computation of the Jacobian.

Now, $\alpha_1({\vec{x}_S})$ and $\alpha_2(\vec{x}_S)$ must be chosen so that:
$\bmat{\nabla h \\ 0}^TF_{\BSigma}({\vec{x}_S})=\bmat{0 \\ \nabla h}^TF_{\BSigma}({\vec{x}_S})=0$.
Let $\alpha(x_S)$ be such that $f_{\Sigma}(x_S)=(1-\alpha(x_S))f^-(x_S)+\alpha(x_S)f^+(x_S)$ as in equation \eqref{SlidingDyn}.
Then, for $i=1,2$, when we impose the tangency conditions, using $M=L \otimes E$, we get:
	\[\alpha_i({\vec{x}_S})=\alpha(x_S)+\frac\sigma{\nabla h(x_S)^T(f^--f^+)(x_S)}\nabla h(x_S)^T(l_{i1}Ex_1+l_{i2}Ex_2)_{x_{1,2}=x_S}=\alpha(x_S),\]
where the last equality follows from the definition of the Laplacian matrix ($l_{i1}=-l_{i2}$ ) and the
fact that the solution is synchronous, i.e. $x_1=x_2=x_S$.
	The gradient of $\alpha_1(\vec{x})$ is then obtained as follows
\begin{equation*}\begin{split}
D_{x_1}\alpha_1(\vec{x}_S)&=\left[\nabla \alpha(x_1)\right]_{x_1=x_S}
+\frac\sigma{\nabla h^T(f^--f^+)(x_S)}l_{11}\nabla h(x_S)^TE+ \\
& \,\ \frac{\nabla h^T(l_{11}Ex_1+l_{12}Ex_2 )(\nabla h^T(Df^+-Df^-))}
{(\nabla h^T(f^--f^+))^2} \vert_{x_1=x_s}\\
		&= \left[\nabla \alpha(x_1)\right]_{x_1=x_S}
		+\frac\sigma{\nabla h^T(f^--f^+)(x_S)}l_{11}\nabla h(x_S)^TE ,
\end{split}\end{equation*}
and similarly
\begin{equation*}
D_{x_2}\alpha_1(\vec{x}_S)=\frac\sigma{\nabla h(x_S)^T(f^--f^+)(x_S)}l_{12}\nabla h(x_S)^TE .
\end{equation*}
Therefore,
$$
\nabla \alpha_1({\vec{x}_S})=\bmat{ \left[\nabla \alpha(x_1)\right]_{x_1=x_S}
+\frac\sigma{\nabla h^T(f^--f^+)}l_{11}\nabla h^TE \\
 \frac\sigma{\nabla h^T(f^--f^+)(x_S)}l_{12}\nabla h^TE}_{x_S}\ .
$$
For $\alpha_2({\vec{x}_S})$ we obtain in a similar way
$$
\nabla \alpha_2({\vec{x}_S})=\bmat{\frac\sigma{\nabla h^T(f^--f^+)}l_{21}\nabla h^TE \\
\left[\nabla \alpha(x_2)\right]_{x_2=x_S}+\frac\sigma{\nabla h^T(f^--f^+)(x_S)}l_{22}\nabla h^TE}_{x_S}\ .
$$
Then
\begin{equation}\begin{split}\label{JacOnSigma}
		DF_{\Sigma}({\vec{x}_S}) & =\left [ \begin{matrix}
			Df_{\Sigma}(x_S)+\sigma l_{11}B &  \sigma l_{12}B \\
			& \\
			\sigma l_{21}B &  Df_{\Sigma}(x_S)+\sigma l_{22}B
		\end{matrix}
		\right ] +\sigma M \\
		& = I_N \otimes Df_{\Sigma}(x)+ \sigma L \otimes B+\sigma L\otimes E,
\end{split}\end{equation}
	where $B=\frac{(f^+-f^-)(x_S)\nabla h(x_S)^T}{\nabla h(x_S)^T(f^--f^+)(x_S)}E$. Then the statement follows at once.
\end{proof}
\begin{rem}
In the case of $N=2$, the Laplacian matrix is trivial: $L=\bmat{-1 & 1 \\ 1 & -1}$.  But. aside
from this simplification,
the proof of Lemma \ref{Kron_slide_prop} is identical for the case of $N>2$ except that we
have to account for more entries in the Laplacian matrix.   In particular, the
expression \eqref{JacOnSigma} remains valid.
\end{rem}

From Lemma  \ref{Kron_slide_prop} and \eqref{Yi_eq1},
with $Df_*=Df^-$ for $i=1$, and $Df_*=Df^+$ for $i=2^N$, we get
\begin{equation}\label{Yi_eq}\begin{split}
	 \frac{d Y_i(t,\tau)}{dt}& = \bigl(I_N \otimes Df_*(x_S(t))+\sigma \Lambda \otimes E\bigr)Y_i(t,\tau), \\ & Y_i(\tau,\tau)=I_{Nn},  \,\ i=1,2^N, \\
	\frac{d Y_{\Sigma}(t,\tau)}{dt}&= \bigl(I_N \otimes Df_{\Sigma}(x_S(t))+\sigma \Lambda \otimes E
	+\sigma \Lambda \otimes B	\bigr)Y_{\Sigma}(t,\tau),  \\ &  Y_{\Sigma}(\tau,\tau)=I_{Nn}.
\end{split}\end{equation}
To sum up
\begin{equation}\label{Y_eq}
Y(T,0)=Y_1(T,t_3)Y_{\bf  \Sigma}(t_3,t_2)(I_N \otimes S_{+,{\bf  \Sigma}})
Y_{2^N}(t_2,t_1)(I_N \otimes S_{-,+})Y_1(t_1,0),
\end{equation}
with $Y_1$, $Y_{2^N}$ and $Y_{\BSigma}$ as in the block diagonal equations \eqref{Yi_eq}.
In conclusion, we proved the following key theorem.
\begin{theorem}\label{main_thm}
The Floquet multipliers of \eqref{PWSnetwork} along the synchronous periodic orbit
${\vec{x}_S}(t)$ are the $nN$ eigenvalues of the $N$ matrices $Z_i(T)$,
$i=1, \ldots, N$, satisfying the following variational equations
\begin{align}\label{Zi_eq}
\dot Z_i=  \left  \{ \begin{matrix}
(Df^-(x_S(t))+\nu E) Z_i, & 0 \leq t<t_1, \\
(Df^+(x_S(t))+\nu E)Z_i,  & t_1<t<t_2, \\
(Df_{\Sigma}(x_S(t))+\nu  [E+B]) Z_i, &  t_2<t<t_3, \\
(Df^-(x_S(t))+\nu E) Z_i,&  t_3<t<T,
\end{matrix} \right .
\end{align}
where $\nu=\sigma\lambda_i$, $i=1,\dots, N$,
$B=\frac{(f^+-f^-)(x_S)\nabla h(x_S)^T}{\nabla h(x_S)^T(f^--f^+)(x_S)}E$, and
subject to the initial conditions: $Z_i(0)=I_n$,
$Z_i(t_1^+)=S_{-,+}Z_i(t_1^-)$,  $Z_i(t_2^+)=S_{+,{\BSigma}}Z_i(t_2^-)$.
\hfill\qed
\end{theorem}
As usual, we call Floquet exponents the logarithms of the multipliers.  For sure there is a $0$
exponent, since $1$ is a multiplier, because $\vec{x}_S$ is a periodic solution of \eqref{PWSnetwork}.
Now, since our network is connected, $L$ has only one
eigenvalue equal to $0$, let it be $\lambda_1=0$, all other eigenvalues of $L$ being
negative.  With this observation we are ready for the following definition.

\begin{definition}\label{msf_def}
Let $\tau_{i,j}(\sigma)$ be the multipliers of \eqref{Zi_eq}, for $\nu=\sigma \lambda_i$, and
$i=2,\dots, N$, and let $l_{i,j}=\log|\tau_{i,j}|$, $j=1,\dots, n$.
The Master Stability Function (MSF)
for \eqref{PWSnetwork}, relative to the synchronous periodic solution $\vec{x}_S$, is the
largest value $l_{i,j}$, call it $\lambda$.  The synchronous manifold
(i.e., the synchronous solution) $\vec{x}_S$ is {\sl transversally stable} for those values
of $\sigma$, if any, for which $\lambda<0$.
\end{definition}

\begin{rem}
In the literature for smooth networks, the MSF is defined in terms of the Lyapunov
exponents of the linearized problem.  Of course, in the case of periodic orbits, these
are the Floquet exponents, and hence our definition is consistent with previous
usage of the MSF.
\end{rem}

Naturally, the value of the MSF $\lambda$ depends on the coupling strength $\sigma$,
as well as on $E$ and the Laplacian $L$.  However, for a given network topology (hence, for
given $L$ and $E$), the MSF depends
only on $\sigma$.   We must further appreciate that the network is synchronizable about
$\vec{x}_S$, if all parameters values $\sigma\lambda_k$, $k=2,\dots, N$, give multipliers
less than $1$ in modulus.
Moreover, this is true regardless of whether or not $x_S$ is an asymptotically
stable periodic orbit of the system \eqref{nonsmooth_eq}.

\section{Periodic orbit of a piecewise smooth mechanical system. \\ Computation of
the MSF}\label{GalvaSection}
Here we study a system of two identical piecewise smooth mechanical oscillators,
first studied in \cite{Galvanetto_95}.
When they are not coupled, the single agents have an asymptotically stable periodic solution that we
denote with $x_S(t)$, and therefore $\vec{x}_S=\bmat{x_S \\ x_S}$ will be a synchronous
solution of the coupled system, for all $\sigma\ge 0$.

\subsection{A piecewise smooth network}
The network equations are
\begin{equation}\begin{split}\label{Galvanetto_eq}
\dot y_1 = & \,\  y_2\\
\dot y_2 = & -y_1-\sigma(y_1-y_3)\pm\frac1{1+\gamma \abs{y_2-\bar v}}\\
\dot y_3 = & \,\ y_4\\
\dot y_4 = & -y_3-\sigma(y_3-y_1)\pm\frac1{1+\gamma \abs{y_4-\bar v}},
\end{split}
\end{equation}
with $\sigma, \gamma\ge 0$.  In the notation of \eqref{nonsmooth_eq}, we have
$$f^\pm(y)=\bmat{y_2 \\ -y_1\mp \frac{1}{1\pm \gamma (y_2-\bar v)}}, \,\ y \in {\mathbb R}^2\ , $$
and the discontinuity surface is the plane
$y_2 -\bar v=0$,
so that $\nabla h=\bmat{0 \\1}$.

Further, with respect to the notation of \eqref{PWSnetwork}, here we have
$$x_1=\bmat{y_1\\  y_2},\quad x_2=\bmat{y_3\\ y_4}, \quad
\vec{x}=\bmat{x_1\\ x_2}, \,\ L=\left ( \begin{matrix} -1 & 1 \\ 1& -1 \end{matrix}\right),\,\
E=\left ( \begin{matrix} 0 & 0 \\ 1& 0 \end{matrix}\right)$$
and rewrite system \eqref{Galvanetto_eq} as
\begin{equation}\label{Galvanetto_2_eq}
	\dot \vec{x}=\left ( \begin{matrix} f^\pm(x_1) \\ f^\pm(x_2) \end{matrix} \right ) +
	\sigma M\vec{x}, \quad M=L \otimes E .
\end{equation}

In the computations below, we fix $\bar v=0.15$, and
$\gamma=3$ as in \cite{Galvanetto_95}, and  use the MSF to
study the stability of the synchronous periodic solution as $\sigma$ varies in $[0,5]$.
For $\sigma=0$, the two oscillators are uncoupled and the synchronous solution
$\vec{x}_S=\bmat{x_S(t) \\ x_S(t)}$
has two Floquet multipliers at $1$ and two identical multipliers less than 1.
Of course, the synchronous solution
persists for $\sigma \ge 0$, though its stability will depend on $\sigma$.
We use
the MSF to compute the Floquet exponents of the synchronous solution for $\sigma \neq 0$.

Now we proceed like we did in Section \ref{MSF}.
Let $V=\begin{pmatrix} -1 & 1 \\1 & 1 \end{pmatrix}$,
and $W=V \otimes I_2$, and let $S=W^{-1}MW$, so that
$S=\begin{pmatrix} -2E & {0} \\ {0} & {0} \end{pmatrix}$ where the $0$'s are
${2 \times 2}$ blocks.
Proceeding like we did to arrive at \eqref{Zi_eq}, we
linearize \eqref{Galvanetto_2_eq} along $\vec{x}_S(t)$,
and end up having to compute the Floquet multipliers of
the linear non autonomous system
\begin{equation}\label{lin_single_eq}
\dot z=\left \{ \begin{matrix} (Df^\pm(x_S(t))-2\sigma E)  z, & {\nabla h^Tx_S(t)\gtrless \bar v},  \\
(Df_{\Sigma}(x_S(t))-2\sigma (E+B))  z, & {\nabla h^Tx_S(t)=\bar v},
\end{matrix} \right .
\end{equation}
where $x_S(t)$ is the periodic solution of the single uncoupled agent $\dot x=f^\pm(x)$
and $B=\frac{(f^+-f^-)(x_S)\nabla h^T(x_S)}{\nabla h(x_S)^T(f^--f^+)(x_S)}E$.

Now, we make the observation that, for our problem \eqref{Galvanetto_2_eq},
we have $E+B=0$, and therefore in the sliding phase of \eqref{lin_single_eq} the
linearized problem is simply $\dot z=Df_{\Sigma}(x_S(t)) z$.

\subsection{Numerical experiments}
Thanks to Theorem \ref{main_thm}, the Floquet exponents of the synchronous solution of
\eqref{Galvanetto_eq} can be computed from \eqref{lin_single_eq}.   Therefore, our task
is to compute the solution of a single oscillator over one period, and then compute the
monodromy matrix of \eqref{lin_single_eq} and extract its Floquet multipliers, of which
we know one has to be $0$ because of sliding.
Computation of the periodic orbit of the single agent is
done with the 4th order event technique of \cite{SlidNoProj} and fixed stepsize
equal to $10^{-4}$ (so to
have a local error per step of size about $\mathtt{eps}$) and the monodromy matrix is
computed on the same mesh at once.  The multiplier at $0$ can always be
recovered exactly due to the structure of the saltation matrix.
In Figure \ref{Mults} we show the other multiplier in function of $\sigma$.
In the intervals where this is less than $1$, the synchronous solution is stable.

\begin{figure}\label{Mults}
		\includegraphics[width=250pt]{Mults_FixedSlideJac.eps}
\caption{Multiplier below 1 indicates stability of synchronized motion}
\end{figure}

To confirm the results of the MSF analysis,
we also integrated the full discontinuous system \eqref{Galvanetto_eq} with a variable
stepsize integrator and event location techniques for sliding along the intersection of two discontinuity manifolds. For $\sigma=4.8$ the MSF plotted in Figure
\ref{Mults} predicts asymptotic stability of the synchronous solution.
Given initial conditions that do not belong to the synchronous
manifold we integrated the full network for sufficiently large time to observe
convergence of the numerical solution to the synchronous periodic orbit. In Figure \ref{galvanetto_4p8_tol1em9_fig} on the left we plot the synchronous periodic
orbit while on the right we plot $(x_1(t)-x_3(t))$, after discarding the transient.

\begin{figure}\label{galvanetto_4p8_tol1em9_fig}
	$
	\begin{matrix}
		\includegraphics[width=200pt]{galvanetto_4p8_tol1em9.eps} &
		\includegraphics[width=200pt]{galvanetto_a4p8_tol1em9_x1mx3.eps}
	\end{matrix}
	$
\caption{Synchronous periodic orbit and network synchronization for $\sigma=4.8$}
\end{figure}

We do not see synchronization for other parameter values such as
$\sigma=1$, or $\sigma=2.6$, as predicted by the MSF,
while we see synchronization for $\sigma=2.7$.
Finally,
for the value of $\sigma=1.2$, our analysis based on the MSF validates the observation
in \cite{Galvanetto_95} that the synchronous manifold is unstable.

\section{Conclusions}
In this work, we extended the Master Stability Function (MSF) tool to networks of
identical Piecewise Smooth  (PWS) differential systems,
in order to infer stability of a synchronous periodic solution of the
network.  Our analysis rested on the appropriate extension of the fundamental matrix
solution in the present PWS case.  We had to overcome several difficulties, in primis
the lack of uniqueness of suitable saltation matrices on the intersection of
several discontinuity manifolds and the possibility to decouple the (large) linearized
$nN$-system into $N$ systems of size $n$, in order to exploit the MSF technique.  We
succeeded in doing this under very general assumptions, for the network synchronizing
along a periodic orbit of a single agent.
We complemented our analysis by a numerical illustration of the use of the MSF
for a PWS system of mechanical oscillators synchronizing (for some values of
the coupling parameter) on a stick-slip oscillatory regime.
The case of synchronization on an orbit different from a periodic one remains
to be analyzed.

\bigskip
{\bf Acknowledgments} \\
This work has been partially supported by the GNCS-Indam group and the PRIN2017 research grant.
The authors gratefully acknowledge the inspiration provided by a series of lectures given
by Mario di Bernardo and Marco Coraggio at the University of Bari in March 2020.
They were the last lectures before universities in Italy had to shut down due to the
pandemic and provided a stimulating diversion in the following months.

redpajama_set_name: RedPajamaArXiv


ALBANY, N.Y. – Bestpass, a company that provides single-source payment and streamlined toll management services to commercial fleets, and the Central Florida Expressway Authority (CFX) today announced a partnership to enhance toll management services for commercial fleets and drivers.
CFX operates and maintains a regional network of expressways in Florida's Lake, Orange, Osceola and Seminole counties. The network spans more than 100 miles and includes seven named expressways, 64 interchanges and 301 bridges.
The Central Florida Expressway Authority (CFX) is an independent agency of the state that operates and maintains a regional network of expressways for 2.3 million residents and more than 60 million visitors in Brevard, Lake, Orange, Osceola and Seminole Counties. Created by the Florida Legislature in 2014, CFX's 118-mile user-funded system includes 815 center lane miles, 69 interchanges, 14 mainline toll plazas, 74 ramp toll plazas and 335 bridges and seven named expressways. On average, more than 1 million toll transactions are recorded daily, 85 percent of them electronically. CFX operates E-PASS, the first electronic toll collection system in Florida, which supports more than 350,000 E-PASS accounts. For more information, visit CFXway.com.


Worldwide Distribution --- USA including states of AK, AL, AR, AZ, CA, CO, CT, FL, GA, HI, IA, ID, IL, IN, KS, KY, LA, MA, MD, ME, MI, MN, MO, MS, NC, NE, NH, NJ, NM, NY, OH, OK, OR, PA, RI, SC, SD, TN, TX, UT, VA, VT, WA, WI, WV and country of Canada.
Sunrise Medical-Hoyer Advance Patient Lifts (Portable and Folding Patient Lift) Sunrise Medical, Inc.
Explore more than 90,000 Recalls, Safety Alerts and Field Safety Notices of medical devices and their connections with their manufacturers.
Do you work in the medical industry? Or have experience with a medical device? Our reporting is not done yet. We want to hear from you.
Medical devices help to diagnose, prevent and treat many injuries and diseases. We are not suggesting or implying that any companies or other entities included in the International Medical Devices Database engaged in unlawful conduct or otherwise acted improperly. The same device may have different names in different countries. This database is not intended to provide medical advice and patients should check with their doctors to determine if it contains relevant information and if such information has medical implications for them.
The International Medical Devices Database is licensed under the Open Database License and its contents under Creative Commons Attribution-ShareAlike license. Always cite the International Consortium of Investigative Journalists when using this data. You can download a raw copy of the database here.


Entrepreneur, futurist, philanthropist, and investor, Jason Hope is more than the sum of his titles. The reason is he is fighting a battle you will not see covered anywhere in the news for some time. It is a battle we all struggle very few, if any, have triumphed successful. The struggle that I am eluding to is that against death, and for Hope, he is not letting the odds stand in his way.
Anti-aging has become one of those things we see more often in ads and on sci-fi flic that go beyond our reality. The creams do not work as we expect, and that flic we find ourselves addicted to is merely a combination of makeup and computer-generated imagery, but what if there was some truth to it. That is where Jason Hope comes in alongside the $500,000 donation he made to the SENS Foundation in 2010. This non-profit organization is hellbent on researching new technologies within the biotech industry to fight the curse of aging.
First, donations like Jason Hope help the organization to take fast action towards finding treatments for diseases like diabetes and Alzheimer's. Others diseases would include arthritis, osteoporosis, cancer and high blood pressure (which can lead to diabetes). As they occur the older we get, the idea is to uncover treatments that reverse their effects and stop them in their tracks. These types of medications called AGE-breakers already exist for small animals, the problem comes when taking the concept and applying it to humans. AGE-breakers work by attempting to obliterate advanced glycation end-products which is the reason as we age, out flesh, becomes veiny and less flexible.
This entry was posted in Business Career, investment expert and tagged continued donations. Bookmark the permalink.


text:
Q: Firefox 4 and 5: invalid 'in' operand this._sandBox - resource://gre/components/nsProxyAutoConfig.js This error drives me insane:
invalid 'in' operand this._sandBox - resource://gre/components/nsProxyAutoConfig.js line:91
in firefox 4 and 5. It started to malfunction when i did the upgrade from 3.6 to 4.
I have search the internet about the error but seems to be an old one, it first occur in Firefox 2.8 and fixed in 3.x and above.
What i have done to try to fix this:
- Upgrade from 4 to 5 - no effect
- Uninstall Firefox and reinstall it - no effect
- Removing caches - no effect
- Modified the resource://gre/components/nsProxyAutoConfig.js, by unpacking it's archive omni.jar (which is actually a zip file), fix the error with a try{} catch(e){} statement - clear cache again - NO EFFECT
It seems to be there is a compiled or cached version of nsProxyAutoConfig.js because the error is refering to the same line, and there is a comment of my fix!
    // Fix by EH, returning error in FF    <- This is now line 91
   try {
   if (!("FindProxyForURL" in this._sandBox))
      return null;
   } catch (e) { return null; }

I need to get rid of this error because there is no fix for it and it stops debugging the scripts of any website so i cannot see the error when there is a error in the javascript. That is very anoying.
Please, does somebody have a solution to this?
Thank you!

A: Alright, tried a new upgrade again from 3.6 to Firefox 6, same error. @Nickolay was right, it has something to do with the proxy configuration, but, i didn't touch it!
To get rid from this message go to menu:
Extra|Options|Advanced|Network Tab and open connection settings.
Set Proxy to "No Proxy"
Error is gone.


text:
Matt Steichen In 2015 Battlefields to Ballfields, Blog
Battlefields To Ballfields Mission Accomplished!
MISSION ACCOMPLISHED Attending 50 professional sporting events in 50 days seems impossible, not only did my dad make it possible, but he honored over 200 military heroes in the process!...
Matt Steichen In 2015 Battlefields to Ballfields, Baseball, Blog
The San Francisco Giants Make Warrior Wishes Come True!
After honoring Army Warrior Brian Bullock at the Giants game on Tuesday night, my dad picked up his luggage that was checked in at the Hyatt hotel & made his...
Warrior Wishes Come True By The Bay
After my Dad Honored a warrior in Oakland thanks to the A's, he made his way across the bay and slept ovenight on the big red couches at San Francisco's...
The Oakland A's Make Warrior Wishes Come True!
After honoring the legacy of Army Sgt Dwight Burn last night at the Dodgers game in LA, my dad made his way to bus stop and took a $1 overnight...
The Dodgers Help Us Honor The Legacy Of A Fallen Army Warrior
My Dad has now been on the road honoring heroes since April 2nd, 2015. He began our Battlefields To Ballfields mission in Chicago & has hosted over 150 warriors at 45...
Making Warrior Wishes Come True & "Staying Classy" In San Diego!
The San Diego Padres have been rained out at home only 1 time in their 46 year existence as a baseball team. Since building Petco Park in 2004, they have...
Rain Rain Go Away, The San Diego Padres Made Warrior Wishes Come True Today!
After spending 2 days in Anaheim making warrior wishes come true at the Angels games,…
There Was An "Angel In The Outfield" At Tonight's Game In Anaheim!
After my dad honored heroes at the Angels game last night, he took a bus for $1.75 back to LA to sleep at the Airport. He headed back to Anaheim this...
California Dreaming… Warrior Wishes Come True Thanks To The Angels!
After spending 2 days in Phoenix making Warrior Wishes come true thanks to the D'Backs, My dad was "California Dreaming" on an overnight greyhound bus to Los Angeles. Over the...
Warrior Wishes Come True From The Snowy Streets Of Denver To The Sunny Streets of Phoenix
After honoring over 100 warriors at 39 professional sporting events in 37 straight days, we finally got rained out. But thank's to Elway's restaurant in downtown Denver, we were still...
Making Some "Mile High" Memories Thanks To The Colorado Rockies
After spending 2 days in Kansas City making Warrior Wishes come true at the Royals Games, My Dad spent the night sleeping at the KC Airport before boarding his 6am flight...
Getting The "Royal" Treatment in Kansas City
After making Warrior Wishes come true at the St. Louis Cardinals game, my dad made his way over to the Greyhound bus station to catch a 3am bus to Kansas...

redpajama_set_name: RedPajamaCommonCrawl


text:
The basketball tournament at the 2017 Jeux de la Francophonie in Abidjan, Ivory Coast took place between 22 July and 29 July. Only women's tournament was held in Palais des Sports de Treichville. France defeated Quebec in the gold medal match 73-53.

Competition schedule

Participating nations

Medalists

Group stage
Times are local UTC+0.

Group A

Group B

Group C

Group D

Placement Round

Final round

Quarterfinals

Semifinals

Bronze medal match

Gold medal match

Final standings

References

External links
 Livre des resultats
Official Website

2017
Jeux de la Francophonie
2017 Jeux de la Francophonie
2017 Jeux de la Francophonie


Made-to-order jeans for the first time in Europe at its London flagship store on Regent Street. The new selection of tailored denim, called Levi's Lot No. 1.
The special models are available in different fits including mid, high or low rise and any style of leg including straight, tapered and skinny. They are available in different denim variants, including five weights and colours of selvedge denim from the best-selected mills in the world. Plus they are completed by different details, that is 20 different thread colours, 12 leather patches and seven button and rivet colours.
Customers can select their desired fit, weight and colour, as well as customize the design details.


text:
Q: How can I change the shell in Windows 98 How do I change the default Windows shell from explore.exe to my own custom shell in Windows 98?

A: Edit %WINDIR%\system.ini using a text editor (or use msconfig.exe). The default shell is defined in section [boot], variable shell.
[boot]
shell=Explorer.exe


text:
How to Crush Any Type of Marathon, According to CrossFit Games Vet Khan Porter
Ahmet Zehir
by Spenser Mestel
On the first day of the CrossFit Games, after biking 12,000 meters, finishing 30 ring muscleups (in three and a half minutes), and maxing out his back squat, strict press, and deadlift, Khan Porter rowed a marathon. Despite the fact the longest he'd ever rowed, uninterrupted, was 5 kilometers, and his abdominals started cramping so badly he had to stop and stand for a minute and a half, Porter completed all 42,195 meters in under three hours.
A few weeks later, the four-time Games athlete decided to do another marathon—this time on a ski erg. (We know, it's masochistic.) Before he started puking (later he would realize it was from food poisoning, not exhaustion), he was on pace to break the world record for his age group. Inspired, Porter says he'd also like to run a marathon for the first time and do a double on the Assault Bike. Below, he shares his training tips for anyone considering an unorthodox kind of marathon.
The 10 Best Workout Classes for Men: 2018
1. Invest in Your Equipment
Before the row, Porter taped his fingertips and wore gloves, which kept him from having any issues with his hands. The same can't be said about the rest of his body…
"I took knee sleeves out with me," he says, "and I figured that once my ass got unbearably sore from sitting on the seat, I would use them for an extra level of padding."
Still, his and everyone else's butts were raw by the end, which hammers home an inescapable reality of marathons: You have to have the right equipment. Wear padded shorts for a rowing marathon; invest in some anti-chafe cream for a burpee 5K; and get comfortable, supportive shoes for a traditional running marathon.
According to Porter, the ski erg was less of a challenge equipment-wise because he was standing and could shift his weight, but he still cramped in his calves, highlighting the importance of staying nourished for any multi-hour event.
The Best Mental Strategies to Power Through Pain
2. Make a Psychological Game Plan
Three hours is a long time to stay engaged on one task, and for that reason, Porter says he'd normally prefer to do the same length workout broken into three movements.
But, at the same time, "I like exploring when I'm in a certain level of discomfort," he says. "I find that challenge enjoyable, having to play different psychological games with myself. For a while, I was doing 800 meters easy and 200 meters harder. Sometimes, I focused on having a quick sip of water or holding a certain pace. You always find something to occupy your thoughts to take them away from the pain."
No matter what your tactic is—counting strokes, singing your favorite song—have it in mind beforehand.
3. Practice When the Stakes Are Low
Porter says the row was so tough that he considered quitting, one of only two times he's done that at the CrossFit Games (the other was during Murph in 2015, when it was over 100 degrees on the field). What helped push him through the hurt locker were the 39 athletes beside him, also competing to be the Fittest Man on Earth. It was a motivating environment, but when Porter did the ski erg, he intentionally kept it low-key.
"I didn't post about it or tell many people," he wrote on his Instagram. "Psychologically I wanted to be able to come up with as many excuses as possible to quit when it got tough, so it would be an exercise in resilience—pushing through simply because I wanted to, not had to."
12 CrossFit Workouts Anyone Can Do
4. Train in Every Time Domain
A basic tenant of Porter's training—and CrossFit in general—is that athletes can work in any time domain. Porter says he's done a few AMRAPs that are around the two-hour mark, so he has experience with endurance events, but he also succeeded because of the foundation of conditioning that he's built, whether that's training for a thirty-minute Murph or a two-minute Fran. According to Porter, that doesn't mean you should attempt a marathon if your only qualification is a solid Murph time, but it does suggest that training at high intensity can carry over into other domains.
5. Play With as Many Movements as Possible
At this year's CrossFit Games, competitors had to run a handstand obstacle course, climb a clean and jerk ladder, carry 445-pound yokes, and drag a weighted dummy across the arena. And that was only four of the 14 events. With his background in surfing and swimming, Porter excelled in the water events, but he also trains in as many movement patterns as he can—an approach that inspired him to tackle marathons in the first place.
"During the row, it sucked, but seeing what my style of training allowed me to do, physically, that's where the idea of doing more marathons came up."
If you're comfortable with running, get in the pool. If you bike every weekend, pick up a barbell. The cross training will help you push past your plateaus.
If you're in Sydney and looking for a gym, check out Porter's Play Movement Co.
7 Brutal CrossFit Workouts That Are Worth the Pain
Topics: CrossFit health & fitness workout