text:
Ка́бадди — древнейшая и популярная командная игра в Азии, включающая в себя элементы борьбы и пятнашек. Родоначальницей кабадди является Индия, а первая игра на международном уровне прошла в Дакке в 1985 году. В 1990 году кабадди среди мужских команд был включен в программу Азиатских игр в Пекине, с тех пор кабадди является регулярной спортивной дисциплиной на Азиатских играх. Первое соревнование по кабадди среди женщин прошло на международных соревнованиях в Гуанчжоу в 2010 году; на Азиатских играх соревнования среди женских команд также стали проводиться с 2010 года.

Правила игры
По правилам игры две команды, в каждой по 12 игроков (7 игроков на поле и 5 игроков в запасе), занимают две противоположные стороны игрового поля размером 12,5 м х 10 м, разделенного посередине линией. Игра начинается с того, что одна команда посылает «захватчика» к разделительной линии, который в подходящий момент перебегает на территорию другой команды (другую половину поля). Пока он находится там, он непрерывно кричит: «Кабадди! Кабадди!» Но на территории противника он может пребывать лишь до тех пор, пока может кричать, не переводя дыхания. Его задача — пока он кричит — рукой или ногой дотронуться до игрока противника (одного или нескольких) и убежать на свою территорию (часть поля). Если ему надо перевести дыхание, он должен бежать, так как команда, на площадке которой он находится, имеет право его схватить. Он сможет освободиться, если сумеет протянуть через разделительную линию руку или ногу. Команда соперников должна заставить его сделать одно из двух: или прикоснуться к земле, или сделать вдох (перевести дыхание). После того, как игрок нападающих успешно вернётся, игрок другой команды, до которого он дотронулся, выбывает из игры. Если нападавший будет схвачен, то один из членов защищавшейся команды становится нападающим. Игра продолжается до тех пор, пока одна из команд не потеряет всех своих участников.

Чемпионат мира по кабадди

Первый чемпионат мира по кабадди прошёл в 2004 году, последующие — в 2007, 2010 и 2011 годах. Победы во всех четырёх турнирах одерживала сборная Индии. Сборная Ирана дважды (в 2004 и 2007 годах) занимала второе место, сборные Пакистана и Канады выигрывали серебряные медали в 2010 и 2011 годах соответственно.

Результаты чемпионатов мира:

Страны-участницы

Ссылки
 Welcome to this Wonderful Game of Kabaddi
 International Kabaddi Federation
 Asian Amateur Kabaddi Federation
 Amateur Kabaddi Federation of India
 Iran Kabaddi Federation
 Japan Kabaddi Association
 International Kabaddi Teams
 A Game Called Kabbadi

В массовой культуре
Хиндиязычный фильм Panga («Второй шанс»), вышедший на экраны в 2020 году, рассказывает историю Джаи Нигам, образцовой жены и заботливой мамы — в прошлом чемпионки по кабадди, — которая, бросая вызов существующему в консервативном индийском обществе эйджизму, решает вернуться в спорт спустя десять лет.

Примечания

Кабадди


The Loveless, Kathryn Bigelow's 1982 debut, shows a lot of promise, but more than anything it shows an incredibly raw filmmaker at a very early phase of her career. Though it has intriguing elements to it, and is filled with style, this is a somewhat shapeless, rambling film, bordering on incoherence.
The Loveless stars a very young Willem Dafoe as the leader of a gang of bikers who come to a small town. The setting isn't firmly established, but the music, the cars, the style, all seem to have come straight out of the '50s. The bikers disrupt small-town life in a myriad of ways, eventually leading to violence, but this film follows a kind of dream logic that makes the action incoherent and verging on irrelevance.
There's the shape of some interesting ideas in here! We see that this film is, at least partially, a deconstruction of the romanticization of '50s biker culture. There are hints that Bigelow is interested in masculinity and exploitation of women, themes she would come back to later in her career. Defoe's character casually mentions that he was imprisoned for breaking the Mann Act – when pressed as to what that means, he declines to comment (The Mann Act made human trafficking, specifically with the aim of prostitution, a felony).
Bigelow doesn't probe these ideas, or go deep with them at all. This is closer a tone poem than a true movie. On the other hand, this film has a commitment to its style and aesthetic that helps to raise it beyond its story and themes (or lack thereof). It's fascinating to consider this in context of Bigelow's latter career, where she's adapted to more of an uncompromisingly austere style.
Overall, I found The Loveless a dull watch, only buoyed by its use as a point of comparison in the larger scope of Bigelow's career. The promise here is clear, and her later career of course proves that the best was yet to come, but I'm just not sure this is a film worth revisiting.


Added on March 3, 2012 by Ben Chapman.
In this episode Don and Ben talk Valentine's day happenings; chocolate and then being famous on the Internet. The guys eventually get to food safety stuff and chat about raw milk risks, inactivation of Salmonella in peanut butter and the risk of someones dirty hands ending up on their toilet paper.
W00t deal, a deal a day where Don and Kristin scored some Valentine's day salted caramels.
Don gets his greeting cards at CVS, Ben wonders if they stock these.
The hip Raleigh restaurant that Ben and Dani went to a couple of years ago for Valentine's Day.
Wine and Design - Dani's gift was to go out without Ben with a bunch of her friends. And then they ate Subway.
American Frozen Food Institute annual meeting, where Don went to hang out after Valentine's Day.
Ben tries to connect chocolate and Valentine's Day and food safety - here is an example.
What someone who is surprised might look like.
The iPhone 5 that cures the flu and makes broccoli taste like chocolate, and made Don famous on the inter webs.
Batz's Double Face Palm post.
What a denominator might look like.
FDA's cold-smoked fish hazards. There's contamination, not many people eat it, total risk is probably pretty low.
There was an outbreak of Salmonella linked to this company in 2009. Tragically these things happened and then this.
Some of the work on the inactivation of Salmonella in peanut butter.
What 1.5 million servings of chocolate might look like.
Don talks about Salmonella in dry spots (not Arizona, but products like hydrolyzed vegetable protein and almonds).
Walmart quietly dropped sprouts from sale.
FDA's sprout guidance document highlights the importance of testing spent irrigation water.
Bob Sanderson, whom Don would eat sprouts from if he liked them.
Galen Weston who thinks farmers markets might kill people soon.
Ben wants to buy food from vendors who think they might kill someone.
What Ben is really talking about -- food safety culture.
One guy who doesn't worry about making waves or causing a stir.
Don's famous internet friend. Not this one.
Merlin put Don on the spot on what his favorite Food Safety Talk episode is. Don picks this one. Ben likes this one.
Quit touching my toilet paper.
How Ben would mitigate this risk.
What Don is worried about at farmers markets.
Added on February 12, 2012 by Don Schaffner.
Ben has McDonalds for lunch, which starts the guys talking about the golden arches, food safety and comfort foods, with brief digressions into banjo music, which is a Deliverance reference. Apparently they have rednecks in Canada too.
From there the guys talk about Don's Extension Stakeholder Review and the trials and tribulations of a life in academe. Food Safety Talk is not a productivity podcast, despite the fact that the guys talked about Getting Things Done by David Allen, and a survey on how people fall off the GTD wagon. Don also mentions The Checklist Manifesto by Atul Gawande that he is listening to as an Audible audiobook.
Ben brings the discussion back to food safety by talking about his food safety infosheets and building a Food safety culture as per Yiannis, Griffiths, Ben and Doug). Ben also brings up Malcolm Gladwell. Don holds his tongue and doesn't mention what others have said about Mr. Gladwell.
Don starts the food safety talk in earnest with a mention of the outbreak linked to Your Family Cow, with an aside to Abbott and Costello. Don also mentions his talk on sampling raw milk for pathogens and the early history of HACCP. This evolves into a discussion of sampling and the uncertainties of epidemiology, and the dangers of possibly getting it wrong.
... we are concerned. We do not take even a suspicion of food-borne illness lightly. We took action accordingly.
After a quick digression on Ben's microbe ornaments (not actual size), Ben moves the discussion to labeling, including meat and the USDA cook to a safe internal temperature label, unpasteurized juice labeling, Pillsbury's polite message, and more terse advice from Betty Crocker, regarding Bisquick. Don chimes in with concerns about the profusion of food allergy labels.
The guys get sidetracked into a discussion of dough and flour related food safety including the E. coli outbreak linked to refrigerated cookie dough, the Aunt Jemima Salmonella in flour mix recall, and the Listeria in Eggo frozen waffle recall.
Ben brings things back to consumers and labeling by mentioning research by Kansas State scientists on consumer preparation of frozen, uncooked, breaded chicken products. Not to be outdone, Don mentions research from his lab on the risk of salmonellosis associated with consumption of raw, frozen chicken products cooked in low-wattage microwave ovens.
Food labels seem to be getting longer and more complicated, but even with increased complexity, managing risks of things like hydrolyzed vegetable protein recall may be problematic give the different ways people eat Ramen Noodles, for example. Clearly the safety of microwaved foods is complicated, as microwaved containers also pose non-microbial risks, like tipping over and burning you.
Finally, no discussion of labeling snafu's would be complete without a mention of those kosher "broiled" chicken livers, which made a bunch or people sick, including one reported case in Minnesota. Why do more people seem to get in Minnesota? Here is a hint, it has a lot to do with the Minnesota's commitment to public health including efforts like Team Diarrhea.
Ben concludes by noting that he is a fan of Canadian Marshall McLuhan, even if he gets the quote wrong.
Top Ten Food Microbiologist pet peeves.
Disgust and it's relationship to food safety.
Talking to Roy Costa about lack of chlorine in melon wash water.
Batz's new restaurant, where employees wear lots of flair, perhaps coming soon to Kickstarter near you.


text:
\section{Introduction}

A classical topic in control theory is the Lur'e problem\cite{Lurie:1944}, which concerns the stability of a feedback interconnection between a linear time-invariant (LTI) system $G$ and any nonlinearity or uncertainty $\Delta$ within some  classes (see Fig.~\ref{fig:lure}\footnote{The Lur'e system or Lur'e problem is originally defined for unforced systems, but it is relaxed to forced systems \cite{Altshuller:2013}.}). The stability of Lur'e systems is mainly studied with two different techniques:  Lyapnov stability and input-output stability.  Lyapnov stability is based on internal state variables of the unforced system, while  input-output stability studies the input-output mapping of the forced system. These two methods are closely related, and sometimes equivalent \cite{Vidyasagar:2002, Hassan:2002}.

\begin{figure}[ht]
	\centering
	\input{lure_pf.tex}
	\caption{The Lur'e system}
	\label{fig:lure}
\end{figure}

The input-output approach splits the stability problem into two steps. Firstly, we need to find a class of LTI systems referred to as multipliers, preserving some properties of the set of nonlinearities $\Delta$. Secondly, we search for a suitable multiplier within the developed class of multipliers for the LTI system $G$. As a result, the stability of the nonlinear system is translated into an LTI design problem. The nonlinearity $\Delta$ is a slope-restricted nonlinearity, where the class of Zames-Falb multipliers $\mathcal{M}$, defined in both continuous-time~\cite{Zames:1968} and discrete-time~\cite{Jan:1971} (see \cite{Joaquin:2016} for a tutorial), is the widest class of LTI multipliers preserving the positivity of the nonlinearity. Note that some other multipliers are phase equivalent to corresponding Zames-Falb multipliers \cite{Joaquin:2013}. Then the absolute stability problem is reduced to a search of $M\in\mathcal{M}$ such that
\begin{equation}\label{eq:re_condition}
Re\{M(z)\left(1+KG(z)\right)\}<0 \quad    \forall |z|=1,
\end{equation}
where $K$ is the maximum slope of the nonlinearity. This condition can be expressed equivalently in the IQC framework \cite{Alexandre:1997},  where the frequency domain condition can be converted to computable linear matrix inequalities (LMIs) by the Kalman-Yakubovich-Popov (KYP) lemma \cite{Rantzer:1996}.
In discrete-time, the search over FIR Zames-Falb multipliers proposed in~\cite{Shuai:2014,Joaquin:2018} provides the less conservative absolute stability results in the literature.

Recently, the analysis of the convergence rates of the Lur'e system has attracted much attention. First-order optimisation algorithms, such as gradient decent method and Nesterov method, are written as Lur'e systems \cite{Lessard:2016}.  Specially, a strongly convex function converges to the optimal point exponentially with a rate $\rho$ ($0<\rho<1$) with first-order optimisation algorithms, which can be considered as the equilibrium point of the corresponding Lur'e system. Less-conservative results are obtained in optimisation \cite{Scoy:2018, Cyrus:2018, Mahyar:2017} and control \cite{Zachary:2017}.

\begin{figure}[b!]
	\centering
	\input{lure_scaled.tex}
	\caption{The scaled system in \cite{Bin:2016}}
	\label{fig:lure_scaled}
\end{figure}
The convergence analysis of Lur'e systems has been presented in two different but equivalent frameworks:
\begin{itemize}
	\item On the one hand, in \cite{Bin:2016}, the (time domain) IQCs are constructed for the scaled uncertainty $\Delta_{\rho}$, and the corresponding exponential stability condition is to search for a multiplier that belongs to a suitable subset of $\mathcal{M}$, such that
	\begin{equation}\label{eq:re_p_condition1}
	Re\{M(z)(1+KG_{\rho}(z))\}<0 \quad \forall |z|=1,
	\end{equation}
	where the multiplier is in the same form of original FIR Zames-Falb multipliers, while the $\ell_1$-norm condition is penalised with $\rho$.
	\item On the other hand, in \cite{Boczar:2015, Boczar:2017, Freeman:2018}, the (frequency domain)  $\rho$-IQCs are constructed for the original uncertainty $\Delta$, and the exponential stability condition is to search for a multiplier that also belongs to a subset of $\mathcal{M}$, such that
	\begin{equation}\label{eq:re_p_condition}
	Re\{M(\rho z)(1+KG(\rho z))\}<0 \quad \forall |z|=1,
	\end{equation}
	where the multiplier is constructed from the original FIR Zames-Falb multiplier $M(z)$ by replacing $z$ by $\rho z$,  and the $\ell_1$-norm condition  is  also penalised with  $\rho$.
\end{itemize}

In both approaches, sound analysis on the causal FIR Zames-Falb multipliers are provided in the literature above.  On contrast, the noncausal FIR Zames-Falb multiplier in the form $M(\rho z)$ is studied in \cite{Freeman:2018}, where its modified $\ell_1$-norm condition is proved, but  details to obtain the stability LMI are not given.

In this technical note, we are concerned with the technique to apply noncausal FIR Zames-Falb multipliers in \cite{Shuai:2014,Joaquin:2018} to estimate exponential convergence rates, and especially focus on the factorisations. The main contribution of this technical note is the development of suitable factorizations for both approaches when noncausal Zames-Falb multipliers are used.
In Section \ref{sec:factorisations_iqc}, the time domain Zames-Falb IQC with causal multipliers in \cite{Bin:2016} for continuous time system is extended to frequency domain with noncausal multipliers for  discrete time system.
Meanwhile, in Section \ref{sec:factorisations_piqc}, we provide a factorisation by lifting \cite{Hosoe:2013} as an unified structure for causal, anticausal and noncausal FIR Zames-Falb multipliers in both the IQC and $\rho$-IQC frameworks. Then, the validity of different factorisations are discussed, which completes the results in \cite{Boczar:2015,Boczar:2017,Freeman:2018}.  Furthermore, we show that the multipliers $M(z)$ in (\ref{eq:re_p_condition1}) and $M(\rho z)$ in (\ref{eq:re_p_condition})  are equivalent by variable conversion, and lead to similar results in numerical examples in Section \ref{sec:results}.

\section{Notations and preliminary results}\label{sec:notations}
Some of the notations and definitions are summarised from \cite{Lessard:2016, Boczar:2015, Boczar:2017, Bin:2016}, which are repeated here for completeness.

\subsection{Notations and Lur'e systems}

Let $\mathbb{Z}$ and $\mathbb{Z}^+$ be the set of integer numbers and positive integer numbers including zero, respectively. The notations $\mathbb{R}$ and $\mathbb{R}^+$ are defined in the same way for real numbers. And let $\mathbb{C}$ be the set of complex numbers.  Let $\ell$ be the space of all real-valued sequences $h: \mathbb{Z}^+\mapsto\mathbb{R}$. Let $\ell_2$ be the space of real-valued square-summable sequences $h:\mathbb{Z}^+\mapsto\mathbb{R}$. For any $\rho\in[0,1]$, we will say that $h\in\ell_2^\rho$ if the sequence $\{\rho^{-k}h_k\}_{k=0}^\infty$ belongs to $\ell_2$. Finally, for absolute-summable sequences $h: \mathbb{Z}^+\mapsto\mathbb{R}$, we define $\|h\|_1=\sum_{k=-\infty}^{\infty}|h_k|$.

Let $\mathbf{RH}_{\infty}$  and $\mathbf{RL}_{\infty}$ be the space consisting of proper real rational transfer functions $G$:  $G\in \mathbf{RH}_{\infty}$ has all poles inside the open unit disk in the complex plane;   $G\in \mathbf{RL}_{\infty}$ has no pole on the unit disk. With the minimal state-space realisation, the transfer function is $G(z) = C(zI -A)^{-1}B+D$, or $G \sim $$\begin{bmatrix}  A\;  B;\; C\; D \end{bmatrix}$ in short.
The expression $G^{*}(z)$ denotes the complex conjugate transpose of $G(z)$ at $|z|=1$, i.e. $G^{*}(z)={G}^T\left(\frac{1}{z}\right)$,
where the superscript $T$ indicates the transpose. Moreover, if a parameter $\rho$ is involved in the variable, the complex conjugate transpose can be expressed as $G^*(\rho, z)=G^T\left(\rho, \frac{1}{z}\right)$.

A nonlinear operator $\Delta: \ell(\mathbb{Z}^+) \mapsto \ell(\mathbb{Z}^+)$ is said to be memoryless if there exists a map $N:  \mathbb{R} \to \mathbb{R} $ such that $(\Delta\upsilon)_k=N(\upsilon_k)$, $\forall k \in \mathbb{Z}$. Assume that $\Delta(0)=0$. The memoryless uncertainty $\Delta$ is said to be (sector) bounded, denoted by $\Delta \in [\underline{k},\overline{k}]$ ($0\le \underline{k}<\overline{k}<\infty$), if $\underline{k} x \le N(x) \le \overline{k} x, \forall x \in \mathbb{R}$. The uncertainty $\Delta$ is said to be slope-restricted, denoted by $\Delta\in S[\underline{k},\overline{k}]$, if $\underline{k}(x_1-x_2) \le N(x_1)-N(x_2) \le \overline{k}(x_1-x_2), \forall x_1, x_2 \in \mathbb{R}$ and $x_1 \ne x_2$. The slope-restricted uncertainty is also sector bounded, but the reverse is not.  Finally, the  uncertainty $\Delta$ is said to be odd if $\Delta(-x)=-\Delta(x)$, $\forall x\in \mathbb{R}$.

Consider the Lur'e system in Figure \ref{fig:lure}. It is expressed as
\begin{equation*}
v=f+Gw, \quad w=g+\Delta v.
\end{equation*}

The feedback interconnection is well-posed if the inverse map $(v,w) \mapsto (g,f)$ is causal in $\ell$.

\begin{definition}
The feedback  interconnection  in Fig. \ref{fig:lure} is  $\ell_2$-stable if it is well-posed, and the signals $(v,w)\in\ell_2$ for any $(g,f)\in\ell_2$.
\end{definition}

\begin{definition} The feedback  interconnection in Fig. \ref{fig:lure} is  globally exponentially stable with convergence rate $\rho$ if there exists some $\rho\in (0,1)$ and $c>0$ such that when $g=0$ and $f=0$,
\begin{equation}\label{eq:exponential_stable}
\|x_k\|\le c\rho^k \|x_0\| \quad \forall k\ge 0,\;\forall x_0\in \mathbb{R}^n .
\end{equation}
\end{definition}

Henceforth, the infimum convergence rate of the feedback  interconnection in Fig. \ref{fig:lure} is referred to as $\rho_{\{G,\Delta\}}$.

\begin{remark}\label{rm:exponential_convergence}
Condition (\ref{eq:exponential_stable}) is  equivalent  to the fact that  the state $x_k$ of $G$ converges to zero exponentially with the rate $\rho$, i.e. $x_k \rho^{-k}\to 0$ as $k\to \infty$.
\end{remark}

As mentioned in the introduction,  $\ell_2$-stability is an input-output relation, while  exponential stability is an internal relation. Therefore, it is not trivial to restate  exponential stability in an input-output manner.

\begin{definition}\label{df:l2p_stability}
The feedback  interconnection  in Fig. \ref{fig:lure} is  $\ell_2^{\rho}$-stable if it is well-posed, and the signals $(v,w)\in\ell_2^{\rho}$ for any $(g,f)\in\ell_2^{\rho}$.
\end{definition}

\begin{theorem}\label{th:stability_relations}
For the Lur'e system in Fig. \ref{fig:lure}, assume $G$ is controllable and observable, and $\Delta$ is  memoryless and slope-restricted. The unforced system is globally exponentially stable with rate $\rho$ if and only if  the forced system is  $\ell_2^{\rho}$-stable.
\end{theorem}

\begin{IEEEproof}
The sufficiency can be proved in a similar way with Proposition 5 in \cite{Boczar:2015,Boczar:2017}, and the necessity  is  proved in outline in Appendix \ref{appendix_proof}.
\end{IEEEproof}

\subsection{Kalman conjecture for convergence analysis}\label{sc:kalman_conjecture}
The Kalman conjecture is a necessary and sufficient condition for stability when it is true, which is stated below.

\begin{definition}[Nyquist value, $K_N$] \label{df:Nyquist}
	The Nyquist value of a stable transfer function $G(z)$ is
	\begin{equation*}\label{Nyquist}
	K_N=\sup_{K}\{ K>0: (1-\tau KG(z))^{-1} \ \textrm{is stable} \ \forall \tau\in[0,1] \}.
	\end{equation*}
\end{definition}

\begin{conjecture}[Kalman conjecture~\cite{Kalman:1957}] Let $\Delta$ be memoryless,  and $\Delta \in S[0, K]$. The feedback interconnection between $G$ and $\Delta$  is asymptotically stable if and only if  $K < K_N$.
\end{conjecture}

We can translate the above definition and conjecture into the convergence analysis. Let us define the absolute convergence rate of the class of systems defined by the Lur'e system as follows
\begin{equation}\label{eq:rho_star}
\rho^*_{\{G,K\}}=\sup_{\Delta\in S[0, K]}\{\rho_{\{G,\Delta\}}\}
\end{equation}
where $K<K_N$.
A lower bound of the $\rho^*_{\{G,K\}}$ is given by
\begin{equation}\label{eq:rho_linear}
\underline{\rho^*_{\{G,K\}}}=\max_{\tau\in[0,1]}\left\{\left|\text{eig}\left(\frac{G}{1-\tau KG}\right)\right|\right\}.
\end{equation}
In some instances, this lower bound is referred to as the theoretical value.  The Kalman conjecture can be restated using the convergence rates defined in (\ref{eq:rho_star}) and (\ref{eq:rho_linear}) as follows.
\begin{conjecture}[Kalman conjecture for convergence analysis]\label{cj:kc_convergence}
	 For any stable $G$, let $\Delta \in S[0, K]$ with $K<K_N$, then
\begin{equation}\label{eq:1}
\rho^*_{\{G,K\}}=\underline{\rho^*_{\{G,K\}}}.
\end{equation}
\end{conjecture}

\subsection{Estimation of upper bound of $\rho^*_{\{G,K\}}$}\label{sc:estimation}
In the last years, Zames-Falb multipliers have been used to estimate an upper bound of $\rho^*_{\{G,K\}}$, denoted by $\overline{\rho^*_{\{G,K\}}}$. As mentioned in the introduction, there are two approaches based on the relation below.

\begin{theorem}[\cite{Boczar:2015,Bin:2016,Boczar:2017}]\label{th:stability_original_sclaed}
		The system in Fig. \ref{fig:lure} is well-posed if and only if the scaled system in Fig. \ref{fig:lure_scaled} is well-posed. Furthermore, the system in Fig. \ref{fig:lure} is $\ell_2^{\rho}$-stable if and only if the scaled system in Fig. \ref{fig:lure_scaled} is $\ell_2$-stable.
\end{theorem}

The two approaches to estimate  $\overline{\rho^*_{\{G,K\}}}$ are reviewed in the following parts.

\subsubsection{Analysis in IQC framework}

In this approach, $\ell_2^{\rho}$ stability of the system in Fig. \ref{fig:lure} is studied by  $\ell_2$ stability of the scaled system in Fig. \ref{fig:lure_scaled}, where an IQC is constructed for the scaled uncertainty $\Delta_{\rho}$ at first.

\begin{definition}[IQC \cite{Alexandre:1997}]\label{df:IQC}
	Let $\Pi(z)$  be a Hermitian (self-adjoint) bounded measurable operator.  Then, for a bounded and causal operator $\Delta_{\rho}: {\ell} \mapsto {\ell}$, it is said to satisfy the IQC defined by $\Pi$, if  for all $v\in {\ell}_2$
	\begin{gather}\label{eq:iqc}
	\int_{|z|=1}
	\begin{bmatrix}
	 \hat{v}(z) \\ \widehat{\Delta_{\rho}v}(z)
	\end{bmatrix}^*
	\Pi(z)
	\begin{bmatrix}
	 \hat{v}(z) \\ \widehat{\Delta_{\rho}v}(z)
	\end{bmatrix}
	dz \ge 0,
	\end{gather}
where $\hat{v}$ and  $\widehat{\Delta_{\rho}v}$ denote the z-transform of $v$ and $\Delta_{\rho}v$ respectively.
\end{definition}

\begin{theorem}[\cite{Alexandre:1997}] \label{th:IQC_discrete}
	For the system in Fig. \ref{fig:lure_scaled}, let $G_{\rho}(z) \equiv G(\rho z) \in \mathbf{RH}_{\infty}$,  and $\Delta_{\rho}$ be a causal bounded operator. Assume that $\forall \tau \in [0,1]$,
	\begin{enumerate}
		\item  the feedback interconnection between $G_{\rho}$ and $\tau \Delta_{\rho}$ is well-posed;
		\item the operator $\tau \Delta_{\rho}$ satisfies the IQC defined by $\Pi$;
		\item there  exists $\epsilon>0$, such that
		\begin{gather}\label{eq:iqc_stability}
		\begin{bmatrix}
		G_{\rho}(z)  \\
		I \\
		\end{bmatrix}^*
		\Pi(z)
		\begin{bmatrix}
		G_{\rho}(z)  \\
		I \\
		\end{bmatrix}
		\le -\epsilon I, \quad \forall |z|=1.
		\end{gather}
	\end{enumerate}
 Then, the system in Fig.\ref{fig:lure_scaled} is $\ell_2$-stable, thus the system in Fig.\ref{fig:lure} is $\ell_2^{\rho}$-stable.
\end{theorem}

In order to make the frequency domain inequality (FDI)  (\ref{eq:iqc_stability}) computable, the Kalman-Yakubovich-Popov (KYP) lemma should be applied.

\begin{lemma}[KYP lemma \cite{Rantzer:1996}]\label{Le:kyp}
	Given $A \in \mathbb{R}^{n \times n}$, $B \in \mathbb{R}^{n \times m}$, $K_p=K_p^{T} \in  \mathbb{R}^{(n+m) \times (n+m)}$, with $ det(z I -A)\ne 0$ for all $|z|=1$, where the pair $(A, B)$ are controllable, the following statements are equivalent:
	\begin{enumerate}
		\item For all $|z|=1$,
		\begin{gather*}
		\begin{bmatrix}
		(zI -A)^{-1}B  \\
		I \\
		\end{bmatrix}^*
		K_p
		\begin{bmatrix}
		(zI -A)^{-1}B  \\
		I \\
		\end{bmatrix}
		\leq 0.
		\end{gather*}
		\item There is a symmetric matrix $P \in \mathbb{R}^{n \times n}$ and
		\begin{gather*}
		\begin{bmatrix}
		A^{T}PA-P & A^TPB   \\
		B^{T}PA & B^TPB \\
		\end{bmatrix}+K_p
		\leq 0.
		\end{gather*}
	\end{enumerate}
\end{lemma}

Generally, the IQC multiplier $\Pi$ is dynamic, and can be factorised as below.
\begin{definition}[\cite{Scherer:2011}]
	Any $\Pi(z)\in\mathbf{RL}_\infty$ has nonunique factorisations $(\Psi,K_p)$ in the form
	\begin{equation}\label{eq:factorisation}
	\Pi(z)=\Psi^*(z)K_p\Psi(z),
	\end{equation}
	where $K_p=K_p^T$ is constant, and $\Psi$ is a stable LTI system with the state-space representation
	\begin{gather}\label{eq:psi_state_space}
	\Psi(z) \sim  \begin{bmatrix}  A_\Psi & B_{\Psi_1} & B_{\Psi_2}\\ C_\Psi & D_{\Psi_1} & D_{\Psi_2} \end{bmatrix}.
	\end{gather}
\end{definition}

Notice that when $G(z) \sim $ $\begin{bmatrix}  A\;  B;\; C\; D \end{bmatrix}$, $G_{\rho}(z)\equiv $ $G(\rho z) $ $\sim \begin{bmatrix}  \rho^{-1}A\;\;  \rho^{-1}B;\;\;  C\;\;  D \end{bmatrix}$. Next, substitute (\ref{eq:factorisation}) into (\ref{eq:iqc_stability}), and apply the KYP lemma, the well-known stability LMI is applied to the scaled system.

\begin{corollary}
	The FDI (\ref{eq:iqc_stability}) is equivalent to the existence of $P=P^T$ such that
	\begin{equation}\label{eq:iqc_lmi}
	\begin{bmatrix} \hat{A}^TP\hat{A}-P  &\hat{A}^TP\hat{B} \\  \hat{B}^TP\hat{A} & \hat{B}^TP\hat{B}\end{bmatrix}+\begin{bmatrix} \hat{C}^T \\\hat{D}^T\end{bmatrix} K_p \begin{bmatrix} \hat{C} &\hat{D}\end{bmatrix}<0,
	\end{equation}
	where $\Psi\begin{bmatrix} G_{\rho}  \\I\end{bmatrix} \sim \begin{bmatrix} \hat{A} & \hat{B}\\ \hat{C} & \hat{D} \end{bmatrix}$, and
	$\hat{A}=\begin{bmatrix}\rho^{-1}A & 0 \\ B_{\Psi_1}C & A_{\Psi} \end{bmatrix}$,
	$\hat{B}=\begin{bmatrix}\rho^{-1}B \\ B_{\Psi_2}+B_{\Psi_1}D \end{bmatrix}$,
	$\hat{C}=\begin{bmatrix}D_{\Psi_1}C & C_{\Psi}\end{bmatrix}$, $ \hat{D}=D_{\Psi_2}+D_{\Psi_1}D$.
\end{corollary}

\subsubsection{Analysis in $\rho$-IQC framework}

In this approach,  $\ell_2^{\rho}$ stability of the system in Fig. \ref{fig:lure} is studied by scaling signals,  where a $\rho$-IQC is defined for the original uncertainty $\Delta$ at first.

\begin{definition}[$\rho$-IQC \cite{Boczar:2015,Boczar:2017}] \label{df:rho_IQC}
	Let $\Pi(\rho, z)$ be a Hermitian (self-adjoint, i.e. $\Pi(\rho, z)=\Pi^*(\rho, z)=\Pi^T(\rho, z^{-1})$) bounded measurable operator.  Then, for a bounded and causal operator $\Delta: {\ell} \mapsto {\ell}$, it is said to satisfy the $\rho$-IQC defined by $\Pi$, if for all $y_2\in {\ell}_2^{\rho}$
			\begin{gather}\label{eq:rho_iqc}
			\int_{|z|=1}
			\begin{bmatrix}
			\widehat{y_2}(\rho, z)\\ \widehat{\Delta y_2}(\rho, z)
			\end{bmatrix}^*
			\Pi(\rho, z)
			\begin{bmatrix}
			\widehat{y_2}(\rho, z)\\ \widehat{\Delta y_2}(\rho, z)
			\end{bmatrix}
			dz \ge 0,
			\end{gather}
	where $\widehat{y_2}(\rho, z) \equiv \widehat{y_2}(\rho z)$ and $\widehat{\Delta y_2}(\rho, z)\equiv \widehat{\Delta y_2}(\rho z)$.
\end{definition}

\begin{remark}
According to the signal relations in the proof of Theorem \ref{th:stability_original_sclaed}, (\ref{eq:iqc}) holds if and only if  (\ref{eq:rho_iqc}) holds, which implies $\Pi(z)$ and $\Pi(\rho,z)$ are  equivalent. It will be shown by the Zames-Falb IQC and $\rho$-IQC in Section \ref{sec:factorisations_iqc} and \ref{sec:factorisations_piqc}.
\end{remark}

\begin{theorem} [\cite{Boczar:2015,Boczar:2017}] \label{th:rho_IQC}
	Fix $\rho\in(0,1)$. For the Lur'e system in Fig. \ref{fig:lure}, let $G(\rho,z) \equiv G(\rho z) \in \mathbf{RH}_{\infty}$, and  $\rho_-\circ(\Delta\circ \rho_+) \equiv \Delta_{\rho}$ be a causal bounded operator. Assume that $\forall \tau \in [0,1]$,
	\begin{enumerate}
		\item  the feedback interconnection between $G$ and $\tau \Delta$ is well-posed;
		\item the operator $\tau \Delta$ satisfies the $\rho$-IQC defined by $\Pi$;
		\item there  exists $\epsilon>0$, such that
		\begin{gather}\label{eq:rho_iqc_stability}
		\begin{bmatrix}
		G(\rho, z)  \\
		I \\
		\end{bmatrix}^*
		\Pi(\rho, z)
		\begin{bmatrix}
		G(\rho, z)  \\
		I \\
		\end{bmatrix}
		\le -\epsilon I, \forall |z|=1.
		\end{gather}
	\end{enumerate}
 Then, the system in Fig.\ref{fig:lure} is $\ell_2^{\rho}$-stable.
\end{theorem}

Similarly, (\ref{eq:rho_iqc_stability}) can be converted to the stability LMI after the factorisation of $\Pi(\rho,z)$ as defined below.

\begin{definition}
	Any $\Pi(\rho,z) \in\mathbf{RL}_\infty$ has nonunique factorisations $(\Psi,K_p)$ in the form
	\begin{equation}\label{eq:factorisation_rho}
	\Pi(\rho,z)=\Psi^*(\rho,z)K_p\Psi(\rho,z),
	\end{equation}
	where $K_p=K_p^T$ is constant, and $\Psi(\rho,z)$ is a stable LTI system with the variable $z$ and the parameter $\rho$.
\end{definition}

Particularly, when the multiplier used in the $\rho$-IQC is causal, $\Psi(\rho,z)=\Psi(\rho z)$  is valid. Additionally, when (\ref{eq:psi_state_space}) holds,
\begin{equation}\label{eq:psi_pz_causal}
\Psi(\rho, z) \sim  \begin{bmatrix}  \rho^{-1}A_\Psi & \rho^{-1}B_{\Psi_1} & \rho^{-1}B_{\Psi_2}\\ C_\Psi & D_{\Psi_1} & D_{\Psi_2} \end{bmatrix}.
\end{equation}

Therefore, $\Psi(\rho, z)\begin{bmatrix} G(\rho, z)  \\I\end{bmatrix}$ $\sim$ $\begin{bmatrix} \rho^{-1}\hat{A} & \rho^{-1}\hat{B}\\ \hat{C} & \hat{D} \end{bmatrix}$, and the LMI   can be further simplified to the form in \cite{Boczar:2015, Lessard:2016, Boczar:2017}: $\exists P=P^T$, such that
\begin{equation}\label{eq:causal_LMI}
\begin{bmatrix} \hat{A}^TP\hat{A}-\rho^2 P  &\hat{A}^TP\hat{B} \\  \hat{B}^TP\hat{A} & \hat{B}^TP\hat{B}\end{bmatrix}+\begin{bmatrix} \hat{C}^T \\\hat{D}^T\end{bmatrix} K_{p} \begin{bmatrix} \hat{C} &\hat{D}\end{bmatrix} <0.
\end{equation}

	\begin{remark}
		In the conventional IQC framework, the variable is $z$ only, and the analysis is conducted on the circle $|z|=1$ in the complex plane. However, in the $\rho$-IQC framework, the variables are $\rho$ and $z$, such as $\rho z$ and $\frac{z}{\rho}$. Then, when the complex conjugate  is used, the analysis is conducted on both circles $|z|=\rho$ and $|z|=\frac{1}{\rho}$, as shown in Fig. \ref{fig:circles}.
	\end{remark}

\begin{figure}[ht]
\centering
\includegraphics[width=\linewidth]{circles.eps}
\caption{Analysis on the circles with radii $\rho$ and $1/\rho$ }
\label{fig:circles}
\end{figure}

\subsection{Zames-Falb IQC with FIR multipliers} \label{sc:ZF-IQC}
In this part, the structure of the Zames-Falb IQC for the class of slope-restricted uncertainties is introduced. Then, three factorisations with FIR multipliers are provided.

\begin{theorem}[Zames-Falb IQC \cite{Heath:2005}]
Assume the uncertainty $\Delta$ is static and $\Delta\in S[0, K]$. It satisfies the Zames-Falb IQC defined by $\Pi$ as
\begin{equation}\label{eq:zf_iqc}
\Pi(z)=
\begin{bmatrix} 0 & K M^*(z) \\ K M(z) & -(M(z)+M^*(z)) \end{bmatrix}.
\end{equation}
\end{theorem}

Here, we focus on the noncausal FIR Zames-Falb multiplier proposed in \cite{Shuai:2014},
\begin{equation}\label{eq:zf_multiplier}
M(z)=-h_{-n_b}z^{-n_b}-\cdots-h_{-1}z^{-1}+h_0-h_{1}z^{1}-\cdots-h_{n_f}z^{n_f},
\end{equation}
where the causal part is with the backward-shift operator $z^{-i_b}$ ($i_b=1,2, \cdots, n_b$), and the anticausal part is with the forward-shift operator $z^{i_f}$ ($i_f=1,2, \cdots, n_f$). In addition, $h_{i_f}>0$ and $h_{-i_{b}}>0$, or $\Delta$ is odd. The $\ell_1$-norm  condition of $M(z)$ is
\begin{equation}\label{eq:FIR_L1}
\sum_{i_b=1}^{n_b}|h_{-i_b}| + \sum_{i_f=1}^{n_f}|h_{i_f}|< h_0,
\end{equation}
where we can set $h_0=1$ without loss of generality.

A standard factorisation of (\ref{eq:zf_iqc}) used in previous literature, such as \cite{Boczar:2015, Lessard:2016}, is (\ref{eq:factorisation}) with
	\begin{equation}\label{eq:zf_factorisation1}
	\Psi_1(z)=\begin{bmatrix} K M(z) & -M(z)  \\0 & 1 \end{bmatrix}, \quad K_{p,1}=\begin{bmatrix} 0 & 1  \\1 & 0 \end{bmatrix},
	\end{equation}
where $M(z)$ must be causal to keep $\Psi_1(z)$ stable, so it  needs  further factorisation for noncausal multipliers. The state-space representation of $\Psi_1(z)$ with causal Zames-Falb multipliers is given in \cite{Lessard:2016}.

Moreover,  the factorisation method called ``lifting factorisation'' is available, which can be treated as the discrete time counterpart of the factorisation for general continuous time multipliers in \cite{Veenman:2016}. One possible lifting factorisation is with
	 {\small{\begin{equation}\label{eq:zf_factorisation2}
 	\Psi_2(z)=\left[\begin{array}{c|c}
 	1 & 0\\ \pmb{Z}^{-i} & \pmb{0}\\ \hline 0 & 1\\\pmb{0} & \pmb{Z}^{-i} \end{array}\right],
 	K_{p,2}=\arraycolsep1pt\left[\begin{array}{cc|cc} 0 & \pmb{0} & Kh_0 & K\pmb{h}^T_{i} \\
 	\pmb{0} & \pmb{0} & K \pmb{h}_{-i} & \pmb{0}\\ \hline
 	Kh_0 & K\pmb{h}^T_{-i} & -2h_0 & -\pmb{h}^T_{-i}-\pmb{h}^T_{i}\\
 	K\pmb{h}_{i} & \pmb{0}  & -\pmb{h}_{-i}-\pmb{h}_{i} & \pmb{0}
 	\end{array}\right],
 	\end{equation}}}
where $\Psi_2(z)$ is called "lifting matrix", whose state-space representation is provided in Appendix \ref{appendix_ss}. Particularly, the causal and the anticausal parts in (\ref{eq:zf_multiplier}) must have the same step with this factorisation, i.e. $n_b=n_f=n_z$. Additionally, $\pmb{Z}^{-i}=[z^{-1} \; z^{-2} \;\cdots \; z^{-n_z}]^T$; $\pmb{h}_{i}=[h_1\; h_2\; \cdots \; h_{n_z}]^T$, $\pmb{h}_{-i}=[h_{-1}\; h_{-2}\; \cdots \; h_{-n_z}]^T$. Moreover, causal multipliers are obtained with $\pmb{h}_{i}=\pmb{0}$; anticausal multipliers are obtained with $\pmb{h}_{-i}=\pmb{0}$. Notice that the causal part and anticausal part share the same base $\begin{bmatrix}
1\\ \pmb{Z}^{-i} \end{bmatrix}$ in $\Psi_2(z)$, so we say they are coupled.

Henceforth, we use the notations $\pmb{0}$ for zero matrices in some proper dimensions, and  $\pmb{I}_{(n)}$ for the $n\times n$ identity matrix.

Finally, another lifting factorisation is defined in (\ref{eq:zf_factorisation3}) on the next page.
\begin{figure*}[t!]
\begin{equation}\label{eq:zf_factorisation3}
\Psi_3(z) =\left[ \begin{array}{c|c}
1 & 0\\ \pmb{Z}^{-i_b} & \pmb{0}\\0 & 1\\\pmb{0} & \pmb{Z}^{-i_b} \\ \hline 1 & 0\\ \pmb{Z}^{-i_f} & \pmb{0}\\0 & 1\\\pmb{0} & \pmb{Z}^{-i_f}
 \end{array} \right],\quad
K_{p,3} =
\left[ \begin{array}{cccc|cccc}
0 & \pmb{0} & K{h_0/2} & \pmb{0} & 0 & \pmb{0} & 0& \pmb{0}\\
\pmb{0} & \pmb{0} & K\pmb{h}_{-i_b} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0}\\
Kh_0/2 & K\pmb{h}_{-i_b}^{T} & -h_0 & -\pmb{h}_{-i_b}^{T} & 0 & \pmb{0} & 0 & \pmb{0}\\
\pmb{0} & \pmb{0} & -\pmb{h}_{-i_b} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0}\\ \hline
0 & \pmb{0} & 0 & \pmb{0} & 0 & \pmb{0} &K{h_0/2} & K\pmb{h}_{i_f}^{T}\\
\pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & \pmb{0}\\
0 & \pmb{0} & 0 & \pmb{0} & K{h_0/2}  & \pmb{0} & -h_0 & -\pmb{h}_{i_f}^{T}\\
\pmb{0} & \pmb{0} & \pmb{0} & \pmb{0} & K\pmb{h}_{i_f} & \pmb{0} & -\pmb{h}_{i_f} & 0\\
\end{array} \right].
\end{equation}
	\hrule
\end{figure*}
In this factorisation, the definitions of matrices are similar with (\ref{eq:zf_factorisation2}) but with possibly different values of $n_b$ and $n_f$. Similarly,  $\pmb{h}_{i_f}=\pmb{0}$ for causal multipliers;  $\pmb{h}_{-i_b}=\pmb{0}$ for anticausal multipliers. Especially, as illustrated in the lifting matrix $\Psi_3(z)$, the bases of the causal part and anticausal part are separated, which brings the flexibility to construct asymmetric noncausal multipliers. We say the causal and anticausal parts are decoupled.

In the following sections \ref{sec:factorisations_iqc} and  \ref{sec:factorisations_piqc}, the validity of these three factorisations will be discussed in IQC and $\rho$-IQC frameworks respectively. In addition, the multipliers $M(z)$ in (\ref{eq:re_p_condition1}) and  $M(\rho z)$ in (\ref{eq:re_p_condition}) will be shown equivalent.

\section{Zames-Falb multipliers for convergence analysis in IQC framework}\label{sec:factorisations_iqc}
In the convergence analysis, the FIR Zames-Falb multipliers belong to a subset of the class of Zames-Falb multipliers ($M\in\mathcal{M}_{\rho}\subset \mathcal{M}$) as their $\ell_1$ norm conditions are penalised with the convergence rate $\rho$.

Consider the noncausal multiplier in the IQC for the scaled uncertainty $\Delta_{\rho}$,
\begin{equation}\label{eq:FIR_noncausal1}
M(z)=-\tilde{h}_{-n_b}z^{-n_b}-\cdots -\tilde{h}_{-1}z^{-1} +h_0-\tilde{h}_{1}z^{1}-\cdots-\tilde{h}_{n_f}z^{n_f},
\end{equation}
where $\tilde{h}_{-i_b}>0$ and $\tilde{h}_{i_f}>0$,  or $\Delta_{\rho}$ is odd. Its $\ell_1$ norm condition is
\begin{equation}\label{eq:FIR_noncausal_L11}
\sum_{i_b=1}^{n_b}|\tilde{h}_{-i_b}|\rho^{-i_b}+\sum_{i_f=1}^{n_f}|\tilde{h}_{i_f}|\rho^{-i_f}<h_0,
\end{equation}
where the proof of the causal part is given in \cite{Bin:2016}, while the proof of the anticausal part will be linked with the anticausal multipliers in the next section.

As mentioned, for causal multipliers, all the factorisations in Section \ref{sc:ZF-IQC} are valid; while for anticausal and noncausal multipliers, (\ref{eq:zf_factorisation2}) and (\ref{eq:zf_factorisation3})  are valid.  Moreover, the analysis is in the IQC-framework, so the LMI also keeps the same form in (\ref{eq:iqc_lmi}).

In short, by this approach, everything  keeps the same as in the conventional IQC analysis except that the $\ell_1$ norm condition of FIR Zames-Falb multipliers are penalised symmetrically on causal and anticausal parts.

\section{Zames-Falb multipliers for convergence analysis in $\rho$-IQC framework}\label{sec:factorisations_piqc}

Different from the IQC analysis in the previous section, the parameter $\rho$ is involved as a variable in the $\rho$-IQC. As a result, the factorisation is restricted in different cases with causal, anticausal and noncausal multipliers.

Firstly, the lifting factorisation (\ref{eq:zf_factorisation3}) can be extended to (\ref{eq:factorisation_rho}) with $K_{p,3}$ being the same, and $\Psi_3$ being modified to

\begin{equation}\label{eq:phi_pz}
\small{\Psi_3(\rho, z) =\left[ \begin{array}{c|c}
	1 & 0\\ \rho^{-i_b}\pmb{Z}^{-i_b} & \pmb{0}\\0 & 1\\\pmb{0} & \rho^{-i_b}\pmb{Z}^{-i_b} \\ \hline 1 & 0\\ \rho^{i_f}\pmb{Z}^{-i_f} & \pmb{0}\\0 & 1\\\pmb{0} & \rho^{i_f}\pmb{Z}^{-i_f}
	\end{array} \right]},
\end{equation}
where $\rho^{-i_b}$ and $\rho^{i_f}$ are multiplied to $\pmb{Z}^{-i_b}$ and $\pmb{Z}^{-i_f}$, respectively. The state-space representation of $\Psi_3(\rho, z)$ is attached in Appendix \ref{appendix_ss}.

In the following, it is straightforward to show that the factorisation (\ref{eq:factorisation_rho}) with $\left(\Psi_3(\rho,z), K_{p,3}\right)$ is an unified  structure in the $\rho$-IQC framework with causal, anticausal and noncausal FIR Zames-Falb multipliers.

With the noncausal multiplier in (\ref{eq:FIR_noncausal1}), setting $h_{-i_b}=\tilde{h}_{-i_b}/\rho^{-i_b}$ and $h_{i_f}=\tilde{h}_{i_f}/\rho^{i_f}$,  the noncausal multiplier $M(\rho,z)$ in the $\rho$-IQC for the original  uncertainty $\Delta$ defined in \cite{Boczar:2015,Boczar:2017,Freeman:2018} is obtained:
\begin{equation}\label{eq:FIR_noncausal}
\begin{split}
M(\rho, z) =& -h_{-n_b}\rho^{-n_b}z^{-n_b} - \cdots - h_{-1}\rho^{-1}z^{-1}\\ & +h_0 - h_{1}\rho z - \cdots - h_{n_f}\rho^{n_f}z^{n_f},
\end{split}
\end{equation}
where $h_{-i_b}>0$ and $h_{i_f}>0$,  or $\Delta$ is odd.  Its $\ell_1$ norm condition is proved in the literature above as
\begin{equation}\label{eq:FIR_noncausal_L1}
	\sum_{i_b=1}^{n_b}|h_{-i_b}|\rho^{-2i_b} + \sum_{i_f=1}^{n_f}|h_{i_f}|< h_0,
\end{equation}
which in turn proves (\ref{eq:FIR_noncausal_L11}).

As the variable conversion is unique, the multipliers $M(\rho,z)$ in (\ref{eq:FIR_noncausal}) and $M(z)$ in (\ref{eq:FIR_noncausal1}) are equivalent and  belong to the same subset $\mathcal{M}_{\rho}$.  Nevertheless, the main issue of the noncausal multiplier $M(\rho,z)$ is the factorisation.

First, for causal multipliers, the factorisations (\ref{eq:zf_factorisation1}), (\ref{eq:zf_factorisation2}) and (\ref{eq:zf_factorisation3}) are all valid with $z$ replaced by $\rho z$. After that the LMI (\ref{eq:causal_LMI}) is obtained with the  KYP lemma.  In addition, when the factorisation (\ref{eq:phi_pz}) is used, the LMI is in the form (\ref{eq:iqc_lmi}).

Second, for anticausal multipliers, the factorisation (\ref{eq:zf_factorisation1}) is invalid. The other two factorisations by lifting are discussed.

The factorisation (\ref{eq:zf_factorisation2}) and  (\ref{eq:zf_factorisation3}) lead to (\ref{eq:factorisation_rho}) with anticausal multipliers by setting $\pmb{h}_{-i}=\pmb{0}$ and $\pmb{h}_{-i_b}=\pmb{0}$, respectively, and replacing $z$ by $\frac{z}{\rho}$ in $\Psi_{2,3}(z)$. The  corresponding state-space representation becomes
\begin{gather*}
\Psi_{2,3}(\rho,z) \sim  \begin{bmatrix}  \rho A_\Psi & \rho B_{\Psi_1} & \rho B_{\Psi_2}\\ C_\Psi & D_{\Psi_1} & D_{\Psi_2} \end{bmatrix}.
\end{gather*}

Notice that $\Psi_{3}(\rho,z)$ here is only valid for anticausal multipliers, while it for noncausal multipliers is (\ref{eq:phi_pz}).

With this factorisation,  the replacements $A_{\Psi} \to \rho A_\Psi $, $B_{\Psi} \to \rho B_\Psi $ are taken in  (\ref{eq:iqc_lmi}) to obtain the LMI in the anticausal case.

The factorisation (\ref{eq:phi_pz}) leads to  (\ref{eq:factorisation_rho}) with anticausal multipliers by setting $\pmb{h}_{-i_b}=\pmb{0}$. Similar to the causal case, the LMI (\ref{eq:iqc_lmi}) will be obtained.

Finally, for noncausal multipliers, the factorisation is more restricted, because it is inconsistent for the causal part and anticausal part. Using the factorisation (\ref{eq:zf_factorisation2}) as an example, it is impossible to replace the variable $z$ in  $\Psi_2$ to $\rho z $ for causal multipliers and to $\frac{z}{\rho}$ for anticausal multipliers simultaneously.  Therefore, the decoupling of causal and anticausal part is significant, and only the factorisation  (\ref{eq:phi_pz}) is valid. Then,  the LMI is (\ref{eq:iqc_lmi}).

In short,  by this approach, the factorisation is more restricted. The factorisation (\ref{eq:factorisation_rho}) with $\left(\Psi_3(\rho,z), K_{p,3}\right)$ is an unified structure in the $\rho$-IQC framework.

\section{Numerical results}\label{sec:results}
In this section, we compare the results by causal, anticausal and noncausal multipliers with different forms and factorisations. The examples are listed in Table \ref{tb:example_plants} with other related information. Here, we set the causal step and anticausal step to be equal in noncausal multipliers ($n_b=n_f=n_z$).  According to the preliminary study, higher order multipliers may not lead to less-conservative results, so the step in each example was tuned in advance to reduce the conservatism.

\begin{table}[ht]
	\centering
	\begin{tabular}{ |c|c|c|c| }
		\hline
		Ex & $G(z)$  &$K$ &$n_z$ \\ \hline
		1 & $-\frac{1}{z-0.4}$& $1$ & $1$ \\  \hline
		2 & $\frac{2z -1}{20z^2-10z+10}$& $9$ & $20$\\ \hline
		3 & $-\frac{10z^2+19z+9}{100z^3 -80z^2+17z-1}$&$3$  & $30$\\ \hline
		4 & $-\frac{0.1z}{z^2-1.8z+0.81}$& $12$  & $20$\\ \hline
	\end{tabular}
	\caption{Examples}
	\label{tb:example_plants}
\end{table}

The best estimates of $\overline{\rho^*_{\{G,K\}}}$ are obtained by the bisection search from the initial range $(max(|eig(A)|), 1)$, where the lower boundary is set to ensure the stability of the scaled plant $G(\rho z)$. The software package CVX with the solver sdpt3 \cite{cvx:2014, cvx:2008} is used to solve the LMI. The results are demonstrated in Table \ref{tb:convergence_results}, where the best result of each example is in bold.

\begin{table}[ht]
	\centering
	\begin{tabular}{ |c|c|c|c|c|c| }
		\hline
		 &   & \multicolumn{4}{|c|}{$\overline{\rho^*_{\{G,K\}}}$} \\ \cline{3-6}

		Ex &  $\underline{\rho^*_{\{G,K\}}}$ & C. (\ref{eq:FIR_noncausal}) &  AC. (\ref{eq:FIR_noncausal})  &  NC. (\ref{eq:FIR_noncausal1})  &   NC. (\ref{eq:FIR_noncausal})\\

	 	&   & with (\ref{eq:zf_factorisation2}) &  with (\ref{eq:zf_factorisation2}) & with (\ref{eq:zf_factorisation2})  & with (\ref{eq:phi_pz}) \\ \hline

		1 & $0.6$ & $0.600037$ &\pmb{$0.600024$} &\pmb{$0.600024$}&\pmb{$0.600024$}\\  \hline
		1a & $0.6$ & $0.600281$ &\pmb{$0.600024$} &\pmb{$0.600024$}&\pmb{$0.600024$}\\  \hline
		2 & $0.974679$& $0.978485$ & $0.975044$ &\pmb{$0.974758$}&$0.974794$\\ \hline
		2a & $0.974679$& $0.998474$ & $0.975044$ &\pmb{$0.974758$}&$0.974794$\\ \hline
		3 & $0.975367$& $0.976437$ & \pmb{$0.975525$}&$0.975815$ &$0.975891$\\ \hline
		3a & $0.975367$& $0.976501$ & \pmb{$0.975769$}&$0.975891$ &$0.975891$\\ \hline
		4 & $0.9$& $0.991760$ & invalid &\pmb{$0.990723$}&\pmb{$0.990723$}\\ \hline
		4a & $0.9$& $0.992794$ & invalid &\pmb{$0.992529$}&\pmb{$0.992529$}\\ \hline
	\end{tabular}
	\caption{Best estimates of $\overline{\rho^*_{\{G,K\}}}$ by causal (C.), anticausal (AC.) and noncausal (NC.) multipliers with different factorisations compared with $\underline{\rho^*_{\{G,K\}}}$}
	\label{tb:convergence_results}
\end{table}

In the table, Ex. 1-4 are with odd uncertainties while Ex. 1a-4a are with non-odd uncertainties. As demonstrated in the table, all the results are valid as they are larger than the theoretical bound $\rho^*$. For the systems that break the Kalman conjecture, such as Ex. 4(a), $\underline{\rho^*_{\{G,K\}}}$ is meaningless. Otherwise, the best results are  close to the theoretical value.

First, as for different types of uncertainties, the optimal convergence rates with odd uncertainties are no greater than those with general uncertainties. which is a natural result and can be reflected by the more strict $\ell_1$ norm conditions on multipliers for general uncertainties.

Then, as for different multipliers, the introduction of anticausal steps are efficient to reduce the conservatism (also see \cite[Fig. 2]{Freeman:2018}). Meanwhile, noncausal multipliers should be less conservative than causal and anticausal multipliers in general. Nevertheless, due to possible numerical problems with noncausal multipliers in MATLAB (e.g. large computational load, complex matrices), anticausal multipliers  provide the optimal results in Ex. 1(1a) and 3(3a).  However, anticausal multipliers are conservative when searching the maximum slope $K$ for  $\ell_2$-stability (also see \cite[Table \rom{2}, \rom{3}]{Shuai:2014}). For example, in Ex. 4(4a), the anticausal multiplier is not sufficient for  stability when $K=12$, while causal and noncausal multipliers are sufficient. In this case, noncausal multipliers are less conservative.

Moreover, the results by the two structures of noncausal multipliers $M(z)$ and $M(\rho, z)$ are almost the same, as they are equivalent. Nevertheless, the computational load with $M(\rho, z)$ would be larger due to its large matrices in the factorisation.

The similar conclusions are also inflected by the convergence rates with uncertainties that have different maximum slopes in Fig. \ref{fig:p_ex2},  \ref{fig:p_ex4}.

\begin{figure}[ht]
	\centering
	\includegraphics[width=\linewidth]{p_ex2.eps}
	\caption{$\overline{\rho^*_{\{G,K\}}}$ with different $K$ and multipliers of Ex.2}
	\label{fig:p_ex2}
\end{figure}
\begin{figure}[ht]
	\centering
	\includegraphics[width=\linewidth]{p_ex4.eps}
	\caption{$\overline{\rho^*_{\{G,K\}}}$ with different $K$ and multipliers of Ex.4}
	\label{fig:p_ex4}
\end{figure}

As illustrated in the figures, anticausal multipliers achieve tight bounds of the convergence rates when they are sufficient for  exponential stability; noncausal multipliers are efficient to achieve less conservative results when the closed-loop system is close to  instability. However, causal multipliers are conservative in general.

In summary, for a general plant whose properties are unknown, it is more reliable to use noncausal multipliers to obtain less conservative bounds of the convergence rates.     Meanwhile, the choice of a specific structure of noncausal multipliers is not crucial as for the conservatism in result.  On the contrary, if the given plant verifies the Kalman conjecture and the feedback uncertainty is odd, Conjecture \ref{cj:kc_convergence} seems true, and the multiplier techniques are not necessary.

\section{Conclusion}\label{sec:conclusion}
In this technical note, we reviewed the stability concepts of Lur'e systems, where exponential stability with the convergence rate $\rho$ is linked with an extension form of  $\ell_2$-stability, defined as $\ell_2^{\rho}$ stability.
On the other hand, we extended the literature results on causal FIR Zames-Falb multipliers to anticausal and noncausal cases. The Zames-Falb IQC and $\rho$-IQC are shown equivalent, where the multipliers can be converted by changing the variables. However, the factorisation of the Zames-Falb $\rho$-IQC is restricted, especially with noncausal multipliers. Hence, an unified factorisation is provided for both Zames-Falb IQC and $\rho$-IQC.
Furthermore, the numerical examples indicated noncausal multipliers are efficient to achieve less conservative estimation of the upper bound convergence rates in general case. However, when the system verifies the Kalman conjecture and the feedback uncertainty is odd, it is reasonable to use the theoretical bound from the linear analysis directly without any multiplier technique.

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


text:
Travel bookings in Cebu, Bohol slightly decline due to terror threats
admin • April 25, 2017	• 3677
MANILA — There is only minimal number of tourists who cancelled their trips to Central Visayas. This is despite a series of travel advisories issued by Japan, China and Korea to their citizens saying it is not safe to travel to Central Visayas due to threats of terrorism in the region.
Based on the records of the travel agency, Nemoto Travel, only 10 tourists have cancelled their travel bookings this month.
Nemoto Travel adds most tourists who availed of travel packages will pursue their vacation plans despite the travel advisories.
"For future bookings, there are some who have cancelled their bookings, but very minimal. Now we are processing the request of the Japanese school to Cebu. They are 30 persons going to Cebu in November. We also have other bookings which have not been cancelled," NAITAS president, Erlinda Tan said.
Tan, the owner of Nemoto Travel and Tours and the president of the National Association of Independent Travel Agencies (NAITAS) said there are about 3,000 Japanese tourists who book flights to Cebu and Bohol monthly.
The Department of Tourism said the same.
"If we talk about cancellations and we talk about those who decided to push through with their trips, it's very clear there a lot more tourists who decided to push through," said DOT Usec. Benito Bengzon Jr.
"Sabi ko nga bababa ang tourism for a while pero but after that it goes up… kasi I've been in travel business for 20 years (As I have said, there will be a decline in the tourism industry for a while. But after that, it goes up. I've been in the travel business for 20 years) ," DOT Secretary Wanda Tulfo Teo said.
Meanwhile, 65 travel agencies from NAITAS will conduct ocular inspection from April 28 to 30.
This is to show tourists that it is safe to travel to Cebu and Bohol. — Aiko Miguel | UNTV News & Rescue
TAGS Benito Bengzon, Bohol, Cebu, DOT, terror threat, tourism, Wanda Tulfo Teo
DOT imposes penalties vs City Garden Grand Hotel
Aileen Cerrudo • January 14, 2021
MANILA, Philippines — The Department of Tourism (DOT) on Thursday (January 14) served a six-month suspension and a P10,000 fine on City Garden Grand Hotel (CGGH) for misrepresenting itself to the public.
The Tourism Department also revoked its Certificate of Authority to Operate.
In a statement, the DOT said the CGGH "misrepresented itself to the public as being allowed to accommodate guests for leisure or staycation purposes despite being a quarantine facility."
Under DOT Administrative Order No. 2020-002-C, hotels operating as quarantine facilities are prohibited from accepting guests for leisure purposes.
Based on the resolution dated January 14, the CGGH in Makati City is liable for the offense of gross and evident bad faith in dealing with clients/fraudulent solicitation of business or making any false, deceptive, or misleading claims or statements.
Aside from the New Year's Eve Party which led to the death of flight attendant Christine Dacera, the hotel also reportedly have marketing packages to accept leisure guests and never indicated that it is a quarantine hotel.
Meanwhile, the CGGH can file an appeal within the period prescribed by the DOT rules and regulations.
The Tourism Department also warned all accredited accommodation establishments (AEs) to confine their operations to the authority given them, limited to just one per given time. AAC (with reports from Asher Cadapan Jr.)
TAGS City Garden Grand Hotel, Department of Tourism, DOT
Result on City Garden Grand Hotel probe out this week – DOT
Robie de Guzman • January 12, 2021
MANILA, Philippines – Results of the investigation into the possible violations of City Garden Grand Hotel in Makati City, where 23-year old flight attendant Christine Dacera died after partying with some friends, is expected to be out this week, the Department of Tourism (DOT) said on Tuesday.
"I was hoping to give a decision today pero hindi pwedeng i-rush. But don't worry, by this week, lalabas na yan," Tourism Secretary Bernadette Romulo-Puyat said in a press briefing.
The Tourism department last week issued a show-cause order against the hotel, directing its management to explain in writing within three days why its accreditation should not be revoked for possible violations for continuing to accept guests for leisure purposes when it is being used as a quarantine facility.
Government regulations state that an establishment being used as a quarantine or isolation facility are not allowed to provide accommodation for leisure purposes.
Puyat said the hotel already responded to the show-cause order and that the department conducted a physical investigation into the establishment last Friday.
She, however, declined to give any information about it but assured they are thoroughly probing the issue.
The tourism chief also assured that the DOT will do what is fair and just.
The hotel previously insisted that it claims no liability in Dacera's death and that it is compliant with government policies. – RRD (with details from Correspondent Asher Cadapan Jr.)
TAGS Bernadette Romulo-Puyat, christine dacera, Department of Tourism, DOT
Only 15 hotels in Metro Manila allowed to accept 'staycation' guests – DOT
Robie de Guzman • January 6, 2021
MANILA, Philippines – The Department of Tourism (DOT) reiterated that not all four- or five-star accommodation establishments (AE) in Metro Manila are allowed to accept guests for leisure purposes.
In a statement on Tuesday, Tourism Secretary Bernadette Romulo-Puyat said only those that have been granted by the DOT a certificate of authority to operate as staycation hotel can provide accommodation for leisure activities.
Puyat said only 15 establishments in the nation's capital have been allowed to operate as "staycation" hotels and these are the following:
Grand Hyatt Hotel
Makati Shangri-La Hotel
Okada Manila Hotel
Shangri-La at the Fort
Nobu Hotel
Joy Nostalg Hotel and Suites Manila
EDSA Shangri-La Manila
Solaire Resort
Hyatt Regency City of Dreams
Nuwa Hotel City of Dreams
The Peninsula Manila
Aruga by Rockwell
Sheraton Manila Hotel
Hilton Manila
Hotel Okura Manila
Accommodation establishments that are not included in the list are either closed or operating as a quarantine facility, the DOT said.
"We would like to correct the misperception that all four- or five-star rated accommodation establishments can operate as staycation hotels. For a hotel to accept guests for leisure purposes, it has to apply for a Certificate of Authority to Operate as Staycation (CAOS) hotel," the tourism chief said.
She also stressed that a hotel operating as a quarantine or isolation facility is prohibited from accepting guests for leisure purposes.
"With the authority is the responsibility to strictly adhere to the minimum health and safety guidelines. One has to pass the rigid inspection to ascertain that measures and systems are put in place and diligently practiced on a daily basis. It cannot, above all things, concurrently operate as a quarantine facility," Puyat said.
Puyat made the statement after the DOT issued a show-cause order against City Garden Grand Hotel in Makati City where a 23-year old flight attendant was found dead in one of its rooms after attending a new year's eve party.
The DOT asked the hotel to explain in three days why it should not be suspended as it is not accredited to operate as a staycation hotel.
"Lives are on the line and our economic recovery depends on everybody being responsible. Let us all perform our duties well. Each has an important part to play in managing this crisis," Puyat said.
"This appeal is also aimed at AEs without CAOS and tour operators and other entities that continue to post invitations or make false, deceptive, and misleading claims or statements for the purpose of soliciting business from clients these non-staycation enterprises have been using social media to carry its ads," she added.
Enterprises that wish to secure accreditation to operate as staycation hotels may submit their application through the department's accreditation portal.
The department also urged the public to confirm via this link the operation status of an establishment before booking or checking-in. – RRD (with details from Correspondent Asher Cadapan Jr.)
TAGS Department of Tourism (DOT), DOT, staycation

redpajama_set_name: RedPajamaCommonCrawl


"Madness & Mayhem" is encouraged!
We like to keep some of the evening details 'under our hat' (excuse the pun).
Keeping up with the Aardvarker Ball tradition, please do drag out your best glad rags and evening wear. Posh frocks, suits, quirkiness and 'down the rabbit hole' or 'Hatter' inspired welcome. The choice is yours!


Every operating environment is different, with various equipment applications and needs.
Fleet management often presents a complicated puzzle, especially when it comes to material handling equipment (MHE), such as forklifts, pallet jacks and scissor lifts. Every operating environment is different, with various equipment applications and needs.
Your first step in purchasing or leasing equipment is to assess how it will be used, where it will be living, and how many assets are required to move products into the hands of your customers. Be careful not to overestimate your equipment usage. You may end up paying more than you should. The Material handling experts at Element can conduct on-site visits to gain an understanding of your operating environment and needs.


"We are so pleased and proud of how this has made our boat look."
We are so pleased with the new enclosure that you have fabricated for our 47 foot Tradewinds sundeck trawler. The original was so rotted that the glass was splitting and crumbling if you tried to touch it. I have no idea how you managed to pattern off of this. We had three quotes done and they all were within $300.00 of each other. We chose your company as we are very familiar with your work and Eunice and I are both in the marine industry. The problem is getting someone to follow up with service after the unit is in and there is a problem. You are usually ignored and if some of them have their money you will even have to pay someone else to repair their mistakes. Not so with your company, two clips malfunctioned and you are all over it. We are so pleased and proud of how this has made our boat look.
Thank you!!!! and well done.
Wildcat - "Don't you love happy endings?"
We are no longer Clewless ! Well, Jan says that I am clueless when it comes to women, but that is different topic.
"The Cheapest and the best"
New sails – the cheapest and the best, both financially and environmentally!
We recently completed installing Shade Sails throughout the beach areas of Necker Island. Made from high quality marine-grade materials, these shades are pre-tensioned and a 90% mesh.
Sailing Anarchy recently sat down with Doyle's Head Designer Richard Bouzaid and Mike Sanderson to talk about everything from the latest Stratis development, to design to Grand Prix programs to Superyachts. Well worth viewing for some insight from two of the top guys in the industry.
The St Barths Bucket is one of the highlights of the superyacht sailing regatta calendar and a fleet of nearly 40 spectacular superyachts are gearing up for four days of racing in the 2013 St Barths Bucket which runs from the 28 – 31 of March.
Following the stepping of a new Southern Spar carbon rig, the 45 metre superyacht "Salperton" returned to Palma to have new Doyle Stratis™ high-performance sails fitted.


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text:
A consumer's guide to the best and worst of sports media and merchandise. Ground rules: If it can be read, played, heard, observed, worn, viewed, dialed or downloaded, it's in play here.
By MIKE PENNER
What: 'Total Sport'
Touting itself as "Britain's Biggest Sports Magazine," Total Sport is a year-old glossy monthly that can be found Stateside, at least locally, at better bookstores, international bookstores, even Tower Records. It is the publication to turn to if your idea of football is played by Manchester United, or if you agree that Britain's Tin Henman is "the rising star of world tennis," or if you're eager for thoughtful analysis on the best rugby union sides ever.
However, Total Sport aspires to live up to its name, so it regularly reaches across the pond to grapple with an ongoing fascination known as American sport.
Occasionally, a thing or two tends to get lost in the translation, or the transmission.
In its November issue, Total Sport features a beginner's guide to baseball, which is "romantically known as 'The Game With No Clock.' " Included is a diagram of a diamond, a brief discussion of such nuances as batting average ("Confusingly measured as a percentage of at-bats against successful hits") and a photograph of "Fred McGriff: ace slugger of the Blue Jays." Which would have been correct, had Total Sport published this primer in 1990.
A photo essay on the phenomenon of sporting mascots ("The Yanks, of course, are just mad about people encased in brightly coloured foam. . . .") runs further adrift. The green-clad spartan of Michigan State is identified as "Michigan's Trojan." USC's Tommy Trojan, meanwhile, is labeled "California['s] bearded bloke in a skirt." Purdue University is re-christened "Perdue College." And Penn State's Nittany Lion is violently slandered as a "rosy-nostrilled bear."
A preview of the 1996-97 NBA season is attempted in the December issue, in which the Atlanta Hawks are thus assessed: "The Georgia side have always lacked a big man in the middle," but that was before "the arrival of Zairean centre Dikembe Mutombo." The Clippers? "Just like Tampa Bay in the NFL and San Marino in international football, the Clippers are the perennial whipping boys of the NBA."
San Marino is ranked 162nd in the latest world soccer poll, just behind Botswana and Kazakhstan. Even without a working knowledge of Donald Sterling, the Total Sport subscriber reading in Middlesex is quickly brought up to speed.


Originally my idea was slightly different (please see below, in the quotation box) - to have a thread for me only. But pauljg jumped in with his shots (nothing wrong about that - it is now more open and we will have more fun with it) and so the thread evolved from "MiPr's random shots" into "Post your random shots".
Recently I decided to move from JAlbum (which I got a bit tired of) to to something more automated: Gallery2. The latter has a nice feature called "random image" - shows just a random shot each time the page is refreshed. Thanks to this I suddenly started discovering my own shots which always have been on my gallery but which I almost have forgotten.
It is a kind refreshing to see your own work from the past. Funny that I can still remember where each shot was taken (a bit harder is to recall "when" ).
Is it your idea that you are going to be the only one who posts pictures in this topic?
In 42 Bob J. challenged others to post pictures too, but that's clearly different.
Pauljg, my initial idea was to start thread with my photos only, however your question triggered a discussion among mods/admins how to organize this if other Dyxumers would like to join. There are a few problems to consider, e.g. possible "pollution" of the RHS "menu" of recent shots (because of this I probably change formulae to "7 random shots once per week" instead of "one shot daily"). I'll let you know the outcome of the discussion.
Might I suggest doing something similar to what I do with my pic-a-day (see my signature)? I have an externally hosted sequence of images and then once a week, I post the best of that week to my thread.
Anyway, bring it on! Or to use the vernacular, MOAR!
#7 And finally a quick glimpse on Technical Museum in Prague. I usually do not do selective colouring but in this case it was unavoidable - simply the exhibition area was so packed with exhibits that there was no other way to bring some separation on the photo.
I hope that you'll enjoy these shots. You are welcome to post comments/opinions/suggestions about them.
Fine series Mipr my favourite is #7 but the others are nice too!
which shows new leaves of a pieris japonica, growing on my balcony.
Please see the first post - I clarified a bit what the thread is about and also the topic changed a bit (again ).
Well, it does say random shots, so here goes.
I still haven't learned how to get them from closer.


A failed sewer line understandably strikes fear in the hearts of homeowners. For one, the high cost of sewer line replacement can make a major impact on most homeowners' bank accounts. For another, a completely failed sewer line means that all waste-water operations in the house must cease: toilets, sinks, showers, and bathtubs.
When a contractor produces an estimate on costs to fix your sewer line, the estimate includes both the cost of digging the trench with earth-moving equipment and the cost of the actual pipe replacement. If you are up to the task of hand-digging the trench required to access the old sewer line and install the new line, you can cut costs dramatically. If you go the next step and replace the line by yourself, the cost of replacing the sewer line comes to cents on the dollar as compared to full-service replacement by a contractor.
Digging and then working in a deep trench can be dangerous. According to the Occupational Safety and Health Administration (OSHA), an average of 70 workers die annually when the earthen walls of trenches collapse on them while they are working within the trenches. An open trench also poses a distinct hazard to children and pets. Make sure to take precautions if your trench needs to be a deep one. Trench shields prevent the walls of the trench from collapse, and OSHA regulations require the use of trench shields for trenches 5 feet or deeper. As a homeowner, you are not bound by OSHA regulations, but it is wise to follow OSHA guidelines for your own safety.
Digging a trench down to your sewer line is mostly just a matter of hard work. It is not technically difficult, but it can be extremely demanding physically. One adult in reasonably good shape may expect to dig a sewer trench 4 feet deep by 8 feet long by 3 feet wide in one weekend of long work days. This estimate is based on the assumption that you have loose soil with a moderate number of roots to cut through. Clay or densely packed soil, as well as the presence of many large roots, can increase digging time significantly.
Cold climates, too, present other complications. First, in these areas, the depth of the sewer pipe can be quite deep because the frost line may be 4 feet or more below the surface. Second, if your sewer line fails in winter, you are faced with the difficult task of breaking through frozen ground in freezing temperatures. This may be impossible to do by hand, and you may need to call in the earth-moving equipment.
One of the most critical steps in digging a trench for sewer line replacement is making sure you are digging in the right location. If you have no idea where your sewer line runs, a video camera inspection can tell you. The technician will run the camera from the sewer clean-out and down the sewer line from your house and can stop at various points. The camera has a radio transmitter that signals its location. At any point, the technician can sweep a locator over the ground so that it reads the signal. Be as accurate as possible because you will pay a high price in useless labor for hand-digging at the wrong spot. Work with the technician until you are certain that you have found the accurate location, then mark it with a line of marking paint.
Call your local utility locator number. This is usually a free service to residents, funded and managed by utility companies. Technicians will mark your yard for electrical, water, gas, and other vital services. Do not forget that there may be other homeowner-installed services such as sprinkler lines and landscape lights in your yard, outside of the purview of utility companies.
It is highly likely that your community requires a permit as a safety measure to ensure that the work is done correctly. An inspector will visit the work site to make sure the work has been done in compliance with the code. In the case of a sewer line replacement, this is normally done after the new sewer line has been installed, but before the trench has been filled.
Before you can get to the soil, you may have to remove obstructions such as concrete or brick sidewalks, driveways, or slabs. Use an 8-pound sledgehammer to break up concrete, starting on the edge. Electric or pneumatic jackhammers can also be leased at tool rental centers.
Dig straight down, minimizing side digging as much as possible for the time being. Later, you will need to extend the trench sideways so that you can stand in the pit and create access to the sewer line. But do as little side digging as possible until you have located the sewer line. As you dig, you may need to clip or saw off tree and bush roots as you encounter them. Some small roots can be severed with your shovel blade. Other roots should be clipped with pruning shears or a hand saw. Heavier roots will require a chainsaw.
The depth at which you reach the sewer line will vary. It can be as shallow as 18 to 30 inches or as much as 5 or 6 feet. In cold climates, the pipe will be buried deeper to prevent the pipe from freezing in the winter. But pipe depth is not always a matter of climate. Even in warm climates, the pipe can sometimes be buried very deep—it may depend on how deep your community's sewer mains are located.
If your sewer problem is an emergency, you may smell the pipe before you see it. You might find that the soil is saturated with waste-water. If the sewer line has been well laid, the appearance of a gravel layer will alert you that you are close to the sewer line. At this point, dig rather carefully as you excavate space around the sewer line, which may be a plastic pipe, clay pipe, or cast-iron pipe. You might even find several types of pipe spliced together. Clear away a good amount of soil all around the pipe to allow the contractor easy access to replace the pipe.
Until the sewer replacement is complete, it is very important to guard the trench against pets or children who might fall in. This can be done by covering the trench or by erecting sturdy temporary fencing to guard against intrusion.
After the sewer replacement is complete, make sure to have the work inspected before filling in the trench with soil. As you fill in the trench, periodically compress the soil to prevent later settling of the earth.
What Are the Steps to Repairing Vinyl Siding?
Is Removing Asbestos Shingles on Wood House Legal?
How Do You Prevent and Remove Ice Dams?


Alpers, W., Ivanov, A.Y. & Dagestad, K.-F. Encounter of Foehn Wind with an Atmospheric Eddy over the Black Sea as Observed by the Synthetic Aperture Radar Onboard Envisat. Monthly Weather Review 139, 3992 - 4000 (2011).


text:
Q: Solving Second Ordinary Differential equations How would we find the general formula for the following Differential equation using undetermined coefficients method or any other method you may know of.
$$y''-y=\cos x+\sin x $$
Thank you☺

A: $$y''-y=\cos x+\sin x $$
The polynomial characteristic is :
$$r^2-1=0 \implies r=\pm 1$$
The solution to the homogeneous euation is :
$$y_h=c_1e^x+c_2e^{-x}$$
For the particular solution, use method of undetermined coefficients and try:
$$y_p=A\cos x +B \sin x$$

A: Solution by Laplace Transform:
$$y'' - y = cosx +sinx$$
$$\mathcal{L}(y'') - \mathcal{L}(y) = \frac{s}{s^2 +1} + \frac{1}{s^2 +1}$$
$$\mathcal{L}(y)=\frac{A}{2}(\frac{1}{s+1} + \frac{1}{s-1}) + \frac{B}{2}(\frac{1}{s-1} - \frac{1}{s+1}) + \frac{1}{(s^2+1)(s-1)}$$
$$\mathcal{L}(y)=\frac{A}{2}(\frac{1}{s+1} + \frac{1}{s-1}) + \frac{B}{2}(\frac{1}{s-1} - \frac{1}{s+1}) -\frac{(s+1)}{2(s^2 +1)} + \frac{1}{2(s-1)}$$
$$y=k_1 e^x +k_2 e^{-x} -\frac{cosx}{2} - \frac{sinx}{2}$$

redpajama_set_name: RedPajamaStackExchange


MP Lobsang Choejor was born in 1982 at Rebkong, Amdo, Tibet. He did his schooling in Tibet.
In exile, he studied at Institute of Buddhist Dialectic, Dharamsala, English at the British Council based in Delhi and went to Taiwan to study Chinese language at Tamkang University.
He worked as a researcher at history section in CTA and was assistant Director at the Central office of Tibet People's Movement for MWA. He also had authored books on history, politics, culture and poetry.
He is a first timer in the 16th TPiE elected from Dhomey Cholkha. He is well informed about the things that are taking place around CTA and Tibetan community and is one of the most outspoken members in the parliament especially on LS.
Most of the Khampa factions both from Cholkha and Choelug seemed at dead end to harass and humiliate him with personal attack on his work before he became a Parliament member, yet he is unyielding.
He takes a clear stand on issues relating to Tibetan community and he was upfront on contempt on the State oracle Nechung and Gaden Phodrang by Tenpa Marphel in the house. He has the courage and the historical background to substantiate his stand also. He should fight to the end of the world.
If not granting his request for delay in going to Australia on family ground is true, how come Bawa Kalsang's request was granted before? Is there a different yardstick from person to person or is there the hand of LS in it as both of them seem playing hare and hound in the parliament?
His loyalty to his root is well displayed in his word and action.
His other name is Sharshok Khutta. He's a chinese spy who carries Chinese flag under MWP banner.
Of course, LS has the authority to stop lobsang from going to australia. If he doesn't have that power, then what is he? a puppet? a joker?
No LS has no authority to stop anyone. He can try but. He tried to have a say on US visa authorization to Tibetans in 2012 but the US embassy said sorry nobody can manipulate their visa decision power. So yes he is a puppet of HH and a joker to the people.


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Author: Bruce
By BruceIn Digital Marketing, Graphic Design, Logos & Branding, Search engine optimization, Social MediaPosted February 8, 2023
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We are proud to introduce the copper push pins with compression springs product series. They are for applications used in markets such as Industrial Computers, Embedded Boards, Medical Computing, Point of Sale, etc. For the heat sink's mechanical fastening, the copper pin is very stable compared to the plastic pin, and the better long-term reliability is determined in many industrial applications.
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Aerozoo Color Clay Soap have the property of clay and it is colorful. Children could knead the favorite shape to achieve more idea.
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This is a set of clay soap. It can also be interpreted as a soap with plasticity. I did not think there was anything special about this clay set until I heard the commentary on the product introduction. For the sake of packaging and alignment of exhibits, it tends to be thought that there is no difference from the clay used in drawing and manual arts classes. In order to encourage children to acquire the habit of washing hands well, the designers hoped to change hand washing into a more interesting thing. Once the clay soap is deformed and finished, it can be used for a long time once it is kept in a shade for a certain time. Children can create "one-of-a-kind" customized soap through their own labor according to their preference. I think that kids in kindergarten would love this fun and unique soap very much. I would like to propose that the characteristics and attributes of goods be emphasized more in packaging parts. Being interesting, unique and novel is the reason this product was awarded.


Many a Revelstoke resident works as a wildland firefighter. For those interested in the job – BC Wildfire Service is accepting applications for wildland firefighter positions for the 2019 fire season.
Online applications can be submitted on the BC Wildfire Service website until Jan. 15, 2019.
While Revelstoke is not specifically listed, the BC Wildfire Service is looking for new recruits to fill between 150 and 200 positions.
Firefighters generally work from May until the end of September, with the potential to work into the autumn.
"After new recruits successfully complete their first fire season, they are eligible for recall rights and will have a job with a four- to eight-month term. If work is available, this could lead to a year-round job with the BC Wildfire Service, allowing people to pursue firefighting as a career," says a press release from the BC Government.


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In 2017, the DFS group generated revenues under IFRS accounting standards in the amount of EUR 1,103.6 million. Despite an increase in flight movements of around 5.7 percent, revenues remained below the level of the previous year of EUR 1,169.7 million. This is due to lower revenues from charges resulting from a reduction of unit rates. DFS reduced its unit rate for en-route flights by 16 percent in 2017. For terminal charges at German airports, airlines have to pay 18 percent less since the beginning of 2017.
In total, the DFS group generated a net income of EUR 30.8 million in 2017. The company's commercial business contributed EUR 66 million to the group's overall performance.


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Special review: The Blinders - Live at the Ritz, Album launch party (Evening performance)
Official poster for the event
Saturday 2nd November at 33 Oldham Street saw the delayed launch party of The Blinders Live at the Ritz album - the original launch having to be postponed due to Thomas being ill.
This was an exclusive event and tickets were very limited so it was a real honour to see them in such a small intimate setting. I don't know how many were there but there couldn't have been many more than 50 there. It was also a great honour to meet the 4th member of The Blinders Katie Macbeth. She's such a darling and is as passionate about live music as I am. This was my 4th time seeing the lads this year but I've got a long way to go to catch up with Katie's circa 70 times!
The exhibition of prints, art and some of the props from the lads video's were on view prior to their live set and it was so great to see all of this in the flesh. I also met Matt on the stairs and after summoning up the courage to talk to him about some of my favourite sets of theirs he asked my name and said yes to a picture.
I took some photos of this but I wasn't there for that. I was there to enjoy a set from one of my favourite bands in an intimate setting.
The Blinders fans are a big family and you genuinely couldn't have shared this magical experience with such a bunch of nicer bunch of people.
The full set list was: L'Etat C'est Moi; Ballad of Winston Smith; Where No Man Comes; I Want Gold (new song); I Can't Breathe Blues; Orbit; Something Wicked (also new); Circle Song; Brave New World and an Encore which you're gonna have to watch the video of to see. I managed to record the whole gig and Q&A session also (see below).
In the interview our host for the evening Dave asked the lads what they thought about the pan-generational appeal of their music. Charlie answered and spoke of older fans relating to the political music of the 70's and 80's. He also said he didn't think that they appealed to many younger fans (Katie disagreed).
I remember when I first heard The Blinders.
I'd seen The Ninth Wave supporting When Young at The Deaf Institute on my birthday weekend February this year. Speaking to Hayden after the gig I said I'd loved their set and he mentioned they were supporting The Blinders at the O2 Ritz (the gig from the live album) and said I should go. He also gave me a flyer for the gig. I bought a ticket that night and thought I'd better listen to The Blinders.
So I went to Spotify and when I heard Brave New World. I was blown away. I was like what the fuck?! I was hearing something current with a heavy influence from the past. The more and more I listened to Columbia the more and more I've fallen in love with this band.
My favourite track from the lads isn't on the album... It's Murder at the Ballet from Hidden Horror Dance.
For me that track encapsulates all that is wonderful about this band. It's Anthemic! Killer riffs and Tom's rich gravely voice. Add to this Charlie's bombastic bass and Matt's dynamic pounding of the skins and you have the perfect cocktail.
I've been lucky enough to see the lads 4 times this year each a very different type of gig too. The first at The O2 Ritz, 2nd Edgeley Park (best band that day); 3rd YNOT Festival and last night at 33 Oldham Street. Each time has been absolutely bloody brilliant. Some bands you just love and the more that you see them, the more that you fall in love with their music.
I also bumped into the lovely Matt after the gig and he asked me if their set was ok?! I told him it was phenomenal and also that I tend to relate to the drummers in bands having played drums as a teenager. I also mentioned I would love to play again and he said he would give me lessons. I would definitely take him up on this! But he was such a kind and humble person. When I met Tom at YNOT as my mate Jack knows him from being a lad, Tom was exactly the same.
Towards the end of the Q&A Dave asked the band also if they were worried about living up Columbia?! If that was the bands peak? The lads replied (this is an abridged version - you can listen to the full answer in the video), that they hoped to make another album they would be proud of. That if they did and no one liked it, then they would still be proud of all they've achieved.
Their music, lyrics and performances are not 3 lads playing instruments. They're incredibly intelligent, kind, humble musicians who when they come together they create something that is so relatable to so many.
Based on tonight's performance and the new songs I heard? The lads are only going to go from strength to strength.
Thanks to Matt, Thomas and Charlie and everyone involved in putting on an evening of pure musical nirvana!
All pics are one's I took and the video's are too - the clip of Orbit is on the full version of the gig. This one is dedicated to you Katie. Up the F***ing Blinders!
Katie this one is dedicated to you - full gig with Q & A below
Special Review
Album launch
Single review - Saytr Play with VCR
Single review The Vitrines with Diamonds in Her Eyes
Gig review - Arlo Parks, Matilda Mann


text:
Борис Борисович Борденюк  (  — ) — військовослужбовець, старший солдат Збройних сил України, учасник російсько-української війни.

Життєпис
Мешкав разом із батьками в селі Благовіщенка Кам'янсько-Дніпровського району Запорізької області. Після 9-го класу вступив у Мелітопольський ліцей залізничного транспорту, закінчивши який здобув фах «оператор поста централізації, складач потягів, регулювальник швидкості руху вагонів». Також закінчив 6-місячні курси провідника. Працював в Енергодарі у залізничній галузі.

У 2016 році підписав контракт на службу у Збройних силах. Старший солдат, старший навідник 1-го мінометного відділення мінометної батареї 1-го батальйону 54-ї окремої механізованої бригади.

Загинув близько 6-ї години ранку 24 лютого 2019 року від осколкового поранення в голову під час мінометного обстрілу 82 мм мінами позицій ЗСУ під с. Катеринівка.

Похований 27 лютого у селі Благовіщенка. Залишились батьки та дві сестри. Раніше померли двоє братів Бориса, на війні мати втратила третього сина.

Нагороди та вшанування
 Указом Президента України № 149/2021 від 07 квітня 2021 року «за особисту мужність і самовіддані дії, виявлені у захисті державного суверенітету та територіальної цілісності України» — нагороджений орденом «За мужність» III ступеня (посмертно).

Примітки

Джерела
 Борденюк Борис Борисович  // Книга пам'яті загиблих
 Бойові втрати лютого 2019-го: імена і фото восьми загиблих  // Новинарня

redpajama_set_name: RedPajamaWikipedia


All first-time hunters, bowhunters or trappers planning to go afield this upcoming hunting and trapping season must first complete a hunter or trapper education course before they can get a license in New York.
Previously licensed hunters who plan to hunt outside of NY, should check with the state you plan to hunt in. All states accept NY's hunter education course but states differ on what is required as proof of completion. Many will request your original "Hunter Education Certificate" in order to purchase their license.
Check to see if you have your certificate, as issuing a replacement will be time consuming. All courses are free of charge, and as hunting seasons approach, courses fill quickly. To locate a nearby course and course list, visit DEC's website. SIGN UP EARLY!