text:
\section{Introduction}
Convergence time is an important performance specification for a controlled system from a practical point of view \cite{lopez2018fixed,dvir2018acceleration}. Indeed, the design of controllers which guarantee predefined-time stability, instead of asymptotic stability, is one of the desired objectives, which appear in many applications such as missile guidance \cite{songtime}, hybrid formation flying \cite{Zuo2018}, group consensus \cite{shang2017fixed}, online differentiators~\cite{Cruz-Zavala2011,Angulo2013,Basin2016}, state observers~\cite{lopez2018fixed}, etc. Furthermore, in the case of switching systems, it is frequently required that the observer (or controller) achieves the stability of the observation error (or tracking error) before the next switching \cite{defoort2011robust,Gomez2015}.

Lack of uniform boundedness of the settling-time function regardless of the initial conditions causes several restrictions to the practical application of finite time observer/controller \cite{Haimo1986,Bhat2000,Moulay2006}. These restrictions can be relaxed using the fixed-time stability concept. It is an extension of global finite-time stability, and guarantees the convergence (settling) time to be globally uniformly bounded, i.e., the bound does not depend on the initial state of the system \cite{Andrieu2008,Polyakov2012,songtime}. To this end, the class of systems
\begin{equation}
\label{Eq:HomFixedPoly}
\dot{x} = -(\alpha|x|^{p} + \beta|x|^{q})^k \hskip2pt{\rm sign}\hskip0pt{x}, \ \ x(0)=x_0,
\end{equation}
where $x$ is a scalar state variable and the real numbers $\alpha,\beta,p,q,k>0$ are system parameters which satisfy the constraints $kp<1$, and $kq>1$, was proposed in~\cite{Andrieu2008,Polyakov2012} and has been extensively used. Indeed, it represents a wide class of systems which present the fixed-time stability property through homogeneity and Lyapunov analysis frameworks. However, it is still difficult to derive a relatively simple relationship between the system parameters and the upper bound of the settling time \cite{Jimenez-Rodriguez2017,Jimenez-Rodriguez2018}. This yields some difficulties in the tuning of the system parameters to achieve a prescribed-time stabilization (see for instance \cite{Cruz-Zavala2011}).

The computation of the least upper bound of the settling time is usually not an easy task. Therefore, it is common to propose an upper bound of the settling time as an attempt to estimate the least upper bound. For instance, in~\cite{Polyakov2012}, it is shown that for system~\eqref{Eq:HomFixedPoly}, the settling-time function $T(x_0)$ is bounded as
\begin{equation}
\label{Eq:UpperEstimate}
T(x_0)\leq \frac{1}{\alpha^k(1-pk)}+\frac{1}{\beta^k(qk-1)}=T_{\text{max}},\ \ \ \forall x_0\in\mathbb{R}.
\end{equation}
However, this bound significantly overestimates the least upper one. This overestimation can lead to restrictions for the practical implementation of prescribed-time observer/controller. In this case, the gains will be over-tuned to achieve a prescribed-time stabilization. It may lead to poor performances in terms of control magnitude or robustness against measurement noise for instance.

Considering the extensive use of the class of systems represented by~\eqref{Eq:HomFixedPoly} and the overestimation exhibited by $T_{\text{max}}$ in~\eqref{Eq:UpperEstimate}, this paper addresses the computation of the least upper bound of the settling-time function for this system and the derivation of a new predefined-time convergent algorithm where the least upper bound of the settling time is set \textit{a priori} as a parameter of the system. Based on this result, new predefined-time controllers for first order and second order systems are introduced to enhance the settling time estimation. In contrast to the common approaches of homogeneity and Lyapunov analysis, the results in this paper are derived using the well-known geometric conditions proposed in~\cite{Haimo1986}.
The overall contribution of this paper is divided into the following four main results:
\begin{enumerate}
    \item The well-known class of fixed-time stable systems~\eqref{Eq:HomFixedPoly} is revisited. For this class of systems, the least upper bound of the settling-time function, \[\gamma=\frac{\Gamma \left(m_p\right) \Gamma \left(m_q\right)}{\alpha^{k}\Gamma (k) (q-p)}\left(\frac{\alpha}{\beta}\right)^{m_p},\] with $m_p=\frac{1-kp}{q-p}$ and $m_q=\frac{kq-1}{q-p}$, is found.

    \item The following new predefined-time convergent system, where the least upper bound of the settling time $T_c$ is set \textit{a priori} as a parameter of the system, is proposed:
    \begin{equation}
    \label{Eq:HomFixed}
    \dot{x} = -\frac{\gamma}{T_c}(\alpha|x|^{p} + \beta|x|^{q})^k \hskip2pt{\rm sign}\hskip0pt{x}, \ \ x(0)=x_0,
    \end{equation}
     where $x$ is a scalar state variable, real numbers $\alpha,\beta,p,q,k>0$ are system parameters which satisfy the constraints $kp<1$ and $kq>1$ and $T_c>0$. Notice that the only difference between \eqref{Eq:HomFixedPoly} and the modified system \eqref{Eq:HomFixed} is the constant gain $\gamma/T_c$. This slight change in the original system represents a considerable improvement in its properties, since the tunable parameter $T_c$ is directly the least upper bound of the convergence time.
     As a consequence of this desirable feature, we say that the origin of system~\eqref{Eq:HomFixed} is predefined-time stable with (strong) predefined-time $T_c$, a notion formally defined in Section \ref{Sec:Preliminaries}.
    \item The bound given in the formula~\eqref{Eq:UpperEstimate} for system~\eqref{Eq:HomFixed} is shown to be a conservative estimation of the settling time $T_f=\sup_{x_0\in\mathbb{R}}T(x_0)=T_c$. Moreover, letting $\alpha=\varrho$ and $\beta=\frac{1}{\varrho}$, it is shown that even if the least upper bound of the convergence time is $T_c$, the upper estimate~\eqref{Eq:UpperEstimate}, given in~\cite{Polyakov2012} goes to infinity as $\varrho\to+\infty$ and as $\varrho\to0$.
\item New predefined-time controllers for first order and second order scalar systems with matched bounded perturbations are introduced to enhance the settling time estimation.
\end{enumerate}

The rest of the manuscript is organized as follows. In Section~\ref{Sec:Preliminaries}, we introduce the preliminaries on finite-time, fixed-time and predefined-time stability. In Section~\ref{Sec:MainResult}, we present the main result on the least upper bound for the settling time and propose a new strongly predefined-time convergent algorithm where the least upper bound of the settling time is set \textit{a priori} as a parameter of the system. We also present the analysis of how conservative the bound provided in~\cite{Polyakov2012} may result and show some numerical results. In Section \ref{Sec:stabilization}, we apply the previous result to derive new predefined-time controllers for first and second order systems. Finally, in Section~\ref{Sec:Conclusion}, we present some concluding remarks.

\section{Preliminaries and Definitions}
\label{Sec:Preliminaries}
Consider the nonlinear system
\begin{equation}\label{Eq:NonSystemDyn}
    \dot{x}=f(x;\rho), \ \ x(0)=x_0,
\end{equation}
where $x\in\mathbb{R}^n$ is the system state, the vector $\rho\in\mathbb{R}^b$ stands for the system~\eqref{Eq:NonSystemDyn} parameters which are assumed to be constant, i.e., $\dot{\rho}=0$. The function $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ is assumed to be nonlinear and continuous, and the origin is assumed to be an equilibrium point of system~\eqref{Eq:NonSystemDyn}, so $f(0;\rho)=0$.

Let us first recall some useful definitions and lemma on finite-time, fixed-time and predefined-time stability.

\begin{definition}(Lyapunov stability~\cite[Definition~4.1]{Khalil2002})\label{def:lyapstab}
The origin of system \eqref{Eq:NonSystemDyn} is said to be \textit{Lyapunov stable} if for all $\epsilon>0$, there is $\delta:=\delta(\epsilon)>0$ such that for all $\norm{x_0}<\delta$, any solution $x(t,x_0)$ of~\eqref{Eq:NonSystemDyn} exists for all $t\geq0$, and $\norm{x(t,x_0)}<\epsilon$ for all $t\geq0$.
\end{definition}

\begin{definition}(Finite-time stability~\cite{Bhat2000})\label{def:finite} The origin of \eqref{Eq:NonSystemDyn} is said to be \textit{globally finite-time stable} if it is Lyapunov stable, and for any $x_0\in\mathbb{R}^n$, there exists $0\leq T < +\infty$ such that the solution $x(t,x_0)=0$ for all $t\geq T$. The function $T(x_0)=\inf\{T:x(t,x_0)=0, \forall t\geq T\}$ is called the \textit{settling-time function}.
\end{definition}

\begin{lemma}(Finite-time stability characterization for scalar systems~\cite[Fact~1]{Haimo1986}) \label{lem:haimo} Let $n=1$ in system~\eqref{Eq:NonSystemDyn} (scalar system). The origin is globally finite-time stable if and only if for all $x\in\mathbb{R}\setminus\{0\}$
\begin{itemize}
    \item[\textit{(i)}] $xf(x;\rho)<0$, and
    \item[\textit{(ii)}] $\int_{x}^{0}\frac{\text{d}z}{f(z;\rho)}<+\infty.$
\end{itemize}
\end{lemma}

\begin{remark} A proof of Lemma~\ref{lem:haimo} shall not be given here, but can be found in \cite[Lemma~3.1]{Moulay2008}. Nevertheless, intuitively, condition \textit{(i)} implies Lyapunov stability. Moreover, under the conditions of Lemma~\ref{lem:haimo}, note that the settling time function is $T(x_0)=\int_{0}^{T(x_0)}\text{d}t$. Since first-order systems do not oscillate, the solution $x(\cdot,x_0):[0,T(x_0))\to[x_0,0)$ of system~\eqref{Eq:NonSystemDyn} as a function of $t$ defines a bijection. Using it as a variable change, the above integral equals (note that $\frac{1}{f(x;\rho)}$ is defined for all $x\in\mathbb{R}\setminus\{0\}$ from condition \textit{(i)}
\begin{equation}\label{eq:settling_scalar}
    T(x_0)=\int_{0}^{T(x_0)}\text{d}t=\int_{x_0}^{0}\frac{\text{d}x}{f(x;\rho)}.
\end{equation}
Thus, condition \textit{(ii)} of Lemma~\ref{lem:haimo} refers to the settling-time function being finite.
\end{remark}

\begin{definition}(Fixed-time stability~\cite{Polyakov2012}) \label{def:fixed} The origin is said to be a \textit{fixed-time stable equilibrium} of~\eqref{Eq:NonSystemDyn} if it is globally finite-time-stable and the settling time function $T(x_0)$ is bounded on $\mathbb{R}^n$, i.e. $\exists T_{\text{max}}>0:\forall x_0\in\mathbb{R}^n:T(x_0)\leq T_{\text{max}}$.
\end{definition}

\begin{remark} Let the origin $x=0$ of  system~\eqref{Eq:NonSystemDyn} be fixed-time stable. Notice that there are multiple upper bounds of the settling-time function $T_\text{max}$; for instance, if $T(x_0)\leq T_\text{max}$, also  $T(x_0)\leq \lambda T_\text{max}$ with $\lambda\geq 1$. However, from this boundedness condition, the least upper bound of the settling-time function $\sup_{x_0 \in \mathbb{R}^n} T(x_0)$ exists.
\end{remark}

\begin{remark}
\label{Rem:Predef}
It has been shown that fixed-time stability is guaranteed if the vector field of the system is homogeneous in the bi-limit, a concept defined in~\cite{Andrieu2008,Polyakov2016}. However, in these cases, the upper bound for the settling time is usually not obtained. To differentiate this case to one where a settling time bound $T_c$ is set in advance as a function of system parameters $\rho$, i.e. $T_c=T_c(\rho)$, we introduce the concept of predefined-time stability. A strong notion of this class of stability is given when $\sup_{x_0 \in \mathbb{R}^n} T(x_0)=T_c$, i.e., $T_c$ is the least upper bound for the settling time.
\end{remark}

\begin{definition}(Predefined-time stability~\cite{Sanchez-Torres2018})\label{def:predefined} For the parameter vector $\rho$ of system~\eqref{Eq:NonSystemDyn} and an arbitrarily selected constant $T_c:=T_c(\rho)>0$, the origin of~\eqref{Eq:NonSystemDyn} is said to be \textit{predefined-time stable} if it is fixed-time stable and the settling-time function $T:\mathbb{R}^n\rightarrow\mathbb{R}$ is such that \[T(x_0)\leq T_c, \quad \forall x_0\in\mathbb{R}^n.\] If this is the case, $T_c$ is called a \textit{predefined time}. Moreover, if the settling-time function is such that $\sup_{x_0 \in \mathbb{R}^n} T(x_0)=T_c$, then $T_c$ is called the \textit{strong predefined time}.
\end{definition}

\begin{remark} The stability property, of any kind, refers to equilibrium points of a system. However, since this study only focuses on the global stability of the origin of the system under consideration, it may be referred hereafter, without ambiguity, to the stability of the system in the respective sense (asymptotic, fixed-time or predefined-time).
\end{remark}

\section{On the least upper bound for the settling time of a class of fixed-time stable systems}
\label{Sec:MainResult}

\subsection{Least upper bound of the settling-time function of system~\eqref{Eq:HomFixedPoly}}
Let us first revisit the well-known class of fixed-time stable systems. In the following theorem, based on an appropriate use of the Gamma function, the least upper bound of the settling-time function of  system~\eqref{Eq:HomFixedPoly} is provided.

\begin{theorem}
\label{th:tf_poly}
Let
\begin{equation}
\label{Eq:MinUpperEstimate}
\gamma=\frac{\Gamma \left(m_p\right) \Gamma \left(m_q\right)}{\alpha^{k}\Gamma (k) (q-p)}\left(\frac{\alpha}{\beta}\right)^{m_p},
\end{equation}
where $\Gamma(\cdot)$ is the Gamma function defined as $\Gamma(z)=\int_0^{+\infty} e^{-t}t^{z-1}dt$~\cite[Chapter~1]{Bateman1953}, and $m_p=\frac{1-k p}{q-p}$ and $m_q=\frac{k q-1}{q-p}$ are positive parameters. The origin $x=0$ of system~\eqref{Eq:HomFixedPoly} is fixed-time stable and the settling time function satisfies $\sup_{x_0 \in \mathbb{R}^n} T(x_0)=\gamma$.
\end{theorem}
\begin{proof} Note that for system~\eqref{Eq:HomFixedPoly}, the field is $f(x;\rho)=-\left(\alpha|x|^{p} + \beta|x|^{q}\right)^k \hskip2pt{\rm sign}\hskip0pt{x}$, where the parameter vector is $\rho=\left[\alpha \, \beta \, p \, q \, k\right]^T\in\mathbb{R}^5$. Furthermore, the product $xf(x;\rho)=- \left(\alpha|x|^{p} + \beta|x|^{q}\right)^k \abs{x}<0$ for all $x\in\mathbb{R}\setminus\{0\}$. Thus, $V(x)=\frac{1}{2}x^2$ is a radially unbounded Lyapunov function for system~\eqref{Eq:HomFixedPoly}, so its origin $x=0$ is Lyapunov stable~\cite[Theorem~4.2]{Khalil2002}.

Now, let $x_0\in\mathbb{R}\setminus\{0\}$ (if $x_0=0$, then $x(t,0)=0$ is the unique solution of~\eqref{Eq:HomFixedPoly} and $T(0)=0$). From~\eqref{eq:settling_scalar}, the settling time function is
\begin{align*}
T(x_0)  &=\int_{x_0}^{0}\frac{\text{d}x}{f(x;\rho)}\\
        &=\int_{0}^{x_0}\frac{\hskip2pt{\rm sign}\hskip0pt{x}\text{d}x}{(\alpha\abs{x}^{p} + \beta\abs{x}^{q})^k}, \qquad z=\abs{x}\\
        &=\int_{0}^{\abs{x_0}}\frac{\text{d}z}{(\alpha z^{p} + \beta z^{q})^k}.
\end{align*}
Since the integrand $\frac{1}{(\alpha z^{p} + \beta z^{q})^k}$ is positive for $z\in\left(0,\abs{x_0}\right)$, the settling time function is increasing with respect to $\abs{x_0}$. Hence, the least upper bound of $T(x_0)$ (in the extended real numbers set, since we do not know yet if it is finite or not) is obtained using Proposition~\ref{prop:integral}, in the Appendix, as \[\sup_{x_0\in\mathbb{R}}T(x_0) = \lim_{|x_0|\to+\infty} T(x_0) = \int_0^{+\infty} \frac{\text{d}z}{(\alpha z^p + \beta z^q)^k}=\gamma,\]
with $\gamma<+\infty$ as in~\eqref{Eq:MinUpperEstimate}. Using Lemma~\ref{lem:haimo} and by the definitions of finite-time and fixed-time stability, the origin $x=0$ of system~\eqref{Eq:HomFixedPoly} is fixed-time stable and $T_f=\gamma$, which completes the proof.
\end{proof}

\subsection{A class of predefined-time stable systems}
Now, the result presented in Theorem~\ref{th:tf_poly} is used to derive a Lyapunov-like condition for characterizing predefined-time stability of a system.

\begin{theorem}(A Lyapunov characterization for predefined-time stable systems)\label{thm:weak_pt}
If there exists a continuous positive definite radially unbounded function $V:\mathbb{R}^n\to\mathbb{R}$ such that any solution $x(t,x_0)$ of~\eqref{Eq:NonSystemDyn} satisfies
\begin{equation}\label{eq:dV_weak}
\dot{V}(x)\leq-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k, \qquad \forall x\in\mathbb{R}^n\setminus\{0\}.
\end{equation}
where $\alpha,\beta,p,q,k>0$, $kp<1$, $kq>1$ and $\gamma$ is given in~\eqref{Eq:MinUpperEstimate}.

Then, the origin of \eqref{Eq:NonSystemDyn} is predefined-time stable and $T_c$ is a predefined time. If in addition, the equality holds in~\eqref{eq:dV_weak}, then $T_c$ is the strong predefined-time.
\end{theorem}
\begin{proof} The negative definiteness of the time derivative of the function $V$ implies Lyapunov stability of the origin of~\eqref{Eq:NonSystemDyn}. Now, suppose that there exists a function $w(t)\geq 0$ that satisfies
\begin{equation*}
    \dot{w} = -(\hat{\alpha} w^p + \hat{\beta} w^q)^k,
\end{equation*}
where $\hat{\alpha} = \alpha\left(\frac{\gamma(\rho)}{T_c}\right)^\frac{1}{k}$ and $\hat{\beta} = \beta\left(\frac{\gamma(\rho)}{T_c}\right)^\frac{1}{k}$, and $V(x_0)\leq w(0)$. Hence, by Theorem \ref{th:tf_poly}, $w(t)$ will converge to the origin in a strong predefined time
\begin{equation*}
\frac{\Gamma \left(m_p\right) \Gamma \left(m_q\right)}{\hat{\alpha}^{k}\Gamma (k) (q-p)}\left(\frac{\hat{\alpha}}{\hat{\beta}}\right)^{m_p}=\frac{T_c}{\gamma(\rho)}\frac{  \Gamma \left(m_p\right) \Gamma \left(m_q\right)}{\alpha^{k}\Gamma (k) (q-p)}\left(\frac{\alpha}{\beta}\right)^{m_p} = T_c
\end{equation*}
which is directly a tunable parameter of the system. Furthermore, by the comparison lemma~\cite[Lemma~3.4]{Khalil2002}, it follows that $V(x(t))\leq w(t)$, with equality only if \eqref{eq:dV_weak} is an equality. Consequently, the origin of system~\eqref{Eq:NonSystemDyn} is predefined-time stable with predefined time $T_c$. Moreover, if \eqref{eq:dV_weak} is an equality, then $\sup_{x_0\in\mathbb{R}^n}T(x_0)=T_c$, i.e., $T_c$ is the strong predefined time.
\end{proof}

\begin{example} Consider system~\eqref{Eq:HomFixed} and the continuous positive definite radially unbounded Lyapunov candidate function $V(x)=\abs{x}$ for this system. The derivative of $V(x)$ along the trajectories of system~\eqref{Eq:HomFixed} is \[\dot{V}(x)=-\hskip2pt{\rm sign}\hskip0pt{x}\frac{\gamma}{T_c}\left(\alpha\abs{x}^p+\beta\abs{x}^q\right)^k\hskip2pt{\rm sign}\hskip0pt{x}=-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k.\] Hence, by Theorem~\ref{thm:weak_pt}, the origin of system~\eqref{Eq:HomFixed} is predefined-time stable with strong predefined time $T_c$.

To illustrate the above, some numerical simulations of system~\eqref{Eq:HomFixed} are conducted, setting the parameters to $\alpha=4$, $\beta=\frac{1}{4}$, $T_c=1$, $p=0.5$, $q=3$, $k=1.5$. The simulations are conducted for several initial conditions $x_0$, as presented in Figure~\ref{Fig:Simu}. It can be seen that $\sup_{x_0\in\mathbb{R}}T(x_0)=T_c=1$ as stated above.
\begin{figure}
    \centering
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    \caption{Trajectories of~\eqref{Eq:HomFixed} for different initial conditions with $\alpha=4$, $\beta=\frac{1}{4}$, $T_c=1$, $p=0.5$, $q=3$, $k=1.5$. The upper estimate in~\eqref{Eq:UpperEstimateRho} is $T_{\text{max}}(4)=4.4331s$. The least upper estimate is $T_c=1s$.}
    \label{Fig:Simu}
\end{figure}

\end{example}

\begin{remark} In~\cite{Parsegov2012}, the least upper estimation of the settling time of~\eqref{Eq:HomFixedPoly} was addressed for the case where $k=1$, $p=1-s$, $q=1+s$, with $0<s<1$, where $\gamma(\rho)$ reduces to $\gamma(\rho)=\frac{\Gamma(\frac{1}{2})^2}{2s\sqrt{\alpha\beta}}=\frac{\pi}{2s\sqrt{\alpha\beta}}$. In~\cite{Jimenez-Rodriguez2018a}, it was shown that, in the case $\alpha=\beta=\frac{\pi}{2sT_c}$, the least upper bound of the settling time is $T_c$. Thus, Theorem~\ref{thm:weak_pt} is a generalization of the results presented in~\cite{Parsegov2012} and~\cite{Jimenez-Rodriguez2018a}. Since only for the case where $k=1$, $p=1-s$ and $q=1+s$, with $0<s<1$, a non-conservative upper bound estimate is provided, many applications, for instance fixed-time consensus protocols~\cite{Zuo2014,Defoort2015,Ning2017b}, have been restricted to this case.
\end{remark}

Given the relevance of fixed-time stability argumented in the introduction, inequality \eqref{eq:dV_weak} is a result of paramount importance, since as we show in the following, the upper estimate~\eqref{Eq:UpperEstimate} is often too conservative. Thus, applications based on the upper estimate of the settling time~\eqref{Eq:UpperEstimate} presented in~\cite[Lemma~1]{Polyakov2012} are often over-engineered.

\subsection{Settling time bound analysis and comparison}
\label{Sec:TmaxAnalysis}
Consider the predefined-time stable system~\eqref{Eq:HomFixed}, with strong predefined-time $T_c$. From~\eqref{Eq:UpperEstimate}, calculated in~\cite{Polyakov2012}, an upper bound for the settling time $T(x_0)$ is
\begin{equation}
\label{Eq:UpperEstimateModified}
T(x_0)\leq \frac{T_c}{\gamma(\rho)}\left(\frac{1}{\alpha^k(1-pk)}+\frac{1}{\beta^k(qk-1)}\right),\ \ \ \forall x_0\in\mathbb{R}.
\end{equation}

Let $\varrho>0$, $\alpha=\varrho$ and $\beta=\frac{1}{\varrho}$. Assuming that $p$, $q$ and $k$ remain constant, and noticing that $\gamma$ is a function of $\varrho$, it can be seen that varying $\varrho$ the least upper bound of the settling time remains constant and equal to $T_c$. However, the bound~\eqref{Eq:UpperEstimateModified} becomes
\begin{equation}
\label{Eq:UpperEstimateRho}
T(x_0)\leq T_{\text{max}}(\varrho):=\frac{T_c}{K} \left(\frac{1}{\varrho^{2m_p}(1-pk)}+\frac{\varrho^{2(k-2m_p)}}{(qk-1)}\right),
\end{equation}
where $K=\frac{  \Gamma \left(m_p\right) \Gamma \left(m_q\right) }{\Gamma (k) (q-p)}$.
It is easy to see that
$$
\lim_{\varrho\to 0}T_{\text{max}}(\varrho)=\lim_{\varrho\to\infty}T_{\text{max}}(\varrho)=+\infty,
$$
i.e., $T_{\text{max}}(\varrho)$ in~\eqref{Eq:UpperEstimateRho} has no upper bound as $\varrho$ increases or is close to zero.

Moreover, the best upper estimate of the bound~\eqref{Eq:UpperEstimateRho} is achieved at $\arg\min_{\varrho>0}T_{\text{max}}(\varrho)=1$, and its value is
\begin{equation*}
\label{Eq:UpperEstimateRhoMin}
\min_{\varrho>0}T_{\text{max}}(\varrho)=T_{\text{max}}(1)=\frac{T_c}{K} \left(\frac{1}{(1-pk)}+\frac{1}{(qk-1)}\right)>T_c.
\end{equation*}

An illustration of this argument, showing $T_{\text{\text{max}}}(\varrho)$ as a function of $\varrho$, with $T_c=1s$, is presented in Figure~\ref{Fig:Convergence}. Although, by Theorem~\ref{thm:weak_pt} the least upper bound of the settling time is $T_c=1s$, it can be seen that in the best case the bound~\eqref{Eq:UpperEstimateModified} provides an overestimation of $\varepsilon T_c$ with $\varepsilon=\left.\frac{1}{K} \left(\frac{1}{(1-pk)}+\frac{1}{(qk-1)}\right)\right|_{p=0.5,q=3,k=1.5}=1.1249$. Moreover, in Figure~\ref{Fig:Simu}, it can be seen that although the least upper bound of the settling-time function is $T_c=1s$, the upper bound estimation provided by~\eqref{Eq:UpperEstimateRho} is $\left.T_{\text{max}}(\varrho)\right|_{\varrho=4}=4.4331s$.
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    \put(0.02996607,0.38){\color[rgb]{0.14901961,0.14901961,0.14901961}\rotatebox{90}{\makebox(0,0)[lt]{\lineheight{1.25}\smash{\begin{tabular}[t]{l}$T_{\text{max}}(\varrho)$\end{tabular}}}}}%
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    \caption{Function $T_{\text{max}}(\varrho)$ for $p=0.5$, $q=3$, $k=1.5$. The least upper bound of the settling time is $T_c=1s$. The smallest value of $T_{\text{max}}(\varrho)$ is $\min_{\varrho>0}T_{\text{max}}(\varrho)=T_{\text{max}}(1)=1.1249s$.}
    \label{Fig:Convergence}
\end{figure}

\section{Application to robust predefined-time stabilization for first and second order systems}\label{Sec:stabilization}

\subsection{First-order predefined-time controllers}
Consider the following perturbed first-order system
\begin{equation}\label{eq:fosys}
\dot{x}(t)=u(t)+\Delta(t),
\end{equation}
where $x\in\mathbb{R}$ is the state variable, $u\in\mathbb{R}$ is the control input and $\Delta(t)\in\mathbb{R}$ is an unknown but bounded perturbation term of the form $\abs{\Delta(t)}\leq\delta$, with $0\leq\delta<\infty$ a known constant.

The objective is to to design the control input $u$ such that the origin $x=0$ of system~\eqref{eq:fosys} is predefined-time stable, in spite of the unknown perturbation term $\bm{\Delta}(t)$.

\begin{theorem} Let $\alpha,\beta,p,q,k>0$, $kp<1$, $kq>1$, $T_c>0$, $\zeta\geq\delta$ and $\gamma$ be as in~\eqref{Eq:MinUpperEstimate}. If the control input $u$ is selected as
\begin{equation}\label{eq:ufo}
u=-\left[\frac{\gamma}{T_c}\left(\alpha\abs{x}^p+\beta\abs{x}^q\right)^k+\zeta\right]\hskip2pt{\rm sign}\hskip0pt{x}
\end{equation}
then, the origin $x=0$ of system~\eqref{eq:fosys} is predefined-time stable with predefined time $T_c$.
\end{theorem}
\begin{proof} Consider the continuous radially unbounded Lyapunov function candidate $V(x)=\abs{x}$. Its derivative along the trajectories of the closed system~\eqref{eq:fosys}-\eqref{eq:ufo} yields
\begin{align*}
\dot{V}(x)&=-\hskip2pt{\rm sign}\hskip0pt{x}\left[\frac{\gamma}{T_c}\left(\alpha\abs{x}^p+\beta\abs{x}^q\right)^k\hskip2pt{\rm sign}\hskip0pt{x}+\zeta\hskip2pt{\rm sign}\hskip0pt{x}-\Delta\right]\\&=-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k-\zeta+\Delta\hskip2pt{\rm sign}\hskip0pt{x}\\
&\leq-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k-\zeta+\abs{\Delta\hskip2pt{\rm sign}\hskip0pt{x}}\\
&\leq-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k-(\zeta-\delta)\\
&\leq-\frac{\gamma}{T_c}\left(\alpha V(x)^p+\beta V(x)^q\right)^k.
\end{align*}
Hence, using Theorem~\ref{thm:weak_pt}, the origin $x=0$ of system~\eqref{eq:fosys} is predefined-time stable with predefined time $T_c$.
\end{proof}
\begin{example}
An example of this approach is shown in Figure \ref{Fig:ScalarControl}, where the control input $u$ given in~\eqref{eq:ufo}, with $\zeta=1$, $p=0.5$, $q=3$, $k=1.5$ and $\alpha=1/\beta=\varrho=4$, is applied to the perturbed system~\eqref{eq:fosys} with disturbance $\Delta(t) = \sin(2\pi t /5)$. Note that although the upper bound of the settling-time function is $T_c=1s$ using the proposed scheme, the upper bound estimation provided by~\cite{Polyakov2012} is $T_{\text{max}}(\varrho)|_{\varrho=4} = 4.4331s$.

\begin{figure}
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\caption{State trajectory of the closed-loop system~\eqref{eq:fosys}-\eqref{eq:ufo} with $T_c=1s$ and the considered perturbation.}
\label{Fig:ScalarControl}
\end{figure}

\end{example}

\subsection{Second-order predefined-time controllers}
Consider the following perturbed second-order system
\begin{align}\label{eq:sosys}
\begin{split}
\dot{x}_1&=x_2\\
\dot{x}_2&=u+\Delta(t),
\end{split}
\end{align}
where $x_1,x_2\in\mathbb{R}$ are the state variables, $u\in\mathbb{R}$ is the control input and $\Delta(t)\in\mathbb{R}$ is an unknown but bounded perturbation term of the form $\abs{\Delta(t)}\leq\delta$, with $0\leq\delta<\infty$ a known constant.

The objective is to design the control input $u$ such that the origin $(x_1,x_2)=(0,0)$ of system~\eqref{eq:sosys} is predefined-time stable, in spite of the unknown perturbation term $\bm{\Delta}(t)$.

The following definition will be useful for stating Theorem~\ref{thm:socont}.

\begin{definition}\label{def:oddpower} For any real number $r$, the function $x\to \barpow{x}^r$ is defined as $\barpow{x}=\abs{x}^r \hskip2pt{\rm sign}\hskip0pt{x}$ for any $x\in \mathbb{R}$ if $r>0$, and for any $x\in \mathbb{R}\setminus\{0\}$ if $r\leq0$.
\end{definition}

\begin{theorem}\label{thm:socont} Let $\alpha_1,\alpha_2,\beta_1,\beta_2,p,q,k>0$, $kp<1$, $kq>1$, $T_{c_1},T_{c_2}>0$, $\zeta\geq\delta$, and \[\gamma_1=\frac{\Gamma \left(\frac{1}{4}\right)^2 }{2\alpha_1^{1/2}\Gamma\left(\frac{1}{2}\right)}\left(\frac{\alpha_1}{\beta_1}\right)^{1/4},\text{ and } \gamma_2=\frac{\Gamma \left(m_{p}\right) \Gamma \left(m_{q}\right)}{\alpha_2^{k}\Gamma (k) (q-p)}\left(\frac{\alpha_2}{\beta_2}\right)^{m_{p}},\] with $m_{p}=\frac{1-kp}{q-p}$ and $m_{q}=\frac{kq-1}{q-p}$. If the control input is selected as
\begin{equation}\label{eq:uso}
u=-\left[\frac{\gamma_2}{T_{c_2}}\left(\alpha_2\abs{\sigma}^{p}+\beta_2\abs{\sigma}^{q}\right)^{k}+\frac{\gamma_1^2}{2T_{c_1}^2}\left(\alpha_1+3\beta_1x_1^2\right)+\zeta\right]\hskip2pt{\rm sign}\hskip0pt{\sigma},
\end{equation}
where the sliding variable $\sigma$ is defined as
\begin{equation}\label{eq:sigmaso}
\sigma=x_2+\barpow{\barpow{x_2}^2+\frac{\gamma_1^2}{T_{c_1}^2}\left(\alpha_1\barpow{x_1}^1+\beta_1\barpow{x_1}^3\right)}^{1/2},
\end{equation}
then the origin $(x_1,x_2)=(0,0)$ of system~\eqref{eq:sosys} is predefined-time stable with predefined time $T_c=T_{c_1}+T_{c_2}$.
\end{theorem}
\begin{proof} The time-derivative of the sliding variable $\sigma$ in~\eqref{eq:sigmaso} is
\begin{align*}
    \begin{split}
        \dot{\sigma}&=u+\Delta+\frac{\abs{x_2}(u+\Delta)+\frac{\gamma_1^2}{2T_{c_1}^2}\left(\alpha_1+3\beta_1x_1^2\right)x_2}{\abs{\barpow{x_2}^2+\frac{\gamma_1^2}{T_{c_1}^2}\left(\alpha_1\barpow{x_1}^1+\beta_1\barpow{x_1}^3\right)}^{1/2}}\\
        &=-\frac{\gamma_2}{T_{c_2}}\left(\alpha_2\abs{\sigma}^{p}+\beta_2\abs{\sigma}^{q}\right)^{k}\hskip2pt{\rm sign}\hskip0pt{\sigma}-\zeta\hskip2pt{\rm sign}\hskip0pt{\sigma}+\Delta-\frac{\gamma_1^2}{2T_{c_1}^2}\left(\alpha_1+3\beta_1x_1^2\right)\hskip2pt{\rm sign}\hskip0pt{\sigma}\\
        &-\frac{\frac{\gamma_2}{T_{c_2}}\left(\alpha_2\abs{\sigma}^{p}+\beta_2\abs{\sigma}^{q}\right)^{k}\hskip2pt{\rm sign}\hskip0pt{\sigma}+\zeta\hskip2pt{\rm sign}\hskip0pt{\sigma}-\Delta}{\abs{\barpow{x_2}^2+\frac{\gamma_1^2}{T_{c_1}^2}\left(\alpha_1\barpow{x_1}^1+\beta_1\barpow{x_1}^3\right)}^{1/2}}-\frac{\frac{\gamma_1^2}{2T_{c_1}^2}\left(\alpha_1+3\beta_1x_1^2\right)}{\abs{\barpow{x_2}^2+\frac{\gamma_1^2}{T_{c_1}^2}\left(\alpha_1\barpow{x_1}^1+\beta_1\barpow{x_1}^3\right)}^{1/2}}\left(\abs{x_2}\hskip2pt{\rm sign}\hskip0pt{\sigma}-x_2\right).
    \end{split}
\end{align*}

Now, considering $V_2(\sigma)=\abs{\sigma}$ as a continuous radially unbounded positive definite Lyapunov candidate function it can be easily checked using the above that \[\dot{V}(\sigma)\leq-\frac{\gamma_2}{T_{c_2}}\left(\alpha_2 V_2(\sigma)^{p}+\beta_2 V_2(\sigma)^{q}\right)^{k},\]
and using Theorem~\ref{thm:weak_pt}, the origin $\sigma=0$ of the sliding variable dynamics is predefined-time stable with predefined time $T_{c_2}$.

Once the system trajectories are constrained to the manifold $\sigma=0$, i.e. for $t\geq T_{c_2}$, the solutions of system~\eqref{eq:sosys} satisfy the following reduced-order dynamics (see~\eqref{eq:sigmaso}): \[\dot{x}_1=x_2=-\frac{\gamma_1}{T_{c_1}}\left(\alpha_1\abs{x_1}+\beta_1\abs{x_1}^3\right)^{1/2}\hskip2pt{\rm sign}\hskip0pt{x_1}.\]
Thus, considering $V_1(x_1)=\abs{x_1}$ as a continuous radially unbounded positive definite Lyapunov candidate function and using Theorem~\ref{thm:weak_pt}, it is concluded that the origin $x_1=0$ of the reduced order system is predefined-time stable with predefined time $T_{c_1}$.
Moreover, from~\eqref{eq:sigmaso}, if $\sigma=0$ and $x_1=0$, then $x_2=0$. Hence, it is concluded that the origin $(x_1,x_2)=(0,0)$ of the closed-loop system~\eqref{eq:sosys}-\eqref{eq:uso} is predefined-time stable with predefined time $T_{c_1}+T_{c_2}$.
\end{proof}

\begin{example}
An example of this approach is shown in Figure~\ref{Fig:SecondOrderControl}, where the control input $u$ given in~\eqref{eq:uso}, with $\zeta=1$, $p=0.5$, $q=3$, $k=1.5$ and $\alpha_1=\alpha_2=1/\beta_1=1/\beta_2=4$ is applied to the perturbed system~\eqref{eq:sosys} with disturbance $\Delta(t) = \sin(2\pi t /5)$. It follows from Theorem~\ref{thm:socont} that with $T_{c_1} = T_{c_2} = 0.5s$, the origin $(x_1,x_2)=(0,0)$ of system~\eqref{eq:sosys} is predefined-time stable with predefined time $T_c=1s$. Note that although the upper bound of the settling-time function is $T_c=1s$ using the proposed scheme, the upper bound estimation provided by~\cite{Polyakov2012} is  $T_{\text{max}}=\frac{T_{c_2}}{\gamma_2}\left(\frac{1}{\alpha_{2}^k(1-pk)} + \frac{1}{\beta_{2}^k(qk-1)}\right) + \frac{2T_{c_1}}{\gamma_1}\left(\frac{1}{\sqrt{\alpha_{1}}} + \frac{1}{\sqrt{\beta_{1}}}\right)  =  5.1073s$.
\begin{figure}
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\caption{State trajectory of the closed-loop system~\eqref{eq:sosys}-\eqref{eq:uso} with $T_c=1s$ and the considered perturbation.}\label{Fig:SecondOrderControl}
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\end{example}

\section{Conclusion}
\label{Sec:Conclusion}
In this paper, we studied the convergence time of a class of fixed-time stable systems with the aim to provide a new non-conservative upper bound for its settling time. We showed that the well-known upper bound condition for the settling time of this class of systems is often too conservative. To illustrate our claim, we showed how by changing one parameter the upper estimate of the settling time tends to infinity even though the actual settling time is always bounded by a constant $T_c$. To address this problem, we proposed a modification to the classical fixed-time algorithm to transform it into a strongly predefined-time (in which the upper bound for the settling time is set in advance as a parameter of the system and is the lowest upper estimate of the settling time) with strong predefined-time $T_c$. With this result, the Lyapunov inequality, which is a sufficient condition for fixed-time stability, was modified in a way that it becomes predefined-time parametrized by $T_c$. When such inequality becomes equality, $T_c$ becomes the lowest upper estimate of the settling time of the system. At last, predefined-time controllers for first order and second order systems were introduced. Some simulation results have shown the performance of the proposed scheme in terms of settling time estimation compared to existing methods. This is an important contribution toward online differentiators, observers and controllers satisfying prescribed-time objectives.

redpajama_set_name: RedPajamaArXiv


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Astana is considered the world's second coldest capital city after Ulan Bator.
Kazakhstan is a secular state where progressive views predominate. Almost any type of clothes is acceptable in public places. It is next to impossible to see a person dressed in traditional costume in the capital or any other large city.
Choosing your clothes you should only be guided by the time of the year. In summer you should wear light clothes from natural textiles, while in winter you will need warm clothes and footwear, in which you will feel comfortable at minus 20°C. In spring and autumn the weather is quite unpredictable.
Visiting a religious place you should be dressed in clothes covering your shoulders and legs.
As well as in other nomadic peoples, the traditional Kazakh cuisine included cooking methods aimed to preserve food for quite a time.
Meat is the key element in the country's kitchen and is used in a wide range of meals and to make various types of sausage, the most popular of which is kazy. Kazy is made of meat from horse's ribs and fat. The piquant taste of this sausage is filled with the warmth of a bright sun and the bitter fragrance of steppe herbs, which makes you feel an ancient nomadic warrior.
Sausage shuzhuk is similar to kazy but has a somewhat more delicate taste. Unlike kazy, shuzhuk is made from any part of horse's body (not only from that near the ribs) with the addition of fat. The ingredients are stuffed into the small intestine.
Sheep's breast on a spit also ranks as one of the most popular Kazakh meals. In this dish the skin is not removed but only singed.
Boiled meat is also a common meal for the Kazakhs. They believe that only boiling allows meat to retain all its taste, aroma and good properties.
Those who have tasted true Kazakh beshbarmak can hardly find a place anywhere in the world where it is so delicious.
Foods from fermented milk also play an important part in Kazakhstan's cuisine. Sour milk tends to retain its properties and does not rot during long journeys across the steppe, supporting and invigorating one's organism and quenching one's thirst. Ayran with its pleasant sour and salty taste, shalap - less savoury but more nourishing through the addition of bread, or koumiss (also spelled kumiss, kumis) making one's body lighter - all these are essential elements on the Kazakh dastarkhan (table).
Ayran is a common drink in some other countries, too. Apart from beverages, sour milk is used to make qurut (also spelled kurut, kurt, qurt, gurt, qort). This food consists of small dried balls or rolls with a particular taste and is perfect with beer.
Kazakhstan can hardly be called a cheap country. Prices for food at shops and restaurants are higher than in the other Central Asian countries. In the capital a substantial lunch at a café would cost you 25USD, a dinner - 30USD. A meal in areas outside large cities would be much cheaper.
The citizens of most of the countries in the former Soviet area are not required to have a Kazakhstan visa to stay in the territory of Kazakhstan.
People from many other countries around the world do not need an invitation to obtain a tourist visa.
The citizens of Austria, Australia, Belgium, Greece, UK, Denmark, Israel, Iceland, Ireland, Italy, Spain, Canada, Luxembourg, Liechtenstein, Monaco, Malaysia, New Zealand, the Netherlands, Norway, USA, Singapore, Portugal, France, Germany, Finland, the Republic of Korea and Japan are not required to obtain a registration while staying in the territory of Kazakhstan, if they have a Kazakh visa.
Hotel workers register their clients in a special book, upon which they must provide their guest with a document confirming that they are staying at this hotel.
The level of criminality in Kazakhstan is very low and tourists can hardly fall victims to criminal offence. Nevertheless, you should take some general safety precautions: do not walk in the backstreets at night time, avoid taking large sums of money with you and showing them and other precious things to other people. You should also follow the recommendations of your guide and hotel workers.
You must retain the customs declaration you filled in when entering the country to the very end of the trip. You do not need to enter into it foreign currency totalling less than 3,000 USD, goods with a total price not exceeding 550 USD, cigarettes under 1,000 pieces and less than 2 litres of strong alcoholic drinks.
Certainly, you may not bring to Kazakhstan weapons, narcotic drugs, ammunition and materials undermining the state's constitutional system.
You may not take from the country hunting birds, rare minerals, animals, gold and precious stones.
You had better save receipts for other goods purchased in Kazakhstan. If you come across illegal actions you should contact the consular department of your country or call the hot-line service of the customs control committee: 8 (7172) 70-98-20.
Purchases and services in Kazakhstan are officially paid only in tenge.
Numerous bank departments and exchange offices are open for 24 hours in Kazakhstan. There are many automated teller machines in large cities. Bank cards are accepted in most trade centres and hotels.
Photographing in Kazakhstan is allowed everywhere but airports. You should avoid taking pictures of military objects and people working there.
The most popular of the traditional Kazakh souvenirs is kashma - a whip of natural leather, which an essential attribute of a nomad.
Women may like saukele - a traditional handmade bridal headdress, or belizik - a massive bracelet decorated with a traditional pattern. They will fill you with a feeling of festivity and eternal youth.
• Date varies - Kurban Ait (Eid-i-Kurbon, Eid ul-Adha), Islamic Holiday.


text:
modified on 15 April 2016 at 17:19 ••• 51,154 views
Edoapra (Talk | contribs)
(→Core Development Team)
(→Authors and Contributors)
==Authors and Contributors ==
E. Apra, E. J. Bylaska, W. A. de Jong, N. Govind,
E. Apra, E. J. Bylaska, W. A. de Jong, N. Govind, K. Kowalski, T. P. Straatsma, M. Valiev, H. J. J. van Dam, D. Wang, T. L. Windus, J. Hammond, J. Autschbach, K. Bhaskaran-Nair, J. Brabec, K. Lopata, S. A. Fischer, S. Krishnamoorthy, W. Ma, M. Klemm, O. Villa, Y. Chen, V. Anisimov, F. Aquino, S. Hirata, M. T. Hackler, T. Risthaus, M. Malagoli, A. Marenich, A. Otero-de-la-Roza, J. Mullin, P. Nichols, R. Peverati, J. Pittner, Y. Zhao, P.-D. Fan, A. Fonari, M. Williamson, R. J. Harrison, J. R. Rehr, M. Dupuis, D. Silverstein, D. M. A. Smith, J. Nieplocha, V. Tipparaju, M. Krishnan, B. E. Van Kuiken, A. Vazquez-Mayagoitia, L. Jensen, M. Swart, Q. Wu, T. Van Voorhis, A. A. Auer, M. Nooijen, L. D. Crosby, E. Brown, G. Cisneros, G. I. Fann, H. Fruchtl, J. Garza, K. Hirao, R. A. Kendall, J. A. Nichols, K. Tsemekhman, K. Wolinski, J. Anchell, D. E. Bernholdt, P. Borowski, T. Clark, D. Clerc, H. Dachsel, M. J. O. Deegan, K. Dyall, D. Elwood, E. Glendening, M. Gutowski, A. C. Hess, J. Jaffe, B. G. Johnson, J. Ju, R. Kobayashi, R. Kutteh, Z. Lin, R. Littlefield, X. Long, B. Meng, T. Nakajima, S. Niu, L. Pollack, M. Rosing, K. Glaesemann, G. Sandrone, M. Stave, H. Taylor, G. Thomas, J. H. van Lenthe, A. T. Wong, Z. Zhang.
K. Kowalski, T. P. Straatsma, M. Valiev, H. J. J. van Dam,
D. Wang, T. L. Windus, J. Hammond, J. Autschbach,
K. Bhaskaran-Nair, J. Brabec, K. Lopata, S. Krishnamoorthy,
W. Ma, M. Klemm, O. Villa, Y. Chen, V. Anisimov, F. Aquino,
S. Hirata, M. T. Hackler, T. Risthaus, M. Malagoli,
A. Marenich, A. Otero-de-la-Roza, J. Mullin, P. Nichols,
R. Peverati, J. Pittner, Y. Zhao, P.-D. Fan, A. Fonari,
R. J. Harrison, M. Dupuis, D. Silverstein, D. M. A. Smith,
J. Nieplocha, V. Tipparaju, M. Krishnan, B. E. Van Kuiken,
A. Vazquez-Mayagoitia, L. Jensen, M. Swart, Q. Wu, T. Van
Voorhis, A. A. Auer, M. Nooijen, L. D. Crosby, E. Brown,
G. Cisneros, G. I. Fann, H. Fruchtl, J. Garza, K. Hirao,
R. A. Kendall, J. A. Nichols, K. Tsemekhman, K. Wolinski,
J. Anchell, D. E. Bernholdt, P. Borowski, T. Clark, D. Clerc,
H. Dachsel, M. J. O. Deegan, K. Dyall, D. Elwood,
E. Glendening, M. Gutowski, A. C. Hess, J. Jaffe,
B. G. Johnson, J. Ju, R. Kobayashi, R. Kutteh, Z. Lin,
R. Littlefield, X. Long, B. Meng, T. Nakajima, S. Niu,
L. Pollack, M. Rosing, K. Glaesemann, G. Sandrone, M. Stave,
H. Taylor, G. Thomas, J. H. van Lenthe, A. T. Wong, Z. Zhang.
1 NWChem Development Consortium
1.1 Core Development Team
1.2 Active Developers
1.3 Authors and Contributors
NWChem Development Consortium
Core Development Team
Current focus
E. J. Bylaska Plane wave development
N. Govind DFT, (LR/RT)-TDDFT, Excited-state methods
K. Kowalski High accuracy methods
M. Valiev QM/MM methodologies
H. J. J. van Dam DFT, Fault tolerance
E. Apra DFT, Parallelization infrastructure
Active Developers
R. J. Harrison (Stony Brook) Multiresolution methods
W. A. de Jong (LBNL) Properties, relativistic effects, basis sets, parallel performance, integrals, HPC
T. P. Straatsma (ORNL) Molecular dynamics
T. L. Windus (Iowa State University and AMES) Dynamic nucleation theory
J. R. Hammond (Intel) Coupled cluster methods, performance optimization and multithreading
J. Autschbach (SUNY Buffalo) Response properties, relativistic effects
L. Jensen (Penn State) DFT functionals, properties
M. Swart (Universitat de Girona, Spain) SSB-D functionals
Q. Wu (BNL) Constrained DFT
T. Van Voorhis (MIT) Constrained DFT
Y. Zhao (Hewlett-Packard) Minnesota (Truhlar) functionals
R. Peverati (UC Berkeley/LBNL) Minnesota (Truhlar) functionals
K. Lopata (LSU) RT-TDDFT development
A. Vazquez-Mayagoitia (ANL) DFT functionals, properties
Kiran Bhaskharan Nair (LSU) Multireference Coupled Cluster (MRCC) Theory development
Jiri Brabec (LBNL) Multireference Coupled Cluster (MRCC) Theory development
Authors and Contributors
Retrieved from "http://www.nwchem-sw.org/index.php/Developer_Team"

redpajama_set_name: RedPajamaCommonCrawl


text:
Omar, Haroub Hamad,et al. "The Effects of Exchange Rate on Export Performance in Tanzania: An Empirical Research for Financial Competition." Handbook of Research on Decision-Making Techniques in Financial Marketing, edited by Hasan Dinçer and Serhat Yüksel, IGI Global, 2020, pp. 47-67. http://doi:10.4018/978-1-7998-2559-3.ch003
Omar, H. H., Ceylan, N. B., & Kapusuzoglu, A. (2020). The Effects of Exchange Rate on Export Performance in Tanzania: An Empirical Research for Financial Competition. In Dinçer, H., & Yüksel, S. (Ed.), Handbook of Research on Decision-Making Techniques in Financial Marketing (pp. 47-67). IGI Global. http://doi:10.4018/978-1-7998-2559-3.ch003
Omar, Haroub Hamad and Nildag Basak Ceylan, and Ayhan Kapusuzoglu. "The Effects of Exchange Rate on Export Performance in Tanzania: An Empirical Research for Financial Competition." In Handbook of Research on Decision-Making Techniques in Financial Marketing. edited by Hasan Dinçer , and Serhat Yüksel, 47-67. Hershey, PA: IGI Global, 2020. http://doi:10.4018/978-1-7998-2559-3.ch003
InfoSci-Media and Communications
The Effects of Exchange Rate on Export Performance in Tanzania: An Empirical Research for Financial Competition
Haroub Hamad Omar (D-Tree International, Tanzania), Nildag Basak Ceylan (Ankara Yildirim Beyazit University, Turkey) and Ayhan Kapusuzoglu (Ankara Yildirim Beyazit University, Turkey)
Source Title: Handbook of Research on Decision-Making Techniques in Financial Marketing
The chapter analyzes the effects of exchange rate of Tanzanian shilling on the country's exports performance applying Vector Auto-Regressive (VAR) model covering the sample period from 1993:Q1 to 2016:Q4. Cointegration and causality tests are performed to investigate the short- and long-term relationships between the variables to evaluate the financial competition. The results show that; there is no long-term relationship (cointegration) between exchange rates and exports and between foreign demand and exports. Moreover, the results of causality test show no short-term relationship (causality) between exchange rates and exports and between foreign demand and exports. As the findings suggest, the exchange rate level of Tanzanian shilling (in nominal terms) does not statistic-significantly affect the country's exports performance.
While in the last touches of World War II, in 1944, when the world was resuming to normality, an international agreement commonly known as Bretton-Woods Agreement was signed for resetting a way through which countries can exchange their currencies and hence carry out trades between them. This came following the collapse of Gold Standard and the financial chaos in Europe and America (Great Depression of 1930s). Under Bretton-Woods system, countries could trade between them using exchange rates maintained within fixed values, once exchange rates between currencies established, governments were intervening to prevent them from deviating for more than one percent (1%) from the original set level. And most of currencies were pegged to either US Dollar or British Pound. Unfortunately, by 1971 it came to light that the US Dollar to which some of currencies were pegged had been overvalued and hence greatly suppressed its demand i.e. more dollars were in supply than what demanders could afford to buy because of overvaluation. This called for a review of the exchange rate regime and finally US Dollar devalued relative to other currencies, and currencies allowed to vary above the initial allowed interval of 1%, hence, this time currencies could vary up to 2.25 percent (2.25%). This was agreed in a common agreement known as Smithsonian Agreement by major trading countries. Very unfortunate, even after being allowed to vary for up to 2.25% below or above the set level, countries were struggling to keep their currencies within the margins of change. Hence, the fixed exchange rate regime proved failure and by March 1973 officially the world wrote a new history as a starting point of "Flexible Exchange Rate Regime" which is in practice until today. (Madura, 2011) By such a new flexible exchange rate system just set in, officially exchange rates among currencies started to fluctuate in response to prevailing market demand and supply conditions – no longer fixed margins of change. The new system got a quick acceptance among many countries fuelled by globalization move. Countries started liberalizing their economies and the trading among them increased in a vacuum. As a result, the level and variability at which one currency trade against such a free globalized world become of paramount importance to determine the country's competitiveness in the world business arena. Not alone to governments' policy makers, exchange rate became of the same importance to rest of actors who use foreign currencies such as investors, individual entities (importers and exporters) and multinational corporations. The fact that exchange rates are now freely adjusting to market conditions, means these stakeholders are not sure of exchange rate movements anymore, so uncertainty set in. The now new worries in the market i.e. uncertainty brought by level as well as frequency of fluctuation of exchange rates became a click to all stakeholders of international business and the same to the world of academia. Many studies so far have been taken but most focusing on effects of exchange rate volatility rather than level. Moreover, they are based on developed countries and recently on BRICS countries as potential power houses of the world, MINT countries and MENA countries. Considering the significance of the proposed topic from both theoretical and practical view, there are very limited studies conducted the impact of exchange rate on exports performance in Tanzania. In this direction, the aim of the study is to analyze the effects of exchange rate of Tanzanian shilling on the country's exports performance.
IMF: International Monetary Fund.
Industrial Production Index: Industrial production refers to the production of industrial corporations. This indicator is measured in an index that expresses the change in production output volume.
Export: Exporting is dispatching products to another country in exchange for payment.
Exchange Rate: The exchange rate is the value of a country's currency in terms of one unit of the other country's currency.


text: Q: NewRelic transactions in Magento 2 Magento Commerce 2 comes with NewRelic module-new-relic-reporting module. It automatically logs transactions. It is possible to add custom parameters by calling the newrelic_add_custom_parameter function. Currently NewRelic displays only those transactions which are "Most time consuming" or with "Slowest Average Response Time" or "Apdex most dissatisfying" or with "Highest throughput". I want the NewRelic dashboard to display all transactions if they have a specific custom parameter. How to achieve that?

redpajama_set_name: RedPajamaStackExchange


text:
package com.planet_ink.coffee_mud.Abilities.Songs;
import com.planet_ink.coffee_mud.core.interfaces.*;
import com.planet_ink.coffee_mud.core.*;
import com.planet_ink.coffee_mud.core.collections.*;
import com.planet_ink.coffee_mud.Abilities.interfaces.*;
import com.planet_ink.coffee_mud.Areas.interfaces.*;
import com.planet_ink.coffee_mud.Behaviors.interfaces.*;
import com.planet_ink.coffee_mud.CharClasses.interfaces.*;
import com.planet_ink.coffee_mud.Commands.interfaces.*;
import com.planet_ink.coffee_mud.Common.interfaces.*;
import com.planet_ink.coffee_mud.Exits.interfaces.*;
import com.planet_ink.coffee_mud.Items.interfaces.*;
import com.planet_ink.coffee_mud.Libraries.interfaces.*;
import com.planet_ink.coffee_mud.Locales.interfaces.*;
import com.planet_ink.coffee_mud.MOBS.interfaces.*;
import com.planet_ink.coffee_mud.Races.interfaces.*;

import java.util.*;

public class Song_Nothing extends Song
{
	@Override
	public String ID()
	{
		return "Song_Nothing";
	}

	private final static String localizedName = CMLib.lang().L("Nothing");

	@Override
	public String name()
	{
		return localizedName;
	}

	@Override
	public int abstractQuality()
	{
		return Ability.QUALITY_INDIFFERENT;
	}

	@Override
	protected boolean skipStandardSongInvoke()
	{
		return true;
	}

	public Song_Nothing()
	{
		super();
		setProficiency(100);
	}

	@Override
	public void setProficiency(final int newProficiency)
	{
		super.setProficiency(100);
	}

	@Override
	public boolean invoke(final MOB mob, final List<String> commands, final Physical givenTarget, final boolean auto, final int asLevel)
	{
		boolean foundOne=false;
		for(int a=0;a<mob.numEffects();a++)
		{
			final Ability A=mob.fetchEffect(a);
			if((A instanceof Song))
				foundOne=true;
		}
		unSingAll(mob,mob,false,true);
		if(!foundOne)
		{
			mob.tell(auto?L("There is no song playing."):L("You aren't singing."));
			return true;
		}

		mob.location().show(mob,null,CMMsg.MSG_NOISE,auto?L("Silence."):L("<S-NAME> stop(s) singing."));
		mob.location().recoverRoomStats();
		return true;
	}
}

redpajama_set_name: RedPajamaGithub


Franks professionalism and easy going manner with all types of audience has been gained through many years of experience and hard work in the Australian and UK music scenes.
Frank was lead vocalist/guitarist and front man with successful Brisbane based bands "SILENT PASSENGER" and critically acclaimed "POETIC LICENCE." Both bands enjoyed great popularity in the grueling Queensland and New South Wales touring circuit between 1984 and 1990. POETIC LICENCE headed their own shows as well as supporting acts such as James Reyne; Joe Camalleri and the Black Sorrows, Kids in the Kitchen, Martha Davis and the Motels and Kiwi acts Marc Hunter and Sharon O'Neil amongst others.
Frank has worked solo for a number of years since 1990 in Australia, London and New Zealand. He is backed by a Midi System using Notater Software, and this gives his performance a full sound, enhancing his vocals and guitar.
Firstly thank you for making our Community New Year a very special one. Frank Durant was fantastic. I would have absolutely no hesitation in recommending Frank to any function whereby the goal was to have people entertained and up dancing. It was a smallish group 45-50 which is hard for a one man performer to motivate but Frank took it in his stride and the majority of the group had sore feet the next day!
The community is looking forward to next years New Years and have also requested a mid year function!
"Great yeah, good turnout. Frank was received really well, I wouldn't hesitate to have him back."


Sweet Mama, our favorite green kabocha.
Full disclosure: We love kabocha squash! Kabochas (and kabocha-type squash) have been a major focus of the WINTER squash project. Kabocha squash typically have deep orange, thick, flaky, nutty and flavorful flesh. Kabochas are not ripe when harvested in September. They need to further ripen for a month or so to increase sugars and flavor.
We have trialed many Kabochas in this project, and none have performed as well as Sweet Mama, a hybrid that was an All American Selections winner in 1979. It stands out from other kabochas as it stores very well into January (most are much more susceptible to the storage rots we have here in western Oregon), and it is beautiful and has a wonderful sweet nutty flavor.
Sweet Mama has also performed well in trials conducted by Fedco Seeds in Maine - thy say it has an outstanding flavor and is very dependable.


text:
Remington Model 870: The Best Pump Shotgun Ever?
From recoil pad to muzzle, it's a simple shotgun that has stood the test of time and proven its versatility and adaptability over more than half a century.
By M.D. Johnson
If you're reading this, there's an awfully good chance you currently own or at one time in the not-too-distant past did own a Remington Model 870, henceforth known simply as the M870, pump gun. Also, it's likely those who currently own one use it quite often and in a variety of situations, including, but not limited to wingshooting, big game hunting, turkey hunting, and/or home defense.
Haven't owned a M870 yet? Or perhaps are currently in the market for your first-ever shotgun and are considering a M870? For many, the pump-action shotgun was, and continues to be, an excellent choice, whether it be as an addition to an already existing firearm collection or as a first gun purchase. Why such a bold statement? Versatility is the M870's most endearing virtue, along with dependability, as we'll explore here.
With the added benefit of affordability, the M870 is an all around excellent choice for young or new shooters alike. The endlessly possible modifications of the M870 by way of choke tube swaps, stock modification, barrel changes, and the addition of sights or sighting devices extends the model's useful life almost indefinite. All this for an initial cost of less than $300, most of the time. Try buying the new iPhone for under $300 (let alone having it work for more than two years)!
To those who sold their M870 at some ill-advised point in the past, allow me to go out on the proverbial limb here and say you've regretted the decision. Maybe not every day, but each and every time you've gone to the gun cabinet reaching in vain for a "One Size Fits All" type of firearm, of which there are actually few.
The Model 870 American Classic model.mfg photo
The M870, however, is one of the few, but is it, as the title might suggest, the 'best ever' in terms of pump-action shotguns? That's saying a lot – 'Best Ever' – especially when you look at the rather lengthy list of popular pump guns, a handful of which might be worthy of the descriptive term, legendary.
Shotguns such as Winchester's Model 12, and the company's M1897. The list goes on: the venerable Mossberg Model 500, Ithaca's Model 37, and more recently, Browning's BPS and Benelli's NOVA.
Best ever? 'Tis truly up to the reader, the M870 fan, even the disgruntled former owner, to decide. There is, as you'll see, plenty of evidence to support the fact the M870 is undeniably good. Perhaps even great. Maybe even, dare we say, the "best ever?"
The M870 walked onto the public stage in 1950, after beginning life two years earlier as the offspring, of Remington's new autoloading shotgun, the M11-48, designed by company engineers Ellis Hailston, C.R. Johnson, and Ray Critendon. Certain design features, like stamped steel parts and component interchangeability, that were key aspects of the then-revolutionary M11-48 made their way to Big Green's new pump-gun.
In the case of the M870, a pair of innovative features would endear the scattergun to the shooting public:
Twin action (slide) arms or bars increased positive lockup and overall smoothness of function; in simpler terms, the M870 became more reliable and less prone to cycling hiccups, we'll call them, than other models. Even the earliest Mossberg 500 shotguns had one action arm, which could bend under heavy stress and bind-up the action, which lead to the company adding an additional action arm to the M500 when Remington's patent expired.
A Model 870 Wingmaster shotgun.mfg photo
Remington's inclusion of a hardened steel locking lug as an integral part of the barrel – as opposed to a mated recess machined into the receiver proper. This not only provided strength at lockup, but allowed barrels to be swapped between like gauges instantly, easily, and positively, without tools, hardship, or the elaborate cursing that commonly accompanied such a task back in the day.
One receiver, three barrels, and a shooter was, within seconds, ready for everything from upland birds and waterfowl to whitetails to a home defense situation, if necessary.
The M870 wasn't Remington's first pump-action shotgun but it is, without a doubt, the company's finest to date, as well as its best-selling.
An example of a Remington Model 10 pump-action, downward ejecting shotgun.photo from tactical-life.com
In 1909, Remington, working through designer John Pedersen, introduced the Model 10, a bottom-ejecting 12-gauge. During the M10's run from '09 to 1929, an updated version of the model, the M29 and later the M31, were brought to market and more than 250,000 were produced. Both updated models featured the now-familiar side ejection port layout. While it was used for various military and law enforcement applications, the M10 and its variants saw heavy use by civilians as well.
Early Variants
Upon its introduction in 1950, some 15 variants of the M870 were available, including the AP Standard Grade, BC Special Grade, ADL Deluxe Grade, BDL Deluxe Special Grade, trap versions, skeet versions, and a Riot Grade, complete with a 20-inch barrel.
There were differences among them, especially in wood quality and barrel length. There were also three gauge options – 12, 16, and 20. The 12- and 20-gauge versions were available with 2-3/4-inch chambers only.
The M870 Express Tactical model with a synthetic stock, extended magazine tube, muzzle brake, and iron sights.mfg photo
Four years later, in 1954, a 3-inch 12-gauge M870 Magnum hit the market. Small bore fans would have to wait until 1969 before 28-gauge and .410 field models sporting downsized true-to-gauge receivers to grace gun-shop shelves.
In 1960, the U.S. military obtained approximately 38,000 M870 riot-style guns—plain-jane five-shot pump actions with either 18- or 20-inch barrels and full length walnut stocks—and shipped them to South Vietnam. Other M870s were set aside for use by the U.S. Marines, Army, and Air Force.
In 1969, Remington signed a contract with the Unites States Marine Corps to produce the M870 MARK (MK) 1, which were modified for military service with the addition of a 7-round extended magazine tube (bringing the total capacity to eight rounds) and a bayonet lug.
Popular among many of the troops, the Mark I would see service until the U.S. withdrawal from Southeast Asia in 1975 and beyond.
The versatility of the M870 is outmatched only by its comparative affordability, making it a shotgun truly anyone can own. More than 11 million have been produced.mfg photo
The M870's milestones over the years are, in a word, numerous. In 1966, Remington announced their one millionth M870; in '73, the two millionth. This number swelled to five million units in 1990, and in April of 2009, the company unveiled the 10 millionth M870.
Eight years later, there are over 11 million M870s out there. Somewhere. Everywhere. A bit of trivia: With an average overall length of 48-1/2 inches, 11 million M870s laid end-to-end would stretch approximately 8,420 miles. That's round-trip from New York City to Los Angeles, with 37 miles to spare. Impressive, eh? Break that one out at your next dinner party.
Then, Much As It Is Now
Since its introduction more than a half century ago, the M870, as a shotgun platform, has largely remained the same—but it has gone through multitudinous cosmetic changes. Features and options such as barrel length, metal finishes, stock and fore-end construction, sights, accessories, and niceties have been offered up.
From recoil pad to muzzle, the elementalism of the M870 is, to many, the firearm's primary allure. It's a simple shotgun built around a simple design, especially by today's standards.
A Model 870 Express ShurShot Synthetic turkey hunting configuration.mfg photo
Stocks have, for the most part, been made of walnut through the shotgun's production; but, synthetic stocks have gained favor in recent years for their extreme weather resistance (important for waterfowls and hunters in general), endurance, and light weight.
Tactical synthetic stocks with pistol grips, as well as folding stocks, have become standard on law enforcement, military, and home defense versions of the 870. Synthetic fore-ends, likewise, have gained popularity for many shooters along with stocks, though wood versions are still popular as well.
Action and Safeties
Operation is straight forward. The aforementioned dual action bars power the bolt assembly; extraction and ejection of the spent hull takes place on the rearward stroke, while elevation and placement into battery of a live round occurs on the forward motion, as the action is closed and locked.
Since the 1970s, nearly all pump-action shotguns, including the M870, include a trigger disconnected safety, which required the trigger to be released before the firing pin can be activated again. On some older pump guns, especially pre-1975 Ithaca 37s, if the user held down the trigger and worked the action, the shotgun would fire instantly as soon as a new round was chambered. This is commonly called slam fire, though an actual slam fire is an accidental discharge that can lead to a runaway. It would be more accurate, in this case, to call it a pump-fire. During both World Wars, soldiers found it an effective technique for clearing trenches and other small areas. Trigger disconnects were later added as a safety feature to prevent pump fires.
The Model 870 Hard Wood Home Defense model with an extended magazine.mfg photo
In contract to the Mossberg M500's top-mounted safety, which is naturally ambidextrous by its positioning, the M870's cross-bolt safety is locate at the rear of the trigger guard.
Southpaws have a couple options for their right-handed M870. If it's an older model, pre-90s, the solution may be as simple as swapping the right-hand safety for a left-hand version. Otherwise, left-hand trigger groups, interchangeable in the case of the M870, are available. Or, Remington makes M870 models that come fully set-up for lefties.
The M870's bolt release on the left side at the leading edge of the trigger guard.
Barrels and the Rem-Choke System
Remington Model 870 Express Fully Rifled Deer model with iron sights.mfg photo
The M870's barrel mates snugly into the receiver and is secured by both a barrel band that wraps completely around the magazine tube, topped with a threaded magazine cap.
Barrel options for M870s have, over the years, been as many as there are stars in the sky. From 18-inch Cylinder Bore shorties meant for home defense to a 32-inch full choke version designed to reach out and touch high-flying ducks and geese, Remington has made available something for every shooter, regardless of their game or target of choice.
Up until 1986, barrels for the M870 had fixed chokes, including improved cylinder, modified, and full.
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Fully-rifled barrels, complete with integral cantilevered scope mounts, were – and still are today – created for deer hunters choosing to hunt in shotgun-only states, e.g. Iowa and Ohio. Most popular among wingshooters, however, are Remington's 26- and 28-inch vent-rib versions, both featuring the Rem-Choke interchangeable choke tube system.
In 1986, Remington introduces it's Rem-Choke interchangeable choke tube system and 870 barrels to go with it. This means shooters were no longer limited to one choke per barrel. Instead of needing at least three barrels to round out an M870 hunting kit, now only one barrel is needed along with a set of choke tubes, making the shotgun even more versatile and affordable. It also opened up the use of more exotic chokes for specific purposes, like XX-Full chokes for turkey hunting, since the purchase of an entire new barrel was no longer required.
Design Changes and Enduring Quality
Model 870 Special Purpose Marine Magnum with an anti-corrosion finish.mfg photo
As for the M870's long list of cosmetic changes, it's tough to imagine a configuration that's not available.
There are turkey gun setups with Extra Full choke tubes and thumbhole stocks; deer guns with cantilevered and fully-rifled barrels along with a top rail for mounting optics; goose-getters chambered for Roman Candle-sized 3.5-inch shotshells; trap guns; skeet guns; setups for sporting clays; left-handed models, and it goes on.
A quick glance at Remington's website revealed some 25 different variants of the M870 ranging from a Plain Jane matte finish/walnut stocked Express to the futuristic-looking Shurshot Synthetic Turkey model featuring a thumbhole stock one could put his head through. Electroless nickel plating covers the Marine Magnum inside and out, while the diminutive all-black TAC-14 home/personal defense variant (introduced in 2017) sports a 14-inch barrel and an overall length of just a shade over 26 inches. And that isn't even taking into account the vast array of aftermarket products available for the 870 from various manufacturers. After all, with 11 million shotguns out there, that's a lot of different tastes that must be catered to.
And finally, affordability, alongside the aforementioned virtues of reliability, dependability, and versatility, help make the M870 what it is, and always has been—you don't have to save for months to afford one, and that's hard to beat.
Prices vary, of course, but Remington lists its least expensive 870, the Express model, at $417. One of the highest-end models, the 200th Anniversary Edition M870 Wingmaster, complete with 24 karat gold inlays, a custom box, and a special serial numbers, rings up at $1,500, which is still extremely affordable.
An Informal Survey
An example of a synthetic stocked model, this M870 has a fully rifled barrel and a cantilever scope mount.mfg photo
Are the results below conclusive or scientific? Not at all. But in the interest of reinforcing my arguments, I hit the inter-webs and set the question to the men and women of Hunting-Washington.com, and gathered their thoughts on the M870, and their opinion as to whether or not it's the best pump-gun ever made.
The comments were indeed interesting, and not, I must say, without a common denominator. Read them as they were posted:
"I love my 870. I have several other (more expensive) shotguns and choose the 870 more often than not. Also bought a youth 20 for my kids...really fun gun to shoot. If I could only have one and use it for everything it would be my 870, no question" —BD1
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"I shot trap when I was in my mid teens, one of the guns that I could always count on to get me back in the game, if/when something would get in my head, was a 870 TC Wingmaster. I put thousands of rounds through one and though it didn't have the "cool" factor of some of the other trap guns I had, I probably shot it as well or better than any I've ever shot. The thing just seemed to always break everything I pointed it at and will forever have a fondness for the 870's" —b23
"I have several. Love them all. My all time favorite gun is the 870 express I was given by my dad in 1990. I have literally over 10,000 rounds through that gun. It started having some jamming issues 2 years ago, but I replaced the trigger and that seams to have fixed it. I have killed ducks, geese, turkey, dove, rabbit, Chucker, quail, pheasant, and sand hill crane. I took it to Africa and used it to kill Dove, Guinea fowl and Francolins. This shotgun is an extension of my body. I can cycle it faster than most semi-autos. I own many firearms worth a lot more than this old 870, but none makes me smile more than this one. After 25 years of hard service, this gun owes me nothing. I call her Heather" —Rob
"I got my old man's 870 in 12 gauge to replace my 20 gauge 870 when I was in middle school. He and all his buddies upgraded to auto loaders from Italy all around the same time. I shot all their guns and never found one I liked more than that twenty year old 870. It's still my only bird gun almost twenty years later" —Jpmiller
If there is a quarry that can be taken with a shotgun, there is an M870 suited to the task.author photo
"An 870 12 gauge was the first gun I bought with my own money. I was 13 years old and saved up the money I earned on my newspaper delivery route. I've had the gun 45 years, shot trap with it for years and killed some birds with it too, never had a single issue with that gun, I'll own it until I die" —bearpaw
There were, certainly, some negative responses as well.
"I currently have an 870 express. POS. It jams more often then it cycles, so I haven't used it in years. I understand that the fix is fairly easy and I need to polish the bore or something?" —WSU
"I had a circa 1970 Wingmaster 20g – it was a solid, reliable gun but it didn't fit well, recoil was harsh, muzzle rise made follow up shots difficult. Newer Express 12g, POS – same issues as the Wingmaster but add poor fit & finish – unreliable jammer" —CP
"I have an 870 express. I have put several thousand rounds through it and it has jammed 10-15 times. Yes, it normally works, but those jams really give a sour taste" —PastorJoel
And then there was this gentleman, whose post definitely falls into the 'And there you have it, folks' category.
"Love them or hate them, they are somewhat iconic, not many guns are produced that long without any major changes" —actionshooter
Truly The Best Ever?
What makes something the best ever? Greatest number sold? Dependability? Versatility? Ease of modification and customization? Price? Maybe a combination of each and every variable here in differing degrees.
There's no denying that Remington's venerable M870 ranks high, if not top of the list, in all of the categories that define greatest, if greatest is indeed defined by the terms mentioned here. Even if we consider only how many M870s have been produced and are in use today, it seems clear the iconic shotgun is a super all-round fit for hundreds of hunters and shooters across the nation.


eh-hem. John Mills AND Dirk Bogarde in one film. JACKPOT!!
In The Singer not the Song, John Mills plays a Catholic priest who arrives in a small Mexican village to take over for the retiring priest. The village is controlled by Anacleto, the town's bad boy played by a leather-clad Dirk Bogarde.
Dirk Bogarde is well known for acting in daring movies, such as Victim - another movie he made the same year as this one, which dealt with homosexuality and actually paved the way for archaic homophobic laws in England to be repealed. This movie deals with perhaps an even touchier subject: atheism. I can't actually recall ever seeing a movie which dealt with this subject before. It's a very taboo topic, even now. While we have quite a few openly gay politicians, there is only one openly atheist politician in the whole country. So for Dirk Bogarde and John Mills to star in a film that dealt frankly with athiesm is, to me at least, as daring as it was for Dirk to star in Victim.
The film is groundbreaking, but it's not entirely without convention. John Mills, as the priest, is still the good man. Dirk Bogarde, as the atheist, is still the bad man. But both actors gave performances that made the one dimensional good vs. evil characters more nuanced. The priest has to fight with his inner demons to remain a chaste, pure man. The atheist who cannot trust anyone places his trust in a man of the cloth.
I think that whether you are an atheist or a devout Christian, you will see your own beliefs reinforced in this film. The film doesn't condemn atheism (this actually took me completely by surprise. I was expecting Anacleto to fall down on his knees, repent, and become a monk by the end of the movie -- something that would be typical in most films from this era) but it also doesn't comdemn Christianity. It forces nothing down your throat, just leaves the matter up in the air for you to decide.
In all honesty, I doubt this film would be released in America today, despite the fact that its message about religion is one of ambiguity not condemnation. I read recently that American theaters won't be distributing a new movie about the life of Charles Darwin for fear of protests. Certainly if a film about evolution (a subject that is discussed in science classrooms across the country) can't make it into theathers, I highly doubt a movie about atheism would!
The one funny thing about the film is that, like in The Spanish Gardener, almost all of the Mexican characters are British. This didn't bother me so much, though, as the fact that the one girl in the story (whose parents are both British and who grew up in Mexico) speaks with a French accent! Where did that come from?!
Oh! One more thing... yesterday I found that one of the movies I reviewed recently, Cast a Dark Shadow, is available in full on YouTube! It was a really great film, but not available on DVD. So if you have some free time you should definitely watch it!
this movie looks very entertaining if not just for the black leather pants! Great post but depressing to hear about the Charles Darwin movie.
Ever since I saw pictures of him in those pants I totally wanted to see the film the pictures were from. This sounds really interesting - I've never really thought about it but I can't think of any atheist characters off-hand. And I agree that it would have been rather a daring role to take! (And, interestingly enough, I'm re-watching Victim at the moment).
I can't believe the film about Charles Darwin isn't going to be released. Frankly, I think that's ridiculous.
I really enjoyed this post, because it reminded me of why I love films. It's always good when you come across a film that reminds you of just how influential and significant of a medium it can be. Anyhow, I just came across your blog, but now I am hooked. And, I must now go discover Dirk Bogarde for myself. Sadly, he is unknown to me. Oh, and I hope you don't mind me linking you!
Odd-- I'd not thought about it until you brought it up, but you're right...don't think I've ever noticed a movie (at least an older movie) dealing with atheism. Interesting.
I have to agree it. Given the reaction to the Darwin movie, I don't think The Singer Not the Song would be released in the States today. That is sad, but I think things have become so polarised that religion has become even more taboo than it once was as a subject.


I've officially been a mom for one month! On April 14, our baby boy arrived. It was by far the most amazing experience of my life. I'll never forget hearing the doctor say, "On the next push, you're going to have a baby!" and then suddenly, he was here—eight pounds and one ounce, twenty-and-a-half inches and totally perfect. We named him William George, after our dads Bill and George, who suggested the name when they first found out I was pregnant. We laughed at them but secretly decided we loved it, which made finally revealing his name one of my all-time favorite moments.
Leaving the hospital two days later was both thrilling and terrifying. At home, there's no button on our bed to summon a nurse when Will spits up or our sleep-deprived brains can't remember how to swaddle. And then suddenly, a month has flown by and somehow, we all survived. It's been a beautiful blur of nursing and cuddling and more nursing, plus bouncing and swinging and shushing and singing. There's been cooing and crying and a constant hum of lullabies that we haven't quite learned the words to yet. There have been times when I desperately wanted Will to sleep somewhere other than my chest but then missed him as soon as he wasn't there.
Nick and Will in our hospital room.
I loved finally putting him in this tiny onesie (previously seen here).
Nick and Daphne didn't hesitate to steal my pregnancy pillow.
Growing faster than a speeding bullet!
After two weeks, Will was an inch taller and had packed on 19 ounces.
To see more of our adventures in parenthood, follow along on Instagram.
Thanks so much to one of my favorite mommies! xo!