On necessary and sufficient conditions for stability for n-input n-output convolution feedback systems with a finite number of unstable poles

1972 ◽  
Author(s):  
Frank Callier ◽  
Charles Desoer
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Feedback Systems ◽  
N Input ◽  
Necessary And Sufficient

On multilattice groups. II

1966 ◽  
Vol 62 (2) ◽  
pp. 149-164 ◽  
Author(s):  
D. B. Mcalister
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Positive Element ◽  
Necessary And Sufficient Conditions ◽  
Countable Number ◽  
Direct Sums ◽  
Lattice Group ◽  
Ordered Groups ◽  
Group A ◽  
Necessary And Sufficient

Conrad ((2)), has shown that any lattice group which obeys (C.F.) each strictly positive element exceeds at most a finite number of pairwise orthogonal elements may be constructed, from a family of simply ordered groups, by carrying out, alternately, the operations of forming finite direct sums and lexico extensions, at most a countable number of times. The main result of this paper, Theorem 3.1, gives necessary and sufficient conditions for a multilattice group, which obeys (ℋ*), to be isomorphic to a multilattice group which is constructed from a family of almost ordered groups, by carrying out, alternately, the operations of forming arbitrary direct sums and lexico extensions, any number of times; we call such a group a lexico sum of the almost ordered groups.

scholarly journals Pointwise stabilization of the Poisson integral for the diffusion type equations with inertia

2016 ◽  
Vol 8 (2) ◽  
pp. 279-283
Author(s):  
H.P. Malytska ◽  
I.V. Burtnyak
Keyword(s):  
Finite Number ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Poisson Integral ◽  
Diffusion Type ◽  
Necessary And Sufficient ◽  
Parabolic Degeneracy

In this paper we consider the pointwise stabilization of the Poisson integral for the diffusion type equations with inertia in the case of finite number of parabolic degeneracy groups. We establish necessary and sufficient conditions of this stabilization for a class of bounded measurable initial functions.

A Note on Quadratic Forms and the u-invariant

1974 ◽  
Vol 26 (5) ◽  
pp. 1242-1244 ◽  
Author(s):  
Roger Ware
Keyword(s):  
Finite Number ◽  
Quadratic Forms ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Real Field ◽  
Witt Ring ◽  
Necessary And Sufficient

The u-invariant of a field F, u = u(F), is defined to be the maximum of the dimensions of anisotropic quadratic forms over F. If F is a non-formally real field with a finite number q of square classes then it is known that u ≦ q. The purpose of this note is to give some necessary and sufficient conditions for equality in terms of the structure of the Witt ring of F.

The Brascamp–Lieb Polyhedron

2010 ◽  
Vol 62 (4) ◽  
pp. 870-888 ◽  
Author(s):  
Stefán Ingi Valdimarsson
Keyword(s):  
Finite Number ◽  
Related Problem ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Linear Maps ◽  
Rank 2 ◽  
Necessary And Sufficient ◽  
Concise Description

AbstractA set of necessary and sufficient conditions for the Brascamp–Lieb inequality to hold has recently been found by Bennett, Carbery, Christ, and Tao. We present an analysis of these conditions. This analysis allows us to give a concise description of the set where the inequality holds in the case where each of the linear maps involved has co-rank 1. This complements the result of Barthe concerning the case where the linear maps all have rank 1. Pushing our analysis further, we describe the case where the maps have either rank 1 or rank 2.A separate but related problem is to give a list of the finite number of conditions necessary and sufficient for the Brascamp–Lieb inequality to hold. We present an algorithm which generates such a list.

scholarly journals Tauberian conditions under which statistical convergence follows from statistical summability $(EC)_{n}^1$

2018 ◽  
Vol 37 (4) ◽  
pp. 9
Author(s):  
Naim L. Braha ◽  
Ismet Temaj
Keyword(s):  
Finite Number ◽  
Statistical Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Complex Numbers ◽  
Real Numbers ◽  
Tauberian Conditions ◽  
Necessary And Sufficient

Let $(x_k)$, for $k\in \mathbb{N}\cup \{0\}$  be a sequence of real or complex numbers and set $(EC)_{n}^{1}=\frac{1}{2^n}\sum_{j=0}^{n}{\binom{n}{j}\frac{1}{j+1}\sum_{v=0}^{j}{x_v}},$ $n\in \mathbb{N}\cup \{0\}.$  We present necessary and sufficient conditions, under which $st-\lim_{}{x_k}= L$ follows from $st-\lim_{}{(EC)_{n}^{1}} = L,$ where L is a finite number. If $(x_k)$ is a sequence of real numbers, then these are one-sided Tauberian conditions. If $(x_k)$ is a sequence of complex numbers, then these are two-sided Tauberian conditions.

Tauberian conditions under which the statistical limit of an integrable function follows from its statistical summability

2006 ◽  
Vol 43 (1) ◽  
pp. 115-129
Author(s):  
Árpád Fekete
Keyword(s):  
Finite Number ◽  
Measurable Function ◽  
Integrable Function ◽  
Sufficient Conditions ◽  
Nondecreasing Function ◽  
Necessary And Sufficient Conditions ◽  
Tauberian Conditions ◽  
Statistical Limit ◽  
Necessary And Sufficient ◽  
Complex Valued

The notions of statistical limit, limit inferior and limit superior of a measurable function at \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\infty\) \end{document} were introduced by Móricz. These notions can be considered as the nondiscrete analogues of those introduced for sequences of numbers by H. Fast, J. A. Fridy and C. Orhan. Let \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(0 \not \equiv p\: \mathbb{R}_+ \to \mathbb{R}_+\) \end{document} be a nondecreasing function such that \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(p(0)=0\) \end{document} and \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mbox{st-\!}\liminf_{t \to \infty} \frac{p(\lambda t)}{p(t)} >1 \ \text{for every} \lambda >1.$$ \end{document} Given a real- or complex-valued function \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(f \in L_{{\rm loc}}^1 (\mathbb{R}_+)\) \end{document}, we define \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$s(x):= \int^x_0 f(u) \, du\ \text{and}\ \sigma(t) := \frac{1}{p(t)} \int^t_0 s(x) d p(x),\quad t>0.$$ \end{document} Our goal is to find necessary and sufficient conditions under which the existence of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\mbox{st-}\lim s(t)=l\) \end{document} follows from that of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\mbox{st-}\lim \sigma(t)=l\) \end{document}, where \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(l\) \end{document} is a finite number. In the case of real-valued functions we present one-sided Tauberian conditions, while in the case of complex-valued functions we present two-sided Tauberian conditions.

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