Necessary and Sufficient Conditions for the Uniform Convergence of Means to their Expectations

1982 ◽  
Vol 26 (3) ◽  
pp. 532-553 ◽  
Author(s):  
V. N. Vapnik ◽  
A. Ya. Chervonenkis
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Necessary And Sufficient

scholarly journals On uniform convergence of some classes of infinite products

1987 ◽  
pp. 107
Author(s):  
K.M. Slepenchuk
Keyword(s):  
Uniform Convergence ◽  
Infinite Product ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We find necessary and sufficient conditions $\{ \alpha_k(x) \}$ must satisfy for the infinite product$$\prod\limits_{k=1}^{\infty} \bigl[ 1 + \alpha_k(x) u_k(x) \bigr]$$to converge uniformly under the condition that:1) the series $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)|$ converges uniformly; 2) $\sum\limits_{k=1}^{\infty} |\Delta u_k(x)| = O(1)$.

scholarly journals On uniform convergence of infinite products

2021 ◽  
pp. 80
Author(s):  
K.M. Slepenchuk ◽  
G.A. Barbashova
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Infinite Products ◽  
Necessary And Sufficient

We establish necessary and sufficient conditions for $\{ \alpha_k(x) \}$ to satisfy such that the product $\prod\limits_{k=1}^{\infty} [1+\alpha_k(x) U_k(x)]$ converges uniformly under the condition that $\{ U_k(x) \}$ belongs to a given class.

Uniform Convergence of Trigonometric Series with General Monotone Coefficients

2019 ◽  
Vol 71 (6) ◽  
pp. 1445-1463
Author(s):  
Mikhail Dyachenko ◽  
Askhat Mukanov ◽  
Sergey Tikhonov
Keyword(s):  
Uniform Convergence ◽  
Rate Of Convergence ◽  
Trigonometric Series ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Monotone Coefficients ◽  
Fourier Sums ◽  
Necessary And Sufficient

AbstractWe study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

On the Convergence of the Density of the Power Normalized Maximum1

1998 ◽  
Vol 48 (1-2) ◽  
pp. 13-20 ◽  
Author(s):  
N. R. Mohan ◽  
U. R. Subramanya
Keyword(s):  
Uniform Convergence ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Stable Laws ◽  
Stable Law ◽  
Necessary And Sufficient ◽  
Local Uniform Convergence

Necessary and sufficient conditions for local uniform convergence of the density of the power normalized maximum to the corresponding density of a p-max stable law is derived for each of the six p-max stable laws. An unified criterian is also obtained.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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