scholarly journals A note on H-convergence

2021 ◽  
Vol 17 ◽  
pp. 150
Author(s):  
O.P. Kogut ◽  
P.I. Kogut ◽  
T.N. Rudyanova
Keyword(s):  
Convergence Property ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
Uniformly Bounded ◽  
Bounded Sequences

In this paper we study the H-convergence property for the uniformly bounded sequences of square matrices $\left\{ A_{\varepsilon} \in L^{\infty} (D; \mathbb{R}^{n \times n}) \right\}_{\varepsilon > 0}$. We derive the sufficient conditions, which guarantee the coincidence of $H$-limit with the weak-* limit of such sequences in $L^{\infty} (D; \mathbb{R}^{n \times n})$.

On the factorable spaces of absolutely p-summable, null, convergent, and bounded sequences

2021 ◽  
Vol 71 (6) ◽  
pp. 1375-1400
Author(s):  
Feyzi Başar ◽  
Hadi Roopaei
Keyword(s):  
Lower Bound ◽  
Sequence Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Matrix Transformations ◽  
Infinite Matrix ◽  
The Matrix ◽  
Alpha Beta ◽  
Necessary And Sufficient ◽  
Bounded Sequences

Abstract Let F denote the factorable matrix and X ∈ {ℓp , c 0, c, ℓ ∞}. In this study, we introduce the domains X(F) of the factorable matrix in the spaces X. Also, we give the bases and determine the alpha-, beta- and gamma-duals of the spaces X(F). We obtain the necessary and sufficient conditions on an infinite matrix belonging to the classes (ℓ p (F), ℓ ∞), (ℓ p (F), f) and (X, Y(F)) of matrix transformations, where Y denotes any given sequence space. Furthermore, we give the necessary and sufficient conditions for factorizing an operator based on the matrix F and derive two factorizations for the Cesàro and Hilbert matrices based on the Gamma matrix. Additionally, we investigate the norm of operators on the domain of the matrix F. Finally, we find the norm of Hilbert operators on some sequence spaces and deal with the lower bound of operators on the domain of the factorable matrix.

scholarly journals A Note on the Large Deviation Principle for Discrete Associated Random Variables

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Przemysław Matuła ◽  
Maciej Ziemba
Keyword(s):  
Large Deviation Principle ◽  
Large Deviation ◽  
Sufficient Conditions ◽  
Random Variables ◽  
Stationary Sequence ◽  
Arithmetic Means ◽  
Associated Random Variables ◽  
Partial Sum ◽  
Deviation Principle ◽  
Uniformly Bounded

We present sufficient conditions under which the sequence of arithmetic means Sn/n, where Sn=X1+⋯+Xn, is the partial sum built on a stationary sequence {Xn}n≥1 of associated integer-valued and uniformly bounded random variables, which satisfy the large deviation principle.

scholarly journals On Extended Branciari b -Distance Spaces and Applications to Fractional Differential Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Reena Jain ◽  
Hemant Kumar Nashine ◽  
Reny George ◽  
Zoran D. Mitrović
Keyword(s):  
Sufficient Conditions ◽  
Integral Boundary Conditions ◽  
Weak Limit ◽  
Nonlinear Fractional Differential Equation ◽  
Integral Boundary ◽  
Shadowing Property ◽  
Underlying Space ◽  
Limit Shadowing ◽  
Fractional Differential ◽  
Well Posed

In this work, we define new α − λ -rational contractive conditions and establish fixed-points results based on aforesaid contractive conditions for a mapping in extended Branciari b -distance spaces. We furnish two examples to justify the work. Further, we discuss results on weak well-posed property, weak limit shadowing property, and generalized w -Ulam-Hyers stability in the underlying space. Finally, as an application of our main result, we obtain sufficient conditions for the existence of solutions of a nonlinear fractional differential equation with integral boundary conditions.

scholarly journals Markov Chains Conditioned Never to Wait Too Long at the Origin

2009 ◽  
Vol 46 (03) ◽  
pp. 812-826
Author(s):  
Saul Jacka
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
State Space ◽  
Sufficient Conditions ◽  
Weak Limit ◽  
The State ◽  
Coin Tossing ◽  
First Time ◽  
Vague Convergence ◽  
Irreducible Markov Chain

Motivated by Feller's coin-tossing problem, we consider the problem of conditioning an irreducible Markov chain never to wait too long at 0. Denoting by τ the first time that the chain,X, waits for at least one unit of time at the origin, we consider conditioning the chain on the event (τ›T). We show that there is a weak limit asT→∞ in the cases where either the state space is finite orXis transient. We give sufficient conditions for the existence of a weak limit in other cases and show that we have vague convergence to a defective limit if the time to hit zero has a lighter tail than τ and τ is subexponential.

Comparison of optimal value and constrained maxima expectations for independent random variables

1986 ◽  
Vol 18 (02) ◽  
pp. 311-340
Author(s):  
Robert P. Kertz
Keyword(s):  
Random Variables ◽  
Poisson Approximation ◽  
Independent Random Variables ◽  
Problem Analysis ◽  
Universal Inequalities ◽  
Optimal Value ◽  
Approximation Type ◽  
Stop Rule ◽  
Uniformly Bounded ◽  
Bounded Sequences

For all uniformly bounded sequences of independent random variablesX1, X2,···, a complete comparison is made between the optimal valueV(X1, X2, ···) = sup {EXt:tis an (a.e.) finite stop rule forX1,X2, ···} and, whereMi(X1,X2, ···) is theith largest order statistic forX1, X2, ··· In particular, fork>1, the set of ordered pairs {(x,y):x=V(X1, X2,···) andfor some independent random variablesX1, X2, ··· taking values in [0, 1]} is precisely the set, whereBk(0) = 0,Bk(1) = 1, and forThe result yields sharp, universal inequalities for independent random variables comparing two choice mechanisms, the mortal&s value of the gameV(X1, X2,···) and the prophet&s constrained maxima expectation of the game. Techniques of proof include probability- and convexity-based reductions; calculus-based, multivariate, extremal problem analysis; and limit theorems of Poisson-approximation type. Precise results are also given for finite sequences of independent random variables.

scholarly journals Uniform convergence of conditional distributions for absorbed one-dimensional diffusions

2018 ◽  
Vol 50 (01) ◽  
pp. 178-203 ◽  
Author(s):  
Nicolas Champagnat ◽  
Denis Villemonais
Keyword(s):  
Sufficient Conditions ◽  
Initial Distribution ◽  
Initial Position ◽  
Local Martingale ◽  
One Dimensional ◽  
Positive Time ◽  
Strict Local Martingale ◽  
Necessary And Sufficient ◽  
Stationary Behavior ◽  
Uniformly Bounded

Abstract In this paper we study the quasi-stationary behavior of absorbed one-dimensional diffusions. We obtain necessary and sufficient conditions for the exponential convergence to a unique quasi-stationary distribution in total variation, uniformly with respect to the initial distribution. An important tool is provided by one-dimensional strict local martingale diffusions coming down from infinity. We prove, under mild assumptions, that their expectation at any positive time is uniformly bounded with respect to the initial position. We provide several examples and extensions, including the sticky Brownian motion and some one-dimensional processes with jumps.

scholarly journals Discrete quadratic estimates and holomorphic functional calculi in Banach spaces

1998 ◽  
Vol 58 (2) ◽  
pp. 271-290 ◽  
Author(s):  
Edwin Franks ◽  
Alan McIntosh
Keyword(s):  
Functional Calculus ◽  
Sufficient Conditions ◽  
Uniform Norm ◽  
Commuting Operators ◽  
Orthonormal Sequence ◽  
Quadratic Estimates ◽  
Holomorphic Functional Calculi ◽  
Necessary And Sufficient ◽  
Dyadic Decomposition ◽  
Uniformly Bounded

We develop a discrete version of the weak quadratic estimates for operators of type w explained by Cowling, Doust, McIntosh and Yagi, and show that analogous theorems hold. The method is direct and can be generalised to the case of finding necessary and sufficient conditions for an operator T to have a bounded functional calculus on a domain which touches σ(T) nontangentially at several points. For operators on Lp, 1 < p < ∞, it follows that T has a bounded functional calculus if and only if T satisfies discrete quadratic estimates. Using this, one easily obtains Albrecht's extension to a joint functional calculus for several commuting operators. In Hilbert space the methods show that an operator with a bounded functional calculus has a uniformly bounded matricial functional calculus.The basic idea is to take a dyadic decomposition of the boundary of a sector Sv. Then on the kth ingerval consider an orthonormal sequence of polynomials . For h ∈ H∞(Sν), estimates for the uniform norm of h on a smaller sector Sμ are obtained from the coefficients akj = (h, ek, j). These estimates are then used to prove the theorems.

scholarly journals A Mathematical Model with Pulse Effect for Three Populations of the Giant Panda and Two Kinds of Bamboo

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Xiang-yun Shi ◽  
Guo-hua Song
Keyword(s):  
Mathematical Model ◽  
Giant Panda ◽  
Floquet Theory ◽  
Sufficient Conditions ◽  
Giant Pandas ◽  
Impulsive Equations ◽  
The Relationship ◽  
Panda Habitat ◽  
Sudden Collapse ◽  
Uniformly Bounded

A mathematical model for the relationship between the populations of giant pandas and two kinds of bamboo is established. We use the impulsive perturbations to take into account the effect of a sudden collapse of bamboo as a food source. We show that this system is uniformly bounded. Using the Floquet theory and comparison techniques of impulsive equations, we find conditions for the local and global stabilities of the giant panda-free periodic solution. Moreover, we obtain sufficient conditions for the system to be permanent. The results provide a theoretical basis for giant panda habitat protection.

scholarly journals On uniformly bounded sequences in Orlicz spaces

1990 ◽  
Vol 41 (3) ◽  
pp. 495-502 ◽  
Author(s):  
Erik J. Balder
Keyword(s):  
Orlicz Spaces ◽  
Young Measures ◽  
Recent Extension ◽  
Uniformly Bounded ◽  
Bounded Sequences

A useful result for uniformly bounded sequences of functions in Orlicz spaces is generalised by means of a recent extension of Komlós' theorem. The same generalisation can also be proven differently, by means of Young measures.

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