On the Borel-Cantelli Problem

Let (Ω, , P) be a probability space, and A1, A2… be a sequence of members of . The classical Borel-Cantelli problem is to determine the probability that infinitely many events Ak occur. The classical results may be found in Feller [2, p. 188]; while related work may be found in Spitzer [3, p. 317], and Dawson and Sankoff [1]. The latter works are generalizations of the Borel-Cantelli lemmas, taken in different directions.In this paper, necessary and sufficient conditions will be given for infinitely many events Ak to occur, with probability 1. A lower bound for the probability that only finitely many Ak occur, is developed. In addition, necessary and sufficient conditions that only finitely many Ak occur, with probability 1, are given.

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

Mathematica Slovaca ◽

10.1515/ms-2021-0059 ◽

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.

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A Network-Flow Feasibility Theorem and Combinatorial Applications

Canadian Journal of Mathematics ◽

10.4153/cjm-1959-045-1 ◽

1959 ◽

Vol 11 ◽

pp. 440-451 ◽

Cited By ~ 21

Author(s):

D. R. Fulkerson

Keyword(s):

Lower Bound ◽

Upper Bound ◽

Network Flow ◽

Present Author ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Private Communication ◽

Circulatory Flow ◽

Necessary And Sufficient ◽

Circulation Theorem

There are a number of interesting theorems, relative to capacitated networks, that give necessary and sufficient conditions for the existence of flows satisfying constraints of various kinds. Typical of these are the supply-demand theorem due to Gale (4), which states a condition for the existence of a flow satisfying demands at certain nodes from supplies at other nodes, and the Hoffman circulation theorem (received by the present author in private communication), which states a condition for the existence of a circulatory flow in a network in which each arc has associated with it not only an upper bound for the arc flow, but a lower bound as well. If the constraints on flows are integral (for example, if the bounds on arc flows for the circulation theorem are integers), it is also true that integral flows meeting the requirements exist provided any flow does so.

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Joint ergodicity for group actions

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700004508 ◽

1988 ◽

Vol 8 (3) ◽

pp. 351-364 ◽

Cited By ~ 4

Author(s):

Vitaly Bergelson ◽

Joseph Rosenblatt

Keyword(s):

Probability Space ◽

Amenable Group ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Compact Sets ◽

Weakly Mixing ◽

Image Position ◽

Necessary And Sufficient ◽

Continuous Representations ◽

Measure Preserving Transformations

AbstractLet T1,…,Tn be continuous representations of a σ-compact separable locally compact amenable group G as measure-preserving transformations of a non-atomic separable probability space (X, β, m). Let (Kn) be a right Følner sequence of compact sets in G. If T1,…,Tn are pairwise commuting in the sense that Ti(g)Tj(h) = Tj(h)Ti(g) for i ≠ j and g, h ∈ G, then necessary and sufficient conditions can be given, in terms of the ergodicity of certain tensor products, for the following to hold: for all F1,…,Fn∈L∞, the sequence AN(x) whereconverges in L2(X) to . The necessary and sufficient conditions are that each of the following representations are ergodic: Tn, Tn−1⊗Tn−1Tn,…,T2⊗T2T3⊗…⊗T2…Tn, T1⊗T1T2⊗…⊗T1…Tn.In order to prove this theorem, specific properties of the decomposition of L2(X) into its weakly mixing and compact subspaces with respect to a representation Ti are needed. These properties are also used to prove some generalizations of wellknown facts from ergodic theory in the case where G is the integer group Z.

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Necessary and sufficient conditions for a maximal ergodic theorem along subsequences

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700003953 ◽

1987 ◽

Vol 7 (2) ◽

pp. 203-210 ◽

Cited By ~ 2

Author(s):

Roger L. Jones

Keyword(s):

Ergodic Theorem ◽

Maximal Function ◽

Probability Space ◽

Weak Type ◽

Sufficient Conditions ◽

Ergodic Measure ◽

Point Transformation ◽

Necessary And Sufficient Conditions ◽

Necessary And Sufficient ◽

Positive Integers

AbstractLet T be an ergodic measure preserving point transformation from a probability space X onto itself. Assume that is an increasing sequence of subsets of the positive integers. Conditions are given which are sufficient for the ergodic maximal function associated with these subsets to be weak type (p, p). These conditions are shown to be both necessary and sufficient for a larger two-sided maximal function. The conditions are in the form of covering lemmas for the integers.

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On γ-Sets in Rings

European Journal of Pure and Applied Mathematics ◽

10.29020/nybg.ejpam.v14i1.3873 ◽

2021 ◽

Vol 14 (1) ◽

pp. 314-326

Author(s):

Eva Jenny C. Sigasig ◽

Cristoper John S. Rosero ◽

Michael Jr. Patula Baldado

Keyword(s):

Lower Bound ◽

Division Ring ◽

Upper Bound ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Minimum Cardinality ◽

Division Rings ◽

Necessary And Sufficient

Let R be a ring with identity 1R. A subset J of R is called a γ-set if for every a ∈ R\J,there exist b, c ∈ J such that a+b = 0 and ac = 1R = ca. A γ-set of minimum cardinality is called a minimum γ-set. In this study, we identified some elements of R that are necessarily in a γ-sets, and we presented a method of constructing a new γ-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique γ-set; an upper bound for the total number of minimum γ-sets in a division ring; a lower bound for the total number of minimum γ-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial γ-set; necessary and sufficient conditions for an image of a γ-set to be a γ-set also; necessary and sufficient conditions for a ring to have a trivial γ-set; and, necessary and sufficient conditions for the families of γ-sets of two division rings to be isomorphic.

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On size multipartite Ramsey numbers for stars

Indonesian Journal of Combinatorics ◽

10.19184/ijc.2019.3.2.4 ◽

2020 ◽

Vol 3 (2) ◽

pp. 109

Author(s):

Anie Lusiani ◽

Edy Tri Baskoro ◽

Suhadi Wido Saputro

Keyword(s):

Lower Bound ◽

Disjoint Union ◽

Sufficient Conditions ◽

Necessary And Sufficient Conditions ◽

Ramsey Number ◽

Complete Graphs ◽

Ramsey Numbers ◽

Simple Graphs ◽

Multipartite Graphs ◽

Necessary And Sufficient

Burger and Vuuren defined the size multipartite Ramsey number for a pair of complete, balanced, multipartite graphs mj(Kaxb,Kcxd), for natural numbers a,b,c,d and j, where a,c >= 2, in 2004. They have also determined the necessary and sufficient conditions for the existence of size multipartite Ramsey numbers mj(Kaxb,Kcxd). Syafrizal et al. generalized this definition by removing the completeness requirement. For simple graphs G and H, they defined the size multipartite Ramsey number mj(G,H) as the smallest natural number t such that any red-blue coloring on the edges of Kjxt contains a red G or a blue H as a subgraph. In this paper, we determine the necessary and sufficient conditions for the existence of multipartite Ramsey numbers mj(G,H), where both G and H are non complete graphs. Furthermore, we determine the exact values of the size multipartite Ramsey numbers mj(K1,m, K1,n) for all integers m,n >= 1 and j = 2,3, where K1,m is a star of order m+1. In addition, we also determine the lower bound of m3(kK1,m, C3), where kK1,m is a disjoint union of k copies of a star K1,m and C3 is a cycle of order 3.

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Convergence for moving averages

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700005381 ◽

1990 ◽

Vol 10 (1) ◽

pp. 43-62 ◽

Cited By ~ 21

Author(s):

Alexandra Bellow ◽

Roger Jones ◽

Joseph Rosenblatt

Keyword(s):

Probability Space ◽

Sufficient Conditions ◽

Ergodic Measure ◽

Point Transformation ◽

Necessary And Sufficient Conditions ◽

Moving Averages ◽

Averaging Operators ◽

Almost Everywhere ◽

Necessary And Sufficient ◽

Positive Integers

AbstractAssume T is an ergodic measure preserving point transformation from a probability space onto itself. Let be a sequence of pairs of positive integers, and define the sequence of averaging operators . Necessary and sufficient conditions are given forthis sequence of averages to converge almost everywhere. Weighted versions are also considered.

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

Journal of Applied Probability ◽

10.1017/s0021900200116031 ◽

1986 ◽

Vol 23 (04) ◽

pp. 851-858 ◽

Cited By ~ 1

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.

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