Comparison of density topologies on the real line

2018 ◽  
Vol 68 (1) ◽  
pp. 173-180
Author(s):  
Renata Wiertelak
Keyword(s):  
Real Line ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
The Real ◽  
Closed Intervals ◽  
The Family ◽  
Necessary And Sufficient

Abstract In this paper will be considered density-like points and density-like topology in the family of Lebesgue measurable subsets of real numbers connected with a sequence 𝓙= {Jn}n∈ℕ of closed intervals tending to zero. The main result concerns necessary and sufficient condition for inclusion between that defined topologies.

scholarly journals The Topological Entropy of Cyclic Permutation Maps and Some Chaotic Properties on Their MPE sets

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Risong Li ◽  
Tianxiu Lu
Keyword(s):  
Dynamical System ◽  
Topological Entropy ◽  
Real Line ◽  
Cyclic Permutation ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Compact Subinterval ◽  
Dimensional Dynamical System ◽  
Necessary And Sufficient

In this paper, we study some chaotic properties of s-dimensional dynamical system of the form Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1, where ak∈Hk for any k∈1,2,…,s, s≥2 is an integer, and Hk is a compact subinterval of the real line ℝ=−∞,+∞ for any k∈1,2,…,s. Particularly, a necessary and sufficient condition for a cyclic permutation map Ψa1,a2,…,as=gsas,g1a1,…,gs−1as−1 to be LY-chaotic or h-chaotic or RT-chaotic or D-chaotic is obtained. Moreover, the LY-chaoticity, h-chaoticity, RT-chaoticity, and D-chaoticity of such a cyclic permutation map is explored. Also, we proved that the topological entropy hΨ of such a cyclic permutation map is the same as the topological entropy of each of the following maps: gj∘gj−1∘⋯∘g1l∘gs∘gs−1∘⋯∘gj+1, if j=1,…,s−1and gs∘gs−1∘⋯∘g1, and that Ψ is sensitive if and only if at least one of the coordinates maps of Ψs is sensitive.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (01) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t −1 ⨍0 t P 0(X s > 0) ds → c as t → ∞ is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds to converge in P 0 law to Fc. Moreover, P 0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t —1 ⨍0 t 1{Xs >0} ds under P 0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

On the arc-sine laws for Lévy processes

1994 ◽  
Vol 31 (1) ◽  
pp. 76-89 ◽  
Author(s):  
R. K. Getoor ◽  
M. J. Sharpe
Keyword(s):  
Random Walks ◽  
Real Line ◽  
Lévy Process ◽  
Elementary Proof ◽  
Levy Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Necessary And Sufficient ◽  
Spitzer's Theorem

Let X be a Lévy process on the real line, and let Fc denote the generalized arcsine law on [0, 1] with parameter c. Then t−1 ⨍0tP0(Xs > 0) ds → c as t → ∞ is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds to converge in P0 law to Fc. Moreover, P0(Xt > 0) = c for all t > 0 is a necessary and sufficient condition for t—1 ⨍0t1{Xs>0}ds under P0 to have law Fc for all t > 0. We give an elementary proof of these results, and show how to derive Spitzer's theorem for random walks in a simple way from the Lévy process version.

Binomial subsampling

2006 ◽  
Vol 462 (2068) ◽  
pp. 1181-1195 ◽  
Author(s):  
Carsten Wiuf ◽  
Michael P.H Stumpf
Keyword(s):  
Mathematical Biology ◽  
Power Series ◽  
Generating Functions ◽  
Binomial Distribution ◽  
Random Variable ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient ◽  
Finite Moments

In this paper, we discuss statistical families with the property that if the distribution of a random variable X is in , then so is the distribution of Z ∼Bi( X ,  p ) for 0≤ p ≤1. (Here we take Z ∼Bi( X ,  p ) to mean that given X = x ,  Z is a draw from the binomial distribution Bi( x ,  p ).) It is said that the family is closed under binomial subsampling. We characterize such families in terms of probability generating functions and for families with finite moments of all orders we give a necessary and sufficient condition for the family to be closed under binomial subsampling. The results are illustrated with power series and other examples, and related to examples from mathematical biology. Finally, some issues concerning inference are discussed.

A continuous generalization of the transversal property

1984 ◽  
Vol 95 (1) ◽  
pp. 21-23
Author(s):  
P. Komjáth
Keyword(s):  
Finite Subset ◽  
Measure Space ◽  
Choice Function ◽  
Finite Measure ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Sets ◽  
Real Numbers ◽  
One To One ◽  
Necessary And Sufficient

A transversal for a set-system is a one-to-one choice function. A necessary and sufficient condition for the existence of a transversal in the case of finite sets was given by P. Hall (see [4, 3]). The corresponding condition for the case when countably many countable sets are given was conjectured by Nash-Williams and later proved by Damerell and Milner [2]. B. Bollobás and N. Varopoulos stated and proved the following measure theoretic counterpart of Hall's theorem: if (X, μ) is an atomless measure space, ℋ = {Hi: i∈I} is a family of measurable sets with finite measure, λi (i∈I) are non-negative real numbers, then we can choose a subset Ti ⊆ Hi with μ(Ti) = λi and μ(Ti ∩ Ti′) = 0 (i ≠ i′) if and only if μ({U Hi: iεJ}) ≥ Σ{λi: iεJ}: for every finite subset J of I. In this note we generalize this result giving a necessary and sufficient condition for the case when I is countable and X is the union of countably many sets of finite measure.

The Inverse Eigenvalue Problem for Symmetric Doubly Arrow Matrices

2013 ◽  
Vol 444-445 ◽  
pp. 625-627
Author(s):  
Kan Ming Wang ◽  
Zhi Bing Liu ◽  
Xu Yun Fei
Keyword(s):  
Eigenvalue Problem ◽  
Special Kind ◽  
Inverse Eigenvalue Problem ◽  
Symmetric Matrices ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Inverse Eigenvalue ◽  
Necessary And Sufficient

In this paper we present a special kind of real symmetric matrices: the real symmetric doubly arrow matrices. That is, matrices which look like two arrow matrices, forward and backward, with heads against each other at the station, . We study a kind of inverse eigenvalue problem and give a necessary and sufficient condition for the existence of such matrices.

scholarly journals Dunkl harmonic analysis and fundamental sets of continuous functions on the unit sphere

2015 ◽  
Vol 6 (3) ◽  
Author(s):  
Roman A. Veprintsev
Keyword(s):  
Continuous Function ◽  
Harmonic Analysis ◽  
Unit Sphere ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Family ◽  
Necessary And Sufficient

AbstractWe establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊

On generalized inverses of matrices

1966 ◽  
Vol 62 (4) ◽  
pp. 673-677 ◽  
Author(s):  
M. H. Pearl
Keyword(s):  
Generalized Inverse ◽  
Sufficient Conditions ◽  
Maximal Rank ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
The Real ◽  
Finite Dimensional ◽  
The Matrix ◽  
Generalized Inverses Of Matrices ◽  
Necessary And Sufficient

The notion of the inverse of a matrix with entries from the real or complex fields was generalized by Moore (6, 7) in 1920 to include all rectangular (finite dimensional) matrices. In 1951, Bjerhammar (2, 3) rediscovered the generalized inverse for rectangular matrices of maximal rank. In 1955, Penrose (8, 9) independently rediscovered the generalized inverse for arbitrary real or complex rectangular matrices. Recently, Arghiriade (1) has given a set of necessary and sufficient conditions that a matrix commute with its generalized inverse. These conditions involve the existence of certain submatrices and can be expressed using the notion of EPr matrices introduced in 1950 by Schwerdtfeger (10). The main purpose of this paper is to prove the following theorem:Theorem 2. A necessary and sufficient condition that the generalized inverse of the matrix A (denoted by A+) commute with A is that A+ can be expressed as a polynomial in A with scalar coefficients.

New Classes of Statistically Pre-Cauchy Triple Sequences of Fuzzy Numbers Defined by Orlicz Function

2018 ◽  
Vol 85 (3-4) ◽  
pp. 411
Author(s):  
Sangita Saha ◽  
Santanu Roy
Keyword(s):  
Cauchy Sequence ◽  
Fuzzy Numbers ◽  
Orlicz Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Fuzzy Real Numbers ◽  
Sequences Of Fuzzy Numbers ◽  
Necessary And Sufficient

In this article, the concept of statistically pre-Cauchy sequence of fuzzy real numbers having multiplicity greater than two defined by Orlicz function is introduced. A characterization of the class of bounded statistically pre-Cauchy triple sequences of fuzzy numbers with the help of Orlicz function is presented. Then a necessary and suffcient condition for a bounded triple sequence of fuzzy real numbers to be statistically pre-Cauchy is proved. Also a necessary and sufficient condition for a bounded triple sequence of fuzzy real numbers to be statistically convergent is derived. Further, a characterization of the class of bounded statistically convergent triple sequences of fuzzy numbers is presented and linked with Cesaro summability.

scholarly journals On sequences of expected maxima and expected ranges

2017 ◽  
Vol 54 (4) ◽  
pp. 1144-1166
Author(s):  
Nickos Papadatos
Keyword(s):  
Bernstein Function ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Real Numbers ◽  
Necessary And Sufficient

Abstract We investigate conditions in order to decide whether a given sequence of real numbers represents expected maxima or expected ranges. The main result provides a novel necessary and sufficient condition, relating an expected maxima sequence to a translation of a Bernstein function through its Lévy–Khintchine representation.

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