The shelahP-point independence theorem

In this paper, we present S. Shelah's example of a model of set theory in which there are noP-points in βN/N. This settles the famous open question: “Is ‘ZFC+there are noP-points in βN/N’ consistent?”     

The central limit theorem without the condition of independence

Necessary and sufficient conditions are presented for sums of asymptotically independent random variables to converge to a normal random variable in the sense of total variation distance, uniform metric for characteristic functions. and mean metric of order q. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models. Moscow. Russia. 1996. Part II.     

Brooks' graph-coloring theorem and the independence number

Let h denote the maximum degree of a connected graph H, and let _χ(H) denote its chromatic, number. Brooks' Theorem asserts that if h ≥ 3, then _χ(H) ≤ h, unless H is the complete graph K_h+1. We show that when H is not K_h+1, there is an h-coloring of H in which a maximum independent set is monochromatic. We characterize those graphs H having an h-coloring in which some color class consists of vertices of degree h in H. Again, without any loss of generality, this color class may be assumed to be maximum with respect to the condition that its vertices have degree h.     

A structure theorem on bivariate positive quadrant dependent distributions and tests for independence in two-way contingency tables

In this paper, the set of all bivariate positive quadrant dependent distributions with fixed marginals is shown to be compact and convex. Extreme points of this convex set are enumerated in some specific examples. Applications are given in testing the hypothesis of independence against strict positive quadrant dependence in the context of ordinal contingency tables. The performance of two tests, one of which is based on eigenvalues of a random matrix, is compared. Various procedures based upon certain functions of the eigenvalues of a random matrix are also proposed for testing for independence in a two-way contingency table when the marginals are random.     

A remark on strict independence relations

We prove that if T is a complete theory with weak elimination of imaginaries, then there is an explicit bijection between strict independence relations for T and strict independence relations for \({T^{\rm eq}}\). We use this observation to show that if T is the theory of the Fraïssé limit of finite metric spaces with integer distances, then \({T^{\rm eq}}\) has more than one strict independence relation. This answers a question of Adler (J Math Log 9(1):1–20, 2009, Question 1.7).     

An informational analog of the theorem of independence of sample mean and sample variance

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei — Trudy Seminara, pp. 63–67, 1985.     

A note on marginal and conditional independence

Some statistical models imply that two random vectors are marginally independent as well as being conditionally independent with respect to another random vector. When the joint distribution of the vectors is normal, canonical correlation analysis may lead to relevant simplifications of the dependence structure. A similar application can be found in elliptical models, where linear independence does not imply statistical independence. Implications for Bayes analysis of the general linear model are discussed.     

A theorem on equidistant codes

It is shown that a nontrivial code consisting of k²+k+2 words at mutual distance 2k (k > 1) exists if and only if there exists a projective plane of order k.     

A remark on Schottky's theorem

In this paper a new explicit upper bound in Schottky's theoremis given, which is sharp in a certain sense.     

A remark on banks theorem

ABSTRACT

For discrete autonomous dynamical systems, it was found that in the three conditions defining Devaney chaos, topological transitivity and dense periodic points together imply sensitive dependence on initial condition (Banks, Brooks, Cairns, Davis and Stacey, On Devaney's definition of chaos, The Amer. Math. Monthly 99 (1992), pp. 332–334). In this paper, we give the definition of finitely generated non-autonomous dynamical systems (NADS) and generalize the Banks et al. theorem to the finitely generated NADS.