A central limit theorem for martingales and an application to branching processes

A functional central limit theorem is obtained for martingales which are not uniformly asymptotically negligible but grow at a geometric rate. The function space is not the usual C[0,1] or D[0,1] but R^N, the space of all real sequences and the metric used leads to a non-separable metric space.The main theorem is applied to a martingale obtained from a supercritical Galton-Watson branching process and as simple corollaries the already known central limit theorems for the Harris and Lotka-Nagaev estimators of the mean of the offspring distribution, are obtained.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous functiong ₁ εC ₀[0,1]² with support in the rectangle [0,1]×[0,1/2] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1]×[1/2,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer.     

Finite Exchangeability, Lévy-Frailty Copulas and Higher-Order Monotonic Sequences

This paper deals with a surprising connection between exchangeable distributions on {0,1} ⁿ and the recently introduced Lévy-frailty copulas, the link being provided by a new class of multivariate distribution functions called linearly order symmetric. The characterisation theorem for Lévy-frailty copulas is given a new and short (non-combinatorial) proof, and a related result is shown for exchangeable Marshall–Olkin distributions. A common thread in all these considerations is higher-order monotonic functions on integer intervals of the form {0,1,…,n}.     

Intersection theorems under dimension constraints

In Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21] we posed a series of extremal (set system) problems under dimension constraints. In the present paper, we study one of them: the intersection problem. The geometrical formulation of our problem is as follows. Given integers 0?t, k?n determine or estimate the maximum number of (0,1)-vectors in a k-dimensional subspace of the Euclidean n-space Rⁿ, such that the inner product (“intersection”) of any two is at least t. Also we are interested in the restricted (or the uniform) case of the problem; namely, the problem considered for the (0,1)-vectors of the same weight ω.The paper consists of two parts, which concern similar questions but are essentially independent with respect to the methods used.In Part I, we consider the unrestricted case of the problem. Surprisingly, in this case the problem can be reduced to a weighted version of the intersection problem for systems of finite sets. A general conjecture for this problem is proved for the cases mentioned in Ahlswede et al. [Discrete Math. 273(1-3) (2003) 9-21]. We also consider a diametric problem under dimension constraint.In Part II, we study the restricted case and solve the problem for t=1 and k<2ω, and also for any fixed 1?t?ω and k large.     

NONEXISTENCE AND EXISTENCE OF POSITIVE SOLUTIONS FOR 2nTH-ORDER SINGULAR SUPERLINEAR PROBLEMS WITH STRUM-LIOUVILLE BOUNDARY CONDITIONS

This paper investigates a class of 2nth-order singular superlinear problems with Strum-Liouville boundary conditions. We obtain a necessary and sufficient condition for the existence of C 2 n- 2 [0, 1] positive solutions, and a sufficient condition, a necessary condition for the existence of C 2 n-1 [0, 1] positive solutions. Relations between the positive solutions and the Green’s functions are depicted. The results are used to judge nonexistence or existence of positive solutions for given boundary value problems.     

Quantitative relation between noise sensitivity and influences

A Boolean function f: {0,1} ⁿ → {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ _p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ^?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand’s lemma, which easily generalizes in various directions, including non-monotone functions.     

A characterization of truth-functions in the nilpotent minimum logic

By introducing a new family of partitions into the n-cube [0,1]ⁿ, the problem of characterizing truth tables of formulas in the nilpotent minimum logic is solved and their normal forms are presented. So far, only this kind of fuzzy truth functions have normal forms among all fuzzy propositional calculi which are based on left-continuous but discontinuous t-norm.     

Divergenzverhalten mehrdimensionaler Shannonscher Abtastreihen

The behaviour of multidimensional Shannon sampling series for continuous functions is examined. A continuous function g ₁∈C ₀ [0,1]² with support in the rectangle [0,1] × [0,?] is indicated in the paper for which the two dimensional Shannon sampling series diverge almost everywhere in the rectangle [0,1] × [?,1]. This shows that the localization principle for Shannon sampling series cannot hold in two dimensions and in higher dimensions. The result solves a problem formulated by P.L. Butzer. Received: 21 December 1995 / Revised version: 5 October 1996     

A Characterization of the free n-generated MV-algebra

An MV-algebra A=(A,0,¬,⊕) is an abelian monoid (A,0,⊕) equipped with a unary operation ¬ such that ¬¬x=x,x⊕¬0=¬0, and y⊕¬(y⊕¬x)=x⊕¬(x⊕¬y). Chang proved that the equational class of MV-algebras is generated by the real unit interval [0,1] equipped with the operations ¬x=1?x and x⊕y=min(1,x+y). Therefore, the free n-generated MV-algebra Free _n is the algebra of [0,1]-valued functions over the n-cube [0,1] ⁿ generated by the coordinate functions ξ _i ,i=1, . . . ,n, with pointwise operations. Any such function f is a McNaughton function, i.e., f is continuous, piecewise linear, and each piece has integer coefficients. Conversely, McNaughton proved that all McNaughton functions f: [0,1] ⁿ →[0,1] are in Free _n . The elements of Free _n are logical equivalence classes of n-variable formulas in the infinite-valued calculus of ?ukasiewicz. The aim of this paper is to provide an alternative, representation-free, characterization of Free _n .     

nth order Stieltjes differential boundary operators and Stieltjes differential boundary systems

This article discusses linear differential boundary systems, which include nth-order differential boundary relations as a special case, in $\text{L}$_n^p[0,1] × $\text{L}$_n^p[0,1], 1 ? p < ∞. The adjoint relation in $\text{L}$_n^q[0,1] × $\text{L}$_n^q[0,1], $\text{1}\text{p}\text{+}\text{1}\text{q}\text{= 1}$, is derived. Green's formula is also found. Self-adjoint relations are found in $\text{L}$_n²[0,1] × $\text{L}$_n²[0,1], and their connection with Coddington's extensions of symmetric operators on subspaces of $\text{L}$_n^p[0,1] × $\text{L}$_n²[0,1] is established.     

Well-posedness of the martingale problem for some degenerate diffusion processes occurring in dynamics of populations

In this paper we establish the well-posedness in C([0,∞);[0,1]^d), for each starting point x∈[0,1]d, of the martingale problem associated with a class of degenerate elliptic operators which arise from the dynamics of populations as a generalization of the Fleming-Viot operator. In particular, we prove that such degenerate elliptic operators are closable in the space of continuous functions on [0,1]^d and their closure is the generator of a strongly continuous semigroup of contractions.     

Counting substructures I: Color critical graphs

Let F be a graph which contains an edge whose deletion reduces its chromatic number. We prove tight bounds on the number of copies of F in a graph with a prescribed number of vertices and edges. Our results extend those of Simonovits (1968) [8], who proved that there is one copy of F, and of Rademacher, Erd?s (1962) [1] and [2] and Lovász and Simonovits (1983) [4], who proved similar counting results when F is a complete graph.One of the simplest cases of our theorem is the following new result. There is an absolute positive constant c such that if n is sufficiently large and 1?q<cn, then every n vertex graph with ⌊n²/4⌋+q edges contains at least

     

On the unimodality of multivariate symmetric distribution functions of class L

A theorem is proved that characterizes multivariate distribution functions of class L. This theorem is used to show that every n-dimensional, symmetric distribution function of class L is unimodal in the sense of Kanter.     

Global classical large solutions to 1D compressible Navier-Stokes equations with density-dependent viscosity and vacuum

In this paper, we investigate an initial boundary value problem for 1D compressible isentropic Navier-Stokes equations with large initial data, density-dependent viscosity, external force, and vacuum. Making full use of the local estimates of the solutions in Cho and Kim (2006) [3] and the one-dimensional properties of the equations and the Sobolev inequalities, we get a unique global classical solution (ρ,u) where ρ∈C¹([0,T];H¹([0,1])) and u∈H¹([0,T];H²([0,1])) for any T>0. As it is pointed out in Xin (1998) [31] that the smooth solution (ρ,u)∈C¹([0,T];H³(R¹)) (T is large enough) of the Cauchy problem must blow up in finite time when the initial density is of nontrivial compact support. It seems that the regularities of the solutions we obtained can be improved, which motivates us to obtain some new estimates with the help of a new test function ρ²utt, such as Lemmas 3.2-3.6. This leads to further regularities of (ρ,u) where ρ∈C¹([0,T];H³([0,1])), u∈H¹([0,T];H³([0,1])). It is still open whether the regularity of u could be improved to C¹([0,T];H³([0,1])) with the appearance of vacuum, since it is not obvious that the solutions in C¹([0,T];H³([0,1])) to the initial boundary value problem must blow up in finite time.     

On multifractality and time subordination for continuous functions

Let Z:[0,1]→R be a continuous function. This paper relates to the existence of a decomposition of Z as Z=g○f, where g:[0,1]→R is a monofractal function with exponent 0<H<1 and f:[0,1]→[0,1] is a time subordinator, i.e. the integral of a positive Borel measure supported by [0,1]. An equivalent question consists of searching for a (multifractal) parametrization of Z which transforms Z into a monofractal function. We establish that such a decomposition can be found for a large class of functions which includes the usual examples of multifractal functions.We find an interesting relationship between self-similar functions and self-similar measures as an application of our results.Our theorems yield new insights in the understanding of the multifractal behaviour of functions, giving a significant role to the regularity analysis of Borel measures.     

Internal normality and internal compactness

This text contains an example which presents a way to modify any Dowker space to get a normal space X such that X×[0,1] is not κ-normal, and a theorem implying the existence of a non-Tychonoff space which is internally compact in a larger regular space. It gives answers to several questions by Arhangel'skii [A.V. Arhangel'skii, Relative normality and dense subspaces, Topology Appl. 123 (2002) 27-36].     

One-dimensional geometric random graphs with nonvanishing densities II: a very strong zero-one law for connectivity

We consider a collection of n independent points which are distributed on the unit interval [0,1] according to some probability distribution function F. Two nodes communicate with each other if their distance is less than some given threshold value. When F admits a density f which is strictly positive on [0,1], we give conditions on f under which the property of graph connectivity for the induced geometric random graph obeys a very strong zero–one law when the transmission range is scaled appropriately with n large. The very strong critical threshold is identified. This is done by applying a version of the method of first and second moments.     

Left mean-ergodicity,fixed points,and invariant means

Recently Lau [15] generalized a result of Yeadon [25]. In the present paper we generalize Yeadon's result in another direction recasting it as a theorem of ergodic type. We call the notion of ergodicity required left mean-ergodicity and show how it relates to the mean-ergodicity of Nagel [21]. Connections with the existence of invariant means on spaces of continuous functions on semitopological semigroups S are made, connections concerning, among other things, a fixed point theorem of Mitchell [20] and Schwartz's property P of W¹-algebras [22]. For example, if $\text{M}$(S) is a certain subspace of C(S) (which was considered by Mitchell and is of almost periodic type, i.e., the right translates of a member of $\text{M}$(S) satisfy a compactness condition), then the assumption that $\text{M}$(S) has a left invariant mean is equivalent to the assumption that every representation of S of a certain kind by operators on a linear topological space X is left mean-ergodic. An analog involving the existence of a (left and right) invariant mean on $\text{M}$(S) is given, and we show our methods restrict in the Banach space setting to give short direct proofs of some results in [4], results involving the existence of an invariant mean on the weakly almost periodic functions on S or on the almost periodic functions on S. An ergodic theorem of Lloyd [16] is generalized, and a number of examples are presented.     

Insertion of lattice-valued and hedgehog-valued functions

Problems of inserting lattice-valued functions are investigated. We provide an analogue of the classical insertion theorem of Lane [Proc. Amer. Math. Soc. 49 (1975) 90-94] for L-valued functions where L is a ?-separable completely distributive lattice (i.e. L admits a countable join-dense subset which is free of completely join-irreducible elements). As a corollary we get an L-version of the Katětov-Tong insertion theorem due to Liu and Luo [Topology Appl. 45 (1992) 173-188] (our proof is different and much simpler). We show that ?-separable completely distributive lattices are closed under the formation of countable products. In particular, the Hilbert cube is a ?-separable completely distributive lattice and some join-dense subset is shown to be both order and topologically isomorphic to the hedgehog J(ω) with appropriately defined topology. This done, we deduce an insertion theorem for J(ω)-valued functions which is independent of that of Blair and Swardson [Indian J. Math. 29 (1987) 229-250]. Also, we provide an iff criterion for inserting a pair of semicontinuous function which yields, among others, a characterization of hereditarily normal spaces.