Measures on independent sets, a quantitative version of Rudin's theorem

We construct measures with independent support whose Fourier coefficients decrease as fast as possible.

     

On some random thin sets of integers

We show how different random thin sets of integers may have different behaviour. First, using a recent deviation inequality of Boucheron, Lugosi and Massart, we give a simpler proof of one of our results in Some new thin sets of integers in harmonic analysis, Journal d'Analyse Mathématique 86 (2002), 105-138, namely that there exist -Rider sets which are sets of uniform convergence and -sets for all but which are not Rosenthal sets. In a second part, we show, using an older result of Kashin and Tzafriri, that, for , the -Rider sets which we had constructed in that paper are almost surely not of uniform convergence.

     

Intersection of sets with -connected unions

We show that if sets in a topological space are given so that all the sets are closed or all are open, and for each every of the sets have a -connected union, then the sets have a point in common. As a consequence, we obtain the following starshaped version of Helly's theorem: If every or fewer members of a finite family of closed sets in have a starshaped union, then all the members of the family have a point in common. The proof relies on a topological KKM-type intersection theorem.

     

Estimating the Hausdorff dimensions of univoque sets for self-similar sets

An approach is given for estimating the Hausdorff dimension of the univoque set of a self-similar set. This sometimes allows us to get the exact Hausdorff dimensions of the univoque sets.     

Exclusion sets in eigenvalue localization sets for tensors

ABSTRACT

By excluding some proper subsets, which do not include any eigenvalues of tensors, from an existing eigenvalue localization set provided by Zhao and Sang, a new eigenvalue localization set for tensors is given. As an application, a sufficient condition such that the determinant of a tensor is not zero is provided.     

Co-countable sets of uniqueness for series of independent random variables

Given a sequence of independent random variables (fk) on a standard Borel space Ω with probability measure μ and a measurable set F, the existence of a countable set S⊂F is shown, with the property that series k_∑ckfk which are constant on S are constant almost everywhere on F. As a consequence, if the functions fk are not constant almost everywhere, then there is a countable set S⊂Ω such that the only series k_∑ckfk which is null on S is the null series; moreover, if there exists b<1 such that for every k and every α, then the set S can be taken inside any measurable set F with μ(F)>b.     

The toric -vectors of partially ordered sets

An explicit formula for the toric -vector of an Eulerian poset in terms of the -index is developed using coalgebra techniques. The same techniques produce a formula in terms of the flag -vector. For this, another proof based on Fine's algorithm and lattice-path counts is given. As a consequence, it is shown that the Kalai relation on dual posets, , is the only equation relating the -vectors of posets and their duals. A result on the -vectors of oriented matroids is given. A simple formula for the -index in terms of the flag -vector is derived.

     

The number of Sidon sets and the maximum size of Sidon sets contained in a sparse random set of integers

A set A of non‐negative integers is called a Sidon set if all the sums , with and a₁, , are distinct. A well‐known problem on Sidon sets is the determination of the maximum possible size F(n) of a Sidon subset of . Results of Chowla, Erd?s, Singer and Turán from the 1940s give that . We study Sidon subsets of sparse random sets of integers, replacing the ‘dense environment’ by a sparse, random subset R of , and ask how large a subset can be, if we require that S should be a Sidon set. Let be a random subset of of cardinality , with all the subsets of equiprobable. We investigate the random variable , where the maximum is taken over all Sidon subsets , and obtain quite precise information on for the whole range of m, as illustrated by the following abridged version of our results. Let be a fixed constant and suppose . We show that there is a constant such that, almost surely, we have . As it turns out, the function is a continuous, piecewise linear function of a that is non‐differentiable at two ‘critical’ points: a = 1/3 and a = 2/3. Somewhat surprisingly, between those two points, the function is constant. Our approach is based on estimating the number of Sidon sets of a given cardinality contained in [n]. Our estimates also directly address a problem raised by Cameron and Erd?s (On the number of sets of integers with various properties, Number theory (Banff, AB, 1988), de Gruyter, Berlin, 1990, pp. 61–79). © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 1–25, 2015     

A construction of interpolating wavelets on invariant sets

We introduce the concept of a refinable set relative to a family of contractive mappings on a metric space, and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure. The notion of a refinable set parallels that of a refinable function, which is the basis of wavelet construction. The interpolation points we recursively generate from a refinable set by a set-theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction. We then use this recursive structure for the points to construct multiscale interpolants. Several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included.

     

On pasting -weights

Theorem 3 gives a condition when two -weights can be ``pasted' together to yield another -weight. It is subsequently used in Example 6 to give an example that shows that a necessary condition by Gohberg, Krupnik, and Spitkovsky is not sufficient.

     

一般随机富里叶级数

人们通常是在随机系数对称条件下，研究随机富里叶级数的，例文南[1]。本文不要对称性假设，研究一般随机富里叶级数一些性质，并得到一些重要成果。     

Stability of Julia sets for a quadratic random dynamical system

For a sequence (cⁿ) of complex numbers, the quadratic polynomials f_cn:= z² + C_n and the sequence (F_n) of iterates F_n: = f_cn ο ⋯ ο f_c1 are considered. The Fatou set F(C_n) is defined as the set of all such that (Fⁿ) is normal in some neighbourhood of z, while the complement J(C_n) of F(c_n) (in ) is called the Julia set. The aim of this paper is to study the stability of the Julia set J(C_n) in the case where (c_n) is bounded. A problem put forward by Brück is solved.     

A remark on intersection of convex sets

Let X be a real Banach space. Let be a family of closed, convex subsets of X. We show that either the intersection ?_γ∈Γ(G_γ)^? of the ?-neighborhood of the sets G_γ is bounded for each ?>0, or it is unbounded for each ?>0. From this we derive a fixed point theorem for suitable maps that move some points along a bounded direction in Hilbert spaces.     

Another construction for large sets of Kirkman triple systems

A large set of Kirkman triple systems of order v, denoted by LKTS(v), is a collection , where every is a KTS(v) and all form a partition of all triples on X. In this article, we give a new construction for LKTS(6v + 3) via OLKTS(2v + 1) with a special property and obtain new results for LKTS, that is there exists an LKTS(3v) for , where p, q ≥ 0, r _i , s _j ≥ 1, q _i is a prime power and mod 12.      

-unisolvent sets and flat incidence structures

For the past forty years or so topological incidence geometers and mathematicians interested in interpolation have been studying very similar objects. Nevertheless no communication between these two groups of mathematicians seems to have taken place during that time. The main goal of this paper is to draw attention to this fact and to demonstrate that by combining results from both areas it is possible to gain many new insights about the fundamentals of both areas. In particular, we establish the existence of nested orthogonal arrays of strength , for short nested -OAs, that are natural generalizations of flat affine planes and flat Laguerre planes. These incidence structures have point sets that are ``flat' topological spaces like the Möbius strip, the cylinder, and strips of the form , where is an interval of . Their circles (or lines) are subsets of the point sets homeomorphic to the circle in the first two cases and homeomorphic to in the last case. Our orthogonal arrays of strength arise from -unisolvent sets of half-periodic functions, -unisolvent sets of periodic functions, and -unisolvent sets of functions , respectively.

Associated with every point of a nested -OA, , is a nested -OA-the derived -OA at the point . We discover that, in our examples that arise from -unisolvent sets of times differentiable functions that solve the Hermite interpolation problem, deriving in our geometrical sense coincides with deriving in the analytical sense.

     

Level sets of a typical function

We determine the level set structure of a typical function.