Riesz-type inequalities on general sets

Sharp Riesz–Bernstein-type inequalities are proven for the derivatives of algebraic polynomials on general subsets of unit circle. The sharp Riesz–Bernstein constant involves the equilibrium density of the set in question.     

On the trace of finite sets

For a family $\text{T}$ of subsets of an n-set X we define the trace of it on a subset Y of X by T_$\text{T}$(Y) = {F∩Y:F?$\text{T}$}. We say that (m,n) → (r,s) if for every $\text{T}$ with |$\text{T| ?m}$ we can find a Y?X|Y| = s such that |T_$\text{T}$(Y)| ? r. We give a unified proof for results of Bollobàs, Bondy, and Sauer concerning this arrow function, and we prove a conjecture of Bondy and Lovász saying $\text{(?}\text{n}{}^2\text{4}\text{? + n + 2,n)→ (3,7)}$, which generalizes Turán's theorem on the maximum number of edges in a graph not containing a triangle.     

Approximation of general zero-range potentials

A norm resolvent convergence result is proved for approximations of general Schrodinger operators with zero-range potentials. An approximation of the δ’-interaction by nonlocal non-Hermitian potentials (without a renormalization of the coupling constant) is also constructed.     

The Janson inequalities for general up‐sets

Janson and Janson, ?uczak and Ruciński proved several inequalities for the lower tail of the distribution of the number of events that hold, when all the events are up‐sets (increasing events) of a special form—each event is the intersection of some subset of a single set of independent events (i.e., a principal up‐set). We show that these inequalities in fact hold for arbitrary up‐sets, by modifying existing proofs to use only positive correlation, avoiding the need to assume positive correlation conditioned on one of the events. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 391–395, 2015     

Transcendental trace formulas for finite-gap potentials

We show that formulas differing from classical analogues of rational trace formulas for algebraic-geometric potentials occur in the theory of finite-gap integration of spectral equations. The new formulas contain transcendental modular functions and hypergeometric series. They result in transcendental relations for theta functions.     

Support functions of general convex sets

No Abstract. .In Memory of Gian-Carlo Rota     

Universality and fine zero spacing on general sets

A recent approach of D. S. Lubinsky yields universality in random matrix theory and fine zero spacing of orthogonal polynomials under very mild hypothesis on the weight function, provided the support of the generating measure μ is [-1,1]. This paper provides a method with which analogous results can be proven on general compact subsets of R. Both universality and fine zero spacing involves the equilibrium measure of the support of μ. The method is based on taking polynomial inverse images, by which results can be transferred from [-1,1] to a system of intervals, and then to general sets.     

The base sets of quasi-primitive zero-symmetric sign pattern matrices with zero trace

Cheng and Liu [Bo Cheng, Bolian Liu, The base sets of primitive zero-symmetric sign pattern matrices, Linear Algebra Appl. 428 (2008) 715-731] showed that the base set of quasi-primitive zero-symmetric (generalized) sign pattern matrices is {1,2,…,2n}. The matrices with zero trace play a prominent role in matrix theory. In this paper, we investigate the bases of quasi-primitive zero-symmetric (generalized) sign pattern matrices with zero trace and prove that the base set of such matrices is {2,3,…,2n-1}.     

A convex sets in general position

The notion of convex cones in general position has turned out to be useful in convex programming theory. In this paper we extend the notion to convex sets and give some characterizations which yield a better insight into this concept. We also consider the case of convex sets in S-general position.     

The Hausdorff dimension of sets related to the general Sierpinski carpets

In this paper we study a class of subsets of the general Sierpinski carpets for which the allowed two digits in the expansions occur with proportional frequency. We calculate the Hausdorff and box dimensions of these subsets and give necessary and sufficient conditions for the corresponding Hausdorff measure to be positive finite.     

A general theory of hypersurface potentials

Summary A general theory of hypersurface potentials in n-dimensional space is proposed. Not only smooth densities but also potentials generated either by L ¹ functions or by measures are considered.     

Non-isotropic hausdorff capacity of exceptional sets of invariant potentials

In the paper we investigate tangential boundary limits of invariant Green potentials on the unit ballB in ? ⁿ ,n≥1. LetG(z, w) denote the Green function for the Laplace-Beltrami operator onB, and let λ denote the invariant measure onB. If μ is a non-negative measure, orf is a non-negative measurable function onB,G _μ andG _f denote the Green potential of μ andf respectively. For ξ∈S=δB, τ≥1, andc>0, let $$\mathcal{T}_{\tau ,c} (\zeta ) = \{ z \in B:\left| {1 - \left\langle {z,\xi } \right\rangle } \right|^\tau< c(1 - \left| z \right|^2 )\} $$ . The main result of the paper is as follows: Letf be a non-negative measurable function onB satisfying $$\int_B {(1 - \left| w \right|^2 )^\beta f^p (w)d\lambda (w)< \infty } $$ for some β, 0<β<n, and somep>n. Then for each τ, 1≤τ<n/β, there exists a setE _t ?S withH _βτ (E _t )=0, such that $$\mathop {\lim }\limits_{\mathop {z \to \zeta }\limits_{z \in \mathcal{T}_{\tau ,c} (\zeta )} } G_f (z) = 0,forall\zeta \in S \sim E_\tau $$ In the above, for 0<α≤n,H _α denotes the non-isotropic α-dimensional Hausdorff capacity onS. We also prove that if {a _k } is a sequence inB satisfying Σ(1?|a _k |²) ^β <∞ for some β, 0 <β<n, and μ=Σδ _ak , where δ _a denotes point mass measure ata, then the same conclusion holds for the potentialG _μ .     

A general duality result for precompact sets

In this paper we present an elementary proof of a general duality result for precompact sets which can be considered as a far-reaching generalization of a well-known result of Grothendieck on precompactness in dual systems. It is then shown that a number of known results can be deduced from it, amongst others a general form of the Arzela-Ascoli theorem and Grothendieck's duality theorem itself.     

Mathematical morphology on bipolar fuzzy sets: general algebraic framework

In many domains of information processing, bipolarity is a core feature to be considered: positive information represents what is possible or preferred, while negative information represents what is forbidden or surely false. If the information is moreover endowed with vagueness and imprecision, as is the case for instance in spatial information processing, then bipolar fuzzy sets constitute an appropriate knowledge representation framework. In this paper, we focus on mathematical morphology as a tool to handle such information and reason on it. Applying mathematical morphology to bipolar fuzzy sets requires defining an appropriate lattice. We extend previous work based on specific partial orderings to any partial ordering leading to a complete lattice. We address the case of algebraic operations and of operations based on a structuring element, and show that they have good properties for any partial ordering, and that they can be useful for processing in particular spatial information, but also other types of bipolar information such as preferences and constraints. Particular cases using Pareto and lexicographic orderings are illustrated. Operations derived from fuzzy bipolar erosion and dilation are proposed as well.     

Finding all closed sets: A general approach

We present a unifying theoretical and algorithmic approach to the problems to determine all closed sets of a closure operator, to do this up to isomorphism, and to determine the elements of certain ideals of a power set. This will be done by generalizing the concept of closure operators using the interplay of several orders of a power set.