Fourth-order difference equation for the first associated of classical discrete orthogonal polynomials

We derive the fourth-order difference equation satisfied by the first associated of classical orthogonal polynomials of a discrete variable. We give it explicitly for first associated of Hahn polynomials from which can be derived by a limiting process the equation satisfied by first associated of all classical families (continuous and discrete).     

An algorithm for the difference between set covers

A set cover for a set S is a collection C of special subsets whose union is S. Given covers A and B for two sets, the set-cover difference problem is to construct a new cover for the elements covered by A but not B. Applications include testing equivalence of set covers and maintaining a set cover dynamically. In this paper, we solve the set-cover difference problem by defining a difference operation A-B, which turns out to be a pseudocomplement on a distributive lattice. We give an algorithm for constructing this difference, and show how to implement the algorithm for two examples with applications in computer science: face covers on a hypercube, and rectangle covers on a grid. We derive an upper bound on the time complexity of the algorithm, and give upper and lower bounds on complexity for face covers and rectangle covers.     

Sharp bounds for the difference between the arithmetic and geometric means

We present sharp bounds for ${\sum_{i=1}^n \alpha_i x_i -\prod_{i=1}^n x_i^{\alpha_i} }$ in terms of the variance of the vector ${(x_1^{1/2},\dots,x_n^{1/2})}$ .     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

On the difference approximations of overdetermined hyperbolic equations of classical mathematical physics

Issues concerning difference approximations of overdetermined systems of hyperbolic equations are examined. The formulations of extended overdetermined systems are given for hydrodynamics equations, magnetohydrodynamics equations, Maxwell equations, and elasticity equations. Some approaches to the construction of difference schemes are discussed for these systems.     

Statistical extension of classical Tauberian theorems in the case of logarithmic summability

Let s: [1,∞) → ? be a locally integrable function in Lebesgue’s sense. The logarithmic (also called harmonic) mean of the function s is defined by $$\tau (t): = \frac{1} {{\log t}}\int_1^t {\frac{{s(x)}} {x}dx, t > 1,}$$ where the logarithm is to the natural base e. Besides the ordinary limit lim _x→∞ s(x), we use the notion of the so-called statistical limit of s at ∞, in notation: st-lim _x→∞ s(x) = l, by which we mean that for every ? > 0, $$\mathop {\lim }\limits_{b \to \infty } \frac{1} {b}\left| {\left\{ {x \in (1,b):\left| {s(x) - \ell } \right| > \varepsilon } \right\}} \right| = 0.$$ We also use the ordinary limit limt→∞ τ (t) as well as the statistical limit st-limt→∞ τ (t). We will prove the following Tauberian theorem: Suppose that the real-valued function s is slowly decreasing or the complex-valued s is slowly oscillating. If the statistical limit st-limtt→∞ τ (t) = l exists, then the ordinary limit limx→∞ s (x) = l also exists.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.     

On the difference between consecutive primes

Inventiones mathematicae -     

Correspondence between fuzzy measures and classical measures

In this paper we study the question whether, given a fuzzy measure (as defined in [3] and [4]). there exists a classical measure such that the fuzzy measure of a measurable fuzzy set μ equals the classical measure of the area below the membership function of μ. The results are that in the case of finite additivity there is a one-to-one correspondence between classical measures and fuzzy measures, whereas in the case of countable additivity this result only holds for generated fuzzy σ-algebras. Finally, some connections of that problem with the existence of an extension of a fuzzy measure defined on an arbitrary fuzzy σ-algebra σ to the generated fuzzy σ-algebra $\text{σ}$ are discussed.     

Bounds for the difference between median and mean of gamma and poisson distributions

We use elementary methods and the Poisson-Gamma relationship to obtain bounds for the difference between median and mean of Gamma and Poisson distributions.     

The relation between intuitionistic and classical modal logics

Intuitionistic propositional logicInt and its extensions, known as intermediate or superintuitionistic logics, in many respects can be regarded as just fragments of classical modal logics containingS4. The main aim of this paper is to construct a similar correspondence between intermediate logics augmented with modal operators—we call them intuitionistic modal logics—and classical polymodal logics We study the class of intuitionistic polymodal logics in which modal operators satisfy only the congruence rules and so may be treated as various sorts of □ and ◇. Supported by the Alexander von Humboldt Foundation. Translated fromAlgebra i Logika, Vol. 36, No. 2, pp. 121–155, March–April, 1997.