A system of functional equations related to plurality functions. A method for the construction of the solutions

Summary In the first part of the present paper we prove some necessary conditions satisfied by the solutions of a system of functional equations related to Plurality Functions. In the second part we describe a geometric-combinatorial procedure for the construction of the solutions of that system. This procedure yields all possible solutions.     

Construction of functions with uniform sublevel sets

In this paper, we study extended real-valued functions with uniform sublevel sets. The sublevel sets are defined by a linear shift of a set in a specified direction. We prove that the class of these functions coincides with the class of Gerstewitz functionals. In this way, we obtain a formula for the construction of such functions. The sublevel sets of Gerstewitz functionals are characterized and illustrated by examples. The results contain statements for translative functions, which are just the functions with uniform sublevel sets considered. The investigated functions are defined on an arbitrary real vector space without assuming any topology or convexity.     

Special functions related to various types of root systems

This paper describes methods that allow one to obtain special functions and study their properties. Using the differential-difference operator theory developed by Ch. Dunkl, the author constructs the family of functions associated with the root system of type G ₂. It is shown that it contains the subset consisting of the well-known Gegenbauer polynomials. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 54, Suzdal Conference–2006, Part 2, 2008.     

Generalized Cheeger sets related to landslides

We study the maximization problem, among all subsets X of a given domain , of the quotient of the integral in X of a given function f by the integral on the boundary of X of another function g. This is a generalization of the well-known Cheeger problem corresponding to constant functions f,g. The non-constant case is motivated by applications to landslides modeling where the the supremum given by a variational blocking problem appears as a safety coefficient. We prove that this coefficient is equal to the supremum of the shape optimization problem formerly mentioned. For constant data, this amounts to studying the first eigenvalue of the 1-laplacian operator.We prove existence of optimal sets, and give some differential characterization of their internal boundary. We study their symmetry properties using the Steiner symmetrization. In dimension two, we give explicit solutions for data depending only on one variable.

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Résumé. Nous étudions le probléme de maximisation, parmi les ensembles X inclus dans un domaine fixé , du quotient de lintégrale dune fonction donnée f dans X par lintégrale dune autre fonction g sur le bord de X. Il sagit donc dune généralisation du célébre probléme de Cheeger (correspondant au cas f, g, constants). Le cas non-constant est motivé par des applications aux glissements de terrain, oú le supremum donné par un probléme variationnel de blocage, apparaít comme un coefficient de sreté. Nous démontrons que ce coefficient est égal á loptimum du probléme doptimisation de formes mentionné précédemment. Dans le cas de données constantes, cela revient á étudier la premiére valeur propre de lopérateur 1-laplacien.Nous démontrons lexistence densembles optimaux, et donnons une caractérisation différentielle de leur bord intérieur. Nous étudions leur symétrie á laide de la symétrisation de Steiner. En dimension deux, nous exhibons des solutions explicites dans le cas oú les données ne dépendent que dune variable.

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Received: 18 June 2004, Accepted: 12 July 2004, Published online: 10 December 2004Mathematics Subject Classification (2000): 49J40, 49Q10     

Exceptional sets related to Hayman's alternative

Let E be a subset of the complex plane C consisting of a countable set of points tending to ∞ and let k?1 be an integer. We derive a spacing condition (dependent on k) on the points of E which ensures that, if f is a function meromorphic in C with sufficiently large Nevanlinna deficiency at the poles, then either f takes every complex value infinitely often, or the kth derivative f(k) takes every non-zero complex value infinitely often, in C−E. This improves a previous result of Langley.     

Construction of Lyapunov functions by the method of localization of invariant compact sets

We suggest a new method for constructing Lyapunov functions for autonomous systems of differential equations. The method is based on the construction of a family of sets whose boundaries have the properties typical of the level surfaces of Lyapunov functions. These sets are found by the method of localization of invariant compact sets. For the resulting Lyapunov function and its derivative, we find analytical expressions via the localizing functions occurring in the specification of the above-mentioned sets. An example of a system with a degenerate equilibrium is considered.     

Connection between sets of uniqueness for multiplicative systems of functions and sets of uniqueness for multiplicative transforms

A problem of uniqueness for series over multiplicative systems of functions and for multiplicative transforms is considered. It is shown that each set of uniqueness for a multiplicative transform is specified by a countable collection of sets of uniqueness for series over the corresponding multiplicative system of functions. Each set of uniqueness for a series over a multiplicative system of functions is a portion on [0, 1) of some set of uniqueness for the corresponding multiplicative transform.     

Critical sets of functions

Summary With the introduction of an alternate definition for critical point, this paper studies critical sets, as defined byW. M. Whyburn, in terms of certain related domains. Critical sets are divided into four classes. Type0 has the limit point property with respect to critical sets which are not type0; type1 and2 critical sets compare, respectively, to classical minimum and maximum points; type3 includes themin-max and flex type. This paper is a result of a study of critical sets made as a dissertation problem under the direction ofW. M. Whyburn, to whom the author is indebted for many helpful suggestions.     

On diagonal common quadratic Lyapunov functions for sets of homogeneous cooperative systems

This paper deals with a class of switched homogeneous cooperative systems. We answer under what conditions there exists a diagonal common quadratic Lyapunov function (DCQLF) for sets of homogeneous cooperative systems. This problem is characterized by easily verifiable algebraic conditions. Namely, the existence of a DCQLF can be related to the properties of some determinants. Finally, some reduced results are presented.     

Threshold functions and Poisson convergence for systems of equations in random sets

Mathematische Zeitschrift - We study threshold functions for the existence of solutions to linear systems of equations in random sets and present a unified framework which includes arithmetic...