Schlicht regions for entire and meromorphic functions

Let f:C →C be a meromorpMc function. We study the size of the maximal disc inC, with respect to the spherical metric, in which a single-valued branch of f^-1 exists. This problem is related to normality and type criteria. Best possible lower estimates of the size of such discs are obtained for entire functions and a class of meromorphic functions containing all elliptic functions. An estimate for the class of rational functions is also given which is best possible for rational functions of degree 7. For algebraic functions of given genus we obtain an estimate which is precise for genera 2 and 5 and asymptotically best possible when the genus tends to infinity. Supported by a Heisenberg fellowship of the DFG. Partially supported by NSF grant DMS-950036 and by the Lady Davis Foundation.     

On the Wilson criterion

The U(1)-gauge theory with the Villain action is considered in a cubic lattice approximation of three-and four-dimensional tori. As the lattice spacing tends to zero, the naturally defined correlation functions converge to the correlation functions of theR-gauge electrodynamics on three- and four-dimensional tori only for a special scaling, which depends on the correlation functions. Another scaling gives degenerate continuum limits. The Wilson criterion for the confinement of charged particles is fulfilled for theR-gauge electrodynamics on a torus. If the radius of the initial torus tends to infinity, then the correlation functions converge to the correlation functions of theR-gauge Euclidean electrodynamics. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 118, No. 1, pp. 67–73, January, 1999.     

Choice Functions and Extensive Operators

The paper puts forth a theory of choice functions in a neat way connecting it to a theory of extensive operators and neighborhood systems. We consider classes of heritage choice functions satisfying conditions M, N, W, and C, or combinations of these conditions. In terms of extensive operators these classes can be considered as generalizations of symmetric, anti-symmetric and transitive binary relations. Among these classes we meet the well-known classes of matroids and convex geometries. Using a ‘topological’ language we discuss these classes of monotone extensive operators (or heritage choice functions) in terms of neighborhood systems. A remarkable inversion on the set of choice functions is introduced. Restricted to the class of heritage choice functions the inversion turns out to be an involution, and under this involution the axiom N is auto-inverse, whereas the axioms W and M change places. This research was supported in part by NWO–RFBR grant 047.011.2004.017, by RFBR grant 05-01-02805 CNRSL_a, and the grant NSh-929.2008.6, School Support. We want to thank B. Monjardet, the editor and a referee for discussions and useful suggestions.     

An Extended Faber Operator

   Abstract. One of the basic tools in the theory of polynomial approximation in the uniform norm on compact plane sets is the Faber operator. Usually, the Faber operator is viewed as an operator acting on functions in the disk algebra, that is, functions which are holomorphic in the open unit disk D and continuous on D. We consider an extended Faber operator acting on arbitrary functions continuous on ; D.     

The formulas for the representation of functions of two variables as a difference of sublinear functions

ABSTRACT

Having a function being a difference of sublinear functions defined on a plane, we present a formula for effective calculation of sublinear functions such that their difference is equal to the given one. Moreover, these newly calculated sublinear functions are minimal and as such unique-up-to-linear-summand. We also provide examples of such functions.     

Manifolds modelled over free modules over the double numbers

Manifolds over the algebra of double numbers, which include the case of manifolds equipped with a pair of equidimensional supplementary foliations, are studied. To this end, B-holomorphic functions and B-analytic functions on B ⁿ, where B denotes the algebra of double numbers, are defined and studied. This revised version was published online in June 2006 with corrections to the Cover Date.     

Characterization of Linear Structures

We study the notionof linear structure of a function defined from F ^mto F ⁿ, and in particular of a Boolean function.We characterize the existence of linear structures by means ofthe Fourier transform of the function. For Boolean functions,this characterization can be stated in a simpler way. Finally,we give some constructions of resilient Boolean functions whichhave no linear structure.     

DISTRIBUTION OF MULTIPLICATIVE FUNCTIONS DEFINED ON SEMIGROUPS

Abstract

The value distribution problem for real-valued multiplicative functions defined on an additive arithmetical semigroup is examined. We prove that, in contrast to the classical theory of number-theoretic functions defined on the semigroup of natural numbers, this problem is equivalent to that for additive functions only under some extra condition. In this way, applying the known results for additive functions we derive general sufficient conditions for the existence of a limit law for appropriately normalized multiplicative functions.     

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ballB⊂C ⁿ with its relative logarithmic capacity inC ⁿ with respect to the same ballB. An analogous comparison inequality for Borel subsets of euclidean balls of any generic real subspace ofC ⁿ is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of plurisubharmonic lemniscates associated to the Lelong class of plurisubharmonic functions of logarithmic singularities at infinity onC ⁿ as well as the Cegrell class of plurisubharmonic functions of bounded Monge-Ampère mass on a hyperconvex domain Ω⊂(C ⁿ . Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of plurisubharmonic functions. This work was partially supported by the programmes PARS MI 07 and AI.MA 180.     

On Semi-fine Limits at Infinity for Riesz Potentials and Monotone BLD Functions

Semi-fine limits at infinity are studied for Riesz potentials of functions on R ⁿ with a certain growth condition. We are also concerned with monotone BLD functions.     

Hölder Continuity of Harmonic Functions with Respect to Operators of Variable Order

ABSTRACT

We consider a class of integrodifferential operators and their corresponding harmonic functions. Under mild assumptions on the family of jump measures we prove a priori estimates and establish Hölder continuity of bounded functions that are harmonic in a domain.     

Hölder Estimates for the ∂¯-Operator on Polycylinders

Hölder estimates for the ?¯-operator are obtained on polycylinders and applied to a number of problems-cohomology with bounds, the corona problem and approximation of Hölder functions by holomorphic functions.     

The q-asymptotic function in c-convex analysis

Abstract

We establish several connections between generalized asymptotic functions and different areas of convexity theory, without coercivity assumptions. Properties and characterizations of abstract subdifferentials, normal cones, conjugates, support functions and optimality conditions for the minimization problem are given. We provide a new result on existence of minimizers for a class of nonconvex functions which is strictly larger than the class of quasiconvex ones.     

On iterative convexity of diffeomorphism

Abstract

A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. We give several constructions of iteratively convex diffeomorphisms and explain the phenomenon that the non-existence of convex iterative roots is a typical property of convex functions. We show also that a slight perturbation of iteratively convex functions causes loss of iterative convexity. However, every convex function can be approximate by iteratively convex functions.     

Relations between threshold and <Emphasis Type="Italic">k</Emphasis>-interval Boolean functions

Every k-interval Boolean function f can be represented by at most k intervals of integers such that vector x is a truepoint of f if and only if the integer represented by x belongs to one of these k (disjoint) intervals. Since the correspondence of Boolean vectors and integers depends on the order of bits an interval representation is also specified with respect to an order of variables of the represented function. Interval representation can be useful as an efficient representation for special classes of Boolean functions which can be represented by a small number of intervals. In this paper we study inclusion relations between the classes of threshold and k-interval Boolean functions. We show that positive 2-interval functions constitute a (proper) subclass of positive threshold functions and that such inclusion does not hold for any k>2. We also prove that threshold functions do not constitute a subclass of k-interval functions, for any k.     

Rational functions and value sharing

This note gives a survey on shared value problems for rational functions. In particular a solution to a problem in [A.K. Pizer (1973). A problem on rational functions. Amer. Math. Mon., 80, 552–553.] is given.     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

Special matrix functions: characteristics,achievements and future directions

ABSTRACT

In recent years, special matrix functions and polynomials of a real or complex variable have been in a focus of increasing attention leading to new and interesting problems. In this work, we present matrix space analogues to generalized some functions and polynomials in the framework of matrix setting. Many of the special matrix functions and polynomials are constructed along standard procedures. Recently published papers are also surveyed and we list the most essential ones.