Integral properties of the classical warping function of a simply connected domain

Let u(z,G) be the classical warping function of a simply connected domain G. We prove that the L ^p -norms of the warping function with different exponents are related by a sharp isoperimetric inequality, including the functional u(G) = sup _x∈G u(x, G). A particular case of our result is the classical Payne inequality for the torsional rigidity of a domain. On the basis of the warping function, we construct a new physical functional possessing the isoperimetric monotonicity property. For a class of integrals depending on the warping function, we also obtain a priori estimates in terms of the L ^p -norms of the warping function as well as the functional u(G). In the proof, we use the estimation technique on level lines proposed by Payne.     

Estimation of the L p -norms of stress functions for finitely connected plane domains

Let u(x, G) be the classical stress function of a finitely connected plane domain G. The isoperimetric properties of the L ^p -norms of u(x, G) are studied. Payne’s inequality for simply connected domains is generalized to finitely connected domains. It is proved that the L ^p -norms of the functions u(x, G) and u ^?1 (x, G) strictly decrease with respect to the parameter p, and a sharp bound for the rate of decrease of the L ^p -norms of these functions in terms of the corresponding L ^p -norms of the stress function for an annulus is obtained. A new integral inequality for the L ^p -norms of u(x, G), which is an analog of the inequality obtained by F. G. Avkhadiev and the author for the L ^p -norm of conformal radii, is proved.     

Embedding of Sobolev spaces and properties of the domain

We establish the embedding of the Sobolev space W _p ^s (G) ? L _q (G) for an irregular domain G in the case of a limit exponent under new relations between the parameters depending on the geometric properties of the domain G.     

Two-point Padé expansions for a family of analytic functions

Each member G(z) of a family of analytic functions defined by Stieltjes transforms is shown to be represented by a positive T-fraction, the approximants of which form the main diagonal in the two-point Padé table of G(z). The positive T-fraction is shown to converge to G(z) throughout a domain D(a, b) = [z: z?[?b, ?a]], uniformly on compact subsets. In addition, truncation error bounds are given for the approximants of the continued function; these bounds supplement previously known bounds and apply in part of the domain of G(z) not covered by other bounds. The proofs of our results employ properties of orthogonal $\text{L}$-polynomials (Laurent polynomials) and $\text{L}$-Gaussian quadrature which are of some interest in themselves. A number of examples are considered.     

On Singular Points of Solutions of the Minimal Surface Equation on Sets of Positive Measure

It is shown that, for any compact set K ? ? ⁿ (n ? 2) of positive Lebesgue measure and any bounded domain G ? K, there exists a function in the Hölder class C^1,1(G) that is a solution of the minimal surface equation in G \ K and cannot be extended from G \ K to G as a solution of this equation.     

Isoperimetric monotony of the L p -norm of the warping function of a plane simply connected domain

Let G be a simply connected domain and let u(x,G) be its warping function. We prove that L ^p -norms of functions u and u ^?1 are monotone with respect to the parameter p. This monotony also gives isoperimetric inequalities for norms that correspond to different values of the parameter p. The main result of this paper is a generalization of classical isoperimetric inequalities of St.Venant-Pólya and the Payne inequalities.     

Hyperbolically convex constricted domains

A subdomain G in the unit disk D is called hyperbolically convex if the non-euclidean segment between any two points in G also lies in G. We introduce the concept of constricted domain relative to the hyperbolic geometry of D and prove that a hyperbolic convex domain is constricted if and only if it is not a quasidisk. Also examples are given to illustrate these ideas.     

Continuous dependence of attractors on the shape of domain

Let Ω₀ be a bounded domain in ? ⁿ , letG be a family of diffeomorphisms, and let Ω _G =G(Ω₀), forG∈G. Denote by Σ _t (G) the semigroup generated by a fixed parabolic PDE with Dirichlet boundary conditions on the boundary of Ω _G . LetA _G be the global attractor of Σ _t (G). Conditions are given under which a generic diffeomorphismG∈G is a continuity point of the mapG »A _G . Bibliography: 12 titles.     

Continuation of separately analytic functions defined on part of a domain boundary

Suppose that D ? ?ⁿ is a domain with smooth boundary ?D, E ? ?D is a boundary subset of positive Lebesgue measure mes(E) > 0, and F ? G is a nonpluripolar compact set in a strongly pseudoconvex domain G ? ?^m. We prove that, under some additional conditions, each function separately analytic on the set X = (D×F)∪(E× G) can be holomorphically continued into the domain where ω* is the P-measure and ω _in ^* is the inner P-measure.     

The hereditary monocoreflective subcategories of Abelian groups and R-modules

The non-trivial hereditary monocoreflective subcategories of the Abelian groups are the following ones: {G ?? Ob Ab | G is a torsion group, and for all g ?? G the exponent of any prime p in the prime factorization of o(g) is at most E(p)}, where E(·) is an arbitrary function from the prime numbers to {0, 1, 2, ??,??}. (o(·) means the order of an element, and n ?? ?? means n < ??.) This result is dualized to the category of compact Hausdorff Abelian groups (the respective subcategories are {G ?? Ob CompAb | G has a neighbourhood subbase {G _?? } at 0, consisting of open subgroups, such that G/G _?? is cyclic, of order like o(g) above}), and is generalized to categories of unitary R-modules for R an integral domain that is a principal ideal domain. For general rings R with 1, an analogous theorem holds, where the hereditary monocoreflective subcategories of unitary left R-modules are described with the help of filters L in the lattice of the left ideals of the ring R. These subcategories consist of those left R-modules, for which the annihilators of all elements belong to L. If R is commutative, then this correspondence between these subcategories and these filters L is bijective.     

On max-stable processes and the functional D-norm

We introduce some mathematical framework for extreme value theory in the space of continuous functions on compact intervals and provide basic definitions and tools. Continuous max-stable processes on [0, 1] are characterized by their “distribution functions” G which can be represented via a norm on function space, called D-norm. The high conformity of this setup with the multivariate case leads to the introduction of a functional domain of attraction approach for stochastic processes, which is more general than the usual one based on weak convergence. We also introduce the concept of “sojourn time transformation” and compare several types of convergence on function space. Again in complete accordance with the uni- or multivariate case it is now possible to get functional generalized Pareto distributions (GPD) W via W?=?1?+?log(G) in the upper tail. In particular, this enables us to derive characterizations of the functional domain of attraction condition for copula processes.     

Large linear manifolds of noncontinuable boundary-regular holomorphic functions

We prove in this paper that if G is a domain in the complex plane satisfying appropriate topological or geometrical conditions, then there exists a large (dense or closed infinite-dimensional) linear submanifold of boundary-regular holomorphic functions on G all of whose nonzero members are not continuable across any boundary point of G.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

Zero Distribution of Bergman Orthogonal Polynomials for Certain Planar Domains

Abstract. Let G be a simply connected domain in the complex plane bounded by a closed Jordan curve L and let P _n , n≥ 0 , be polynomials of respective degrees n=0,1,··· that are orthonormal in G with respect to the area measure (the so-called Bergman polynomials). Let ? be a conformal map of G onto the unit disk. We characterize, in terms of the asymptotic behavior of the zeros of P _n 's, the case when ? has a singularity on L . To investigate the opposite case we consider a special class of lens-shaped domains G that are bounded by two orthogonal circular arcs. Utilizing the theory of logarithmic potentials with external fields, we show that the limiting distribution of the zeros of the P _n 's for such lens domains is supported on a Jordan arc joining the two vertices of G . We determine this arc along with the distribution function.     

Approximation by rational functions in Smirnov-Orlicz classes

Let G be a doubly-connected domain bounded by Dini-smooth curves. In this work, the approximation properties of the Faber-Laurent rational series expansions in Smirnov-Orlicz classes EM(G) are studied.     

The Zassenhaus variety of a reductive Lie algebra in positive characteristic

Let g be the Lie algebra of a connected reductive group G over an algebraically closed field k of characteristic p>0. Let Z be the centre of the universal enveloping algebra U=U(g) of g. Its maximal spectrum is called the Zassenhaus variety of g. We show that, under certain mild assumptions on G, the field of fractions Frac(Z) of Z is G-equivariantly isomorphic to the function field of the dual space g^∗ with twisted G-action. In particular Frac(Z) is rational. This confirms a conjecture of J. Alev. Furthermore we show that Z is a unique factorisation domain, confirming a conjecture of A. Braun and C. Hajarnavis. Recently, A. Premet used the above result about Frac(Z), a result of Colliot-Thelene, Kunyavskii, Popov and Reichstein and reduction mod p arguments to show that the Gelfand-Kirillov conjecture cannot hold for simple complex Lie algebras that are not of type A, C or G₂.     

The interval function of a connected graph and road systems

Let V be a finite nonempty set. In this paper, a road system on V (as a generalization of the set of all geodesics in a connected graph G with V(G)=V) and an intervaloid function on V (as a generalization of the interval function (in the sense of Mulder) of a connected graph G with V(G)=V) are introduced. A natural bijection of the set of all intervaloid functions on V onto the set of all road systems on V is constructed. This bijection enables to deduce an axiomatic characterization of the interval function of a connected graph G from a characterization of the set of all geodesics in G.     

Vector spaces of non-extendable holomorphic functions

In this paper, we analyze the linear structure of the family H _e (G) of holomorphic functions on a domain G of the complex plane that are not analytically continuable beyond the boundary of G. We prove that H _e (G) contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of H _e (G) consisting of boundary-regular functions contains dense vector spaces with maximal dimension as well as infinite dimensional closed vector spaces and large algebras. We also consider the case in which G is a domain of existence in a complex Banach space. The results obtained complete or extend a number of previous results by several authors.     

On the Galois cohomology of dedekind rings

Let R be a Dedekind domain, G a finite group of automorphisms of R, and A an ambiguous ideal of R i.e., σA = A for all σ ∈ G. The Tate groups Hⁿ(G, A) are considered as R^G-modules. A localization theorem is proved and the precise R^G-module structure determined in a particular case. In addition some remarks are made concerning cohomological triviality.