On Ω-closed sets and Ωs-closed sets in topological spaces

Summary New classes of sets called -closed sets and s-closed sets are introduced and studied. Also, we introduce and study -continuous functions and s-continuous functions and prove pasting lemma for these functions. Moreover, we introduce classes of topological spaces -T_1/2 and -T_s.     

Extensions from the Sobolev Spaces H 1 Satisfying Prescribed Dirichlet Boundary Conditions

Extensions from H ¹(_P) into H ¹() (where _P ) are constructed in such a way that extended functions satisfy prescribed boundary conditions on the boundary of . The corresponding extension operator is linear and bounded.     

Partially ordered permutation groups

In this note we discuss permutation groups (G, ) in which the set admits aG-invariant order. By aG-invariant partial order (G-partial order) we mean a partial order < of such that < implies g<g, for all and in andg inG. If the set admits aG-partial order which is a total order, then (G, ) is an O-permutation group (orderable permutation group).The main concern of this paper is the development of a foundation for partially ordered permutation groups analogous to the existing one for partially ordered groups, as found in Fuchs [2].     

Graphs of Constant Mean Curvature in Hyperbolic Space

We study the problem of finding constant mean curvature graphsover a domain of a totally geodesic hyperplane andan equidistant hypersurface Q of hyperbolic space. We findthe existence of graphs of constant mean curvature H overmean convex domains Q and with boundary for –H < H |h|, where H > 0 is the mean curvature of the boundary . Here h is the mean curvature respectively of the geodesic hyperplane (h= 0) and of the equidistant hypersurface (0 < |h|< 1). The lower bound on H is optimal.     

Mengenwertige Masse und Fortsetzungen

Let (, _i) be a probability space for i=1,2 with and : ^m a correspondence, i.e. () is a non-void subset of ^m for all . We give necessary and sufficient conditions under which it holds, that ₂ extends ₁. iff _A d₂ is equal to _A d₁ for all A, where _A d_i is the set of all integrals _A f d_i of functions f: ^m with f()() _i.-a.e.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

Positive Solutions for a Singular Nonlinear Problem on a Bounded Domain in R 2

For a bounded regular Jordan domain in R ², we introduce and study a new class of functions K() related on its Green function G. We exploit the properties of this class to prove the existence and the uniqueness of a positive solution for the singular nonlinear elliptic equation u+(x,u)=0, in D(), with u=0 on and uC(), where is a nonnegative Borel measurable function in ×(0,) that belongs to a convex cone which contains, in particular, all functions (x,t)=q(x)t ^–,>0 with nonnegative functions qK(). Some estimates on the solution are also given.     

Decktransformationen transnormaler Mannigfaltigkeiten

For every transnormal m-manifold V (see [3] or [7]) in ⁿ :VW, mapping pV into its normal plane (p) is a covering map onto a submanifold W of the open Grassmannian H_n,n–m of all (n–m)-dimensional planes in ⁿ. The transnormal frame T:=^–1((p)) admits a transitive operation by a group J of isometries. The group action of the covering transformations of (V,,W) on T commutes with the action of J. The elements of J, which are restrictions of covering transformations to T, are exactly the elements of the centre of J. This property is applied to show the existence of nontrivial covering transformations of (V,,W) for n–m3.

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Diese Arbeit faßt die Kapitel 5, 6 und 7 der von der Fakultät für Allgemeine Ingenieurwissenschaften der TU Berlin genehmigten Dissertation [6] zusammen.     

Some overdetermined boundary value problems for harmonic functions

Summary In this paper we study various overdetermined problems involving harmonic functions. In particular, we show that if the second eigenfunctionu ₂ of the Stekloff eigenvalue problem in a bounded simply connected plane domain has a constant value of u ₂ on , then is a disk

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Résumé Cet article est consacré à l'étude de certains problèmes surdéterminés pour des fonctions harmoniques. En particulier, nous montrons que si le gradient de la seconde fonction propre du problème de Stekloff défini dans un domaine borné, simplement connexe du plan, a son module constant sur la frontière , alors est nécessairement un disque.

     

Operatori lineariT-invarianti rispetto ad un gruppo di congruenze

Summary Let be an open subset of ⁿ, W_m() the linear space of m-vector valued functions defined on , G{} a group of orthogonal matrices mapping onto itself and T{T()} a linear representation of order m of G. A suitable groupC(G,T) of linear operators of W_m(), which leads to a general definition of T-invariant linear operator with respect to G, is here introduced. Characterization theorems concerning the linear differential and integral T-invariant operators are also given. When G is a finite group, projection operators are explicitly obtained; they define a «maximal» decomposition of W_m() into a direct sum of subspaces each of them invariant with respect to any T-invariant linear operator of W_m(). Some examples are givenc.

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Lavoro eseguito nell'ambito del progetto nazionale di ricerca «Analisi numerica e matematica computazionale» nell'anno 1985–86.     

Extremal feedback perturbations for families of closed operators

Let T- S, be a family of not necessarily bounded semi-Fredholm operators, where T and S are operators acting between Banach spaces X and Y, and where S is bounded with D(S) D(T). For compact sets , as well as for certain open sets , we investigate existence and minimal rank of bounded feedback perturbations of the form F=BE such that min.ind (T-S+F)=0 for all . Here B is a given operator from a linear space Z to Y and E is some operator from X to Z.We give a simple characterization of that situation, when such regularizing feedback perturbations exist and show that for compact sets the minimal rank never exceeds max { min.ind (T-S) }+1. Moreover, an example shows that the minimal rank, in fact, may increase from max {...} to max {...}+1, if the given B enforces a certain structure of the feedbachk perturbation F.However, the minimal rank is equal to max { min.ind (T-S) }, if is an open set such that min.ind (T-S) already vanishes for all but finitely many points . We illustrate this result by applying it to the stabilization of certain infinite-dimensional dynamical systems in Hilbert space.     

Effilement minimal en une singularité isolée de l'équation de Schrödinger et application au principe de Picard

In this paper, the generalized Schrödinger equation (–)u=0 on the punctured unit disk of ² is investigated. If is rotation free and satisfies the Picard principle at the origin, it is shown that if a setE is minimal thin relatively to an extremal harmonic functionh with zero boundary values at {|x|=1}, there exists a sequence (r _n ) converging to zero such that B(O,r _n ) C E. Lete be the -unit. It is proved that if a measure satisfies _\E e h d<, for a minimal thin, relatively toh , setE then the Picard principle is valid for the measure + .

     

Brownian Motion in a Hilbert Space with a Semipermeable Membrane on a Hyperplane

In a separable Hilbert space, we construct a continuous Markov process whose behavior coincides everywhere, except for a hyperplane S orthogonal to a given unit vector , with the behavior of a homogeneous Gaussian process with a given correlation operator tB, where B is a nonsingular nuclear operator. As the process hits the hyperplane, it receives an impulse infinite in modulus in the direction A such that |(A, )| (B, ).We obtain a stochastic differential equation whose solutions are trajectories of the process constructed.     

Bounce problem with weak hypotheses of regularity

Summary In this paper the elastic bounce problem is formulated in very general hypotheses. More precisely we consider the motion of a material point constrained to move in a domain R ⁿ, bouncing against its boundary, and we suppose that is neither regular nor convex. Assuming that is in the class of p-convex sets introduced in [4] and C^0,1, an existence theorem is stated.     

Linearly similar models of Toeplitz operators

Let be a simply connected domain with a piecewise smooth boundary and assume that the function F is meromorphic in , does not have poles on , and the index of each point \F() with respect to the curve F() is nonnegative (at the positive traversal of the curve ). Under these assumptions, for a class of Banach spaces (including the Hardy-Smirnov spaces, the analytic spacesLip , the Bergman, Bloch spaces, etc.) one defines the Toeplitz operator T_F and one establishes its similarity to the operator of multiplication f·f by the projection of a specially selected Riemann surface _*=_*(T_F called the ultraspectrum of the operator T_F.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 113–123, 1987.The author is deeply grateful to his scientific advisor N. K. Nikol'skii for the formulation of the problem and for his constant interest in the paper, and also to A. A. Borichev, A. L. Vol'berg, and B. M. Solomyak for useful discussions.     

Comparative estimation of the degree of crosslinking of styrene-unsaturated polyester copolymers

Copolymers of polydiethylene glycol maleate-sebacates and styrene have been investigated. The degree of crosslinking () has been estimated by a physical method (_p) from the Mooney — Rivlin equation and Wall's formula and also by a chemical method (_c) based on hydrolysis of the copolymer in an alkaline medium. In most cases the difference between the values of calculated from the Mooney — Rivlin equation and Wall's formula does not exceed 20%. It is shown that a linear correlation exists between _p and _c. The fact that the values of _p are lower than those of _c is attributable to the specific characteristics of the molecular network — the presence of short chain segments between neighboring network points and nonload-bearing free ends.Scientific-Research Institute of Plastics, Moscow. Translated from Mekhanika Polimerov, No. 6, pp. 963–968, November–December, 1971.     

Complete Subobjects of Fuzzy Sets Over MV-Algebras

A subobjects structure of the category -FSet of -fuzzy sets over a complete MV-algebra is investigated, where an -fuzzy set is a pair A = (A, ) such that A is a set and : A × A is a special map. Special subobjects (called complete) of an -fuzzy set A which can be identified with some characteristic morphisms A ^* = (L × L, ) are then investigated. It is proved that some truth-valued morphisms are characteristic morphisms of complete subobjects.     

Equivalence between the dilation and lifting properties of an ordered group through multiplicative families of isometries. A version of the commutant lifting theorem on some lexicographic groups

Let be a locally compact abelian ordered group. has the dilation property if a special extension of the Naimark dilation theorem holds for and it has the commutant lifting property if a natural extension of the Sz.-Nagy — Foias commutant lifting theorem holds for .We prove that these two conditions are equivalent and we give another necessary and sufficient condition in terms of unitary extensions of multiplicative families of partial isometries.A version of the commutant lifting theorem is given for the groups ⁿ and × ⁿ with the lexicographic order and the natural topologies.Both authors were partially supported by the CDCH of the Universidad Central de Venezuela, and by CONICIT grant G-97000668.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Homogenization of a Singularly Perturbed Parabolic Problem in a Thick Periodic Junction of the Type 3:2:1

We prove a convergence theorem and obtain asymptotic (as 0) estimates for a solution of a parabolic initial boundary-value problem in a junction that consists of a domain ₀ and a large number N ² of -periodically located thin cylinders whose thickness is of order = O(N ^–1).