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.     

Diameters of classes of continuous functions in the Lp space

In the L_p(a, b) space the exact values of n-diameters (n=1, 2, ...) are found of the class H[a, b] of the functions f(x) such that ¦f(x)-f(x)¦ (¦x-x¦), where (t) is a given continuity module which is convex upwards.Translated from Matematicheskie Zametki, Vol. 10, No. 5, pp. 493–500, November, 1971.     

Theory of multiple j-elementary matrix-valued functions with a pole at the boundary of the unit circle

The theory of a matrix-valued function (), j-elementary in the unit circle and having a pole at the point ₀, ¦₀¦=1, on the boundary of the unit circle, is considered. The structure of () is determined, conditions for the splittingoff of () from an arbitrary matrix-valued function W(), j-expanding in ¦|< 1, are formed, a theorem on the parametrization of a j-elementary matrix-valued function () of full rank is proved, and a decomposition of () of full rank into the product of parametrized j-elementary factors of full rank with simple poles at the point ₀ is found.Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 50, pp. 62–74, 1988.     

Triangular extension identities

It is shown that if the - -bimodule M generates a category of - -bimodules, then the ideal of identities of the triangular extension of the direct sum of algebras and by means of the bimodule M is equal to the product of ideals of identities of the algebras and .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 132, pp. 5–11, 1983.     

The normal subgroup lattice of 2-transitive automorphism groups of linearly ordered sets

Using combinatorial and model-theoretic means, we examine the structure of normal subgroup lattices N(A()) of 2-transitive automorphism groups A() of infinite linearly ordered sets (, ). Certain natural sublattices of N(A()) are shown to be Stone algebras, and several first order properties of their dense and dually dense elements are characterized within the Dedekind-completion of (, ). As a consequence, A() has either precisely 5 or at least 2²¹ (even maximal) normal subgroups, and various other group- and lattice-theoretic results follow.     

Uniqueness of kernel functions of the heat equation

We consider the heat equation on ={(x,t) R ²;t<0, ¦x¦<(–t)} and give the uniqueness of kernel functions at the infinity (see Theorem 5). For the proof, we examine the continuity of the density of the parabolic measure onD ={(x,t);t>x}, closely related to . By this theorem, we can decide the Martin boundary of (<1) with respect to the heat equation.     

Estimates of the solution of the Dirichlet problem for the laplace operator in exterior domains

Let u be the solution of the equation u=f with a finite function f in the exterior domain R^u and satisfying the conditions u¦=0, u0 as ¦x¦ . It is shown that the coercive estimate is valid only for p < n/2. For p n/2 it holds for the solution of the exterior Dirichlet problem, which does not vanish at infinity, and may tend to constant or even to a linear (for p > n) function.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 146, pp. 92–101, 1985.     

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.     

Some Geometric Inequalities for a Tangent Point Simplex

The tangent point simplex of a simplex is the pedal simplex of the incenter of . In this paper we obtain some geometric inequalities between and .     

Polyhedral polarity defined by a general bilinear inequality

LetX andY be finite dimensional vector spaces over the real numbers. Let be a binary relation betweenX andY given by a bilinear inequality. The-polar of a subsetP ofX is the set of all elements ofY which are related by to all elements ofP. The-polar of a subset ofY is defined similarly. The-polar of the-polar ofP is called the-closure ofP andP is called-closed ifP equals its-closure. We describe the-polar of any finitely generated setP as the solution set of a finite system of linear inequalities and describe the-closure ofP as a finitely generated set. The-closed polyhedra are characterized in terms of defining systems of linear inequalities and also in terms of the relationship of the polyhedronP with its recessional cone and with certain subsets ofX andY determined by the relation. Six classes of bilinear inequalities are distinguished in the characterization of-closed polyhedra.     

New lower semicontinuity results for polyconvex integrals

Summary We study integral functionals of the formF(u, )= f(u)dx, defined foru C¹(;R ^k), R ⁿ . The functionf is assumed to be polyconvex and to satisfy the inequalityf(A) c₀¦(A)¦ for a suitable constant c₀ > 0, where (A) is then-vector whose components are the determinants of all minors of thek×n matrixA. We prove thatF is lower semicontinuous onC ¹(;R ^k) with respect to the strong topology ofL ¹(;R ^k). Then we consider the relaxed functional , defined as the greatest lower semicontinuous functional onL ¹(;R ^k ) which is less than or equal toF on C¹(;R ^k). For everyu BV(;R ^k) we prove that (u,) f(u)dx+c₀¦D^su¦(), whereDu=u dx+D^su is the Lebesgue decomposition of the Radon measureDu. Moreover, under suitable growth conditions onf, we show that (u,)= f(u)dx for everyu W^1,p(;R ^k), withp min{n,k}. We prove also that the functional (u, ) can not be represented by an inte- gral for an arbitrary functionu BV_loc(R ⁿ;R ^k). In fact, two examples show that, in general, the set function (u, ) is not subadditive whenu BV_loc(R ⁿ;R ^k), even ifu W _loc ^1,p (R ⁿ;R ^k) for everyp < min{n,k}. Finally, we examine in detail the properties of the functionsu BV(;R ^k) such that (u, )= f(u)dx, particularly in the model casef(A)=¦(A)¦.     

An estimate for the solutions of stokes equations in exterior domains

A boundary value problem for the Stokes equations is examined in an exterior domain ⁿ with a uniform Dirichlet condition on the boundary and a homogeneous condition at infinity. It is shown that estimating the norm L _p() of the second derivatives of the velocity vector field by the same norm of the exterior forces vector field is correct for p < n/2, but not for p n/2. This estimate is valid also for p n/2 if the boundary conditions are modified at infinity. Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Akademii Nauk SSSR, Vol. 180, pp. 105–120, 1990.     

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.     

Convexity theories III. Classification of certain real convexity theories

In this paper we classify all real convexity theories that contain the standard convexity theory _c. For this purpose we consider three subcases: finitary; infinitary and (_sc\_c)Ø; infinitary and _sc=_c. In each of these subcases one encounters a phenomenon resembling bifurcation.This research was supported by the Deutsche Forschungsgemeinschaft.     

Vector Valued Differentiation Theorems for Multiparameter Additive Processes in Lp Spaces

Let X be a Banach space and (,,µ) be a -finite measure space. We consider a strongly continuous d-dimensional semigroup T={T(u):u=(u₁,..., u_d, u_i >0, 1 i d} of linear contractions on L_p((,,µ); X), with 1 p<. In this paper differentiation theorems are proved for d-dimensional bounded processes in L_p((,,µ); X) which are additive with respect to T. In the theorems below we assume that each T(u) possesses a contraction majorant P(u) defined on L_p((,,µ); R), that is, P(u) is a positive linear contraction on L_p((,,µ); R) such that T(u)f(w) P(u)f(·)() almost everywhere on for all f L_p((,,µ); X).     

Isoperimetric inequalities in a boundary value problem on a Riemannian manifold

In this paper the problem u+1=0 in ,u=0 on is considered. Here is a finite domain on a Riemannian manifold and the associated Laplace-Beltrami operator. By means of maximum principles isoperimetric bounds for the maximum ofu and the maximum of the absolute value of the gradient ofu, as well as some related bounds are derived.

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Zusammenfassung Diese Arbeit behandelt das Problem u+1=0 in ,u=0 auf , wobei ein Gebiet auf einer zweidimensionalen Riemann'schen Mannigfaltigkeit ist, und der zugehörige Laplace-Beltrami Operator. Es werden isoperimetrische Schranken für das Maximum vonu und |u| aus gewissen Maximumsprinzipien hergeleitet, sowie einige verwandte Resultate.

     

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.     

Reinforcement problems for elliptic equations and variational inequalities

Summary Consider the Dirichlet problem for an elliptic equation in a domain , with coefficients having discontinuity on a surface . Suppose divides into _{1 2}(₂ the inner core), the thickness of ₁ is of order of magnitude , and the modulus of ellipticity in ₁ is of order magnitude ₁. The asymptotic behavior of the solution is studied as 0, ₁ 0, provided lim (/₁) exists. Other questions of this type are studied both for elliptic equations and for elliptic variational inequalities.The second author is partially supported by National Science Foundation Grant 7406375 A01. The third author is partially supported by National Science Foundation Grant MC575-21416 A01.     

Central orderings in fields of real meromorphic function germs

Let X_oR ⁿ be an irreducible analytic germ and the order space of its field of meromorphicfunetion germs. A formal half-branch in X_o is a kind of C-map germ c[0,)X_o; an ordering is centered at c if it contains the functions which are positive on c. We obtain a partition ¹,...,^d, d=dim X_o, of the set ^* of central (i.e.: centered at some half-branch) orderings, according to the dimension of half-branches. Then we show that all ^e, e= 1,.,d, as well as the set \^* of noncentral orderings, are dense in . Finally, we solve the 17th Hubert Problem for analytic germs.     

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.