Lattice Fully Orderable Groups

Let be a linearly ordered set, A() be the group of all order automorphisms of , and L() be a normal subgroup of A() consisting of all automorphisms whose support is bounded above. We argue to show that, for every linearly ordered set such that: (1) A() is an o-2-transitive group, and (2) contains a countable unbounded sequence of elements, the simple group A()/L() has exactly two maximal and two minimal non-trivial (mutually inverse) partial orders, and that every partial order of A()/L() extends to a lattice one (Thm. 2.1). It is proved that every lattice-orderable group is isomorphically embeddable in a simple lattice fully orderable group (Thm. 2.2). We also state that some quotient groups of Dlab groups of the real line and unit interval are lattice fully orderable (Thms. 3.1 and 3.2).     

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.     

The Laplacian with Robin Boundary Conditions on Arbitrary Domains

Using a capacity approach, we prove in this article that it is always possible to define a realization of the Laplacian on L ²() with generalized Robin boundary conditions where is an arbitrary open subset of R ⁿ and is a Borel measure on the boundary of . This operator generates a sub-Markovian C ₀-semigroup on L ²(). If d=d where is a strictly positive bounded Borel measurable function defined on the boundary and the (n–1)-dimensional Hausdorff measure on , we show that the semigroup generated by the Laplacian with Robin boundary conditions has always Gaussian estimates with modified exponents. We also obtain that the spectrum of the Laplacian with Robin boundary conditions in L ^p () is independent of p[1,). Our approach constitutes an alternative way to Daners who considers the (n–1)-dimensional Hausdorff measure on the boundary. In particular, it allows us to construct a conterexample disproving Daners' closability conjecture.     

Cohomological characterization of pseudoconvexity in a Banach space

Let X be a Banach space with a countable unconditional basis (e.g., X=₂), X open. We show that is pseudoconvex if and only if for each affine complex line L in X the sheaf cohomology group H ¹ (,I) vanishes, where I is the ideal sheaf of all holomorphic functions on that vanish on L. We also give an example that the condition H ^q (,)=0 for all q1 unlike in finite dimensions does not imply the pseudoconvexity of . Lastly, we prove an interpolation result. Mathematics Subject Classification (2002): 32T05, 46G20.     

On the diameters of some classes of analytic functions. I

In the spaces of analytic functionsE _q (), q1, introduced by V. I. Smirnov, where is a bounded simply connected domain in the plane with sufficiently smooth boundary , we obtain order estimates of diameters of the classesW ^r E _p () (p1, and r is a natural number 2) for distinct p and q.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 3, pp. 324–333, March, 1992.     

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.     

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 .     

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.     

On Some Isomorphism on the Category of b-Spaces

Given a nuclear b-space N, we show that if is a finite or -finite measure space and 1p, then the functors L _loc ^p (,N.) and NL ^p (,.) are isomorphic on the category of b-spaces of L. Waelbroeck.     

Weighted Composition Operators on Bergman and Dirichlet Spaces

Let H() denote a functional Hilbert space of analytic functions on a domain . Let w : C and : be such that w f is in H() for every f in H(). The operator wC given by f w f is called a weighted composition operator on H(). In this paper we characterize such operators and those for which (wC )* is a composition operator. Compact weighted composition operators on some functional Hilbert spaces are also characterized. We give sufficient conditions for the compactness of such operators on weighted Dirichlet spaces.     

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.     

An embedding theorem for a limiting exponent

We consider the function space B _p ^l () of functionsf(x), defined on the domain of a certain class and characterized by specific differential-difference properties in L_p(). We prove a theorem on the embedding B _p,q ^l () L_q in the case whenl=n/p –n/q >0 and its generalization for vectorl, p, q.Translated from Matematicheski Zametki, Vol. 6, No. 2, pp. 129–138, August, 1969.     

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)¦.     

On Central Endomorphisms of a Group with Operators

Let G be a group with a set of operators such that Z(G) is -admissible. Central -automorphisms occur in the Krull-Remak-Schmidt Theorem. We discuss the existence of a central -endomorphism of G that is not an automorphism.2000 Mathematics Subject Classification: 20E36     

Schatten class composition operators on weighted Bergman spaces of bounded symmetric domains

Summary We obtain trace ideal criteria for 0A ² () of a Bounded symmetric diomain in _n.     

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.     

Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

We consider multidimensional weak and strong Hele-Shaw dynamics (t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain (0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions (t), and show that if (0) is starshaped with respect to a small ball centered on the point of injection, then the evolution (t) exists for all time.Received: 5 March 2004     

On a Generalization of Nikolskij's Extension Theorem in the Case of Two Variables

A modification of the Nikolskij extension theorem for functions from Sobolev spaces H ^k() is presented. This modification requires the boundary to be only Lipschitz continuous for an arbitrary k however, it is restricted to the case of two-dimensional bounded domains.     

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].     

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.