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     

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     

Spectral pairs in cartesian coordinates

Let ^d have finite positive Lebesgue measure, and let () be the corresponding Hilbert space of -functions on . We shall consider the exponential functionse on given bye (x)=e ^i2·x . If these functions form an orthogonal basis for (), when ranges over some subset in ^d , then we say that (, ) is a spectral pair, and that is a spectrum. We conjecture that (, ) is a spectral pair if and only if the translates of some set by the vectors of tile ^d. In the special case of =I^d, the d-dimensional unit cube, we prove this conjecture, with =I^d, for d3, describing all the tilings by I^d, and for all d when is a discrete periodic set. In an appendix we generalize the notion of spectral pair to measures on a locally compact abelian group and its dual.Acknowledgements and Notes, Work supported by the National Science Foundation.Dedicated to the memory of Irving E. Segal     

Holomorphe Funktionen auf Gebieten über Banach-Räumen zu vorgegebenen Konvergenzradien

Given a function: ₊ on a domain spread over an infinite dimensional complex Banach space E with a Schauder basis such that -log is plurisubharmonic and d (d denotes the boundary distance on ) one can find a holomorphic function f: with _f, where _f is the radius of convergence of f. If, in addition, is locally Lipschitz continuous with constant 1, f can be chosen so that (3M)^–1 _f, where M is the basis constant of E. In the particular case of E= ₁ there are holomorphic functions f on with= _f.     

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.     

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.     

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.     

Compactness method in the finite element theory of nonlinear elliptic problems

Summary Finite element approximation of a nonlinear elliptic pseudomonotone second-order boundary value problem in a bounded nonpolygonal domain with mixed Dirichlet-Neumann boundary conditions is studied. In the discretization we approximate the domain by a polygonal one, use linear conforming triangular elements and evaluate integrals by numerical quadratures. We prove the solvability of the discrete problem and on the basis of compactness properties of the corresponding operator (which is not monotone in general) we prove the convergence of approximate solutions to an exact weak solutionuH ¹ ). No additional assumption on the regularity of the exact solution is needed.     

Total flux estimates for a finite element approximation of the Dirichlet problem using the boundary penalty method

Summary This paper considers a fully practical piecewise linear finite element approximation of the Dirichlet problem for a second order self-adjoint elliptic equation,Au=f, in a smooth region< ⁿ (n=2 or 3) by the boundary penalty method. Using an unfitted mesh; that is ^h , an approximation of with dist (, ^h )Ch ² is not in general a union of elements; and assuminguH ⁴ () we show that one can recover the total flux across a segment of the boundary of with an error ofO(h ²⁾. We use these results to study a fully practical piecewise linear finite element approximation of an elliptic equation by the boundary penalty method when the prescribed data on part of the boundary is the total flux.Supported by a SERC research studentship     

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.     

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).     

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.

     

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.     

Continuation of <Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>-holomorphic functions

Let be a bounded domain in Cⁿ and let L²_h() be the L²-holomorphic functions on . We show that the envelope of holomorphy and the L²_h()-envelope of holomorphy of differ by at most a pluripolar set.Mathematics Subject Classification (2000): 32D10, 32A36, 32U30This article is based on work the author did during his Ph.D. studies at the University of Michigan under the supervision of John Erik Fornæss.     

On polynomial variance functions

Summary Let be a natural exponential family on and (V, ) be its variance function. Here, is the mean domain of andV, defined on , is the variance of . A problem of increasing interest in the literature is the following: Given an open interval and a functionV defined on , is the pair (V, ) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, ) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.     

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 .     

Identities of an algebra of triangular matrices

This paper deals with the ideals of identities of certain associative algebras over a field F of characteristic zero. An algebra W of matrices of the form ,,,M, where and , are F-algebras with unity and M is a (,)-bimodule, is considered. Under certain natural restrictions on M one obtains the equality of ideals of identities T(W)=T()T(), if [[x₁,x₂], x₃[x₄,x₅]]T().Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 7–27, 1982.     

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.     

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.