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.     

Subsets with Restricted Movement

Let G be a finite permutation group on a set with no fixed points in and let m and k be integers with 0 < m < k. For a finite subset of the movement of is defined as move() = max_gG| ^g \ |. Suppose further that G is not a 2-group and that p is the least odd prime dividing |G| and move() m for all k-element subsets of . Then either || k + m or k (7m – 5) / 2, || (9m – 3)/2. Moreover when || > k + m, then move() m for every subset of .     

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].     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

Homogenization of Some Elasticity Problems with Rapidly Oscillating Convex Constraints on Displacements

The problem of homogenization is considered for an elastic body occupying a perforated domain = obtained from a fixed domain and an -contraction of a 1-periodic domain .     

Dimension de Hausdorff de la Nervure

For a separable Hilbert space E whose dimension is 2 and for an open subset of E, not empty and different from E, let be the set of all points of which have at least two projections on the close set E\, and let be the set of all the centres of the open balls contained in and which are maximal for inclusion. We show that the Hausdorff dimension dim_H( ) of may be any real value s such that 0sdim E; we also show that can be chosen so that is everywhere dense in and so that we have dim_H( )=1.Associons à un ouvert d'un espace de Hilbert séparable E de dimension 2, non vide et distinct de E, l'ensemble des points de admettant plusieurs projections sur le fermé E\, et l'ensemble des centres des boules ouvertes inclues dans et maximales pour l'inclusion. Nous montrons d'une part que la dimension de Hausdorff dim_H( ) de peut prendre toute valeur réelle s telle que 0sdim E, et d'autre part qu'on peut choisir de sorte que soit dense dans et qu'on ait dim_H( )=1.     

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.     

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.     

Spatial dynamics of time periodic solutions for the Ginzburg-Landau equation

We consider the dynamics of the Ginzburg-Landau equation in a small neighborhood of a known pulse solution by studying a Poincaré map,P: _{T T} , where _T is a section which is transverse to the pulse. Due to the fact that the Ginzburg-Landau equation possesses both a rotational symmetry and a spatial symmetry, we are able to conduct a detailed analytical study of this map in neighborhoods arbitrarily close to the pulse solution. Thus, we are able to complement the work of Holmes [8], who conducted an analytical study of the Poincaré map in a punctured neighborhood of the pulse. We find that the Poincaré map contains an invariant set _itT, where is not necessarily a Cantor set of points, such thatP: is homeomorphic to a shift map on (at least) two symbols. Furthermore, we find that for eachm 1 the mapP _itm possesses a fixed point. Since is not necessarily a Cantor set, this is not immediately clear. Finally, we find that when the pulse solution is broken, for eachm1 there exist parameter values such that pulses possessingm maxima appear.On leave at the University of Utah during 1993/94. Supported by the DFG, Habilitationsstipendium Ma 1587/1-1.     

Cofibrations of algebras

Summary The chain complex of a twisted free product A_*t, FK, is chain homotopy equivalent to a differential graded algebra, which is identified to be a confibration of algebras as defined by Quillen. Under certain connectivity conditions we obtain a long exact sequence connecting the homologies of A, K, and A_*t FK. In particular we derive a long exact sequence connecting the homologies of Y, X and (Y U_g CX) (, C, are the loop, the cone and the suspension constructions respectively). A chain complex equivalent to the chain complex of the Milnor free group FX is recognized, from which results a theorem of Bott and Samelson that H(X) is freely generated as a graded algebra by H(X).     

Approximation of differentiable functions by polynomial splines in domains with external peaks

For unbounded domains with external power-type peaks, we propose a method for the approximation of functionsf(x) w _p ^r () by polynomial splines in the metricw _p ^r (), 1pq, and present the corresponding estimates.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 9, pp. 1224–1233, September, 1994.     

On monotone extensions of boundary data

Summary A functionf C (), is called monotone on if for anyx, y the relation x – y ₊ ^s impliesf(x)f(y). Given a domain with a continuous boundary and given any monotone functionf on we are concerned with the existence and regularity ofmonotone extensions i.e., of functionsF which are monotone on all of and agree withf on . In particular, we show that there is no linear mapping that is capable of producing a monotone extension to arbitrarily given monotone boundary data. Three nonlinear methods for constructing monotone extensions are then presented. Two of these constructions, however, have the common drawback that regardless of how smooth the boundary data may be, the resulting extensions will, in general, only be Lipschitz continuous. This leads us to consider a third and more involved monotonicity preserving extension scheme to prove that, when is the unit square [0, 1]² in ², strictly monotone analytic boundary data admit a monotone analytic extension.Research supported by NSF Grant 8922154Research supported by DARPA: AFOSR #90-0323     

Parametric and non-parametric minima

We prove that if uBV() is a local minimum for the non-parametric functional F(u) associated to a parametric integrand g××ⁿ⁺¹[0,+), then the hypograph of u is locally minimizing for the parametric functionalG associated to g, among the sets of finite perimeter in ×.     

On Compensated Compactness for Nonlinear Elliptic Problems in Perforated Domains

We consider a sequence of Dirichlet problems for a nonlinear divergent operator A: W _m ¹( _s ) [W _m ¹( _s )]^* in a sequence of perforated domains _s . Under a certain condition imposed on the local capacity of the set \ _s , we prove the following principle of compensated compactness: , where r _s(x) and z _s(x) are sequences weakly convergent in W _m ¹() and such that r _s(x) is an analog of a corrector for a homogenization problem and z _s(x) is an arbitrary sequence from whose weak limit is equal to zero.     

Existence of quadrature surfaces for positive measures with finite support

Let be a positive measure with finite support in ⁿ (n2). Then we show that there is a bounded open set , containing the support of , whose single-layer potential coincides with the potential of outside .     

Computing a family of reproducing kernels for statistical applications

For an open subset of , an integer,m, and a positive real parameter , the Sobolev spacesH ^m () equipped with the norms: u²=u(t)²dt+(1/^2m u ^(m)(t)² constitute a family of reproducing kernel Hilbert spaces. When is an open interval of the real line, we describe the computation of their reproducing kernels. We derive explicit formulas for these kernels for all values ofm in the case of the whole real line, and form=1 andm=2 in the case of a bounded open interval.This research was partly supported by NSF Grant DMS-9002566.     

Strong solution for an elastic-plastic plate

Given a regular bounded open set R ²,, >0 andg L ^q () withq>2, we prove, under compatibility and safe load conditions ong, the existence of a minimizing pair for the functional, over closed setsK ² and functionsu C⁰( ) C²(/K); here ¦[Du]¦ denotes the jump ofDu acrossK and ¹ is the 1-dimensional Hausdorff measure.Dedicated to Enrico Magenes for his 70th birthday     

Linearization of operator functions on arbitrary open sets

In this paper it is shown that after a suitable extension an operator function A, which is holomorphic on an open set in , is equivalent on to a linear pencil S-V. If is bounded, then V turns out to be right invertible, and in that case a further extension is equivalent on to a linear pencil T-I.     

A characterization ofL n(K) as a permutation group

In 1972 M. O'Nan proved thatL _n (q),h 3; can be characterized as a doubly-transitive groupG on a finite set , whereG _a has an Abelian normal subgroup acting not semi-regularly on -a. In the Main Theorem we show that a similar statement holds if is infinite. Our result implies O'Nan's theorem.This paper is part of the author's Ph.D. thesis written under supervision of Prof. F. G. Timmesfeld.