A New Critical Behavior for Nonlinear Wave Equations

We study the inhomogeneous semilinear wave equations on with initial values and ,where is a noncompact, complete manifold. We founda new critical behavior in the following sense. There exists ap* > 0. When 1 < p p*, the above problem hasno global solution for any nonnegative not identicallyzero and for any and ; when the problem has a global solution for some and some and . If , which is equipped with the Euclideanmetric, then . If we show that belongs to the blow upcase. Although homogeneous semilinear wave equations are known to exhibit acritical behavior for a long time, this seems to be the first result oninhomogeneous ones.     

Simultaneous Preservation of Orthogonality of Polynomials by Linear Operators Arising from Dilation of Orthogonal Polynomial Systems

For an orthogonal polynomial system and a sequence of nonzero numbers,let be the linear operator defined on the linear spaceof all polynomials via for all .We investigate conditions on and under which can simultaneously preserve the orthogonality ofdifferent polynomial systems. As an application, we get that for , a generalized Laguerre polynomial system, no can simultaneously preserve the orthogonality of twoadditional Laguerre systems, and , where and . On the other hand, for ,the Chebyshev polynomial system and , simultaneously preserves the orthogonality of uncountablymany kernel polynomial systems associated with p. We study manyother examples of this type.     

The Order of Best Approximation in Some Classes of Functions

Starting from the equivalence between the Ditzian–Totik modulus and , where , in this article large classes of functions are introduced for which the modulus can be easily calculated. As a consequence, very good estimates for the bestapproximation are obtained. The attempts to estimate or calculate themodulus can be a very intricateproblem.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

A Comparison on the Commutative Neutrix Convolution of Distributions and the Exchange Formula

Let , be ultradistributions in and let and where is a sequence in which converges to the Dirac-delta function . Then the neutrix product is defined on the space of ultradistributions as the neutrix limit of the sequence provided the limit exist in the sense that

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

A Jackson Type Estimate for Shepard Operators in L p Spaces for p≧ 1

This paper considers the approximation of the Kantorovich–Shepard operators in spaces for . For the Kantorovich–Shepard operators are defined by (1.1). Then

A Canonical Directly Infinite Ring

Let be the set of nonnegative integers and the ring of integers. Let be the ring of N × N matrices over generated by the following two matrices: one obtained from the identity matrix by shifting the ones one position to the right and the other one position down. This ring plays an important role in the study of directly finite rings. Calculation of invertible and idempotent elements of yields that the subrings generated by them coincide. This subring is the sum of the ideal consisting of all matrices in with only a finite number of nonzero entries and the subring of generated by the identity matrix. Regular elements are also described. We characterize all ideals of , show that all ideals are finitely generated and that not all ideals of are principal. Some general ring theoretic properties of are also established.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation of an even codimension is the bundle of the complex structures of the fibers of the transversalbundle of . On there exists a foliation by covering spaces of the leaves of , and any Bottconnection of produces an ordered pair of transversal almost complex structures of . The existence of a Bott connection which yields a structure ₁ that is projectable to the space of leaves isequivalent to the fact that is a transversallyprojective foliation. A Bott connection which yields a projectablestructure ₂ exists iff isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, ₁is projectable as well. ₂ is never integrable.The essential integrability condition of ₁ isthe flatness of the transversal projective structure of .     

An Extension of a Class of Iterative Procedures for Nonlinear Quasivariational Inequalities

Consider the convergence of the projection methods based on an extension of a special class of algorithms for the approximation--solvability of the following class of nonlinear quasivariational inequality (NQVI) problems: find an element such that and

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .     

The Spectrum of the C*-Algebra of Pseudodifferential Boundary Value Problems on a Manifold with Smooth Edges

The C ^*-algebra generated by the operators of pseudodifferential boundary value problems on a manifold with smooth closed disjoint edges and boundary is studied. The operators act in the space L ₂( ) L ₂( ). The goal of this paper is to describe all (up to an equivalence) irreducible representations of the algebra Bibliography: 12 titles.     

Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

A Note on Extremes of Concomitants of Order Statistics

Let be a sequence of identically distributed variables. We study the asymptotic distribution of , where Y _[r:n] denotes the concomitant of the rth order statistic X _r:n , corresponding to , and is held fixed while . Conditions are given for the and to have the same asymptotic behavior as that we would apply if were i.i.d. The result is illustrated with a simple linear regression model , where is a stationary sequence with extremal index .     

K-Theory for the Banach Algebra of Operators on James''s Quasi-reflexive Banach Spaces

We prove that the K-groups of the Banach algebra of bounded, linear operators on the pth James space , where 1 < p < , are given by and . Moreover, for each Banach space and each non-zero, closed ideal contained in the ideal of inessential operators, we show that and . This enables us to calculate the K-groups of for each Banach space which is a direct sum of finitely many James spaces and -spaces.     

On the Spectrum of Degenerate Operator Equations

We study the distribution in the complex plane of the spectrum of the operator , generated by the closure in of the operation originally defined on smooth functions with values in a Hilbert space satisfying the Dirichlet conditions . Here and A is a model operator acting in . Criterial conditions on the parameter for the eigenfunctions of the operator to form a complete and minimal system as well as a Riesz basis in the Hilbert space H are given.     

On the Existence of Relative Injective Covers

Let be a hereditary torsion theory for the category -mod of unital left -modules over an associative ring with an identity element. The purpose of this note is to prove that if the associated Gabriel filter consists of finitely presented left ideals, then every module has a -injective cover and if contains a cofinal subset of finitely presented left ideals, then every module has a -torsionfree -injective cover. The methods used working with pure submodules contained in ``large" submodules also allow to unify the proofs of some previously known results.     

Test Rank for Some Free Polynilpotent Groups

We prove a theorem on possible test rank values for groups of the form . It is shown that test rank of a free polynilpotent group is equal to or , for any and every collection of classes. Moreover, for and .