Some Landau's type inequalities for infinitesimal generators

Summary Lett T(t) be a strongly continuous contraction semigroup on a complex Banach space and letA be its infinitesimal generator. We prove that, forx D(A ³), the following inequalities hold true: Ax³ 243/8 x²A ³ x, A ² x 24 xA ³ x². Ift T(t) is a contraction group (resp. cosine function) we get the analogous but better inequalities with constants 9/8 and 3 (resp. 81/40 and 72/25) instead of 243/8 and 24. We consider also uniformly bounded semigroups, groups and cosine functions.     

Maximum monoreflections

It is shown that, in a category with a specified class of monics and under some mild hypothesis,there is a monoreflection maximum among those whose reflection maps lie in . Thus, for example, any variety, and most SP-classes in a variety, have both amaximum monoreflection and amaximum essential reflection (which might be the same, but frequently aren't, and which might be the identity functor, but frequently aren't). And, for example, under some mild hypotheses, beneath each completion lies a maximum monoreflection, so that, for example, any category of rings has amaximum functorial ring of quotients.     

Logarithms and imaginary powers of closed linear operators

The imaginary powersA ^it of a closed linear operatorA, with inverse, in a Banach spaceX are considered as aC ₀-group {exp(itlogA);t R} of bounded linear operators onX, with generatori logA. Here logA is defined as the closure of log(1+A) – log(1+A ^–1). LetA be a linearm-sectorial operator of typeS(tan ), 0(/2), in a Hilbert spaceX. That is, |Im(Au, u)| (tan )Re(Au, u) foru D(A). Then ±ilog(1+A) ism-accretive inX andilog(1+A) is the generator of aC ₀-group {(1+A) ^it ;t R} of bounded imaginary powers, satisfying the estimate (1+A) ^it exp(|t|),t R. In particular, ifA is invertible, then ±ilogA ism-accretive inX, where logA is exactly given by logA=log(1+A)–log(1+A ^–1), and {A ^it;t R} forms aC ₀-group onX, with the estimate A ^it exp(|t|),t R. This yields a slight improvement of the Heinz-Kato inequality.     

Weyl's theorem for operator matrices

Weyl's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison Browder's theorem holds for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl's theorem and Browder's theorem are liable to fail for 2×2 operator matrices. In this paper we explore how Weyl's theorem and Browder's theorem survive for 2×2 operator matrices on the Hilbert space.Supported in part by BSRI-97-1420 and KOSEF 94-0701-02-01-3.     

О предельном распределении для сумм независимых слууайных велиуин на бикомпактных коммутативных топологиуеских группах

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Solution of the Truncated Parabolic Moment Problem

Given real numbers with ₀₀ >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that We prove that admits a representing measure (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card In this case, admits a rank -atomic (minimal) representing measure.Submitted: August 25, 2003     

A Strong Maximum Principle for some quasilinear elliptic equations

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain ⁿ ,n 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of – u + (u) = f with a nondecreasing function ,(0)=0, andf0 a.e. in if and only if the integral((s)s) ^–1/2 ds diverges ats=0+. We extend the result to more general equations, in particular to – _p u + (u) =f where _p (u) = div(|Du| ^p-2 Du), 1 <p < . Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.This work was partly done while the author was visiting the University of Minnesota as a Fulbright Scholar.     

Joint torsion of Toeplitz operators with H∞ symbols

We give a new² index theorem for the basic example of Toeplitz operators on the circle. The joint torsion, a non zero complex valued analytic index, of a pair of Fredholm Toeplitz operatorsT andT withH symbols is computed by residues in the disk, and is determined by a monodromy integral which specifies the isomorphism class of a flat line bundle on the circle. When the symbols and are rational a product of joint torsions identifies the isomorphism class of the bundle inH ¹ (S ¹,C ^*), and the identification extends by rational approximation to the case of smooth symbols defined on the circle.Partially supported by National Science Foundation grants to both authors.     

Compact and weakly compact weighted composition operators on weighted spaces of continuous functions

In this note we characterize the compact and weakly compact weighted composition operatorsW _, on certain weighted locally convex spacesCV _o(X, E) of vector-valued continuous functions induced by self maps ofX and the operator-valued mappings XB(E).The work of this author was supported in part by CSIR Grant 9/100/92-EMR-IThe work of this author was supported in part by UGC Grant F.8-7/91 (RBB-II)     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

Harmonic analysis on tori

We consider measurable subsets {ofR}ⁿ with 0<m()<, and we assume that has a spectral set . (In the special case when is also assumed open, may be obtained as the joint spectrum of a family of commuting self-adjoint operators {H _k: 1kn} in L ² () such that each H _k is an extension of i(/x _k) on C _c (), k=1, ..., n.)It is known that is a fundamental domain for a lattice if is itself a lattice. In this paper, we consider a class of examples where is not assumed to be a lattice. Instead is assumed to have a certain inhomogeneous form, and we prove a necessary and sufficient condition for to be a fundamental domain for some lattice in {ofR}ⁿ. We are thus able to decide the question, fundamental domain or not, by considering only properties of the spectrum . Our criterion is obtained as a corollary to a theorem concerning partitions of sets which have a spectrum of inhomogeneous form.Work supported in part by the NSF.Work supported in part by the NSRC, Denmark.     

A two‐level queueing network model with blocking and non‐blocking messages

We analyze a novel twolevel queueing network with blocking, consisting of N level1 parallel queues linked to M level2 parallel queues. The processing of a customer by a level1 server requires additional services that are exclusively offered by level2 servers. These level2 servers are accessed through blocking and nonblocking messages issued by level1 servers. If a blocking message is issued, the level1 server gets blocked until the message is fully processed at the level2 server. The queueing network is analyzed approximately using a decomposition method, which can be viewed as a generalization of the wellknown twonode decomposition algorithm used to analyze tandem queueing networks with blocking. Numerical tests show that the algorithm has a good accuracy.     

Produktsätze für Verfahren zur Limitierung von Doppelfolgen

{p _mn } - ₀₀>0, (1, 1) (1.1) (1.2). {s _mn } J _p - ( bJ _p -lims _mn =), (1.3) 0<x,y<1 p _s (, )/p(x, y) x, y 1-. {r _mn } - , (1.5) 0<, <1. N _rp - , (1.6). , bJ _p -lims _mn = bJ _q -lim(N _rps) _mn =. J _p - . , .     

Stochastic analysis of the fracture of solids with microcracks

The present study is concerned with the stochastic formulation of the failure of solids with microcracks. The analysis is directed towards the development of microcracks in a given material structure due to the application of an external constant load. The formulation is based on the theory of stochastic population processes and that of point processes. The consideration of certain aspects of the extreme value theory is however necessary in characterizing the evolution of the microcrack population. It is shown to be convenient to first consider a collection of microcracks in a specific material domain (microdomain) which is regarded as a family of objects and represented by its state so that: {z₁, z₂ ..., z _n }=z Z , where Z is the state space and a general probabilistic function space. Governing equations for the development of microcracks are derived and the results of the analysis are then compared with experimental observations obtained from short-time creep tests on a class of high temperature materials.     

Isomorphismen von rechtseiträumen

Spaces called rectangular spaces were introduced in [5] as incidence spaces (P,G) whose set of linesG is equipped with an equivalence relation and whose set of point pairs P² is equipped with a congruence relation , such that a number of compatibility conditions are satisfied. In this paper we consider isomorphisms, automorphisms, and motions on the rectangular spaces treated in [5]. By an isomorphism of two rectangular spaces (P,G, , ) and (P,G, , ) we mean a bijection of the point setP onto P which maps parallel lines onto parallel lines and congruent points onto congruent points. In the following, we consider only rectangular spaces of characteristic 2 or of dimension two. According to [5] these spaces can be embedded into euclidean spaces. In case (P,G, , ) is a finite dimensional rectangular space, then every congruence preserving bijection ofP onto P is in fact an isomorphism from (P,G, , ) onto (P,G, , ) (see (2.4)). We then concern ourselves with the extension of isomorphisms. Our most important result is the theorem which states that any isomorphism of two rectangular spaces can be uniquely extended to an isomorphism of the associated euclidean spaces (see (3.2)). As a consequence the automorphisms of a rectangular space (P,G, , ) are precisely the restrictions (onP) of the automorphisms of the associated euclidean space which fixP as a whole (see (3.3)). Finally we consider the motions of a rectangular space (P,G, , ). By a motion of(P. G,, ) we mean a bijection ofP which maps lines onto lines, preserves parallelism and satisfies the condition((x), (y)) (x,y) for allx, y P. We show that every motion of a rectangular space can be extended to a motion of the associated euclidean space (see (4.2)). Thus the motions of a rectangular space (P,G, , ) are seen to be the restrictions of the motions of the associated euclidean space which mapP into itself (see (4.3)). This yields an explicit representation of the motions of any rectangular plane (see (4.4)).

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Herrn Professor Burau zum 85. Geburtstag gewidmet     

Waterman classes and spherical partial sums of double Fourier series

f(x,y) ₀BV(T²), ={1/n} _n=1 .

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Dedicated to Professor Károly Tandori, the outstanding mathematician and academician on his seventieth birthday

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This work was done under the financial support of the Russian Foundation for Fundamental Scientific Research, Grant 93-01-00240.     

Intrinsic ultracontractivity of Dirichlet Laplacians in non-smooth domains

We give a general criterion for the intrinsic ultracontractivity of Dirichlet Laplacians – _D on domainsD ofR ^d d 3, based on the Lieb's formula. It applies to various classes of domains (e.g. John, Hölder andL ^p-averaging domains) and gives new conditions for intrinsic ultracontractivity in terms of the Minkowski dimension of the boundary D. In particular, isotropic self-similar fractals and domains satisfying a c-covering condition are considered.     

Circularity of Finite Groups without Fixed Points

Let be a fixed point free group given by the presentation where and are relative prime numbers, t = /s and s = gcd( – 1,), and is the order of modulo . We prove that if (1) = 2, and (2) is embeddable into the multiplicative group of some skew field, then is circular. This means that there is some additive group N on which acts fixed point freely, and |((a)+b)((c)+d)| 2 whenever a,b,c,d N, a0c, are such that (a)+b(c)+d.     

Complete stability of large order statistics

Fix an integerr1. For eachnr, letM _nr be the rth largest ofX ₁,...,X _n, where {X _n,n1} is a sequence of i.i.d. random variables. Necessary and sufficient conditions are given for the convergence of _n=r n P[|M _nr /a _n –1|<] for every >0, where {a _n} is a real sequence and –1. Moreover, it is shown that if this series converges for somer1 and some >–1, then it converges for everyr1 and every >–1.     

On the limit superior of analytic sets

. , A ₀,A ₁,— - lim supA _j - H, . , - - . , , ; , , . - . - .