On the admissibility of function spaces of typeB p,q s andF p,q s,and boundary value problems for non-linear partial differential equations

u=f(x)+S(u), S — , u-G(u), G — . B _p,q ^s () -F _p,q ^s (). R _n — . — . _p,q ^s F _p,q ^s .     

The state density of elliptic operators with random potentials

Summary The existence of the state density N() is established for certain linear elliptic coercive selfadjoint operators subject to perturbation by a unbounded random potential and its asymptotic properties discussed as (and for certain Gaussian perturbations as — ). The estimates depend on recent worlc by Métivier on the approximation numbers of the classical Sobolev spaces.The research of the author was supported by F.I.N.E.P. at U.F.R.G.S. under contract B/31/79/183/00/00 and by the CNPq.     

Cycles and measure of bifurcation sets for two-dimensional diffeomorphisms

Summary We describe a large class of one-parameter families , {}, , of two-dimensional diffeomorphisms which arestable for <0, exhibit acycle for =0, and thereafter have a bifurcation set of positive but arbitrarily smallrelative measure for in small intervals [0, ]. A main assumption is that the basic sets involved in the cycle havelimit capacities that are not too large.The second author acknowledges hospitality and financial support from IMPA/CNPq during the period this paper was prepared     

Differentiability properties of the minimal average action

Given aZ ⁿ⁺¹-periodic variational principle onR ⁿ⁺¹ we look for solutionsu:R ⁿ R minimizing the variational integral with respect to compactly supported variations. To every vector R ⁿ we consider a subset of solutions which have an average slope when averaging overR ⁿ. The minimal average action A() is defined by the average value of the variational integral given by a solution with average slope . Our main result is:A is differentiable at if and only if the set is totally ordered (in the natural sense). In case that is not totally ordered,A is differentiable at in some direction R ⁿ{0} if and only if is orthogonal to the subspace defined by the rational dependency of . Assuming that the i^th component of is rational with denominator sⁱ N in lowest terms, we show: The difference of right- and left-sided derivative in the i^th standard unit direction is bounded by const · .     

Moser-type inequalities in Lorentz spaces

Moser-type estimates for functions whose gradient is in the Lorentz space L(n, q), 1q, are given. Similar results are obtained for solutions uH _inf0 ^sup1 of Au=(f _i ) _{x i} , where A is a linear elliptic second order differential operator and |f|L(n, q), 2q.Work partially supported by MURST (40%).     

On strong summability of polynomial expansions

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Dedicated to Professor L. Leindler on his 50^th birthday     

Reiter’s Condition <Emphasis Type="Italic">P</Emphasis><Subscript>1</Subscript> and Approximate Identities for Polynomial Hypergroups

Let K be a commutative hypergroup with the Haar measure . In the present paper we investigate whether the maximal ideals in L¹(K,) have bounded approximate identities. We will show that the existence of a bounded approximate identity is equivalent to the existence of certain functionals on the space L(K,). Finally we apply the results to polynomial hypergroups and obtain a rather complete solution for this class.The third author was partially supported by KBN (Poland) under grant 5 P03A 034 20 and by Research Training Network Harmonic Analysis and Related Problems Contract HPRN-CT-2001-00273.     

Detection of a signal of known shape in a multichannel system

We consider an n-channel signal detection system. Each of its channels may contain (or not contain) a signal. We assume that the signal is a function of known shape observed in the white Gaussian noise of level > 0. Let k be the number of channels containing signals. We study the asymptotically minimax variant of the detection problem as n depending on k and on the relation signal-noise in the channels. Bibliography: 9 titles.Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 294, 2002, pp. 88–112.This research was partially supported by the Russian Foundation for Basic Research, grants 00-01-00111, 00-15-96019, 02-01-00262, and 02-01-04001.Translated by Yu. I. Ingster and I. A. Suslina.     

A Hubert space characterization among function spaces

— [0,1] ,E — - e=1 [0,1]. I — _E =1, E=L ₂ x _e =xL ₂ x E.

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This work was prepared when the second author was a visiting professor of the CNR at the University of Firenze. He was supported by the Soros International Fund.     

The Feynman-Kac formula with a Lebesgue-Stieltjes measure: An integral equation in the general case

Let u(t) be the operator associated by path integration with the Feynman-Kac functional in which the time integration is performed with respect to an arbitrary Borel measure instead of ordinary Lebesgue measurel. We show that u(t), considered as a function of time t, satisfies a Volterra-Stieltjes integral equation, denoted by (*). We refer to this result as the Feynman-Kac formula with a Lebesgue-Stieltjes measure. Indeed, when n=l, we recover the classical Feynman-Kac formula since (*) then yields the heat (resp., Schrödinger) equation in the diffusion (resp., quantum mechanical) case. We stress that the measure is in general the sum of an absolutely continuous, a singular continuous and a (countably supported) discrete part. We also study various properties of (*) and of its solution. These results extend and use previous work of the author dealing with measures having finitely supported discrete part (Stud. Appl. Math.76 (1987), 93–132); they seem to be new in the diffusion (or imaginary time) as well as in the quantum mechanical (or real time) case.Research partially supported by the National Science Foundation under Grant DMS 8703138. This work was also supported in part by NSF Grant 8120790 at the Mathematical Sciences Research Institute in Berkeley, U.S.A., the CNPq and the Organization of Latin American States at theInstituto de Matemática Pura E Aplicada (IMPA) in Rio de Janeiro, Brazil, as well as theUniversité Pierre et Marie Curie (Paris VI) and the Université Paris Dauphine in Paris, France.     

Orthogonal systems invariant under an automorphism

, , - ( ) . , - , ( ) .     

Einschließungsaussagen für gewöhnliche Differentialoperatoren

Summary For differential operatorsM of second order (as defined in (1.1)) we describe a method to prove Range-Domain implications—Mu–u and an algorithm to construct these functions , , , . This method has been especially developed for application to non-inverse-positive differential operators. For example, for non-negativea ₂ and for given functions = we require =C ₀[0, 1] C ₂([0, 1]–T) whereT is some finite set), (M) (t)(t), (t[0, 1]–T) and certain additional conditions for eachtT. Such Range-Domain implications can be used to obtain a numerical error estimation for the solution of a boundary value problemMu=r; further, we use them to guarantee the existence of a solution of nonlinear boundary value problems between the bounds- and .     

Асимптоический Анализ Распределения Статистики Диксона

. u 60-u u     

Compressing Coefficients While Preserving Ideals in K-Theory for C*-Algebras

An invariant based on orderedK-theory with coefficients in _n>1 /n and an infinite number of natural transformations has proved to be necessary and sufficient to classify a large class of nonsimple C* -algebras. In this paper, we expose and explain the relations between the order structure and the ideals of the C* -algebras in question.As an application, we give a new complete invariant for a large class of approximately subhomogeneous C*-algebras. The invariant is based on ordered K-theory with coefficients in /. This invariant is more compact (hence, easier to compute) than the invariant mentioned above, and its use requires computation of only four natural transformations.     

Determinants in net subgroups

Suppose R is a commutative ring with 1, =( _ij ) is a fixedD-net of ideals of R of ordern, and G is the corresponding net subgroup of the general linear group GL (n, R). There is constructed for a homomorphismdet of the subgroup G() into a certain Abelian group (). Let I be the index set {1...,n}. For each subset I let ()= _{ij ji} , wherei, ranges over all indices in and j independently over the indices in the complement I ((I) is the zero ideal). Letdet (a) denote the principal minor of order ||n of the matrixa G () corresponding to the indices in , and let' () be the Cartesian product of the multiplicative groups of the quotient rings R/() over all subsets I. The homomorphismdet is defined as follows: It is proved that if R is a semilocal commutative Bezout ring, then the kernelKer det coincides with the subgroup E() generated by all transvections in G(). For these R is also definedTm det .Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 114, pp. 37–49, 1982.     

On the exactness of a theorem of G.A. Fomin

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Divergence of orthogonal series to +∞

An overall bounded orthonormal set of functions _n (x) is constructed for which there exists a series _m =1 a_nn(x) with coefficients a_n=o(ln'n/n), which diverges to + almost everywhere. The bibliography contains 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 483–494, November, 1967.     

Approximation of functions of several variables by trigonometric polynomials with given number of harmonics,and estimates of ε-entropy

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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.     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.