On the Uniform Summability of Multiple Walsh-Fourier Series

In this paper we prove that if f C (0, 1 ^N ) and the function f is of bounded partial variation, then the N-dimensional Walsh-Fourier series of the function f is uniformly (C,–) summable (₁ +...+ _N < 1, _i > 0, i = 1,...,N) in the sense of Pringsheim. If ₁ +...+ _N = 1, _i > 0, i = 1,2,...,N, then there exists a continuous function f ₀ of bounded partial variation on [0, 1] ^N such that the Cesàro (C,–) means _m ^– (f₀,Õ) of the N-dimensional Walsh-Fourier series of f ₀ diverge over cubes.     

Turán's extremal problem in random graphs: Forbidding odd cycles

For 0<1 and graphsG andH, we writeGH if any -proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC ^k for a cycle of lengthk. We show that, for every fixed integerl1 and real >0, there exists a real constantC=C(l, ), such that almost every random graphG _{n, p} withp=p(n)Cn ^–1+1/2l satisfiesG _n,p1/2+ C ^2l+1. In particular, for any fixedl1 and >0, this result implies the existence of very sparse graphsG withG _1/2+ C ^2l+1.The first author was partially supported by NSERC. The second author was partially supported by FAPESP (Proc. 93/0603-1) and by CNP_q (Proc. 300334/93-1). The third author was partially sopported by KBN grant 2 1087 91 01.     

Lawless order

R. Baer asked whether the group operation of every (totally) ordered group can be redefined, keeping the same ordered set, so that the resulting structure is an Abelian ordered group. The answer is no. We construct an ordered set (G, ) which carries an ordered group (G, , ) but which islawless in the following sense. If (G, *, ) is an ordered group on the same carrier (G, ), then the group (G, *) satisfies no nontrivial equational law.Research partially supported by NSERC of Canada Grants #A4044 and A3040.Research partially supported by NSERC of Canada Grant #U0075.Research partially supported by a grant from the BSF.     

Criteria of regularity for norm-sequences

General sufficient conditions are given for a regularity property of norm-sequences. This regularity of the norm-sequence {T ⁿ } _nN makes possible to associate an isometryV with the operatorT in a similar way as it has been known in the power bounded case.Research partially supported by Hungarian NFS Research grant no. T 022920.     

Multilevel large deviations and interacting diffusions

Summary Let ( ^N ) be a sequence of random variables with values in a topological space which satisfy the large deviation principle. For eachM and eachN, let ^{M, N} denote the empirical measure associated withM independent copies of ^N . As a main result, we show that ( ^{M, N} ) also satisfies the large deviation principle asM,N. We derive several representations of the associated rate function. These results are then applied to empirical measure processes ^{M, N} (t) =M ^–1 _i=1 ^N _i ^N (t) 0tT, where ( ₁ ^N ,..., _M ^N (t)) is a system of weakly interacting diffusions with noise intensity 1/N. This is a continuation of our previous work on the McKean-Vlasov limit and related hierarchical models ([4], [5]).Research partially supported by a Natural Science and Engineering Research Council of Canada operating grant     

Convergence in distribution and Skorokhod Convergence for the general theory of processes

Summary This paper proves some Skorokhod Convergence Theorems for processes with filtration. Roughly, these are theorems which say that if a family of processes with filtration (X ⁿ , ⁿ ),n, converges in distribution in a suitable sense, then there exists a family of equivalent processes (Y ⁿ , ⁿ ),n, which converges almost surely. The notion of equivalence used is that of adapted distribution, which guarantees that each (X ⁿ , ⁿ ) has the same stochastic properties as (X ⁿ , ⁿ ), with respect to its filtration, such as the martingale property or the Markov property. The appropriate notion of convergence in distribution is convergence in adapted distribution, which is developed in the paper. Fortunately, any tight sequence of processes has a subsequence which converges in adapted distribution. For discrete time processes, (Y ⁿ , ⁿ ),n, and their limit (Y, ) may be taken as all having the same fixed filtration ⁿ =. In the continuous time case, theY ⁿ , ⁿ may require different filtrations ⁿ , which converge to. To handle this, convergence of filtrations is defined and its theory developed.During part of the time this work was in progress, it was supported by an NSERC operating grant, and the author was an NSERC University Research Fellow. The author wishes to thank the Steklov Mathematical Institute of the Soviet Academy of Sciences for its hospitality while the principle research in this paper was being begun, A.N. Shiryaev and P.C. Greenwood, who made the author's visit there possible, and Ján Miná for his hospitality while that research was being finished. We thank the referee who suggested the results in Sect. 12     

Round-off error analysis of iterations for large linear systems

Summary We deal with the rounding error analysis of successive approximation iterations for the solution of large linear systemsA _x =b. We prove that Jacobi, Richardson, Gauss-Seidel and SOR iterations arenumerically stable wheneverA=A ^*>0 andA has PropertyA. This means that the computed resultx _k approximates the exact solution with relative error of order A·A ^–1 where is the relative computer precision. However with the exception of Gauss-Seidel iteration the residual vector Ax _k –b is of order A² A ^–1 and hence the remaining three iterations arenot well-behaved.This work was partly done during the author's visit at Carnegie-Mellon University and it was supported in part by the Office of Naval Research under Contract N00014-76-C-0370; NR 044-422 and by the National Science Foundation under Grant MCS75-222-55     

Quasianapole Interaction of Charged Fermions

We obtain the analytic expression for the total cross section of the reaction e ^– e ⁺l ^– l ⁺ (l=,) taking possible quasianapole interaction effects into account. We find numerical restrictions on the interaction parameter value from data for the reaction e ^– e ^+–+ in the energy domain below the Z ⁰ peak.     

A note on the classification of holomorphic harmonic morphisms

In this note we give a complete classification of those holomorphic maps :U ⁿ defined on open and connected subsets of ^m which are harmonic morphisms.The first author was supported by the Icelandic Science Fund.     

Cartan-Grauert theorem for tuboids with “curvilinear” edge

Tuboids are tube type domains with totally real edge that are asymptotically approximated near the edge points by local tubes over convex cones. For these domains we prove an analog of the Cartan-Grauert theorem on holomorphic convexity of domains in ^{n n} .Translated fromMatematicheskie Zametki, Vol. 64, No. 6, pp. 888–901, December, 1998.The author wishes to express his gratitude to Professor A. G. Sergeev for setting the problem and for constant attention to this work.This research was supported by the Russian Foundation for Basic Research under grant No. 98-01-00640 and by the program Leading Scientific Schools under grant No. 96-15-96131.     

Extension of the Zorn lemma to general nontransitive binary relations

Let be an irreflexive (strict) binary relation on a nonempty setX. Denote the completion of by , i.e.,yx ifxy does not hold. An elementx ^* X is said to be a maximal element of onX ifx ^* x, xX. In this paper, an extension of the Zorn lemma to general nontrasitive binary relations (may lack antisymmetry) is established and is applied to prove existence of maximal elements for general nontrasitive (reflexive or irreflexive) binary relations on nonempty sets without assuming any topological conditions or linear structures. A necessary and sufficient condition has been also established to completely characterize the existence of maximal elements for general irreflexive nontrasitive binary relations. This is the first such result available in the literature to the best of our knowledge. Many recent known existence sults in the literature for vector optimization are shown to be special cases of our result.This work was supported in part by AFSOR Grant 91-0097.The author is grateful to the referees and Professor P. L. Yu for their comments and suggestions that led to this improved paper.     

Ergodic properties of semi-Markovian operators on the Z1-part

Summary In this note we consider a semi-Markovian operator, that is a positive linear mapping T: L ¹ L ¹ such that sup T ⁿ <. We study the behavior of T ⁿ on the Z ¹-part of the space (the disappearing part in Sucheston's terminology). We show in particular, that if the operator T has a non-trivial conservative part in Z ¹, then the ratio theorem must fail.Research supported by the U.S.Army Research Office (Durham) under contract DA-31-124-ARO(D)-288.     

Some remarks on the modulus of continuity of a conformal mapping of the disk onto a Jordan domain

Letd(;z, t) be the smallest diameter of the arcs of a Jordan curve with endsz andt. Consider the rapidity of decreasing ofd(;)=sup{d(;z, t):z, t , ¦z–t¦} (as 0,0) as a measure of nicety of . Letg(x) (x0) be a continuous and nondecreasing function such thatg(x)x,g(0)=0. Put¯g(x)=g(x)+x, h(x)=(¯g(x))². LetH(x) be an arbitrary primitive of 1/h ^–1(x). Note that the functionH ^–1 x is positive and increasing on (–, +),H ^–1 0 asx– andH ^–1+ asx +. The following statement is proved in the paper.Translated fromMatematicheskie Zametki, Vol. 60, No. 2, pp. 176–184, August, 1996.This research was supported by the Russian Foundation for Basic Research under grant No. 93-01-00236 and by the International Science Foundation under grant No. NCF000.     

Schwache Teilbarkeitshalbgruppen

In this paper we continue the study of Tbk-semigroups [1]. We define a weak Tbk-semigroup as an algebra (s,·,) which satisfies (A1) a(bc) = (ab)c, (A2) (S, ) is a semilattice, (A3) × (a b) y = xay xby, (A4) a (a b) S S (a b) . We shall prove that every weak Tbk-semigroup can be embedded into a Tbk-semigroup with 1, and we shall show that the theory of [1] can be developed already in weak Tbk-semigroups.

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Über den Inhalt dieser Note berichtete der Autor verschiedene Male , zuletzt im Juni 1974 an der T.U. München.     

On the problem of two linearized wells

We study the behaviour of sequences of elastic deformationsy ^{n n} whose gradients approach two linearized wells, and give an application to magnetostriction.This article was processed by the author using the style filepljour1m from Springer-Verlag.     

Problems with one quarter

In this paper two sequences of oscillation criteria for the self-adjoint second order differential equation (r(t)u(t)) + p(t)u(t) = 0 are derived. One of them deals with the case dt/r(t) = , and the other with the case dt/r(t) < .This work was supported by the grant VGA of Slovak Republic No. 1/7466/20.     

Sur certaines algèbres associées à une mesure de Guelfand

Let be a Guelfand measure (cf. [A, B]) on a locally compact groupG DenoteL ₁ (G)=*L ₁(G)* the commutative Banach algebra associated to . We show thatL ₁ (G) is semi-simple and give a characterization of the closed ideals ofL ₁ (G). Using the -spherical Fourier transform, we characterize all linear bounded operators inL ₁ (G) which are invariants by -translations (i.e. such that ¹(( _x f) )=( _x ((f)) for eachxG andfL ₁ (G); where _x f(y)=f(xy); x,y G). WhenG is compact, we study the algebraL ₁ (G) and obtain results analogous to ones obtained for the commutative case: we show thatL ₁ (G) is regular, all closed sets of its Guelfand spectrum are sets of synthesis and establish theorems of harmonic synthesis for functions inL _p (G) (p=1,2 or +).

     

The structure of crossed productC *-algebras: A proof of the generalized Effros-Hahn conjecture

IfG is a second countable locally compact group acting continuously on a separableC ^*-algebraA, then every primitive ideal of the crossed productC ^* (G, A) is contained in an induced primitive ideal, and ifG is amenable, equality holds. Thus ifG is amenable and acts freely on Prim(A), the generalized Effros-Hahn conjecture holds: there is a canonical bijection between primitive ideals ofC ^* (G, A) andG-quasi-orbits in Prim(A). Applications to the Mackey machine for a non-regularly embedded normal subgroup of a locally compact group are discussed. The proof of the theorem is based on a local cross-section result together with Mackey's original methods.The authors were partially supported by National Science Foundation Research GrantsThe first-named author would like to thank the Department of Mathematics, University of Pennsylvania, for its warm hospitality during his 1977–78 stay, during which time this research was conducted     

Path properties of kernel generated two-time parameter Gaussian processes

Summary We study path properties of two-parameter Gaussian processes {X(t,v),t R} of the formX(t,v)= _{0 –} (t,v,x,y)dW(x,y), where the kernel function (t, v, x, y) is assumed to be square integrable in (x, y) onR×R ⁺, andW(x, y) is a standard two-parameter Wiener process.Work partially supported by an NSERC Canada Operating Grant at Carleton UniversityWork supported by an NSERC Canada International Scientific Exchange Award at Carleton University and by National Natural Science Foundation of China     

A variational principle for eigenvalues of pencils of Hermitian matrices

LetH=(A, B) be a pair of HermitianN×N matrices. A complex number is an eigenvalue ofH ifdet(A–B)=0 (we include = ifdetB=0). For nonsingularH (i.e., for which some is not an eigenvalue), we show precisely which eigenvalues can be characterized as _k ⁺ =sup{inf{*A:*B=1,S},SS _k},S _k being the set of subspaces of C ^N of codimensionk–1.Dedicated to the memory of our friend and colleague Branko NajmanResearch supported by NSERC of Canada and the I.W.Killam FoundationProfessor Najman died suddenly while this work was at its final stage. His research was supported by the Ministry of Science of CroatiaResearch supported by NSERC of Canada