On a theorem of lovász on covers inr-partite hypergraphs

A theorem of Lovász asserts that (H)/*(H)r/2 for everyr-partite hypergraphH (where and * denote the covering number and fractional covering number respectively). Here it is shown that the same upper bound is valid for a more general class of hypergraphs: those which admit a partition (V ₁, ...,V _k ) of the vertex set and a partitionp ₁+...+p _k ofr such that |eV _i |p _i r/2 for every edgee and every 1ik. Moreover, strict inequality holds whenr>2, and in this form the bound is tight. The investigation of the ratio /* is extended to some other classes of hypergraphs, defined by conditions of similar flavour. Upper bounds on this ratio are obtained fork-colourable, stronglyk-colourable and (what we call)k-partitionable hypergraphs.Supported by grant HL28438 at MIPG, University of Pennsylvania, and by the fund for the promotion of research at the Technion.This author's research was supported by the fund for the promotion of research at the Technion.     

Covering times of random walks on bounded degree trees and other graphs

The motivating problem for this paper is to find the expected covering time of a random walk on a balanced binary tree withn vertices. Previous upper bounds for general graphs ofO(|V| |E|)⁽¹⁾ andO(|V| |E|/d _min)⁽²⁾ imply an upper bound ofO(n ²). We show an upper bound on general graphs ofO( |E| log |V|), which implies an upper bound ofO(n log² n). The previous lower bound was (|V| log |V|) for trees.⁽²⁾ In our main result, we show a lower bound of (|V| (log _{d max} |V|)²) for trees, which yields a lower bound of (n log² n). We also extend our techniques to show an upper bound for general graphs ofO(max{E T_i} log |V|).     

Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

In this paper we study the relationship _V (M)=E(1 _M dV _S ) between operatorvalued processesV with finite variation V and operator-valued stochastic measures _V with finite variation | _V |. The variations satisfy the inequality | _V | _|V|, which, under certain conditions, is an equality (for example, ifV is measurable).     

Carleson measure characterization of Bloch functions

We give several equivalences of Bloch functions and little Bloch functions. Using these results we obtain the generalized Carleson measure characterization of Bloch functions and the generalized vanishing Carleson measure characterization of little Bloch functions, that is,f B if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized Carleson measure;f B ₀ if and only if |D f(z)| ^p (1-|z|²)^p-1 dm(z) is a generalized vanishing Carleson measure, whereD f( > 0) is the fractional derivative of analytic functionf of order, m denotes the normalised Lebesgue measure.Supported partly by the Young Teacher Natural Science Foundation of Shandong Province.     

Factorization intok-dimensional linear bijections

LetV be a vector space,k withkdimV andS _k{GL(V)|dimV(–1)=k}. ThenS _k generates GL _f (V){GL(V)|V(-1) is finite-dimensional} (with the exception that dimV=2=k and the field is GF2). We study the length problem in GL _f (V) withS _k as set of generators.     

Bijectivity of -endomorphisms of ℬ(H) and the unilateral shift

LetH be a separable infinite-dimensional complex Hilbert space. We prove that if : (H)(H) is a^*-preserving ring homomorphism whose range contains a rank-one operator and an operator with dense range, then is an isometric linear or conjugate-linear algebra automorphism of (H). In particular, if the unilateral shift is contained in the range of a^*-endomorphism of (H), then is bijective.Research partially supported by the Hungarian National Research Science Foundation, Operating Grant Number OTKA 1652 and K&H Bank Ltd., Universitas Foundation.     

The endomorphism spectrum of an ordered set

Theendomorphism spectrum of an ordered setP, spec(P)={|f(P)|:f End(P)} andspectrum number, sp(P)=max(spec(P)\{|P|}) are introduced. It is shown that |P|>(1/2)n(n – 1) ^{n – 1} implies spec(P) = {1, 2, ...,n} and that if a projective plane of ordern exists, then there is an ordered setP of size 2n ²+2n+2 with spec(P)={1, 2, ..., 2n+2, 2n+4}. Lettingh(n)=max{|P|: sp(P)n}, it follows thatc ₁ n ²h(n)c ₂ n ⁿ⁺¹ for somec ₁ andc ₂. The lower bound disproves the conjecture thath(n)2n. It is shown that if |P| – 1 spec(P) thenP has a retract of size |P| – 1 but that for all there is a bipartite ordered set with spec(P) = {|P| – 2, |P| – 4, ...} which has no proper retract of size|P| – . The case of reflexive graphs is also treated.Partially supported by a grant from the NSERC.Partially supported by a grant from the NSERC.     

Two combinatorial problems on posets

Bialostocki proposed the following problem: Let nk2 be integers such that k|n. Let p(n, k) denote the least positive integer having the property that for every poset P, |P|p(n, k) and every Z _k -coloring f: P Z _k there exists either a chain or an antichain A, |A|=n and _aA f(a) 0 (modk). Estimate p(n, k). We prove that there exists a constant c(k), depends only on k, such that (n+k–2)²–c(k) p(n, k) (n+k–2)²+1. Another problem considered here is a 2-dimensional form of the monotone sequence theorem of Erdös and Szekeres. We prove that there exists a least positive integer f(n) such that every integral square matrix A of order f(n) contains a square submatrix B of order n, with all rows monotone sequences in the same direction and all columns monotone sequences in the same direction (direction means increasing or decreasing).     

Length functions of lemniscates

We study metric and analytic properties of generalized lemniscates E _t (f)={z:ln|f(z)|=t}, where f is an analytic function. Our main result states that the length function |E _t (f)| is a bilateral Laplace transform of a certain positive measure. In particular, the function ln|E _t (f)| is convex on any interval free of critical points of ln|f|. As another application we deduce explicit formulae of the length function in some special cases.The author was supported the Göran Gustafsson foundation and grant RFBR no. 03-01-00304.The author was supported by Russian President grant for young doctorates no. 00-15-99274 and grant RFBR no. 03-01-00304. Mathematics Subject Classification (2000):30E05, 42A82, 44A10     

Formule d'Itô pour des Diffusions Uniformément Elliptiques,et Processus de Dirichlet

Soit X une diffusion uniformément elliptique sur R ^d ,F une fonction dans H _loc ¹(R ^d ) et la loi initiale de la diffusion. On montre que si l'intégrale |F|²(x)U(x)dx est finie, oùU désigne le potentiel de la mesure , alors F(X) est un processus de Dirichlet. Si de plus, F appartient àH ² _loc(R ^d ) et si les intégrales |F|²(x)U(x)dx et |f _k |²(x)U(x)dx sont finies, pour les dérivées faibles f _k de F, alors on peut écrire une formule d'Itô. En particulier, on définit l'intégrale progressive F(X)dX et on prouve l'existence des covariations quadratiques [f _k (X),X ^k ].     

A Comparative Result for Multiplicative Functions

In this paper, we compare the asymptotic behavior of _nx f(n) and _nx g(n) for multiplicative functions f and g, respectively, where |f| g. Our results extend relevant theorems by E. Wirsing and G. Hal@aacute;sz. The methods we use are elementary.     

Codings of graphs with binary edge labels

LetG(V,E) be a graph. A mappingf:E{0,1} ^m is called a (binary) coding ofG, if the induced mapping , assigns different vectors to the vertices. For the Boolean sum,f is called aB-code, and for the mod 2 sum anM-code. Letm _B (G) resp.m _M (G) be the smallest lengthm for whichB-codes resp.M-codes are possible. Trivially,m _B (G),m _M (G) log₂|V|. Improving results of Z. Tuza we showm _B (G)log₂|V| + 1,m _M (G)log₂|V|+4.     

The erdös-dushnik-miller theorem for topological graphs and orders

A topological graph is a graph G=(V, E) on a topological space V such that the edge set E is a closed subset of the product space V x V. If the graph contains no infinite independent set then, by a well-known theorem of Erdös, Dushnik and Miller, for any infinite set LV, there is a subset LL of the same oardinality |L| = |L| such that the restriction G L is a complete graph. We investigate the question of whether the same conclusion holds if we weaken the hypothesis and assume only that some dense subset AV does not contain an infinite independent set. If the cofinality cf (|L|)>|A|, then there is an L as before, but if cf (|L|)<-|A|, then some additional hypothesis seems to be required. We prove that, if the graph GA is a comparability graph and A is a dense subset, then for any set LV such that cf (|L|)>, there is a subset LL of size |L|=|L| such that GL is complete. The condition cf (|L|)> is needed.Research supported by NSERC grant #A5198.     

Smoothness of solutions of a nonlinear ode

Smoothness of aC -functionf is measured by (Carleman) sequence {M _k} ₀ ; we sayfC _M [0, 1] if|f ^(k) (t)|CR ^k M _k,k=0, 1, ... withC, R>0. A typical statement proven in this paper isTHEOREM: Let u, b be two C -functions on [0, 1]such that (a) u=u ²+b, (b) |b ^(k) (t)|CR ^k (k!) , >1,k_–.Then |u^(k)(t)|C₁R^k((k–1)!),k_–.The first author acknowledges the hospitality of Mathematical Research Institute of the Ohio State University during his one month visit there in the spring of 1999     

The Lê varieties,I

Summary For a complex polynomial,f:( ⁿ⁺¹ ,0) (, 0), with a singular set of complex, dimensions at the origin, we define a sequence of varieties—the Lê varieties, _f ^(k) , off at 0. The multiplicities of these varieties, _f ^(k) , generalize the Milnor number for an isolated singularity. In particular, we show that ifsn-2, the Milnor, fibre off is obtained fromB ²ⁿ by successively attaching _f ^{(n – k)} k-handles, wheren-skn Ifs=n-1, the Milnor fibre off is obtained from a2n-manifold with the homotopy type of a bouquet of _f ^{(n – 1)} circles by successively attaching _f ^{(n – k)} k-handles, where 2kn.The author is a National Science Foundation, Postdoctoral Research Fellow supported by grant # DMS-8807216     

Topological classification of linear endomorphisms

Letf=f ₊f _–f f ₀ be a linear direct sum decomposition of a real endomorphismf End ⁿ , where the components correspond respectively to absolute values of eigenvalues ||: 0<||<1(f ₊); ||>1(f _–); ||=0(f ); and ||=1(f ₀). We conjecture that a complete set of invariants with respect to topological equivalence consists of the dimensions and orientations off ₊ and off _–, together with the linear isomorphy classes off andf ₀. We prove that this is true in case ⁿ contains nof-periodic points of periods 5 or 7. The conjecture is also true in case it is true for all periodic rotations. In §9 we make some comments on this unsolved case. The main interest of the paper is in §6 and 7.Research supported by the National Science Foundation (Contract number 144-B 695), the Wisconsin Alumni Research Foundation (project number 120432), and l'Institut des Hautes Etudes Scientifiques.     

The inclusion of certain classes of functions

For any sequence {N_k} with {N_k} O we find sharp theorems on the inclusion of the classes {f: f l (0, 2),e _k ⁽¹⁾ (f) = O(N_k)¦, whereE _k ⁽¹⁾ (f) is the best approximation (in L) of f by trigonometric polynomials of order no greater than k, in the classL with slowly growing and in the class Lv, 1 <v < .Translated from Matematicheskie Zametki, Vol. 20, No. 6, pp. 835–841, December, 1976.     

The steinitz lemma inl inf∞ sup2

Let (a, b) be a pair of non-negative numbers such that (1)a, b1 and (2)a+b3. Letu ₁,...,u _n be a sequence of vectors from the set {(x, y)R ²: |x|, |y|1}, withu ₁+...+u _n =0. It is shown that there is a permutation of indices such that all partial sumsu ₍₁₎+...+u _(k) lie in the rectangle |x|a, |y|b. Conditions (1) and (2) are also necessary.     

Multipliers on the Set of Rademacher Series in Symmetric Spaces

Let E be a symmetric space on [0,1]. Let (,E) be the space of measurable functions f such that fg E for every almost everywhere convergent series g=b _n r _n E, where (r _n) are the Rademacher functions. It was shown that, for a broad class of spaces E, the space (,E) is not order isomorphic to a symmetric space, and we study the conditions under which such an isomorphism exists. We give conditions on E for (,E) to be order isomorphic to L . This includes some classes of Lorentz and Marcinkiewicz spaces. We also study the conditions under which (,E) is order isomorphic to a symmetric space that differs from L . The answer is positive for the Orlicz spaces E=L _q with _q(t)=exp|t|^q-1 and 0相似文献