Dimension de Hausdorff de la Nervure

For a separable Hilbert space E whose dimension is 2 and for an open subset of E, not empty and different from E, let be the set of all points of which have at least two projections on the close set E\, and let be the set of all the centres of the open balls contained in and which are maximal for inclusion. We show that the Hausdorff dimension dim_H( ) of may be any real value s such that 0sdim E; we also show that can be chosen so that is everywhere dense in and so that we have dim_H( )=1.Associons à un ouvert d'un espace de Hilbert séparable E de dimension 2, non vide et distinct de E, l'ensemble des points de admettant plusieurs projections sur le fermé E\, et l'ensemble des centres des boules ouvertes inclues dans et maximales pour l'inclusion. Nous montrons d'une part que la dimension de Hausdorff dim_H( ) de peut prendre toute valeur réelle s telle que 0sdim E, et d'autre part qu'on peut choisir de sorte que soit dense dans et qu'on ait dim_H( )=1.     

Strong topologies on vector-valued function spaces

Let be a real Banach space and let E be an ideal of L ⁰ over a -finite measure space (, , ). Let (X) be the space of all strongly -measurable functions f: X such that the scalar function , defined by , belongs to E. The paper deals with strong topologies on E(X). In particular, the strong topology the order continuous dual of E(X)) is examined. We generalize earlier results of [PC] and [FPS] concerning the strong topologies.     

On Kolmogorov-Type Inequalities Taking into Account the Number of Changes in the Sign of Derivatives

For 2-periodic functions and arbitrary q [1, ] and p (0, ], we obtain the new exact Kolmogorov-type inequality which takes into account the number of changes in the sign of the derivatives (x ^(k)) over the period. Here, = (r – k + 1/q)/(r + 1/p), _r is the Euler perfect spline of degree r, and . The inequality indicated turns into the equality for functions of the form x(t) = a _r (nt + b), a, b R, n N. We also obtain an analog of this inequality in the case where k = 0 and q = and prove new exact Bernstein-type inequalities for trigonometric polynomials and splines.     

Random fixed point theorems for a certain class of mappings in banach spaces

Let (, ) be a measurable space and C a nonempty bounded closed convex separable subset of p-uniformly convex Banach space E for some p > 1. We prove random fixed point theorems for a class of mappings T: × C C satisfying: for each x, y C, and integer n 1,

Augmentation and Modification Problems for Hermitian Matrices

We obtain necessary and sufficient conditions for the solvability of the augmentation and modification problems of order for Hermitian matrices. The augmentation problem consists in the construction of a Hermitian -matrix with a given -block in block -representation and with the prescribed eigenvalues. The modification problem consists in the construction of a Hermitian -matrix of rank not greater than so that the obtained matrix, being added to a given Hermitian -matrix , will have the required spectrum. We give an estimate for the minimal number of different eigenvalues of the solutions to these problems.     

Contractive Automorphisms of Locally Compact Groups and the Concentration Function Problem

Let G be a noncompact locally compact group. We show that a necessary and sufficient condition in order that G support an adapted probability measure whose concentration functions fail converge to zero is that G be the semidirect product , where is an automorphism of N contractive modulo a compact subgroup. Any adapted a probability measure whose concentration functions fail to converge to zero has the form =v×₁ where v is a probability measure on N. If G is unimodular then the concentration functions of an adapted probability measure fail to converge to zero if and only if is supported on a coset of a compact normal subgroup.     

A horn sentence for involution lattices of quasiorders

The quasiorders of a setA form a lattice Quord(A) with an involution ^–1={x, y: y, x}. Some results in [1] and Chajda and Pinus [2] lead to the problem whether every lattice with involution can be embedded in Quord(A) for some setA. Using the author's approach to the word problem of lattices (cf. [3]), which also applies for involution lattices, it is shown that the answer is negative.Research supported by the Hungarian National Foundation for Scientific Research (OTKA), under grant no. T 7442.     

Operator stable laws: Series representations and domains of normal attraction

LetX,X ₁,X ₂,... be i.i.d. random vectors in ^d. The limit laws that can arise by suitable affine normalizations of the partial sums,S _n=X ₁+...+X _n, are calledoperator-stable laws. These laws are a natural extension to ^d of the stable laws on. Thegeneralized domain of attraction of [GDOA()] is comprised of all random vectorsX whose partial sums can be affinely normalized to converge to . If the linear part of the affine transformation is restricted to take the formn ^–B for some exponent operatorB naturally associated to thenX is in thegeneralized domain of normal attraction of [GDONA()]. This paper extends the theory of operator-stable laws and their domains of attraction and normal attraction.     

Best Khintchine Type Inequalities for Sums of Independent,Rotationally Invariant Random Vectors

Let be an i.i.d. sequence of rotationally invariant random vectors in . If X ₁² is dominated (in the sense defined below) by Z² for a rotationally invariant normal random vector Z in , then for each k and

Weighted inequalities for integral operators with some homogeneous kernels

In this paper we study integral operators of the form

The algebra of implicit operations

For an ordinal and a class of topological algebras of a given type (which may be infinite and may contain inflnitary operations), an-aryimplicit operation on is any new-ary operation whose introduction does not eliminate any continuous homomorphisms between members of . The set of all-ary implicit operations on is denoted by and forms an algebra of the given type which is endowed with the least topology making continuous all homomorphisms into members of . With this topology, is a topological algebra in which the subalgebra of all-ary operations on which are induced by terms is dense, provided that is closed under the formation of closed subalgebras and finitary direct products. This is obtained by realizing as an inverse limit of-generated members of . These results are applied to pseudovarieties of topological and finite algebras.This work was supported, in part, by INIC grant 85/CEX/4. This paper was written while the author was a faculty member at the Universidade do Minho.Presented by Ralph McKenzie.     

Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2-periodic functions x with bounded variation of the derivative x, namely

Orthomodular lattices from 3-dimensional quadratic spaces

If E is a vector space over a field K, then any regular symmetric bilinear form on E induces a polarity on the lattice of all subspaces of E. In the particular case where E is 3-dimensional, the set of all subspaces M of E such that both M and are not N-subspaces (which, in most cases, is equivalent to saying that M is nonisotropic), ordered by inclusion and endowed with the restriction of the above polarity, is an orthomodular lattice T(E, ). We show that if K is a proper subfield of K, with K F₂, and E a 3-dimensional K -subspace of E such that the restriction of to E × E is, up to multiplicative constant, a bilinear form on the K -space E , then T(E , ) is isomorphic to an irreducible 3-homogeneous proper subalgebra of T(E, ). Our main result is a structure theorem stating that, when K is not of characteristic 3, the converse is true, i.e., any irreducible 3-homogeneous proper subalgebra of T(E, ) is of this form. As a corollary, we construct infinitely many finite orthomodular lattices which are minimal in the sense that all their proper subalgebras are modular. In fact, this last result was our initial aim in this paper.Received June 4, 2003; accepted in final form May 18, 2004.     

Singular limits in Liouville-type equations

We consider the boundary value problem in a bounded, smooth domain in with homogeneous Dirichlet boundary conditions. Here 0,k(x) $$ " align="middle" border="0"> is a non-negative, not identically zero function. We find conditions under which there exists a solution which blows up at exactly m points as and satisfies . In particular, we find that if , 0 $" align="middle" border="0"> and is not simply connected then such a solution exists for any given Received: 11 February 2004, Accepted: 17 August 2004, Published online: 22 December 2004     

Finitely subadditive and countably subadditive outer measures

Results are given comparing countably subadditive (csa) outer measures and finitely subadditive (fsa) outer measures, especially relating to regularity and measurability conditions such as (*) condition:A setE (of an arbitrary setX), is measurable ( an outer measure),ES (the collection of measurable sets) iff (X)=(E)+(E). Specific examples are given contrasting csa and fsa outer measures. In particular fsa and csa outer measures derived from finitely additive measures defined on an algebra of sets generated by a lattice of sets, are investigated in some detail.     

On oscillation and asymptotic property of a class of third order differential equations

In this paper, oscillation and asymptotic behaviour of solutions of have been studied under suitable assumptions on the coefficient functions such that a(t 0, b(t) 0 and c(t) < 0.     

On the Boundedness of a Recurrence Sequence in a Banach Space

We investigate the problem of the boundedness of the following recurrence sequence in a Banach space B: where |y _n} and | _n } are sequences bounded in B, and A _k, k 1, are linear bounded operators. We prove that if, for any > 0, the condition is satisfied, then the sequence |x _n} is bounded for arbitrary bounded sequences |y _n} and | _n } if and only if the operator has the continuous inverse for every z C, |z| 1.