Showing papers in "Israel Journal of Mathematics in 1975"

Saddle points and instability of nonlinear hyperbolic equations

Abstract: A number of authors have investigated conditions under which weak solutions of the initial-boundary value problem for the nonlinear wave equation will blow up in a finite time. For certain classes of nonlinearities sharp results are derived in this paper. Extensions to parabolic and to abstract operator equations are also given.

Martingales with values in uniformly convex spaces

Abstract: Using the techniques of martingale inequalities in the case of Banach space valued martingales, we give a new proof of a theorem of Enflo: every super-reflexive space admits an equivalent uniformly convex norm. Letr be a number in ]2, ∞[; we prove moreover that if a Banach spaceX is uniformly convex (resp. ifδ x(ɛ)/ɛ r whenɛ → 0) thenX admits for someq<∞ (resp. for someq

An example concerning fixed points

Abstract: An example is given of a contractionT defined on a bounded closed convex subset of Hilbert space for which ((I+T)/2) n does not converge.

A compactness theorem for singular cardinals, free algebras, Whitehead problem and tranversals

Abstract: We prove, in an axiomatic way, a compactness theorem for singular cardinals. We apply it to prove that, for singular λ, every λ-free algebra is free; and similar compactness results for transversals and colouring numbers. For the general result on free algebras, we develop some filters onSk(A). As an application we conclude thatV=L implies that every Whitehead group is free.

A double-dual characterization of separable Banach spaces containingl 1

Abstract: It is proved that a separable Banach spaceB contains a subspace isomorphic tol 1 if (and only if) there exists an element inB**, the double-dual ofB, which is not a weak* limit of a sequence of elements inB. ConsequentlyB contains an isomorph ofl 1 if (and only if) the cardinality ofB** is greater than that of the continuum.

Quelques proprietes des systemes dynamiques qui se decomposent en un produit de deux systemes dont l’un est un schema de Bernoulli

Abstract: The purpose of this paper is to investigate some properties of dynamical systems that split into the product of two systems, one of them being a Bernoulli shift. With this view in mind, a relative version of the main results of D. S. Ornstein’s theory of Bernoulli shifts is developed here.

A class of semi-linear equations of evolution

Abstract: In this paper we prove existence, and study the asymptotic behavior of mild solutions of a class of semi-linear equations of evolution which are characterized by the fact that the associated homogeneous linear problem “generates” a strongly continuous semi-group of compact operators. We also prove a regularity result for which the associated homogeneous linear problem generates a holomorphic semi-group.

Some nonlinear parabolic variational inequalities

Abstract: In this paper we study initial value problems for nonlinear parabolic variational inequalities involving time-dependent subdifferentials of convex functions on a Hilbert space. We shall show the existence of a solution by a semi-discretisation method with respect to the time.

On the relation of the operator ∂/∂s+∂/∂τ to evolution governed by accretive operators

Abstract: Existence theorems for the equationdu/dt+Au=f(t), whereA is an accretive operator in a general Banach spaceX, are typically proved by showing that limits of solutions of discrete approximations to the equation exist. Here the estimates required to show this convergence are exhibited as special cases of estimates relating solutions of difference schemes for ∂/∂s+∂/∂τ to exact solutions.

The central limit theorem for Carleman’s equation

Abstract: The purpose of this paper is to discuss the analogy between the law of large numbers and the central limit theorem of classical probability theory on the one hand and the hydrodynamical approximations in the statistical mechanics of gases on the other. The chief illustration is provided by Carleman's model [2] for which the central limit approximation is a kind of non-linear Brownian motion regulated by ∂n/∂t=(n'/n)′.

Unique ergodicity of the horocycle flow: Variable negative curvature case

Abstract: H. Furstenberg showed that horocycle flows on compact manifolds of constant negative curvature are uniquely ergodic. This paper generalizes his result to the case of variable negative curvature, in the more general context of flows whose orbits are the unstable manifolds of certain Anosov flows.

w* sequential convergence

Abstract: LetX be an infinite dimensional Banach space, andX* its dual space. Sequences {χ n * } n=1 ∞ ⊆X* which arew* converging to 0 while inf n ‖x* n ‖>0, are constructed.

Uniform convexity in Lorentz sequence spaces

Abstract: Necessary and sufficient conditions for Lorentz sequence spacesd(a,p) (1

Hamiltonian circuits in polytopes with even sided faces

Abstract: It has been conjectured that ifP is a simple 3-polytope all of whose faces have an even number of sides, thenP has a Hamiltonian circuit. In this paper it is shown that, if all the faces ofP are either quadrilaterals or hexagons, thenP does have a Hamiltonian circuit.

Spaces of compact operators and their dual spaces

Abstract: LetE andF be reflexive Banach spaces andC the space of all compact linear operators fromE toF. A representation of the dual space ofC is given and it is proved thatC is either reflexive or nonconjugate. Applications of these results are also given.

Categoricity in ℵ1 of sentences in $$L_{\omega _1 ,\omega } (Q)$$

Abstract: We investigate the categoricity and number of non-isomorphic models in ℵ1 of sentences in\(L_{\omega _1 ,\omega } (Q)\). AssumingV=L we prove that no sentence in\(L_{\omega _1 ,\omega } (Q)\) has exactly one uncountable model. Thus partially answering problem 24 of a problem list by Friedman.

Farthest points in weakly compact sets

Abstract: LetS be a weakly compact subset of a Banach spaceB. We show that of all points inB which have farthest points inS contains a denseG 5 ofB. Also, we give a necessary and sufficient condition for bounded closed convex sets to be the closed convex hull of their farthest points in reflexive Banach spaces.

The isometries ofCp

Abstract: Theorem.Let 1≦p≦∞,p ≠ 2, and let V be an isometry of Cp onto itself. Then there exist two unitary operators u and w on l2 so that V acts on Cp in one of the following forms:\((i) Vx = u \cdot x \cdot w; (ii) Vx = u \cdot x^T \cdot w\) (where xT is the transpose of x).

Every transformation is bilaterally deterministic

Abstract: Every ergodic transformationT with a finite generator α, has another finite generator β, which refines α, and is bilaterally deterministic, i.e. V|i|>n T iβ is the full σ-algebra for everyn.

Normal numbers and selection rules

Abstract: Given anormal number x=0,x 1 x 2 ··· to base 2 and aselection rule S ⊂{0, 1}*=∪ n=0 /t8 {0, 1} n , we define a subsequencex,=0,\(\chi _{t_1 } \chi _{t_2 } \)·· where {t 1

Numbertheoretical endomorphisms with σ-finite invariant measure

Abstract: A class of measurable transformations which serves as a model for severalf-expansions is discussed. Sufficient conditions for ergodicity and the existence of a σ-finite invariant measure are given.

Une classe de systemes pour lesquels la conjecture de pinsker est vraie

Abstract: In this paper it is proved that any factor of a dynamical system which is isomorphic to the direct product of a Bernoulli shift by a zero entropy system splits in the same way.

On factors ofC([0,1]) with nonseparable dual

Abstract: LetC denote the Banach space of scalar-valued continuous functions defined on the closed unit interval. It is proved that ifX is a Banach space andT:C→X is a bounded linear operator withT * X * non-separable, then there is a subspaceY ofC, isometric toC, such thatT|Y is an isomorphism. An immediate consequence of this and a result of A. Pelczynski, is that every complemented subspace ofC with non-separable dual is isomorphic (linearly homeomorphic) toC.

When do two Banach spaces have isometrically isomorphic nonstandard hulls

Abstract: The answer to the title question is given in terms of the elementary properties of Banach spaces regarded as structures for a certain first-order language. The same question for Banach space ultrapowers is also considered. The connection between nonstandard hulls and Banach space ultrapowers derives in part from the following fact, of independent interest in nonstandard analysis: for each cardinal number κ there exist ultrapower enlargements which are κ-saturated and which have the κ-isomorphism property.

Models of Arithmetic and recursive functions

Abstract: We investigate homomorphic images of the semiring of recursive functions as models of the Π2 fragment of Arithmetic, and some relations between this fragment, its models and recursion theory.

Finite group extensions of irrational rotations

Abstract: The ergodicity of certain skew products of irrational rotations of the circle with finite groups is established with application to the construction of “well-distributed sequence generators” for finite groups.

The Artin-Rees property and homology

Abstract: The Artin-Rees property for a finitely generated nilpotent groupG is used to prove that\(H_ * (G, M) \approx H_ * (G, \hat M)\) for any finitely generatedG-moduleM, where\(\hat M\) is the completion ofM with respect to the augmentation ideal ofZ[G]. Applications to topology are given.

The quantum mechanical virial theorem and the absence of positive energy bound states of Schrödinger operators

Abstract: For a class of potentialsq which may become as singular asr−2 at the origin we show that the eigenfunctionsu of the Friedrichs extensionTFof (−Δ+q)↾C0∞(Rn/{0}) satisfy the virial theorem 2(−Δu,u)=(u,rq,u) Two proofs of this relation are given One extends the Fock-Weidmann method of scale transformations, the other makes an argument by Finkelstein rigorous and can be viewed of as a substitute of the “commutator proof” of (*) employed in the physical literature The virial theorem provides a convenient tool for proving the nonexistence of eigenvalues embedded in the continuum or the total absence of eigenvalues ofTF

On pelczynski’s paper “universal bases”

Abstract: A short proof is given to a theorem of Pelczynski concerning universal bases.

Remarques sur les systemes dynamiques donnes avec plusieurs facteurs

Abstract: Two Bernoulli shifts are given, (X, T) and (X′, T′), with independent generatorsR=P ∨Q andR′=P′ ∨Q′ respectively. (R andR′ are finite). One can chooseR such that if (X′, T′) can be made a factor of (X, T) in such a way that (P′) T′ and (Q′) T′ are full entropy factors of (P) T and (Q) T respectively thend (P ∨Q)=d(P′ ∨Q′). In addition it is proved that if (X, T) is a Bernoulli shift and ifS is a measure preserving transformation ofX that has the same factor algebras asT thenS=T orS=T −1. A tool for this proof, which may be of independent interest is a relative version for very weak Bernoullicity.