Showing papers in "Israel Journal of Mathematics in 1976"

Provability interpretations of modal logic

Abstract: We consider interpretations of modal logic in Peano arithmetic (P) determined by an assignment of a sentencev * ofP to each propositional variablev. We put (⊥)*=“0 = 1”, (χ → ψ)* = “χ* → ψ*” and let (□ψ)* be a formalization of “ψ)* is a theorem ofP”. We say that a modal formula, χ, isvalid if ψ* is a theorem ofP in each such interpretation. We provide an axiomitization of the class of valid formulae and prove that this class is recursive.

On locally conformal almost Kähler manifolds

Abstract: In the first section of this note, we discuss locally conformal symplectic manifolds, which are differentiable manifoldsV 2n endowed with a nondegenerate 2-form Ω such thatdΩ=θ ∧ Ω for some closed form θ. Examples and several geometric properties are obtained, especially for the case whendΩ ≠ 0 at every point. In the second section, we discuss the case when Ω above is the fundamental form of an (almost) Hermitian manifold, i.e. the case of the locally conformal (almost) Kahler manifolds. Characterizations of such manifolds are given. Particularly, the locally conformal Kahler manifolds are almost Hermitian manifolds for which some canonically associated connection (called the Weyl connection) is almost complex. Examples of locally conformal (almost) Kahler manifolds which are not globally conformal (almost) Kahler are given. One such example is provided by the well-known Hopf manifolds.

A non-standard representation for Brownian Motion and Itô integration

Abstract: In a recent paper [10], Peter A. Loeb showed how to convert non-standard measure spaces into standard ones and gave applications to probability theory. We apply these results to Brownian Motion and Ito integration. We first develop a number of new tools about Loeb spaces. We then show that Brownian Motion can be obtained as the Loeb process corresponding to a non-standard random walk obtained from a*-finite number of coin tosses. This permits a very constructive proof of a special case of Donsker's Theorem. The Ito integral with respect to this Brownian Motion is a non-standard Stieltjes integral with respect to the random walk. As a consequence, an easy proof of Ito’s Lemma is possible. The results in this paper were announced in [1].

Remarks on the Korteweg-de Vries equation

Abstract: We show for the Korteweg-de Vries equation an existence uniqueness theorem in Sobolev spaces of arbitrary fractional orders≧2, provided the initial data is given in the same space.

Image d'une somme d'operateurs monotones et applications

Abstract: LetA andB be monotone (multivalued) operators in a Hilbert spaceH. The paper deals with the relations between the rangeR(A+B) ofA+B and the algebraic sum of the ranges ofA andB, R(A)+R(B).

NewK-automorphisms and a problem of Kakutani

Abstract: A property is introduced, for 1-1 measure-preserving transformations of probability spaces, calledloose Bernoulliness (LB), which is invariant under taking factors, inducing, and tower-building. It amounts to replacing, in Ornstein’s definition ofvery weak Bernoulli, the Hamming distance on strings by a coarser metric. The main result is the construction of a transformationT0 which is ergodic and of entropy 0 butnot LB. On the other hand, any irrational rotationis LB. Consequently, the equivalence relation generated by inducing and tower-building (which I callKakutani equivalence, and the Russians callmonotone equivalence) has at least two distinct equivalence classes among the ergodic entropy zero transformations. A similar situation exists for ergodic positive-entropy transformations: on the one hand, any Bernoulli shift is LB, while on the other hand a non LBK-automorphism\(\hat T_0 \) can be made by skewingT0 over a Bernoulli base.

Nonstandard hulls of Banach spaces

Abstract: The main theme of this paper is the relationship between a Banach spaceE and its nonstandard hullsE (including ultrapowers ofE). Emphasis is placed on the ways in which the general structure ofE is determined by the approximate shape and arrangement of the finite dimensional subspaces ofE.

Alternating hamiltonian cycles

Abstract: Coloar the edges of a complete graph with n vertices in such a way that no vertex is on more than k edges of the same colour . We prove that for every k there is a constant c ksuch that if n > ck then there is a Hamiltonian cycle with adjacent edges having different colours . We prove a number of other results in the same vein and mention some unsolved problems .

Ranks and definability in superstable theories

Abstract: We study the notion of definable type, and use it to define theproduct of types and theheir of a type. Then, in the case of stable and superstable theories, we make a general study of the notion of rank. Finally, we use these techniques to give a new proof of a theorem of Lachlan on the number of isomorphism types of countable models of a superstable theory.

Potentials and isometric embeddings inL 1

Abstract: lp/3 does not embed isometrically inL 1, forp>2. The question whenl r n embeds isometrically inL pis completely answered.

Elimination of quantifiers for modules

Abstract: Every first-order formula in the language ofR-modules (R an associative ring) is equivalent relative to the theory ofR-modules to a boolean combination of positive primitive formulas and ∀∃-sentence.

Centrally symmetric convex bodies and distributions

Abstract: To each centrally symmetric convex body is assigned a distribution on the sphere. As applications, geometric formulas and a characterization of zonoids are obtained.

A new proof of the Amitsur-Levitski identity

Abstract: We give a simple proof of the Amitsur-Levitzki identity by analysing the powers of matrices with “differential 1-forms” as entries. Using the fact that 2-forms are central the identity is seen to follow from the Cayley-Hamilton theorem.

Model-companions and definability in existentially complete structures

Abstract: The theory of model-companions and existentially complete structures is both reviewed and developed further. The review begins with A. Robinson’s work in the fifties and continues through the definability of second order structures in existentially complete groups. New results include necessary and sufficient conditions for the existence of a model-companion in terms of the definability of general elementary properties. The main theorem of the paper gives necessary and sufficient conditions for the existence of a model-companion for universal theories with finite presentations and the amalgamation property. This result generalizes the result of P. Eklof and G. Sabbagh that the theory ofR-modules has a model-completion if and only ifR is coherent.

Embedding weakly compact sets into Hilbert space

Abstract: We give an example of a weakly compact set in a Banach space, which does not embed topologically as a weakly compact subset of Hilbert space. We also show that a weakly compact set embeds in a super-reflexive space iff it embeds in Hilbert space.

Applications of nonstandard analysis to ideal boundaries in potential theory

Abstract: A solution is given of the generalized Dirichlet problem for an arbitrary compactification of a Brelot harmonic space. A method of obtaining the Martin-Choquet integral representation of positive harmonic functions is given, and the existence is established of an ideal boundary Δ supporting the maximal representing measures for positive bounded and quasibounded harmonic functions with almost all points of Δ being regular for the Dirichlet problem.

Some applications of Henkin quantifiers

Abstract: We show how to approximate a Henkin formula by first order formulas. This method of approximation is then applied to problems of axiomatizing classes of structures.

Weak mixing and unique ergodicity on homogeneous spaces

Abstract: Under certain conditions the weak mixing of a translation onG/Γ implies that the action of an associated subgroup ofG onG/Γ is uniquely ergodic. This result generalizes earlier theorems of Furstenberg and Veech.

The model theory of differential fields revisited

Abstract: The intent of this article is to provide a general and elementary account of the model theory of differential fields, collecting together various results (many without proof) and offering a few algebraic details for the logician reader. The first model-theoretic look at differential fields was taken by Abraham Robinson in the context of model completeness, while later developments have served to illustrate concepts developed by Morley and Shelah.

The uniform approximation property in Orlicz spaces

Abstract: It is proved that for every reflexive Orlicz spaceX there is a functionn(k,e) so that wheneverE is ak-dimensional subspace ofX there exists an operatorT: X→X such thatT 1E=identity, ‖T‖≦1+e and dimTX≦n(k,e). Some general facts concerning the uniform approximation property are also presented.

General decomposition theorems form-convex sets in the plane

Abstract: A setS inRdis said to bem-convex,m≧2, if and only if for everym distinct points inS, at least one of the line segments determined by these points lies inS. Clearly any union ofm−1 convex sets ism-convex, yet the converse is false and has inspired some interesting mathematical questions: Under what conditions will anm-convex set be decomposable intom−1 convex sets? And for everym≧2, does there exist aσ(m) such that everym-convex set is a union ofσ(m) convex sets? Pathological examples convince the reader to restrict his attention to closed sets of dimension≦3, and this paper provides answers to the questions above for closed subsets of the plane.

Une méthode topologique pour L'etude de la propriété de ramsey

Abstract: The paper is devoted to the study of some particular subclasses of the class of Ramsey sets, each one associated with an ultrafilter onΝ. By topological methods, we show that every such class contains the analytic sets. This generalizes the results of Silver and Mathias on this subject. Furthermore some applications to functional analysis are given, and a discussion of the additivity of these classes and of the Ramsey property of the ultrafilters is presented, under the hypothesis of Martin's Axiom.

On a class of valuation fields introduced by A. Robinson

Abstract: It is shown that the nonarchimedean valuation fieldsρ R introduced by A. Robinson are not only complete but are also spherically complete. Further-more, it is shown that to every normed linear space over the reals there exists a nonarchimedean normed linear spaceρ E overρ R in the sense of Monna which is spherically complete and extendsE.

Torsion-free covers ii

Abstract: This paper continues the study of the existence of torsion-free covers with respect to a faithful hereditary torsion theory (ℑ,F) of left modules over a ringR with unity. If the filter of left ideals associated with (ℑ,F) has a cofinal subset of finitely generated left ideals, then every leftR-module has a torsion-free cover. An example is given to illustrate how this result generalizes all previously known existence theorems for torsion-free covers.

A chain compact space which is not strongly scattered

Abstract: A compactT2 spaceX which is separable, scattered and uncountable, but still so thatXα−Xα+1is countable for all α∈[1, ω) is constructed. This answers one of the problems presented by M. E. Rudin in a conference as an open problem and attributed by her to Telgarsky.

A banach lattice without the approximation property

Abstract: A Banach latticeL without the approximation property is constructed. The construction can be improved so thatL is, in addition, uniformly convex. These results yield the existence of a uniformly convex Banach space with symmetric basis and without the uniform approximation property.

Generalized hexagons of order (t,t)

Abstract: A generalized hexagon of order (t,t) in which certain subsets are maximal may be characterized as the generalized hexagon associated with Dickson’s groupG2(t). From this geometric result, it follows that ifG is a group of automorphisms of a generalized hexagon of order (p,p) for a primep and ifG has rank 4 on points, thenG ⊵G2(p).

Complex Banach spaces whose duals areL 1-spaces

Abstract: We prove that for a complex Banach spaceA the following properties are equivalent: i) A* is isometric to anL1(μ)-space; ii) every family of 4 balls inA with the weak intersection property has a non-empty intersection; iii) every family of 4 balls inA such that any 3 of them have a non-empty intersection, has a non-empty intersection.

A functional characterization of certain mixed volumes

Abstract: A function over the convex coneK{inn}, of convex bodiesK in Euclideann-space (where addition is vector addition, positive scalar multiplication is dilatation), which is linear overK{inn}, increasing with respect to set inclusion, and zero at point bodies must be a mixed volumeV(K; đ, p−1;σ 1, …,σ n−p). Heređ, takenp−1 times, is inK{inn} andσ 1, …,σ n−pare pairwise orthogonal unit segments spanning the orthogonal complement of the affine hull ofđ.

On the length of linked curves

Abstract: If two curves inR 3 are linked and at distance 1 from each other then each is of length 2π at least.