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Showing papers in "Israel Journal of Mathematics in 1974"


Journal ArticleDOI
TL;DR: In this article, it was shown that the answer to Whitehead's problem is independent from the usual axioms of set theory, and the existence of large rigid systems for groups of power λ, with no restriction on λ.
Abstract: We solve here some problems from Fuchs’ book. We show that the answer to Whitehead’s problem (for groups of power ℵ1) is independent from the usual axioms of set theory. We prove the existence of large rigid systems for groups of power λ, with no restriction on λ. We also prove that there are many non-isomorphic reduced separablep-groups.

228 citations


Journal ArticleDOI
TL;DR: In this paper, the authors generalize a theorem of Goebel and Kirk for asymptotically nonexpansive mappings, and define mappingsT:K→K satisfying, for sufficiently large, "Tix−Tiy∼−‖x−y∼,x,y∈K, wherek ≥ 1 asi→∞".
Abstract: LetX be a Banach space,K a nonempty, bounded, closed and convex subset ofX, and supposeT:K→K satisfies: for eachx∈K, lim sup i→∞{sup y∈K ‖t ix−Tiy∼−‖x−y‖}≦0. IfT N is continuous for some positive integerN, and if either (a)X is uniformly convex, or (b)K is compact, thenT has a fixed point inK. The former generalizes a theorem of Goebel and Kirk for asymptotically nonexpansive mappings. These are mappingsT:K→K satisfying, fori sufficiently large, ‖Tix−Tiy‖≦k i‖x−y∼,x,y∈K, wherek i→1 asi→∞. The precise assumption in (a) is somewhat weaker than uniform convexity, requiring only that Goebel’s characteristic of convexity, ɛ0 (X), be less than one.

168 citations


Journal ArticleDOI
TL;DR: A simple example is given in this article, where it is shown that a non-WCG space whose dual is a WCG space with an unconditional basis is a subspace of a WcG space.
Abstract: A simple example is given of a non WCG space whose dual is a WCG space with an unconditional basis. It is proved that ifX* is WCG andX is a subspace of a WCG space thenX itself is WCG.

130 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that for every separable metric space X, there is a mapT:X →co satisfyingd(x, y)≦Tx−Ty≦Kd(k, k) for every x, y ∈ X.
Abstract: It is shown that there is a constantK so that, for every separable metric spaceX, there is a mapT:X →co satisfyingd(x, y)≦‖Tx−Ty‖≦Kd(x, y) for everyx, y ∈ X.

106 citations


Journal ArticleDOI
TL;DR: In this paper, the authors make use of the theory of rings with polynomial identities as a tool for attacking the problems stated before, and derive the following results: (a) classifying the representations of R in (hO, (or n dimensional representations). (b) Classifying n-dimensional representations up to the natural equivalence, and (c) equivalence classes of irreducible and semisimple representations.
Abstract: Let R be a ring, K a field, and n a natural number. We will be concerned with the following type of questions: (a) Classify the representations of R in (hO, (or n dimensional representations). (b) Classify n dimensional representations up to the natural equivalence. If ~bt, ~b2 : R ~ (K), we say ~bx is equivalent to ~b 2 if there is a K automorphism 7 of (K), such that 7q~1 = 4)2. (c) Classify equivalence classes of irreducible and semisimple representations. For the sake of simplicity we will restrict ourselves mainly to the case that K is algebraically closed and R is a finitely generated K algebra. In fact, for most of this paper, we will assume that R = K{xa, ..., x,) is a free algebra; at the end we will deduce from the results in this ease the more general theorems for not necessarily free algebras. Before describing the theorems that we will obtain, let us digress in order to motivate the use that we will make of the theory of rings with polynomial identities as a tool for attacking the problems stated before. Let us recall, therefore, some of the basic structural results of this theory and interpret them in terms of representation theory.

85 citations


Journal ArticleDOI
TL;DR: In this paper, a special flowS t over a shift in the space of sequences (X, μ) constructed using a continuous f with {fx380-1} was considered and a condition for μ such that the K-flowS t is aB-flow was formulated.
Abstract: We consider a special flowS t over a shift in the space of sequences (X, μ) constructed using a continuousf with {fx380-1} We formulate a condition for μ such that theK-flowS t is aB-flow.

81 citations


Journal ArticleDOI
TL;DR: The fundamental constant of Grothendieck's inequality was shown to be less than π/2 for the positive definite case as discussed by the authors, and it was shown that π /2 suffices.
Abstract: The fundamental constant of Grothendieck's inequality, defined below, was shown by Grothendieck to be less than sinh π/2=2.301+. We improve the bound slightly, and show that for the positive definite case π/2 suffices.

68 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that the number of non-isomorphic countable models of a complete theory of linear order is either finite or 2ω, depending on the language of the theory.
Abstract: LetT be a complete theory of linear order; the language ofT may contain a finite or a countable set of unary predicates. We prove the following results. (i) The number of nonisomorphic countable models ofT is either finite or 2ω. (ii) If the language ofT is finite then the number of nonisomorphic countable models ofT is either 1 or 2ω. (iii) IfS 1(T) is countable then so isS n(T) for everyn. (iv) In caseS 1(T) is countable we find a relation between the Cantor Bendixon rank ofS 1(T) and the Cantor Bendixon rank ofS n(T). (v) We define a class of modelsL, and show thatS 1(T) is finite iff the models ofT belong toL. We conclude that ifS 1(T) is finite thenT is finitely axiomatizable. (vi) We prove some theorems concerning the existence and the structure of saturated models.

61 citations


Journal ArticleDOI
TL;DR: Skew products of the powers of an ergodic measure preserving transformation with a Bernoulli base are shown to be k-automorphisms in this paper, where the authors show that they can be expressed as
Abstract: Skew-products of the powers of an ergodic measure preserving transformation with a Bernoulli base are shown to bek-automorphisms.

55 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that if a process is finitely determined then it is very weak Bernoulli (VWB) and combined with known results, it is shown that a process can be isomorphic to a Bernoullis shift if and only if it satisfies an asymptotic independence condition.
Abstract: It is shown that if a process is finitely determined then it is very weak Bernoulli (VWB). Combined with known results this says that a process is isomorphic to a Bernoulli shift if and only if it satisfies an asymptotic independence condition, namely that of being VWB.

52 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that for every space with an unconditional basis, there exists a uniformly bounded sequence of projections whose ranges are uniformly isomorphic to either forp = 1, orp=2, or forp=∞.
Abstract: It is shown that for every space with an unconditional basis there exists a uniformly bounded sequence of projectionsP n;n=1, 2, ... whose ranges are uniformly isomorphic tol p n ;n=1, 2, ... either forp=1, orp=2, or forp=∞.

Journal ArticleDOI
TL;DR: In this article, a non-reflexive Banach space is defined, where there is a λ > 1 such that there is no isomorphismT ofl1(3) into the space.
Abstract: An example is given of a nonreflexive Banach space\(\tilde X\) that is uniformly nonoctahedral (or uniformly non-l1(3)), in the sense that there is a λ>1 such that there is no isomorphismT ofl1(3) into\(\tilde X\) for which $$\lambda ^{ - 1} \left\| x \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \left\| {T(x)} \right\|\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \leqslant } \lambda \left\| x \right\|{\text{ }}if x \in l_1^{(3)}$$ .

Journal ArticleDOI
TL;DR: In this article, a finite PI ring which cannot be embedded in matrices over any commutative ring was constructed, including a semiprime PI ring with no classical ring of quotients, showing that the property of having all regular elements invertible is not inherited by matrix rings.
Abstract: Several examples are constructed, including a finite ring which cannot be embedded in matrices over any commutative ring, a semiprime PI ring with no classical ring of quotients, an example showing that the property of having all regular elements invertible is not inherited by matrix ringsMn(R), and a prime PI ringR with an idempotente such thatR/ReR has finitely generated projective modules not induced by any finitely-generated projective R-module.


Journal ArticleDOI
TL;DR: In this paper, it was shown that a Banach space X is isomorphic to a subspace of Cp (1
Abstract: Let (Gn) be a sequence which is dense (in the sense of the Banach-Mazur distance coefficient) in the class of all finite dimensional Banach spaces. Set\(C_p = (\Sigma G_n )_{l_p } (1< p< \infty ) = (\Sigma G_n )_{c_0 } \). It is shown that a Banach spaceX is isomorphic to a subspace ofCp (1

Journal ArticleDOI
TL;DR: In this paper, the authors considered a hilbertian field and its Galois group and showed that almost all (σ) ∈ fixme (k/k)e have the following properties.
Abstract: We consider here a hilbertian fieldk and its Galois group (k s/k). For a natural numbere we prove that almost all (σ) ∈ (ks/k)e have the following properties. (1) The closedsubgroup 〈σ〉 which is generated by σ1, …, σe is a free pro-finite group withe generators. (2) LetK be a proper subfield of the fixed fieldk s (σ) of 〈σ〉, …, σe ink s, which containsk. Then the group (k s/K) cannot be topologically generated by less thene+1 elements. (3) There does not exist a τ ∈ (k/k), τ≠1, of finite order such that [k s (σ):k s (σ, τ)]<∞. (4) Ife=1, there does not exist a fieldk⊆K⊆k s (σ) such that 1<[k s (σ):K]<∞. Here “almost all” is used in the sense of the Haar measure of the compact group (ks/k)e.

Journal ArticleDOI
TL;DR: In this paper, it was shown that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts, and that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.
Abstract: We show that ergodic algebraic automorphisms of the infinite torus are measure isomorphic to Bernoulli shifts. Using the same techniques, we also show that the existence of such an automorphism with finite entropy is equivalent to an open problem in algebraic number theory.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a graph G has all matchings of equal size if and only if for every matching setλ in G, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle.
Abstract: It is shown that a graphG has all matchings of equal size if and only if for every matching setλ inG, G\V(λ) does not contain a maximal open path of odd length greater than one, which is not contained in a cycle. (V(λ) denotes the set of vertices incident with some edge ofλ.) Subsequently edge-coverings of graphs are discussed. A characterization is supplied for graphs all whose minimal covers have equal size.



Journal ArticleDOI
TL;DR: A preliminary study on permanents of (1, − 1)-matrices is given in this paper, where some inequalities are derived and a few unsolved problems are mentioned, believed to be new.
Abstract: A preliminary study on permanents of (1, − 1)-matrices is given. Some inequalities are derived and a few unsolved problems, believed to be new, are mentioned.

Journal ArticleDOI
TL;DR: In this paper, the authors consider uniform ultrafilters with (k, k, k +)-regularity and show that uniform ultra-filters are (k and k +)regular.
Abstract: If there arek ++ eventually functions fromk + intok or if there arek ++ eventually different functions fromk + then uniform ultrafilters onk + are (k, k +)-regular.

Journal ArticleDOI
TL;DR: In this paper, it was shown that for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen−k.
Abstract: We consider interpolation of Hermite data by splines of degreen withk given knots, satisfying boundary conditions which may involve derivatives at both end points (e.g., a periodicity condition). It is shown that, for a certain class of boundary conditions, a necessary and sufficient condition for the existence of a unique solution is that the data points and knots interlace properly and that there does not exist a polynomial solution of degreen−k. The method of proof is to show that any spline interpolating zero data vanishes identically, rather than the usual determinantal approach.

Journal ArticleDOI
TL;DR: In this article, the structure of invariant measures on a transformation group which is a free compact abelian group extension of another transformation group is studied from both the geometric and analytic viewpoints.
Abstract: In this paper, the structure of the set of invariant measures on a transformation group which is a free compact abelian group extension of another transformation group is studied from both the geometric and analytic viewpoints. It is shown in general that genuine ergodic decompositions are obtained in the non-metric setting for measures that project onto an ergodic measure. In addition, when all the spaces involved are metric, there is a structure theorem for all ergodic measures in terms of the ergodic measures on the base and naturally defined subgroups.

Journal ArticleDOI
TL;DR: A complete and rather simple proof of the famous Dvoretzky theorem is presented in this paper, which is a proof of a theorem that is also related to our work.
Abstract: A complete and rather simple proof of the famous Dvoretzky’s theorem is presented.

Journal ArticleDOI
TL;DR: The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the sign changes in the sequence of successive derivatives evaluated at the end-points as mentioned in this paper.
Abstract: The Budan-Fourier theorem for polynomials connects the number of zeros in an interval with the number of sign changes in the sequence of successive derivatives evaluated at the end-points. An extension is offered to splines with knots of arbitrary multiplicities, in which case the connection involves the number of zeros of the highest derivative. The theorem yields bounds on the number of zeros of splines and is a valuable tool in spline interpolation and approximation with boundary conditions.

Journal ArticleDOI
TL;DR: In this article, the triples on ν marks were partitioned into q sets, each forming a BIBD of parameters (ν,3,2). But the results of the partitioning were not discussed.
Abstract: Letq=6t±1, ν=2q+2. The (ν/3) tri ples on ν marks may be partitioned intoq sets, each forming a BIBD of parameters (ν,3,2). Related results, some of them known, are also discussed briefly.

Journal ArticleDOI
TL;DR: In this article, the authors considered the case where the division ring generated by generic n×n matrices over a field k is not a crossed product in the following cases: (i) there exists a primeq such thatq>>\s 3ℛn; (ii)[k:Q]=m, whereQ is the field of rationals.
Abstract: LetA=k (X 1, X2..., Xm) be the division ring generated by genericn×n matrices over a fieldk; thenA is not a crossed product in the following cases: (i) there exists a primeq such thatq 3ℛn;(ii)[k:Q]=m, whereQ is the field of rationals, then if eitherq 3ℛn for someq for whichq-1ℛm, orq 2/nn for some other prime; (iii)k=Z p r a finite field ofp r elements and eitherq 3ℛn for sameqℛp r-1 orq 2ℛn for some other primes. Other cases are also considered.

Journal ArticleDOI
TL;DR: In this paper, the authors construct examples of 3-dimensional polytopes which are not subpolytopes of stack-polytopes, which are called asubpolytope of a polytope if vertP ⊆ vertQ.
Abstract: A polytopeP is called asubpolytope of a polytopeQ if vertP ⊆ vertQ. The purpose of this paper is to construct examples of 3-dimensional polytopes which are not subpolytopes of stack polytopes. Previously no such examples were known.

Journal ArticleDOI
TL;DR: In this article, a complete Boolean algebra of cardinality κ is defined, and if V = L, and κ ≥ 0, then there exists a simple complete boolean algebra κ with cardinality ≥ 0.
Abstract: IfV=L, and κ is an uncountable regular non weakly compact cardinal, then there exists a simple complete Boolean algebra of cardinality κ.