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Showing papers in "Israel Journal of Mathematics in 1980"


Journal ArticleDOI
TL;DR: In this paper, sufficient conditions for supersolvability of finite groups are given considering the maximal subgroups of Sylow subgroups, and an example is given to show that the conditions given are not necessary or sufficient.
Abstract: In this paper two sufficient conditions for supersolvability of finite groups are given considering the maximal subgroups of Sylow subgroups of the group. An example is given to show that the conditions given are not necessary for supersolvability.

206 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that any I.E.T. is not mixing with respect to any Borel invariant measure, and the same is true for any special flow constructed by any I E.T and any "roof" function of bounded variation.
Abstract: An interval exchange transformation (I.E.T.) is a map of an interval into itself which is one-to-one and continuous except for a finite set of points and preserves Lebesgue measure. We prove that any I.E.T. is not mixing with respect to any Borel invariant measure. The same is true for any special flow constructed by any I.E.T. and any “roof” function of bounded variation. As an application of the last result we deduce that in any polygon with the angles commensurable with π the billiard flow is not mixing on two-dimensional invariant manifolds.

158 citations


Journal ArticleDOI
TL;DR: In this paper, the invariant densities of numbertheoretical endomorphisms with indifferent fixed points are considered, whose invariant density can be determined up to multiplication by functions bounded away from zero and infinity.
Abstract: Numbertheoretical endomorphisms with indifferent fixed points are considered, whose invariant densities can be determined up to multiplication by functions bounded away from zero and infinity.

154 citations


Journal ArticleDOI
TL;DR: A countable group has property T of Kazhdan if and only if no measure preserving ergodic action of Γ has non-trivial asymptotically invariant sets as mentioned in this paper.
Abstract: A countable group Γ has property T of Kazhdan if and only if no measure preserving ergodic action of Γ has non-trivial asymptotically invariant sets.

143 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that a variant of ♣ called ♣ (club) is consistent with 2ℵ 0<⵵ 1.1. And also that ♣ is consistent in the sense that it can be expressed as a diamond suit.
Abstract: We prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of $$ \diamondsuit _{\omega _1 } $$ called ♣ (club), is consistent with 2ℵ 0<ℵ1.

107 citations


Journal ArticleDOI
TL;DR: In this article, the authors give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval, and give a priori bound for the number of different ergodic measures.
Abstract: We give sufficient conditions for the existence of absolutely continuous invariant measures, finite or σ-finite, for maps on the interval. We givea priori bound for the number of different ergodic measures. The results are obtained via the first return map.

106 citations


Journal ArticleDOI
TL;DR: In this article, the existence of asymptotically invariant sequences for a natural action of SL (2,Z) on the 2-sphere has been shown.
Abstract: LetG be a countable group which acts non-singularly and ergodically on a Lebesgue space (X, ȑ, μ). A sequence (Bn) in ℒ is calledasymptotically invariant in limn μ (BnΔgBn)=0 for everygeG. In this paper we show that the existence of such sequences can be characterized by certain simple assumptions on the cohomology of the action ofG onX. As an explicit example we prove that a natural action of SL (2,Z) on the 2-sphere has no asymptotically invariant sequences. The last section deals with a particular cocycle for this action which has an interpretation as a random walk on the integers with “time” in SL (2,Z).

105 citations


Journal ArticleDOI
TL;DR: In this article, the authors used functional analytic methods to solve a fully nonlinear second order elliptic equation, a Hamilton-Jacobi equation, and a functional/partial differential equation from plasma physics.
Abstract: We use similar functional analytic methods to solve (a) a fully nonlinear second order elliptic equation, (b) a Hamilton-Jacobi equation, and (c) a functional/partial differential equation from plasma physics The technique in each case is to approximate by the solutions of simpler problems, and then to pass to limits using a modification of G Minty’s device to the spaceL ∞

97 citations


Journal ArticleDOI
TL;DR: In this article, the authors prove the consistency of the existence of arbitrarily large strongly compact cardinals with respect to ZF of the statement that every infinite set is a countable union of sets of smaller cardinality.
Abstract: Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of smaller cardinality. Some other statements related to this one are investigated too.

86 citations


Journal ArticleDOI
TL;DR: In this paper, a Banach subspace E of L 1 is constructed such that E fails the Radon-Nikodym property, yet has no bounded dyadic δ-tree for any δ > 0.
Abstract: A Banach subspaceE ofL1 is constructed such thatE fails the Radon-Nikodym property, yetE has no bounded dyadicδ-tree for anyδ>0. The unit ball ofE is also relatively compact in the topology of convergence in probability, which implies thatE has the strong Schur property. The construction uses martingales and probabilistic techniques.

75 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied Krull domains that are half-factorial domains (HFDs) and provided examples of HFDs that contradict a conjecture of Narkiewicz.
Abstract: LetR be a commutative domain with 1. We termR an HFD (Half-Factorial-Domain) provided the equality Π i=1 n χi=Π{f=1/m}y f impliesm=n, whenever thex’s and they’s are non-zero, non-unit and irreducible elements ofR. The purpose of this note is to study HFD’s, in particular, Krull domains that are HFD’s, and to provide examples of HFD’s, that contradict a conjecture of Narkiewicz.

Journal ArticleDOI
TL;DR: In this article, isoperimetric inequalities are used to obtain measure estimates on almost constancy sets of functions on product spaces, which are applied to produce almost unconditional or symmetric block sequences from given sequences.
Abstract: Isoperimetric inequalities are used to obtain measure estimates on almost constancy sets of functions on product spaces. These are applied to produce almost unconditional or symmetric block sequences from given sequences. Their length, which is (logn)1/2 in the general case, improves ton a where a cotype condition is imposed or when the given sequences arep-type attaining for somep<2. In thep-type attaining case, block sequences (1+e)-equivalent to the unit vector basis ofl p m can be obtained when log logm ∼ log logn.

Journal ArticleDOI
TL;DR: In this paper, sufficient conditions for the orbit of a (euclidean)p-frame over a vector spaceV,p
Abstract: We obtain certain sufficient conditions for the orbit of a (euclidean)p-frame over a vector spaceV,p

Journal ArticleDOI
TL;DR: In this paper, the unconditional basis constants of subspaces of finite-dimensional Banach spaces of dimension n were studied and a new isometric characterization of l ∞n is given, some new information is obtained concerning p-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern 1/2.
Abstract: Lower bounds are obtained for thegl constants and hence also for the unconditional basis constants of subspaces of finite dimensional Banach spaces. Sharp results are obtained for subspaces ofl∞n, while in the general case thegl constants of “random large” subspaces are related to the distance of “random large” subspaces to Euclidean spaces. In addition, a new isometric characterization ofl∞n is given, some new information is obtained concerningp-absolutely summing operators, and it is proved that every Banach space of dimensionn contains a subspace whose projection constant is of ordern1/2.

Journal ArticleDOI
TL;DR: Using the duality between Dunford-Pettis operators and Pettis-Cauchy martingales, this paper showed that a Banach space X has the Radon-Nikodým property provided the members of ℒ(L1,X) are representable.
Abstract: Using the duality between Dunford-Pettis operators onL1 and Pettis-Cauchy martingales, we prove that the Dunford-Pettis operators fromL1 intoL1 form a lattice. We show also that a Banach spaceX has the Radon-Nikodým property provided the Dunford-Pettis members of ℒ(L1,X) are representable. The lifting of dual valued Dunford-Pettis operators is investigated. Some remarks are included.

Journal ArticleDOI
TL;DR: In this paper, the Grothendieck property was used to construct a compact space K such that the unit ball of K does not contain βN, and that ∞(N) is neither a subspace nor quotient of l(K).
Abstract: Using the continuum hypothesis, we construct a compact spaceK such that l(K) possesses the Grothendieck property, but such that the unit ball of l(K)′ does not containβN, and hence, in particular, such thatl∞(N) is neither a subspace nor quotient of l(K). In particular,K does not contain a convergent sequence but does not containβN.

Journal ArticleDOI
TL;DR: In this article, it was shown that if G is a solvable group such that G and R + have disjoint torsions, then J(R)=0 impliesJ(R(G))=0.
Abstract: LetR be a ring and σ an automorphism ofR. We prove the following results: (i)J(R σ[x])={Σiri x i:r0∈I∩J(R]), r i∈I for allie 1} whereI↪ {r∈R:rx ∈J(R Σ[x])|s= (ii)J(R σ )=(J(R σ )∩R)σ . As an application of the second result we prove that ifG is a solvable group such thatG andR, + have disjoint torsions thenJ(R)=0 impliesJ(R(G))=0.

Journal ArticleDOI
TL;DR: An expository account of Wolff's recent elementary proof of Carleson's Corona Theorem (1962) is given in this paper, which answers a question raised by S. Kakatani (1957) as to whether the open unit disc in the complex plane is dense in the maximal ideal space of the Banach algebra of bounded analytic functions thereon.
Abstract: An expository account is given of T. Wolff’s recent elementary proof of Carleson’s Corona Theorem (1962). The Corona Theorem answers affirmatively a question raised by S. Kakatani (1957) as to whether the open unit disc in the complex plane is dense in the maximal ideal space of the Banach algebra of bounded analytic functions thereon.

Journal ArticleDOI
TL;DR: In this paper, it was shown that any space isomorphic to the Banach space of all bounded real functions on ω1 with countable support (with the supremum norm) contains a subspace isometric tom ≥ ω(ω1).
Abstract: Let ω1 denote the first uncountable ordinal,m ω(ω1) the Banach space of all bounded real functions on ω1 with countable support (with the supremum norm). It is shown that any space isomorphic tom ω(ω1) contains a subspace isometric tom ω(ω1). Several similar results concerning higher cardinals are obtained.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every Σ13 set is Lebesgue measurable, has the Baire property and is either countable or contians a perfect subset.
Abstract: We prove under the assumption of the existence of a measurable, cardinal and precipitous ideal onw1 that every Σ13 set is Lebesgue measurable, has the Baire property and is either countable or contians a perfect subset. We get similar results for Σ14 sets, if we add the additional assumptions of C. H. and that\(P_w (2^{2w_1 } )\) carries a normal precipitous ideal.

Journal ArticleDOI
TL;DR: In this paper, the authors give sufficient conditions for the differential operatorP(λ = P0+λP1++λm−1Pm− 1+λm, depending polynomially on the complex parameter λ, to verify the following statement: there exists a Schwartz space of rapidly decreasing functions such that P(λ0)u0=0-
Abstract: In this paper, we give some sufficient conditions for which the differential operatorP(λ=P0+λP1++λm−1Pm−1+λm, depending polynomially on the complex parameter λ, verifies the following statement: there exists λ0 ∈ ℂ,uo=0,u0 ∈ ℒ(ℝn) a Schwartz space of rapidly decreasing functions, such thatP(λ0)u0=0-

Journal ArticleDOI
TL;DR: In this article, the unconditional structure of a space constructed by Kalton and Peck is investigated and it is proved that even though this space has an unconditional decomposition into subspaces of dimension two, it does not have G.L.u.st.
Abstract: The unconditional structure of a space constructed by Kalton and Peck is investigated. Among other things it is proved that even though this space has an unconditional decomposition into subspaces of dimension two, it does not have G.L. l.u.st.

Journal ArticleDOI
TL;DR: In this paper, facts derived from an elementary analysis of the bahavior of {Jλx}λ≧0 for fixedxeD(T) are applied to obtain theorems concerning existence of zeros and surjectivity.
Abstract: An operatorT defined on a subsetD(T) of a Banach spaceE and taking values in 2E is said to bem-accretive if for each λ>0, the mappingJλ=(I+λT)−1 is a single-valued nonexpansive mapping defined on all ofE. In this paper, facts derived from an elementary analysis of the bahavior of {Jλx}λ≧0 for fixedxeD(T) are applied to obtain theorems concerning existence of zeros and surjectivity.

Journal ArticleDOI
TL;DR: In this paper, a new approach is given to the entropy of a probability-preserving group action by defining an approximate "r-entropy", 0
Abstract: A new approach is given to the entropy of a probability-preserving group action (in the context ofZ and ofR n ), by defining an approximate “r-entropy”, 0

Journal ArticleDOI
TL;DR: In this paper it was shown that if C(K) is isomorphic to a subspace of C (K), then the subspace C(k) is positively isometric to subspace c (k) of c(k), where k is a measure.
Abstract: Using the continuum hypothesis we construct a compact spaceK such that the spaceM (K)) of measures onK is vaguely separable, i.e., thatC (K) is injected intol ∞, but thatC (K) is not isomorphic to a subspace ofl ∞. It is shown that ifC(K) is isomorphic to a subspace ofC (K) is positively isometric to a subspace ofl ∞(⌈). Nevertheless, under the continuum hypothesis one can construct a compact spaceL such that the spaceM 1 + (L) of probabilities onL is vaguely separable, butL cannot be the support of a measureμ withL 1(μ) separable in the norm.

Journal ArticleDOI
TL;DR: In this article, it was shown that Gregory's result is the best possible for well-chosen sequences of ordinals converging to a given ordinal subsequence, which implies that GCH does not have the uniformization property.
Abstract: We present S. Shelah’s result thatS 1 2 ={δ<ω 2: cf(δ)=ω 1} may have the uniformization property (cf., §1, or [3] for a definition) for “well-chosen sequences”, 〈η δ:δ∈S 1 2 ^η δ an increasingω 1-sequence of ordinals converging to δ〉. This implies that $$GCH ot \to \diamondsuit _{s_1^2 } $$ , which shows that Gregory’ result (cf., [2]), $$GCH \to \diamondsuit _{s_0^2 } $$ , is the best possible.

Journal ArticleDOI
TL;DR: In this paper, it was shown that if T :L 1 → L 1 is a bounded linear operator failing the Dunford-Pettis property, then T fixes a copy of ⊕ll 1 (l 2).
Abstract: It is shown that ifT :L 1 →L 1 is a bounded linear operator failing the Dunford-Pettis property, thenT fixes a copy of ⊕ll1 (l 2).

Journal ArticleDOI
TL;DR: In this paper, a simpler, alternate approach to the prime ideals in crossed products of finite groups was proposed. But this approach requires the use of the induced ideal mapG instead of theν map.
Abstract: In this note, we offer a simpler, alternate approach to the work of Section 3 of “Prime ideals in crossed products of finite groups.” Indeed, by using the induced ideal mapG instead of theν map, we have eliminated many of the unpleasant computations of the original argument.

Journal ArticleDOI
TL;DR: In this article, the authors investigated the combinatorial structure of quotient polytopes, where C is a cyclic polytope and F is a face of C. They showed that every simpliciald-polytope with + 3 vertices is isomorphic to a quotient of a simplicial-d polytop.
Abstract: We investigate the quotient polytopesC/F, whereC is a cyclic polytope andF is a face ofC. We describe the combinatorial structure of such quotients, and show that under suitable restrictions the pair (C, F) is determined by the combinatorial type ofC/F. We describe alternative constructions of these quotients by “splitting vertices” of lower-dimensional cyclic polytopes. Using Gale diagrams, we show that every simpliciald-polytope withd+3 vertices is isomorphic to a quotient of a cyclic polytope.

Journal ArticleDOI
TL;DR: In this article, a quotient space of the ordinals not greater than the order topology is constructed which is not isomorphic to a subspace of C(α),a < w.............. 1.
Abstract: A quotient space ofC (w w ), the continuous functions on the ordinals not greater thanW w with the order topology, is constructed which is not isomorphic to a subspace ofC(α),a < w 1.