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Showing papers in "Israel Journal of Mathematics in 1970"


Journal ArticleDOI
TL;DR: In this article, the linear topological properties of the Banach space spanned by a sequence of independent random variables in L 1, L 2, L 3, L 4, L 5, L 6, L 7, L 8, L 9, L 10, L 11, L 12, L 1, L 2
Abstract: Let 2

498 citations


Journal ArticleDOI
TL;DR: In this paper, fixed point theorems for multi-valued global and local contraction mappings are proved for a certain type of single-valued locally expansive mapping, and these results concerning contractions are then applied to obtain a fixed point theorem for a specific type of local expansive mapping.
Abstract: Several fixed point theorems for multi-valued global and local contraction mappings are proved. These results concerning contractions are then applied to obtain a fixed point theorem for a certain type of single-valued locally expansive mapping.

413 citations


Journal ArticleDOI
TL;DR: In this paper, the probability distribution of the point sought in the real line is not known to the searcher, and the situation is treated as a game and obtain minimax type solutions.
Abstract: The linear search problem has been discussed previously by one of the present authors. In this paper, the probability distribution of the point sought in the real line is not known to the searcher. Since there is noa priori choice of distribution which recommends itself above all others, we treat the situation as a game and obtain minimax type solutions. Different minimaxima apply depending on the factors which one wishes to minimize (resp. maximize). Certain criteria are developed which help the reader judge whether the results obtained can be considered “good advice” in the solution of real problems analogous to this one.

147 citations


Journal ArticleDOI
Burgess Davis1
TL;DR: In this paper, it was proved that the norms of sup n isEnabled |f petertodd n |f n¯¯ |f fixme n | and of ¯¯ ¯¯ $ (€ 1.2 ) are equivalent.
Abstract: Letf=(f 1,f 2,..) be a martingale. It is proved that theL 1 norms of sup n |f n | and of $$\left( {\sum {\left( {f_n - f_{n - 1} } \right)^2 } } \right)^{\tfrac{1}{2}} $$ are equivalent. This result completes results of D. L. Burkholder and R. F. Gundy on operators on martingales.

138 citations


Journal ArticleDOI
TL;DR: In this article, the authors considered the problem of finding global solutions of the initial value problem (1) in the Banach space, and showed that various approximation processes converge to the solution (whenever it exists).
Abstract: This paper is concerned with global solutions of the initial value problem (1)du/dt +Au∋0,u(0)=x whereA is a (nonlinear) accretive set in a Banach spaceX. We show that various approximation processes converge to the solution (whenever it exists). In particular we obtain an exponential formula for the solutions of (1).

68 citations


Journal ArticleDOI
Per Enflo1
TL;DR: In this article, the relation between the structure of the topological group and the Banach space was investigated and the results on the relationship between topological groups and Banach spaces were obtained.
Abstract: Commutative groups uniformly homeomorphic to certain Banach spaces are considered. Results on the relation between the structure of the topological group and the Banach space are obtained.

66 citations


Journal ArticleDOI
TL;DR: In this article, the existence of subprocesses and ergodicity of conservative processes are studied and the necessary and sufficient conditions for a σ-finite invariant measure are given for discrete and continuous time processes.
Abstract: A Markov operator preservingC(X) is known to induce a decomposition of the locally compact spaceX to conservative and dissipative parts. Two notions of ergodicity are defined and the existence of subprocesses is studied. A sufficient condition for the existence of a conservative subprocess is given, and then the process is assumed to be conservative. When it has no subprocesses, sufficient conditions for the existence of a σ-finite invariant measure are given, and are extended to continuous-time processes. When the invariant measure is unique, ratio limit theorems are proved for the discrete and continuous time processes. Examples show that some combinations of conservative processes are not necessarily conservative.

39 citations


Journal ArticleDOI
TL;DR: A new and conceptually simpler proof of Aleksandrov and Shephard's theorem is given in this paper, showing that ad-polytope (d ≥ 3) is itself centrally symmetric.
Abstract: A new and conceptually simpler proof is given of the theorem of A D Aleksandrov and G C Shephard, that ad-polytope (d≧3), all of whose facets are centrally symmetric, is itself centrally symmetric

36 citations


Journal ArticleDOI
TL;DR: The propertyPm (directly analogous to Valentine's propertyP3) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) if a closed setS has propertyPM and contains k points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets as discussed by the authors.
Abstract: The propertyPm (directly analogous to Valentine’s propertyP3) is used to prove several curious results concerning subsets of a topological linear space, among them the following: (a) If a closed setS has propertyPm and containsk points of local nonconvexity no distinct pair of which can see each other viaS, thenS is the union ofm − k − 1 or fewer starshaped sets (b) Any closed connected set with propertyPm is polygonally connected (c) A closed connected setS with propertyPm is anLm−1 set (each pair of points may be joined by a polygonal arc ofm − 1 of fewer sides inS) (d) A finite-dimensional set with propertyPm is anL2m − 3 set A new proof of Tietze’s theorem on locally convex sets is given, and various examples refute certain plausible conjectures

32 citations


Journal ArticleDOI
TL;DR: In this paper, a generalization of Rellich's theorem on the Helmholz equation was obtained for a large class of higher order equationsP(1/i∂/∂x)u=f, subject to the condition that the Gaussian curvature ofP(ξ)=0 never vanish.
Abstract: In an earlier paper a generalization of Rellich’s theorem on the Helmholz equation was obtained for a large class of higher order equationsP(1/i∂/∂x)u=f, subject to the condition that the Gaussian curvature ofP(ξ)=0 never vanish. This restriction is removed in this note.

27 citations


Journal ArticleDOI
TL;DR: In this article, it was shown that if X, Y are compact countable metric spaces such that Y contains no subset homeomorphic to X, then for any isomorphism Φ of C(X) into C(Y), Φ φ ‖ ‖ φ−1 ‖≧3.
Abstract: IfX, Y are compact countable metric spaces such thatY contains no subset homeomorphic toX, then for any isomorphismΦ ofC(X) intoC(Y), ‖ φ ‖ ‖ φ−1 ‖≧3. This result and some variants of it are established here, and prove a special case of a conjecture raised in [1].

Journal ArticleDOI
TL;DR: In this paper, it was shown that the space of measures on the circle T is continuous with respect to each other whenμ varies on bounded subsets of M(T), and that measures which are almost idempotent (i.e.
Abstract: For measuresμ on the circleT the quantities\(lim sup_{n \to + \infty } \left| {\hat \mu (n)} \right|\),\(lim sup\left| {\hat \mu (n)} \right|_{_{n \to - \infty } } \) need not be equal; it is shown, however, that they are continuous with respect to each other whenμ varies on bounded subsets ofM(T), the space of measures onT. It is also shown that measuresμ which areɛ-almost idempotent (i.e.\(lim sup_{\left| n \right| \to \infty } \left| {\hat \mu (n) - \hat \mu (n)^2 } \right|< \varepsilon \)) are the sum of an idempotent measure and of a measureυ satisfying\(lim sup_{\left| n \right| \to \infty } \left| {\hat u (n)} \right|< 2\varepsilon \) providedɛ is small enough (as a function of ‖μ‖).

Journal ArticleDOI
TL;DR: In this article, the set of polynomial identities of a ring A is considered, as well as some types of minimal identities, and the change which occurs in these identities upon passage to related rings is then studied.
Abstract: The set of polynomial identities of a ringA is considered, as well as some types of minimal identities. The change which occurs in these identities upon passage to related rings is then studied.

Journal ArticleDOI
TL;DR: In this paper, a combinatorial theorem is established, which states that if a family consisting of subsets of a set M contains every subset of each member, then the complements inM of the A's have a permutation such that each subset has at most two of them being coprime.
Abstract: A combinatorial theorem is established, stating that if a familyA 1,A 2, …,A s of subsets of a setM contains every subset of each member, then the complements inM of theA’s have a permutationC 1,C 2, …,C s such thatC i ⊃A i . This is used to determine the minimal size of a maximal set of divisors of a numberN no two of them being coprime.

Journal ArticleDOI
TL;DR: In this paper, necessary and sufficient conditions for ω to be thepth order, elementary symmetric function of the prncipal radii on the boundary of a sufficiently smooth convex body of revolution in Euclidean space are described.
Abstract: Necessary and sufficient conditions are described on ap function ω over the unit sphere in Euclideann-spaceE n in order for ω to be thepth order, elementary symmetric function of the prncipal radii on the boundary of a sufficiently smooth convex body of revolution inE n ; here these radii are taken as functions of the outer unit normal direction on the bounding surface;p satisfies 1≦p

Journal ArticleDOI
TL;DR: Ben-Israel and Charnes as discussed by the authors gave explicit noniterative solutions of interval linear programs, which extend earlier results of Ben-Israel et al. They considered the problem of maximizing {(c, x):a≦A x≦b} for given matrixA and vectorsa, b andc.
Abstract: An interval linear program is the problem of maximizing {(c, x):a≦A x≦b} for given matrixA and vectorsa, b andc. The explicit (noniterative) solutions of interval programs given here, extend earlier results of Ben-Israel and Charnes.

Journal ArticleDOI
TL;DR: In this article, a non-simpliciald-polytope with strictly fewer k-faces is shown to have strictly fewerk-faces than a simpliciald polytope, where k is the number of vertices.
Abstract: A non-simpliciald-polytope is shown to have strictly fewerk-faces ([(d−1)/2]≦k≦d−1) then some simpliciald-polytope with the same number of vertices; actual numerical bounds are given. This provides a strong affirmative answer to a problem of Klee.

Journal ArticleDOI
TL;DR: In this paper, the sets of integers weakly and strongly representable in a system of second-order arithmetic with an infinity rule of inference are given. But the results generalize to each of a family of such systems.
Abstract: We answer two questions posed in a recent paper by H. B. Enderton by giving characterizations of the sets of integers weakly and strongly representable in a system of second order arithmetic with an infinity rule of inference. The results generalize to each of a family of such systems.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the joint spectrum of the tensor product of several operators is the cartesian product of their spectra, i.e., it is the product of the spectra of all the operators' tensor products.
Abstract: We prove that the joint spectrum of the tensor product of several operators is the cartesian product of their spectra.

Journal ArticleDOI
TL;DR: A new proof of Levy's version of the Absoluteness Lemma is given in this paper, a proof which avoids dependent choices and leads to stronger versions of the lemma.
Abstract: A new proof of Levy’s version of the Absoluteness Lemma is given—a proof which avoids dependent choices and leads to stronger versions of the lemma.

Journal ArticleDOI
TL;DR: A notion of a probabilistic tree automaton is defined and a condition is given under which it is equivalent to a usual tree Automaton, and a theorem about context free languages is stated.
Abstract: A notion of a probabilistic tree automaton is defined and a condition is given under which it is equivalent to a usual tree automaton. A theorem about context free languages is stated.

Journal ArticleDOI
TL;DR: In this article, a 3-dimensional convex polytopeP and a simple circuitC in the 1-skeleton of P are shown to be combinatorially equivalent to a convex 2-dimensional circuit in P and a direction such that if P′ is projected orthogonally in this direction, then the inverse image of the boundary of the projection is the circuit inP′ corresponding to the circuit C inP.
Abstract: Given any 3-dimensional convex polytopeP, and any simple circuitC in the 1-skeleton ofP, there is a convex polytopeP′ combinatorially equivalent toP, and a direction such that ifP′ is projected orthogonally in this direction, then the inverse image of the boundary of the projection is the circuit inP′ corresponding to the circuitC inP.

Journal ArticleDOI
TL;DR: There are infinitely many isomorphism types of separable infinite-dimensional spaces as discussed by the authors, and isomorphisms can be expressed in terms of a set of isomorphic types of spaces.
Abstract: There are infinitely many isomorphism types of separable infinite-dimensional Ł1 spaces.

Journal ArticleDOI
TL;DR: Semisimple, semiperfect, and perfect rings are characterized by quasiprojective modules and covers over them as mentioned in this paper, where the modules are composed of quasimple modules and cover is composed of covers.
Abstract: Semisimple, semiperfect, and perfect rings are characterized by quasiprojective modules and quasiprojective covers over them.

Journal ArticleDOI
TL;DR: In this paper, it was shown that for a given univalent function such that |a 2| < 0.867, |a n ≥ n ≥ 1 for each n for eachn.
Abstract: In the following we prove that for a given univalent function such that |a 2| <0.867, |a n |≦n for eachn. The method of proof is closely related to Milin’s method.

Journal ArticleDOI
TL;DR: In this paper, it was shown that every locally compact non-discrete abelian group G contains a compact subset E such that A(E) admits spectral synthesis, although it contains a closed, regular, self-adjoint subalgebra which is isomorphic to an algebra of infinitely differentiable functions on [−1, 1].
Abstract: We prove that every locally compact non-discrete abelian groupG contains a compact subsetE such thatA(E) — the restriction algebra ofA(G) toE — admits spectral synthesis, although it contains a closed, regular, self-adjoint subalgebra which is isomorphic to an algebra of infinitely differentiable functions on [−1, 1]. We also give some general results concerning the failure of spectral synthesis in regular Banach algebras.

Journal ArticleDOI
TL;DR: In this article, two general criterias for the union of topological spaces of a given type to be also of the same type, and two general criteria for topological topology unions are presented.
Abstract: Two general criterias for the union of topological spaces of a given type to be also of the same type, are presented.

Journal ArticleDOI
TL;DR: In this paper, it was shown that a series, Σx n, in a complete linear metric space with a basis is sub-series convergent if and only if it is weakly subseries convergegent.
Abstract: In this note, we show that a series, Σx n , in a complete linear metric space with a basis is subseries convergent if and only if it is weakly subseries convergent.

Journal ArticleDOI
TL;DR: In this paper, a conjecture of S. Znam was disproved by means of a counterexample, and the conjecture was shown to be a conjecture that cannot be proven.
Abstract: A conjecture of S. Znam [1] is disproved by means of a counterexample.

Journal ArticleDOI
TL;DR: In this article, a large number of examples are given to show that such a presentation does exist for nearly all such groups for which r(G) has been calculated, for which a presentation is known to exist.
Abstract: It is an open question whether or not every finitep-groupG has a presentation withd(G)=dimH1(G,Zp) generators andr(G)=dimH2(G, Zp) relations; in this article, a large number of examples are given to show that such a presentation does exist for nearly all such groups for whichr(G) has been calculated.