Showing papers in "Israel Journal of Mathematics in 1965"
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TL;DR: In this article, the maximum number of cliques possible in a graph with n nodes is determined and bounds are obtained for the number of different sizes of clique possible in such a graph.
Abstract: A clique is a maximal complete subgraph of a graph. The maximum number of cliques possible in a graph withn nodes is determined. Also, bounds are obtained for the number of different sizes of cliques possible in such a graph.
907 citations
TL;DR: In this article, it was shown that if C is a sufficiently large constant, then every graph of n vertices and [Cn====== 3/2] edges contains a hexagon.
Abstract: The author proves that ifC is a sufficiently large constant then every graph ofn vertices and [Cn
3/2] edges contains a hexagonX
1,X
2,X
3,X
4,X
5,X
6 and a seventh vertexY joined toX
1,X
3 andX
5. The problem is left open whether our graph contains the edges of a cube, (i.e. an eight vertexZ joined toX
2,X
4 andX
6).
119 citations
TL;DR: In this article, disconnected remarks on number theory are made in the context of the paper of the same title, and the disconnected remarks are shown to be disconnected from the original paper. [5]
Abstract: Like the previous paper of the same title [5] this note contains disconnected remarks on number theory.
101 citations
TL;DR: In this article, the following theorem was proved: if there is an isomorphism φ of C(X) onto C(Y) (X, Y-compact) with ∥ φ ∥ · ∥π−1 ∥ < 2, then X and Y are homeomorphic.
Abstract: The following theorem, which strengthens the classical theorem of Stone, is proved: If there is an isomorphism φ ofC(X) ontoC(Y) (X, Y-compact) with ∥ φ ∥ · ∥ φ−1 ∥ < 2, thenX andY are homeomorphic.
79 citations
TL;DR: In this paper, it was shown that if Euclidean is replaced by n-dimensional Banach space, the direct statement is true, but the converse statement is false.
Abstract: In Euclidean space a set of constant width has the property that it is not a proper subset of any set of the same diameter. The converse implication is also true. Here we show that if Euclidean is replaced byn-dimensional Banach space the direct statement is true, but the converse statement is false. Attention is drawn to the problem of characterising those Banach spaces of finite dimension for which the converse is true.
73 citations
TL;DR: In this article, the authors define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity, and under them, there is always a minimizing search procedure.
Abstract: The linear search problem concerns a search made in the real line for a point selected according to a given probability distribution. The search begins at zero and is made by continuous motion with constant speed along the line, first in one direction and then the other. The problem is to search in such a manner that the expected time required for finding the point according to the chosen plan of search is a minimum. This plan of search is usually conceived of as having a first step, a second,etc., and in that case, this author has previously shown a necessary and sufficient condition on the probability distribution for the existence of a search plan which minimizes the expected searching time. In this paper, we define a notion of search in which there is no first step, but the steps are instead numbered from negative to positive infinity. These new rules change the problem, and under them, there is always a minimizing search procedure. In those cases which satisfy the earlier criterion, the solutions obtained are essentially the same as those obtained previously.
57 citations
TL;DR: An implicit function theorem and a resulting modified Newton-Raphson method for roots of functions between finite dimensional spaces, without assuming non-singularity of the Jacobian at the initial approximation are given in this paper.
Abstract: An implicit function theorem and a resulting modified Newton-Raphson method for roots of functions between finite dimensional spaces, without assuming non-singularity of the Jacobian at the initial approximation.
39 citations
TL;DR: In this paper, the problem of the partition-numbersJ ℱ(p, q), considered by Hadwiger and Debrunner for the family C n of convex bodies, is extended to simplicial complexes and arbitrary families assuming only the validity of Helly's theorem.
Abstract: The problem of the partition-numbersJ ℱ(p, q), considered by Hadwiger and Debrunner for the family ℱ=C n of convex bodies, is extended to simplicial complexes and arbitrary families assuming only the validity of Helly’s theorem. We obtain results similar to those of Hadwiger and Debrunner. Further we show the existence of all partition-numbers for the familyℱ = H nC of homothets of a convex body and we get new informations on the partition-numbers for the family of parallel rectangles.
32 citations
TL;DR: In this article, a characterisation of uniformly non-1n/(1)Orlicz space is obtained intrinsically in terms of the Young function determining the Orlicz spaces, which is shown to be reflexive.
Abstract: A characterisation of uniformly non-1n/(1)Orlicz space is obtained intrinsically in terms of the Young function determining the Orlicz space. It is shown that a uniformally non-1n/(1)Orlicz space is reflexive.
25 citations
TL;DR: In this paper, it was proved that some conjectures concerning non-separable reflexive Banach spaces are true for reflexive spaces having the metric approximation property, and that the same conjecture holds also for nonseparable Banach Spaces.
Abstract: It is proved that some conjectures concerning non-separable reflexive Banach spaces are true for reflexive spaces having the metric approximation property.
23 citations
TL;DR: In this article, a complete proof for Schnirelmann's theorem on the existence of a square in C====== 2 Jordan curves is given, using the same method as in this paper.
Abstract: A complete proof is given for Schnirelmann’s theorem on the existence of a square inC
2 Jordan curves. The following theorems are then proved, using the same method: 1. On every hypersurface inR
n,C
3-diffeomorphic toS
n−1, there exist 2n points which are the vertices of a regular 2
n
-cellC
n. 2. Every planeC′ Jordan curve can beC′ approximated by a curve on which there are 2N distinct points which are the vertices of a centrally symmetric 2N-gon (anglesπ not excluded). 3. On every planeC
2 curve there exist 5 distinct points which are the vertices of an axially symmetric pentagon with given base anglesa, π/2≦a<π. (The angle at the vertex on the axis of symmetry might beπ).
TL;DR: In this paper, necessary and sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found.
Abstract: Sufficient conditions for generalized absolutely monotone functions to possess a Taylor-type expansion in terms of the corresponding Extended Tchebycheff systems were found by Karlin and Ziegler. The question of necessary conditions, however, was left open. In this paper we solve this question by finding necessary and sufficient conditions for the validity of the expansion. The structure of the cone of generalized absolutely monotone functions and its extreme rays are also discussed.
TL;DR: In this paper, it was shown that an ann-dimensional convex body is the affine image of a body of constant breadth if and only if it has constant diagonal, where the main diagonal of the circumscribed boxes has constant length.
Abstract: A convex body is said to have constant diagonal if and only if the main diagonal of the circumscribed boxes has constant length. It is shown that ann-dimensional convex body,n≧3, is the affine image of a body of constant breadth if and only if it has constant diagonal. Affine images of bodies of constant breadth are also characterized by the property that the orthogonal projection on each hyperplane is the affine image of a body of constant breadth in that hyperplane.
TL;DR: In this article, the structure of the kernel of a composition of two games is investigated, and a comparison with the results for Von-Neumann and Morgenstern solutions is made.
Abstract: The structure of the kernel of a composition of two games is investigated; a comparison with the results for Von-Neumann and Morgenstern solutions is included.
TL;DR: In this article, a bound of (2/m)m −1)consuming m−1 for normed linear spaces was obtained for |p(z 0)|, where p(z) = \prod\limits_{i = 1}^m {(z - z_i ),\left| {z_i } \right| \leqq 1, i = 1,...m,} $$====== andp′(z ≥ 0.
Abstract: Letx 1,...,x
m be points in the solid unit sphere ofE
n and letx belong to the convex hull ofx
1,...,x
m. Then
$$\prod\limits_{i = 1}^m {\left| {x - x_i } \right.\left\| \leqq \right.(1 - \left\| x \right\|)(1 + \left\| x \right\|)m^{ - 1} } $$
. This implies that all such products are bounded by (2/m)
m
(m −1)
m−1. Bounds are also given for other normed linear spaces. As an application a bound is obtained for |p(z
0)| where
$$p(z) = \prod\limits_{i = 1}^m {(z - z_i ),\left| {z_i } \right| \leqq 1,i = 1,...m,} $$
andp′(z
0)=0.
TL;DR: In this paper, it was shown that the Mellin transform does not satisfy the regularity conditions of the theorems of Landau Polya, and corresponding conclusions were obtained when F(s) did not satisfy Landau and Polya.
Abstract: If F(s), the Mellin transform ƒ(x) (real valued), satisfies certain regularity conditions and if its behaviour on the abscissa of convergence is known, then theorems of Landau and Polya give information concerning sign changes of ƒ(x). In the present paper, corresponding conclusions are obtained when F(s) does not satisfy the regularity conditions of the theorems of Landau Polya.
TL;DR: In this article, a classification of closed subsets Θ ofT, the reals modulo 1, as sets of (p,α) multiplicity (or (p α uniqueness) as Θ is (or is not) the support of a non-zero distribdtion ψ such that ψ4 (n) ∈ χp, α.
Abstract: The spaces χp,α induce a classification of the closed subsets Θ ofT, the reals modulo 1, as sets of (p,α) multiplicity (or (p, α uniqueness) as Θ is (or is not) the support of a non-zero distribdtion ψ such that ψ4 (n) ∈ χp,α.
TL;DR: It is proved in this note that formulas giving the probabilities of various states of the system hold as well when the incoming stream is not orderly.
Abstract: A service system consisting ofn = ∞ lines, is considered Formulas (2) giving the probabilities of various states of the system were obtained by the assumption that the incoming stream is stationary, orderly, and without after-effects It is proved in this note that these formulas hold as well when the incoming stream is not orderly
TL;DR: In this paper, the results of Rosenblatt on almost periodic transition operators are extended to the reducible case, where the transition operator can be reduced to a reducible transition operator.
Abstract: Results of Rosenblatt on almost periodic transition operators are extended to the reducible case.
TL;DR: In this paper, a proof is given for the equivalence of Polya's W-property of a linear differential equation to the possibility of decomposing it into a Chebyshev system in a given interval.
Abstract: A proof is given for the equivalence of Polya’sW- property of a linear differential equationL
n (D) y=0 to the possibility of decomposingL
n (D) ≡ Π
1
[D+λi(x)] in a given interval In this case a set ofn independent solutions form a Chebyshev system in the interval An application determines intervals of non-oscillation for solutions of linear equations of the second order
TL;DR: In this article, the authors considered the compound decision problem under the assumption that nature plays the role of an opponent, and showed that in many instances sequential compound rules with good asymptotic properties for the fixed case have similar good properties in the opponent case.
Abstract: The compound decision problem is considered under the assumption that Nature plays the role of an opponent. The empirical Bayes approach is a particular case of this approach. It is shown that in many instances sequential compound rules with good asymptotic properties for the fixed case have similar good properties in the opponent case.
TL;DR: A short proof of the Levy continuity theorem in Hilbert space is given in this paper, where it is shown that the characteristic function of a probability distribution is a function that is uniformly τ-continuous near zero.
Abstract: A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e
i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ
j be the characteristic function of the probability measurem
j onH, Then necessary and sufficient that ∫f dm
j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ
j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem.
TL;DR: On demontre quelques resultats concernant les ensembles de synthese ou de resolution spectrale dans les groupes abeliens localement compacts.
Abstract: On demontre quelques resultats concernant les ensembles de synthese ou de resolution spectrale dans les groupes abeliens localement compacts.
TL;DR: In this article, a necessary and sufficient condition for an operator to be normalizable in terms of Dunford's spectral theory is given in the context of normalization of an operator.
Abstract: A necessary and sufficient condition for an operator to be normalizable is given in terms of Dunford’s spectral theory.
TL;DR: The exact solution of number of problems in quantum mechanics has been given in terms of Appell's functionF2; in an extension of this work I have given here a summation formula, which is as follows:======¯¯¯¯¯¯�$$\begin{gathered} \sum\limits_{n = 0}^m {F_2 (a,} - n, - n;1;x,y) \hfill \\ = \frac{{(m + 1)(x - y)^{ - 1} }}{a}[F_ 2 (a
Abstract: The exact solution of number of problems in quantum mechanics has been given in terms of Appell’s functionF2; in an extension of this work I have given here a summation formula, which is as follows:
$$\begin{gathered} \sum\limits_{n = 0}^m {F_2 (a,} - n, - n;1;x,y) \hfill \\ = \frac{{(m + 1)(x - y)^{ - 1} }}{a}[F_2 (a - 1, - m, - m - 1;1,1;x,y) - \rightleftharpoons ] \hfill \\ \end{gathered} $$
, where ⇌ shows the presence of a similar term withx andy interchanged.
TL;DR: The main result of as mentioned in this paper states that if the expected number of visits of a vector random walk to a bounded region is finite, then all the moments of the number of visit are finite.
Abstract: The main result of this note states that if the expected number of visits of a vector random walk to a bounded region is finite than all the moments of the number of visits are finite.
TL;DR: In this article, the common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces, and the authors show that the properties of the Banach space can be characterized by the following properties:
Abstract: The common Banach spaces are investigated with respect to some properties of their finite dimensional subspaces.
TL;DR: In this article, the authors characterized a class of simple closed polygonal regions which are starshaped, and proved that these regions can be represented by simple closed connected polygonals with local nonconvexity.
Abstract: Previously [7] we proved among other results that a closed connected set inEn which has a unique point of local nonconvexity is starshaped. Here we characterize a fairly large class of plane sets whose points of local nonconvexity are so arranged that starshapedness follows. This theory determines as a special case the simple closed polygonal regions which are starshaped. In order to proceed simply we utilize the following notations and definitions.
TL;DR: In this paper, a recent result of Ceder yields easily converses to well known theorems of Wallace and Birkhoff and thus provides two new characterazations of compactness.
Abstract: The purpose of this note is to point out that a recent result of Ceder yields easily converses to well known theorems of Wallace and Birkhoff and thus provides two new characterazations of compactness as well as specifying the class of spaces for which the theorems are true. A very slight extension of Ceder’s theorem is also obtained, as well as new and simple proofs.
TL;DR: In this paper, it was shown that the symmetry condition is necessary and sufficient for these theorems to hold, and the following Theorem 1 and Theorem 2 were established.
Abstract: The following Theorem 1 and Theorem 2 are established. The proof utilizes the elementary properties of symmetric operators. It is shown that the symmetry condition is necessary for these theorems to hold.