Showing papers in "Israel Journal of Mathematics in 1968"
TL;DR: In this paper, it was shown that weak solutions of mixed elliptic problems are Holder continuous of any order less than 1/2 and that they possess higher regularity in noncritical directions.
Abstract: It is shown that weak solutions of mixed elliptic problems are Holder continuous of any order less than 1/2 and that they possess higher regularity in non-critical directions.
143 citations
TL;DR: In this paper, the point arboricity ρ(G) of a graph G is defined as the minimum number of subsets in a partition of the point set of G so that each subset induces an acyclic subgraph.
Abstract: The point-arboricity ρ(G) of a graphG is defined as the minimum number of subsets in a partition of the point set ofG so that each subset induces an acyclic subgraph Dually, the tuleity τ(G) is the maximum number of disjoint, point-induced, non-acyclic subgraphs contained inG Several results concerning these numbers are presented, among which are formulas for the point arboricity and tulgeity of the class of completen-partite graphs
124 citations
TL;DR: In this article, it was shown that for a strongly transient random walk with p < 1, the distribution converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant.
Abstract: Let {S
n
, n=0, 1, 2, …} be a random walk (S
n
being thenth partial sum of a sequence of independent, identically distributed, random variables) with values inE
d
, thed-dimensional integer lattice. Letf
n
=Prob {S
1 ≠ 0, …,S
n
−1 ≠ 0,S
n
=0 |S
0=0}. The random walk is said to be transient if
$$p = 1 - \sum {_{k = 1}^\infty f_k } > 0$$
and strongly transient if
$$\sum {_{n = 1}^\infty } \sum {_{k = n + 1}^\infty f} _k< \infty $$
. LetR
n
=cardinality of the set {S
0,S
1, …,S
n
}. It is shown that for a strongly transient random walk with p<1, the distribution of [R
n
−np]/σ √n converges to the normal distribution with mean 0 and variance 1 asn tends to infinity, where σ is an appropriate positive constant. The other main result concerns the “capacity” of {S
0, …,S
n
}. For a finite setA inE
d
, let C(A=Σ
x∈A
) Prob {S
n
∉A, n≧1 |S
0=x} be the capacity ofA. A strong law forC{S
0, …,S
n
} is proved for a transient random walk, and some related questions are also considered.
120 citations
TL;DR: In this paper, the structure theorems for rings R with involution whose symmetric elements satisfy a polynomial identity are obtained, and it is shown that such rings satisfy polynomially identities.
Abstract: Structure theorems for ringsR with involution whose symmetric elements satisfy a polynomial identity are obtained. In particular, it is shown that such rings satisfy polynomial identities.
69 citations
TL;DR: In this paper, a study of topological and cardinality properties of the set of extreme points of bounded closed convex sets with interior in reflexive Banach spaces is presented.
Abstract: A study is made of topological and cardinality properties of the set of extreme points of bounded closed convex sets with interior in reflexive Banach spaces. Some related results, and applications to earlier work, are obtained as corollaries.
56 citations
TL;DR: In this article, it is proved that if all bases of a Banach space are shrinking, or all bases are boundedly complete, then the space is reflexive, i.e., it is not shrinking.
Abstract: LetX be a Banach space with a basis. It is proved that if (a) all bases ofX are shrinking, or (b) all bases ofX are boundedly complete, thenX is reflexive.
50 citations
TL;DR: Model Theoretic methods are used to extend models of set theory while leaving specified sets fixed as mentioned in this paper, where every countable modelU of ZF has an extension leaving every set in U fixed, and for each (inU) regular cardinala an extension enlarginga but leaving each cardinal less than a fixed.
Abstract: Model Theoretic methods are used to extend models of set theory while leaving specified sets fixed. In particular, every countable modelU of ZF has: (i) an extension leaving every set inU fixed, and (ii) for each (inU) regular cardinala an extension enlarginga but leaving each cardinal less thana fixed.
46 citations
TL;DR: In this paper, it was shown that if p ≥ 6 ≈ 8, then every valuep ≥ 6≧8 is suitable for a simple 3-polytope.
Abstract: Letp k
denote the number ofk-gonal faces of a simple 3-polytope. Euler’s relation leads to an equation between thep
k
’s which does not involvep
6. Eberhard proved in 1891 that every sequence of non-negative integers (p
3,p
4,…) satisfying this equation corresponds to a polytope for suitable values ofp
6. In the present paper it is established that ifp
3=p
4=0 then every valuep
6≧8 is suitable.
36 citations
TL;DR: In this paper, it was shown that if ZF + the axiom of choice + "there is a measurable cardinal" is consistent thenZF + "ω 1 is measurable" is also consistent.
Abstract: It is shown that ifZF + the axiom of choice + “there is a measurable cardinal” is consistent thenZF + “ω
1 is measurable” is consistent. The corresponding model is a symmetric submodel of the Cohen-type extension which collapses the first measurable cardinal onto ω0.
29 citations
TL;DR: The Hausdorff dimension of the range of an arbitrary subordinator is exactly determined in terms of the rate of linear drift and the Levy measure of the subordinator.
Abstract: The Hausdorff dimension of the range of an arbitrary subordinator is exactly determined in terms of the rate of linear drift and the Levy measure of the subordinator. This generalizes the result of Blumenthal and Getoor: that for a stable subordinator of indexσ, the dimension of the range isσ.
28 citations
TL;DR: In this paper, a homomorphic map is defined from the algebra of norm bounded analytic N-operator valued functions in the unit disc into an algebra of bounded operators in Hilbert spaces represented as left invariant subspaces of H 2(N), and the spectral properties of the map are studied.
Abstract: A homomorphic map is defined from the algebra of norm bounded analyticN-operator valued functions in the unit disc into the algebra of bounded operators in Hilbert spaces represented as left invariant subspaces ofH 2(N), and the spectral properties of the map are studied.
TL;DR: Finite simple groups of order divisible by three primes only and the odd Sylow subgroups of which are cyclic are clssified as mentioned in this paper, which is a special case of finite simple groups.
Abstract: Finite simple groups of order divisible by three primes only and the odd Sylow subgroups of which are cyclic are clssified.
TL;DR: The main theorem in this article states that if a certan bound is imposed on the associated pressure pertaining to a weak solution of the Navier-Stokes equation then the solution is actually smooth.
Abstract: The main theorem in this paper states that if a certan bound is imposed on the associated pressure pertaining to a weak solution of the Navier-Stokes equation then the solution is actually smooth. The proof uses the fact that such a bound implies a bound on the first derivatives of the solution which, in turn, leads to smoothness.
TL;DR: In this paper, a criterion is given for deciding whether a given subset of the vertices of a cyclic polytope is the set of vertices in some face of the polytopes.
Abstract: LetC(ν, d) represent a cyclic polytope withν vertices ind dimensions. A criterion is given for deciding whether a given subset of the vertices ofC(ν, d) is the set of vertices of some face ofC(ν, d). This enables us to determine, in a simple manner, the number ofj-faces ofC(ν, d) for each value ofj (1≦j≦d−1).
Abstract: A σ finite invariant measure is found, for a Markov process, on a locally compact space, which maps continuous functions to continuous functions.
TL;DR: In this paper, it was proved that for uncountable ε,c 0(Γ) has no quasicomplement in m(ε) for ε ≥ 0.
Abstract: It is proved that for uncountableΓ,c
0(Γ) has no quasicomplement inm(Γ).
TL;DR: In this paper, the existence of noncomplemented subspaces and Lozinsky-Kharshilade systems in Banach spaces not isomorphic to Hilbert spaces is studied.
Abstract: In this paper we study the problems of existence of noncomplemented subspaces and Lozinsky-Kharshilade systems [10] in Banach spaces not isomorphic to Hilbert spaces.
TL;DR: In this article, the projection and Macphail constants of real Banach spaces are shown to be asymptotically equivalent to 1 − 1/2 and − 1 /2 respectively.
Abstract: We prove that the projection and Macphail constants ofl
n
p
(1≦p≦2) are asymptotically equivalent ton
1/2 andn
−1/2 respectively. We also obtain some relations linking certain parameters of general finite dimensional real Banach spaces.
TL;DR: In this article, the authors prove limit theorems for Markov processes with no finite invariant measure using only functional analytic methods, which is similar to the approach used in this paper.
Abstract: The aim of this paper is to prove some limit theorems for Markov processes using only functional analytic methods. Some of our results were proved in [7], [8] and [5] by probabilistic methods. We prove in the Appendix a theorem on Markov processes that have no finite invariant measure.
TL;DR: In this paper, the structure of groups with dense normal subgroups is investigated and it is shown that an infinite group with dense ascendant subgroup is locally nil-potent.
Abstract: A group is said to have dense normal subgroups, if each non-empty open interval in its lattice of subgroups contains a normal subgroup. The structure of this and related classes of groups is investigated. Typical results are: an infinite group with dense ascendant subgroups is locally nilpotent: a nontorsion group with dense normal subgroups is abelian, etc.
TL;DR: In this article, a method for successive approximations for partial differential equations is presented, where the limiting function in the process depends on the initial value and convergence is proved by means of some identities and inequalities involving tensor products between multi-linear functions on a vector space.
Abstract: A method for successive approximations for partial differential equations is presented The limiting function in the process depends on the initial value Convergence is proved by means of some identities and inequalities involving tensor products between multi-linear functions on a vector space
TL;DR: In this article, sufficient conditions for the existence of σ-finite invariant measure for conservative and ergodic Markov processes are given for both types of Markov Processes.
Abstract: Sufficient conditions are given for the existence ofσ-finite invariant measure for conservative and ergodic Markov processes.
TL;DR: In this article, the maximum number of cut-vertices in a connected graph withn vertices andm edges is investigated and it is proved that the maximum cut vertices in such graphs can be found in all extremal graphs.
Abstract: It is proved that the maximum number of cut-vertices in a connected graph withn vertices andm edges is
$$max\left\{ {q:m \leqq (_2^{n - q} ) + q} \right\}$$
All the extremal graphs are determined and the corresponding problem for cut-edges is also solved.
TL;DR: The main result of as discussed by the authors is a complete characterization of the compact, metrically homogeneous subsets of the Euclidean plane of a metric space, where the set of distances from a chosen point of a subset to all the other points of the subset is independent of the chosen point.
Abstract: A subset of a metric space is called metrically homogeneous if the set of distances from a chosen point of the subset to all the other points of the subset is independent of the chosen point. The main result of the paper is a complete characterization of the compact, metrically homogeneous subsets of the Euclidean plane.
TL;DR: In this paper, the solutions of p(T)=S, whereS is normal and p is a polynomial, are described, where T is the length of the solution.
Abstract: The solutions ofp(T)=S, whereS is normal andp a polynomial, are described.
TL;DR: In this article, it was shown that the first two formal perturbation equations corresponding to the smallest unperturbed eigenvalue do admit solutions in the appropriate Hilbert space, which implies that the phenomenon of spectral concentration holds.
Abstract: Let the unperturbed operator be the helium-Schrodinger operator and let the perturbation be the homogeneous-electric-field operator. It is shown that the first two formal perturbation equations corresponding to the smallest unperturbed eigenvalue do admit solutions in the appropriate Hilbert space. According to general considerations this implies that for this perturbation problem, the phenomenon of spectral concentration holds.
TL;DR: In this paper, it was shown that if for some fixed interval I, constant 1
Abstract: LetX 1,X 2,... be independent random variables, all with the same distribution symmetric about 0;
$$S_n = \sum\limits_{i = 1}^n {X_i } $$
It is shown that if for some fixed intervalI, constant 1
TL;DR: In this paper, the necessary and sufficient conditions for a Markov process to be conservative were studied and it was shown that if P is conservative then Pk is conservative if and only if the probability of returning to A, from a point in A, is one.
Abstract: Necessary and sufficient conditions, for a Markov process to be conservative, are studied. As a consequence it is proved: 1. IfP is conservative so isPk. 2.P is conservative if and only if the probability of returning toA, from a point inA, is one.
TL;DR: In this paper, the problem of obtaining sufficient conditions in order that the pth derivative of Rn(f, x) converges uniformly to f(p)(x) on the real line was addressed.
Abstract: LetRn(f; x) be a trigonometric polynomial of ordern satisfying Eqs. (1.1) and (1.2). The object of this note is to obtain sufficient conditions in order that thepth derivative ofRn(f, x) converges uniformly tof(p)(x) on the real line. The sufficient conditions turns out to bef(p)(x) ∈ Lipα, α>0 with the restrictions of Eq. (1.3).
TL;DR: In this article, the clasical Hausdorff-young theorem is extended in two ways for certain Orlicz spacesLΦ, where LΦ is based on an arbitrary compact topological group with Haar measure.
Abstract: In this paper the clasical Hausdorff-Young theorem, which states that iff ∈Lp, 1≦p≦2, on the line and\(\hat f\) is its Fourier transform, then\(\left\| {\hat f} \right\|_q \leqq \left\| f \right\|_p \) whereq−1+p−1=1, is extended in two ways for certain Orlicz spacesLΦ. IfLΦ is based on (G, μ), (1) an arbitrary compact topological group with Haar measure, and (2) a locally compact abelian topological group andμ is again the Haar measure, then the above inequality is extended to these cases. Various other related results and remarks are also included.