Showing papers in "Israel Journal of Mathematics in 1988"
TL;DR: In this article, it was shown that the set of squares {n 2|n=1, 2,…} satisfies the pointwise ergodic theorem for L 2-functions.
Abstract: It is shown that the set of squares {n
2|n=1, 2,…} or, more generally, sets {n
t|n=1, 2,…},t a positive integer, satisfies the pointwise ergodic theorem forL
2-functions. This gives an affirmative answer to a problem considered by A. Bellow [Be] and H. Furstenberg [Fu]. The previous result extends to polynomial sets {p(n)|n=1, 2,…} and systems of commuting transformations. We also state density conditions for random sets of integers in order to be “good sequences” forL
p-functions,p>1.
245 citations
TL;DR: In this paper, the space of Iwahori subalgebras containing a given element of a semisimple Lie algebra over C is studied, and a map from nilpotent orbits in a semi-simplified Lie algebra to conjugacy classes in the Weyl group is studied.
Abstract: We study the space of Iwahori subalgebras containing a given element of a semisimple Lie algebra over C((ɛ)). We also define and study a map from nilpotent orbits in a semisimple Lie algebra over C to conjugacy classes in the Weyl group.
169 citations
TL;DR: The linear arboricity conjecture for graphs with maximum degree Δ ≥ 0 was proved in this paper, where it was shown that for every δ > 0, there is a Δ 0=Δ 0(Δ0(δ) so that δ ≥ 1/2+δ √ √ Δ √ δ for every graph δ with an even maximum degree δ and with girth g≧50Δ.
Abstract: Alinear forest is a forest in which each connected component is a path. Thelinear arboricity la(G) of a graphG is the minimum number of linear forests whose union is the set of all edges ofG. Thelinear arboricity conjecture asserts that for every simple graphG with maximum degree Δ=Δ(G),\(la(G) \leqq [\frac{{\Delta + 1}}{2}].\). Although this conjecture received a considerable amount of attention, it has been proved only for Δ≦6, Δ=8 and Δ=10, and the best known general upper bound for la(G) is la(G)≦⌈3Δ/5⌉ for even Δ and la(G)≦⌈(3Δ+2)/5⌉ for odd Δ. Here we prove that for everyɛ>0 there is a Δ0=Δ0(ɛ) so that la(G)≦(1/2+ɛ)Δ for everyG with maximum degree Δ≧Δ0. To do this, we first prove the conjecture for everyG with an even maximum degree Δ and withgirth g≧50Δ.
160 citations
TL;DR: In this paper, a construction for a general class of measure-valued Markov branching processes is given, where the underlying spatial motion process is an arbitrary Borel right Markov process and state-dependent offspring laws are allowed.
Abstract: A construction is given for a general class of measure-valued Markov branching processes. The underlying spatial motion process is an arbitrary Borel right Markov process, and state-dependent offspring laws are allowed. It is shown that such processes are Hunt processes in the Ray weak* topology, and have continuous paths if and only if the total mass process is continuous. The entrance spaces of such processes are described explicitly.
151 citations
TL;DR: The pointwise ergodic theorem for prime powers for functions in L ε, p,p>1 was proved in this article for functions with p ≥ 1. This extends a result of Bourgain where he proved a similar theorem for p>1.
Abstract: The pointwise ergodic theorem is proved for prime powers for functions inL
p,p>1. This extends a result of Bourgain where he proved a similar theorem forp>(1+√3)/2.
110 citations
TL;DR: In this paper, the convergence of the pointwise ergodic theorem for 2-functions was shown to be almost sure in the case of dynamical systems under certain conditions.
Abstract: The purpose of this note is to show how the results of [B] on the pointwise ergodic theorem forL
2-functions may be extended toL
p for certainp<2. More precisely, we give a proof of the almost sure convergence of the means
$$\frac{1}{N}\sum\limits_{1\underline \le n\underline \le N} {T^{(n1)} } $$
(t≧1) given a dynamical system (Ω,B, μ, T) andf of classL
p(Ω,μ),p>(√5+1)/2.
110 citations
TL;DR: In this article, it was shown that there is a continuous map of a Banach space into itself such that the orbit of any nonzero vectorx in X is dense in X. This is an interesting result even if stated in a form which disregards the linearity of T.
Abstract: We continue here the line of investigation begun in [7], where we showed that on every Banach spaceX=l1⊗W (whereW is separable) there is an operatorT with no nontrivial invariant subspaces. Here, we work on the same class of Banach spaces, and produce operators which not only have no invariant subspaces, but are also hypercyclic. This means that for every nonzero vectorx inX, the translatesTr x (r=1, 2, 3,...) are dense inX. This is an interesting result even if stated in a form which disregards the linearity ofT: it tells us that there is a continuous map ofX{0\{ into itself such that the orbit {Trx :r≧0{ of anyx teX \{0\{ is dense inX \{0\{. The methods used to construct the new operatorT are similar to those in [7], but we need to have somewhat greater complexity in order to obtain a hypercyclic operator.
96 citations
TL;DR: In this paper, it was shown that under additional hypotheses, this is true under the assumption that the only continuous probability measure on the circleT = R/Z which is invariant under bothx ↦ 2x and x ↦ 3x is Lebesgue measure.
Abstract: Furstenberg has conjectured that the only continuous probability measure on the circleT=R/Z which is invariant under bothx ↦ 2x andx ↦ 3x is Lebesgue measure. We shall show that under additional hypotheses, this is true. We also discuss related conjectures and theorems.
67 citations
TL;DR: It is shown that the almost sure convergence property for certain sequences of operators {Sn{ implies a uniform bound on the metrical entropy of the sets {Snf|n=1, 2, ...{, wheref is taken in the L2-unit ball.
Abstract: It is shown that the almost sure convergence property for certain sequences of operators {Sn{ implies a uniform bound on the metrical entropy of the sets {Snf|n=1, 2, ...{, wheref is taken in theL2-unit ball. This criterion permits one to unify certain counterexamples due to W. Rudin [Ru] and J.M. Marstrand [Mar] and has further applications. The theory of Gaussian processes is crucial in our approach.
66 citations
TL;DR: In this paper, the equivalence of the square function and maximal function of a martingale is extended to the setting of rearrangement invariant function spaces, where the maximal function is defined as a function of the set of martingales.
Abstract: The Burkholder-Davis-Gundy equivalence of the square function and maximal function of a martingale is extended to the setting of rearrangement invariant function spaces.
63 citations
TL;DR: In this article, it was shown that for any symmetric compact convex body B ⊂Rn, there is an affine transformation T satisfying Q ⊆T(B) ⊈λnQ, whereQ is then-dimensional cube.
Abstract: We show the existence of a sequence (λn) of scalars withλn=o(n) such that, for any symmetric compact convex bodyB ⊂Rn, there is an affine transformationT satisfyingQ ⊂T(B) ⊂λnQ, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλn. We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l2m→X andβ:X→l∞m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum.
TL;DR: In this article, it was shown that if we approximate the Euclidean ball by a Minkowski sum of n segments, then the smallest possible n is equal (up to a possible logarithmic factor) toc(n)e −2(n−1)/(n+2).
Abstract: It is proved that if we approximate the Euclidean ballB
n in the Hausdorff distance up toɛ by a Minkowski sum ofN segments, then the smallest possibleN is equal (up to a possible logarithmic factor) toc(n)e
−2(n−1)/(n+2). A similar result is proved ifB
n is replaced by a general zonoid inR
n
.
TL;DR: In this article, the authors study two well-known distortion functions, λ(K) and ϕ K(r), of the theory of plane quasiconformal mappings and obtain several new inequalities for them.
Abstract: The authors study two well-known distortion functions, λ(K) andϕ
K(r), of the theory of plane quasiconformal mappings and obtain several new inequalities for them. The proofs make use of some properties of elliptic integrals.
TL;DR: In this paper, it was shown that the set of valuesp for which the p-space is isomorphic to a complemented subspace of the φ φ (resp.
Abstract: Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel
φ (resp. an Orlicz function spaceL
φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel
p-space is isomorphic to a complemented subspace ofl
φ (resp.L
φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernandez and Peirats [1] is extended showing that, even for the case in which the indices satisfy αφ
∞<2<βφ
∞, there exist minimal Orlicz function spacesL
φ(μ) with no complemented copy ofl
p for anyp ≠ 2.
TL;DR: In this paper it was shown that for each n the class of graphs omitting all paths of length n and all circuits of length at least n possess universal elements in all infinite powers.
Abstract: It is shown that various classes of graphs have universal elements. In particular, for eachn the class of graphs omitting all paths of lengthn and the class of graphs omitting all circuits of length at leastn possess universal elements in all infinite powers.
TL;DR: In this paper, the connectivity of infinite random graphs is investigated and necessary and sufficient conditions for a zero-one law to hold are given when the edges are chosen independently, and specific examples are treated including one where the vertex set is N and the probability that an edge joiningi toj is present depends only on |i−j|.
Abstract: Several results concerning the connectivity of infinite random graphs are considered. A necessary sufficient condition for a zero–one law to hold is given when the edges are chosen independently. Some specific examples are treated including one where the vertex set isN and the probability that an edge joiningi toj is present depends only on |i−j|.
TL;DR: In this article, it was shown that in the core model every uniform ultrafilter is regular and that the existence of a non-regular ultrafilter on a singular cardinal implies the existence in the inner model of an inner model with a measurable cardinal.
Abstract: We show that in the core model every uniform ultrafilter is regular. In addition, we prove that the existence of a nonregular uniform ultrafilter on a singular cardinal implies the existence of an inner model with a measurable cardinal.
TL;DR: In this paper, strong elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators on L 1 ≤ p(R n ≥ 0.
Abstract: Strongly elliptic differential operators with (possibly) unbounded lower order coefficients are shown to generate analytic semigroups of linear operators onL
p(R
n
), 1≦p≦∞. An explicit characterization of the domain is given for 1
TL;DR: In this article, the authors studied the class of Riesz subsets of abelian discrete groups, that is, the sets for which the F. and M.Riesz theorem extends.
Abstract: We study the class of the Riesz subsets of abelian discrete groups, that is, the sets for which the F. and M. Riesz theorem extends. We show that the “classical” tools of the theory — Riesz projections, localization in the Bohr sense, products — are leading to Riesz sets which are satisfying nice additional properties, e.g., the Mooney-Havin result extends to this class. We give an alternative proof of a result of A. B. Alexandrov, and we improve a construction of H. P. Rosenthal. The connection is made between this class and theM-structure theory. We show a result of convergence at the boundary for holomorphic functions on the polydisc. The Bourgain-Davis result on convergence of analytic martingales is improved.
TL;DR: In this article, the authors define a Z/2Z-graded cocharacter for algebras with graded identities and use it to study graded tensor products and a graded version of the Capelli identity.
Abstract: We define aZ/2Z-graded cocharacter forZ/2Z-graded algebras with graded identities. We relate this cocharacter to the ordinary cocharacter and use it to study graded tensor products and a graded version of the Capelli identity.
TL;DR: For any countable group G, there is a norm Θ(G) equivalent to the original one such that the group of isometries of G is isomorphic to G × {− 1, + 1}.
Abstract: For any Banach spaceX there is a norm |||·||| onX, equivalent to the original one, such that (X, |||·|||) has only trivial isometries. For any groupG there is a Banach spaceX such that the group of isometries ofX is isomorphic toG × {− 1, 1}. For any countable groupG there is a norm ‖ · ‖
G
onC([0, 1]) equivalent to the original one such that the group of isometries of (C([0, 1]), ‖ · ‖
G
) is isomorphic toG × {−1, + 1}.
TL;DR: In this article, it was shown that for every orientable 2-manifold there is a finite set of triangulations from which all other triangulation can be generated by sequences of vertex splittings.
Abstract: We show that for every orientable 2-manifold there is a finite set of triangulations from which all other triangulations can be generated by sequences of vertex splittings.
TL;DR: In this paper, it was shown that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations, even in the nonamenable, ergodically, measure-preserving case.
Abstract: We define rigorously a “treed” equivalence relation, which, intuitively, is an equivalence relation together with a measurably varying tree structure on each equivalence class. We show, in the nonamenable, ergodic, measure-preserving case, that a treed equivalence relation cannot be stably isomorphic to a direct product of two ergodic equivalence relations.
TL;DR: In this article, the authors obtained explicit formulae for degrees on diagonals, Hadamard products and Lamperti products, and showed that they can be expressed as follows:
Abstract: We obtain explicit formulae for degrees on diagonals, Hadamard products and Lamperti products.
TL;DR: In this article, it was shown that every bounded and "asymptotically isometric" almost-orbit of a contraction semigroup on a closed convex subset of a uniformly convex Banach space is weakly almost periodic in the sense of Eberlein.
Abstract: We show that if (S(t))t≧0 is a contraction semigroup on a closed convex subset of a uniformly convex Banach space, then every bounded and “asymptotically isometric” almost-orbit of (S(t))t≧0 is weakly almost periodic in the sense of Eberlein. As one consequence, results on the existence of almost periodic solutions to the abstract Cauchy problem are obtained without the need fora priori compactness assumptions. As a further consequence, the known strong ergodic limit theorems for (almost-) orbits of nonlinear contraction semigroups turn out to be special cases of Eberlein’s classical ergodic theorem for weakly almost periodic functions.
TL;DR: In this article, the 2-circle and 2-disk problems were studied on a homogeneous tree and the authors obtained results which have remarkably the same flavor as classical theorems.
Abstract: Our purpose here is to consider on a homogeneous tree two Pompeiutype problems which classically have been studied on the plane and on other geometric manifolds. We obtain results which have remarkably the same flavor as classical theorems. Given a homogeneous tree, letd(x, y) be the distance between verticesx andy, and letf be a function on the vertices. For each vertexx and nonnegative integern let Σnf(x) be the sum Σd(x, y)=nf(y) and letBnf(x)=Σd(x, y)≦nf(y). The purpose is to study to what extent Σnf andBnf determinef. Since these operators are linear, this is really the study of their kernels. It is easy to find nonzero examples for which Σnf orBnf vanish for one value ofn. What we do here is to study the problem for two values ofn, the 2-circle and the 2-disk problems (in the cases of Σn andBn respectively). We show for which pairs of values there can exist non-zero examples and we classify these examples. We employ the combinatorial techniques useful for studying trees and free groups together with some number theory.
TL;DR: In this paper, a conservative ergodic invertible natural extension for the Maharam skew product of nonsingular endomorphisms is presented, and the construction of such an extension is shown.
Abstract: We show that the Maharam skew product ofμ-recurrent nonsingular endomorphisms is conservative and give some applications. Among them is the construction of a conservative ergodic invertible natural extension forμ-recurrent ergodic nonsingular endomorphisms.
TL;DR: In this paper, the fractional Helly theorem for families of boxes with edges parallel to the coordinate axes was proved for each k. The result is best possible for eachk, where k denotes the number of subfamilies ofP of sizek+1 with non-empty intersection.
Abstract: LetP be a family ofn boxes inR
d
(with edges parallel to the coordinate axes). Fork=0, 1, 2, …, denote byf
k
(P) the number of subfamilies ofP of sizek+1 with non-empty intersection. We show that iff
r
(P)=0 for somer≦n, then where thef
k
(n, d, r) are ceg196rtain definite numbers defined by (3.4) below. The result is best possible for eachk. Fork=1 it was conjectured by G. Kalai (Israel J. Math.48 (1984), 161–174). As an application, we prove a ‘fractional’ Helly theorem for families of boxes inR
d
.
TL;DR: In this paper, the authors investigated the cofinalities of reduced products of regular cardinals and gave some applications, such as the nonproductiveness of δ + δ when δ > 2cfλ; and δ − δ with δ ≥ 2c.c.
Abstract: We continue here our investigation of cofinalities of reduced products of regular cardinals and give some applications, such as the non-productiveness ofλ
+-c.c. whenλ>2cfλ;.
TL;DR: In this paper, it was shown that if R is a commutative ring such that each localization at a prime ideal has finite weak global dimension, then every R-module has a flat envelope if and only ifR is coherent and has weak global dimensions less than or equal to two.
Abstract: We prove that ifR is a commutative ring such that each localization at a prime ideal has finite weak global dimension then everyR-module has a flat envelope if and only ifR is coherent and has weak global dimension less than or equal to two