Showing papers in "Israel Journal of Mathematics in 1983"
TL;DR: In this article, it was shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups on L.............. p, 1≦p < ∞.
Abstract: It is shown that general second order elliptic boundary value problems on bounded domains generate analytic semigroups onL
1. The proof is based on Phillips’ theory of dual semigroups. Several sharp estimates for the corresponding semigroups inL
p, 1≦p<∞, are given.
212 citations
TL;DR: In this article, a complete system of relations is determined for these generators and the presentations of M n,0 and M n,1 obtained in this way are much simpler than the known presentations.
Abstract: LetF n.k be an orientable compact surface of genusn withk boundary components. For a suitable choice of 2n + 1 simple closed curves onF n,1 the corresponding Dehn twists generate bothM n,o andM n,1. A complete system of relations is determined for these generators and the presentations ofM n,0 andM n,1 obtained in this way are much simpler than the known presentations.
206 citations
TL;DR: In this article, the asymptotic behavior of independent matrix-valued random variables is studied for auxiliary measures in projective space and the study of markov processes on projective spaces.
Abstract: The asymptotic behavior of ‖X
n
X
n
−1…X
1υ‖ is studied for independent matrix-valued random variablesX
n
. The main tool is the use of auxiliary measures in projective space and the study of markov processes on projective space.
191 citations
TL;DR: In this article, a generalization of the matrix tree theorem was proposed, and it was shown that a simplicial complex with complete (k−1)-skeleton and k-faces is a finite group.
Abstract: Let (n, k) be the class of all simplicial complexesC over a fixed set ofn vertices (2≦k≦n) such that: (1)C has a complete (k−1)-skeleton, (2)C has precisely (
−1
)k-faces, (3)H
k
(C)=0. We prove that for
,H
k−1(C) is a finite group, and our main result is:
. This formula extends to high dimensions Cayley’s formula for the number of trees onn labelled vertices. Its proof is based on a generalization of the matrix tree theorem.
175 citations
TL;DR: Under certain regularity conditions, a real transformation with indifferent fixed points has an infinite invariant measure equivalent to Lebesgue measure as mentioned in this paper, and several ergodic properties of such transformations are established.
Abstract: Under certain regularity conditions a real transformation with indifferent fixed points has an infinite invariant measure equivalent to Lebesgue measure. In this paper several ergodic properties of such transformations are established.
170 citations
TL;DR: In this article, it was shown that every Ψ has at least 2 N ≥ 1 non-isomorphic uncountable models of power n ≥ 0 and has models in all cardinalities.
Abstract: Assuming that 2Nn < 2Nn+1 forn < ω, we prove that everyψ ∈L
ω_1, ω has many non-isomorphic models of powerN
n for somen>0or has models in all cardinalities. We can conclude that every such Ψ has at least 2
N
1 non-isomorphic uncountable models. As for the more vague problem of classification, restricting ourselves to the atomic models of some countableT (we can reduce general cases to this) we find a cutting line named “excellent”. Excellent classes are well understood and are parallel to totally transcendental theories, have models in all cardinals, have the amalgamation property, and satisfy the Los conjecture. For non-excellent classes we have a non-structure theorem, e.g., if they have an uncountable model then they have many non-isomorphic ones in someN
n
(provided {ie212-7}).
123 citations
TL;DR: The SMB theorem for amenable groups that possess Folner sets is proved in this article for groups with the property that for some constantM, and all,n, |A� −1>>\s A� n>>\s | ≦M· |A>>\s n |A¯¯¯¯ n |
Abstract: We prove the SMB theorem for amenable groups that possess Folner sets {A
n
} with the property that for some constantM, and all,n, |A
−1
A
n
| ≦M· |A
n
|
113 citations
TL;DR: In this paper, the authors characterized distributions of selections of a random set in terms of inequalities, similar to the marriage problem, and showed that the ensemble of such distributions is convex compact and depends continuously on the distribution of the random set.
Abstract: Distributions of selections of a random set are characterized in terms of inequalities, similar to the marriage problem. A consequence is that the ensemble of such distributions is convex compact and depends continuously on the distribution of the random set.
79 citations
TL;DR: In this paper, the main results of Part A were proved and discussed in n-dimensional diagrams of models, and the main result of Part B was proved in the same way.
Abstract: We continue here Part A, and the main results are proved here. This part deals withn-dimensional diagrams of models.
79 citations
TL;DR: In this article, the authors define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation of l andc o. They show that many Banach spaces may (3+e)-equivalently be renormed to have property α.
Abstract: We define two geometric concepts of a Banach space, property α and β, which generalize in a certain way the geometric situation ofl andc
o. These properties have been used by J. Lindenstrauss and J. Partington in the study of norm attaining operators. J. Partington has shown that every Banach space may (3+e)-equivalently be renormed to have property β. We show that many Banach spaces (e.g., every WCG space) may (3+e)-equivalently be renormed to have property α. However, an example due to S. Shelah shows that not every Banach space is isomorphic to a Banach space with property α.
65 citations
TL;DR: In this paper, it was shown that polyhedral 2-manifolds in the class ℳp,q contain infinitely many distinct combinatorial types and that for eachg ≥ 2, there is a constant k-approximation such that there exists a closed polyhedron with at mostkg/logg vertices.
Abstract: An equivelar polyhedral 2-manifold in the class ℳp,q is one embedded inE
3 in which every face is a convexp-gon and every vertex isq-valent In this paper, examples are constructed, to show that each of the classes ℳ3,q (q≧7), ℳ4,q (q≧5) and ℳp,4 (p≧5) contains infinitely many distinct combinatorial types As particular examples, there are polyhedral 2-manifolds with 576 vertices and genus 577, and with 4096 faces and genus 4097 A modification of one construction shows that there is a constantk, such that for eachg≧2, there exists a closed polyhedral 2-manifold inE
3 of genusg with at mostkg/logg vertices
TL;DR: If a separable Banach spaceX admits a real valued function with bounded nonempty support, φ is locally Lipschitzian and if no subspace of X is isomorphic to c o, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschnitzian on the unit sphere of X.
Abstract: If a separable Banach spaceX admits a real valued function ф with bounded nonempty support, φ 艂 is locally Lipschitzian and if no subspace ofX is isomorphic toc
o, thenX admits an equivalent twice Gateaux differentiable norm whose first Frechet differential is Lipschitzian on the unit sphere ofX.
TL;DR: In this paper, it was shown that the Schur index of an irreducible character of a finite group can be computed from a knowledge of the character table of the group.
Abstract: Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm
p (χ) denote the Schur index of χ overQ
p
, the completion ofQ atp. It is shown that ifx is ap′-element ofG such that
$$X_u \left( x \right) \in Q_p \left( X \right)$$
for all irreducible charactersX
u ofG thenm
p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E
n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz).
TL;DR: In this article, the authors presented an asymptotic formula for the number of closed orbits of a suspension of a shift of finite type when the suspended flow is topologically weak mixinging and when the suspending function is locally constant.
Abstract: Following the classical procedure developed by Wiener and Ikehara for the proof of the prime number theorem we find an asymptotic formula for the number of closed orbits of a suspension of a shift of finite type when the suspended flow is topologically weak-mixing and when the suspending function is locally constant.
TL;DR: In this paper, it was shown that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free (a group virtually has a property if some subgroup of finite index has it).
Abstract: We show first that certain automorphism groups of algebraic varieties, and even schemes, are residually finite and virtually torsion free (A group virtually has a property if some subgroup of finite index has it) The rest of the paper is devoted to a study of the groups of automorphisms Aut(Γ) and outer automorphisms Out(Γ) of a finitely generated group Γ, by using the finite-dimensional representations of Γ This is an old idea (cf the discussion of Magnus in [11]) In particular the classes of semi-simplen-dimensional representations of Γ are parametrized by an algebraic varietySn(Γ) on which Out(Γ) acts We can apply the above results to this action and sometimes conclude that Out(Γ) is residually finite and virtually torsion free This is true, for example, when Γ is a free group, or a surface group In the latter case Out(Γ) is a “mapping class group”
TL;DR: An arithmetic variety is the quotient space of a symmetric space with complex structure by an arithmetic subgroup of the associated algebraic Lie group as mentioned in this paper, and it is shown that the variety obtained from an arithmetic variety by a base change corresponding to any automorphism of C is again an algebraic variety.
Abstract: An arithmetic variety is the quotient space of a symmetric space with complex structure by an arithmetic subgroup of the associated algebraic Lie group. It is shown that the variety obtained from an arithmetic variety by a base change corresponding to any automorphism ofC is again an arithmetic variety.
TL;DR: In this article, it was shown that for all but a small number of co-compact groups, the Martin boundary of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ.
Abstract: Let Γ be a finitely generated non-elementary Fuchsian group, and let μ be a probability measure with finite support on Γ such that supp μ generates Γ as a semigroup. If Γ contains no parabolic elements we show that for all but a small number of co-compact Γ, the Martin boundaryM of the random walk on Γ with distribution μ can be identified with the limit set Λ of Γ. If Γ has cusps, we prove that Γ can be deformed into a group Γ', abstractly isomorphic to Γ, such thatM can be identified with Λ', the limit set of Γ'. Our method uses the identification of Λ with a certain set of infinite reduced words in the generators of Γ described in [15]. The harmonic measure ν (ν is the hitting distribution of random paths in Γ on Λ) is a Gibbs measure on this space of infinite words, and the Poisson boundary of Γ, μ can be identified with Λ, ν.
TL;DR: The results of [12, VIII] on constructing models more available for application, by separating the combinatorial parts, have been strengthened in this paper by showing that there are many isomorphism types of models of an unsuperstable theory.
Abstract: This article has three aims: (1) To make the results of [12, VIII] on constructing models more available for application, by separating the combinatorial parts. Thus in applications one will only need the relevant things from the area of application. (2) To strengthen the results there. In particular, we were mainly interested in [12, VIII] in showing that there are many isomorphism types of models of an unsuperstable theory, with results about the number of models not elementarily embeddable in each other being a side benefit. Here we consider the latter case in more detail, getting more cases. We also consider some more complicated constructions along the same lines % MathType!MTEF!2!1!+-\((K_{ptr}^\omega )\). (3) To solve various problems from the list of van Dowen, Monk and Rubin [3] on Boolean algebras, which was presented at a conference on Boolean algebra in Oberwolfach January 1979 (most of the solutions are mentioned in the final version). Some of them are not related to (1) and (2). This continues [10, §2] in which the existence of a rigid B.A. in every uncountable power was proved. There (and also here) we want to demonstrate the usefulness of the methods developed in [12, VIII] (and §§ 1,2) for getting many (rigid) non-embeddable models in specific classes.
TL;DR: In this paper, the question of existence of a universal object in the category of universal locally finite groups was dealt with, and the answer is negative for many uncountable cardinalities; for example, for 2ℵ 0, and assuming GCH for every cardinal whose confinality is > ℵ
Abstract: We deal with the question of existence of a universal object in the category of universal locally finite groups; the answer is negative for many uncountable cardinalities; for example, for 2ℵ 0, and assuming GCH for every cardinal whose confinality is >ℵ0 However, if λ>κ when κ is strongly compact and of λ=ℵ0, then there exists a universal locally finite group of cardinality λ The idea is to use the failure of the amalgamation property in a strong sense We shall also prove the failure of the amalgamation property for universal locally finite groups by transferring the kind of failure of the amalgamation property from LF into ULF
TL;DR: In this paper, it was shown that the mod 1 circle group is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals.
Abstract: LetT be the mod 1 circle group, α∈T be irrational and 0<β<1. LetE be the closed subgroup ofR generated by β and 1. DefineX=T×E andT:X→X byT(x, t)=(x+α,t+1[0,β](x)−β). Then we have the theorem:T is ergodic if and only if β is rational or 1, α and β are linearly independent over the rationals.
TL;DR: The notion of superstable Banach spaces was introduced by Krivine and Maurey as mentioned in this paper, who showed that the unit ball of c 0 does not uniformly imbed into any stable Banach space.
Abstract: We introduce here the notion of superstable Banach space, as the superproperty associated with the stability property of J. L. Krivine and B. Maurey. IfE is superstable, so are theL p (E) for eachp∈[1, +∞[. If the Banach spaceX uniformly imbeds into a superstable Banach space, then there exists an equivalent invariant superstable distance onX; as a consequenceX contains subspaces isomorphic tol p spaces (for somep∈[1, ∞[). We give also a generalization of a result of P. Enflo: the unit ball ofc 0 does not uniformly imbed into any stable Banach space.
TL;DR: In this paper, the POD condition in topological dynamics was established for counterexample construction, and a connection was established with the Proximal Orbit Dense (POD) condition.
Abstract: LetTα be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=Tα(x−1) forx∈[1,3/2) andTx = Tαx forx∈[1/2, 1), i.e.T isTα with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {pn/qn} with |α−pn/qn|=o(qn−2)) and λ is a measure onXk all of whose one-dimensional marginals are Lebesgue and which is ⊗i − 1kT1 invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction. A connection is established with the POD (proximal orbit dense) condition in topological dynamics.
TL;DR: In this article, it was shown that for every ∈>0, there exists a constantC(∈) such that every normed space Xn of dimensionn either contains a (1+∈)-isomorphic copy ofl2m for somem satisfying ln lnm≧ 1/2 lnn/n lnn or contains a 1 + ∈-isomorphism ofl∞k for somek satisfying ll lnk>1/2 Lnn−C( ∈).
Abstract: Letx1,x2, ...,xn ben unit vectors in a normed spaceX and defineMn=Ave{‖Σi=1ne1xi‖:e1=±1}. We prove that there exists a setA⊂{1, ...,n} of cardinality\(\left| A \right| \geqq \left[ {\sqrt n /\left( {2^7 M_n } \right)} \right]\) such that {xi}i∈A is 16Mn-isomorphic to the natural basis ofl∞k. This result implies a significant improvement of the known results concerning embedding ofl∞k in finite dimensional Banach spaces. We also prove that for every ∈>0 there exists a constantC(∈) such that every normed spaceXn of dimensionn either contains a (1+∈)-isomorphic copy ofl2m for somem satisfying ln lnm≧1/2 ln lnn or contains a (1+∈)-isomorphic copy ofl∞k for somek satisfying ln lnk>1/2 ln lnn−C(∈). These results follow from some combinatorial properties of vectors with ±1 entries.
TL;DR: In this article, the existence result for maximal monotone maps with images not necessarily convex is given, where A is a maximal nonconvex map and F is a set-valued map.
Abstract: We give an existence result for
$$\dot x \in -- Ax + F(x)$$
whereA is a maximal monotone map andF is a set-valued map, with images not necessarily convex.
TL;DR: For graphs, multigraphs and pseudographs, the corresponding weights are at most about 48g, 8g + 7g and 24g−9, respectively as mentioned in this paper, and if a (multi, pseudo) graph triangulates, then its weight is at most 15.
Abstract: The weight of a graphG is the minimum sum of the two degrees of the end points of edges ofG Kotzig proved that every graph triangulating the sphere has weight at most 13, and Grunbaum and Shephard proved that every graph triangulating the torus has weight at most 15 We extend these results for graphs, multigraphs and pseudographs “triangulating” the sphere withg handlesS
g
,g≧1, showing that the corresponding weights are at most about
$$\sqrt {48g} ,8g + 7$$
and 24g−9, respectively; if a (multi, pseudo) graph triangulatesS
g
and it is big enough, then its weight is at most 15
TL;DR: It is proved that with the standard duality, also the consistency of “ℬ/Itc does not split” is proved and the method is the oracle chain condition.
Abstract: We prove the consistency of “ℬ/I
mz does not split” (see Notation). We write the proof so that with the standard duality, also the consistency of “ℬ/I
tc does not split” (i.e., replacing measure zero by first category, random by generic, etc.) is proved. The method is the oracle chain condition.
TL;DR: The simple relation between representations of the covering groups of SL2 and GL2 makes it possible to fuse and extend the recent metaplectic results of Shimura, Waldspurger, Flicker, and ourselves as discussed by the authors.
Abstract: The simple relation between representations of the covering groups of SL2 and GL2 makes it possible to fuse and extend the recent metaplectic results of Shimura, Waldspurger, Flicker, and ourselves. By giving a new (purely local andL-function theoretic) treatment of the Waldspurger-Shintani correspondence, we also simplify some of Waldspurger’s original results.
TL;DR: In this article, a Banach space is constructed such that it has the Schur property and the Dunford-Pettis property, and such that C ([0, 1],E) and L 1 ([ 0, 1, E] and L 2 [0, 0], E] fail the Dunman-Petts property.
Abstract: We construct a Banach spaceE such thatE′ has the Schur property (henceE′ has the Dunford-Pettis property) but such thatC ([0, 1],E) andL
1 ([0, 1],E′) fail the Dunford-Pettis property.
TL;DR: In this article, the authors show that a unitary Banach ideal has uniform convexity and smoothness, type and cotype information from properties of the symmetric Banach sequence space.
Abstract: Information about geometric properties, such as uniform convexity and smoothness, type and cotype, of a unitary Banach idealSE is obtained from properties of the symmetric Banach sequence spaceE. In particularSE has cotype 2 ifE does. The proofs use real interpolation and complex geometry.
TL;DR: In this article, the notion of quasi-factor in topological dynamics has been extended to the context of ergodic theory, and it is shown that a process with minimal self-joinings is of this type.
Abstract: Motivated by the notion of quasi-factor in topological dynamics, we introduce an analogous notion in the context of ergodic theory. For two processes,X andY , we haveX∡Y if and only ifY has a factor which is isomorphic to a quasi-factor ofX. On the other hand, weakly mixing processes can have nontrivial quasifactors which are not w.m. We characterize those ergodic processes which admit only trivial continuous ergodic quasi-factors, and use this characterization to conclude that a process with minimal selfjoinings is of this type. From this we derive the fact that for every suchX and any ergodicY eitherX ⊥Y orY extends some symmetric product ofX.