Showing papers in "Israel Journal of Mathematics in 1991"
TL;DR: In this article, a Tsirelson-like Banach space is constructed, which is arbitrarily distortable and can be used to construct arbitrary Banach spaces, such as the one in this paper.
Abstract: In this work we construct a “Tsirelson like Banach space” which is arbitrarily distortable.
159 citations
TL;DR: In this article, it was shown that there exists a function ∈ (k) which tends to 0 ask tends to infinity, such that any k-regular graph onn vertices contains at most 2(1/2+∈(k))n independent sets.
Abstract: It is shown that there exists a function∈(k) which tends to 0 ask tends to infinity, such that anyk-regular graph onn vertices contains at most 2(1/2+∈(k))n independent sets. This settles a conjecture of A. Granville and has several applications in Combinatorial Group Theory.
126 citations
TL;DR: Theorem 4.12 as discussed by the authors shows that the standard action of F on T n is locally rigid, i.e., every action that is conjugate to the standard actions by C °o diffeomorphisms which are sufficiently close to F is also locally rigid.
Abstract: Let F ---- SL(n, 7.) or any subgroup of finite index, n > 4. We show that the standard action of F on T n is locally rigid, i.e., every action of F on T n by C °o diffeomorphisms which is sufficiently close to the standard action is conjugate to the standard action by a Coo diffeomorphism. In the course of the proof, we obtain a global rigidity result (Theorem 4.12) for actions of free abelian subgroups of maximal rank in SL(n, Z).
111 citations
TL;DR: In this paper, it was shown that every pairwise independent joining of weakly mixing systems with purely singular spectrum is independent, and that every mixing system with pure singular spectrum (SIS) is mixing of all orders.
Abstract: We prove that every pairwise independent joining of weakly mixing systems with purely singular spectrum is independent; it follows that every mixing system with purely singular spectrum is mixing of all orders.
92 citations
TL;DR: The set of invariant measures of a compact dynamical system is well known to be a nonempty compact metrizable Choquet simplex as mentioned in this paper, and it is shown that all such simplices are realized already for the class of minimal flows.
Abstract: The set of invariant measures of a compact dynamical system is well known to be a nonempty compact metrizable Choquet simplex. It is shown that all such simplices are realized already for the class of minimal flows. Moreover, sufficient is the class of 0–1 Toeplitz flows. Previously, it is proved that the set of invariant measures of the regular Toeplitz flows contains homeomorphic copies of all metric compacta.
82 citations
54 citations
TL;DR: In this paper, it was shown that the normalization of the closure of the adjoint action of a complex connected semi-simple Lie group is Gorenstein and has rational singularities.
Abstract: Let\(\mathcal{O}\) be a nilpotent orbit of the adjoint action of a complex connected semi-simple Lie group on its Lie algebra. We prove that the normalization of the closure of\(\mathcal{O}\) is Gorenstein and has rational singularities.
48 citations
TL;DR: In this article, the authors give a one-dimensional version of the Whitney trick, by means of which any graphK1 satisfyingo(K1)=0 can be, step by step, embedded in R 2.
Abstract: Forn≥3, the ordinary Whitney trick shows that a simplicial complexKn having van Kampen obstruction classo(kn)=0 embeds inR2n. We give a one-dimensional version of the Whitney trick, by means of which any graphK1 satisfyingo(K1)=0 can be, step by step, embedded inR2. We then deduce some other planarity criteria, including Kuratowski’s, from this result. As a byproduct we also obtain a fascinating description of the mod 2 homology of the deleted product of a graph.
47 citations
TL;DR: In this article, generalizations of the Andreev-Thurston circle packing theorem are proved, and one such result is the following, which is also related to our work.
Abstract: Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following.
46 citations
TL;DR: In this article, the authors prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes, making crucial use of percolation on trees.
Abstract: We prove amenability for a broad class of equivalence relations which have trees associated to the equivalence classes. The proof makes crucial use of percolation on trees. We also discuss related concepts and results, including amenability of automorphism groups. A second main result is that no discrete subgroup of the automorphism group of a tree is isomorphic to the fundamental group of any closed manifoldM admitting a nontrivial connection-preserving, volume-preserving action of a noncompact, simply connected, almost simple Lie group having Kazhdan’s property (T). The technique of proof also shows that M does not admit a hyperbolic structure.
44 citations
TL;DR: In this article, an extension of Burnside's Theorem for a general Banach space is given, which also gives new results on invariant subspaces. But this result is restricted to a special case of the problem.
Abstract: The classical Burnside’s Theorem guarantees in a finite dimensional space the existence of invariant subspaces for a proper subalgebra of the matrix algebra. In this paper we give an extension of Burnside’s Theorem for a general Banach space, which also gives new results on invariant subspaces.
TL;DR: In this article, it was shown that a semi-parallel hypersurface of a sphere and a hyperbolic space is either flat, parallel or a rotation hypersuran whose profile curve is a helix.
Abstract: We show that a semi-parallel hypersurface of a sphere and a hyperbolic space is either flat, parallel or a rotation hypersurface whose profile curve is a helix.
TL;DR: In this paper, the convex floating body of a polytope and polyhedral approximation of a convex body is considered, and the authors consider convex bodies of polytopes and polytopes.
Abstract: We consider the convex floating body of a polytope and polyhedral approximation of a convex body.
TL;DR: In this article, it was proved that the finiteness length of a Bruhat-tits filtration building is at most 2 ifn ≥ 2 and q ≥ 2 n−2.
Abstract: It is proved that the finiteness length of Γ=SL
n
(ℱ
q
[t]) isn−2 ifn≥2 andq≥2
n−2. The proof consists in studying the homotopy type of a certain Γ-invariant filtration of an appropriate Bruhat-Tits building on which Γ acts.
TL;DR: In this article, the required results from Young measure theory are derived in a new fashion, based on pointwise averages, from well-known results on weak convergence of probability measures.
Abstract: Under suitable extreme point conditions weak convergence can imply strong convergence inL 1-spaces [28, 31, 12, 26] Here a number of such results are generalized by means of a unifying, very general approach using Young measures. The required results from Young measure theory are derived in a new fashion, based on pointwise averages [6], from well-known results on weak convergence of probability measures.
TL;DR: In this paper, the authors describe groups with finite measure preserving actions that are stably orbit equivalent to such an action of a higher rank simple Lie group, and obtain information on the question of when ergodic equivalence relations are generated by a free action.
Abstract: We describe those discrete groups with finite measure preserving actions that are stably orbit equivalent to such an action of a higher rank simple Lie group. This is applied to obtain information on the question of when ergodic equivalence relations are generated by a free action of a group.
TL;DR: A survey of the rationality problem of the center of generic division algebras is given in this paper, where connections with Brauer groups of fields, geometric moduli problems and representation theory are made.
Abstract: A survey is given of the rationality problem of the center of generic division algebras. Connections are given with Brauer groups of fields, geometric moduli problems and representation theory. An outline is given of recent results.
TL;DR: In this article, a tight connection between infinite paths in recursive trees and Hamiltonian path in recursive graphs is established, and a corollary is that determining Hamiltonicity is highly undecidable.
Abstract: A tight connection is exhibited between infinite paths in recursive trees and Hamiltonian paths in recursive graphs. A corollary is that determining Hamiltonicity in recursive graphs is highly undecidable, viz, Σ
1
1
-complete. This is shown to hold even for highly recursive graphs with degree bounded by 3. Hamiltonicity is thus an example of an interesting graph problem that is outside the arithmetic hierarchy in the infinite case.
TL;DR: This paper gives a classification of the infinite behavior of clusters of a dependent percolation model, showing that the lattice is divided into disjoint infinite strips, lying adjacently.
Abstract: A dependent percolation model is a random coloring of the two-lattice. It is assumed that there are a finite number of colors and that the coloring is translation invariant. Each color defines a random subset of the lattice. Connected components of this subset are called clusters. This paper gives a classification of the infinite behavior of these clusters. In particular, it is shown that the lattice is divided into disjoint infinite strips, lying adjacently. Each strip is either composed of an infinite cluster together with isolated finite clusters or else is entirely composed of finite clusters. Examples of the various types of behavior are constructed.
TL;DR: In this article, the ingredients of an L-function machine for the quasi-split group U n, n +1 × Res GL n are treated, following similar theories of P. Shapiro and S. Gelbart.
Abstract: The ingredients of an “L-function machine” for the quasi-split groupU
n, n
+1 × Res GL
n
are treated here, following similar theories of P. Shapiro and S. Gelbart. We start with a known Rankin-Selberg type integral having an Euler product. In section 2 we compute the local integral to get a localL function. This is done by working with an “L group” related to
L
G and the relative root system. All computations are carried out for the split and the non-split case. In section 3 we address the problem of analytic continuation of the Eisenstein series. This involves computation of poles of intertwining operators.
TL;DR: In this paper, the authors show that the optimal choice of n = 2n is optimal for n ≥ 24 when the norm is the sup norm and the set of nonnegative vectors in the Banach space is ℝn.
Abstract: Suppose thatE is a finite-dimensional Banach space with a polyhedral norm ‖·‖, i.e., a norm such that the unit ball inE is a polyhedron. ℝn with the sup norm or ℝn with thel1-norm are important examples. IfD is a bounded set inE andT:D→D is a map such that ‖T(y)−T(z)‖≤ ‖y−z‖ for ally andz inE, thenT is called nonexpansive with respect to ‖·‖, and it is known that for eachx ∈D there is an integerp=p(x) such that limj→∞Tjp(x) exists. Furthermore, there exists an integerN, depending only on the dimension ofE and the polyhedral norm onE, such thatp(x)≤N: see [1,12,18,19] and the references to the literature there. In [15], Scheutzow has raised a question about the optimal choice ofN whenE=ℝn,D=Kn, the set of nonnegative vectors in ℝn, and the norm is thel1-norm. We provide here a reasonably sharp answer to Scheutzow’s question, and in fact we provide a systematic way to generate examples and use this approach to prove that our estimates are optimal forn≤24. See Theorem 2.1, Table 2.1 and the examples in Section 3. As we show in Corollary 2.3, these results also provide information about the caseD=ℝn, i.e.,T:ℝn→ℝn isl1-nonexpansive. In addition, it is conjectured in [12] thatN=2n whenE=ℝn and the norm is the sup norm, and such a result is optimal, if true. Our theorems here show that a sharper result is true for an important subclass of nonexpansive mapsT:(ℝn,‖ · ‖∞)→(ℝn,‖ · ‖∞).
TL;DR: In this article, explicit rational forms for the Poincare series of the commutative and non-commutative trace rings (5.3 and 5.5) were derived.
Abstract: In this paper we derive explicit rational forms for the Poincare series of the commutative and the non-commutative trace rings (5.3 and 5.5). To this end, we use the Molien-Weyl formula to reduce the question to a problem about flows in a particular graph.
TL;DR: In this paper, it was shown that the set of all prime divisors of all the co-degrees of the nonlinear irreducible characters of a nonlinear character of a finite group G is not a set of n-potent groups unless G is nilpotent-by-abelian.
Abstract: LetG be a finite group. The co-degree of an irreducible character χ ofG is defined to be the number |G|/χ(1). The set of all prime divisors of all the co-degrees of the nonlinear irreducible characters ofG is denoted by Σ(G). First we show that Σ(G)=π(G) (the set of all prime divisors of |G|) unlessG is nilpotent-by-abelian. Then we make Σ(G) a graph by adjoining two elements of Σ(G) if and only if their product divides a co-degree of some nonlinear character ofG. We show that the graph Σ(G) is connected and has diameter at most 2. Additional information on the graph is given. These results are analogs to theorems obtained for the graph corresponding to the character degrees (by Manz, Staszewski, Willems and Wolf) and for the graph corresponding to the class sizes (by Bertram, Herzog and Mann). Finally, we investigate groups with some restriction on the co-degrees. Among other results we show that ifG has a co-degree which is ap-power for some primep, then the corresponding character is monomial andOp(G)≠1. Also we describe groups in which each co-degree of a nonlinear character is divisible by at most two primes. These results generalize results of Chillag and Herzog. Other results are proved as well.
TL;DR: In this article, all possible relations between the height (Weyr) characteristic and the level characteristic of an M-matrix were determined under the assumption that the two characteristics have the same number of elements.
Abstract: We determine all possible relations between the height (Weyr) characteristic and the level characteristic of anM-matrix. Under the assumption that the two characteristics have the same number of elements, we determine the possible relations between the two characteristics for a wider class of matrices, which also contains the class of strictly triangular matrices over an arbitrary field. Given two sequences which satisfy the above condition, we construct a loopless acyclic graphG with the following property: Every matrix whose graph isG has its height characteristic equal to the first sequence and its level characteristic equal to the second. We give several counterexamples to possible extensions of our results, and we raise some open problems.
TL;DR: In this article, the subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered and the almost sure convergence in Egorov's sense for such sequences is proved.
Abstract: The subadditive sequences of operators which belong to a von Neumann algebra with a faithful normal state and a given positive linear kernel are considered. We prove the almost sure convergence in Egorov’s sense for such sequences.
TL;DR: In this article, it was shown that the Hausdorff dimension of the Julia set J(R) of a Misiurewicz rational map is finite, positive and the onlyh-conformal measure for R :\(R:\bar {C}} \to \bar {\mathbb{C}}\) up to a multiplicative constant.
Abstract: Leth be the Hausdorff dimension of the Julia setJ(R) of a Misiurewicz’s rational mapR :\(R:\bar {\mathbb{C}} \to \bar {\mathbb{C}}\) (subexpanding case). We prove that theh-dimensional Hausdorff measure H h onJ(R) is finite, positive and the onlyh-conformal measure forR :\(R:\bar {\mathbb{C}} \to \bar {\mathbb{C}}\) up to a multiplicative constant. Moreover, we show that there exists a uniqueR-invariant measure onJ(R) equivalent to H h .
TL;DR: In this article, it was shown that theσ-idealUcffff 0 of closed sets of extended uniqueness in T is hereditarily non-Borel, i.e. every "non-trivial"σideal of a closed setI⊆U>>\s 0 is not Borel.
Abstract: It is shown that theσ-idealU
0 of closed sets of extended uniqueness inT is hereditarily non-Borel, i.e. every “non-trivial”σ-ideal of closed setsI⊆U
0 is non-Borel. This implies both the result of Solovay, Kaufman that bothU
0 andU (theσ-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset ofT of extended uniqueness is of the first category. A further extension to ideals contained inU
0 is given.
TL;DR: In this paper, the authors studied finite-valued invariants of knots and links in the 3-sphere via the Markov theorem, where the kernel of the corresponding epimorphism is the normal closure of just four elements.
Abstract: Forn even andp an odd prime a symplectic group Sp(n, p) is a quotient of the Artin braid groupB
n+1. Ifs
1, …,s
n
are standard generators ofB
n+1 then the kernel of the corresponding epimorphism is the normal closure of just four elements:s
1
,(s
1
s
2)6,s
1
(+1)/2
s
2
4
s
1
(−1)/2
s
2
−2
s
1
−1
s
2
2
and (s
1
s
2
s
3)4
A
−1
s
1
−2
A, whereA=s
2
s
3
−1
s
2
(−1)/2
s
4
s
3
2
s
4, all of them lying in the subgroupB
5. Sp(n, p) acts on a vector space and the image of the subgroupB
n
ofB
n+1 in Sp(n, p), denoted Sp(n−1,p), is a stabilizer of one vector. A sequence of inclusions …B
k+1·B
k
… induces a sequence of inclusions …Sp(k,p)·Sp(k−1,p)…, which can be used to study some finite-valued invariants of knots and links in the 3-sphere via the Markov theorem.
TL;DR: In this paper, the implications of superrigidity and Ratner's theorem on invariant measures on homogeneous spaces for understanding the fundamental group of manifolds with an action of a semisimple Lie group are discussed.
Abstract: We discuss the implications of superrigidity and Ratner’s theorem on invariant measures on homogeneous spaces for understanding the fundamental group of manifolds with an action of a semisimple Lie group.
TL;DR: In this article, it was shown that T is lightly mixing, not partially mixing, and not lightly 2-mixing, and that T can be constructed by Chacon that is weakly mixing but not mixing.
Abstract: A rank one transformationT was constructed by Chacon that is weakly mixing but not mixing. We will show thatT is lightly mixing, not partially mixing, and not lightly 2-mixing.