Results 1  10
of
120
Rigidity Of Commensurators And Irreducible Lattices
 Invent. Math
"... this paper. We shall address here many ingredients of the well developed linear rigidity theory, such as superrigidity, strongrigidity, arithmeticity, normal subgroup structure and others, in two general situations: lattices in products of (general) topological groups, and commensurators of lattice ..."
Abstract

Cited by 76 (3 self)
 Add to MetaCart
(Show Context)
this paper. We shall address here many ingredients of the well developed linear rigidity theory, such as superrigidity, strongrigidity, arithmeticity, normal subgroup structure and others, in two general situations: lattices in products of (general) topological groups, and commensurators of lattices in topological groups. The approach we take for this purpose is, naturally, dierent from previous ones. Of course, our results apply also in the conventional linear framework. Some of them are new even in that case, and others provide an alternative, sometimes more natural approach, to several wellknown results (including a new proof of Margulis' arithmeticity and commensuratorarithmeticity theorems for a certain class of lattices).
Rates of Convex Approximation in NonHilbert Spaces
 CONSTRUCTIVE APPROXIMATION
, 1994
"... This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approx ..."
Abstract

Cited by 26 (0 self)
 Add to MetaCart
(Show Context)
This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subset S of a function space. Specifically, the focus is on the rate at which the lowest achievable error can be reduced as larger subsets of S are allowed when constructing an approximant. The new results extend those given for Hilbert spaces by Jones and Barron, including in particular a computationally attractive incremental approximation scheme. Bounds are derived for broad classes of Banach spaces; in particular, for Lp spaces with 1 < p < ∞, the O(n −1/2) bounds of Barron and Jones are recovered when p = 2. One motivation for the questions studied here arises from the area of “artificial neural networks, ” where the problem can be stated in terms of the growth in the number of “neurons ” (the elements of S) needed in order to achieve a desired error rate. The focus on nonHilbert spaces is due to the desire to understand approximation in the more “robust” (resistant to exemplar noise) Lp, 1 ≤ p < 2 norms. The techniques used borrow from results regarding moduli of smoothness in functional analysis as well as from the theory of stochastic processes on function spaces.
A quantitative Mean Ergodic Theorem for uniformly convex Banach spaces
, 2008
"... We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by ..."
Abstract

Cited by 18 (10 self)
 Add to MetaCart
(Show Context)
We provide an explicit uniform bound on the local stability of ergodic averages in uniformly convex Banach spaces. Our result can also be viewed as a finitary version in the sense of T. Tao of the Mean Ergodic Theorem for such spaces and so generalizes similar results obtained for Hilbert spaces by Avigad, Gerhardy and Towsner [1] and T. Tao [10]. 1
An isoperimetric inequality for uniformly logconcave measures and uniformly convex bodies
, 2008
"... ..."
Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces
, 2008
"... This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the KrasnoselskiMann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the g ..."
Abstract

Cited by 17 (10 self)
 Add to MetaCart
This paper provides a fixed point theorem for asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces as well as new effective results on the KrasnoselskiMann iterations of such mappings. The latter were found using methods from logic and the paper continues a case study in the general program of extracting effective data from primafacie ineffective proofs in the fixed point theory of such mappings.
definite distributions and subspaces of L−p with applications to stable processes
 Canad. Math. Bull
, 1999
"... Abstract. We define embedding of an ndimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lpspaces. The wellknown connection between embeddings into Lp and positive definite functions is extended to the case of ..."
Abstract

Cited by 15 (6 self)
 Add to MetaCart
(Show Context)
Abstract. We define embedding of an ndimensional normed space into L−p, 0 < p < n by extending analytically with respect to p the corresponding property of the classical Lpspaces. The wellknown connection between embeddings into Lp and positive definite functions is extended to the case of negative p by showing that a normed space embeds in L−p if and only if ‖x ‖ −p is a positive definite distribution. Using this criterion, we generalize the recent solutions to the 1938 Schoenberg’s problems by proving that the spaces ℓ n q, 2 < q ≤ ∞ embed in L−p if and only if p ∈ [n − 3, n). We show that the technique of embedding in L−p can be applied to stable processes in some situations where standard methods do not work. As an example, we prove inequalities of correlation type for the expectations of norms of stable vectors. In particular, for every p ∈ [n−3, n), E(maxi=1,...,n Xi  −p) ≥ E(maxi=1,...,n Yi  −p), where X1,..., Xn and Y1,...,Yn are jointly qstable symmetric random variables, 0 < q ≤ 2, so that, for some k ∈ N, 1 ≤ k < n, the vectors (X1,...,Xk) and (Xk+1,...,Xn) have the same distributions as (Y1,...,Yk) and (Yk+1,...,Yn), respectively, but Yi and Yj are independent for every choice of 1 ≤ i ≤ k, k + 1 ≤ j ≤ n. 1.
On the computational content of the Krasnoselski and Ishikawa fixed point theorems
, 2000
"... This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general ..."
Abstract

Cited by 13 (10 self)
 Add to MetaCart
This paper is a case study in proof mining applied to noneffective proofs in nonlinear functional analysis. More specifically, we are concerned with the fixed point theory of nonexpansive selfmappings f of convex sets C in normed spaces. We study the Krasnoselski iteration as well as more general socalled KrasnoselskiMann iterations. These iterations converge to fixed points of f under certain compactness conditions. But, as we show, already for uniformly convex spaces in general no bound on the rate of convergence can be computed uniformly in f . This is related to the nonuniqueness of fixed points. However, the iterations yield even without any compactness assumption and for arbitrary normed spaces approximate fixed points of arbitrary quality for bounded C (asymptotic regularity, Ishikawa 1976). We apply proof theoretic techniques (developed in previous papers of us) to noneffective proofs of this regularity and extract effective uniform bounds on the rate of the asymptotic re...
Effective uniform bounds from proofs in abstract functional analysis
 CIE 2005 NEW COMPUTATIONAL PARADIGMS: CHANGING CONCEPTIONS OF WHAT IS COMPUTABLE
, 2005
"... ..."
(Show Context)
Convergence and stability of balanced implicit methods for systems of SDEs, Int
 J. Numer. Anal. Model
, 2005
"... This paper is dedicated to academic fairness and honesty. Abstract. Several convergence and stability issues of the balanced implicit methods (BIMs) for systems of realvalued ordinary stochastic differential equations are thoroughly discussed. These methods are linearimplicit ones, hence easily i ..."
Abstract

Cited by 10 (3 self)
 Add to MetaCart
(Show Context)
This paper is dedicated to academic fairness and honesty. Abstract. Several convergence and stability issues of the balanced implicit methods (BIMs) for systems of realvalued ordinary stochastic differential equations are thoroughly discussed. These methods are linearimplicit ones, hence easily implementable and computationally more efficient than commonly known nonlinearimplicit methods. In particular, we relax the so far known convergence condition on its weight matrices cj. The presented convergence proofs extend to the case of nonrandom variable step sizes and show a dependence on certain Lyapunovfunctionals V: IRd → IR1+. The proof of L2convergence with global rate 0.5 is based on the stochastic KantorovichLaxRichtmeyer principle proved by the author (2002). Eventually, pth mean stability and almost sure stability results for martingaletype test equations document some advantage of BIMs. The problem of weak convergence with respect to the test class C2 b(κ) (IRd, IR1) and with global rate 1.0 is tackled too. Key Words. Balanced implicit methods, linearimplicit methods, conditional mean consistency, conditional mean square consistency, weak Vstability, stochastic KantorovichLaxRichtmeyer principle, L2convergence, weak convergence, almost sure stability, pth mean stability. 1.