Random permutations of countable sets

Summary The main objective of this paper is a study of random decompositions of random point configurations onR ^d into finite clusters. This is achieved by constructing for each configurationZ a random permutation ofZ with finite cycles; these cycles then form the cluster decomposition ofZ. It is argued that a good candidate for a random permutation ofZ is a Gibbs measure for a certain specification, and conditions are given for the existence and uniqueness of such a Gibbs measure. These conditions are then verified for certain random configurationsZ.     

Energy of Flows on Percolation Clusters

It is well known for which gauge functions H there exists a flow in Z ^d with finite H energy. In this paper we discuss the robustness under random thinning of edges of the existence of such flows. Instead of Z ^d we let our (random) graph cal C cal (Z ^d,p) be the graph obtained from Z ^d by removing edges with probability 1–p independently on all edges. Grimmett, Kesten, and Zhang (1993) showed that for d3,p>p _c(Z ^d), simple random walk on cal C cal (Z ^d,p) is a.s. transient. Their result is equivalent to the existence of a nonzero flow f on the infinite cluster such that the x ² energy _e f(e)² is finite. Levin and Peres (1998) sharpened this result, and showed that if d3 and p>p _c(Z ^d), then cal C cal (Z ^d,p) supports a nonzero flow f such that the x ^q energy is finite for all q>d/(d–1). However, for general gauge functions, there is a gap between the existence of flows with finite energy which results from the work of Levin and Peres and the known results on flows for Z ^d. In this paper we close the gap by showing that if d3 and Z ^d supports a flow of finite H energy then the infinite percolation cluster on Z ^d also support flows of finite H energy. This disproves a conjecture of Levin and Peres.     

Bahadur representation of nonparametric M-estimators for spatial processes

Under some mild conditions, we establish a strong Bahadur representation of a general class of nonparametric local linear M-estimators for mixing processes on a random field. If the socalled optimal bandwidth hn = O（｜n｜^-1/5）, n ∈ Z^d, is chosen, then the remainder rates in the Bahadur representation for the local M-estimators of the regression function and its derivative are of order O（｜n｜^-4/5 log ｜n｜）. Moreover, we derive some asymptotic properties for the nonparametric local linear M-estimators as applications of our result.     

Tiling Lattices with Sublattices,I

Call a coset C of a subgroup of Z^d{\bf Z}^{d} a Cartesian coset if C equals the Cartesian product of d arithmetic progressions. Generalizing Mirsky–Newman, we show that a non-trivial disjoint family of Cartesian cosets with union Z^d{\bf Z}^{d} always contains two cosets that differ only by translation. Where Mirsky–Newman’s proof (for d=1) uses complex analysis, we employ Fourier techniques. Relaxing the Cartesian requirement, for d>2 we provide examples where Z^d{\bf Z}^{d} occurs as the disjoint union of four cosets of distinct subgroups (with one not Cartesian). Whether one can do the same for d=2 remains open.     

Geometry of Periodic Modules over Tame Concealed and Tubular Algebras

Let A be a tame concealed or tubular algebra and d the dimension-vector of a periodic module with respect to the action of the Auslander–Reiten translation. We prove that the affine variety mod _A (d) of all A-modules of dimension-vector d is a normal complete intersection. Moreover, we show that a module M in mod _A (d) is nonsingular if and only if Ext _A ²(M,M)=0.     

Fitting ideals and the Gorenstein property

Let p be a prime number and G be a finite commutative group such that p ² does not divide the order of G. In this note we prove that for every finite module M over the group ring Z _p [G], the inequality #M  ￡  #Z_p[G]/Fit_Zp[G](M){\#M\,\leq\,\#{\bf Z}_{p}[G]/{{\rm Fit}}_{{\bf Z}_{p}[G]}(M)} holds. Here, Fit_Zp[G](M){\rm Fit}_{{\bf Z}_{p}[G]}(M) is the Z _p [G]-Fitting ideal of M.     

On nonnegatively curved 4-manifolds with discrete symmetry

Let M be the closed, simply connected, 4-manifold with nonnegative sectional curvature, called a nonnegatively curved 4-manifold, with an effective and isometric Z _m -action for a positive integer m ≧ 61⁷. Assume that Z _m acts trivially on the homology of M. The goal of this short paper is to prove that if the fixed point set of any nontrivial element of Z _m has at most one two-dimensional component, then M is homeomorphic to S ⁴, # _i ^l =1S ² × S ², l = 1, 2, or # _j ^k = 1 ± CP ², k = 1, 2, 3, 4, 5. The main strategy of this paper is to give an upper bound of the Euler characteristic χ(M) under the homological assumption of a Z _m -action as above by using the Lefschetz fixed point formula.     

Cobordism of immersions of surfaces¶in non-orientable 3-manifolds

Some properties of non-orientable 3-manifolds are shown. In particular, for a connected, non-orientable 3-manifold M, the group of cobordism clases of immersions of surfaces in M is isomorphic to a group structure on the set H ₂(M,Z/2Z)×H ₁(M,Z/2Z)×Z/2Z. Received: 8 June 2000 / Revised version: 2 October 2000     

Integral Loop Rings of Loops Whose Derived Subloop Is Central

It is proven that finite loops whose derived subloop is central are determined by Z, the ring of integers, that is, if L and M are two such loops and Z L ? Z M, then L ? M.     

On the mixing time of a simple random walk on the super critical percolation cluster

 We study the robustness under perturbations of mixing times, by studying mixing times of random walks in percolation clusters inside boxes in Z ^d . We show that for d≥2 and p>p _c (Z ^d ), the mixing time of simple random walk on the largest cluster inside is Θ(n ² ) – thus the mixing time is robust up to a constant factor. The mixing time bound utilizes the Lovàsz-Kannan average conductance method. This is the first non-trivial application of this method which yields a tight result. Received: 16 December 2001 / Revised version: 13 August 2002 / Published online: 19 December 2002     

Degrees of categoricity of computable structures

Defining the degree of categoricity of a computable structure M{\mathcal{M}} to be the least degree d for which M{\mathcal{M}} is d-computably categorical, we investigate which Turing degrees can be realized as degrees of categoricity. We show that for all n, degrees d.c.e. in and above 0 ⁽ⁿ⁾ can be so realized, as can the degree 0 ^(ω).     

Mixing and linear equations over groups in positive characteristic

We prove a result on linear equations over multiplicative groups in positive characteristic. This is applied to settle a conjecture about higher order mixing properties of algebraicZ ^d -actions.     

Difference Families in Z2d+1⊕Z2d+1 and Infinite Translation Designs in Z⊕Z

We analyse 3-subset difference families of Z_2d+1⊕Z_2d+1 arising as reductions (mod 2d+1) of particular families of 3-subsets of Z⊕Z. The latter structures, namely perfect d-families, can be viewed as 2-dimensional analogues of difference triangle sets having the least scope. Indeed, every perfect d-family is a set of base blocks which, under the natural action of the translation group Z⊕Z, cover all edges {(x,y),(x′,y′)} such that |x−x′|, |y−y′|≤d. In particular, such a family realises a translation invariant (G,K₃)-design, where V(G)=Z⊕Z and the edges satisfy the above constraint. For that reason, we regard perfect families as part of the hereby defined translation designs, which comprise and slightly generalise many structures already existing in the literature. The geometric context allows some suggestive additional definitions. The main result of the paper is the construction of two infinite classes of d-families. Furthermore, we provide two sporadic examples and show that a d-family may exist only if d≡0,3,8,11 (mod 12).     

Some dimension-free features of vector-valued martingales

Summary Given any local maringaleM inR ^d orl ², there exists a local martingaleN inR ², such that |M|=|N|, [M]=[N], and «M»=«N». It follows in particular that any inequality for martingales inR ² which involves only the processes |M|, [M] and «M» remains true in arbitrary dimension. WhenM is continuous, the processes |M|² and |M| satisfy certain SDE's which are independent of dimension and yield information about the growth rate ofM. This leads in particular to tail estimates of the same order as in one dimension. The paper concludes with some new maximal inequalities in continuous time.Research supported by NSF grant DMS-9002732 and by AFOSR Contract F49620 85C 0144     

Projections in the space<Emphasis Type="Italic">H</Emphasis><Superscript>∞</Superscript> and the corona theorem for subdomains of coverings of finite bordered Riemann surfaces

LetM be a non-compact connected Riemann surface of a finite type, andR⋐M be a relatively compact domain such thatH ₁(M,Z)=H ₁(R,Z). Let be a covering. We study the algebraH ^∞(U) of bounded holomorphic functions defined in certain subdomains . Our main result is a Forelli type theorem on projections inH ^∞(D). Research supported in part by NSERC.     

Zero-sum problems with subgroup weights

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider ℤ _p ^d for a general d and subgroups of Z* _p as weights.     

Distribution of values of bounded generalized polynomials

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Z ^d  → R ^l has a representation u(n) = f(ϕ(n)x), n ∈ Z ^d , where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Z ^d -action by translations on X. This fact is used to show that the sequence u(n), n ∈ Z ^d , is well distributed on a piecewise polynomial surface (with respect to the Borel measure on that is the image of the Lebesgue measure under the piecewise polynomial function defining ). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.     

Mean dimension, small entropy factors and an embedding theorem

In this paper we show how the notion of mean dimension is connected in a natural way to the following two questions: what points in a dynamical system (X, T) can be distinguished by factors with arbitrarily small topological entropy, and when can a system (X, T) be embedded in (([0, 1] ^d ) ^Z , shift). Our results apply to extensions of minimalZ-actions, and for this case we also show that there is a very satisfying dimension theory for mean dimension.     

Anderson Localization for Time Periodic Random Schrödinger Operators

Abstract

We prove that at large disorder, Anderson localization in Z ^d is stable under localized time-periodic perturbations by proving that the associated quasi-energy operator has pure point spectrum. The formulation of this problem is motivated by questions of Anderson localization for non-linear Schrödinger equations.     

Reduced bodies in normed planes

We say that a convex body R of a d-dimensional real normed linear space M ^d is reduced, if Δ(P) < Δ(R) for every convex body P ⊂ R different from R. The symbol Δ(C) stands here for the thickness (in the sense of the norm) of a convex body C ⊂ M ^d . We establish a number of properties of reduced bodies in M ². They are consequences of our basic Theorem which describes the situation when the width (in the sense of the norm) of a reduced body R ⊂ M ² is larger than Δ(R) for all directions strictly between two fixed directions and equals Δ(R) for these two directions.