On isomorphity of measure-preserving ℤ<Superscript>2</Superscript>-actions that have isomorphic Cartesian powers

Assume that Δ and Π are representations of the group ℤ² by operators on the space L ₂(X, μ) that are induced by measure-preserving automorphisms, and for some d, the representations Δ^⨂d and Π^⨂d are conjugate to each other, Δ(ℤ² \(0, 0)) consists of weakly mixing operators, and there is a weak limit (over some subsequence in ℤ² of operators from Δ(ℤ²)) which is equal to a nontrivial, convex linear combination of elements of Δ(ℤ²) and of the projection onto constant functions. We prove that in this case, Δ and Π are also conjugate to each other. Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 8, pp. 193–212, 2007.     

Successive minima,intrinsic volumes,and lattice determinants

In Euclideand-spaceE ^d we prove a lattice-point inequality for arbitrary lattices and for the intrinsic volumesV _i (i.e., normalized quermassintegrals) of convex bodies. TheV _i are not equi-affine invariant (except the volume), hence suitable functionals of the lattice have to be introduced. The result generalizes an earlier result of Henk for the integer lattice ℤ ^d .     

On lattice points in polyhedral cross-sections

(a) We prove that the convex hull of anyk ^d +1 points of ad-dimensional lattice containsk+1 collinear lattice points. (b) For a convex polyhedron we consider the numbers of its lattice points in consecutive parallel lattice hyperplanes (levels). We prove that if a polyhedron spans no more than 2 ^d−1 levels, then this string of numbers may be arbitrary. On the other hand, we give an example of a string of 2 ^d−1+1 numbers to which no convex polyhedron corresponds inR ^d .     

Isomorphism rigidity of irreducible algebraic ℤd-actions

An irreducible algebraic ℤ ^d -actionα on a compact abelian group X is a ℤ ^d -action by automorphisms of X such that every closed, α-invariant subgroup Y⊊X is finite. We prove the following result: if d≥2, then every measurable conjugacy between irreducible and mixing algebraic ℤ ^d -actions on compact zero-dimensional abelian groups is affine. For irreducible, expansive and mixing algebraic ℤ ^d -actions on compact connected abelian groups the analogous statement follows essentially from a result by Katok and Spatzier on invariant measures of such actions (cf. [4] and [3]). By combining these two theorems one obtains isomorphism rigidity of all irreducible, expansive and mixing algebraic ℤ ^d -actions with d≥2. Oblatum 30-IX-1999 & 4-V-2000?Published online: 16 August 2000     

Classification of Refinable Splines in ℝ d

A refinable spline in ℝ ^d is a compactly supported refinable function whose support can be decomposed into simplices such that the function is a polynomial on each simplex. The best-known refinable splines in ℝ ^d are the box splines. Refinable splines play a key role in many applications, such as numerical computation, approximation theory and computer-aided geometric design. Such functions have been classified in one dimension in Dai et al. (Appl. Comput. Harmon. Anal. 22(3), 374–381, 2007), Lawton et al. (Comput. Math. 3, 137–145, 1995). In higher dimensions Sun (J. Approx. Theory 86, 240–252, 1996) characterized those splines when the dilation matrices are of the form A=mI, where m∈ℤ and I is the identity matrix. For more general dilation matrices the problem becomes more complex. In this paper we give a complete classification of refinable splines in ℝ ^d for arbitrary dilation matrices A∈M _d (ℤ).     

Mixing properties of the generalized T, T-1-process

Consider a general random walk on ℤ^d together with an i.i.d. random coloring of ℤ^d. TheT, T ^-1-process is the one where time is indexed by ℤ, and at each unit of time we see the step taken by the walk together with the color of the newly arrived at location. S. Kalikow proved that ifd = 1 and the random walk is simple, then this process is not Bernoulli. We generalize his result by proving that it is not Bernoulli ind = 2, Bernoulli but not Weak Bernoulli ind = 3 and 4, and Weak Bernoulli ind ≥ 5. These properties are related to the intersection behavior of the past and the future of simple random walk. We obtain similar results for general random walks on ℤ^d, leading to an almost complete classification. For example, ind = 1, if a step of sizex has probability proportional to l/|x|^α (x ⊋ 0), then theT, T ^-1-process is not Bernoulli when α ≥2, Bernoulli but not Weak Bernoulli when 3/2 ≤α < 2, and Weak Bernoulli when 1 < α < 3/2. Research partially carried out while a guest of the Department of Mathematics, Chalmers University of Technology, Sweden in January 1996. Research supported by grants from the Swedish Natural Science Research Council and from the Royal Swedish Academy of Sciences.     

The relative trace formula for groups with involution

A theorem of Bourgain states that the harmonic measure for a domain in ℝ ^d is supported on a set of Hausdorff dimension strictly less thand [2]. We apply Bourgain’s method to the discrete case, i.e., to the distribution of the first entrance point of a random walk into a subset of ℤ ^d ,d≥2. By refining the argument, we prove that for allβ>0 there existsρ(d,β)<d andN(d,β), such that for anyn>N(d,β), anyx ∈ ℤ ^d , and anyA ⊂ {1,…,n} ^d •{y∈ℤ whereν _A,x (y) denotes the probability thaty is the first entrance point of the simple random walk starting atx intoA. Furthermore,ρ must converge tod asβ → ∞. Supported by Swiss NF grant 20-55648.98.     

Tube Formulas and Complex Dimensions of Self-Similar Tilings

We use the self-similar tilings constructed in (Pearse in Indiana Univ. Math J. 56(6):3151–3169, 2007) to define a generating function for the geometry of a self-similar set in Euclidean space. This tubularzeta function encodes scaling and curvature properties related to the complement of the fractal set, and the associated system of mappings. This allows one to obtain the complex dimensions of the self-similar tiling as the poles of the tubularzeta function and hence develop a tube formula for self-similar tilings in ℝ^d. The resulting power series in εis a fractal extension of Steiner’s classical tube formula for convex bodies K⊆ℝ ^d . Our sum has coefficients related to the curvatures of the tiling, and contains terms for each integer i=0,1,…,d−1, just as Steiner’s does. However, our formula also contains a term for each complex dimension. This provides further justification for the term “complex dimension”. It also extends several aspects of the theory of fractal strings to higher dimensions and sheds new light on the tube formula for fractals strings obtained in (Lapidus and van Frankenhuijsen in Fractal Geometry, Complex Dimensions and Zeta Functions: Geometry and Spectra of Fractal Strings, 2006).     

On weak mixing in lattice models

Summary. For lattice models on ℤ ^d , weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region. For a wide class of models on ℤ², including all finite range models, we show that weak mixing is a consequence of Gibbs uniqueness, exponential decay of an appropriate form of connectivity, and a natural coupling property. In particular, on ℤ², the Fortuin-Kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially, and the q-state Potts model above the critical temperature is weak mixing whenever correlations decay exponentially, a hypothesis satisfied if q is sufficiently large. Ratio weak mixing is the property that uniformly over events A and B occurring on subsets Λ and Γ, respectively, of the lattice, |P(A∩B)/P(A)P(B)−1| decreases exponentially in the distance between Λ and Γ. We show that under mild hypotheses, for example finite range, weak mixing implies ratio weak mixing. Received: 27 August 1996 / In revised form: 15 August 1997     

Realizable Quadruples for Hex-polygons

There are many interesting combinatorial results and problems dealing with lattice polygons, that is, polygons in ℝ² with vertices in the integral lattice ℤ². Geometrically, ℤ² is the set of corners of a tiling of ℝ² by unit squares. Denote by H the set of corners of a tiling of the plane by regular hexagons of unit area and call a polygon P a Hex-polygon or an H-polygon if all vertices of P are in H. Our purpose is to study several combinatorial properties of H-polygons that are analogous to properties of lattice polygons. In particular we aim to find some relationships between the numbers b and i of points from H on the boundary and in the interior of an H-polygon P with the numbers v and c of vertices and the so-called boundary characteristic of P. We also pose three open problems dealing with convex Hex-polygons.     

Families of m-convex polygons:

Polygons are described as almost-convex if their perimeter differs from the perimeter of their minimum bounding rectangle by twice their ‘concavity index’, m. Such polygons are called m-convex polygons. We first use the inclusion–exclusion principle to rederive the known generating function for 1-convex self-avoiding polygons (SAPs). We then use our results to derive the exact anisotropic generating functions for osculating and neighbour-avoiding 1-convex SAPs, their isotropic form having recently been conjectured.     

Scaling limits for internal aggregation models with multiple sources

We study the scaling limits of three different aggregation models on ℤ ^d : internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which particles perform deterministic analogues of random walks; and the divisible sandpile, in which each site distributes its excess mass equally among its neighbors. As the lattice spacing tends to zero, all three models are found to have the same scaling limit, which we describe as the solution to a certain PDE free boundary problem in ℝ ^d . In particular, internal DLA has a deterministic scaling limit. We find that the scaling limits are quadrature domains, which have arisen independently in many fields such as potential theory and fluid dynamics. Our results apply both to the case of multiple point sources and to the Diaconis-Fulton smash sum of domains.     

On polynomial symbols for subdivision schemes

Given a dilation matrix A :ℤ^d→ℤ^d, and G a complete set of coset representatives of 2π(A ^−Tℤ^d/ℤ^d), we consider polynomial solutions M to the equation ∑ _g∈G M(ξ+g)=1 with the constraints that M≥0 and M(0)=1. We prove that the full class of such functions can be generated using polynomial convolution kernels. Trigonometric polynomials of this type play an important role as symbols for interpolatory subdivision schemes. For isotropic dilation matrices, we use the method introduced to construct symbols for interpolatory subdivision schemes satisfying Strang–Fix conditions of arbitrary order. Research partially supported by the Danish Technical Science Foundation, Grant No. 9701481, and by the Danish SNF-PDE network.     

On transfer inequalities in Diophantine approximation, II

Let Θ be a point in R ⁿ . We are concerned with the approximation to Θ by rational linear subvarieties of dimension d for 0 ≤ d ≤ n−1. To that purpose, we introduce various convex bodies in the Grassmann algebra Λ(R ⁿ⁺¹). It turns out that our convex bodies in degree d are the dth compound, in the sense of Mahler, of convex bodies in degree one. A dual formulation is also given. This approach enables us both to split and to refine the classical Khintchine transference principle.     

Haar Type Orthonormal Wavelet Bases in R2

K.-H. Grochenig and A. Haas asked whether for every expanding integer matrix A ∈ M_n(ℤ) there is a Haar type orthonormal wavelet basis having dilation factor A and translation lattice ℤⁿ. They proved that this is the case when the dimension n = 1. This article shows that this is also the case when the dimension n = 2.     

Some zero-sum constants with weights

For an abelian group G, the Davenport constant D(G) is defined to be the smallest natural number k such that any sequence of k elements in G has a nonempty subsequence whose sum is zero (the identity element). Motivated by some recent developments around the notion of Davenport constant with weights, we study them in some basic cases. We also define a new combinatorial invariant related to (ℤ/nℤ) ^d , more in the spirit of some constants considered by Harborth and others and obtain its exact value in the case of (ℤ/nℤ)² where n is an odd integer.     

Discrete Convexity

In the paper, we explain what subsets of the lattice ℤ ⁿ and what functions on the lattice ℤ ⁿ could be called convex. The basis of our theory is the following three main postulates of classical convex analysis: concave functions are closed under sums; they are also closed under convolutions; and the superdifferential of a concave function is nonempty at each point of the domain. Interesting (and even dual) classes of discrete concave functions arise if we require either the existence of superdifferentials and closeness under convolutions or the existence of superdifferentials and closeness under sums. The corresponding classes of convex sets are obtained as the affinity domains of such discretely concave functions. The classes of the first type are closed under (Minkowski) sums, and the classes of the second type are closed under intersections. In both classes, the separation theorem holds true. Unimodular sets play an important role in the classification of such classes. The so-called polymatroidal discretely concave functions, most interesting for applications, are related to the unimodular system . Such functions naturally appear in mathematical economics, in Gelfand-Tzetlin patterns, play an important role for solution of the Horn problem, for describing submodule invariants over discrete valuation rings, and so on. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 312, 2004, pp. 86–93.     

Existence and analyticity of eigenvalues of a two-channel molecular resonance model

We consider a family of operators H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d := (−π,π]^d, associated with the Hamiltonian of a system consisting of at most two particles on a d-dimensional lattice ℤ^d, interacting via both a pair contact potential (μ > 0) and creation and annihilation operators (γ > 0). We prove the existence of a unique eigenvalue of H_γμ(k), k ∈ \mathbbT^d \mathbb{T}^d , or its absence depending on both the interaction parameters γ,μ ≥ 0 and the system quasimomentum k ∈ \mathbbT^d \mathbb{T}^d . We show that the corresponding eigenvector is analytic. We establish that the eigenvalue and eigenvector are analytic functions of the quasimomentum k ∈ \mathbbT^d \mathbb{T}^d in the existence domain G ⊂ \mathbbT^d \mathbb{T}^d .     

The Discrete and Classical Dirichlet Problem

Let W ì \mathbbR^d{\Omega \subset \mathbb{R}^d} be some bounded domain with reasonable boundary and let f be a continuous function on the complement Ω ^c . We can construct an unique continuous function u that is harmonique on Ω and u = f on Ω ^c . Similarly, u _d is the unique function on the lattice points such that for each lattice point of Ω satisfies the “average” property with respect to its nearest neighbours and u _d = f on Ω ^c . In this paper when ∂Ω is Lipschitz I give a “best possible” estimate of ||u − u _d ||_∞.     

Fluctuation–dissipation equation of asymmetric simple exclusion processes

Summary. We consider asymmetric simple exclusion processes on the lattice Zopf; ^d in dimension d≥3. We denote by L the generator of the process, ∇ the lattice gradient, η the configuration, and w the current of the dynamics associated to the conserved quantity. We prove that the fluctuation–dissipation equation w=Lu+D∇η has a solution for some function u and some constant D identified to be the diffusion coefficient. Intuitively, Lu represents rapid fluctuation and this equation describes a decomposition of the current into fluctuation and gradient of the density field, representing the dissipation. Using this result, we proved rigorously that the Green-Kubo formula converges and it can be identified as the diffusion coefficient. Received: 14 May 1996 / In revised form: 20 February 1997