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.     

Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ ^d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ ^d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ ^d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ ^d is always a quotient ofdifferentiably finite power series.     

The limit shape of convex lattice polygons

It is proved here that, asn→∞, almost all convex (1/n)ℤ²-lattice polygons lying in the square [−1, 1]² are very close to a fixed convex set. This research was partially supported by Hungarian Science Foundation Grants 1907 and 1909.     

Invariant measures for certain expansive ℤ2-actions

Letp>1 be prime, and letY ⊂X=(ℤ/pℤ)^{ℤ 2}) be an infinite, closed, shift-invariant subgroup with the following properties: the restriction toY of the shift-actionσ of ℤ² onX is mixing with respect to the Haar measureλ _Y ofY, and every closed, shift-invariant subgroupZ ⊂Y is finite. We prove that every sufficiently mixing, non-atomic, shift-invariant probability measureμ onY is equal toλ _Y . The author would like to thank the Department of Mathematics, University of Vienna, for hospitality while this work was done.     

On Jackson's theorem in the space ℓ2(ℤ 2 n )

Estimates of Jackson's constants in the space ℓ₂(ℤ ₂ ⁿ ) are given for the case of approximation by sums of subspaces on which irreducible representations of the isometry group of (ℤ ₂ ⁿ ) act and for the case in which the modulus of continuity is defined using generalized translations. Coding theory results on efficiency estimates for binaryd-codes with respect to the Hamming distance are used. Translated fromMatematicheskie Zametki, Vol. 60, No. 3, pp. 390–405, September, 1996.     

Some results on quasipolyhedral convexity

The objective of this paper is to analyze under what well-known operations the class of quasipolyhedral convex functions, which can be regarded as an extension of that of polyhedral convex functions, is closed. The operations that will be considered are those that preserve polyhedral convexity, such that the image and the inverse image under linear transformations, right scalar multiplication (including the case where λ=0⁺) and pointwise addition.      

Quotients of the unit ball of ℂn for a free action of ℤ

LetB _n be the unit ball of ℂⁿ and ℤ ≅ Γ ⊂ AutB _n be generated by a parabolic element of AutB _n. We show that the quotientB _n/Γ is biholomorphic to a holomorphically convex domain of ℂⁿ, whose automorphism group is explicity described. It follows thatB _n/ℤ is Stein for any free action of ℤ. Investigation partially supported by University of Bologna. Funds for selected research topics. The second author was supported by an Instituto Nazionale di Alta Matematica grant.     

Best Approximations and Widths of Classes of Convolutions of Periodic Functions of High Smoothness

We consider classes of 2π-periodic functions that are represented in terms of convolutions with fixed kernels Ψ _β ⁻ whose Fourier coefficients tend to zero at exponential rate. We determine exact values of the best approximations of these classes in the uniform and integral metrics. In several cases, we determine the exact values of the Kolmogorov, Bernstein, and linear widths for these classes in the metrics of the spaces C and L. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 7, pp. 946–971, July, 2005.     

Bounds for exponential sums modulo <Emphasis Type="Italic">p</Emphasis><Superscript>2</Superscript>

In this paper we consider exponential sums over subgroups G ⊂ ℤ _q ^* . Using Stepanov’s method, we obtain nontrivial bounds for exponential sums in the case where q is a square of a prime number. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 11, No. 6, pp. 81–94, 2005.     

On the pontryagin-steenrod-wu theorem

We present a short and direct proof (based on the Pontryagin-Thom construction) of the following Pontryagin-Steenrod-Wu theorem: (a) LetM be a connected orientable closed smooth (n + 1)-manifold,n≥3. Define the degree map deg: π ⁿ (M) →H ⁿ (M; ℤ) by the formula degf =f*[S ⁿ ], where [S ⁿ ] εH ⁿ (M; ℤ) is the fundamental class. The degree map is bijective, if there existsβ εH ₂(M, ℤ/2ℤ) such thatβ ·w ₂(M) ε 0. If suchβ does not exist, then deg is a 2-1 map; and (b) LetM be an orientable closed smooth (n+2)-manifold,n≥3. An elementα lies in the image of the degree map if and only ifρ ₂ α ·w ₂(M)=0, whereρ ₂: ℤ → ℤ/2ℤ is reduction modulo 2.     

Threefolds of order one in the six-quadric

Consider the smooth quadric Q ₆ in ℙ⁷. The middle homology group H ₆(Q ₆, ℤ) is isomorphic to ℤ ⊕ ℤ, with a basis given by two classes of linear subspaces. We classify all threefolds of bidegree (1, p) inside Q ₆.     

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     

Some properties of certain subclasses of analytic functions with negative coefficients by using generalized ruscheweyh derivative operator

By making use of the known concept of neighborhoods of analytic functions we prove several inclusions associated with the (j, δ)-neighborhoods of various subclasses of starlike and convex functions of complex order b which are defined by the generalized Ruscheweyh derivative operator. Further, partial sums and integral means inequalities for these function classes are studied. Relevant connections with some other recent investigations are also pointed out.     

Continuous sums of ridge functions on a convex body and the class VMO

Sums of ridge functions on convex bodies in the space ? ⁿ are studied. It is established that, under sufficiently general constraints on the functions of one variable generating the sums, each of these sums must belong to the class VMO on each finite closed interval of its domain.     

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 a cardinal group invariant related to decompositions of Abelian groups

For each Abelian groupG, a cardinal invariant χ(G) is introduced and its properties are studied. In the special caseG = ℤ ⁿ , the cardinalχ(ℤ ⁿ ) is equal to the minimal cardinality of an essential subset of ℤ ⁿ , i.e., a of a subsetA ⊂ ℤ ⁿ such that, for any coloring of the group ℤ ⁿ inn colors, there exists an infinite one-color subset that is symmetric with respect to some pointα ofA. The estimaten( ^{n +} l)/2 ≤χ(ℤ ⁿ ) < 2n is proved for alln and the relationχ(ℤ ⁿ ) =n(n + 1)/2 forn ≤ 3. The structure of essential subsets of cardinalityχ(ℤ ⁿ ) in ℤ ⁿ is completely described forn ≤ 3. Translated fromMatematicheskie Zametki, Vol. 64, No. 3, pp. 341–350, September, 1998.     

Extreme points for convex mappings ofB n

We exhibit a collection of extreme points of the family of normalized convex mappings of the open unit ball of ℂ ⁿ forn≥2. These extreme points are defined in terms of the extreme points of a closed ball in the Banach space of homogeneous polynomials of degree 2 in ℂ ⁿ⁻¹, which are fully classified. Two examples are given to show that there are more convex mappings than those contained in the closed convex hull of the set of extreme points here exhibited.     

Optimization for products of concave functions

Ifh denotes the product of finitely many concave non-negative functions on a compact interval [a, b], then it is shown that there exist pointsα andβ witha≤α≤β≤b such thath is strictly increasing on [α, α), constant on (α, β), and strictly decreasing on (β, b]. This structure theorem leads to an extension of several classical optimization results for concave functions on convex sets to the case of products of concave functions.     

Two-side estimates for sums of absolute values of Fourier coefficients of functions from Hω(Tm) with the use of partial moduli of smoothness

Two-side estimates of sums of absolute values of Fourier coefficients for functions of many variables from the classes H ^ω (T ^m ) are established in the paper with the help of partial moduli of smoothness.     

Totally Disconnected and Locally Compact Heisenberg-Weyl Groups

Harmonic analysis on ℤ(p ^ℓ ) and the corresponding representation of the Heisenberg-Weyl group HW[ℤ(p ^ℓ ),ℤ(p ^ℓ ),ℤ(p ^ℓ )], is studied. It is shown that the HW[ℤ(p ^ℓ ),ℤ(p ^ℓ ),ℤ(p ^ℓ )] with a homomorphism between them, form an inverse system which has as inverse limit the profinite representation of the Heisenberg-Weyl group \mathfrak HW[\mathbbZ_p,\mathbbZ_p,\mathbbZ_p]\mathfrak {HW}[{\mathbb{Z}}_{p},{\mathbb{Z}}_{p},{\mathbb{Z}}_{p}]. Harmonic analysis on ℤ _p is also studied. The corresponding representation of the Heisenberg-Weyl group HW[(ℚ _p /ℤ _p ),ℤ _p ,(ℚ _p /ℤ _p )] is a totally disconnected and locally compact topological group.