Affine linear sieve, expanders, and sum-product

Let O\mathcal{O} be an orbit in ℤ ⁿ of a finitely generated subgroup Λ of GL _n (ℤ) whose Zariski closure Zcl(Λ) is suitably large (e.g. isomorphic to SL₂). We develop a Brun combinatorial sieve for estimating the number of points on O\mathcal{O} at which a fixed integral polynomial is prime or has few prime factors, and discuss applications to classical problems, including Pythagorean triangles and integral Apollonian packings. A fundamental role is played by the expansion property of the “congruence graphs” that we associate with O\mathcal{O} . This expansion property is established when Zcl(Λ)=SL₂, using crucially sum-product theorem in ℤ/qℤ for q square-free.     

Classification of quasifinite\mathcal{W}_\infty -modules-modules

It is proved that an irreducible quasifinite -module is a highest or lowest weight module or a module of the intermediate series; a uniformly bounded indecomposable weight -module is a module of the intermediate series. For a nondegenerate additive subgroup Λ ofF _n, whereF is a field of characteristic zero, there is a simple Lie or associative algebraW(Λ,n)⁽¹⁾ spanned by differential operatorsuD ₁ ^m …D ₁ ^m foru ∈F[Γ] (the group algebra), andm _i≥0 with , whereD _i are degree operators. It is also proved that an indecomposable quasifinite weightW(Λ,n)⁽¹⁾-module is a module of the intermediate series if Λ is not isomorphic to ℤ. Supported by NSF grant no. 10471091 of China and two grants “Excellent Young Teacher Program” and “Trans-Century Training Programme Foundation for the Talents” from the Ministry of Education of China.     

A weighted estimate for the square function on the unit ball in ℂ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show that the Luzin area integral or the square function on the unit ball of ℂ ⁿ , regarded as an operator in the weighted space L ²(w) has a linear bound in terms of the invariant A ₂ characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted L ²(w) norm of the square function. We prove the equivalence of the classical and the invariant A ₂ classes.     

A unified characterization of reproducing systems generated by a finite family,II

By a “reproducing” method forH =L ²(ℝ ⁿ ) we mean the use of two countable families {e _α : α ∈A}, {f _α : α ∈A}, inH, so that the first “analyzes” a function h ∈H by forming the inner products {<h,e _α >: α ∈A} and the second “reconstructs” h from this information:h = Σα∈A <h,e _α >:f _α. A variety of such systems have been used successfully in both pure and applied mathematics. They have the following feature in common: they are generated by a single or a finite collection of functions by applying to the generators two countable families of operators that consist of two of the following three actions: dilations, modulations, and translations. The Gabor systems, for example, involve a countable collection of modulations and translations; the affine systems (that produce a variety of wavelets) involve translations and dilations. A considerable amount of research has been conducted in order to characterize those generators of such systems. In this article we establish a result that “unifies” all of these characterizations by means of a relatively simple system of equalities. Such unification has been presented in a work by one of the authors. One of the novelties here is the use of a different approach that provides us with a considerably more general class of such reproducing systems; for example, in the affine case, we need not to restrict the dilation matrices to ones that preserve the integer lattice and are expanding on ℝ ⁿ . Another novelty is a detailed analysis, in the case of affine and quasi-affine systems, of the characterizing equations for different kinds of dilation matrices.     

Penultimate limiting forms in extreme value theory

Summary Let {X _n}_n≧1 be a sequence of independent, identically distributed random variables. If the distribution function (d.f.) ofM _n=max (X ₁,…,X _n), suitably normalized with attraction coefficients {α_n}_n≧1(α_n>0) and {b _n}_n≧1, converges to a non-degenerate d.f.G(x), asn→∞, it is of interest to study the rate of convergence to that limit law and if the convergence is slow, to find other d.f.'s which better approximate the d.f. of(M _n−b_n)/a_n thanG(x), for moderaten. We thus consider differences of the formF ⁿ(a_nx+b_n)−G(x), whereG(x) is a type I d.f. of largest values, i.e.,G(x)≡Λ(x)=exp (-exp(−x)), and show that for a broad class of d.f.'sF in the domain of attraction of Λ, there is a penultimate form of approximation which is a type II [Ф _α(x)=exp (−x^−α), x>0] or a type III [Ψ _α(x)= exp (−(−x)^α), x<0] d.f. of largest values, much closer toF ⁿ(a_nx+b_n) than the ultimate itself.     

On the covering multiplicity of lattices

Let the lattice Λ have covering radiusR, so that closed balls of radiusR around the lattice points just cover the space. The covering multiplicityCM(Λ) is the maximal number of times the interiors of these balls overlap. We show that the least possible covering multiplicity for ann-dimensional lattice isn ifn≤8, and conjecture that it exceedsn in all other cases. We determine the covering multiplicity of the Leech lattice and of the latticesI _n, A_n, D_n, E_n and their duals for small values ofn. Although it appears thatCM(I _n)=2 ⁿ⁻¹ ifn≤33, asn → ∞ we haveCM(I _n)∼2.089... ⁿ . The results have application to numerical integration.     

Completeness of the system {z^λn} in L^2[B,α0]

Let B be an unbounded domain located outside an angle domain with vertex at the origin, A ={λn}（n = 1,2,...） be a sequence of complex numbers satisfying sup ｜ arg（λn）｜ 〈 α 〈 π/2 and denote by M（∧） = {z^λ, λ ∈ ∧} the corresponding system of functions z^λ（λ∈∧）. Let α0（z） be a weight function defined on B. We obtain a completeness theorem for the system M（∧） in the Hilbert space L^2 [B, α0].     

On Λ(p)-subsets of squares

This paper is a follow up of [B₁]. It is shown that the sequence of squares {n ²|n=1, 2, ...} contains Λ(p)-subsets of “maximal density”, for any givenp>4. The proof is based on the probabilistic method developed in [B₁] and a precise estimate of the Λ(p)-constant for the sequence of squares itself. Analogues of this phenomenon are obtained for other arithmetic sets, such as the sequence ofkth powers {n ^k |n=1, 2, ...} or the sequence of prime numbers. Sections 2 and 3 of the paper are of independent interest to orthogonal system theory.     

Enveloping Σ-C *-algebra of a smooth Frechet algebra crossed product by ℝ,K-theory and differential structure inC *-algebras

Given anm-tempered strongly continuous action α of ℝ by continuous^*-automorphisms of a Frechet^*-algebraA, it is shown that the enveloping ↡-C ^*-algebraE(S(ℝ, A^∞, α)) of the smooth Schwartz crossed productS(ℝ,A ^∞, α) of the Frechet algebra A^∞ of C^∞-elements ofA is isomorphic to the Σ-C ^*-crossed productC ^*(ℝ,E(A), α) of the enveloping Σ-C ^*-algebraE(A) ofA by the induced action. WhenA is a hermitianQ-algebra, one getsK-theory isomorphismRK _*(S(ℝ, A^∞, α)) =K ^*(C ^*(ℝ,E(A), α) for the representableK-theory of Frechet algebras. An application to the differential structure of aC ^*-algebra defined by densely defined differential seminorms is given.     

Suites de longueur minimale associees a un ensemble normal donne

For a given subsetA of the set of real numbers, we search a sequence Λ=(λ _n) of real numbers such that bothA is the normal setB(Λ) associated to Λ, and Λ takes its values in a bounded interval, with a minimal lengthM. A lower bound ofM is obtained, which gives some necessary conditions of existency of such a bounded sequence Λ. More details are given whenA is a subset of the set of integers. In this case, the problem is to find a polynomialQ of lowest degree such that the productP.Q has non-negative coefficients, for some special given polynomialP.      

Distribution modulo 1 of some oscillating sequences

Sequences of the form (P(n)f(Q(n))) _n=1 ^∞ ,P andQ polynomials,f a “highly differentiable” periodic function, are considered. The results of [3] concerning the recurrence of this sequence to its value forn=0 are given a quantitative form. Density and uniform distribution modulo 1 are studied for specialQ’s.     

The moduli space of curves of genus two covering elliptic curves

We consider the moduli spaceS _n of curvesC of genus 2 with the property:C has a “maximal” mapf of degreen to an elliptic curveE. Here, the term “maximal” means that the mapf∶C→E doesn't factor over an unramified cover ofE. By Torelli mapS _n is viewed as a subset of the moduli spaceA ₂ of principally polarized abelian surfaces. On the other hand the Humbert surfaceH _Δ of invariant Δ is defined as a subvariety ofA ₂(C), the set of C-valued points ofA ₂. The purpose of this paper is to releaseS _n withH _Δ.     

Degenerate triply nonlinear problems with nonhomogeneous boundary conditions

The paper addresses the existence and uniqueness of entropy solutions for the degenerate triply nonlinear problem: b(v) _t − div α(v, ▽g(v)) = f on Q:= (0, T) × Ω with the initial condition b(v(0, ·)) = b(v ₀) on Ω and the nonhomogeneous boundary condition “v = u” on some part of the boundary (0, T) × ∂Ω”. The function g is continuous locally Lipschitz continuous and has a flat region [A ₁, A ₂,] with A ₁ ≤ 0 ≤ A ₂ so that the problem is of parabolic-hyperbolic type.     

The Banach-Mazur distance to the cube and the Dvoretzky-Rogers factorization

We show the existence of a sequence (λ _n ) of scalars withλ _n =o(n) such that, for any symmetric compact convex bodyB ⊂R ⁿ , there is an affine transformationT satisfyingQ ⊂T(B) ⊂λ _n Q, whereQ is then-dimensional cube. This complements results of the second-named author regarding the lower bound on suchλ _n . We also show that ifX is ann-dimensional Banach space andm=[n/2], then there are operatorsα:l ₂ ^m →X andβ:X→l _∞ ^m with ‖α‖·‖β‖≦C, whereC is a universal constant; this may be called “the proportional Dvoretzky-Rogers factorization”. These facts and their corollaries reveal new features of the structure of the Banach-Mazur compactum. Research performed while this author was visiting IHES. Supported in part by the NSF Grant DMS-8702058 and the Sloan Research Fellowship.     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.     

On linguistic dynamical systems,families of graphs of large girth,and cryptography

The paper is devoted to the study of a linguistic dynamical system of dimension n ≥ 2 over an arbitrary commutative ring K, i.e., a family F of nonlinear polynomial maps f _α : K ⁿ → K ⁿ depending on “time” α ∈ {K − 0} such that f _α ⁻¹ = f _−αM, the relation f _α1 (x) = f _α2 (x) for some x ∈ Kⁿ implies α₁ = α₂, and each map f _α has no invariant points. The neighborhood {f _α (υ)∣α ∈ K − {0}} of an element v determines the graph Γ(F) of the dynamical system on the vertex set Kⁿ. We refer to F as a linguistic dynamical system of rank d ≥ 1 if for each string a = (α₁, υ, α₂), s ≤ d, where α_i + α_i+1 is a nonzero divisor for i = 1, υ, d − 1, the vertices υ _a = f _α1 × ⋯ × f _αs (υ) in the graph are connected by a unique path. For each commutative ring K and each even integer n ≠= 0 mod 3, there is a family of linguistic dynamical systems L_n(K) of rank d ≥ 1/3n. Let L(n, K) be the graph of the dynamical system L_n(q). If K = F_q, the graphs L(n, F_q) form a new family of graphs of large girth. The projective limit L(K) of L(n, K), n → ∞, is well defined for each commutative ring K; in the case of an integral domain K, the graph L(K) is a forest. If K has zero divisors, then the girth of K drops to 4. We introduce some other families of graphs of large girth related to the dynamical systems L_n(q) in the case of even q. The dynamical systems and related graphs can be used for the development of symmetric or asymmetric cryptographic algorithms. These graphs allow us to establish the best known upper bounds on the minimal order of regular graphs without cycles of length 4n, with odd n ≥ 3. Bibliography: 42 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 326, 2005, pp. 214–234.     

Everywhere H?lder continuity of the spatial gradient of weak solutions to nonlinear parabolic systems

We consider weak solutions to the parabolic system ∂u ⁱ∂t−D _α A _i ^α (∇u)=B _i(∇u) in (i=1,...,) (Q=Ω×(0,T), R ⁿ a domain), where the functionsB _i may have a quadratic growth. Under the assumptionsn≤2 and ∇u ɛL _loc ^4+δ (Q; R ^nN ) (δ>0) we prove that ∇u is locally H?lder continuous inQ.     

Representation of measurable functions by series in Walsh subsystems

For any sequence {ω(n)} _n∈ℕ tending to infinity we construct a “quasiquadratic” representation spectrum Λ = {n ² + o(ω(n))} _n∈ℕ: for any almost everywhere (a. e.) finite measurable function f(x) there exists a series in the form $ \mathop \sum \limits_{k \in \Lambda } $ \mathop \sum \limits_{k \in \Lambda } α _k ω _k (x) that converges a. e. to this function, where {w _k (x)} _k∈ℕ is the Walsh system. We find representation spectra in the form {n ^l + o(n ^l )} _n∈ℕ, where l ∈ {2 ^k } _k∈ℕ.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Sulle equazioni differenziali lineari soddisfatte dal prodotto di integrali particolari di due equazioni differenziali lineari omogenee assegnate e su alcune formule integrali dei polinomi di Laguerre e di Hermite

Sunto Stabilita l'equazione soddisfatta dal prodotto degli integrali particolari di due generali equazioni differenziali lineari omogenee del 2° ordine assegnate, con procedimento che ha carattere generale, si deducono poi particolari equazioni soddisfatte da cL _n ^(α )(x)L _m ^(β )(x), cH_n(x)H_m(x), ecc., (c=costante). Infine si utilizzano alcune di esse per determinare delle formule integrali degli L _n ^(α) (x), H_n(x).