Riesz means for the sublaplacian on the Heisenberg group

The uniform boundedness of the Riesz means for the sublaplacian on the Heisenberg groupH ⁿ is considered. It is proved thatS _R ^α are uniformly bounded onL ^p(Hⁿ) for 1≤p≤2 provided α>α(p)=(2n+1)[(1/p)−(1/2)].     

A phase transition for the metric distortion of percolation on the hypercube

Let H _n be the hypercube {0, 1} ⁿ , and denote by H _n,p Bernoulli bond percolation on H _n , with parameter p = n ^−α . It is shown that at α = 1/2 there is a phase transition for the metric distortion between H _n and H _n,p . For α < 1/2, the giant component of H _n,p is likely to be quasi-isometric to H _n with constant distortion (depending only on α). For 1/2 < α < 1 the minimal distortion tends to infinity as a power of n. We argue that the phase 1/2 < α < 1 is an analogue of the non-uniqueness phase appearing in percolation on non-amenable graphs.     

Global existence and time decay of small solutions to the landau-ginzburg type equations

We study the Cauchy problem for the nonlinear dissipative equations (0.1) uo∂u-αδu + Β|u|^2/n u = 0,x ∃ Rⁿ,t } 0,u(0,x) = u0(x),x ∃ Rⁿ, where α,Β ∃ C, ℜα 0. We are interested in the dissipative case ℜα 0, and ℜδ(α,Β)≥ 0, θ = |∫ u0(x)dx| ⊋ 0, where δ(α, Β) = ##|α|^n-1n^n/2 / ((n + 1)|α|² + α^{2 n/2}. Furthermore, we assume that the initial data u₀ ∃ L^p are such that (1 + |x|)^αu0 ∃ L¹, with sufficiently small norm ∃ = (1 + |x|)^α u₀ 1 + u₀ p, wherep 1, α ∃ (0,1). Then there exists a unique solution of the Cauchy problem (0.1)u(t, x) ∃ C ((0, ∞); L^∞) ∩ C ([0, ∞); L¹ ∩ L^p) satisfying the time decay estimates for allt0 u(t)||_∞ Cɛt^-n/2(1 + η log 〈t〉)^-n/2, if hg = θ^2/n 2_π ℜδ(α, Β) 0; u(t)||_∞ Cɛt^-n/2(1 + Μ log 〈t〉)^-n/4, if η = 0 and Μ = θ^4/n 4π)² (ℑδ(α, Β))² ℜ((1 + 1/n) υ1-1 υ2) 0; and u(t)||_∞ Cɛt^-n/2(1 + κ log 〈t〉)^-n/6, if η = 0, Μ = 0, κ 0, where υl,l = 1,2 are defined in (1.2), κ is a positive constant defined in (2.31).     

Quintuplets of orthoprojectors associated by a linear relation

We consider the equation α₁ P ₁ + α₂ P ₂ + … α _n P _n = I over orthoprojectors P ₁, … ,P _n in a Hilbert space. We show that the set of real parameters (α₁, … α _n ) for which there exists a solution of this equation in orthoprojectors contains an open set from ℝ⁵.     

L p − Lq estimates for convolution operators with n-Dimensional singular measures

Let S ⊂ ℜⁿ⁺¹ be the graph of the function ϕ :[−1, 1] ⁿ → ℜ defined by ϕ (x ₁ , …, x_n) = ∑ _j=1 ⁿ |x_j|^αj, with1<α ₁ ≤ … ≤ α_n, let σ the Euclidean area measure on S. In this article we study the L^p − L^q boundedness of convolution operators with the singular Borel measure on Rⁿ⁺¹ given by μ (E)=σ (E ∩ S)     

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.     

Boundedness of Multilinear Operators in Herz-type Hardy Space

Let κ∈ℕ. We prove that the multilinear operators of finite sums of products of singular integrals on ℝⁿ are bounded from HK ^α1,p1 _q1 (ℝⁿ) ×···×HK ^αk,pk _qk (ℝⁿ) into HK ^α,p _q (ℝⁿ) if they have vanishing moments up to a certain order dictated by the target spaces. These conditions on vanishing moments satisfied by the multilinear operators are also necessary when α_j≥ 0 and the singular integrals considered here include the Calderón-Zygmund singular integrals and the fractional integrals of any orders. Received September 6, 1999, Revised November 17, 1999, Accepted December 9, 1999     

The best Lipschitz constants of Bernstein polynomials and Bezier nets over a given triangle

This paper proved the following three facts about the Lipschitz continuous property of Bernstein polynomials and Bezier nets defined on a triangle: suppose f(P) is a real valued function defined on a triangle T, (1) If f(P) satisfies Lipschitz continuous condition, i.e. f(P)∃Lip_Aα, then the corresponding Bernstein Bezier net f_n∃Lip _Asec ^αφα, here φ is the half of the largest angle of triangle T; (2) If Bernstein Bezier net f_n∃Lip _Bα, then its elevation Bezier net Ef_n∃Lip _Bα; and (3) If f(P)∃Lip _Aα, then the corresponding Bernstein polynomials B_n(f;P)∃Lip _Asec ^αφα, and the constant Asec^αφ is best in some sense. Supported by NSF and SF of National Educational Committee     

Self-similar extremal processes

Given an extremal process X: [0,∞)→[0,∞)^d with lower curve C and associated point process N={(t_k, X_k):k≥0}, t_k distinct and X_k independent, given a sequence ζ _n =(τ _n , ξ _n ), n≥1, of time-space changes (max-automorphisms of [0,∞)^d+1), we study the limit behavior of the sequence of extremal processes Y_n(t)=ξ _n ^-1 ○ X ○ τ_n(t)=C_n(t) V max {ξ _n ^-1 ○ X_k: t_k ≤ τ_n(t){ ⇒ Y under a regularity condition on the norming sequence ζ_n and asymptotic negligibility of the max-increments of Y_n. The limit class consists of self-similar (with respect to a group η_α=(σ_α, L_α), α>0, of time-space changes) extremal processes. By self-similarity here we mean the property L_α ○ Y(t) ₌ ^d Y ○ α_α(t) for all α>0. The univariate marginals of Y are max-self-decomposable. If additionally the initial extremal process X is assumed to have homogeneous max-increments, then the limit process is max-stable with homogeneous max-increments. Supported by the Bulgarian Ministry of Education and Sciences (grant No. MM 234/1996). Proceedings of the Seminar on Stability Problems for Stochastic Models, Hajdúszoboszló, Hungary, 1997, Part I.     

Varieties with at most quadratic growth

Let V be a variety of non-necessarily associative algebras over a field of characteristic zero. The growth of V is determined by the asymptotic behavior of the sequence of codimensions c _n (V), n = 1, 2, …, and here we study varieties of polynomial growth. Recently in [16], for any real number α, 3 < α < 4, a variety V was constructed satisfying C ₁ n ^α < c _n (V) < C ₂ n ^α , for some constants C ₁, C ₂. Motivated by this result here we try to classify all possible growth of varieties V such that c _n (V) < C _n ^α , with 0 < α < 2, for some constant C. We prove that if 0 < α < 1 then, for n large, c _n (V) ≤ 1, whereas if V is a commutative variety and 1 < α < 2, then lim _n→∞ log _n c _n (V) = 1 or c _n (V) ≤ 1 for n large enough.     

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.     

A unified approach for univalent functions with negative coefficients using the Hadamard product

For given analytic functions ϕ(z) = z + Σ _n=2^∞ λ _n z ⁿ , Ψ(z) = z + Σ _n=2^∞ μ with λ _n ≥ 0, μ _n ≥ 0, and λ _n ≥ μ _n and for α, β (0≤α<1, 0<β≤1), let E(φ,ψ; α, β) be of analytic functions ƒ(z) = z + Σ _n=2^∞ a _n z ⁿ in U such that f(z)*ψ(z)≠0 and

A family of counterexamples in ergodic theory

LetT _α be the translationx↦x+α (mod 1) of [0, 1), α irrational. LetT be the Lebesgue measure-preserving automorphism ofX=[0, 3/2) defined byTx = x + 1 forx∈[0, 1/2),Tx=T _α(x−1) forx∈[1,3/2) andTx = T _α x forx∈[1/2, 1), i.e.T isT _α with a tower of height one built over [0, 1/2). If α is poorly approximable by rationals (there does not exist {p _n /q _n } with |α−p _n /q _n |=o(q _n ⁻²)) and λ is a measure onX ^k all of whose one-dimensional marginals are Lebesgue and which is ⊗ _{i − 1} ^k T ¹ invariant and ergodic (l>0) then λ is a product of off-diagonal measures. This property suffices for many purposes of counterexample construction. A connection is established with the POD (proximal orbit dense) condition in topological dynamics. Research supported in part by NSF contract MCS-8003038.     

Classification ofS n -normal semigroups

IfS _n andC _n denote, respectively, the symmetric group and inverse semigroup onn symbols, thenS _n⊂C_n and a semigroupT⊂C_n isS _n -normal ifα ⁻¹ Tα ⊂Tfor every α∈S _n . TheS _n -normal semigroups are classified.     

Classical operators on BLOCH spaces

Let \mathbbD \mathbb{D} ⁿ denote the unit polydisk and let B _n denote the unit ball in \mathbbC \mathbb{C} ⁿ , n ≥1. We study weighted composition operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } ( \mathbbD \mathbb{D} ⁿ ), α > 1. We also study Cesàro type operators on the α-Bloch spaces B^a {\mathcal{B}^\alpha } (B _n ), α > 0. Bibliography: 15 titles.     

The weighted Herz-type Hardy spaces hK q α,p (ω1,ω2)

Let 1<q<∞, n(1−1/q)≤α<∞, 0<p<∞ and ω₁,ω₂ ɛA ₁(R ⁿ ) (the Muckenhoupt class). In this paper, the author introduce the weighted Herz-type Hardy spaces hk _q ^α,p (gw₁,ω₂) and present their atomic decomposition. Using the atomic decomposition, the author find out their dual spaces, establish the boundedness on these spaces of the pseudo-differential operators of order zero and show thatD(R ⁿ ), the class of C^∞(Rⁿ)-functions with compactly support, is dense inhK _q ^α,p (ω₁,ω₂) and there is a subsequence, which converges in distrbutional sense to some distribution ofhK _q ^α,p (ω₁,ω₂), of any bounded sequence inhK _q ^α,p (ω₁,ω₂). In addition, the author also set up the boundedness of some non-linear quantities in compensated compactness. Supported by the NECF and the NECF and the NNSF of China.     

Irreducible characters with equal roots in the groups Sn and An

We show that treating of (non-trivial) pairs of irreducible characters of the group S_n sharing the same set of roots on one of the sets A_n and S_n \ A_n is divided into three parts. This, in particular, implies that any pair of such characters χ^α and χ^β (α and β are respective partitions of a number n) possesses the following property: lengths d(α) and d(β) of principal diagonals of Young diagrams for α and β differ by at most 1. Supported by RFBR grant No. 04-01-00463 and by RFBR-NSFC grant No. 05-01-39000. __________ Translated from Algebra i Logika, Vol. 46, No. 1, pp. 3–25, January–February, 2007.     

Note on the concentration-compactness principle for generalized Moser-Trudinger inequalities

Let Ω ⊂ ℝ ⁿ , n ≥ 2, be a bounded domain and let α < n − 1. Motivated by Theorem I.6 and Remark I.18 of [Lions P.-L., The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, 1985, 1(1), 145–201] and by the results of [Černy R., Cianchi A., Hencl S., Concentration-Compactness Principle for Moser-Trudinger inequalities: new results and proofs, Ann. Mat. Pura Appl. (in press), DOI: 10.1007/s10231-011-0220-3], we give a sharp estimate of the exponent concerning the Concentration-Compactness Principle for the embedding of the Orlicz-Sobolev space W ₀¹ L ⁿ log ^α L(Ω) into the Orlicz space corresponding to a Young function that behaves like exp t ^{n/(n−1−α)} for large t. We also give the result for the case of the embedding into double and other multiple exponential spaces.     

Moments and Distribution of the Local Times of a Transient Random Walk on ℤ d

Consider an arbitrary transient random walk on ℤ ^d with d∈ℕ. Pick α∈[0,∞), and let L _n (α) be the spatial sum of the αth power of the n-step local times of the walk. Hence, L _n (0) is the range, L _n (1)=n+1, and for integers α, L _n (α) is the number of the α-fold self-intersections of the walk. We prove a strong law of large numbers for L _n (α) as n→∞. Furthermore, we identify the asymptotic law of the local time in a random site uniformly distributed over the range. These results complement and contrast analogous results for recurrent walks in two dimensions recently derived by Černy (Stoch. Proc. Appl. 117:262–270, 2007). Although these assertions are certainly known to experts, we could find no proof in the literature in this generality.      

An asymptotic property of gap series. II

An upper bound estimate in the law of the iterated logarithm for Σf(n _k ω) where n_k+1∫n_k≧ 1 + ck ^-α (α≧0) is investigated. In the case α<1/2, an upper bound had been given by Takahashi [15], and the sharpness of the bound was proved in our previous paper [8]. In this paper it is proved that the upper bound is still valid in case α≧1/2 if some additional condition on {n _k} is assumed. As an application, the law of the iterated logarithm is proved when {n _k} is the arrangement in increasing order of the set B(τ)={₁ ⁱ ₁...q_τ ⁱ _τ|i₁,...,i_τ∈N ₀}, where τ≧ 2, N ₀=NU{0}, and q ₁,...,q _τ are integers greater than 1 and relatively prime to each others. This revised version was published online in June 2006 with corrections to the Cover Date.