On Recognition by Spectrum of Finite Simple Linear Groups over Fields of Characteristic 2

A finite group G is said to be recognizable by spectrum, i.e., by the set of element orders, if every finite group H having the same spectrum as G is isomorphic to G. We prove that the simple linear groups L _n (2^k) are recognizable by spectrum for n = 2^m ≥ 32.Original Russian Text Copyright © 2005 Vasil’ev A. V. and Grechkoseeva M. A.The authors were supported by the Russian Foundation for Basic Research (Grant 05-01-00797), the State Maintenance Program for the Leading Scientific Schools of the Russian Federation (Grant NSh-2069.2003.1), the Program “ Development of the Scientific Potential of Higher School” of the Ministry for Education of the Russian Federation (Grant 8294), the Program “Universities of Russia” (Grant UR.04.01.202), and a grant of the Presidium of the Siberian Branch of the Russian Academy of Sciences (No. 86-197).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 749–758, July–August, 2005.     

A new approach to Euler splines

Starting from the exponential Euler polynomials discussed by Euler in “Institutions Calculi Differentialis,” Vol. II, 1755, the author introduced in “Linear operators and approximation,” Vol. 20, 1972, the so-called exponential Euler splines. Here we describe a new approach to these splines. Let t be a constant such that t=|t|e^iα, −π<α<π,t≠0,t≠1.. Let S₁(x:t) be the cardinal linear spline such that S₁(v:t) = t^v for all v ε Z. Starting from S₁(x:t) it is shown that we obtain all higher degree exponential Euler splines recursively by the averaging operation . Here S_n(x:t) is a cardinal spline of degree n if n is odd, while is a cardinal spline if n is even. It is shown that they have the properties S_n(v:t) = t^v for v ε Z.     

On the Behavior of a Nonstationary Poiseuille Solution as <Emphasis Type="Italic">t</Emphasis> → ∞

A nonstationary Poiseuille solution describing the flow of a viscous incompressible fluid in an infinite cylinder is defined as a solution to an inverse problem for the heat equation. The behavior as t → ∞ of the nonstationary Poiseuille solution corresponding to the prescribed flux F(t) ofthe velocity field is studied. In particular, it is proved that if the flux F(t) tends exponentially to a constant flux F _* then the nonstationary Poiseuille solution tends exponentially as t → ∞ to the stationary Poiseuille solution having the flux F _*.Original Russian Text Copyright © 2005 Pileckas K.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 890–900, July–August, 2005.     

On improved estimators of the generalized variance

Treated in this paper is the problem of estimating with squared error loss the generalized variance | Σ | from a Wishart random matrix S: p × p W_p(n, Σ) and an independent normal random matrix X: p × k N(ξ, Σ I_k) with ξ(p × k) unknown. Denote the columns of X by X₍₁₎ ,…, X_(k) and set ψ⁽⁰⁾(S, X) = {(n − p + 2)!/(n + 2)!} | S |, ψ⁽ⁱ⁾(X, X) = min[ψ⁽ⁱ⁻¹⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!} | S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |] and Ψ⁽ⁱ⁾(S, X) = min[ψ⁽⁰⁾(S, X), {(n − p + i + 2)!/(n + i + 2)!}| S + X₍₁₎ X′₍₁₎ + + X_(i) X′_(i) |], i = 1,…,k. Our result is that the minimax, best affine equivariant estimator ψ⁽⁰⁾(S, X) is dominated by each of Ψ⁽ⁱ⁾(S, X), i = 1,…,k and for every i, ψ⁽ⁱ⁾(S, X) is better than ψ⁽ⁱ⁻¹⁾(S, X). In particular, ψ^(k)(S, X) = min[{(n − p + 2)!/(n + 2)!} | S |, {(n − p + 2)!/(n + 2)!} | S + X₍₁₎X′₍₁₎|,…,| {(n − p + k + 2)!/(n + k + 2)!} | S + X₍₁₎X′₍₁₎ + + X_(k)X′_(k)|] dominates all other ψ's. It is obtained by considering a multivariate extension of Stein's result (Ann. Inst. Statist. Math. 16, 155–160 (1964)) on the estimation of the normal variance.     

Passage times of random walks and Lévy processes across power law boundaries

We establish an integral test involving only the distribution of the increments of a random walk S which determines whether limsup _n→∞(S_n/n^κ) is almost surely zero, finite or infinite when 1/2<κ<1 and a typical step in the random walk has zero mean. This completes the results of Kesten and Maller [9] concerning finiteness of one-sided passage times over power law boundaries, so that we now have quite explicit criteria for all values of κ≥0. The results, and those of [9], are also extended to Lévy processes.This work is partially supported by ARC Grant DP0210572.     

On the Self-Similar Jordan Arcs Admitting Structure Parametrization

We study the attractors γ of a finite system of contraction similarities S _j (j = 1,..., m) in ℝ^d which are Jordan arcs. We prove that if a system possesses a structure parametrization (ℐ,ϕ) and ℱ(ℐ) is the associated family of ℐ then we have one of the following cases:1. The identity mapping Id does not belong to the closure of ℱ(ℐ). Then (if properly rearranged) is a Jordan zipper.2. The identity mapping Id is a limit point of ℱ(ℐ). Then the arc γ is a straight line segment.3. The identity mapping Id is an isolated point of .We construct an example of a self-similar Jordan curve which implements the third case.Original Russian Text Copyright © 2005 Aseev V. V. and Tetenov A. V.The authors were supported by the Program “Universities of Russia” (Grant UR.04.01.456).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 733–748, July–August, 2005.     

The Degree Spectra of Definable Relations on Boolean Algebras

We study some questions concerning the structure of the spectra of the sets of atoms and atomless elements in a computable Boolean algebra. We prove that if the spectrum of the set of atoms contains a 1-low degree then it contains a computable degree. We show also that in a computable Boolean algebra of characteristic (1, 1, 0) whose set of atoms is computable the spectrum of the atomless ideal consists of all Π ₀ ² degrees.Original Russian Text Copyright © 2005 Semukhin P. M.The author was supported by the Russian Foundation for Basic Research (Grant 02-01-00593), the Leading Scientific Schools of the Russian Federation (Grant NSh-2112.2003.1), and the Program “Universities of Russia” (Grant UR.04.01.013).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 928–941, July–August, 2005.     

Oscillations of first-order neutral delay differential equations

Consider the neutral delay differential equation (*) (d/dt)[y(t) + py(t − τ)] + qy(t − σ) = 0, t t₀, where τ, q, and σ are positive constants, while p ε (−∞, −1) (0, + ∞). (For the case p ε [−1, 0] see Ladas and Sficas, Oscillations of neutral delay differential equations (to appear)). The following results are then proved. Theorem 1. Assume p < − 1. Then every nonoscillatory solution y(t) of Eq. (*) tends to ± ∞ as t → ∞. Theorem 2. Assume p < − 1, τ > σ, and q(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Theorems 3. Assume p > 0. Then every nonoscillatory solution y(t) of Eq. (*) tends to zero as t → ∞. Theorem 4. Assume p > 0. Then a necessary condition for all solutions of Eq. (*) to oscillate is that σ > τ. Theorem 5. Assume p > 0, σ > τ, andq(σ − τ)/(1 + p) > (1/e). Then every solution of Eq. (*) oscillates. Extensions of these results to equations with variable coefficients are also obtained.     

Root Decomposition of the Ternary Malcev Algebra <Emphasis Type="Italic">M</Emphasis><Subscript>8</Subscript>

A root decomposition is constructed of the simple eight-dimensional ternary Malcev algebra M ₈. In result, M ₈ is equipped with a structure of a Z ₃-graded ternary algebra.Original Russian Text Copyright © 2005 Pozhidaev A. P.The author was supported by the Russian Science Support Foundation and partially by the Russian Foundation for Basic Research (Grant 05-01-00230).__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 4, pp. 901–906, July–August, 2005.     

A construction for large sets of disjoint Kirkman triple systems

A Steiner system S(t, k, v) is called i-resolvable, 0 < i < t, if its block set can be partitioned into S(i, k, v). In this paper, a 2-resolvable S(3, 4, v) is used to construct a large set of disjoint Kirkman triple systems of order 3v − 3 (briefly LKTS) and some new orders for LKTS are then obtained. Research supported by Tianyuan Mathematics Foundation of NSFC Grant 10526032 and Natural Science Foundation of Universities of Jiangsu Province Grant 05KJB110111.     

Age-dependent branching processes in random environments

We consider an age-dependent branching process in random environments. The environments are represented by a stationary and ergodic sequence ξ = (ξ0,ξ1,...) of random variables. Given an environment ξ, the process is a non-homogenous Galton-Watson process, whose particles in n-th generation have a life length distribution G(ξn) on R , and reproduce independently new particles according to a probability law p(ξn) on N. Let Z(t) be the number of particles alive at time t. We first find a characterization of the conditional probability generating function of Z(t) (given the environment ξ) via a functional equation, and obtain a criterion for almost certain extinction of the process by comparing it with an embedded Galton-Watson process. We then get expressions of the conditional mean EξZ(t) and the global mean EZ(t), and show their exponential growth rates by studying a renewal equation in random environments.     

The Existence of Positive Solutions to Neutral Differential Equations

In this paper, we shall consider a class of neutral differential equations of the form

Full-size image (1K)

where τ (0, ∞), σ [0, ∞), Q(t) C([t₀, ∞), R ⁺ ), r(t) C([t₀, ∞), (0, ∞)) with r(t) nondecreasing on [t₀ − τ, ∞). We shall show that all positive solutions of ( * ) can be classified into four types, A, B, C, and D, and we shall obtain sufficient and necessary conditions for the existence of A-type, B-type, and D-type positive solutions of ( * ), respectively. A sufficient condition for the existence of C-type positive solutions of ( * ) is also given. Finally, we shall offer a sharp oscillation result for all solutions of ( * ). Our results generalize and improve those established in B. Yang and B. G. Zhang (Funkcial. Ekvac.39 (1996), 347–362).     

Numerical Schottky Uniformizations

A real algebraic curve of algebraic genus g ≥ 2 is a pair (S, τ), where S is a closed Riemann surface of genus g and τ is a reflection on S (anticonformal involution with fixed points). In this note, we discuss a numerical (Burnside) program which permits to obtain a Riemann period matrix of the surface S in terms of an uniformizing real Schottky group. If we denote by Aut₊(S, τ) the group of conformal automorphisms of S commuting with the real structure τ, then it is a well known fact that |Aut₊(S,τ)| ≥ 12(g−1). We say that (S,τ) is maximally symmetric if |Aut₊(S,τ)|=12(g−1). We work explicitly such a numerical program in the case of maximally symmetric real curves of genus two. We construct a real Schottky uniformization for each such real curve and we use the numerical program to obtain a real algebraic curve, a Riemann period matrix and the accessory parameters in terms of the corresponding Schottky uniformization. In particular, we are able to give for Bolza’s curve a Schottky uniformization (at least numerically), providing an example for which the inverse uniformization theorem is numerically solved.Partially supported by Projects Fondecyt 1030252 1030373 and UTFSM 12.03.21     

On Equivalence of Moduli of Smoothness

It is known that iffW^k_p, thenω_m(f, t)_ptω_m−1(f′, t)_p…. Its inverse with any constants independent offis not true in general. Hu and Yu proved that the inverse holds true for splinesSwith equally spaced knots, thusω_m(S, t)_ptω_m−1(S′, t)_pt²ω_m−2(S″, t)_p…. In this paper, we extend their results to splines with any given knot sequence, and further to principal shift-invariant spaces and wavelets under certain conditions. Applications are given at the end of the paper.     

Damped Oscillatory Integrals and the Boundedness Problem for Maximal Operators

Let S ⊂ ℝR ^{n +1} be a real-analytic hypersurface with surface measure dσ, and let ψ be a smooth nonnegative compactly supported cutoff function. Consider the surface measure dμ _q = ψ|Λ(X)|^q dσ, where Λ(X) is a damping factor determined by the matrices of the first and second fundamental forms of the surface. We show that its Fourier transform decays for large |ξ| as O (|ξ|^−(1/2+ε)), ε > 0, provided that q > 3/2. We also consider applications involving maximal operators associated with means of functions over hypersurfaces.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 70–74, 2005Original Russian Text Copyright © by I. A. Ikromov     

The Number of Steps in the Robinson-Schensted Algorithm

Suppose that a permutation σ ∈ S _n is chosen at random (n is large) and the Robinson-Schensted algorithm is applied to compute the associated Young diagram. Then for almost all permutations the number of bumping operations performed by the algorithm is about (128/27π²)n ^3/2, and the number of comparison operations is about (64/27π²)n ^3/2 log₂ n.__________Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 39, No. 2, pp. 82–86, 2005Original Russian Text Copyright © by D. Romik     

On the Rate of Almost Everywhere Convergence of Certain Classical Integral Means

The main purpose of this article is to establish nearly optimal results concerning the rate of almost everywhere convergence of the Gauss–Weierstrass, Abel–Poisson, and Bochner–Riesz means of the one-dimensional Fourier integral. A typical result for these means is the following: If the function f belongs to the Besov spaceB^s_{p, p}, 1<p<∞, 0<s<1, thenT_mtf (x)−f(x)=o_x(t^s) a.e. ast→0+.     

Best Polynomial Approximations in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript> and Widths of Some Classes of Functions

We obtain the exact values of extremal characteristics of a special form that connect the best polynomial approximations of functions f(x) ∈ L ₂ ^r (r ∈ ℤ₊) and expressions containing moduli of continuity of the kth order ω_k(f(^r), t). Using these exact values, we generalize the Taikov result for inequalities that connect the best polynomial approximations and moduli of continuity of functions from L ₂. For the classes (k, r, Ψ_*) defined by ω _k(f ^(r), t) and the majorant , we determine the exact values of different widths in the space L₂.__________Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 11, pp. 1458–1466, November, 2004.     

On stochastic integral representation of stable processes with sample paths in Banach spaces

Certain path properties of a symmetric α-stable process X(t) = ∫_Sh(t, s) dM(s), t T, are studied in terms of the kernel h. The existence of an appropriate modification of the kernel h enables one to use results from stable measures on Banach spaces in studying X. Bounds for the moments of the norm of sample paths of X are obtained. This yields definite bounds for the moments of a double α-stable integral. Also, necessary and sufficient conditions for the absolute continuity of sample paths of X are given. Along with the above stochastic integral representation of stable processes, the representation of stable random vectors due to[13], Ann. Probab.9, 624–632) is extensively used and the relationship between these two representations is discussed.     

On deviations between empirical and quantile processes for mixing random variables

Let {X_n} be a strictly stationary φ-mixing process with Σ_j=1^∞ φ^1/2(j) < ∞. It is shown in the paper that if X₁ is uniformly distributed on the unit interval, then, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = (O(n^−3/4(log n)^1/2(log log n)^1/4) a.s., where F_n and F_n⁻¹(t) denote the sample distribution function and tth sample quantile, respectively. In case {X_n} is strong mixing with exponentially decaying mixing coefficients, it is shown that, for any t [0, 1], |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)^1/2(log log n)^3/4) a.s. and sup_0≤t≤1 |F_n⁻¹(t) − t + F_n(t) − t| = O(n^−3/4(log n)(log log n)^1/4) a.s. The results are further extended to general distributions, including some nonregular cases, when the underlying distribution function is not differentiable. The results for φ-mixing processes give the sharpest possible orders in view of the corresponding results of Kiefer for independent random variables.