A limit theorem for random coverings of a circle

LetN _{α, m} equal the number of randomly placed arcs of length α (0<α<1) required to cover a circleC of unit circumferencem times. We prove that lim_α→0 P(N_α,m≦(1/α) (log (1/α)+mlog log(1/α)+x)=exp ((−1/(m−1)!) exp (−x)). Using this result for m=1, we obtain another derivation of Steutel's resultE(N_α,1)=(1/α) (log(1/α)+log log(1/α)+γ+o(1)) as α→0, γ denoting Euler's constant.     

Rauzy tilings and bounded remainder sets on the torus

For the two-dimensional torus , we construct the Rauzy tilings d⁰ ⊃ d¹ ⊃ … ⊃ d^m ⊃ …, where each tiling d^m+1 is obtained by subdividing the tiles of d^m. The following results are proved. Any tiling d^m is invariant with respect to the torus shift S(x) = x+ mod ℤ², where ζ⁻¹ > 1 is the Pisot number satisfying the equation x³− x²−x-¹ = 0. The induced map is an exchange transformation of B^md ⊂ , where d = d⁰ and . The map S^(m) is a shift of the torus , which is affinely isomorphic to the original shift S. This means that the tilings d^m are infinitely differentiable. If Z_N(X) denotes the number of points in the orbit S¹(0), S²(0), …, S^N(0) belonging to the domain B^md, then, for all m, the remainder r_N(B^md) = Z_N(B^md) − N ζ^m satisfies the bounds −1.7 < r_N(B^md) < 0.5. Bibliography: 10 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 322, 2005, pp. 83–106.     

Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid

Let _N+2m ={−m, −m+1, …, −1, 0, 1, …,N−1,N, …,N−1+m}. The present paper is devoted to the approximation of discrete functions of the formf : _N+2m → ℝ by algebraic polynomials on the grid Ω _N ={0, 1, …,N−1}. On the basis of two systems of Chebyshev polynomials orthogonal on the sets Ω _N+m and Ω _N , respectively, we construct a linear operatorY _{n+2m, N} =Y _{n+2m, N} (f), acting in the space of discrete functions as an algebraic polynomial of degree at mostn+2m for which the following estimate holds (x ε Ω _N ):

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Concentration and influences

Consider the discrete cube Ω={0,1} ^N , provided with the uniform probabilityP. We denote byd(x, A) the Hamming distance of a pointx of Ω and a subsetA of Ω. We define the influenceI(A) of theith coordinate onA as follows. Forx in Ω, consider the pointT _i (x) obtained by changing the value of theith coordinate. Then We prove that we always have Since it is easy to see that , this recovers the well known fact that ∫_Ω d(x, A)dP(x) is at most of order whenP(A)≥1/2. The new information is that ∫_Ω d(x, A)dP(x) can be of order only ifA reassembles the Hamming ball {x; ∑_1≤N x _i ≥N/2}.     

Problem with periodicity conditions for a degenerating equation of mixed type

For the equation K(t)u _xx + u _tt − b ² K(t)u = 0 in the rectangular domain D = “(x, t)‖ 0 < x < 1, −α < t < β”, where K(t) = (sgnt)|t| ^m , m > 0, and b > 0, α > 0, and β > 0 are given real numbers, we use the spectral method to obtain necessary and sufficient conditions for the unique solvability of the boundary value problem u(0, t) = u(1, t), u _x (0, t) = u _x (1, t), −α ≤ t ≤ β, u(x, β) = φ(x), u(x,−α) = ψ(x), 0 ≤ x ≤ 1.     

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.     

Random matrix products and applications to cellular automata

Let λ be the upper Lyapunov exponent corresponding to a product of i.i.d. randomm×m matrices (X _i) _i ^0/∞ over ℂ. Assume that theX _i's are chosen from a finite set {D ₀,D ₁...,D _t-1(ℂ), withP(X _i=D_j)>0, and that the monoid generated byD ₀, D₁,…, D_q−1 contains a matrix of rank 1. We obtain an explicit formula for λ as a sum of a convergent series. We also consider the case where theX _i's are chosen according to a Markov process and thus generalize a result of Lima and Rahibe [22]. Our results on λ enable us to provide an approximation for the numberN _≠0(F(x)ⁿ,r) of nonzero coefficients inF(x) ⁿ.(modr), whereF(x) ∈ ℤ[x] andr≥2. We prove the existence of and supply a formula for a constant α (<1) such thatN _≠0(F(x)ⁿ,r) ≈n ^α for “almost” everyn. Supported in part by FWF Project P16004-N05     

Distance-Regular Graph with <Emphasis Type="Italic">c</Emphasis><Subscript>2</Subscript> > 1 and <Emphasis Type="Italic">a</Emphasis><Subscript>1</Subscript> = 0 < <Emphasis Type="Italic">a</Emphasis><Subscript>2</Subscript>

Let Γ be a distance-regular graph of diameter d ≥ 3 with c ₂ > 1. Let m be an integer with 1 ≤ m ≤ d − 1. We consider the following conditions:

On a class of nonlinear problems involving a p(x)-Laplace type operator

We study the boundary value problem in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ ^N . Our attention is focused on two cases when , where m(x) = max{p ₁(x), p ₂(x)} for any x ∈ or m(x) < q(x) < N · m(x)/(N − m(x)) for any x ∈ . In the former case we show the existence of infinitely many weak solutions for any λ > 0. In the latter we prove that if λ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ℤ₂-symmetric version for even functionals of the Mountain Pass Theorem and some adequate variational methods.     

On the Spectral Function of Some Higher Order Elliptic or Degenerate-elliptic Operators

^m σ/|α|=|β|=m D^α(a _αβD^β) is a higher order degenerate-elliptic operator on L²(R^N) and V ∈ L¹(R^N) we show in particular that s(λ) > 0 for all λ > 0 provided that ∫_RN V(x)dx≥ 0, V not equivalent to 0 and N ≤ 2m.     

On pair correlations and Hausdorff dimension

We consider a system of “generalised linear forms” defined at a point x = (x _{(i, j)}) in a subset of R ^d by

Discrete spectrum in the gaps of a perturbed pseudorelativistic hamiltonian

The pseudorelativistic Hamiltonian is considered under wide conditions on potentials A(x), W(x). It is assumed that a real point λ is regular for G_1/2. Let G_1/2(α)=G_1/2−αV, where α>0, V(x)≥0, and V ∈L _d(ℝ^d). Denote by N(λ, α) the number of eigenvalues of G_1/2(t) that cross the point λ as t increases from 0 to α. A Weyl-type asymptotics is obtained for N(λ, α) as α→∞. Bibliography: 5 titles. To O. A. Ladyzhenskaya Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 249, 1997. pp. 102–117. Translated by A. B. Pushnitskii.     

Center conditions III: Parametric and model center problems

We consider an Abel equation (*)y’=p(x)y ² +q(x)y ³ withp(x), q(x) polynomials inx. A center condition for (*) (closely related to the classical center condition for polynomial vector fields on the plane) is thaty ₀=y(0)≡y(1) for any solutiony(x) of (*). Folowing [7], we consider a parametric version of this condition: an equation (**)y’=p(x)y ² +εq(x)y ³ p, q as above, ε ∈ ℂ, is said to have a parametric center, if for any ɛ and for any solutiony(ɛ,x) of (**)y(ɛ, 0)≡y(ɛ, 1).. We give another proof of the fact, shown in [6], that the parametric center condition implies vanishing of all the momentsm _k (1), wherem _k (x)=∫ ₀ ^{x pk} (t)q(t)(dt),P(x)=∫ ₀ ^x p(t)dt. We investigate the structure of zeroes ofm _k (x) and generalize a “canonical representation” ofm _k (x) given in [7]. On this base we prove in some additional cases a composition conjecture, stated in [6, 7] for a parametric center problem. The research of the first and the third author was supported by the Israel Science Foundation, Grant No. 101/95-1 and by the Minerva Foundation.     

An estimator of the tail index based on increment ratio statistics

In this paper, we introduce an increment ratio statistic (IR ^N,m ) based estimator for estimation of the tail index of a heavy-tailed distribution. For i.i.d. observations depending on the zone of attraction of an α-stable law (0 < α < 2), the IR ^N,m statistic converges to a decreasing function L(α) as both the sample size N and bandwidth parameter m tend to infinity. We obtain a rate of decay of the bias EIR ^N,m −L(α) and mean square error E(IR ^N,m −L(α))². A central limit theorem (IR ^N,m −EIR ^N,m )⟹ (0,σ²(α)) is also obtained. Monte Carlo simulations show that our tail index estimator has quite good empirical mean square error and, unlike the Hill estimator, is not so sensitive to a change of bandwidth parameter m. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-25/08.     

Weighted <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> boundedness for multilinear fractional integral on product spaces

For 0 < α < mn and nonnegative integers n ≥ 2, m ≥ 1, the multilinear fractional integral is defined by

On the Problem of Parameter Estimation in Exponential Sums

Let I≥1 be an integer, ω ₀=0<ω ₁<⋯<ω _I ≤π, and for j=0,…,I, a _j ∈ℂ, a_-j=[`(a_j)]a_{-j}={\overline{{a_{j}}}}, ω _−j =−ω _j , and a_j 1 0a_{j}\not=0 if j 1 0j\not=0. We consider the following problem: Given finitely many noisy samples of an exponential sum of the form

Existence of Positive Solutions for Singular Second-order<Emphasis Type="Italic">m</Emphasis>-Point Boundary Value Problems

Abstract   The singular second-order m-point boundary value problem

Irreducibility criteria of Schur-type and Pólya-type

Let f(x)=(x-a₁)?(x-a_m){f(x)=(x-a_1)\cdots (x-a_m)}, where a ₁, . . . , a _m are distinct rational integers. In 1908 Schur raised the question whether f(x) ± 1 is irreducible over the rationals. One year later he asked whether (f(x))^2k+1{(f(x))^{2^k}+1} is irreducible for every k ≥ 1. In 1919 Pólya proved that if P(x) ? \mathbbZ[x]{P(x)\in\mathbb{Z}[x]} is of degree m and there are m rational integer values a for which 0 < |P(a)| < 2^−N N! where N=ém/2ù{N=\lceil m/2\rceil}, then P(x) is irreducible. A great number of authors have published results of Schur-type or Pólya-type afterwards. Our paper contains various extensions, generalizations and improvements of results from the literature. To indicate some of them, in Theorem 3.1 a Pólya-type result is established when the ground ring is the ring of integers of an arbitrary imaginary quadratic number field. In Theorem 4.1 we describe the form of the factors of polynomials of the shape h(x) f(x) + c, where h(x) is a polynomial and c is a constant such that |c| is small with respect to the degree of h(x) f(x). We obtain irreducibility results for polynomials of the form g(f(x)) where g(x) is a monic irreducible polynomial of degree ≤ 3 or of CM-type. Besides elementary arguments we apply methods and results from algebraic number theory, interpolation theory and diophantine approximation.     

Nonlocal initial-boundary-value problems for a degenerate hyperbolic equation

We consider the equation y ^m u _xx − u _yy − b ² y ^m u = 0 in the rectangular area {(x, y) | 0 < x < 1, 0 < y < T}, where m < 0, b ≥ 0, T > 0 are given real numbers. For this equation we study problems with initial conditions u(x, 0) = τ(x), u _y (x, 0) = ν(x), 0 ≤ x ≤ 1, and nonlocal boundary conditions u(0, y) = u(1, y), u _x (0, y) = 0 or u _x (0, y) = u _x (1, y), u(1, y) = 0 with 0≤y≤T. Using the method of spectral analysis, we prove the uniqueness and existence theorems for solutions to these problems