<Emphasis Type="Italic">L</Emphasis>-Approximation of a linear combination of the poisson kernel and its conjugate kernel by trigonometric polynomials

A linear combination Π _q,α = cos(απ/2)P + sin(απ/2)Q of the Poisson kernel P(t) = 1/2 + q cos t + q ² cos 2t + ... and its conjugate kernel Q(t) = q sin t + q ² sin 2t + ... is considered for α ∈ ? and |q| < 1. A new explicit formula is found for the value E _n?1(Π _q,α ) of the best approximation in the space L = L _2π of the function Π _q,α by the subspace of trigonometric polynomials of order at most n ? 1. More exactly, it is proved that \(E_{n - 1} \left( {\prod _{q,\alpha } } \right) = \left. {\frac{{\left| q \right|^n \left( {1 - q^2 } \right)}}{{1 - q^{4n} }}} \right\|\left. {\frac{{\cos \left( {nt - {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right) - q^{2n} \cos \left( {nt + {{\alpha \pi } \mathord{\left/ {\vphantom {{\alpha \pi } 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}{{1 - q^2 - 2q \cos t}}} \right\|_L\). In addition, the value E _n?1(Π _q,α ) is represented as a rapidly convergent series.     

On solutions of three-dimensional systems describing the transition from an unstable equilibrium to a stable cycle

Given a three-dimensional dynamical system on the interval t ₀ < t < +∞, the transition from the neighborhood of an unstable equilibrium to a stable limit cycle is studied. In the neighbor-hood of the equilibrium, the system is reduced to a normal form. The matrix of the linearized system is assumed to have a complex eigenvalue λ = ? + iβ, with β ? ? > 0 and a real eigenvalue with δ < 0 with |δ| ? ?. On the arbitrary interval [t ₀, +∞), an approximate solution is sought as a polynomial P _N (?) in powers of the small parameter with coefficients from Hölder function spaces. It is proved that there exist ? _N and C _N depending on the initial data such that, for 0 < ? < ? _N , the difference between the exact and approximate solutions does not exceed C _N ? ^N+1.     

Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Let X ₁,..., X _n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H ₀: X ₁,..., X _n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H ₀ versus the “at most one change (AMOC) in the mean” alternative hypothesis H _A: there is an integer k*, 1 ≤ k* < n, such that EX ₁ = · · · = EX_k* ≠ EX_k*+1 = ··· = EX _n. A natural way of testing for H ₀ versus H _A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ_k = |S _k/k - (S _n - S _k)/(n - k)|, where S _k:= X ₁ + ··· + X _k. Asymptotic distributions of these test statistics under H ₀ as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z _n(t)|, where {Z _n(t), 0 ≤ t ≤ 1}_n≥1 are the tied-down partial sums processes such that

$${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$

if 0 ≤ t < 1, and Z _n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H ₀ versus H _A via sup-functionals of appropriately weighted |Z _n(t)|. Simply considering max_1?k<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z _n(t)|/(t(1 - t)) under H ₀ and assuming also that E|X|^r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z _n(t)| has not been considered before.

     

Kazhdan-Lusztig cells in some weighted Coxeter groups

Let(W,S) be a Coxeter group with S = I■J such that J consists of all universal elements of S and that I generates a finite parabolic subgroup W_I of W with w_0 the longest element of W_I. We describe all the left cells and two-sided cells of the weighted Coxeter group(W,S,L) that have non-empty intersection with W_J,where the weight function L of(W, S) is in one of the following cases:(i) max{L(s) | s ∈J} min{L(t)|t∈I};(ii) min{L(s)|s ∈J} ≥L(w_0);(iii) there exists some t ∈ I satisfying L(t) L(s) for any s ∈I-{t} and L takes a constant value L_J on J with L_J in some subintervals of [1, L(w_0)-1]. The results in the case(iii) are obtained under a certain assumption on(W, W_I).     

Integral averaging technique for oscillation of damped half-linear oscillators

This paper is concerned with the oscillatory behavior of the damped half-linear oscillator (a(t)?_p(x′))′ + b(t)?_p(x′) + c(t)?_p(x) = 0, where ?p(x) = |x|^p?1 sgn x for x ∈ ? and p > 1. A sufficient condition is established for oscillation of all nontrivial solutions of the damped half-linear oscillator under the integral averaging conditions. The main result can be given by using a generalized Young’s inequality and the Riccati type technique. Some examples are included to illustrate the result. Especially, an example which asserts that all nontrivial solutions are oscillatory if and only if p ≠ 2 is presented.     

The Joint Distribution of Running Maximum of a Slepian Process

Consider the Slepian process S defined by S(t) = B(t +?1) ? B(t),t ∈ [0, 1] with B(t), t ∈ ? a standard Brownian motion. In this contribution we analyze the properties between the maximum \(m_{s}=\max \limits _{0\leq u\leq s}S(u)\) and the maximum \(m_{t}=\max \limits _{0\leq u\leq t}S(u)\) for 0 ≤ s < t ≤?1 fixed. Explicit integral expressions are obtained for the joint distribution function between m _s and m _t and the distribution function of the partial maximum m _s . Further, we apply our results for the determination of the moments of m _s .     

Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone

In this paper, we study the initial-boundary value problem of porous medium equation ρ(x)u _t  = Δu ^m  + V(x)h(t)u ^p in a cone D = (0, ∞) × Ω, where \({V(x)\,{\sim}\, |x|^\sigma, h(t)\,{\sim}\, t^s}\). Let ω ₁ denote the smallest Dirichlet eigenvalue for the Laplace-Beltrami operator on Ω and let l denote the positive root of l ² + (n ? 2)l = ω ₁. We prove that if \({m < p \leq 1+(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has no global nonnegative solutions for any nonnegative u ₀ unless u ₀ = 0; if \({p >1 +(m-1)(1+s)+\frac{2(s+1)+\sigma}{n+l}}\), then the problem has global solutions for some u ₀ ≥ 0.     

On a result by geronimus

In 1935, Ya.L. Geronimus found the best integral approximation on the period [?π,π) of the function sin(n + 1)t ? 2q sin nt, q ∈ ?, by the subspace of trigonometric polynomials of degree at most n ? 1. This result is an integral analog of the known theorem by E.I. Zolotarev (1868). At present, there are several methods of proving this fact. We propose one more variant of the proof. In the case |q| ≥ 1, we apply the (2π/n)-periodization and the fact that the function | sin nt| is orthogonal to the harmonic cos t on the period. In the case |q| < 1, we use the duality relations for Chebyshev’s theorem (1859) on a rational function least deviating from zero on a closed interval with respect to the uniform metric.     

On a question of Sárközy on gaps of product sequences

Motivated by a question of Sárközy, we study the gaps in the product sequence B = A · A = {b ₁ < b ₂ < …} of all products a _i a _j with a _i , a _j ∈ A when A has upper Banach density α > 0. We prove that there are infinitely many gaps b _n+1 ? b _n ? α ^?3 and that for t ≥ 2 there are infinitely many t-gaps b _n+t ? b _n ? t ² α ^?4. Furthermore, we prove that these estimates are best possible.We also discuss a related question about the cardinality of the quotient set A/A = {a _i /a _j , a _i , a _j ∈ A} when A ? {1, …, N} and |A| = αN.     

Inverse problem with nonlocal observation of finding the coefficient multiplying <Emphasis Type="Italic">u</Emphasis> <Subscript> <Emphasis Type="Italic">t</Emphasis> </Subscript> in the parabolic equation

We study the inverse problem of the reconstruction of the coefficient ?(x, t) = ?⁰(x, t) + r(x) multiplying u_t in a nonstationary parabolic equation. Here ?⁰(x, t) ≥ ?⁰ > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L_∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫₀^Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.     

Diophantine inequality involving binary forms

Let d ? 3 be an integer, and set r = 2^d?1 + 1 for 3 ? d ? 4, \(\tfrac{{17}}{{32}} \cdot 2^d + 1\) for 5 ? d ? 6, r = d²+d+1 for 7 ? d ? 8, and r = d²+d+2 for d ? 9, respectively. Suppose that Φ _i (x, y) ∈ ?[x, y] (1 ? i ? r) are homogeneous and nondegenerate binary forms of degree d. Suppose further that λ₁, λ₂,..., λ _r are nonzero real numbers with λ₁/λ₂ irrational, and λ₁Φ₁(x₁, y₁) + λ₂Φ₂(x₂, y₂) + · · · + λ _r Φ _r (x _r , y _r ) is indefinite. Then for any given real η and σ with 0 < σ < 2^2?d, it is proved that the inequality

$$\left| {\sum\limits_{i = 1}^r {{\lambda _i}\Phi {}_i\left( {{x_i},{y_i}} \right) + \eta } } \right| < {\left( {\mathop {\max \left\{ {\left| {{x_i}} \right|,\left| {{y_i}} \right|} \right\}}\limits_{1 \leqslant i \leqslant r} } \right)^{ - \sigma }}$$

has infinitely many solutions in integers x₁, x₂,..., x _r , y₁, y₂,..., y _r . This result constitutes an improvement upon that of B. Q. Xue.

     

Connecting orbits of Lagrangian systems in a nonstationary force field

We study connecting orbits of a natural Lagrangian system defined on a complete Riemannian manifold subjected to the action of a nonstationary force field with potential U(q, t) = f(t)V(q). It is assumed that the factor f(t) tends to ∞ as t→±∞ and vanishes at a unique point t ₀ ∈ ?. Let X ₊, X _? denote the sets of isolated critical points of V (x) at which U(x, t) as a function of x distinguishes its maximum for any fixed t > t ₀ and t < t ₀, respectively. Under nondegeneracy conditions on points of X _± we prove the existence of infinitely many doubly asymptotic trajectories connecting X _? and X ₊.     

On the number of nonstationary bounded trajectories of a class of autonomous systems on the plane

We study the number of nonstationary bounded trajectories of autonomous systems of the form z′ = \(\overline {P_n (z)} \), z = x + iy ∈ C, where P _n (z) is a polynomial of degree n with complex coefficients that has k distinct roots, n, k > 1. We prove that the number N of nonstationary bounded trajectories of this system satisfies the following assertions (Theorem 1): (a) N = n + k ? N ₊, N ₊ = N _?, n + 1 ≤ N ₊ ≤ n + k, where N ₊ and N _? are the numbers of system trajectories unbounded as t → +∞ and t → ?∞, respectively; (b) if some r distinct roots \(c_{j_1 } \), ..., \(c_{j_r } \) of the polynomial P _n satisfy the relations V _n+1 (\(c_{j_1 } \)) = ··· = V _n+1 (\(c_{j_r } \)), where V _n+1 is the imaginary part of the indeterminate integral of P _n , then N ≥ \(m_{j_1 } \) + ··· + \(m_{j_r } \) + r ? n ? 1; (c) if k = 2, then the conditions N = 1 and V _n+1 (c ₁) = V _n+1 (c ₂) are equivalent. For n = k = 3, we derive a formula for the number of nonstationary bounded trajectories (Theorem 2).     

On decompositions of trigonometric polynomials

Let R _t [θ] be the ring generated over R by cosθ and sinθ, and R _t (θ) be its quotient field. In this paper we study the ways in which an element p of R _t [θ] can be decomposed into a composition of functions of the form p = R ? q, where R ∈ R(x) and q ∈ R _t (θ). In particular, we describe all possible solutions of the functional equation R ₁ ? q ₁ = R ₂ ? q ₂, where R ₁,R ₂ ∈ R[x] and q ₁, q ₂ ∈ R _t [θ].     

Turán-Type Results for Complete <Emphasis Type="Italic">h</Emphasis>-Partite Graphs in Comparability and Incomparability Graphs

We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <₁,...,< _r be partial orders on P. Let G(P, <₁,...,< _r ) be the graph whose vertices are the elements of P, and x, y ∈ P are joined by an edge if x< _i y or y< _i x holds for some 1 ≤ i ≤ r. We show that if the edge density of G(P, <₁, ... , < _r ) is strictly larger than 1 ? 1/(2h ? 2) ^r , then P contains h disjoint sets A ₁, ... , A _h such that A ₁ < _j ... < _j A _h holds for some 1 ≤ j ≤ r, and |A ₁| = ... = |A _h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 ? 1/(3h ? 3), then P contains h disjoint sets A ₁, ... , A _h such that the elements of A _i are incomparable with the elements of A _j for 1 ≤ i < j ≤ h, and |A ₁| = ... = |A _h | = |P|^1?o(1). Finally, we prove that if the edge density of the complement of G(P, <₁, <₂) is α, then there are disjoint sets A, B ? P such that any element of A is incomparable with any element of B in both <₁ and <₂, and |A| = |B| > n ^1?γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.     

Conditions for the existence of a classical solution of a cauchy type problem for the diffusion equation with a Riemann-Liouville partial derivative

We study a Cauchy type problem for a differential equation containing a fractional Riemann-Liouville partial derivative of order α, 0 < α < 2. Conditions under which the solution of the problem tends to zero as |x| → ∞ are obtained. We prove an existence theorem for a classical solution of the Cauchy type problem and show that the solution has a singularity as t → 0 of order 1 ? α if 0 < α ≤ 1 and of order 2 ? α if 1 < α < 2.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

Stabilization by output of continuous and pulse-modulated uncertain systems

The system ? _i = ? _i (?) + x _i+2, \(i \in \overline {1,n - 2} \), ? _n?1 = ? _n?1(?) + u ₁, ? _n = ? _n (?) + u ₂,where ? _i (·) are nonanticipating functionals of an arbitrary nature with the following properties—\(\left| {{\varphi _i}\left( \cdot \right)} \right| \leqslant c\sum\nolimits_{k = 1}^i {\left| {{x_k}\left( t \right)} \right|} \), \(i \in \overline {1,n} \), c = const—and u ₁ and u ₂ are the controls is considered. It is assumed that only the outputs x ₁ and x ₂ are measurable. The problem of synthesis of both continuous and impulsive controls u1 and u2, which make the system globally asymptotically stable, is solved. The solution of the problem is based on the construction of the observer-based equations, the quadratic Lyapunov function, and the averaging method.     

Systems that generate solutions with a small period

Let (j₁,..., j_n) be a permutation of the n-tuple (1, ..., n). A system of differential equations \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) in which each function f_i is continuous on ? is considered. This system is said to have the property of generation of solutions with a small period if, for any number M > 0, there exists a number ω₀ = ω₀(M) > 0 such that if 0 < ω ≤ ω₀ and h_i(t, x₁, ..., x_n) are continuous functions on ? × ?ⁿ ω-periodic in t that satisfy the inequalities |h_i| ≤ M the system \(\dot x = {f_i}\left( {{x_{{j_i}}}} \right),i = 1, \ldots ,n\) has an ω-periodic solution. It is shown that a system has the property of generation of solutions with a small period if and only if f_i(?) = ? for i = 1,..., n. It is also shown that the smallness condition on the period is essential.     

On the problem of determining the structure of a layered medium and the shape of an impulse source

For the equation of wave propagation in the half-space ? ₊ ² + = {(x, y) ∈ ?² | y > 0} we consider the problem of determining the speed of wave propagation that depends only on the variable y and the shape of a point impulse source on the boundary of the half-space. We show that, under some assumptions on the shape of the source and the structure of the medium, both unknown functions of one variable are uniquely determined by the displacements of boundary points of the medium. We estimate stability of a solution to the problem.