Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ?², where ? takes arbitrary values in the interval (0, 1]. When ? vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ? tends to zero, a parabolic boundary layer with a characteristic width ? appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ?-uniformly at a rate of O(N ^?2ln² N + N ₀ ^?1 , where \(N = \mathop {\min }\limits_s N_s \), N ^s + 1 and N ₀ + 1 are the numbers of mesh points on the axes x _s and t, respectively.     

The Fourier–Faber–Schauder Series Unconditionally Divergent in Measure

We prove that, for every ε ∈ (0, 1), there is a measurable set E ? [0, 1] whose measure |E| satisfies the estimate |E| > 1?ε and, for every function f ∈ C_[0,1], there is ? f ∈ C_[0,1] coinciding with f on E whose expansion in the Faber–Schauder system diverges in measure after a rearrangement.     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

The Exact Value of Normal Structure Coefficients and WCS coefficients in a Class of Orlicz Function Spaces

Let φ be an N-function. Then the normal structure coefficients N and the weakly convergent sequence coefficients WCS of the Orlicz function spaces L ^φ[0, 1] generated by φ and equipped with the Luxemburg and Orlicz norms have the following exact values. (i) If F _φ(t) = t _?(t)/φ(t) is decreasing and 1 < C _φ < 2 (where \(C_\Phi = \lim _{t \to + \infty } t\varphi (t)/\Phi (t)\)), then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1?1/Cφ. (ii) If F _φ(t) is increasing and C _φ > 2, then N(L ^(φ)[0, 1]) = N(L ^φ[0, 1]) = WCS(L ^(φ)[0, 1]) = WCS(L ^φ[0, 1]) = 2^1/Cφ.     

On the limiting behavior of the characteristic function of the ergodic distribution of the semi-Markov walk with two boundaries

The semi-Markov walk (X(t)) with two boundaries at the levels 0 and β > 0 is considered. The characteristic function of the ergodic distribution of the processX(t) is expressed in terms of the characteristics of the boundary functionals N(z) and S _N(z), where N(z) is the firstmoment of exit of the random walk {Sn}, n ≥ 1, from the interval (?z, β ? z), z ∈ [0, β]. The limiting behavior of the characteristic function of the ergodic distribution of the process W _β (t) = 2X(t)/β ? 1 as β → ∞ is studied for the case in which the components of the walk (η _i) have a two-sided exponential distribution.     

Two-Sided Estimates of Fourier Sums Lebesgue Functions with Respect to Polynomials Orthogonal on Nonuniform Grids

Let Ω = {t₀, t₁, …, t_N} and Ω_N = {x₀, x₁, …, x_N–1}, where x_j = (t_j + t_{j + 1})/2, j = 0, 1, …, N–1 be arbitrary systems of distinct points of the segment [–1, 1]. For each function f(x) continuous on the segment [–1, 1], we construct discrete Fourier sums S_{n, N}( f, x) with respect to the system of polynomials {p?_k,N(x)} _k=0 ^N–1 , forming an orthonormal system on nonuniform point systems Ω_N consisting of finite number N of points from the segment [–1, 1] with weight Δt_j = t_{j + 1}–t_j. We find the growth order for the Lebesgue function L_n,N (x) of the considered partial discrete Fourier sums S_n,N ( f, x) as n = O(δ _N ^?2/7 ), δ_N = max_{0≤ j≤N?1} Δt_j More exactly, we have a two-sided pointwise estimate for the Lebesgue function L_{n, N}(x), depending on n and the position of the point x from [–1, 1].     

Relationships between generalized Wiener integrals and conditional analytic Feynman integrals over continuous paths

Let C[0, t] denote a generalized Wiener space, the space of real-valued continuous functions on the interval [0, t], and define a random vector Z _n: C[0, t] → R ⁿ⁺¹ by \({Z_n}\left( x \right) = \left( {x\left( 0 \right) + a\left( 0 \right),\int_o^{{t_1}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_1}} \right),...,\int_0^{{t_n}} {h\left( s \right)dx\left( s \right) + x\left( 0 \right) + a\left( {{t_n}} \right)} } } \right)\), where a ∈ C[0, t], h ∈ L ₂[0, t], and 0 < t ₁ <... < t _n ≤ t is a partition of [0, t]. Using simple formulas for generalized conditional Wiener integrals, given Z _n we will evaluate the generalized analytic conditional Wiener and Feynman integrals of the functions F in a Banach algebra which corresponds to Cameron-Storvick’s Banach algebra S. Finally, we express the generalized analytic conditional Feynman integral of F as a limit of the non-conditional generalized Wiener integral of a polygonal function using a change of scale transformation for which a normal density is the kernel. This result extends the existing change of scale formulas on the classical Wiener space, abstract Wiener space and the analogue of the Wiener space C[0, t].     

Extracting outer function part from Hardy space function

Any analytic signal fa(e~(it)) can be written as a product of its minimum-phase signal part(the outer function part) and its all-phase signal part(the inner function part). Due to the importance of such decomposition, Kumarasan and Rao(1999), implementing the idea of the Szeg?o limit theorem(see below),proposed an algorithm to obtain approximations of the minimum-phase signal of a polynomial analytic signal fa(e~(it)) = e~(iN0t)M∑k=0a_k~(eikt),(0.1)where a_0≠ 0, a_M≠ 0. Their method involves minimizing the energy E(f_a, h_1, h_2,..., h_H) =1/(2π)∫_0~(2π)|1+H∑k=1h_k~(eikt)|~2|fa(e~(it))|~2dt(0.2) with the undetermined complex numbers hk's by the least mean square error method. In the limiting procedure H →∞, one obtains approximate solutions of the minimum-phase signal. What is achieved in the present paper is two-fold. On one hand, we rigorously prove that, if fa(e~(it)) is a polynomial analytic signal as given in(0.1),then for any integer H≥M, and with |fa(e~(it))|~2 in the integrand part of(0.2) being replaced with 1/|fa(e~(it))|~2,the exact solution of the minimum-phase signal of fa(e~(it)) can be extracted out. On the other hand, we show that the Fourier system e~(ikt) used in the above process may be replaced with the Takenaka-Malmquist(TM) system, r_k(e~(it)) :=((1-|α_k|~2e~(it))/(1-α_ke~(it))~(1/2)∏_(j=1)~(k-1)(e~(it)-α_j/(1-α_je~(it))~(1/2), k = 1, 2,..., r_0(e~(it)) = 1, i.e., the least mean square error method based on the TM system can also be used to extract out approximate solutions of minimum-phase signals for any functions f_a in the Hardy space. The advantage of the TM system method is that the parameters α_1,..., α_n,...determining the system can be adaptively selected in order to increase computational efficiency. In particular,adopting the n-best rational(Blaschke form) approximation selection for the n-tuple {α_1,..., α_n}, n≥N, where N is the degree of the given rational analytic signal, the minimum-phase part of a rational analytic signal can be accurately and efficiently extracted out.     

Difference scheme for an initial–boundary value problem for a singularly perturbed transport equation

An initial–boundary value problem for a singularly perturbed transport equation with a perturbation parameter ε multiplying the spatial derivative is considered on the set ? = G ∪ S, where ? = D? × [0 ≤ t ≤ T], D? = {0 ≤ x ≤ d}, S = S ^l ∪ S, and S ^l and S₀ are the lateral and lower boundaries. The parameter ε takes arbitrary values from the half-open interval (0,1]. In contrast to the well-known problem for the regular transport equation, for small values of ε, this problem involves a boundary layer of width O(ε) appearing in the neighborhood of S ^l ; in the layer, the solution of the problem varies by a finite value. For this singularly perturbed problem, the solution of a standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm. Convergence occurs only if h=dN^-1 ? ε and N₀^-1 ? 1, where N and N₀ are the numbers of grid intervals in x and t, respectively, and h is the mesh size in x. The solution of the considered problem is decomposed into the sum of regular and singular components. With the behavior of the singular component taken into account, a special difference scheme is constructed on a Shishkin mesh, i.e., on a mesh that is piecewise uniform in x and uniform in t. On such a grid, a monotone difference scheme for the initial–boundary value problem for the singularly perturbed transport equation converges ε-uniformly in the maximum norm at an ?(N^?1 + N₀^?1) rate.     

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.     

Weakly nonlinear Schrödinger equation with random initial data

It is common practice to approximate a weakly nonlinear wave equation through a kinetic transport equation, thus raising the issue of controlling the validity of the kinetic limit for a suitable choice of the random initial data. While for the general case a proof of the kinetic limit remains open, we report on first progress. As wave equation we consider the nonlinear Schrödinger equation discretized on a hypercubic lattice. Since this is a Hamiltonian system, a natural choice of random initial data is distributing them according to the corresponding Gibbs measure with a chemical potential chosen so that the Gibbs field has exponential mixing. The solution ψ _t (x) of the nonlinear Schrödinger equation yields then a stochastic process stationary in x∈? ^d and t∈?. If λ denotes the strength of the nonlinearity, we prove that the space-time covariance of ψ _t (x) has a limit as λ→0 for t=λ ^?2 τ, with τ fixed and |τ| sufficiently small. The limit agrees with the prediction from kinetic theory.     

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).     

On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations

For the first-order ordinary delay differential equation

$$u'(t) + p(t)u(r(t)) = 0,$$

where p ∈ L _loc(?₊; ?₊), τ ∈ C(?₊; ?₊), τ(t) ≤ t for t ∈ ?₊, lim_t→+∞ τ(t) = +∞, and ?₊:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.

     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.     

On nonstationary radiation fields in an infinite one-dimensional homogeneous medium

This paper considers nonstationary monochromatic radiative transfer in an infinite onedimensional homogeneous medium. The medium is considered to be illuminated by a momentary isotropic point energy source. The optical properties of the medium are characterized by the absorption coefficient α, the single-scattering albedo λ, the mean time t ₁ of photon stay in the absorbed state, and the mean time t ₂ of its stay on the path between two consecutive scatterings. The exact solution of the nonstationary radiative transfer equation has been obtained for the case t ₁ = t ₂. Asymptotic expressions have been derived for the source function, for the average intensity, and for radiation flux when points of the medium are located at large optical distances from the power source |τ| ? 1 and for small absorption of light in the medium (1 ? λ ? 1), assuming that t ₁ ? t ₂, t ₁ ? t ₂, or t ₁ = t ₂. These expressions are more precise than the ones previously known.     

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 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.     

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.

     

On Excess-time Correlated Cumulative Processes

In this paper a class of correlated cumulative processes, B _s (t) = ∑^N(t)_i=1 H _s (X _i )X _i , is studied with excess level increments X _i ?s, where {N(t), t ?0} is the counting process generated by the renewal sequence T _n , T _n and X _n are correlated for given n, H _s (t) is the Heaviside function and s?0 is a given constant. Several useful results, for the distributions of B _s (t), and that of the number of excess (non-excess) increments on (0, t) and the corresponding means, are derived. First passage time problems are also discussed and various asymptotic properties of the processes are obtained. Transform results, by applying a flexible form for the joint distribution of correlated pairs (T _n , X _n ) are derived and inverted. The case of non-excess level increments, X _i < s, is also considered. Finally, applications to known stochastic shock and pro-rata warranty models are given.     

Global Asymptotics of Orthogonal Polynomials Associated with a Generalized Freud Weight

In this paper,the authors consider the asymptotic behavior of the monic polynomials orthogonal with respect to the weight function w(x) = |x|~(2α)e~(-(x~4+tx~2)),x ∈ R,where α is a constant larger than -1/2 and t is any real number. They consider this problem in three separate cases:(i) c -2,(ii) c =-2,and(iii) c -2,where c := t N~(-1/2) is a constant,N = n + α and n is the degree of the polynomial. In the first two cases,the support of the associated equilibrium measure μ_t is a single interval,whereas in the third case the support of μ_t consists of two intervals. In each case,globally uniform asymptotic expansions are obtained in several regions. These regions together cover the whole complex plane. The approach is based on a modified version of the steepest descent method for Riemann-Hilbert problems introduced by Deift and Zhou(1993).