Blow-up analysis for a system of heat equations with nonlinear flux which obey different laws

We consider a system of heat equations u_t=Δu and v_t=Δv in Ω×(0,T) completely coupled by nonlinear boundary conditions

We prove that the solutions always blow up in finite time for non-zero and non-negative initial values. Also, the blow-up only occurs on ∂Ω with

for p,q>0, 0≤α<1 and 0≤β<p.     

Riemann–Hilbert analysis for Jacobi polynomials orthogonal on a single contour

Classical Jacobi polynomials , with α,β>-1, have a number of well-known properties, in particular the location of their zeros in the open interval (-1,1). This property is no longer valid for other values of the parameters; in general, zeros are complex. In this paper we study the strong asymptotics of Jacobi polynomials where the real parameters α_n,β_n depend on n in such a way that

with . We restrict our attention to the case where the limits A,B are not both positive and take values outside of the triangle bounded by the straight lines A=0, B=0 and A+B+2=0. As a corollary, we show that in the limit the zeros distribute along certain curves that constitute trajectories of a quadratic differential.The non-hermitian orthogonality relations for Jacobi polynomials with varying parameters lie in the core of our approach; in the cases we consider, these relations hold on a single contour of the complex plane. The asymptotic analysis is performed using the Deift–Zhou steepest descent method based on the Riemann–Hilbert reformulation of Jacobi polynomials.     

Large time behavior for nonlinear higher order convection–diffusion equations

We study the large time asymptotic behavior, in L^p (1p∞), of higher derivatives D^γu(t) of solutions of the nonlinear equation

(1)

where the integers n and θ are bigger than or equal to 1, a is a constant vector in with . The function ψ is a nonlinearity such that and ψ(0)=0, and is a higher order elliptic operator with nonsmooth bounded measurable coefficients on . We also establish faster decay when .     

Lp Markov–Bernstein Inequalities on All Arcs of the Circle

Let 0<p<∞ and 0α<β2π. We prove that for n1 and trigonometric polynomials s_n of degree n, we have

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cn^p ∫^β_α |s_n(θ)|^p dθ, where c is independent of α, β, n, s_n. The essential feature is the uniformity in [α,β] of the estimate and the fact that as [α,β] approaches [0,2π], we recover the L_p Markov inequality. The result may be viewed as the complete L_p form of Videnskii's inequalities, improving earlier work of the second author.     

Constant-sign solutions for systems of singular integral equations of Hammerstein type

We consider the system of Hammerstein integral equations

where T>0 is fixed, ρ_i’s are given functions and the nonlinearities f_i(t,x₁,x₂,…,x_n) can be singular at t=0 and x_j=0 where j{1,2,,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ_iu_i(t)≥0 for t[0,T] and 1≤i≤n, where θ_i{1,−1} is fixed. The tools used are a nonlinear alternative of Leray–Schauder type, Krasnosel’skii’s fixed point theorem in a cone and Schauder’s fixed point theorem. We also include examples and applications to illustrate the usefulness of the results obtained.     

Quadrature formulae connected to σ-orthogonal polynomials

Let dλ(t) be a given nonnegative measure on the real line , with compact or infinite support, for which all moments exist and are finite, and μ₀>0. Quadrature formulas of Chakalov–Popoviciu type with multiple nodes

where σ=σ_n=(s₁,s₂,…,s_n) is a given sequence of nonnegative integers, are considered. A such quadrature formula has maximum degree of exactness d_max=2∑_ν=1ⁿs_ν+2n−1 if and only if

The proof of the uniqueness of the extremal nodes τ₁,τ₂,…,τ_n was given first by Ghizzetti and Ossicini (Rend. Mat. 6(8) (1975) 1–15). Here, an alternative simple proof of the existence and the uniqueness of such quadrature formulas is presented. In a study of the error term R(f), an influence function is introduced, its relevant properties are investigated, and in certain classes of functions the error estimate is given. A numerically stable iterative procedure, with quadratic convergence, for determining the nodes τ_ν, ν=1,2,…,n, which are the zeros of the corresponding σ-orthogonal polynomial, is presented. Finally, in order to show a numerical efficiency of the proposed procedure, a few numerical examples are included.     

A note on asymptotic behavior for negative drift random walk with dependent heavy-tailed steps and its application to risk theory

In this article, the dependent steps of a negative drift random walk are modelled as a two-sided linear process Xn =-μ ∞∑j=-∞ψn-jεj, where {ε, εn; -∞＜ n ＜ ∞}is a sequence of independent, identically distributed random variables with zero mean, μ＞0 is a constant and the coefficients {ψi;-∞＜ i ＜∞} satisfy 0 ＜∞∑j=-∞|jψj| ＜∞. Under the conditions that the distribution function of |ε| has dominated variation and ε satisfies certain tail balance conditions, the asymptotic behavior of P{supn≥0(-nμ ∞∑j=-∞εjβnj) ＞ x}is discussed. Then the result is applied to ultimate ruin probability.     

An approximation algorithm for the m-machine permutation flow shop scheduling problem with controllable processing times

The paper deals with the m-machine permutation flow shop scheduling problem in which job processing times, along with a processing order, are decision variables. It is assumed that the cost of processing a job on each machine is a linear function of its processing time and the overall schedule cost to be minimized is the total processing cost plus maximum completion time cost. A algorithm for the problem with m = 2 is provided; the best approximation algorithm until now has a worst-case performance ratio equal to . An extension to the m-machine (m ≥2) permutation flow shop problem yields an approximation algorithm with a worst-case bound equal to

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, where is the worst-case performance ratio of a procedure used, in the proposed algorithm, for solving the (pure) sequencing problem. Moreover, examples which achieve this bound for = 1 are also presented.     

On recurrence coefficients for rapidly decreasing exponential weights

Let, for example,

where α>0, k1, and exp_k=exp(exp(…exp())) denotes the kth iterated exponential. Let {A_n} denote the recurrence coefficients in the recurrence relation

xp_n(x)=A_np_n+1(x)+A_n-1p_n-1(x)

for the orthonormal polynomials {p_n} associated with W². We prove that as n→∞,

where log_k=log(log(…log())) denotes the kth iterated logarithm. This illustrates the relationship between the rate of convergence to of the recurrence coefficients, and the rate of decay of the exponential weight at ±1. More general non-even exponential weights on a non-symmetric interval (a,b) are also considered.     

Some oscillation criteria for second order nonlinear functional ordinary differential equations

The main objective of this article is to study the oscillatory behavior of the solutions of the following nonlinear functional differential equations (a(t)x'(t))' δ1p(t)x'(t) δ2q(t)f(x(g(t))) = 0,for 0 ≤ t0 ≤ t, where δ1 = ±1 and δ2 = ±1. The functions p,q,g : [t0, ∞) → R, f :R → R are continuous, a(t) ＞ 0, p(t) ≥ 0,q(t) ≥ 0 for t ≥ t0, limt→∞ g(t) = ∞, and q is not identically zero on any subinterval of [t0, ∞). Moreover, the functions q(t),g(t), and a(t) are continuously differentiable.     

On weighted Lebesgue function type sums

Let I be a finite or infinite interval and dμ a measure on I. Assume that the weight function w(x)>0, w^′(x) exists, and the function w^′(x)/w(x) is non-increasing on I. Denote by ℓ_k's the fundamental polynomials of Lagrange interpolation on a set of nodes x₁<x₂<<x_n in I. The weighted Lebesgue function type sum for 1≤i<j≤n and s≥1 is defined by

In this paper the exact lower bounds of S_n(x) on a “big set” of I and are obtained. Some applications are also given.     

On the distribution of running times of certain integer factoring algorithms

There are several algorithms for computing the prime decomposition of integers whose running times essentially depend on the size of the second largest prime factor of the input. For several such algorithms, we give uniform estimates for the number of inputs n with 1 ≤ n ≤ x for which the algorithm will halt in at most t steps. As a consequence we derive the best known lower bound for the number of integers n ≤ x that can be completely factored in random polynomial time.     

Prophet inequalities for parallel processes

Generalizations of prophet inequalities for single sequences are obtained for optimal stopping of several parallel sequences of independent random variables. For example, if {X_{i, j}, 1 ≤ i ≤ n, 1 ≤ j < ∞} are independent non-negative random variables, then

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and this bound is best possible. Applications are made to comparisons of the optimal expected returns of various alternative methods of stopping of parallel processes.     

Bounds on margin distributions in learning problems

Let be a probability space and let P_n be the empirical measure based on i.i.d. sample (X₁,…,X_n) from P. Let be a class of measurable real valued functions on For define F_f(t):=P{ft} and F_n,f(t):=P_n{ft}. Given γ(0,1], define _n,γ(δ):=1/(n^1−γ/2δ^γ). We show that if the L₂(P_n)-entropy of the class grows as ^−α for some α(0,2), then, for all and all δ(0,Δ_n), Δ_n=O(n^1/2),

and

where and c(σ)↓1 as σ↓0 (the above inequalities hold for any fixed σ(0,1] with a high probability). Also, define

Then for all

uniformly in and with probability 1 (for the above ratio is bounded away from 0 and from ∞). The results are motivated by recent developments in machine learning, where they are used to bound the generalization error of learning algorithms. We also prove some more general results of similar nature, show the sharpness of the conditions and discuss the applications in learning theory.     

Permanence and global stability in a Lotka-Volterra predator-prey system with delays

Consider the permanence and global asymptotic stability of models governed by the following Lotka-Volterra-type system:

, with initial conditions

x_i(t) = φ_i(t) ≥ o, t ≤ t₀, and φ_i(t₀) > 0. 1 ≤ i ≤n

. We define x₀(t) = x_n+1(t)≡0 and suppose that φ_i(t), 1 ≤ i ≤ n, are bounded continuous functions on [t₀, + ∞) and γ_i, α_i, c_i > 0,γ_i,j ≥ 0, for all relevant i,j.Extending a technique of Saito, Hara and Ma[1] for n = 2 to the above system for n ≥ 2, we offer sufficient conditions for permanence and global asymptotic stability of the solutions which improve the well-known result of Gopalsamy.     

On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables

Let {Xni, 1 ≤ n,i ＜∞} be an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 ＜ an ↑∞. The limiting behavior of maximum partial sums 1/an max 1≤k≤n| kΣi=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].     

RELATIVE WIDTH OF SMOOTH CLASSES OF MULTIVARIATE PERIODIC FUNCTIONS WITH RESTRICTIONS ON ITERATED LAPLACE DERIVATIVES IN THE L2-METRIC

For two subsets W and V of a Banach space X, let Kn(W, V, X) denote the relative Kolmogorov n-width of W relative to V defined by Kn (W, V, X) := inf sup Ln f∈W g∈V∩Ln inf ‖f-g‖x,where the infimum is taken over all n-dimensional linear subspaces Ln of X. Let W2(△r) denote the class of 2w-periodic functions f with d-variables satisfying ∫[-π,π]d |△rf(x)|2dx ≤ 1,while △r is the r-iterate of Laplace operator △. This article discusses the relative Kolmogorov n-width of W2(△r) relative to W2(△r) in Lq([-r, πr]d) (1 ≤ q ≤∞), and obtain its weak asymptotic result.     

The Asymptotic Worst-Case Behavior of the FFD Heuristic for Small Items

The First-Fit-Decreasing (FFD) algorithm is one of the most famous and most studied methods for an approximative solution of the bin-packing problem. The question on the parametric behavior of the FFD heuristic for small items was raised in D. S. Johnson's thesis (1973, MIT, Cambridge, MA) and in E. G. Coffman et al. (1987, SIAM J. Comput.7, 1–17): what is the asymptotic worst-case ratio for FFD when restricted to lists with item sizes in the interval (0, α] for α ≤ . Let R^∞_FFD(α) denote the asymptotic worst-case ratio for these lists. In his thesis, Johnson gave the values of R^∞_FFD(α) for and he conjectured that

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for all integers m ≥ 4. J. Csirik (1993, J. Algorithms15, 1–28) proved that, for all integers m ≥ 5, this conjecture is true when m is even. When m is odd, he further showed where G_m ≡ 1 + (m² + m − 1)/(m(m + 1)(m + 2)) = F_m + 1/(m(m + 1)(m + 2)). These results leave open the values of R^∞_FFD(α) for 0 < α < 1/5 that are not the reciprocals of integers. In this paper we resolve the remaining open cases.     

Inequalities for Cyclic Functions

The nth cyclic function is defined by

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We prove that if k is an integer with 1kn−1, then

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holds for all positive real numbers x with the best possible constantsα=1 and β= 2n-k over n.     

Decomposable graphs and definitions with no quantifier alternation

Let D(G) be the minimum quantifier depth of a first order sentence Φ that defines a graph G up to isomorphism. Let D₀(G) be the version of D(G) where we do not allow quantifier alternations in Φ. Define q₀(n) to be the minimum of D₀(G) over all graphs G of order n.We prove that for all n we have

log^*n−log^*log^*n−2≤q₀(n)≤log^*n+22,