Extended directed triple systems

Let {n;b₂,b₁} denote the class of extended directed triple systems of the order n in which the number of blocks of the form [a,b,a] is b₂ and the number of blocks of the form [b,a,a] or [a,a,b] is b₁. In this paper, we have shown that the necessary and sufficient condition for the existence of the class {n;b₂,b₁} is b₁≠1, 0?b₂+b₁?n and

(1)

for ;

(2)

for .

     

Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

Existence-uniqueness results for a class of singular boundary value problems-II

In this paper we establish existence-uniqueness of solution of a class of singular boundary value problem −(p(x)y^′′(x))=q(x)f(x,y) for 0<x?b and y(0)=a, α₁y(b)+β₁y^′(b)=γ₁, where p(x) satisfies (i) p(x)>0 in (0,b), (ii) p(x)∈C¹(0,r), and for some r>b, (iii) is analytic in and q(x) satisfies (i) q(x)>0 in (0,b), (ii) q(x)∈L¹(0,b) and for some r>b, (iii) is analytic in with quite general conditions on f(x,y). Region for multiple solutions have also been determined.     

On the properties of Newton-Euler pairs

If and are two sequences such that a₁=b₁ and , then we say that (a_n,b_n) is a Newton-Euler pair. In the paper, we establish many formulas for Newton-Euler pairs, and then make use of them to obtain new results concerning some special sequences such as and B_n, where p(n) is the number of partitions of n, σ(n) is the sum of divisors of n, and B_n is the nth Bernoulli number.     

A unique solution to a nonlinear elliptic equation

The purpose of this paper is to prove the existence of a unique, classical solution to the nonlinear elliptic partial differential equation −∇⋅(a(u(x))∇u(x))=f(x) under periodic boundary conditions, where u(x₀)=u₀ at x₀∈Ω, with Ω=TN, the N-dimensional torus, and N=2,3. The function a is assumed to be smooth, and a(u(x))>0 for , where G⊂R is a bounded interval. We prove that if the functions f and a satisfy certain conditions, then a unique classical solution u exists. The range of the solution u is a subset of a specified interval . Applications of this work include stationary heat/diffusion problems with a source/sink, where the value of the solution is known at a spatial location x₀.     

Regularity theorems for parabolic equations

We study regularity properties of solutions of a parabolic equation in R₊×R^d, d?3 under fairly general conditions on the drift term coefficients. The results are already new for the case a=I, , b=b(x) and .     

Exponential stability of wave equations with potential and indefinite damping

First, we consider the linear wave equation utt−uxx+a(x)ut+b(x)u=0 on a bounded interval (0,L)⊂R. The damping function a is allowed to change its sign. If is positive and the spectrum of the operator (_∂xx−b) is negative, exponential stability is proved for small . Explicit estimates of the decay rate ω are given in terms of and the largest eigenvalue of (_∂xx−b). Second, we show the existence of a global, small, smooth solution of the corresponding nonlinear wave equation utt−σx(ux)+a(x)ut+b(x)u=0, if, additionally, the negative part of a is small enough compared with ω.     

On the Weyl-Heisenberg frames generated by simple functions

Let with , and let (?,a,1), 0<a?1 be a Weyl-Heisenberg system {e2πimx?(x−na):m,n∈Z}. We show that if E=[0,1] (and some modulo extension of E), then (?,a,1) is a frame for each 0<a?1 (for certain a, respectively) if and only if the analytic function has no zero on the unit circle {z:|z|=1}. These results extend the case of Casazza and Kalton (2002) [6] that and a=1, which brought together the frame theory and the function theory on the closed unit disk. Our techniques of proofs are based on the Zak transform and the distribution of fractional parts of {na}_n∈Z.     

Invariant tori of nonlinear Schrödinger equation

In this paper, we consider one-dimensional nonlinear Schrödinger equation iut−uxx+V(x)u+f(²|u|)u=0 on [0,π]×R under the boundary conditions a₁u(t,0)−b₁ux(t,0)=0, a₂u(t,π)+b₂ux(t,π)=0, , for i=1,2. It is proved that for a prescribed and analytic positive potential V(x), the above equation admits small-amplitude quasi-periodic solutions corresponding to d-dimensional invariant tori of the associated infinite-dimensional dynamical system.     

Maps of several variables of finite total variation. II. E. Helly-type pointwise selection principles

Given a=(a₁,…,an), b=(b₁,…,bn)∈Rn with a<b componentwise and a map f from the rectangle into a metric semigroup M=(M,d,+), denote by the Hildebrandt-Leonov total variation of f on , which has been recently studied in [V.V. Chistyakov, Yu.V. Tretyachenko, Maps of several variables of finite total variation. I, J. Math. Anal. Appl. (2010), submitted for publication]. The following Helly-type pointwise selection principle is proved: If a sequence{fj}_j∈Nof maps frominto M is such that the closure in M of the set{fj(x)}_j∈Nis compact for eachandis finite, then there exists a subsequence of{fj}_j∈N, which converges pointwise onto a map f such that. A variant of this result is established concerning the weak pointwise convergence when values of maps lie in a reflexive Banach space (M,‖⋅‖) with separable dual M^∗.     

An application of anisotropic regularization to the existence of weak Pareto minimal points

Given a function f on Rⁿ, we introduce the concept of anisotropic regularization as a generalization of Tikhonov regularization f_ε(x)=f(x)+εx. When f is a continuous -function on Rⁿ and K is a box in Rⁿ, we study the properties of and the limiting behavior of solutions of a regularized box variational inequality problem , with emphasis on the existence of weak Pareto minimal points with respect to K. This work generalizes results of Sznajder and Gowda (1998) proved in the setting of nonlinear complementarity problems.     

Diophantine equations with products of consecutive values of a quadratic polynomial

Let a, b, c, d be given nonnegative integers with a,d?1. Using Chebyshev?s inequalities for the function π(x) and some results concerning arithmetic progressions of prime numbers, we study the Diophantine equation

     

Quartic, octic residues and Lucas sequences

Let be a prime and a,b∈Z with a²+b²≠p. Suppose p=x²+(a²+b²)y² for some integers x and y. In the paper we develop the calculation technique of quartic Jacobi symbols and use it to determine . As applications we obtain the congruences for modulo p and the criteria for (if ), where {Un} is the Lucas sequence given by U₀=0, U₁=1 and Un+1=bUn+k²Un−1(n?1). We also pose many conjectures concerning , or .     

Log-concavity and LC-positivity

A triangle {a(n,k)}_0?k?n of nonnegative numbers is LC-positive if for each r, the sequence of polynomials is q-log-concave. It is double LC-positive if both triangles {a(n,k)} and {a(n,n−k)} are LC-positive. We show that if {a(n,k)} is LC-positive then the log-concavity of the sequence {xk} implies that of the sequence {zn} defined by , and if {a(n,k)} is double LC-positive then the log-concavity of sequences {xk} and {yk} implies that of the sequence {zn} defined by . Examples of double LC-positive triangles include the constant triangle and the Pascal triangle. We also give a generalization of a result of Liggett that is used to prove a conjecture of Pemantle on characteristics of negative dependence.     

A generalization of Barbashin-Krasovski theorem

The classical criterion of asymptotic stability of the zero solution of equations x^′=f(t,x) is that there exists a function V(t,x), a(‖x‖)?V(t,x)?b(‖x‖) for some a,b∈K, such that for some c∈K. In this paper we prove that if f(t,x) is bounded, is uniformly continuous and bounded, then the condition that can be weakened and replaced by and contains no complete trajectory of , t∈[−T,T], where , uniformly for (t,x)∈[−T,T]×BH.     

Mapping properties for oscillatory integrals in d-dimensions

For aj,bj?1, j=1,2,…,d, we prove that the operator maps into itself for , where , and k(x,y)=φ(x,y)eig(x,y), φ(x,y) satisfies (1.2) (e.g. φ(x,y)=|x−y|^iτ,τ real) and the phase g(x,y)=xa⋅yb. We study operators with more general phases and for these operators we require that aj,bj>1, j=1,2,…,d, or al=bl?1 for some l∈{1,2,…,d}.     

Positive periodic solutions of functional differential equations

We consider the existence, multiplicity and nonexistence of positive ω-periodic solutions for the periodic equation x′(t)=a(t)g(x)x(t)−λb(t)f(x(t−τ(t))), where are ω-periodic, , , f,g∈C([0,∞),[0,∞)), and f(u)>0 for u>0, g(x) is bounded, τ(t) is a continuous ω-periodic function. Define , , i₀=number of zeros in the set and i_∞=number of infinities in the set . We show that the equation has i₀ or i_∞ positive ω-periodic solution(s) for sufficiently large or small λ>0, respectively.     

The Cameron-Erd?s conjecture

A subset A of integers is said to be sum-free if a+b∉A for any a,b∈A. Let s(n) be the number of sum-free sets in interval [1,n] of integers. P. Cameron and P. Erd?s conjectured that s(n)=O(2^n/2). We show that for even n and for odd n, where are absolute constants, thereby proving the conjecture.     

The structure of max-min hyperplanes

In this article, continuing [12,13], further contributions to the theory of max-min convex geometry are given. The max-min semiring is the set endowed with the operations ⊕=max,⊗=min in . A max-min hyperplane (briefly, a hyperplane) is the set of all points satisfying an equation of the form

a₁⊗x₁⊕…⊕a_n⊗x_n⊕a_n+1=b₁⊗x₁⊕…⊕b_n⊗x_n⊕b_n+1,