The elementary symmetric functions of a reciprocal polynomial sequence

Erdös and Niven proved in 1946 that for any positive integers m and d, there are at most finitely many integers n   for which at least one of the elementary symmetric functions of 1/m,1/(m+d),…,1/(m+(n−1)d)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ are integers. Recently, Wang and Hong refined this result by showing that if n?4$n?4$, then none of the elementary symmetric functions of 1/m,1/(m+d),…,1/(m+(n−1)d)$1/m,1/(m+d),\dots ,1/(m+(n-1)d)$ is an integer for any positive integers m and d. Let f   be a polynomial of degree at least 2 and of nonnegative integer coefficients. In this paper, we show that none of the elementary symmetric functions of 1/f(1),1/f(2),…,1/f(n)$1/f(1),1/f(2),\dots ,1/f(n)$ is an integer except for f(x)=x^m$f(x)=x^m$ with m?2$m?2$ being an integer and n=1$n=1$.     

On Waringʼs problem for systems of skew-symmetric forms

This paper is devoted to a problem of finding the smallest positive integer s(m,n,k)$s(m,n,k)$ such that (m+1)$(m+1)$ generic skew-symmetric (k+1)$(k+1)$-forms in (n+1)$(n+1)$ variables as linear combinations of the same s(m,n,k)$s(m,n,k)$ decomposable skew-symmetric (k+1)$(k+1)$-forms.     

Re-nnd generalized inverses

In this paper, we give some necessary and sufficient conditions for the existence of Re-nnd and nonnegative definite {1,3}$\{1,3\}$- and {1,4}$\{1,4\}$-inverses of a matrix A∈C^n×n$A\in {\mathbb{C}}^{n\times n}$ and completely described these sets. Moreover, we prove that the existence of nonnegative definite {1,3}$\{1,3\}$-inverse of a matrix A   is equivalent with the existence of its nonnegative definite {1,2,3}$\{1,2,3\}$-inverse and present the necessary and sufficient conditions for the existence of Re-nnd {1,3,4}$\{1,3,4\}$-inverse of A.     

Random partitions and edge-disjoint Hamiltonian cycles

Nash-Williams [1] proved that every graph with n   vertices and minimum degree n/2$n/2$ has at least ⌊5n/224⌋$\lfloor 5n/224\rfloor$ edge-disjoint Hamiltonian cycles. In [2], he raised the question of determining the maximum number of edge-disjoint Hamiltonian cycles, showing an upper bound of ⌊(n+4)/8⌋$\lfloor (n+4)/8\rfloor$.     

Arithmetic properties of ℓ-regular partitions

For a given prime p, by studying p  -dissection identities for Ramanujan?s theta functions ψ(q)$\psi (q)$ and f(−q)$f(-q)$, we derive infinite families of congruences modulo 2 for some ?  -regular partition functions, where ?=2,4,5,8,13,16$?=2,4,5,8,13,16$.     

Lê–Greuel type formula for the Euler obstruction and applications

The Euler obstruction of a function f   can be viewed as a generalization of the Milnor number for functions defined on singular spaces. In this work, using the Euler obstruction of a function, we establish several Lê–Greuel type formulas for germs f:(X,0)→(C,0)$f:(X,0)\to (\mathbb{C},0)$ and g:(X,0)→(C,0)$g:(X,0)\to (\mathbb{C},0)$. We give applications when g is a generic linear form and when f and g have isolated singularities.     

New approach for the existence of pseudo almost periodic solutions for some second order differential equation with piecewise constant argument

The following equation d²/dt²(x(t)+px(t-1))=qx(2[(t+1)/2])+f(t)${\mathrm{d}}^2/\mathrm{d}{t}^2(x(t)+\mathit{px}(t-1))=\mathit{qx}(2[(t+1)/2])+f(t)$ is considered and necessary and sufficient conditions are given in order to ensure the existence and uniqueness of pseudo almost periodic solutions.     

Identities for the Ramanujan zeta function

We prove formulas for special values of the Ramanujan tau zeta function. Our formulas show that L(Δ,k)$L(\mathrm{\Delta },k)$ is a period in the sense of Kontsevich and Zagier when k?12$k?12$. As an illustration, we reduce L(Δ,k)$L(\mathrm{\Delta },k)$ to explicit integrals of hypergeometric and algebraic functions when k∈{12,13,14,15}$k\in \{12,13,14,15\}$.     

Classification of invariant valuations on the quaternionic plane

We describe the orbit space of the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$ on the real Grassmann manifolds _Grk(H²)${\mathrm{Gr}}_k({\mathbb{H}}^2)$ in terms of certain quaternionic matrices of Moore rank not larger than 2. We then give a complete classification of valuations on the quaternionic plane H²${\mathbb{H}}^2$ which are invariant under the action of the group Sp(2)Sp(1)$\mathrm{Sp}(2)\mathrm{Sp}(1)$.     

A linearized compact difference scheme for an one-dimensional parabolic inverse problem

The parabolic equation with the control parameter is a class of parabolic inverse problems and is nonlinear. While determining the solution of the problems, we shall determinate some unknown control parameter. These problems play a very important role in many branches of science and engineering. The article is devoted to the following parabolic initial-boundary value problem with the control parameter: ∂u/∂t=∂²u/∂x²+p(t)u+?(x,t),0<x<1,0<t?T$\mathrm{\partial }u/\mathrm{\partial }t={\mathrm{\partial }}^{2}u/\mathrm{\partial }{x}^2+p(t)u+?(x\text{,}t)\text{,}0<x<1\text{,}0<t?T$ satisfying u(x,0)=f(x),0<x<1$u(x\text{,}0)=f(x)\text{,}0<x<1$; u(0,t)=_g0(t)$u(0\text{,}t)=g_0(t)$, u(1,t)=_g1(t)$u(1\text{,}t)=g_1(t)$, u(x^∗,t)=E(t),0?t?T$u(x^{\ast }\text{,}t)=E(t)\text{,}0?t?T$ where ?(x,t),f(x),_g0(t),_g1(t)$?(x\text{,}t)\text{,}f(x)\text{,}{g}_0(t)\text{,}{g}_1(t)$ and E(t)$E(t)$ are known functions, u(x,t)$u(x\text{,}t)$ and p(t)$p(t)$ are unknown functions. A linearized compact difference scheme is constructed. The discretization accuracy of the difference scheme is two order in time and four order in space. The solvability of the difference scheme is proved. Some numerical results and comparisons with the difference scheme given by Dehghan are presented. The numerical results show that the linearized difference scheme of this article improve the accuracy of the space and time direction and shorten computation time largely. The method in this article is also applicable to the two-dimensional inverse problem.