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.     

Inequalities for ranks of partitions and the first moment of ranks and cranks of partitions

We prove two monotonicity properties of N(m,n)$N(m,n)$, the number of partitions of n with rank m. They are (i) for any nonnegative integers m and n,

N(m,n)?N(m+2,n),$N(m,n)?N(m+2,n),$

and, (ii) for any nonnegative integers m and n   such that n?12$n?12$, n≠m+2$n\ne m+2$,

N(m,n)?N(m,n−1).$N(m,n)?N(m,n-1).$

G.E. Andrews, B. Kim, and the first author introduced ospt(n)$\mathrm{ospt}(n)$, a function counting the difference between the first positive rank and crank moments. They proved that ospt(n)>0$\mathrm{ospt}(n)>0$. In another article, K. Bringmann and K. Mahlburg gave an asymptotic estimate for ospt(n)$\mathrm{ospt}(n)$. The two monotonicity properties for N(m,n)$N(m,n)$ lead to stronger inequalities for ospt(n)$\mathrm{ospt}(n)$ that imply the asymptotic estimate.     

Dyadic harmonic analysis beyond doubling measures

We characterize the Borel measures μ   on R$\mathbb{R}$ for which the associated dyadic Hilbert transform, or its adjoint, is of weak-type (1,1)$(1,1)$ and/or strong-type (p,p)$(p,p)$ with respect to μ  . Surprisingly, the class of such measures is strictly bigger than the traditional class of dyadically doubling measures and strictly smaller than the whole Borel class. In higher dimensions, we provide a complete characterization of the weak-type (1,1)$(1,1)$ for arbitrary Haar shift operators, cancellative or not, written in terms of two generalized Haar systems and these include the dyadic paraproducts. Our main tool is a new Calderón–Zygmund decomposition valid for arbitrary Borel measures which is of independent interest.     

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.     

Singular points of cyclic weighted shift matrices

Let S be an n-by-n   cyclic weighted shift matrix, and _FS(t,x,y)=det(tI+xℜ(S)+yℑ(S))$F_S(t,x,y)=\det (tI+x\Re (S)+y\Im (S))$ be a ternary form associated with S  . We investigate the number of singular points of the curve _FS(t,x,y)=0$F_S(t,x,y)=0$, and show that the number of singular points of _FS(t,x,y)=0$F_S(t,x,y)=0$ associated with a cyclic weighted shift matrix whose weights are neither 1-periodic nor 2-periodic is less than or equal to n(n−3)/2$n(n-3)/2$. Furthermore, we verify the upper bound n(n−3)/2$n(n-3)/2$ is sharp for 4?n?7$4?n?7$.