Related papers: Sign variation and descents

Varieties of minimal degree and del Pezzo varieties are basic objects in projective algebraic geometry. Those varieties have been characterized and classified for a long time in many aspects. Motivated by the question "which varieties are…

Let $\mathcal{A}$ be a unital algebra, $\delta$ be a linear mapping from $\mathcal{A}$ into itself and $m$, $n$ be fixed integers. We call $\delta$ an (\textit{m, n})-derivable mapping at $Z$, if…

We study the microlocal regularity of the analytic/Gevrey vectors for the following class of second order partial differential equations \begin{align*} P(x,D) = \sum_{\ell,j=1}^{n} a_{\ell,j}(x) D_{\ell} D_{j} + \sum_{\ell=1}^{n} i…

In this paper, first we prove that all finite dimensional special Heisenberg Lie superalgebras with even center have same dimension, say $(2m+1\mid n)$ for some non-negative integers $m,n$ and are isomorphism with them. Further, for a…

Let k be an algebraically closed field of characteristic p > 0. Let H be a subgroup of GL(n,k). We are interested in the determination of the vector invariants of H. When the characteristic of k is 0, it is known that the invariants of d…

On a smooth complex projective variety $X$ of dimension $n$, consider an ample vector bundle $\mathcal{E}$ of rank $r \leq n-2$ and an ample line bundle $H$. A numerical character $m_2=m_2(X,\mathcal{E},H)$ of the triplet…

Let $q:=e^{2 \pi iz}$, where $z \in \mathbb{H}$. For an even integer $k$, let $f(z):=q^h\prod_{m=1}^{\infty}(1-q^m)^{c(m)}$ be a meromorphic modular form of weight $k$ on $\Gamma_0(N)$. For a positive integer $m$, let $T_m$ be the $m$th…

In order to study signed Eulerian numbers, we introduce permutations of a particular type, called parity-alternate permutations, because they take even and odd entries alternately. The objective of this paper is twofold. The first is to…

For integer $n$ and real $u$, define $\Delta(n,u):= |\{d : d \mid n,\,{\rm e}^u <d\leqslant {\rm e}^{u+1} \}|$. Then, put $ \Delta(n):=\max_{u\in{\mathbb R}} \Delta(n,u).$ We provide uniform upper and lower bounds for the mean-value of…

In this paper, we evaluate the sum $\sum_{m,n}\leg {m}{n}d(n)$, where $\leg {m}{n}$ is the Kronecker symbol and $d(n)$ is the divisor function.

Let $P^k_\ell$ denote the loose $k$-path of length $\ell$ and let define $f^k_\ell(n,m)$ as the minimum value of $\Delta(H)$ over all $P^k_\ell$-free $k$-graphs $H$ with $n$ vertices and $m$ edges. In the paper we study the behavior of…

We study higher order determinantal varieties obtained by considering generic $m\times n$ ($m \le n$) matrices over rings of the form $F[t]/(t^k)$, and for some fixed $r$, setting the coefficients of powers of $t$ of all $r \times r$ minors…

Let $\text{Tr}(n,m,k)$ denote the largest number of distinct projections onto $k$ coordinates guaranteed in any family of $m$ binary vectors of length $n$. The classical Sauer-Perles-Shelah Lemma implies that $\text{Tr}(n, n^r, k) = 2^k$…

We study the problem of finding the largest number $T(n, m)$ of ternary vectors of length $n$ such that for any three distinct vectors there are at least $m$ coordinates where they pairwise differ. For $m = 1$, this is the classical…

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine $h$-vector of balanced semi-Eulerian complexes…

For every positive integer $n$, consider the linear operator $\U_{n}$ on polynomials of degree at most $d$ with integer coefficients defined as follows: if we write $\frac{h(t)}{(1 - t)^{d + 1}} = \sum_{m \geq 0} g(m) t^{m}$, for some…

We show in this article that if a holomorphic vector bundle has a nonnegative Hermitian metric in the sense of Bott and Chern, which always exists on globally generated holomorphic vector bundles, then some special linear combinations of…

Let $n, r, k$ be positive integers such that $3\leq k < n$ and $2\leq r \leq k-1$. Let $m(n, r, k)$ denote the maximum number of edges an $r$-uniform hypergraph on $n$ vertices can have under the condition that any collection of $i$ edges,…

Let $T(n,m)$ be the set of all plane labelled bipartite trees with $n$ white vertices and $m$ black. If the number $n+m$ of vertices is even, then the set $T(n,m)$ is a union of two disjoint subsets --- subset od "even" trees and subset of…

For every integer $n \geq 3$, we prove that the n-sublinear generalization of the Bi-Carleson operator of Muscalu, Tao, and Thiele given by nC^{\vec{\alpha}} :(f_1,..., f_n) \mapsto \sup_{M} \left| \int_{\vec{\xi} \cdot \vec{\alpha} >0,…