相关论文: Towards Heim and Neuhauser's Unimodality Conjectur…

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are "strictly" log-concave for…

There are multiple conjectures relating the cohomological Hall algebras (CoHAs) of certain substacks of the moduli stack of representations of a quiver $Q$ to the Yangian $Y^{Q}_{MO}$ by Maulik-Okounkov, whose construction is based on the…

Our main result here is that the specialization at $t=1/q$ of the $Q_{km,kn}$ operators studied in [4] may be given a very simple plethystic form. This discovery yields elementary and direct derivations of several identities relating these…

In this paper, by constructing some identities, we prove some $q$-analogues of some congruences. For example, for any odd integer $n>1$, we show that \begin{gather*} \sum_{k=0}^{n-1} \frac{(q^{-1};q^2)_k}{(q;q)_k} q^k \equiv (-1)^{(n+1)/2}…

We prove that the (B) conjecture and the Gardner-Zvavitch conjecture are true for all log-concave measures that are rotationally invariant, extending previous results known for Gaussian measures. Actually, our result apply beyond the case…

We use K-area homology to summarize some results about the Novikov conjecture and the Hirzebruch L-class. In fact, we provide necessary and sufficient conditions for closed manifolds to have a homotopy invariant L-class. In order to obtain…

Previous work has conjectured that the graded $\mathfrak{S}_n$-representations $H^\bullet(\operatorname{Conf}(n,\mathbb{R}^d);\mathbb{Q})$ are strongly equivariantly log concave, and has proven this conjecture in low degrees. By leveraging…

The relative Novikov conjecture states that the relative higher signatures of manifolds with boundary are invariant under orientation-preserving homotopy equivalences of pairs. In this paper, we study the relative Baum-Connes assembly map…

The Boros-Moll polynomials $P_m(a)$ arise in the evaluation of a quartic integral. It has been conjectured by Boros and Moll that these polynomials are infinitely log-concave. In this paper, we show that $P_m(a)$ is 2-log-concave for any…

We consider the problem of constructing a conjugate $(1/q, q)$-harmonic homogeneous polynomial $V_k$ of degree $k$ to a given $(1/q, q)$-harmonic homogeneous polynomial $U_k$ of degree $k.$ The conjugated harmonic polynomials $V_k$ and…

In this paper we prove a $q$-analogue of Koshy's formula in terms of the Narayana polynomial due to Lassalle and a $q$-analogue of Koshy's formula in terms of $q$-hypergeometric series due to Andrews by applying the inclusion-exclusion…

We study multivariate entire functions and polynomials with non-negative coefficients. A class of {\bf Strongly Log-Concave} entire functions, generalizing {\it Minkowski} volume polynomials, is introduced: an entire function $f$ in $m$…

Let $\mu$ be a non-trivial probability measure on the unit circle $\partial\bbD$, $w$ the density of its absolutely continuous part, $\alpha_n$ its Verblunsky coefficients, and $\Phi_n$ its monic orthogonal polynomials. In this paper we…

In their study of a quartic integral, Boros and Moll discovered a special class of Jacobi polynomials, which we call the Boros-Moll polynomials. Kauers and Paule proved the conjecture of Moll that these polynomials are log-concave. In this…

Let $1<q<2$ and \[ \Lambda(q)={\sum_{k=0}^n a_kq^k\mid a_k\in\{-1,0,1\}, n\ge1}. \] It is well known that if $q$ is not a root of a polynomial with coefficients $0,\pm1$, then $\Lambda(q)$ is dense in $\mathbb{R}$. We give several…

Ruzsa's conjecture asserts that any sequence $(a_n)_{n \geq 0}$ of integers that preserves congruences, $\textit{i.e.}$, satisfies $ a_{n+k} \equiv a_n \mod k $, and has the growth condition $\limsup_{n \to +\infty} |a_n|^{1/n} < e$, must…

For every multivariable polynomial $p$, with $p(0)=1$, we construct a determinantal representation $$p=\det (I - K Z),$$ where $Z$ is a diagonal matrix with coordinate variables on the diagonal and $K$ is a complex square matrix. Such a…

The Kahane--Salem--Zygmund inequality for multilinear forms in $\ell_{\infty}$ spaces claims that, for all positive integers $m,n_{1},...,n_{m}$, there exists an $m$-linear form $A\colon\ell_{\infty}^{n_{1}}\times\cdots\times…

Let r be an orbit of the quiver representation of type A_n (equioriented case). In this paper we study the Poincare dual of the closure of r (a.c.a. Thom polynomial/degeneracy loci formula) in equivariant cohomology. Using general Thom…

We test the predictions of the L-functions Ratios Conjecture for the family of cuspidal newforms of weight k and level N, with either k fixed and N --> oo through the primes or N=1 and k --> oo. We study the main and lower order terms in…