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Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted $\mathcal{A}$ and $\mathcal{B}$) that are constructed from a nondegenerate quasihomogeneous polynomial $W$ and a related group of symmetries…
In this note, we derive non trivial sharp bounds related to the weighted harmonic-geometric-arithmetic means inequalities, when two out of the three terms are known. As application, we give an explicit bound for the trace of the inverse of…
We offer a conjecture on sharp estimation of a definite improper integral depend on a parameter $\lambda \in (0,+\infty)$ by means of given estimate of other definite integral depend on parameters $t\in [0,+\infty)$ and $\lambda$. Such…
Let $M$ be an $n\times n$ matrix with iid subgaussian entries with mean $0$ and variance $1$ and let $\sigma_n(M)$ denote the least singular value of $M$. We prove that \[\mathbb{P}\big( \sigma_{n}(M) \leq \varepsilon n^{-1/2} \big) =…
The upper bound inequality for variance of weighted sum of correlated random variables is derived according to Cauchy-Schwarz's inequality, while the weights are non-negative with sum of 1. We also give a novel proof with positive…
In this paper we give alternate proofs of some well-known matrix inequalities. In particular, we show that under certain conditions the inequality holds \begin{align}\sum \limits_{\lambda_i\in \mathrm{Spec}(ab^{T})}\mathrm{min}\{\log…
We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover…
Let $Q_n$ denote a random symmetric $n$ by $n$ matrix, whose upper diagonal entries are i.i.d. Bernoulli random variables (which take values 0 and 1 with probability 1/2). We prove that $Q_n$ is non-singular with probability…
A precise tie between a univariate spline's knots and its zeros abundance and dissemination is formulated. As an application, a conjecture formulated by De Concini and Procesi is shown to be true in the special univariate, unimodular case.…
We consider a polynomial $P\in \mathbb{R}[x_{1},\cdots, x_{d}]$ of degree $ \delta $ that depends non-trivially on each of $x_1,...,x_d$ with $d\geq 2$. For any integer $t$ with $2\leq t\leq d$, any natural number $n \in \mathbb{N}$, and…
Conjectured links between the distribution of values taken by the characteristic polynomials of random orthogonal matrices and that for certain families of L-functions at the centre of the critical strip are used to motivate a series of…
This dissertation presents new results on three different themes all related to matroid polytopes. First we investigate properties of Ehrhart polynomials of matroid polytopes, independence matroid polytopes, and polymatroids. We prove that…
The Bell inequality constrains the outcomes of measurements on pairs of distant entangled particles. The Bell contradiction states that the Bell inequality is inconsistent with the calculated outcomes of these quantum experiments. This…
We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…
Let A be a (central) arrangement of hyperplanes in a finite dimension complex vector space V. Let M(A) be the dependence matroid determined by A. The Orlik-Solomon algebra OS(M) of a matroid M is the exterior algebra on the points modulo…
We investigate the strong Rayleigh property of matroids for which the basis enumerating polynomial is invariant under a Young subgroup of the symmetric group on the ground set. In general, the Grace-Walsh-Szeg\H{o} theorem can be used to…
We prove Dual Smale's mean value conjecture for all odd polynomials with nonzero linear term. Precisely, if $P$ is an odd polynomial of degree $d\ge3$ with $P(0)=0$ and $P'(0)=1$, then there exists a critical point $\zeta$ of $P$ such that…
We prove analogues for elliptic interpolation functions of Macdonald's version of the Littlewood identity for (skew) Macdonald polynomials, in the process developing an interpretation of general elliptic "hypergeometric" sums as skew…
A simple classical, deterministic, local situation violating the Bell inequality is described. The detectors used in the experiment are ideal and the observers who decide which pair of measuring devices to choose for a given pair of…
For finite sets A and B in the plane, we write A+B to denote the set of sums of the elements of A and B. In addition, we write tr(A) to denote the common number of triangles in any triangulation of the convex hull of A using the points of A…