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We prove in a direct fashion that a multidimensional probability measure is determinate if the higher dimensional analogue of Carleman's condition is satisfied. In that case, the polynomials, as well as certain proper subspaces of the…

Under binary matrices we mean matrices whose entries take one of two values. In this paper, explicit formulae for calculating the determinant of some type of binary Toeplitz matrices are obtained. Examples of the application of the…

Using Casorati determinants of Hahn polynomials $(h_n^{\alpha,\beta,N})_n$, we construct for each pair $\F=(F_1,F_2)$ of finite sets of positive integers polynomials $h_n^{\alpha,\beta,N;\F}$, $n\in \sigma _\F$, which are eigenfunctions of…

Barvinok introduced the symmetrized determinant ($\sdet$) as a \emph{non-commutative} analogue of the determinant. Intuitively, given a square matrix over an associative algebra, we can obtain the symmetrized determinant by averaging over…

We consider the moment space $\mathcal{M}^{p}_{2n+1}$ of moments up to the order $2n + 1$ of $p_n\times p_n$ real matrix measures defined on the interval $[0,1]$. The asymptotic properties of the Hankel determinant $\{\log\det…

Let R be a complete discrete valuation ring with maximal ideal generated by pi. Let f(X) in R[X] be a monic polynomial with nonzero discriminant Delta(f). Let s >= v_pi(Delta(f)) + 1. Suppose given a factorisation of f(X) in (R/pi^s R)[X]…

We study the Gram matrix determinants for the groups $S_n,O_n,B_n,H_n$, for their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and for the half-liberated versions $O_n^*,H_n^*$. We first collect all the known computations of such determinants,…

The primary purpose of this note is to prove two recent conjectures concerning the $n$ body matrix that arose in recent papers of Escobar-Ruiz, Miller, and Turbiner on the classical and quantum $n$ body problem in $d$-dimensional space.…

We consider the Bochner-Schr\"odinger operator $H_{p}=\frac 1p\Delta^{L^p}+V$ on tensor powers $L^p$ of a Hermitian line bundle $L$ on a Riemannian manifold $X$ of bounded geometry under the assumption of non-degeneracy of the curvature…

Effective computation of resultants is a central problem in elimination theory and polynomial system solving. Commonly, we compute the resultant as a quotient of determinants of matrices and we say that there exists a determinantal formula…

From the irreducible decompositions' point of view, the structure of the cyclic $GL_n$-module generated by the $\alpha$-determinant degenerates when $\alpha=\pm \frac1k (1\leq k\leq n-1)$. In this paper, we show that $-\frac1k$-determinant…

We prove, by simple manipulation of commutators, two noncommutative generalizations of the Cauchy-Binet formula for the determinant of a product. As special cases we obtain elementary proofs of the Capelli identity from classical invariant…

In this paper we evaluate some Toeplitz-type determinants. Let $n>1$ be an integer. We prove the following two basic identities: \begin{align*} \det{[j-k+\delta_{jk}]_{1\leq j,k\leq n}}&=1+\frac{n^2(n^2-1)}{12}, \\…

We obtain an asymptotic formula for the number of integer $2\times 2$ matrices that have determinant $\Delta$ and whose absolute values of the entries are at most $H$. The result holds uniformly for a large range of $\Delta$ with respect to…

The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…

For a prime number p, we construct a generating set for the ring of invariants for the p+1 dimensional indecomposable modular representation of a cyclic group of order p^2. We then use the constructed invariants to describe the…

Let $m\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\binom n2+n$ $(n=0,1,2,\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers…

We study sets of the form $A = \big\{ n \in \mathbb N \big| \lVert p(n) \rVert_{\mathbb R / \mathbb Z} \leq \varepsilon(n) \big\}$ for various real valued polynomials $p$ and decay rates $\varepsilon$. In particular, we ask when such sets…

Let $p$ be a prime integer and $n,i$ be positive integers such that \linebreak $S=\{-1, \ \theta, \ \mu_i=1+\theta+... + \theta^{i-1} \ \mid 1 < i < \frac{p^n}{2}, \ gcd(p^n,i)=1 \}$ generates the group of units of $\mathbb{Z}[\theta],$…

Let $\gamma_n=[x_1,\dots,x_n]$ be the $n$th lower central word. Denote by $X_n$ the set of $\gamma_n$-values in a group $G$ and suppose that there is a number $m$ such that $|g^{X_n}|\leq m$ for each $g\in G$. We prove that…