Related papers: Strictly Hermitian Positive Definite Functions

Let $\{\hat{P}_{n}(x)\}$ be an orthonormal polynomial sequence and denote by $\{w_{n}(x)\}$ the respective sequence of functions of the second kind. Suppose the Hamburger moment problem for $\{\hat{P}_{n}(x)\}$ is determinate and denote by…

The main result of this note essentially is that if the base and fibers of a compact fibration carry Hermitian metrics of positive holomorphic sectional curvature, then so does the total space of the fibration. The proof is based on the use…

A formula for the Hurwitz zeta function at the positive integers $k$, $\zeta(k,b)$, is created by solving the real and the imaginary parts separately and then combining them. A few different formulae for the Hurwitz zeta function are known…

We prove that the category of systems of sesquilinear forms over a given hermitian category is equivalent to the category of unimodular 1-hermitian forms over another hermitian category. The sesquilinear forms are not required to be…

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A \in {}_2Br(F)$ be a central simple algebra with an involution $\sigma$ of any kind and $F_0 =F^{\sigma}$. Let $h$ be…

A hermitian algebra is a unital associative ${\mathbb C}$-algebra endowed with an involution such that the spectra of self-adjoint elements are contained in ${\mathbb R}$. In the case of an algebra ${\mathcal A}$ endowed with a…

A hermitian matrix can be parametrized by a set consisting of its determinant and the eigenvalues of its submatrices. We established a group of equations which connect these variables with the mixing parameters of diagonalization. These…

Let k be a an algebraically closed field of arbitrary characteristic, and we let h be the usual Weil height for the n-dimensional affine space corresponding to the function field k(t) (extended to its algebraic closure). We prove that for…

This paper show that two doubly infinite series generate a totally nonnegative Hurwitz-type matrix if and only if their ratio represents an S-functions of a certain kind. The doubly infinite case needs a specific approach, since the ratios…

Let $(F_n)_{n\geq 0}$ be the Fibonacci sequence given by $F_{n+2}=F_{n+1}+F_n$, for $n\geq 0$, where $F_0=0$ and $F_1=1$. There are several interesting identities involving this sequence such as $F_n^2+F_{n+1}^2=F_{2n+1}$, for all $n\geq…

Let Q(u,v) be a positive definite binary quadratic form with arbitrary real coefficients. For large real x, one may ask for the number B(x) of primitive lattice points (integer points (m,n) with gcd(m,n) = 1) in the ellipse disc Q(u,v) < x,…

We study ranks of the $r\textrm{th}$ Hadamard powers of doubly nonnegative matrices and show that the matrix $A^{\circ r}$ is positive definite for every $n\times n$ doubly nonnegative matrix $A$ and for every $r>n-2$ if and only if no…

Consider the set of scalars $\alpha$ for which the $\alpha$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup…

We present, among other things, a necessary and sufficient condition for the strict positive definiteness of an isotropic and positive definite kernel on the cartesian product of a circle and a higher dimensional sphere. The result…

For two continuous and isotropic positive definite kernels on the same compact two-point homogeneous space, we determine necessary and sufficient conditions in order that their product be strictly positive definite. We also provide a…

Let $H$ be a monoid, $\mathscr F(X)$ be the free monoid on a set $X$, and $\pi_H$ be the unique extension of the identity map on $H$ to a monoid homomorphism $\mathscr F(H) \to H$. Given $A \subseteq H$, an $A$-word $\mathfrak z$ (i.e., an…

A polynomial is real-rooted if all of its roots are real. This note gives a simple proof of the Hermite-Sylvester theorem that a polynomial $f(x) \in {\mathbf R}[x]$ is real-rooted if and only if an associated quadratic form is positive…

Motivated by quantum thermodynamics we first investigate the notion of strict positivity, that is, linear maps which map positive definite states to something positive definite again. We show that strict positivity is decided by the action…

It is well-known that $AB$ and $BA$ are similar when $A$ and $B$ are complex square Hermitian matrices. In this note we answer a question of F. Zhang by demonstrating that similarity can fail if $A$ is Hermitian and $B$ is normal. Perhaps…

Finite dimensional linear spaces (both complex and real) with indefinite scalar product [.,.] are considered. Upper and lower bounds are given for the size of an indecomposable matrix that is normal with respect to this scalar product in…