Related papers: Pick matricies and quaternionic power series

Structural properties are given for $D(K)$, the Banach algebra of (complex) differences of bounded semi-continuous functons on a metric space $K$. For example, it is proved that if all finite derived sets of $K$ are non-empty, then a…

We analyse a special case of the robust stabilization problem under structured uncertainty. We obtain a new criterion for the solvability of the spectral Nevanlinna-Pick problem, which is a special case of the $\mu$-synthesis problem of…

In this article, I introduce a group-theoretical method to prove positivity of certain linear combinations (with coefficients generally lying in $\mathbb{C}$) of exponential functions under a set of semidefinite linear constraints. The…

Let K be an algebraic number field of degree d and discriminant D over Q. Let A be an associative algebra over K given by structure constants such that A is isomorphic to the algebra M_n(K) of n by n matrices over K for some positive…

P. V. Chung showed that there are many multiplicative functions $f$ which satisfy $f(m^2+n^2) = f(m^2)+f(n^2)$ for all positive integers $m$ and $n$. In this article, we show that if more than $2$ squares in the additive condition are…

Let $H^d$ be the set of all rational maps of degree $d\ge 2$ on the Riemann sphere which are expanding on Julia set. We prove that if $f\in H^d$ and all or all but one critical points (or values) are in the immediate basin of attraction to…

We introduce an algebraic model, based on the determinantal expansion of the product of two matrices, to test combinatorial reductions of set functions. Each term of the determinantal expansion is deformed through a monomial factor in d…

We introduce determinantal sieving, a new, remarkably powerful tool in the toolbox of algebraic FPT algorithms. Given a polynomial $P(X)$ on a set of variables $X=\{x_1,\ldots,x_n\}$ and a linear matroid $M=(X,\mathcal{I})$ of rank $k$,…

An integer-valued multiplicative function $f$ is said to be polynomially-defined if there is a nonconstant separable polynomial $F(T)\in \mathbb{Z}[T]$ with $f(p)=F(p)$ for all primes $p$. We study the distribution in coprime residue…

For each odd prime power q, we construct an infinite sequence of rational functions f(X) in F_q(X), each of which is exceptional, which means that for infinitely many n the map c-->f(c) induces a bijection of P^1(F_{q^n}). Moreover, each of…

A matrix is called a P-matrix if all its principal minors are positive. P-matrices have found important applications in functional analysis, mathematical programming, and dynamical systems theory. We introduce a new class of real matrices…

Taking matrix as a synonym for a numerical function on the Cartesian product of two (in general, infinite) sets, a simple purely algebraic "reciprocity property" says that the set of rows spans a finite-dim space iff the set of columns does…

Let $p$ be a multilinear polynomial in several non-commuting variables with coefficients in an arbitrary field $K$. Kaplansky conjectured that for any $n$, the image of $p$ evaluated on the set $M_n(K)$ of $n$ by $n$ matrices is either…

In 1942 I. J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg's theorem for…

A Pick function of $d$ variables is a holomorphic map from $\Pi^d$ to $\Pi$, where $\Pi$ is the upper halfplane. Some Pick functions of one variable have an asymptotic expansion at infinity, a power series $\sum_{n=1}^\infty \rho_n z^{-n}$…

Using Zeilberger generating function formula for the values of a discrete analytic function in a quadrant we make connections with the theory of structured reproducing kernel spaces, structured matrices and a generalized moment problem. An…

Let $X^N = (X_1^N,\dots, X^N_d)$ be a d-tuple of $N\times N$ independent GUE random matrices and $Z^{NM}$ be any family of deterministic matrices in $\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a self-adjoint…

A classical theorem of Hutchinson asserts that if an iterated function system acts on $\mathbb{R}^d$ by similitudes and satisfies the open set condition then it admits a unique self-similar measure with Hausdorff dimension equal to the…

It is known that a $2\times 2$ quaternionic matrix has one, two or an infinite number of left eigenvalues, but the available algebraic proofs are difficult to generalize to higher orders. In this paper a different point of view is adopted…

If $A$ is a $2n \times 2n$ real positive definite matrix, then there exists a symplectic matrix $M$ such that $M^TAM = \left [ \begin{array}{cc} D & O \\ O & D \end{array} \right ]$ where $D= \diag (d_1 (A), \ldots, d_n(A))$ is a diagonal…