We examine the electrostatic properties of exceptional and regular zeros of Xm-Laguerre and Xm-Jacobi polynomials. Since there is a close connection between the electrostatic properties of the zeros and the stability of interpolation on the system of zeros, we can deduce an Egerváry-Turán type result as well. The limit of the energy on the regular zeros is also investigated.

Abstract: Necessary and sufficient conditions for the reachability and observability of the positive electrical circuits composed of resistors, coils, condensators and voltage sources are established. Definitions of the input-decoupling zeros, output-decoupling zeros and input-output decoupling zeros of the positive electrical circuits are proposed. Some properties of the decoupling zeros of po...

For lossless periodic structures with a proper symmetry, the transmission and reflection spectra often have peaks dips that are truly $100\%$ $0\%$, respectively. The full zero typically appear near resonant frequencies, they robust respect to structural perturbations preserve required symmetry. However, current theories on existence of incomplete difficult use. bound state in continuum (BIC), ...

For every $1leq s< n$, the $s^{th}$ derivative of a polynomial $P(z)$ of degree $n$ is a polynomial $P^{(s)}(z)$ whose degree is $(n-s)$. This paper presents a result which gives generalizations of some inequalities regarding the $s^{th}$ derivative of a polynomial having zeros outside a circle. Besides, our result gives interesting refinements of some well-known results.

We study convexity properties of the zeros of some special functions that follow from the convexity theorem of Sturm. We prove results on the intervals of convexity for the zeros of Laguerre, Jacobi and ultraspherical polynomials, as well as functions related to them, using transformations under which the zeros remain unchanged. We give upper as well as lower bounds for the distance between con...

This note is concerned with the zeros of holomorphic Hecke cusp forms of large weight on the modular surface. The zeros of such forms are symmetric about three geodesic segments and we call those zeros that lie on these segments, real. Our main results give estimates for the number of real zeros as the weight goes to infinity. Mathematics Subject Classification (2010). Primary: 11F11, 11F30. Se...

Monotonicity of zeros of orthogonal Laurent polynomials associated with a strong distribution with respect to a parameter is discussed. A natural analog of a classical result of A. Markov is proved. Recent results of Ismail and Muldoon based on the Hellman-Feynman theorem are also extended to a monotonicity criterion for zeros of Laurent polynomials. Results concerning the behaviour of extreme ...

A theorem of Stieltjes proves that, given any sequence {pn}n=0 of orthogonal polynomials, there is at least one zero of pn between any two consecutive zeros of pk if k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends, under certain conditions, to the zeros of Jacobi polynomials from different sequences. In particular, we prove that the zeros of P n+1 inte...

Outlier detection is an important task for the statistical analysis of multivariate data, because often the outliers contain important information about the data structure. In compositional data, represented usually as proportions subject to a unit sum constraint, the ratios between the parts (variables) contain the essential information. This inherent property is, however, incompatible with th...

The minimum phase systems have ultimately been redefined for LTI discrete-time systems, at first SISO and later square MIMO ones, as those systems for which minimum variance control (MVC) is asymptotically stable, or in other words those systems who have ‘stable’ zeros, or at last as ‘stably invertible’ systems. The redefinition has soon been extended to nonsquare discrete-time LTI MIMO systems...