The zeros of semi-orthogonal functions with respect to a probability measure µ supported on the unit circle can be applied to obtain Szeg˝ o quadrature formulas. The discrete measures generated by these formulas weakly converge to the orthogonality measure µ. In this paper we construct families of semi-orthogonal functions with interlacing zeros, and give a representation of the support of µ in...

The inverse scattering transformation for a novel nonlocal Lakshmanan–Porsezian–Daniel (LPD) equation with rapidly decaying initial data is studied in the framework of Riemann–Hilbert problem. Firstly, integrable LPD corresponding to $$3\times 3$$ Lax pair proposed. Secondly, process left-right matrix Riemann–Hilbert(RH) problem constructed. analytical properties and symmetry relations Jost fun...

Rankin and Swinnerton-Dyer [R, S-D] prove that all zeros of the Eisenstein series Ek in the standard fundamental domain for Γ lie on A := {eiθ : π 2 ≤ θ ≤ 2π 3 }. In this paper we generalize their theorem, providing conditions under which the zeros of other modular forms lie only on the arc A. Using this result we prove a speculation of Ono, namely that the zeros of the unique “gap function” in...

We present a general, rigorous theory of partition function zeros for lattice spin models depending on one complex parameter. First, we formulate a set of natural assumptions which are verified for a large class of spin models in a companion paper [5]. Under these assumptions, we derive equations whose solutions give the location of the zeros of the partition function with periodic boundary con...

There are several numerical methods for computing approximate zeros of a given univariate polynomial. In this paper, we develop a simple and novel method for determining sharp upper bounds on errors in approximate zeros of a given polynomial using Rouche's theorem from complex analysis. We compute the error bounds using non-linear optimization. Our bounds are scalable in the sense that we compu...

We give a numerical criterion for a badly conditioned zero of a system of analytic equations to be part of a cluster of two zeros.

We dealt in a series of previous publications with some geometric aspects of the mappings by functions obtained as analytic continuations to the whole complex plane of general Dirichlet series. Pictures illustrating those aspects contain a lot of other information which has been waiting for a rigorous proof. Such a task is partially fulfilled in this paper, where we succeeded among other things...

1 We investigate the zeros of polynomial solutions to the differential-difference equation P n+1 (x) = A n (x)P ′ n (x) + B n (x)P n (x), n = 0, 1,. .. where A n and B n are polynomials of degree at most 2 and 1 respectively. We address the question of when the zeros are real and simple and whether the zeros of polynomials of adjacent degree are interlac-ing. Our result holds for general classe...