On the poles of topological zeta functions

We study the topological zeta function associated to a polynomial with complex coefficients. This is a rational function in one variable, and we want to determine the numbers that can occur as a pole of some topological zeta function; by definition these poles are negative rational numbers. We deal with this question in any dimension. Denote has a pole in . We show that is a subset of ; for and , the last two authors proved before that these are exactly the poles less than . As the main result we prove that each rational number in the interval is contained in .

     

On the poles of the resolvent in Calkin algebra

In the present note, we study the problem of lifting poles in Calkin algebra on a separable infinite-dimensional complex Hilbert space . We show by an example that such lifting is not possible in general, and we prove that if zero is a pole of the resolvent of the image of an operator in the Calkin algebra, then there exists a compact operator for which zero is a pole of if and only if the index of is zero on a punctured neighbourhood of zero. Further, a useful characterization of poles in Calkin algebra in terms of essential ascent and descent is provided.

     

On the smallest poles of Igusa's p-adic zeta functions

Let K be a p-adic field. We explore Igusa's p-adic zeta function, which is associated to a K-analytic function on an open and compact subset of Kⁿ. First we deduce a formula for an important coefficient in the Laurent series of this meromorphic function at a candidate pole. Afterwards we use this formula to determine all values less than −1/2 for n=2 and less than −1 for n=3 which occur as the real part of a pole.     

On the poles of regular differentials of singular curves

We describe the pole behaviour of the regular differentials of projective algebraic curves in terms of discrete invariants of the singular points.     

On the zeros and poles of Padé approximants toe z

Summary In this paper, we study the location of the zeros and poles of general Padé approximats toe ^z. The location of these zeros and poles is useful in the analysis of stability for related numerical methods for solving systems of ordinary differential equations.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688, and by the University of South Fla. Research Council.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Atomic Energy Commission under Grant AT(11-1)-2075.     

On the orthogonal rational functions with arbitrary poles and interpolation properties

Let = z_n,mm=1ⁿ with |z_n,m| < 1, n = 1,2,…, be an arbitrary sequence of complex numbers. We generalize the orthogonal rational functions with poles at . We study the weak convergence and the interpolation properties of the orthogonal rational functions.     

On the structure invariants of proper rational matrices with prescribed finite poles

Abstract

The algebraic structure of matrices defined over arbitrary fields whose elements are rational functions with no poles at infinity and prescribed finite poles is studied. Under certain very general conditions, they are shown to be matrices over an Euclidean domain that can be classified according to the corresponding invariant factors. The relationship between these invariants and the local Wiener–Hopf factorization indices will be clarified. This result can be seen as an extension of the classical theorem on pole placement by Rosenbrock in control theory.     

On the q-Bernstein polynomials of rational functions with real poles

The paper aims to investigate the convergence of the q  -Bernstein polynomials _Bn,q(f;x)$B_{n,q}(f;x)$ attached to rational functions in the case q>1$q>1$. The problem reduces to that for the partial fractions (x−α)^−j${(x-\alpha )}^{-j}$, j∈N$j\in \mathbb{N}$. The already available results deal with cases, where either the pole α   is simple or α≠q^−m$\alpha \ne q^{-m}$, m∈_N0$m\in {\mathbb{N}}_0$. Consequently, the present work is focused on the polynomials _Bn,q(f;x)$B_{n,q}(f;x)$ for the functions of the form f(x)=(x−q^−m)^−j$f(x)={(x-q^{-m})}^{-j}$ with j?2$j?2$. For such functions, it is proved that the interval of convergence of {_Bn,q(f;x)}$\{B_{n,q}(f;x)\}$ depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously.     

On the stability of rational approximations to the cosine with only imaginary poles

Letr(z) be a rational approximation to cosz with only imaginary poles ±i ₁ ^–1/2 , ±i ₂ ^–1/2 , ..., ±i _m ^–1/2 such that |cozz –r(z)| C|z|^2m+2 as |z| 0. If the degree of the numerator ofr(z) is less than or equal to 2m and _i m/4,i=1, ...,m, then we show that |r(z)|1 for all realz.     

On existence of quadratic differentials with poles of high orders

Conditions of existence of a quadratic differential which has poles of high orders at the points c_k, k=1,...,p, and is the limit of a sequence of quadratic differentials of a special type are established. The quadratic differential mentioned has no poles of order greater than 2 and has poles of order 2 at the points situated in a suitable uniform way on the circles |z-c_k|=ɛ_k,ɛ_k→0, k=1, ..., p. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 212, 1994, pp. 129–138 Translated by G. V. Kuz'mina     

On the zeros and poles of Padé approximants toe z . III

Summary In this paper, we continue our study of the location of the zeros and poles of general Padé approximants toe ^z . We state and prove here new results for the asymptotic location of the normalized zeros and poles for sequences of Padé approximants toe ^z , and for the asymptotic location of the normalized zeros for the associated Padé remainders toe ^z . In so doing, we obtain new results for nontrivial zeros of Whittaker functions, and also generalize earlier results of Szegö and Olver.Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2688Research supported in part by the Air Force Office of Scientific Research under Grant AFOSR-74-2729, and by the Energy Research and Development Administration (ERDA) under Grant EY-76-S-02-2075     

On L-functions with poles satisfying Maass's functional equation

We prove a generalisation of the Converse Theorem of Maass for Dirichlet series with a finite number of poles.     

On dominant poles and model reduction of second order time-delay systems

The method known as the dominant pole algorithm (DPA) has previously been successfully used in combination with model order reduction techniques to approximate standard linear time-invariant dynamical systems and second order dynamical systems. In this paper, we show how this approach can be adapted to a class of second order delay systems, which are large scale nonlinear problems whose transfer functions have an infinite number of simple poles. Deflation is a very important ingredient for this type of methods. Because of the nonlinearity, many deflation approaches for linear systems are not applicable. We therefore propose an alternative technique that essentially removes computed poles from the system?s input and output vectors. In general, this technique changes the residues, and hence, modifies the order of dominance of the poles, but we prove that, under certain conditions, the residues stay near the original residues. The new algorithm is illustrated by numerical examples.