Set Of Rational Functions Research Articles

On the uniqueness of two different classes of meromorphic functions under the sharing of two sets of rational functions

We study the uniqueness problem of two special classes of meromorphic functions sharing two sets of small functions. One of the considered classes has the property to include the Selberg class L functions, while the other class is comprising of arbitrary meromorphic functions having finitely many poles. We obtain a number of results which extend and improve a number of earlier results such as Li [ Proc. Amer. Math. Soc., 138 (2010), 2071-2077], Lin and Lin [Filomat 30 (2016), 3795-3806] and others. We have also been able to replace the strict CM (IM) sharing of the sets in our theorems to almost CM (almost IM) sharing.

Counting points of bounded height in monoid orbits

Given a set of endomorphisms on $$\mathbb {P}^N$$ , we establish an upper bound on the number of points of bounded height in the associated monoid orbits. Moreover, we give a more refined estimate with an associated lower bound when the monoid is free. Finally, we show that most sets of rational functions in one variable satisfy these more refined bounds.

Circuit Model Extraction of Parallel-Connected Dual-Passband Coupled-Resonator Filters

In this paper, an analytical coupling matrix extraction scheme is proposed for parallel-connected dual-passband coupled-resonator filters for the first time. Model-based vector fitting technique is adopted to fit the measured responses of the dual-passband filter (DPBF) with a set of rational functions, from which a transversal coupling matrix can be constructed. A novel coupling matrix transformation strategy is developed to transform the transversal coupling matrix to the one corresponding to the topology of a parallel-connected DPBF. A practical filter is used as a test example to demonstrate the extraction procedure. In diagnosing the coupling matrix of the test filter, two kinds of ambiguity problems are discussed in detail and are solved by a perturbation technique. This paper provides a general theoretic framework for computer-aided tuning of a parallel-connected dual-passband or multiple-passband filter.

On nonlinear simultaneous approximation problems

This paper deals with best nonlinear simultaneous approximation problems to two continuous functions under a general monotonic norm on . Characterizations results of Kolmogorov type and of alternation type and uniqueness results for the best simultaneous approximation are established. Applications are given to the set of rational functions.

Uniform perfectness of Julia sets of rational functions over P<sup>1</sup>(C<sub><italic>p</italic></sub>)

In this paper,we show that the Julia set of rational functions over P1(Cp) is uniformly perfect.

Epsilon Systems on Geometric Crystals of type An

We introduce an epsilon system on a geometric crystal of type $A_n$, which is a certain set of rational functions with some nice properties. We shall show that it is equipped with a product structure and that it is invariant under the action of tropical R maps.

On algebraic automorphisms and their rational invariants

Let X be an affine irreducible variety over an algebraically closed field k of characteristic zero. Given an automorphism Φ, we denote by k(X)Φ its field of invariants, i.e., the set of rational functions f on X such that f º Φ = f. Let n(Φ) be the transcendence degree of k(X)Φ over k. In this paper we study the class of automorphisms Φ of X for which n(Φ) = dim X - 1. More precisely, we show that under some conditions on X, every such automorphism is of the form Φ = ϕg, where ϕ is an algebraic action of a linear algebraic group G of dimension 1 on X, and where g belongs to G. As an application, we determine the conjugacy classes of automorphisms of the plane for which n(Φ) = 1.

Comparison of the best uniform approximations of analytic functions in the disk and on its boundary

Denote by C A the set of functions that are analytic in the disk |z| < 1 and continuous on its closure |z| ≤ 1; let ℜ n , n = 0, 1, 2, ..., be the set of rational functions of degree at most n. Denote by R n (f) (R n (f) A ) the best uniform approximation of a function f ∈ C A on the circle |z| = 1 (in the disk |z| ≤ 1) by the set ℜ n . The following equality is proved for any n ≥ 1: sup{R n (f) A /R n (f): f ∈ C A ∖ ℜ n } = 2. We also consider a similar problem of comparing the best approximations of functions in C A by polynomials and trigonometric polynomials.

On the Probability of the Number of Solutions for the P4P Problem

This paper studies the multi-solution phenomenon for the perspective four point (P4P) problem from geometric and algebraic aspects. We give a pure geometric proof that the P4P problem could have up to five solutions. We also give a clear picture on how these five solutions could be realized. We prove that with probability one, the P4P problem has a unique solution which can be represented by a set of rational functions in the parameters. The simulant experiments show that to solve the P4P problem with the rational functions is stable and accurate.

Impact of the shape of functions on the orders of piecewise polynomial and rational approximation

Let be the set of functions on a finite interval such that the divided differences of order of these functions are non-negative for all systems of distinct points . Let be the set of piecewise polynomial splines of order with free knots, and the set of rational functions , where and are polynomials of order . For the classes , where is the unit ball in , the precise orders and of the best approximations in the metrics are found for .

Exact evaluation of the simple cubic lattice Green function for a general lattice point

The simple cubic lattice Green function is investigated, where {l, m, n} denotes a set of integers and w = u + iv is a complex variable in the (u, v) plane. In particular, it is shown that the modified Green function can be expressed in the ξ-parametric form where K(k) and E(k) are complete elliptic integrals of the first and second kind respectively, with a modulus The connection between the parameter ξ and the variable w is given by and {Rj(l, m, n; ξ) : j = 0, 1, 2, 3} is a set of rational functions of ξ. It is found that the complete elliptic integral formulae for the Green functions {(n, n, n; w) : n 1, 2, 3, 4} and {(2n, n, n; w) : n = 1, 2, 3, 4} can be factorized as a product of two linear forms in K(k) and E(k) whose coefficients are rational functions of the parameter ξ. On the basis of these explicit results it is conjectured that this factorization propery is valid for all integer values of n.

On the Multiplicative Complexity of the Inversion and Division of Hamiltonian Quaternions

The multiplicative complexity of a finite set of rational functions is the number of essential multiplications and divisions that are necessary and sufficient to compute these rational functions. We prove that the multiplicative complexity of inversion in the division algebra \H of Hamiltonian quaternions over the reals, that is, the multiplicative complexity of the coordinates of the inverse of a generic element from \H , is exactly eight. Furthermore, we show that the multiplicative complexity of the left and right division of Hamiltonian quaternions is at least eleven.

Connectedness of Julia Sets of Rational Functions

For a polynomial P it is well known that its Julia set \({\cal J_P}\) is connected if and only if the orbits of the finite critical points are bounded. But there is no such simple criteria for the connectedness of the Julia set of a rational function. Indeed, up to the very nice result of Shishikura that any rational function which has one repelling fixed point only has a connected Julia set almost nothing is known on the connectivity. In the first part of the paper we give constructive sufficient conditions for a basin of attraction to be completely invariant and the Julia set to be connected. Then it is shown that the connectedness of a basin of attraction depends heavily on the fact whether the critical points from the basin tend to the attracting fixed point z0 via a preimage of z0 or not. As a consequence we obtain for instance that rational functions with a finite postcritical set or with a Fatou set which contains no Herman rings and each component of which contains at most one critical point, counted without multiplicity, have a connected Julia set.

Uniform approximations of Stieltjes functions by orthogonal projection on the set of rational functions

Let μ be a positive Borel measure having support supp μ ⊂ [1, ∞) and satisfying the conditionf(t−1)−1dμ(t)<∞. In this paper we study the order of the uniform approximation of the function $$\widehat\mu = \smallint \tfrac{{d\mu (t)}}{{t - z}}, z \in \mathbb{C},$$ on the disk |z|≤1 and on the closed interval [−1, 1] by means of the orthogonal projection of\(\widehat\mu \) on the set of rational functions of degreen. Moreover, the poles of the rational functions are chosen depending on the measure μ. For example, it is shown that if supp μ is compact and does not contain 1, then this approximation method is of best order. But if supp μ=[1,a],a>1, the measure μ is absolutely continuous with respect to the Lebesgue measure, and\(\mu '\left( t \right) _\frown ^\smile \left( {t - 1} \right)^\alpha \) fort∈[1,a] and some α>0, then the order of such an approximation differs from the best only by\(\sqrt n \).

Optimum parameters for the generalized ADI method

In the context of the generalized ADI method, we are concerned with the problem of finding in the set of rational functions r with numerator degree m and denominator degree n an element \(r^*\) that minimizes \(\) where E,F are disjoint real intervals. By extending a recent analysis by Levin and Saff, we present an explicit formula for choosing the pair (m,n) for given m +n. Furthermore, we provide a characterization of \(r^*\) and a Remes type algorithm for its determination. Extensive numerical computations furnish some comparison of \(r^*\) with asymptotically optimal solutions based on Fejer-Walsh and Leja-Bagby points.

On the residual Julia sets of rational functions

We consider the subset of the Julia set called the residual Julia set, which comes from an analogy of the residual limit set of a Kleinian group. We give a necessary and sufficient condition in order that the residual Julia set is empty in the case of hyperbolic rational functions.

GaAs MESFET characterization using least squares approximation by rational functions

The authors propose a method of characterizing active devices such as the FET by describing S-parameters with a set of rational functions of angular frequency. The set of rational functions is uniquely determined by only 27 coefficients, while the conventional method using tabulated S-parameters requires eight times the number of sampling points (a typical case might require 404 data points in floating-point notation). This drastically reduces the database size required to give adequate information for circuit design. A method for determining the equivalent circuit is described. Since the equivalent circuit is determined from the set of rational functions, no additional measurements are needed to determine extrinsic elements. In conventional methods, selection of initial values affects the final results. In the present method, reliable initial values are extracted from the rational functions' coefficients. The calculated S-parameters of three GaAs MESFETs having different gate widths agree closely with those measured by wafer-probe. >

Capacities of Julia sets of rational functions

For any polynomialp(z)=a0zn+a|zn−1+⋯+an, a0≠0, n⩾2,F is the Julia set andμ* is the equilibrium distribution onF. Hans Brolin[1] proved thatγ (F)>0, andSμ*=F. Up to now, we know nothing about rational functions. The aim of this paper is to discuss the case of rational functions.

Robust controller design methodology for multivariable chemical processes

In this work, a methodology for a controller design of multivariable linear time invariant systems is presented. The singular values of the return difference operator and the time constant for each control loop can be easily specified through a set of rational functions in the Laplace domain. The frequency response of an ideal controller is directly obtained, allowing the implementation of different controller structures (PI, PID etc.). First, the discussion is focused on some new results of the perturbation theory which constitute the basis of the proposed methodology. Later, a complete analysis of the resulting control scheme is presented. Its robustness with respect to stability, performance and interactions as well as the normality of the closed-loop system is demonstrated. Some implementation aspects for the formulation of the ideal controller matrix are discussed. Approximation of this ideal controller to different realizable structures is then discussed. Finally, a case study of the dual-composition control of two typical distillation columns currently found in the literature is presented.

Convergence of a class of Borel-Padé-type approximants

In this paper we propose the use of a recently introduced set of rational functions, the so-called Pade-type approximants, to perform the analytic continuation of Borel transforms into a cut complex plane. It is shown that, by choosing a suitable set of orthogonal generating polynomials, the convergence of the Borel-Pade-type approximation method can be rigorously proven.