System Of Rational Functions Research Articles

Periodic points of polynomials over finite fields

Fix an odd prime p p . If r r is a positive integer and f f is a polynomial with coefficients in F p r \mathbb {F}_{p^r} , let P p , r ( f ) P_{p,r}(f) be the proportion of P 1 ( F p r ) \mathbb {P}^1\left (\mathbb {F}_{p^r}\right ) that is periodic with respect to f f . We show that as r r increases, the expected value of P p , r ( f ) P_{p,r}(f) , as f f ranges over quadratic polynomials, is less than 22 / ( log ⁡ log ⁡ p r ) 22/\left (\log {\log {p^r}}\right ) . This result follows from a uniformity theorem on specializations of dynamical systems of rational functions over residually finite Dedekind domains. The specialization theorem generalizes previous work by Juul et al. that holds for rings of integers of number fields. Moreover, under stronger hypotheses, we effectivize this uniformity theorem by using the machinery of heights over general global fields; this version of the theorem generalizes previous work of Juul on polynomial dynamical systems over rings of integers of number fields. From these theorems we derive effective bounds on image sizes and periodic point proportions of families of rational functions over finite fields.

Functional Feature Extraction for Hyperspectral Image Classification With Adaptive Rational Function Approximation

A functional feature extraction method based on rational function approximation for hyperspectral image (HSI) classification is proposed. In digital imagery, the spectral information of a pixel can be regarded as a 1-D signal. An HSI is composed of these 1-D signals arranged in a certain spatial structure. According to the functional characteristic of hyperspectral data, 1-D signals can be approximated by a linear combination of basis functions. Thus, a joint rational basis function system (JRBFS) based on class adaptivity is here first built for an HSI by adaptive Fourier decomposition (AFD). Second, the functional representations (FRs) and corresponding reconstructed spectral curves are obtained by decomposing the original spectral information in a JRBFS. Furthermore, the functional spectral-spatial features are extracted on the basis of FRs by an edge-preserving filtering method, FR-EPFs. Finally, the functional spectral-spatial features are used for HSI classification by SVM. Experimental results for five commonly used HSI data sets demonstrate the effectiveness and advantages of the proposed method FR-EPFs.

Generalized Rational Variable Projection With Application in ECG Compression

In this paper we develop an adaptive transform-domain technique based on rational function systems. It is of general importance in several areas of signal theory, including filter design, transfer function approximation, system identification, control theory etc. The construction of the proposed method is discussed in the framework of a general mathematical model called variable projection. First we generalize this method by adding dimension type free parameters. Then we deal with the optimization problem of the free parameters. To this order, based on the well-known particle swarm optimization (PSO) algorithm, we develop the multi-dimensional hyperbolic PSO algorithm. It is designed especially for the rational transforms in question. As a result, the system along with its dimension is dynamically optimized during the process. The main motivation was to increase the adaptivity while keeping the computational complexity manageable. We note that the proposed method is of general nature. As a case study the problem of electrocardiogram (ECG) signal compression is discussed. By means of comparison tests performed on the PhysioNet MIT-BIH Arrhythmia database we demonstrate that our method outperforms other transformation techniques.

Factorial-type Schur functions, orthogonal rational functions, and discrete dressing chains

A relationship between orthogonal rational functions and discrete integrable systems is studied by an approach based on Schur-type symmetric functions. A system of orthogonal rational functions is constructed using a multiparameter deformation of the Schur functions. Spectral equations for the orthogonal rational functions are derived by using properties of the Schur-type symmetric functions. The compatibility condition of the spectral equations induces a discrete dressing chain which is a Toda-type discrete integrable system describing dressing transformations for orthogonal rational functions.

Convergence of Fourier series on the systems of rational functions on the real axis

The systems of rational functions {Φn(z)}, n ∈ ℤ; that are orthonormalized on the real axis ℝ and are defined by the fixed set of points a := {ak}k = 0∞, (Im ak > 0) and b := {bk}k = 1∞, (Im bk 1; and the pointwise convergence of Fourier series on the systems {Φn(t)}, n ∈ ℤ; are studied under the certain restrictons on the sequences of poles of these systems. Some analogs of the classical Jordan–Dirichlet and Dini–Lipschitz criteria of convergence of Fourier series in a trigonometric system are constructed.

Backward shift invariant subspaces with applications to band preserving and phase retrieval problems

The band preserving and phase retrieval problems have long been interested and studied. In this paper, we, for the first time, give solutions to these problems in terms of backward shift invariant subspaces. The backward shift method among other methods seems to be direct and natural. We show that a function , with , that makes the band of fg to be within that of f if and only if g divided by an inner function related to f, belongs to some backward shift invariant subspace in relation to f. By the construction of backward shift invariant space, the solution g is further explicitly represented through the span of the rational function system whose zeros are those of the Laplace transform of f. As an application, we also use the backward shift method to give a characterization for the solutions of the phase retrieval problem. Copyright © 2015 John Wiley & Sons, Ltd.

Common composites of triangular polynomial systems and hash functions

We study common composites of triangular polynomial and rational function systems with favorable effects under composition: polynomial degree growth. We construct classes of such systems that do not have common composites. This property makes them suitable for the construction of a recently proposed hash function. We give estimates for the number of collisions of this hash function using these systems. We also mention as future work the study of common composites of systems with sparse representation and pose an open problem related to their usability as hash functions.

Finiteness of fixed equilibrium configurations of point vortices in the plane with a background flow

For a dynamic system consisting of n point vortices in an ideal plane fluid with a steady, incompressible and irrotational background flow, a more physically significant definition of a fixed equilibrium configuration is suggested. Under this new definition, if the complex polynomial w that determines the aforesaid background flow is non-constant, we have found an attainable generic upper bound for the number of fixed equilibrium configurations. Here, m = deg w, i0 is the number of species, and each ni is the number of vortices in a species. We transform the rational function system arising from fixed equilibria into a polynomial system, whose form is good enough to apply the BKK theory (named after Bernshtein (1975 Funct. Anal. Appl. 9 183–5), Khovanskii (1978 Funct. Anal. Appl. 12 38–46) and Kushnirenko (1976 Funct. Anal. Appl. 10 233–5)) to show the finiteness of its number of solutions. Having this finiteness, the required bound follows from Bézout's theorem or the BKK root count by Li and Wang (1996 Math. Comput. 65 1477–84).

Best two‐parameter rational approximation algorithm of given order: a variation of adaptive Fourier decomposition

The concept of mono‐component is widely used in nonstationary signal processing and time‐frequency analysis. In this paper, we construct several classes of complete rational function systems in the Hardy space, whose boundary values are mono‐components. Then, we propose a best approximation algorithm (BAA) based on optimal selections of two parameters in the orthonormal bases according to the approximation error. Effectiveness of BAA is evaluated by comparison experiments with the classical Fourier decomposition algorithm. It is also shown that BAA has promising results for filtering out noises and dealing with real‐world signals. Copyright © 2014 John Wiley & Sons, Ltd.

Numerical approach of flow and mass transfer on nonlinear stretching sheet with chemically reactive species using rational Jacobi collocation method

PurposeThe aim is to present in this paper an effective strategy in dealing with a semi‐infinite interval by using a suitable mapping that transforms a semi‐infinite interval to a finite interval.Design/methodology/approachThe authors introduce a new orthogonal system of rational functions induced by general Jacobi polynomials with the parameters alpha and beta. It is more flexible in applications. In particular, alpha and beta could be regulated, so that the systems are mutually orthogonal in certain weighted Hilbert spaces.FindingsThis approach is applied for solving a non‐linear system two‐point boundary value problem (BVP) on semi‐infinite interval, describing the flow and diffusion of chemically reactive species over a nonlinearly stretching sheet immersed in a porous medium. The new approach reduces the solution of a problem to the solution of a system of algebraic equations.Originality/valueThe paper presents an effective strategy in dealing with a semi‐infinite interval by using a suitable mapping that transforms a semi‐infinite interval to a finite interval.

Rational Gauss-Chebyshev Quadratures for Wireless Performance Analysis

The convergence rate of the classical Gauss-Chebyshev quadrature (GCQ) rule for wireless performance as a function of the signal-to-noise ratio (SNR) is analyzed. The convergence rate is found to decline as the SNR varies from high to low. Thus, at low SNR, the number of nodes needed to achieve the desired accuracy is extremely high. A generalized rational GCQ rule is thus adopted. The nodes and weights are then computed from a system of orthogonal rational functions. The rational GCQ is much more accurate than the classical GCQ over the entire SNR range. Especially at low SNR, the accuracy is extremely high.

KMS STATES ON FINITE-GRAPH C∗-ALGEBRAS

We study Kubo-Martin-Schwinger(KMS) states on finite-graph C∗-algebras with sinks and sources. We compare finite-graph C∗-algebras with C∗-algebras associated with complex dynamical systems of rational functions. We show that if the inverse temperature β is large, then the set of extreme β-KMS states is parametrized by the set of sinks of the graph. This means that the sinks of a graph correspond to the branched points of a rational function from the point of KMS states. Since we consider graphs with sinks and sources, left actions of the associated bimodules are not injective. Then the associated graph C∗-algebras are realized as (relative) Cuntz-Pimsner algebras studied by Katsura. We need to generalize Laca-Neshvyev's theorem of the construction of KMS states on Cuntz-Pimsner algebras to the case that left actions of bimodules are not injective.

RAIT: the Rational Approximation and Interpolation Toolbox for Matlab, with Experiments on ECG Signals

There is a wide range of applications of rational function systems. Including in system, control theories and signal processing. A special class of rational functions, the so-called Blaschke functions and the orthonormal Malmquist--Takenaka (MT) systems are effectively used for representing signals especially electrocardiograms. We present our project on a general Matlab library for rational function systems and their applications. It contains Blaschke functions, MT systems and biorthogonal systems. We implemented not only the continuous but the discrete versions as well, since in applications the latter one is needed. The complex and real interpretations are both available. We also built in methods for finding the poles automatically. Also, some interactive GUIs were implemented for visual demonstration that help the users in understanding the roles of certain parameters such as poles, multiplicity etc.

Discrete Symbol Calculus

This paper deals with efficient numerical representation and manipulation of differential and integral operators as symbols in phase-space, i.e., functions of space x and frequency $\xi$. The symbol smoothness conditions obeyed by many operators in connection to smooth linear partial differential equations allow fast-converging, nonasymptotic expansions in adequate systems of rational Chebyshev functions or hierarchical splines to be written. The classical results of closedness of such symbol classes under multiplication, inversion, and taking the square root translate into practical iterative algorithms for realizing these operations directly in the proposed expansions. Because symbol-based numerical methods handle operators and not functions, their complexity depends on the desired resolution N very weakly, typically only through $\log N$ factors. We present three applications to computational problems related to wave propagation: (1) preconditioning the Helmholtz equation, (2) decomposing wave fields into one-way components, and (3) depth extrapolation in reflection seismology. The software is made available in the software sections of math.mit.edu/~laurent and www.math.utexas.edu/users/lexing.

The free interpolation problem and basis property of a system of rational functions in weighted Hardy spaces

The paper considers the free interpolation problem in the Hardy weighted space Hp(ρ). It is assumed that the weight ρ has unique singularity of the order αonly at the point 1. Particularly, for α − p[(α + 1)p−1] > 0 the corresponding free interpolation problem is stated and its solvability is proved.

A family of tridiagonal pairs and related symmetric functions

A family of tridiagonal pairs which appear in the context of quantum integrable systems is studied in details. The corresponding eigenvalue sequences, eigenspaces and the block tridiagonal structure of their matrix realizations with respect the dual eigenbasis are described. The overlap functions between the two dual basis are shown to satisfy a coupled system of recurrence relations and a set of discrete second-order $q-$difference equations which generalize the ones associated with the Askey-Wilson orthogonal polynomials with a discrete argument. Normalizing the fundamental solution to unity, the hierarchy of solutions are rational functions of one discrete argument, explicitly derived in some simplest examples. The weight function which ensures the orthogonality of the system of rational functions defined on a discrete real support is given.

Dual pairs of orthogonal systems of rational matrix-valued functions on the unit circle

SummaryWe analyse certain dual pairs of orthogonal systems of rational matrix-valued functions with poles not located on the unit circle. The rational matrix-valued functions belonging to such kind of pairs are recursively connected similar as the functions of the first and the second kind in the classical theory of orthogonal polynomials on the unit circle. We present some basic facts on these dual pairs and, in particular, we show that the rational matrix-valued functions in question admit specific integral representations.

A moment problem and a family of integral evaluations

We study the Al-Salam-Chihara polynomials when q > 1. Several solutions of the associated moment problem are found, and the orthogonality relations lead to explicit evaluations of several integrals. The polynomials are shown to have raising and lowering operators and a second order operator equation of Sturm-Liouville type whose eigenvalues are found explicitly. We also derive new measures with respect to which the Ismail-Masson system of rational functions is biorthogonal. An integral representation of the right inverse of a divided difference operator is also obtained.

Elements of Prescribed Order, Prescribed Traces and Systems of Rational Functions Over Finite Fields

Let k ≥ 1 and f1 .... , fr ∈ Fqk (x) be a system of rational functions forming a strongly linearly independent set over a finite field Fq. Let γ1 ..... γr ∈ Fq be arbitrarily prescribed elements. We prove that for all sufficiently large extensions Fqkm, there is an element ξ ∈ Fqkm of prescribed order such that TrFqkm/Fq (fi (ξ))= γi for i = 1, ..., r, where TrFqkm/Fq is the relative trace map from Fqkm onto Fq. We give some applications to BCH codes, finite field arithmetic and ordered orthogonal arrays. We also solve a question of Helleseth et al. (Hypercubic 4 and 5-designs from Double-Error-Correcting codes, Des. Codes. Cryptgr. 28(2003). pp. 265-282) completely.

A rational approximation and its applications to nonlinear partial differential equations on the whole line

An orthogonal system of rational functions is discussed. Some inverse inequalities, imbedding inequalities and approximation results are obtained. Two model problems are considered. The stabilities and convergences of proposed rational spectral schemes and rational pseudospectral schemes are proved. The techniques used in this paper are also applicable to other problems on the whole line. Numerical results show the efficiency of this approach.