Polynomial Decomposition Research Articles

FACTORIZATION OF SOME COMPOSITE POLYNOMIALS OVER FINITE FIELDS

In this paper we study the factorization of composite polynomials, constructed with some polynomial composition methods, into irreducible factors over finite fields. The main aim of this paper is to provide a proof of a useful theorem of Varshamov (1973), which had been stated by him, without proof.

Pseudospectra of exponential matrix polynomials

This paper extends a result of Graillat, which was based on Rump’s Theorem, to show that pseudospectra of matrix polynomials expressed in other bases, and also some nonlinear matrix functions, are unaffected by drawing the matrix coefficients from certain structured families. We show by example that this behaviour is not universal. This result is of interest because computation of structured pseudospectra is much more expensive in general than computation of unstructured pseudospectra, and in the cases covered by Rump’s Theorem there is no point going to the extra effort.The paper also contains some examples that may be of interest to the community, including an analytical solution of a nested exponential equation that shows why exponential polynomial decomposition algorithms may be of interest.

Piecewise polynomial chaos expansion with an application to brake squeal of a linear brake system

This paper proposes numerical developments based on polynomial chaos (PC) expansions to process stochastic eigenvalue problems efficiently. These developments are applied to the problem of linear stability calculations for a simplified brake system: the stability of a finite element model of a brake is investigated when its friction coefficient or the contact stiffness are modeled as random parameters. Getting rid of the statistical point of view of the PC method but keeping the principle of a polynomial decomposition of eigenvalues and eigenvectors, the stochastic space is decomposed into several elements to realize a low degree piecewise polynomial approximation of these quantities. An approach relying on continuation principles is compared to the classical dichotomy method to build the partition. Moreover, a criterion for testing accuracy of the decomposition over each cell of the partition without requiring evaluation of exact eigenmodes is proposed and implemented.Several random distributions are tested, including a uniform-like law for description of friction coefficient variation. Results are compared to Monte Carlo simulations so as to determine the method accuracy and efficiency. Some general rules relative to the influence of the friction coefficient or the contact stiffness are also inferred from these calculations.

Dynamical Genetic Programming in XCSF

A number of representation schemes have been presented for use within learning classifier systems, ranging from binary encodings to artificial neural networks. This paper presents results from an investigation into using a temporally dynamic symbolic representation within the XCSF learning classifier system. In particular, dynamical arithmetic networks are used to represent the traditional condition-action production system rules to solve continuous-valued reinforcement learning problems and to perform symbolic regression, finding competitive performance with traditional genetic programming on a number of composite polynomial tasks. In addition, the network outputs are later repeatedly sampled at varying temporal intervals to perform multistep-ahead predictions of a financial time series.

Guaranteed error bounds for structured complexity reduction of biochemical networks

Biological systems are typically modelled by nonlinear differential equations. In an effort to produce high fidelity representations of the underlying phenomena, these models are usually of high dimension and involve multiple temporal and spatial scales. However, this complexity and associated stiffness makes numerical simulation difficult and mathematical analysis impossible. In order to understand the functionality of these systems, these models are usually approximated by lower dimensional descriptions. These can be analysed and simulated more easily, and the reduced description also simplifies the parameter space of the model. This model reduction inevitably introduces error: the accuracy of the conclusions one makes about the system, based on reduced models, depends heavily on the error introduced in the reduction process.In this paper we propose a method to calculate the error associated with a model reduction algorithm, using ideas from dynamical systems. We first define an error system, whose output is the error between observables of the original and reduced systems. We then use convex optimisation techniques in order to find approximations to the error as a function of the initial conditions. In particular, we use the Sum of Squares decomposition of polynomials in order to compute an upper bound on the worst-case error between the original and reduced systems. We give biological examples to illustrate the theory, which leads us to a discussion about how these techniques can be used to model-reduce large, structured models typical of systems biology.

On functional decomposition of multivariate polynomials with differentiation and homogenization

This paper gives a theoretical analysis for the algorithms to compute functional decomposition for multivariate polynomials based on differentiation and homogenization which were proposed by Ye, Dai, and Lam (1999) and were developed by Faugère, Perret (2006, 2008, 2009). The authors show that a degree proper functional decomposition for a set of randomly decomposable quartic homogenous polynomials can be computed using the algorithm with high probability. This solves a conjecture proposed by Ye, Dai, and Lam (1999). The authors also propose a conjecture which asserts that the decomposition for a set of polynomials can be computed from that of its homogenization and show that the conjecture is valid with high probability for quartic polynomials. Finally, the authors prove that the right decomposition factors for a set of polynomials can be computed from its right decomposition factor space.

Algorithmic approach to simulate Hamiltonian dynamics and an NMR simulation of quantum state transfer

We propose an iterative algorithm to simulate the dynamics generated by any $n$-qubit Hamiltonian. The simulation entails decomposing the unitary time evolution operator $U$ (unitary) into a product of different time-step unitaries. The algorithm product-decomposes $U$ in a chosen operator basis by identifying a certain symmetry of $U$ that is intimately related to the number of gates in the decomposition. We illustrate the algorithm by first obtaining a polynomial decomposition in the Pauli basis of the $n$-qubit Quantum State Transfer unitary by Di Franco et. al. (Phys. Rev. Lett. 101, 230502 (2008)) that transports quantum information from one end of a spin chain to the other; and then implement it in Nuclear Magnetic Resonance to demonstrate that the decomposition is experimentally viable and well-scaled. We furthur experimentally test the resilience of the state transfer to static errors in the coupling parameters of the simulated Hamiltonian. This is done by decomposing and simulating the corresponding imperfect unitaries.

Effects of starter diet supplementation with arginine on broiler production performance and on small intestine morphometry

The effects of starter diet (days 1 to 21) supplemented with arginine (Arg) on the production performance and duodenum and jejunum mucosa morphometry of broilers were studied. Male Cobb broiler chickens (990) were randomly assigned to one of five treatments in a complete random design. Measurements of 33 chicks per treatment were made in six repetitions. The treatments consisted of a basal diet with 1.390% digestible Arg (no supplementation) and four dietary levels (1.490%, 1.590%, 1.690%, and 1.790%), providing a relationship with lysine of 1.103; 1.183; 1.262; 1.341 and 1.421%, respectively. From the age of 22 days on, all birds received conventional grower diet. The data were submitted to regression analysis by polynomial decomposition of the degrees of freedom in relation to the levels of Arg. The Arg supplementation increased (P<0.05) the live weight and the feed conversion ratio without increasing the feed intake of the birds. However, no effect was observed (P>0.05) in the growth phase (days 22 to 42) in the absence of the Arg supplementation. The supplementation of Arg over of NRC recommendation during the starter phase may be necessary for the expression of the maximal weight gain potential in birds. No effect (P<0.05) of Arg dietary supplementation was observed either on small intestine weight and length at any age. However, the duodenum villus:crypt ratio increased and the crypt depth decreased in the first week in response to increasing dietary Arg. It is concluded that broiler Arg dietary supplementation in the starter diet improved production performance and small intestine morphometry, especially in the first week.

Method for Expressing Clinical and Statistical Significance of Ocular and Corneal Wave Front Error Aberrations

The significance of ocular or corneal aberrations may be subject to misinterpretation whenever eyes with different pupil sizes or the application of different Zernike expansion orders are compared. A method is shown that uses simple mathematical interpolation techniques based on normal data to rapidly determine the clinical significance of aberrations, without concern for pupil and expansion order. Corneal topography maps (TOMEY, Inc, Nagoya, Japan) from 30 normal corneas were collected, and the corneal wave front error was analyzed by Zernike polynomial decomposition into specific aberration types for pupil diameters of 3, 5, 7, and 10 mm and Zernike expansion orders of 6, 8, 10, and 12. Using this 4 × 4 matrix of pupil sizes and fitting orders, the best-fitting 3-dimensional functions were determined for the mean and standard deviation of the root-mean-square error for specific aberrations. The functions were encoded into a software application to determine the significance of data acquired from nonnormal cases. The best-fitting functions for 6 types of aberrations were determined: defocus, astigmatism, prism, coma, spherical aberration, and all higher-order aberrations. A clinical screening method of color coding the significance of aberrations in normal, postoperative laser in situ keratomileusis, and keratoconus cases having different pupil sizes and different expansion orders is demonstrated. A method to calibrate wave front aberrometry devices using a standard sample of normal cases was devised. This method could be potentially useful in clinical studies involving patients with uncontrolled pupil sizes or in studies that compare data from aberrometers that use different Zernike fitting-order algorithms.

Algebraic geometry tools for the study of entanglement: an application to spin squeezed states

A short review of algebraic geometry tools for the decomposition of tensors and polynomials is given from the point of view of applications to quantum and atomic physics. Examples of application to assemblies of indistinguishable two-level bosonic atoms are discussed using modern formulations of the classical Sylvester algorithm for the decomposition of homogeneous polynomials in two variables. In particular, the symmetric rank and symmetric border rank of spin squeezed states are calculated as well as their Schrödinger-cat-like decomposition as the sum of macroscopically different coherent spin states; Fock states provide an example of states for which the symmetric rank and the symmetric border rank are different.

Research on a Composite Polynomial Model for Magnetorheological Damper

The reported mathematical models of magnetorheological (MR) damper cannot make a good tradeoff among reflecting the damper’s nonlinear behavior and controllability. Damping characteristic experiments have been conducted on a MR damper. A composite polynomial model has been proposed integrating the experimental investigation and the polynomial model, in which the plot of polynomial coefficient vs. current is divided into two sections to reflect the property of the current saturation, meanwhile, the affections of exciting amplitude and frequency are considered in this model. The reverse model of the proposed model is easy to be obtained, so it is convenient to realize an open-loop control system to achieve a desirable damping force. The parameters of this model are identified using experimental data in a certain frequency and amplitude, as well as diverse currents. Compared numerical simulation with experimental data, it is verified that the proposed model can accurately predict the damping force without modifying the parameters of the model when frequency, amplitude and current changed.

Lifted generalized permutahedra and composition polynomials

We introduce a "lifting'' construction for generalized permutohedra, which turns an $n$-dimensional generalized permutahedron into an $(n+1)$-dimensional one. We prove that this construction gives rise to Stasheff's multiplihedron from homotopy theory, and to the more general "nestomultiplihedra,'' answering two questions of Devadoss and Forcey. We construct a subdivision of any lifted generalized permutahedron whose pieces are indexed by compositions. The volume of each piece is given by a polynomial whose combinatorial properties we investigate. We show how this "composition polynomial'' arises naturally in the polynomial interpolation of an exponential function. We prove that its coefficients are positive integers, and conjecture that they are unimodal. Nous introduisons une construction de "lifting'' (redressement) pour permutaèdres généralisés, qui transforme un permutaèdre généralisé de dimension $n$ en un de dimension $n+1$. Nous démontrons que cette construction conduit au multiplièdre de Stasheff à partir de la théorie d'homotopie, et aux "nestomultiplièdres", ce qui répond à deux questions de Devadoss et Forcey. Nous construisons une subdivision de n'importe quel permutaèdre généralisé dont les pièces sont indexées par compositions. La volume de chaque pièce est donnée par un polynôme dont nous recherchons les propriétés combinatoires. Nous montrons comment ce "polynôme de composition'' surgit naturellement dans l'interpolation d'une fonction exponentielle. Nous démontrons que ses coefficients sont strictement positifs, et nous conjecturons qu'ils sont unimodaux.

Hölder Metric Subregularity with Applications to Proximal Point Method

This paper is mainly devoted to the study and applications of Hölder metric subregularity (or metric $q$-subregularity of order $q\in(0,1]$) for general set-valued mappings between infinite-dimensional spaces. Employing advanced techniques of variational analysis and generalized differentiation, we derive neighborhood and point-based sufficient conditions as well as necessary conditions for $q$-metric subregularity with evaluating the exact subregularity bound, which are new even for the conventional (first-order) metric subregularity in both finite and infinite dimensions. In this way we also obtain new fractional error bound results for composite polynomial systems with explicit calculating fractional exponents. Finally, metric $q$-subregularity is applied to conduct a quantitative convergence analysis of the classical proximal point method (PPM) for finding zeros of maximal monotone operators on Hilbert spaces.

On the Irreducibility of Some Composite Polynomials

In this paper we study the irreducibility of some compos- ite polynomials, constructed by a polynomial composition method over finite fields. Finally, a recurrent method for constructing families of irre- ducible polynomials of higher degree from given irreducible polynomials over finite fields is given.

Methodology investigations on uncertainties propagation in nuclear data evaluation

Abstract In this paper, we discuss distinct ways of calculating nuclear cross-section variance–covariance matrices issued from the spherical optical model such as the deterministic sensitivity method and Monte Carlo method. We also discuss the selection of uncertain parameters using both the local sensitivity and global sensitivity methods (based on spectral methods using a polynomial decomposition) to determine the importance of the input parameters with respect to the final uncertainty. Finally, we attempt to evaluate the differences between the uncertainties generated by methods like least squares method and Bayesian method, both necessitating the use of experimental data. Discussing the results of simulation for the cross-section calculations of the major neutron reactions on 89 Y, our interest was particularly focused on the mathematical rigour and quality of the uncertainties that we obtained.

Computation of connection coefficients and measure modifications for orthogonal polynomials

We observe that polynomial measure modifications for families of univariate orthogonal polynomials imply sparse connection coefficient relations. We therefore propose connecting L (2) expansion coefficients between a polynomial family and a modified family by a sparse transformation. Accuracy and conditioning of the connection and its inverse are explored. The connection and recurrence coefficients can simultaneously be obtained as the Cholesky decomposition of a matrix polynomial involving the Jacobi matrix; this property extends to continuous, non-polynomial measure modifications on finite intervals. We conclude with an example of a useful application to families of Jacobi polynomials with parameters (gamma,delta) where the fast Fourier transform may be applied in order to obtain expansion coefficients whenever 2 gamma and 2 delta are odd integers.

Decompositions of trigonometric polynomials with applications to multivariate subdivision schemes

We study multivariate trigonometric polynomials satisfying the sum-rule conditions of a certain order. Based on the polyphase representation of these polynomials relative to a general dilation matrix, we develop a simple constructive method for a special type of decomposition of such polynomials. These decompositions are of interest in the analysis of convergence and smoothness of multivariate subdivision schemes associated with general dilation matrices. The approach presented in this paper leads directly to constructive algorithms, and is an alternative to the analysis of multivariate subdivision schemes in terms of the joint spectral radius of certain operators. Our convergence results apply to arbitrary dilation matrices, while the smoothness results are limited to two classes of dilation matrices.

On the reducibility of some composite polynomials over finite fields

Let g(x) = x n + a n-1 x n-1 + . . . + a 0 be an irreducible polynomial over $${\mathbb{F}_q}$$ . Varshamov proved that for a = 1 the composite polynomial g(x p ?ax?b) is irreducible over $${\mathbb{F}_q}$$ if and only if $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a = 1 and $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$$ and for the case a ? 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form $${g(x^{r^kp}-x^{r^k})}$$ are also presented. Moreover, Cohen's method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q 3 ? q) = 1.

A complete classification of quintic space curves with rational rotation-minimizing frames

An adapted orthonormal frame ( f 1 , f 2 , f 3 ) on a space curve r ( t ) , where f 1 = r ′ / | r ′ | is the curve tangent, is rotation-minimizing if its angular velocity satisfies ω ⋅ f 1 ≡ 0 , i.e., the normal-plane vectors f 2 , f 3 exhibit no instantaneous rotation about f 1 . The simplest space curves with rational rotation-minimizing frames (RRMF curves) form a subset of the quintic spatial Pythagorean-hodograph (PH) curves, identified by certain non-linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials u ( t ) , v ( t ) , p ( t ) , q ( t ) and their derivatives to be reducible to an analogous expression in just two polynomials a ( t ) , b ( t ) . This condition has been analyzed, thus far, in the case where a ( t ) , b ( t ) are also quadratic, the corresponding solutions being called Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a ( t ) , b ( t ) are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6–as compared to degree 8 for Class I curves–their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.

The basic element method

The basic element method (BEM) for decomposition of the algebraic polynomial via one cubic and three quadratic parabolas (basic elements) is developed within the four-point transformation technique. Representation of the polynomial via basic elements gives a lever for solving various tasks of applied mathematics. So, in the polynomial approximation and smoothing problems, the BEM allows one to reduce the computational complexity of algorithms and increase their resistance to errors by choosing an internal relationship structure between variable and control parameters.