Ordinary Polynomials Research Articles

Vertex-weighted Wiener polynomials of subdivision-related graphs

Singly and doubly vertex-weighted Wiener polynomials are generalizations of both vertex-weighted Wiener numbers and the ordinary Wiener polynomial. In this paper, we show how the vertex-weighted Wiener polynomials of a graph change with subdivision operators, and apply our results to obtain vertex-weighted Wiener numbers.

Some Applications of Generalized Char-Sets of Ordinary Differential Polynomial Sets

The notion of characteristic sets, which are a special kind of triangular sets, is introduced by J. F Ritt and W.T. Wu. Wu extended Ritt’s work and developed the characteristic set method not only in theory but in algorithms, efficiency and its numerous applications. Triangular sets are widely considered as a good representation for the solution of polynomial systems. After the introduction of characteristic sets by Ritt, triangular sets have become an alternative tool for representing the ideal besides the Grobner bases. This paper is about implementation and applications of generalized characteristic sets of ordinary differential polynomial sets defined by author.

Random regression models for the estimation of genetic and environmental covariance functions for growth traits in Santa Ines sheep.

Polynomial functions of different orders were used to model random effects associated with weight of Santa Ines sheep from birth to 196 days. Fixed effects included in the models were contemporary groups, age of ewe at lambing, and fourth-order Legendre polynomials for age to represent the average growth curve. In the random part, functions of different orders were included to model variances associated with direct additive and maternal genetic effects and with permanent environmental effects of the animal and mother. Residual variance was fitted by a sixth-order ordinary polynomial for age. The higher the order of the functions, the better the model fit the data. According to the Akaike information criterion and likelihood ratio test, a continuous function of order, five, five, seven, and three for direct additive genetic, maternal genetic, animal permanent environmental, and maternal permanent environmental effects (k = 5573), respectively, was sufficient to model changes in (co)variances with age. However, a more parsimonious model of order three, three, five, and three (k = 3353) was suggested based on Schwarz's Bayesian information criterion for the same effects. Since it was a more flexible model, model k = 5573 provided inconsistent genetic parameter estimates when compared to the biologically expected result. Predicted breeding values obtained with models k = 3353 and k = 5573 differed, especially at young ages. Model k = 3353 adequately fit changes in variances and covariances with time, and may be used to describe changes in variances with age in the Santa Ines sheep studied.

Approximation Multivariate Distribution with Pair Copula Using the Orthonormal Polynomial and Legendre Multiwavelets Basis Functions

We concentrate on constructing higher dimensional distributions using a fast growing graphical model called Vine/ pair-copula model which has been introduced and developed by Joe, Cooke, Bedford, Kurowica, Daneshkhah, and others. They first construct a n-dimensional copula density by stacking together n(n − 1)/2 bivariate copula density, and they then approximate arbitrarily well these bivariate copulas and the corresponding multivariate distribution using a semi-parametric method. One constructive approach involves the use of minimum information copulas that can be specified to any required degree of precision based on the available data (or possibly based on the experts’ judgments). By using this method, one is able to use a fixed finite dimensional family of copulas to be employed in terms of a vine construction, with the promise of a uniform level of approximation.The basic idea behind this method is to use a two-dimensional ordinary polynomial series to approximate any log-density of a bivariate copula function by truncating the series at an appropriate point. We make this approximation method more accurate and computationally faster by using the orthonormal polynomial and Legendre multiwavelets (LMW) series as the basis functions. We show the derived approximations are more precise and computationally faster with better properties than the one proposed previous method in the literature. We then apply our method to modeling a dataset of Norwegian financial data that was previously analyzed in the series of articles, and finally compare our results by them. At the end, we present a method to simulate from the approximated models, and validate our approximation using the simulation results to recover the same dependency structure of the original data.

Annihilator Ideals of Noncommutative Ring Constructions

Hirano studied the quasi-Armendariz property of rings, and then this concept was generalized by some authors, defining quasi-Armendariz property for skew polynomial rings and monoid rings. In this article, we consider unified approach to the quasi-Armendariz property of skew power series rings and skew polynomial rings by considering the quasi-Armendariz condition in mixed extension ring [R; I][x; σ], introducing the concept of so-called (σ, I)-quasi Armendariz ring, where R is an associative ring equipped with an endomorphism σ and I is an σ-stable ideal of R. We study the ring-theoretical properties of (σ, I)-quasi Armendariz rings, and we obtain various necessary or sufficient conditions for a ring to be (σ, I)-quasi Armendariz. Constructing various examples, we classify how the (σ, I)-quasi Armendariz property behaves under various ring extensions. Furthermore, we show that a number of interesting properties of an (σ, I)-quasi Armendariz ring R such as reflexive and quasi-Baer property transfer to its mixed extension ring and vice versa. In this way, we extend the well-known results about quasi-Armendariz property in ordinary polynomial rings and skew polynomial rings for this class of mixed extensions. We pay also a particular attention to quasi-Gaussian rings.

Parabolic Partial Differential Equations with Nonlocal Initial and Boundary Values

Parabolic partial differential equations possessing nonlocal initial and boundary specifications are used to model some real-life applications. This paper focuses on constructing fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on a double power series representation of the solution via ordinary polynomials. Consequently, a parabolic partial differential equation is reduced to a system involving algebraic equations. Exact solutions are obtained when the solutions are themselves polynomials. Some parabolic partial differential equations are treated by the technique to judge its validity and also to measure its accuracy as compared to the existing methods.

On the complexity of skew arithmetic

In this paper, we study the complexity of several basic operations on linear differential operators with polynomial coefficients. As in the case of ordinary polynomials, we show that these complexities can be expressed almost linearly in terms of the cost of multiplication.

Equivalent Dynamic Stiffness Mapping technique for identifying nonlinear structural elements from frequency response functions

A simple and general Equivalent Dynamic Stiffness Mapping technique is proposed for identifying the parameters or the mathematical model of a nonlinear structural element with steady-state primary harmonic frequency response functions (FRFs). The Equivalent Dynamic Stiffness is defined as the complex ratio between the internal force and the displacement response of unknown element. Obtained with the test data of responses׳ frequencies and amplitudes, the real and imaginary part of Equivalent Dynamic Stiffness are plotted as discrete points in a three dimensional space over the displacement amplitude and the frequency, which are called the real and the imaginary Equivalent Dynamic Stiffness map, respectively. These points will form a repeatable surface as the Equivalent Dynamic stiffness is only a function of the corresponding data as derived in the paper. The mathematical model of the unknown element can then be obtained by surface-fitting these points with special functions selected by priori knowledge of the nonlinear type or with ordinary polynomials if the type of nonlinearity is not pre-known. An important merit of this technique is its capability of dealing with strong nonlinearities owning complicated frequency response behaviors such as jumps and breaks in resonance curves. In addition, this technique could also greatly simplify the test procedure. Besides there is no need to pre-identify the underlying linear parameters, the method uses the measured data of excitation forces and responses without requiring a strict control of the excitation force during the test. The proposed technique is demonstrated and validated with four classical single-degree-of-freedom (SDOF) numerical examples and one experimental example. An application of this technique for identification of nonlinearity from multiple-degree-of-freedom (MDOF) systems is also illustrated.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined for a system P of n ordinary Laurent differential polynomials in n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P) of L polynomials in L−1 algebraic variables is obtained, which is nonsparse in the order of derivation, as defined in this paper. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal generated by P. If the system satisfies certain conditions, then the formulas obtained are multiples of sparse differential resultants, defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Differential elimination by differential specialization of Sylvester style matrices

Differential resultant formulas are defined, for a system P of n ordinary Laurent differential polynomials in n−1 differential variables. These are determinants of coefficient matrices of an extended system of polynomials obtained from P through derivations and multiplications by Laurent monomials. To start, through derivations, a system ps(P) of L polynomials in L− 1 algebraic variables is obtained, which is non sparse in the order of derivation. This enables the use of existing formulas for the computation of algebraic resultants, of the multivariate sparse algebraic polynomials in ps(P), to obtain polynomials in the differential elimination ideal generated by P. The formulas obtained are multiples of the sparse differential resultant defined by Li, Yuan and Gao, and provide order and degree bounds in terms of mixed volumes in the generic case.

Mellin transforms with only critical zeros: Legendre functions

We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Res=1/2. The polynomials with zeros only on the critical line are identified in terms of certain F23(1) hypergeometric functions. These polynomials possess the functional equation pn(s)=(−1)⌊n/2⌋pn(1−s). Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory.

Five axis machine tool volumetric error prediction through an indirect estimation of intra- and inter-axis error parameters by probing facets on a scale enriched uncalibrated indigenous artefact

The volumetric accuracy of five-axis machine tools is affected by intra-axis geometric errors (error motions) and inter-axis geometric errors (axes relative position and orientation errors). Self-probing of uncalibrated facets on the existing machine tool table is proposed to provide the necessary data for the self-calibration of the machine error parameters and of the artefact geometry using an indirect approach. A set of 86 non-confounded coefficients are selected from the ordinary cubic polynomials used to model both the intra- and inter-axis errors. A scale bar is added to provide the isotropic scale factor. The estimated model is then used to predict the actual tool to workpiece position. Experimental trials are conducted on a five-axis horizontal machining centre using its original unmodified machine table as an artefact. For validation purposes only, the estimated artefact geometry is compared to accurate coordinate measuring machine (CMM) measurements. A study of the volumetric error predictive capability of the model for selected subsets of estimated error coefficients is also conducted.

Finitely generated algebras with involution and multiplicities bounded by a constant

Let A be an algebra with involution ⁎ over a field F of characteristic zero, and let χn⁎(A), n=1,2,⋯ , be the sequence of ⁎-cocharacters of A. For every n≥1, let ln⁎(A) denote the nth ⁎-colength of A which is the sum of the multiplicities in χn⁎(A). In this article, we classify in two different ways the finitely generated ⁎-algebras satisfying an ordinary polynomial identity whose multiplicities of the ⁎-cocharacters χn⁎(A) are bounded by a constant. As a consequence this also yields a characterization of the ⁎-varieties whose ⁎-colength ln⁎(A), n=1,2,… , is bounded by a constant.

Strong Asymptotics of the Orthogonal Polynomials with Respect to a Measure Supported on the Plane

We consider the orthogonal polynomials with respect to the measure urn:x-wiley::media:cpa21541:cpa21541-math-0002 over the whole complex plane. We obtain the strong asymptotic of the orthogonal polynomials in the complex plane and the location of their zeros in a scaling limit where n grows to infinity with N.The asymptotics are described in terms of three (probability) measures associated with the problem. The first measure is the limit of the counting measure of zeros of the polynomials, which is captured by the g‐function much in the spirit of ordinary orthogonal polynomials on the real line. The second measure is the equilibrium measure that minimizes a certain logarithmic potential energy, supported on a region K of the complex plane. The third measure is the harmonic measure of Kc with a pole at ∞. This appears as the limit of the probability measure given (up to the normalization constant) by the squared modulus of the nth orthogonal polynomial times the orthogonality measure, i.e., The compact region K that is the support of the second measure undergoes a topological transition under the variation of the parameter in a double scaling limit near the critical point given by we observe the Hastings‐McLeod solution to Painlevé II in the asymptotics of the orthogonal polynomials. © 2014 Wiley Periodicals, Inc.

Towards ℛ-matrix construction of Khovanov-Rozansky polynomials I. Primary T-deformation of HOMFLY

We elaborate on the simple alternative from arXiv:1308.5759 to the matrix-factorization construction of Khovanov-Rozansky (KR) polynomials for arbitrary knots and links in the fundamental representation of arbitrary SL(N). Construction consists of 2 steps: first, with every link diagram with m vertices one associates an m-dimensional hypercube with certain q-graded vector spaces, associated to its 2^m vertices. A generating function for q-dimensions of these spaces is what we suggest to call the primary T-deformation of HOMFLY polynomial -- because, as we demonstrate, it can be explicitly reduced to calculations of ordinary HOMFLY polynomials, i.e. to manipulations with quantum R-matrices. The second step is a certain minimization of residues of this new polynomial with respect to T+1. Minimization is ambiguous and is actually specified by the choice of commuting cut-and-join morphisms, acting along the edges of the hypercube -- this promotes it to Abelian quiver, and KR polynomial is a Poincare polynomial of associated complex, just in the original Khovanov's construction at N=2. This second step is still somewhat sophisticated -- though incomparably simpler than its conventional matrix-factorization counterpart. In this paper we concentrate on the first step, and provide just a mnemonic treatment of the second step. Still, this is enough to demonstrate that all the currently known examples of KR polynomials in the fundamental representation can be easily reproduced in this new approach. As additional bonus we get a simple description of the DGR relation between KR polynomials and superpolynomials and demonstrate that the difference between reduced and unreduced cases, which looks essential at KR level, practically disappears after transition to superpolynomials. However, a careful derivation of all these results from cohomologies of cut-and-join morphisms remains for further studies.

Nonlinear Multivariate Spline-Based Control Allocation for High-Performance Aircraft

High-performance flight control systems based on the nonlinear dynamic inversion principle require highly accurate models of aircraft aerodynamics. In general, the accuracy of the internal model determines to what degree the system nonlinearities can be canceled; the more accurate the model, the better the cancellation, and with that, the higher the performance of the controller. In this paper, a new control system is presented that combines nonlinear dynamic inversion with multivariate simplex spline-based control allocation. Three control allocation strategies that use novel expressions for the analytic Jacobian and Hessian of the multivariate spline models are presented. Multivariate simplex splines have a higher approximation power than ordinary polynomial models and are capable of accurately modeling nonlinear aerodynamics over the entire flight envelope of an aircraft. This nonlinear spline based controller is applied to control a high-performance aircraft (F-16) with a large flight envelope. The simulation results indicate that perfect feedback linearization can be achieved throughout the entire flight envelope, leading to a significant increase in tracking performance compared with ordinary polynomial-based nonlinear dynamic inversion.

Axis Location Errors and Error Motions Calibration for a Five-axis Machine Tool Using the SAMBA Method

Positioning accuracy is one of the most important factors influencing a machine tool's ability to manufacture parts meeting the required tolerances. Thus, regular check-ups followed by geometric compensation or mechanical adjustments are necessary to prevent accuracy degradation on such machines. This paper presents an enhanced measurement strategy to extend the capability of the Scale and Master Balls Artefact (SAMBA) method to the estimation of not only the axis location errors but also error motion parameters modeled as ordinary polynomials. This indirect measuring method uses on-machine probing of a scale enriched uncalibrated master balls artefact to gather observations of the machine volumetric behaviour. The analysis of the kinematic model and its associated Jacobian matrix which characterizes the sensitivity of the volumetric errors, as detected by the SAMBA method, to the axis location errors and error motions provide the mathematical basis for the probing strategy design. The simulation and experimental results presented demonstrate the contribution of the applied strategy in enriching substantially the machine tool error model.

Medidas repetidas no estudo de características de crescimento e carcaça avaliadas por ultrassom em novilhas de corte cruzadas

Com o objetivo de estudar o desenvolvimento de características de carcaça obtidas por ultrassom, peso vivo, perímetro torácico e altura do posterior, dados longitudinais foram analisados por diferentes estruturas de (co)variâncias residuais que consideram homogeneidade ou heterogeneidade de (co)variâncias, para melhor representar a variabilidade entre observações e dentro de indivíduo, ao longo do tempo. As medidas foram obtidas em 120 novilhas, sendo 60 ½ Braunvieh x ¼ Santa Gertrudes ¼ x Nelore e 60 ½ Santa Gertrudes x ½ Nelore. As características foram avaliadas em intervalos de 28 dias, totalizando até cinco medidas em cada grupo. Foram comparados diversos modelos com o intuito de identificar estruturas de (co)variâncias residuais adequadas para melhor representar a variação das medidas dentro de indivíduos, com base no critério Schwarz’s Bayesian Criterion. Posteriormente, avaliou-se o modelo mais adequado para regredir as características em função da idade, usando polinômios ordinários e verificando a necessidade de diferentes curvas para cada grupo genético. Diferentes estruturas de (co)variância residual devem ser consideradas em análise de dados longitunais, visando melhor representar a variabilidade entre observações e dentro de indivíduo. Os grupos genéticos estudados diferem entre si, ao longo do tempo, para as características peso, altura do posterior, perímetro torácico e área de olho de lombo.

Generalized matrix functions and determinants

Abstract In this paper we prove that, up to a scalar multiple, the determinant is the unique generalized matrix function that preserves the product or remains invariant under similarity. Also, we present a new proof for the known result that, up to a scalar multiple, the ordinary characteristic polynomial is the unique generalized characteristic polynomial for which the Cayley-Hamilton theorem remains true.

An effective approach for numerical solutions of high-order Fredholm integro-differential equations

We propose an effective method to solve high-order linear Fredholm integro-differential equations having a weak or strong kernel. The target is to construct fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on the power series representation via ordinary polynomials. Employing such polynomials leads to algebraic equations to be solved regarding the treated integro-differential equations. A mathematical proof for the numerical analysis is also provided. In practice, the convergence of the power series solution can be easily pursued via the ratio test. Exact solutions are obtained when the solutions are themselves polynomials. Better accuracies are also achieved within the method by increasing the number of polynomials. The introduced approach is applied to already worked problems in the literature by means of different numerical methods. Comparisons clearly show that our scheme is better and even more superior as compared to the existing ones.