Kalman particle swarm optimized polynomials for data classification

Data classification is an important area of data mining. Several well known techniques such as decision tree, neural network, etc. are available for this task. In this paper we propose a Kalman particle swarm optimized (KPSO) polynomial equation for classification for several well known data sets. Our proposed method is derived from some of the findings of the valuable information like number of terms, number and combination of features in each term, degree of the polynomial equation etc. of our earlier work on data classification using polynomial neural network. The KPSO optimizes these polynomial equations with a faster convergence speed unlike PSO. The polynomial equation that gives the best performance is considered as the model for classification. Our simulation result shows that the proposed approach is able to give competitive classification accuracy compared to PNN in many datasets.     

The use of the stochastic arithmetic to estimate the value of interpolation polynomial with optimal degree

One of the considerable discussions in data interpolation is to find the optimal number of data which minimizes the error of the interpolation polynomial. In this paper, first the theorems corresponding to the equidistant nodes and the roots of the Chebyshev polynomials are proved in order to estimate the accuracy of the interpolation polynomial, when the number of data increases. Based on these theorems, then we show that by using a perturbation method based on the CESTAC method, it is possible to find the optimal degree of the interpolation polynomial. The results of numerical experiments are presented.     

Direct and inverse polynomial perturbations of hermitian linear functionals

This paper is devoted to the study of direct and inverse (Laurent) polynomial modifications of moment functionals on the unit circle, i.e., associated with hermitian Toeplitz matrices. We present a new approach which allows us to study polynomial modifications of arbitrary degree.The main objective is the characterization of the quasi-definiteness of the functionals involved in the problem in terms of a difference equation relating the corresponding Schur parameters. The results are presented in the general framework of (non-necessarily quasi-definite) hermitian functionals, so that the maximum number of orthogonal polynomials is characterized by the number of consistent steps of an algorithm based on the referred recurrence for the Schur parameters.The non-uniqueness of the inverse problem makes it more interesting than the direct one. Due to this reason, special attention is paid to the inverse modification, showing that different approaches are possible depending on the data about the polynomial modification at hand. These different approaches are translated as different kinds of initial conditions for the related inverse algorithm.Some concrete applications to the study of orthogonal polynomials on the unit circle show the effectiveness of this new approach: an exhaustive and instructive analysis of the functionals coming from a general inverse polynomial perturbation of degree one for the Lebesgue measure; the classification of those pairs of orthogonal polynomials connected by a certain type of linear relation with constant polynomial coefficients; and the determination of those orthogonal polynomials whose associated ones are related to a degree one polynomial modification of the original orthogonality functional.     

Representations of skew polynomial algebras

C. De Concini and C. Procesi have proved that in many cases the degree of a skew polynomial algebra is the same as the degree of the corresponding quasi polynomial algebra. We prove a slightly more general result. In fact we show that in case the skew polynomial algebra is a P.I. algebra, then its degree is the degree of the quasi polynomial algebra.

Our argument is then applied to determine the degree of some algebras given by generators and relations.

     

A filtered polynomial approach to density estimation

In this paper, a little known computational approach to density estimation based on filtered polynomial approximation is investigated. It is accompanied by the first online available density estimation computer program based on a filtered polynomial approach. The approximation yields the unknown distribution and density as the product of a monotonic increasing polynomial and a filter. The filter may be considered as a target distribution which gets fixed prior to the estimation. The filtered polynomial approach then provides coefficient estimates for (close) algebraic approximations to (a) the unknown density function and (b) the unknown cumulative distribution function as well as (c) a transformation (e.g., normalization) from the unknown data distribution to the filter. This approach provides a high degree of smoothness in its estimates for univariate as well as for multivariate settings. The nice properties as the high degree of smoothness and the ability to select from different target distributions are suited especially in MCMC simulations. Two applications in Sects. 1 and 7 will show the advantages of the filtered polynomial approach over the commonly used kernel estimation method.      

关于n维单形保多项式超限插值的表示问题

以R~n表示n维欧氏空间,Z_+~n是R~n中坐标均为非负整数的全体,e~s为Z_+~(n+1)中第s个坐标为1其余坐标为0的单位向量;π_d(R~n)为全次数不大于d的n元多项式全体,     

NP-hardness of deciding convexity of quartic polynomials and related problems

We show that unless P = NP, there exists no polynomial time (or even pseudo-polynomial time) algorithm that can decide whether a multivariate polynomial of degree four (or higher even degree) is globally convex. This solves a problem that has been open since 1992 when N. Z. Shor asked for the complexity of deciding convexity for quartic polynomials. We also prove that deciding strict convexity, strong convexity, quasiconvexity, and pseudoconvexity of polynomials of even degree four or higher is strongly NP-hard. By contrast, we show that quasiconvexity and pseudoconvexity of odd degree polynomials can be decided in polynomial time.     

The degree of the E-characteristic polynomial of an even order tensor

The E-characteristic polynomial of an even order supersymmetric tensor is a useful tool in determining the positive definiteness of an even degree multivariate form. In this paper, for an even order tensor, we first establish the formula of its E-characteristic polynomial by using the classical Macaulay formula of resultants, then give an upper bound for the degree of that E-characteristic polynomial. Examples illustrate that this bound is attainable in some low order and dimensional cases.     

The Composition Structure of Alternative and Malcev Algebras

We construct a polynomial of degree 5 from the associative nucleus (kernel) of the free alternative algebra. We show that this polynomial is of minimal degree. Using this polynomial, we obtain decompositions of the varieties of alternative and Malcev algebras.     

三奇次散乱点多项式自然样条插值

为解决较为复杂的三变量散乱数据插值问题,提出了一种三元多项式自然样条插值方法.在使得对一种带自然边界条件的目标泛函极小的情况下,用Hilbert空间样条函数方法,构造出了插值问题的解,并可表为一个分块三元三奇次多项式.其表示形式简单,且系数可由系数矩阵对称的线性代数方程组确定.     

五次键合多项式P-不可约的两个定理

在生物化学研究领域,对键合多项式P-不可约性的判定是一个重要问题.已有结果主要考虑四次或四次以下的多项式,而对五次或五次以上键合多项式的讨论尚未见报道.文章在这方面作了一定的探索,给出了五次键合多项式P-不可约的两组充分条件,这两组条件均是用多项式的系数构成的等式或不等式组显式表示的.     

Non-isochronicity of the center at the origin in polynomial Hamiltonian systems with even degree nonlinearities

In 2002 X. Jarque and J. Villadelprat proved that no center in a planar polynomial Hamiltonian system of degree 4 is isochronous and raised a question: Is there a planar polynomial Hamiltonian system of even degree which has an isochronous center? In this paper we give a criterion for non-isochronicity of the center at the origin of planar polynomial Hamiltonian systems. Moreover, the orders of weak centers are determined. Our results answer a weak version of the question, proving that there is no planar polynomial Hamiltonian system with only even degree nonlinearities having an isochronous center at the origin.     

Polar Varieties,Real Equation Solving,and Data Structures: The Hypersurface Case

In this paper we apply for the first time a new method for multivariate equation solving which was developed for complex root determination to therealcase. Our main result concerns the problem of finding at least one representative point for each connected component of a real compact and smooth hypersurface. The basic algorithm yields a new method for symbolically solving zero-dimensional polynomial equation systems over the complex numbers. One feature of central importance of this algorithm is the use of a problem-adapted data type represented by the data structures arithmetic network and straight-line program (arithmetic circuit). The algorithm finds the complex solutions of any affine zero-dimensional equation system in nonuniform sequential time that ispolynomialin the length of the input (given in straight-line program representation) and an adequately definedgeometric degree of the equation system.Replacing the notion of geometric degree of the given polynomial equation system by a suitably definedreal (or complex) degreeof certain polar varieties associated to the input equation of the real hypersurface under consideration, we are able to find for each connected component of the hypersurface a representative point (this point will be given in a suitable encoding). The input equation is supposed to be given by a straight-line program and the (sequential time) complexity of the algorithm is polynomial in the input length and the degree of the polar varieties mentioned above.     

Application of degree reduction of polynomial Bézier curves to rational case

An algorithmic approach to degree reduction of rational Bézier curves is presented. The algorithms are based on the degree reduction of polynomial Bézier curves. The method is introduced with the following steps: (a) convert the rational Bézier curve to polynomial Bézier curve by using homogenous coordinates, (b) reduce the degree of polynomial Bézier curve, (c) determine weights of degree reduced curve, (d) convert the Bézier curve obtained through step (b) to rational Bézier curve with weights in step (c).     

Isochronous centers of polynomial Hamiltonian systems and a conjecture of Jarque and Villadelprat

We study the conjecture of Jarque and Villadelprat stating that every center of a planar polynomial Hamiltonian system of even degree is nonisochronous. This conjecture has already been proved for quadratic and quartic systems. Using the correction of a vector field to characterize isochronicity and explicit computations of this quantity for polynomial vector fields, we are able to describe a very large class of nonisochronous Hamiltonian systems of even arbitrarily large degree.     

The group and the minimal polynomial of a graph

This paper presents some results linking the minimal polynomial of the adjacency matrix of a graph with its group structure. An upper bound on the order of the group is derived for graphs whose minimal and characteristic polynomials are identical. It is also shown that for a graph with transitive group, the degree of the minimal polynomial is bounded above by the number of orbits of the stabilizer of any given element. Finally, the order of the group of a point-symmetric graph with a prime number of points is shown to depend on the degree of the minimal polynomial, and an algorithm for constructing such a group is given.     

Degree elevation from Bzier curve to C-Bzier curve with corner cutting form

The existing results of curve degree elevation mainly focus on the degree of algebraic polynomials. The paper considers the elevation of degree of the trigonometric polynomial, from a Bzier curve on the algebraic polynomial space, to a C-B′ezier curve on the algebraic and trigonometric polynomial space. The matrix of degree elevation is obtained by an operator presentation and a derivation pyramid. It possesses not a recursive presentation but a direct expression. The degree elevation process can also be represented as a corner cutting form.     

Strong tractability of multivariate integration of arbitrary high order using digitally shifted polynomial lattice rules

In this paper we prove the existence of digitally shifted polynomial lattice rules which achieve strong tractability results for Sobolev spaces of arbitrary high smoothness. The convergence rate is shown to be the best possible up to a given degree of smoothness of the integrand. Indeed we even show the existence of polynomial lattice rules which automatically adjust themselves to the smoothness of the integrand up to a certain given degree.Further we show that strong tractability under certain conditions on the weights can be obtained and that polynomial lattice rules exist for which the worst-case error can be bounded independently of the dimension. These results hold independent of the smoothness.     

The numerical experiment in the study of polynomial quasisolutions of linear differential-difference equations

In this paper we consider the initial problem with an initial point for a scalar linear inhomogeneous differential-difference equation of neutral type. For polynomial coefficients in the equation we introduce a formal solution, representing a polynomial of a certain degree (“a polynomial quasisolution”); substituting it in the initial equation, one obtains a residual. The work is dedicated to the definition and the analysis (on the base of numerical experiments) of polynomial quasisolutions for the solutions of the initial problem under consideration.     

The Complexity of Factors of Multivariate Polynomials

The existence of string functions, which are not polynomial time computable, but whose graph is checkable in polynomial time, is a basic assumption in cryptography. We prove that in the framework of algebraic complexity, there are no such families of polynomial functions of polynomially bounded degree over fields of characteristic zero. The proof relies on a polynomial upper bound on the approximative complexity of a factor g of a polynomial f in terms of the (approximative) complexity of f and the degree of the factor g. This extends a result by Kaltofen. The concept of approximative complexity allows us to cope with the case that a factor has an exponential multiplicity, by using a perturbation argument. Our result extends to randomized (two-sided error) decision complexity.