A Parallel Algorithm for Finding Roots of a Complex Polynomial

A distribution theory of the roots of a polynomial and a parallel algorithm for finding roots of a complex polynomial based on that theory are developed in this paper. With high parallelism, the algorithm is an im- provement over the Wilf algorithm.

Surface contact behavior of functionally graded thermoelectric materials indented by a conducting punch

The contact problem for thermoelectric materials with functionally graded properties is considered.The material properties,such as the electric conductivity,the thermal conductivity,the shear modulus,and the thermal expansion coefficient,vary in an exponential function.Using the Fourier transform technique,the electro-thermoelastic problems are transformed into three sets of singular integral equations which are solved numerically in terms of the unknown normal electric current density,the normal energy flux,and the contact pressure.Meanwhile,the complex homogeneous solutions of the displacement fields caused by the gradient parameters are simplified with the help of Euler’s formula.After addressing the non-linearity excited by thermoelectric effects,the particular solutions of the displacement fields can be assessed.The effects of various combinations of material gradient parameters and thermoelectric loads on the contact behaviors of thermoelectric materials are presented.The results give a deep insight into the contact damage mechanism of functionally graded thermoelectric materials(FGTEMs).

Natural variation of hormone levels in Arabidopsis roots and correlations with complex root architecture

Studies on natural variation are an important tool to unravel the genetic basis of quantitative traits in plants. Despite the significant roles of phytohormones in plant development, including root architecture, hardly any studies have been done to investigate natural variation in endogenous hormone levels in plants. Therefore, in the present study a range of hormones were quantified in root extracts of thirteen Arabidopsis thaliana accessions using a ultra performance liquid chromatography triple quadrupole mass spectrometer. Root system architecture of the set of accessions was quantified, using a new parameter （mature root unit） for complex root systems, and correlated with the phytohormone data. Significant variations in phytohormone levels among the accessions were detected, but were remarkably small, namely less than three-fold difference between extremes. For cytokinins, relatively larger variations were found for ribosides and glucosides, as compared to the free bases. For root phenotyping, length-related traits--lateral root length and total root length--showed larger variations than lateral root number-related ones. For root architecture, antagonistic interactions between hormones, for example, indole-3-acetic acid to trans-zeatin were detected in correlation analysis. These findings provide conclusive evidence for the presence of natural variation in phytohormone levels in Arabidopsis roots, suggesting that quantitative genetic analyses are feasible.

A Sufficient Statistical Test for Dynamic Stability

In the existing Statistics and Econometrics literature, there does not exist a statistical test which may test for all kinds of roots of the characteristic polynomial leading to an unstable dynamic response, i.e., positive and negative real unit roots, complex unit roots and the roots lying inside the unit circle. This paper develops a test which is sufficient to prove dynamic stability (in the context of roots of the characteristic polynomial) of a univariate as well as a multivariate time series without having a structural break. It covers all roots (positive and negative real unit roots, complex unit roots and the roots inside the unit circle whether single or multiple) which may lead to an unstable dynamic response. Furthermore, it also indicates the number of roots causing instability in the time series. The test is much simpler in its application as compared to the existing tests as the series is strictly stationary under the null (C01, C12).

Finding roots of arbitrary high order polynomials based on neural network recursive partitioning method

This paper proposes a novel recursive partitioning method based on constrained learning neural networks to find an arbitrary number (less than the order of the polynomial) of (real or complex) roots of arbitrary polynomials. Moreover, this paper also gives a BP network constrained learning algorithm (CLA) used in root-finders based on the constrained relations between the roots and the coefficients of polynomials. At the same time, an adaptive selection method for the parameter d P with the CLA is also given. The experimental results demonstrate that this method can more rapidly and effectively obtain the roots of arbitrary high order polynomials with higher precision than traditional root-finding approaches.

Parametric“non-nested”discriminants for multiplicities of univariate polynomials

We consider the problem of complex root classification,i.e.,finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots.It is well known that such conditions can be written as conjunctions of several polynomial equalities and one inequality in the coefficients.Those polynomials in the coefficients are called discriminants for multiplicities.It is also known that discriminants can be obtained using repeated parametric greatest common divisors.The resulting discriminants are usually nested determinants,i.e.,determinants of matrices whose entries are determinants,and so on.In this paper,we give a new type of discriminant that is not based on repeated greatest common divisors.The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees.