Resultant-based methods for plane curves intersection problems

http://www.springeronline.com/3-540-28966-6We present an algorithm for solving polynomial equations, which uses generalized eigenvalues and eigenvectors of resultant matrices. We give special attention to the case of two bivariate polynomials and the Sylvester or Bezout resultant constructions. We propose a new method to treat multiple roots, detail its numerical aspects and describe experiments on tangential problems, which show the efficiency of the approach. An industrial application of the method is presented at the end of the paper. It consists in recovering cylinders from a large cloud of points and requires intensive resolution of polynomial equations

Recursive Polynomial Remainder Sequence and the Nested Subresultants

We give two new expressions of subresultants, nested subresultant and reduced nested subresultant, for the recursive polynomial remainder sequence (PRS) which has been introduced by the author. The reduced nested subresultant reduces the size of the subresultant matrix drastically compared with the recursive subresultant proposed by the authors before, hence it is much more useful for investigation of the recursive PRS. Finally, we discuss usage of the reduced nested subresultant in approximate algebraic computation, which motivates the present work.Comment: 12 pages. Presented at CASC 2005 (Kalamata, Greece, Septermber 12-16, 2005

Transfer Matrices and Partition-Function Zeros for Antiferromagnetic Potts Models. V. Further Results for the Square-Lattice Chromatic Polynomial

We derive some new structural results for the transfer matrix of square-lattice Potts models with free and cylindrical boundary conditions. In particular, we obtain explicit closed-form expressions for the dominant (at large |q|) diagonal entry in the transfer matrix, for arbitrary widths m, as the solution of a special one-dimensional polymer model. We also obtain the large-q expansion of the bulk and surface (resp. corner) free energies for the zero-temperature antiferromagnet (= chromatic polynomial) through order q^{-47} (resp. q^{-46}). Finally, we compute chromatic roots for strips of widths 9 <= m <= 12 with free boundary conditions and locate roughly the limiting curves.Comment: 111 pages (LaTeX2e). Includes tex file, three sty files, and 19 Postscript figures. Also included are Mathematica files data_CYL.m and data_FREE.m. Many changes from version 1: new material on series expansions and their analysis, and several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

The Renormalization Group with Exact beta-Functions

The perturbative $\beta$-function is known exactly in a number of supersymmetric theories and in the 't Hooft renormalization scheme in the $\phi_4^4$ model. It is shown how this allows one to compute the effective action exactly for certain background field configurations and to relate bare and renormalized couplings. The relationship between the MS and SUSY subtraction schemes in $N = 1$ super Yang-Mills theory is discussed

Transfer matrices and partition-function zeros for antiferromagnetic Potts models. VI. Square lattice with special boundary conditions

We study, using transfer-matrix methods, the partition-function zeros of the square-lattice q-state Potts antiferromagnet at zero temperature (= square-lattice chromatic polynomial) for the special boundary conditions that are obtained from an m x n grid with free boundary conditions by adjoining one new vertex adjacent to all the sites in the leftmost column and a second new vertex adjacent to all the sites in the rightmost column. We provide numerical evidence that the partition-function zeros are becoming dense everywhere in the complex q-plane outside the limiting curve B_\infty(sq) for this model with ordinary (e.g. free or cylindrical) boundary conditions. Despite this, the infinite-volume free energy is perfectly analytic in this region.Comment: 114 pages (LaTeX2e). Includes tex file, three sty files, and 23 Postscript figures. Also included are Mathematica files data_Eq.m, data_Neq.m,and data_Diff.m. Many changes from version 1, including several proofs of previously conjectured results. Final version to be published in J. Stat. Phy

Some Applications of the Lambert W Function to Physics

Two standard physics problems are solved in terms of the Lambert W  function, to show the applicability of this recently defined function to physics. Other applications of the function are cited, but not described. The problems solved concern Wien\u27s displacement law and the fringing fields of a capacitor, the latter problem being representative of some problems solved using conformal transformations. The physical content of the solutions remains unchanged, but they gain a new elegance and convenience

On the Lambert W Function

The Lambert W function is defined to be the multivalued inverse of the function w 7! we w . It has many applications in pure and applied mathematics, some of which are briefly described here. We present a new discussion of the complex branches of W , an asymptotic expansion valid for all branches, an efficient numerical procedure for evaluating the function to arbitrary precision, and a method for the symbolic integration of expressions containing W . On the Lambert W function 2 1. Introduction In 1758, Lambert [47] solved the trinomial equation x = q + x m by giving a series development for x in powers of q. Later [48], he extended the series to give powers of x as well. Euler [28] transformed Lambert&apos;s equation into a more symmetric form by substituting x \Gammafi for x and setting m = fffi and q = (ff \Gamma fi)v. His version of the equation was x ff \Gamma x fi = (ff \Gamma fi)vx ff+fi ; (1:1) and his version of Lambert&apos;s series solution was x n = 1 + nv + 1 2 ..

Passive Fault-Tolerant Control Design for near-space Hypersonic Vehicle Dynamical System

In this paper, an observer-based passive fault-tolerant control (FTC) scheme is proposed for a near-space hypersonic vehicle (NSHV) dynamical system with both parameter uncertainty and actuator faults. The parameter uncertainty is assumed to be norm-bounded, and the possible fault of each actuator is described by a variable varying within a given interval. Our aim is to design an observer-based FTC law such that, for the admissible parameter uncertainty and possible actuator faults, the resulting closed-loop system is asymptotically stable with a given disturbance attenuation level γ . The unknown gain matrices are characterized in terms of the solutions to some linear matrix inequalities (LMIs) which can be readily solved using standard software packages. The FTC scheme presented in this study is finally demonstrated via simulation on a linearized NSHV dynamical system to illustrate the effectiveness