Abstract: In this note, Kharitonov's theorem on robust Hurwitz polynomials is simplified for low-order polynomials. Specifically, for n = 3, 4 , and 5, the number of polynomials required to check robust stability is one, two, and three, respectively, instead of four. Furthermore, it is shown that for n \geq 6 , the number of polynomials for robust stability checking is necessarily four, thus further simplification is not possible. The same simplifications arise in robust Schur polynomials by using the bilinear transformation. Applications of these simplifications to two-dimensional polynomials as well as to robustness for single parameters are indicated.

Abstract: It is shown that the Kharitonov test for Hurwitz stability of an interval polynomial does not extend in general to the discrete-time case, unless the degree n of the polynomial is not greater than two. For n \leq 3 a given monic interval polynomial has all roots inside the unit disk if all 2^{3} = 8 extreme polynomials have that property (instead of only four polynomials in Kharitonov's test). For n = 4 it is shown by a counterexample that discrete-time stability of all extreme polynomials does not guarantee the stability of the interval polynomial.

Abstract: 2. Introduction.- 2.1 Classical Pade approximation.- 2.2 Toeplitz and Hankel systems.- 2.3 Continued fractions.- 2.4 Orthogonal polynomials.- 2.5 Rhombus algorithms and convergence.- 2.6 Block structure.- 2.7 Laurent-Pade approximants.- 2.8 The projection method.- 2.9 Applications.- 2.10 Outline.- 3. Moebius transforms, continued fractions and Pade approximants.- 3.1 Moebius transforms.- 3.2 Flow graphs.- 3.3 Continued fractions (CF).- 3.4 Formal series.- 3.5 Pade approximants.- 4. Two algorithms.- 4.1 Algorithm 1.- 4.2 Algorithm 2.- 5. All kinds of Pade Approximants.- 5.1 Pade approximants.- 5.2 Laurent-Pade approximants.- 5.3 Two-point Pade approximants.- 6. Continued fractions.- 6.1 General observations.- 6.2 Some special cases.- 7. Moebius transforms.- 7.1 General observations.- 7.2 Some special cases.- 8. Rhombus algorithms.- 8.1 The ab parameters (sawtooth path).- 8.2. The FG parameters (row path).- 8.3. A staircase path.- 8.4 ?? paramaters (diagonal path).- 8.5 Some dual results.- 8.6 Relation with classical algorithms.- 9. Biorthogonal polynomials, quadrature and reproducing kernels.- 9.1 Biorthogonal polynomials.- 9.2 Interpolatory quadrature methods.- 9.3 Reproducing kernels.- 9.4 Other orthogonality relations.- 10. Determinant expressions and matrix interpretations.- 10.1 Determinant expressions.- 10.2 Matrix interpretations.- 10.2.1 Toeplitz matrices.- 10.2.2 Hankel matrices.- 10.2.3 Tridiagonal matrices.- 11. Symmetry Properties.- 11.1 Symmetry for F(z) and $$\hat F$$(z) = F(1/z).- 11.2 Symmetry for F(z) and G(z) = 1/F(z).- 12. Block structures.- 12.1 Pade forms, Laurent-Pade forms and two-point Pade forms.- 12.2 The T-table.- 12.3 The Pade, Laurent-Pade, and two-point Pade tables.- 13. Meromorphic functions and asymptotic behaviour.- 13.1 The function F(z).- 13.2 Asymptotics for finite Toeplitz determinants.- 13.3 Asymptotics for infinite Toeplitz determinants.- 13.4 Consequences for the T-table.- 14. Montessus de Ballore theorem for Laurent-Pade approximants.- 14.1 Semi infinite Laurent series.- 14.2 Bi-infinite Laurent series.- 15. Determination of poles.- 15.1 Rutishauser polynomials of type 1 and type 2.- 15.2 Rutishauser polynomials of type 3.- 15.3 Rutishauser polynomials and Laurent series.- 15.4 Convergence of parameters.- 16. Determination of zeros.- 16.1 Dual Rutishauser polynomials and semi-infinite series.- 16.2 From semi-infinite to bi-infinite series.- 16.3 Convergence of parameters.- 17. Convergence in a row of the Laurent-Pade table.- 17.1 Toeplitz operators and the projection method.- 17.2 Convergence of the denominator.- 17.3 Convergence of the numerator.- 18. The positive definite case and applications.- 18.1 Function classes.- 18.2 Connection with the previous results.- 18.3 Stochastic processes and systems.- 18.4 Lossless inverse scattering and transmission lines.- 18.5 Laurent-Pade approximation and ARMA-filtering.- 18.6 Concluding remarks.- 19. Examples.- 19.1 Example 1.- 19.2 Example 2.- 19.3 Example 3.- References.- List of symbols.

Abstract: A simple tight-binding Hamiltonian is used in the band model to calculate binding energies of crystal structures in silicon using the recursion method. The use of matrix orthogonal polynomials is described for efficient computation of the symmetry-preserving matrix Green functions. This provides the basis for a proposed rotationally invariant interatomic force algorithm.

Abstract: Properties of various multidimensional polynomials arising in studies on discrete multidimensional systems are investigated. Reactance Schur polynomials and immittance Schur polynomials occurring, respectively, as the denominators (and numerators) of discrete reactance functions and discrete positive functions are introduced and their properties studied. The role of these polynomials in scattering or immittance descriptions of passive discrete-time domain multiports are brought out. The interrelations between various classes of multidimensional polynomials arising in studies on discrete systems and the corresponding classes of polynomials in the context of continuous systems are also studied via the bilinear transformation.

Abstract: Etude de l'evaluation numerique d'une integrale #7B-F −1 1 exp(iωx)f(x)dx ou f possede un pole simple dans l'intervalle [−1, 1]. Application de la formule d'interpolation de Lagrange. Estimation de l'erreur et exemples numeriques

Abstract: We study the spectrum of the Jacobi matrix $(\delta _{m,n + 1} + \delta _{m,n - 1} + aq^n \delta _{m,n} )$, m, $n = 0,1, \cdots $ and the corresponding orthogonal polynomials. The spectral measure is computed when $q \in ( - 1,1)$ and sufficient conditions are given to guarantee the absolute continuity of the spectral measure. When $ q > 1$ or < - 1 the measure is purely discrete. The case $q = - 1$ leads to a set of polynomials orthogonal on the union of two disjoint intervals. When $q = 1$, the polynomials are essentially the Chebyshev polynomials $\{ U_n (x)\} $.

Abstract: Dickson polynomials over finite fields are familiar examples of permutation polynomials, i.e. of polynomials for which the corresponding polynomial mapping is a permutation of the finite field. We prove that a Dickson polynomial can be a complete mapping polynomial only in some special cases. Complete mapping polynomials are of interest in combinatorics and are defined as polynomials f(x) over a finite field for which both f(x) and f(x) + x are permutation polynomials. Our result also verifies a special case of a conjecture of Chowla and Zassenhaus on permutation polynomials.

Abstract: Fast DecoIIlposition in the tame case 2 polynomials over F. We obtain a range of results, trom Ulldecidability over sufficiently general fields to fast sequential and parallel algorithms over finite fields. A version of the algorithm of Theorem 1 below has beel implemented [2,6J and compares favorably with [3J. Dick erson [9J has extended some of these results to multivariate polynomials. We should give a brief history of the research behind this joint paper. Kozen and Landau [18] gave the first polynomial-time sequential and NCalgorithms for this problem in the tame case. The time hounds were O(n3 ) sequential, O(n ) if F supports an FFT, and 0(1og2 n) parallel. They also presented the structure theorem (Theorem 9), reducing the problem in the wild case to factorization, and gave an O(n ) algorithm for the decomposition of irreducible polynomials over general fields admitting a polynomial-time factorization algorithm, and an NC algorithm for irreducible polynomials over finite fields. Based on the algorithm of [18], von zur Gathen [17] improved the bounds in the tame case to those stated above. These results are presented in §2. He also gave an improved algorithm for the wild case, yielding a polynomial-time reduction to factorization of polynomials, and observed undecidability over sufficiently general fields. These results are presented in §3. Introduction 1

Abstract: Least-square approximation by polynomials, with respect to area measure on the regular hexagon, is useful in the construction and analysis of hexagonal optical elements. Notably the Keck Ten-Meter Telescope will utilize a main mirror composed of thirty-six hexagonal segments. By use of the representation theory of the symmetry group of the hexagon a method for constructing a family of orthogonal polynomials is derived. The computations can be carried out in rational (exact) arithmetic, and an appendix lists the polynomials and orthogonal expansions of monomials up to degree six.

Abstract: It is shown that the zeros of polynomials orthogonal with respect to certain positive measures on the unit circle have an asymptotic distribution which is uniform on a circle of radius less than or equal to one. These results explain some phenomena observed when various linear prediction methods are used to estimate sinusoidal frequencies. They also describe the asymptotic zero distribution of the prediction error filter polynomials for a class of time series including autoregressive moving average (ARMA) models.

Abstract: The q-Pollaczek polynomials are orthogonal polynomials having a generating function of the form $A(t)\prod _{k = o}^\infty {F(x,tq^k ) = \sum _{n = 0}^\infty {P_n (x)t^n } } $, where $F(x,t) = {{[1 - xH(t)]} / {[1 - xK(t)]}}$ and $A(t)$, $H(t)$ and $K(t)$ are formal power series with $H(0) = K(0) = 0$, $A(0)H'(0)K'(0) e 0$. We determine all orthogonal polynomials having generating functions of this form. We find that in addition to the q-Pollaczek polynomials, there are two other sets that are closely related to the q-Pollaczek polynomials.

Abstract: The invariant polynomials null (Davis [8] and Chikuse [2] with r ( r ≥ 2) symmetric matrix arguments have been defined, extending the zonal polynomials, and applied in multivariate distribution theory. The usefulness of the polynomials has attracted the attention of econometricians, and some recent papers have applied the methods to distribution theory in econometrics (e.g., Hillier [14] and Phillips [22]).

Abstract: In this note a class of interesting generating relation, which is stated in the form of theorem, involving Laguerre polynomials is derived. Some applications of the theorem are also given here.

Abstract: One method for constructing cubature formulae of a given degree of precision consists of using the common zeros of a finite set F of polynomials as nodes. The formula exists if and only if F is an H-basis and some well-defined orthogonality conditions hold. Groebner bases and especially Buchberger’s algorithm for their computation allow an effective calculation of H-bases and easy proofs and generalizations of known methods based on H-bases. Groebner bases, a powerful tool in Computer Algebra for analyzing ideals and solving systems of algebraic equations, allow in addition the calculation of the common zeros of the polynomials in F also in cases, where the number of unknowns is different from the number of equations.

Abstract: The technique of using Tschirnhaus transformations to generate invariants was used successfully in the theory of binary quantics. These transformations led to systems of semi-invariants, called protomorphs, which then were used to construct invariants, themselves [2;Chapter X]

Abstract: A new modification to the Ansell's Hermite matrix method for the testing of very strict Hurwitz polynomials is presented. In this modification, the testing at infinite distant points is carried out as an integral part of the testing of the determinant of the Hermite matrix. Hence, the additional 1-D tests required to determine the behavior of the polynomial at infinite distant points are eliminated.

Abstract: The severity of the underlying assumptions of the line-spring model (LSM) are such that verification with three-dimensional solutions is necessary. Such comparisons show that the model is quite accurate, and therefore, its use in extensive parameter studies is justified. Investigations into the endpoint behavior of the line-spring model have led to important conclusions about the ability of the model to predict stresses in front of the crack tip. An important application of the LSM was to solve the contact plate bending problem. Here the flexibility of the model to allow for any crack shape is exploited. The use of displacement quantities as unknowns in the formulation of the problem leads to strongly singular integral equations, rather than singular integral equations which result from using displacement derivatives. The collocation method of solving the integral equations was found to be better and more convenient than the quadrature technique. Orthogonal polynomials should be used as fitting functions when using the LSM as opposed to simpler functions such as power series.

Abstract: For an example where the functions have different branch points we derive the asymptotics of diagonal Hermite-Pade polynomials of type I. The method uses an integral equation obtained by approximating a reproducing kernel. The results are consistent with a new conjecture on the asymptotics of the polynomials associated with more general functions with different branch points.