Scalar Polynomial Research Articles

Deflating quadratic matrix polynomials with structure preserving transformations

Given a pair of distinct eigenvalues ( λ 1 , λ 2 ) of an n × n quadratic matrix polynomial Q ( λ ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q ( λ ) into a quadratic of the form Q d ( λ ) 0 0 q ( λ ) having the same eigenvalue s as Q ( λ ) , with Q d ( λ ) an ( n - 1 ) × ( n - 1 ) quadratic matrix polynomial and q ( λ ) a scalar quadratic polynomial with roots λ 1 and λ 2 . This block diagonalization cannot be achieved by a similarity transformation applied directly to Q ( λ ) unless the eigenvectors corresponding to λ 1 and λ 2 are parallel. We identify conditions under which we can construct a family of 2 n × 2 n elementary similarity transformations that (a) are rank-two modifications of the identity matrix, (b) act on linearizations of Q ( λ ) , (c) preserve the block structure of a large class of block symmetric linearizations of Q ( λ ) , thereby defining new quadratic matrix polynomials Q 1 ( λ ) that have the same eigenvalue s as Q ( λ ) , (d) yield quadratics Q 1 ( λ ) with the property that their eigenvectors associated with λ 1 and λ 2 are parallel and hence can subsequently be deflated by a similarity applied directly to Q 1 ( λ ) . This is the first attempt at building elementary transformations that preserve the block structure of widely used linearizations and which have a specific action.

Kundt spacetimes as solutions of topologically massive gravity

We obtain new solutions of topologically massive gravity. We find the general Kundt solutions, which in three dimensions are spacetimes admitting an expansion-free null geodesic congruence. The solutions are generically of algebraic type II, but special cases are types III, N or D. Those of type D are the known spacelike-squashed AdS3 solutions and of type N are the known AdS pp-waves or new solutions. Those of types II and III are the first known solutions of these algebraic types. We present explicitly the Kundt solutions that are constant scalar invariant (CSI) spacetimes, for which all scalar polynomial curvature invariants are constant, whereas for the general case, we reduce the field equations to a series of ordinary differential equations. The CSI solutions of types II and III are deformations of spacelike-squashed AdS3 and the round AdS3, respectively.

Curvature operators and scalar curvature invariants

We continue the study of the question of when a pseudo-Riemannain manifold can be locally characterized by its scalar polynomial curvature invariants (constructed from the Riemann tensor and its covariant derivatives). We make further use of the alignment theory and the bivector form of the Weyl operator in higher dimensions and introduce the important notions of diagonalizability and (complex) analytic metric extension. We show that if there exists an analytic metric extension of an arbitrary dimensional space of any signature to a Riemannian space (of Euclidean signature), then that space is characterized by its scalar curvature invariants. In particular, we discuss the Lorentzian case and the neutral signature case in four dimensions in more detail.

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients D i ( u i ) may have the property D i ( 0 ) = 0 for some or all i = 1 , … , N , and the boundary condition is u i = 0 . Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.

Fiedler Companion Linearizations and the Recovery of Minimal Indices

A standard way of dealing with a matrix polynomial $P(\lambda)$ is to convert it into an equivalent matrix pencil—a process known as linearization. For any regular matrix polynomial, a new family of linearizations generalizing the classical first and second Frobenius companion forms has recently been introduced by Antoniou and Vologiannidis, extending some linearizations previously defined by Fiedler for scalar polynomials. We prove that these pencils are linearizations even when $P(\lambda)$ is a singular square matrix polynomial, and show explicitly how to recover the left and right minimal indices and minimal bases of the polynomial $P(\lambda)$ from the minimal indices and bases of these linearizations. In addition, we provide a simple way to recover the eigenvectors of a regular polynomial from those of any of these linearizations, without any computational cost. The existence of an eigenvector recovery procedure is essential for a linearization to be relevant for applications.

Jordan structures of alternating matrix polynomials

Alternating matrix polynomials, that is, polynomials whose coefficients alternate between symmetric and skew-symmetric matrices, generalize the notions of even and odd scalar polynomials. We investigate the Smith forms of alternating matrix polynomials, showing that each invariant factor is an even or odd scalar polynomial. Necessary and sufficient conditions are derived for a given Smith form to be that of an alternating matrix polynomial. These conditions allow a characterization of the possible Jordan structures of alternating matrix polynomials, and also lead to necessary and sufficient conditions for the existence of structure-preserving strong linearizations. Most of the results are applicable to singular as well as regular matrix polynomials.

Note on the invariant classification of vacuum type D spacetimes

We illustrate the fact that the class of vacuum type D spacetimes which are -non-degenerate are invariantly classified by their scalar polynomial curvature invariants.

Polynomial ultradistributions on [formula omitted

Let G + ′ = G ′ ( R + d ) stand for Roumieu ultradistributions with supports in the positive cone R + d . Throughout P ( G + ′ ) denotes the algebra of continuous scalar polynomials on the space G + ′ . We investigate the dual pair 〈 P ′ ( G + ′ ) ∣ P ( G + ′ ) 〉 generated by the algebra P ( G + ′ ) and by its strong dual P ′ ( G + ′ ) . Properties of the polynomially extended operational calculus and the semigroups of shifts along the cone R + d are considered.

Relativistic Gravitational Collapse of a Cylindrical Shell of Dust. II: -- Settling Down Boundary Condition --

We numerically study the dynamics of an imploding hollow cylinder composed of dust. Since there is no cylindrical black hole in 4-dimensional spacetime with physically reasonable energy conditions, a collapsed dust cylinder involves a naked singularity accompanied by its causal future, or a fatal singularity which terminates the history of the whole universe. In a previous paper, the present authors have shown that if the dust is assumed to be composed of collisionless particles such that these particles go through the symmetry axis of the cylinder, then the scalar polynomial singularity formed on the symmetry axis is so weak that almost all of geodesics are complete, and thus effectively no singularity forms by the collapse of a hollow dust cylinder. By contrast, in this paper, we assume that whole of the collapsed dust settles down on the symmetry axis by changing its equation of state. Obtained solutions are the straightforward extension of Morgan's null dust solution, in which no gravitational radiation is emitted. However, in the present case with timelike dust, infinite amount of $C$-energy initially stored in the system is released through gravitational radiation. We also show that the gravitational waves asymptotically behave in a self-similar manner.

Kundt spacetimes

Kundt spacetimes are of great importance in general relativity in four dimensions and have a number of physical applications in higher dimensions in the context of string theory. The degenerate Kundt spacetimes have many special and unique mathematical properties, including their invariant curvature structure and their holonomy structure. We provide a rigorous geometrical kinematical definition of the general Kundt spacetime in four dimensions; essentially a Kundt spacetime is defined as one admitting a null vector that is geodesic, expansion-free, shear-free and twist-free. A Kundt spacetime is said to be degenerate if the preferred kinematic and curvature null frames are all aligned. The degenerate Kundt spacetimes are the only spacetimes in four dimensions that are not -non-degenerate, so that they are not determined by their scalar polynomial curvature invariants. We first discuss the non-aligned Kundt spacetimes, and then turn our attention to the degenerate Kundt spacetimes. The degenerate Kundt spacetimes are classified algebraically by the Riemann tensor and its covariant derivatives in the aligned kinematic frame; as an example, we classify Riemann type D degenerate Kundt spacetimes in which ∇(Riem), ∇(2)(Riem) are also of type D. We discuss other local characteristics of the degenerate Kundt spacetimes. Finally, we discuss degenerate Kundt spacetimes in higher dimensions.

A two-dimensional matrix Padé-type approximation in the inner product space

By introducing a bivariate matrix-valued linear functional on the scalar polynomial space, a general two-dimensional (2-D) matrix Padé-type approximant ( B M P T A ) in the inner product space is defined in this paper. The coefficients of its denominator polynomials are determined by taking the direct inner product of matrices. The remainder formula is developed and an algorithm for the numerator polynomials is presented when the generating polynomials are given in advance. By means of the Hankel-like coefficient matrix, a determinantal expression of B M P T A is presented. Moreover, to avoid the computation of the determinants, two efficient recursive algorithms are proposed. At the end the method of B M P T A is applied to partial realization problems of 2-D linear systems.

LMI Tests for Positive Definite Polynomials: Slack Variable Approach

The considered problem is assessing non-negativity of a function's values when indeterminates are in domains constrained by scalar polynomial inequalities. The tested functions are multi-indeterminates polynomial matrices which are required to be positive semi-definite. New tests based on linear matrix inequalities are provided in a Slack Variables type approach. The results are compared to those obtained via the Sum-Of-Squares approach. They are proved to be equivalent in case of unbounded domains and less conservative if polytopic-type bounds are known.

Spacetimes characterized by their scalar curvature invariants

In this paper, we determine the class of four-dimensional Lorentzian manifolds that can be completely characterized by the scalar polynomial curvature invariants constructed from the Riemann tensor and its covariant derivatives. We introduce the notion of an -non-degenerate spacetime metric, which implies that the spacetime metric is locally determined by its curvature invariants. By determining an appropriate set of projection operators from the Riemann tensor and its covariant derivatives, we are able to prove a number of results (both in the algebraically general and in algebraically special cases) of when a spacetime metric is -non-degenerate. This enables us to prove our main theorem that a spacetime metric is either -non-degenerate or a Kundt metric. Therefore, a metric that is not characterized by its curvature invariants must be of degenerate Kundt form. We then discuss the inverse question of what properties of the underlying spacetime can be determined from a given a set of scalar polynomial invariants, and some partial results are presented. We also discuss the notions of strong and weak non-degeneracy.

Matrix differential equations and scalar polynomials satisfying higher order recursions

We show that any scalar differential operator with a family of polynomials as its common eigenfunctions leads canonically to a matrix differential operator with the same property. The construction of the corresponding family of matrix valued polynomials has been studied in [A. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993) 83–109; A. Durán, On orthogonal polynomials with respect to a positive definite matrix of measures, Canad. J. Math. 47 (1995) 88–112; A. Durán, W. van Assche, Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995) 261–280] but the existence of a differential operator having them as common eigenfunctions had not been considered. This correspondence goes only one way and most matrix valued situations do not arise in this fashion. We illustrate this general construction with a few examples. In the case of some families of scalar valued polynomials introduced in [F.A. Grünbaum, L. Haine, Bispectral Darboux transformations: An extension of the Krall polynomials, Int. Math. Res. Not. 8 (1997) 359–392] we take a first look at the algebra of all matrix differential operators that share these common eigenfunctions and uncover a number of phenomena that are new to the matrix valued case.

The Invar tensor package: Differential invariants of Riemann

Abstract The long standing problem of the relations among the scalar invariants of the Riemann tensor is computationally solved for all 6 ⋅ 10 23 objects with up to 12 derivatives of the metric. This covers cases ranging from products of up to 6 undifferentiated Riemann tensors to cases with up to 10 covariant derivatives of a single Riemann. We extend our computer algebra system Invar to produce within seconds a canonical form for any of those objects in terms of a basis. The process is as follows: (1) an invariant is converted in real time into a canonical form with respect to the permutation symmetries of the Riemann tensor; (2) Invar reads a database of more than 6 ⋅ 10 5 relations and applies those coming from the cyclic symmetry of the Riemann tensor; (3) then applies the relations coming from the Bianchi identity, (4) the relations coming from commutations of covariant derivatives, (5) the dimensionally-dependent identities for dimension 4, and finally (6) simplifies invariants that can be expressed as product of dual invariants. Invar runs on top of the tensor computer algebra systems xTensor (for Mathematica ) and Canon (for Maple ). Program summary Program title: Invar Tensor Package v2.0 Catalogue identifier: ADZK_v2_0 Program summary URL: http://cpc.cs.qub.ac.uk/summaries/ADZK_v2_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 3 243 249 No. of bytes in distributed program, including test data, etc.: 939 Distribution format: tar.gz Programming language: Mathematica and Maple Computer: Any computer running Mathematica versions 5.0 to 6.0 or Maple versions 9 and 11 Operating system: Linux, Unix, Windows XP, MacOS RAM: 100 Mb Word size: 64 or 32 bits Supplementary material: The new database of relations is much larger than that for the previous version and therefore has not been included in the distribution. To obtain the Mathematica and Maple database files click on this link. Classification: 1.5, 5 Does the new version supersede the previous version?: Yes. The previous version (1.0) only handled algebraic invariants. The current version (2.0) has been extended to cover differential invariants as well. Nature of problem: Manipulation and simplification of scalar polynomial expressions formed from the Riemann tensor and its covariant derivatives. Solution method: Algorithms of computational group theory to simplify expressions with tensors that obey permutation symmetries. Tables of syzygies of the scalar invariants of the Riemann tensor. Reasons for new version: With this new version, the user can manipulate differential invariants of the Riemann tensor. Differential invariants are required in many physical problems in classical and quantum gravity. Summary of revisions: The database of syzygies has been expanded by a factor of 30. New commands were added in order to deal with the enlarged database and to manipulate the covariant derivative. Restrictions: The present version only handles scalars, and not expressions with free indices. Additional comments: The distribution file for this program is over 53 Mbytes and therefore is not delivered directly when download or Email is requested. Instead a html file giving details of how the program can be obtained is sent. Running time: One second to fully reduce any monomial of the Riemann tensor up to degree 7 or order 10 in terms of independent invariants. The Mathematica notebook included in the distribution takes approximately 5 minutes to run.

To solving multiparameter problems of algebra. 11. Computing the regular spectrum of a polynomial matrix

For a general multiparameter polynomial matrix \(F(\bar \lambda )\), solution of the equation \(F(\bar \lambda )x = 0\) at points of the finite spectrum of \(F(\bar \lambda )\) is considered. Points of the spectrum of \(F(\bar \lambda )\) are classified in terms of solutions of the determinantal system of nonlinear algebraic equations and also in terms of zeros of certain scalar polynomials (in particular, the characteristic polynomial of the matrix). Algorithms for computing points of the finite regular spectrum of the matrix and the corresponding vectors are presented. Bibliography: 7 titles.

To solving multiparameter problems of algebra. 10. Computing zeros of a scalar polynomial

The paper considers the problem of computing zeros of scalar polynomials in several variables. The zeros of a polynomial are subdivided into the regular (eigen-and mixed) zeros and the singular ones. An algorithm for computing regular zeros, based on a decomposition of a given polynomial into a product of primitive polynomials, is suggested. The algorithm is applied to solving systems of nonlinear algebraic equations. Bibliography: 5 titles.

Scalar polynomial singularities in power-law spacetimes

Recently, Helliwell and Konkowski have examined the quantum “healing” of some classical singularities in certain power-law spacetimes. Here I further examine classical properties of these spacetimes and show that some of them contain naked strong curvature singularities.

ORTHOGONALLY ADDITIVE POLYNOMIALS ON C*-ALGEBRAS

We show that for every orthogonally additive scalar n-homogeneous polynomial P on a C*-algebra A there exists φ in A* satisfying P(x)=φ (xn), for each element x in A. The vector-valued analogue follows as a corollary.

Generalized Homogeneous Multivariate Matrix PadÉ-Type Approximants and PadÉ Approximants

Generalized homogeneous multivariate matrix Pade-type approximants (GHMPTA) and Pade approximants are studied in ways similar to those of Brezinski and Kida in the scalar cases. By choosing an arbitrary monic bivariate scalar polynomial from the triangular form as the generating one of the approximant, we discuss their several typical important properties and study the connection between generalized homogeneous bivariate matrix Pade-type approximants and Pade approximants. The arguments given in detail in two variables can be extended directly to the case of d variables (d ges 2).