Family Of Pencils Research Articles

Some Families of Pencils with a Unique Base Point and Their Associated Foliations

Some Families of Pencils with a Unique Base Point and Their Associated Foliations

Structured strong linearizations from Fiedler pencils with repetition I

In many applications, the polynomial eigenvalue problem, P(λ)x=0, arises with P(λ) being a structured matrix polynomial (for example, palindromic, symmetric, skew-symmetric). In order to solve a structured polynomial eigenvalue problem it is convenient to use strong linearizations with the same structure as P(λ) to ensure that the symmetries in the eigenvalues due to that structure are preserved in numerical computations. In this paper we characterize all the pencils in the family of the Fiedler pencils with repetition, introduced by Vologiannidis and Antoniou [25], associated with a square matrix polynomial P(λ) that are block-symmetric for every matrix polynomial P(λ). We show that this family of pencils is precisely the set of all Fiedler pencils with repetition that are symmetric when P(λ) is. When some generic nonsingularity conditions are satisfied, these pencils are strong linearizations of P(λ). In particular, our family strictly contains the standard basis for DL(P), a k-dimensional vector space of symmetric pencils introduced by Mackey, Mackey, Mehl, and Mehrmann [20].

Palindromic companion forms for matrix polynomials of odd degree

The standard way to solve polynomial eigenvalue problems P(λ)x=0 is to convert the matrix polynomial P(λ) into a matrix pencil that preserves its spectral information — a process known as linearization. When P(λ) is palindromic, the eigenvalues, elementary divisors, and minimal indices of P(λ) have certain symmetries that can be lost when using the classical first and second Frobenius companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure. Recently new families of pencils have been introduced with the goal of finding linearizations that retain whatever structure the original P(λ) might possess, with particular attention to the preservation of palindromic structure. However, no general construction of palindromic linearizations valid for all palindromic polynomials has as yet been achieved. In this paper we present a family of linearizations for odd degree polynomials P(λ) which are palindromic whenever P(λ) is, and which are valid for all palindromic polynomials of odd degree. We illustrate our construction with several examples. In addition, we establish a simple way to recover the minimal indices of the polynomial from those of the linearizations in the new family.

The integral geometry boundary determination problem for a pencil of straight lines

Under consideration is the problem of integrating finitely many functions over straight lines. Each function as well as the corresponding line is assumed unknown. The available information is the sum of integrals over all straight lines of a family of pencils in each of which the intersection of lines is a point of a given bounded open set in a finite-dimensional Euclidean space. Each integrand depends on a greater number of variables than the sum of the integrals. Hence, the conventional statement of the problem of determining the integrands becomes underspecified. In this situation we pose and study the problem of determining the discontinuity surfaces of the integrands. The uniqueness theorem is proven under the condition that these surfaces exist. The present article is a refinement of the previous studies of the authors and differs from them in [1–6] by not only some technical improvements but also the principally new fact that the integration is performed over an unknown set.

Free pencils on divisors

Let X be a smooth projective variety defined over an algebraically closed field, and let Y in X be a reduced and irreducible ample divisor in X. We give a numerical sufficient condition for a base point free pencil on $Y$ to be the restriction of a base point free pencil on $X$. This result is then extended to families of pencils and to morphisms to arbitrary smooth curves. Serrano had already studied this problem in the case n=2 and 3, and Reider had then attacked it in the case $n=2$ using vector bundle methods based on Bogomolov's instability theorem on a surface (char(k)=0). The argument given here is based on Bogomolov's theorem on an n-dimensional variety, and on its recent adaptations to the setting of prime charachterstic (due to Shepherd-Barron and Moriwaki).

On the Segré, Weyr characteristics of right (left) regular matrix pencils

The classical Segré, Weyr characteristic theory of the standard eigenvalue-eigenvector problem defined on A ϵ ℝn×n is extended to the case of right (left) regular pencils sF — G, F, G ϵ ℝm×n. The notions of α-Toeplitz matrices and of the right (left) α. — (F, G) sequences Jα r(F,G) (Jα 1(F,G)) respectively are introduced. For right (left) regular pairs it is shown that Jα r(F,G) (Jα 1(F,G)) are piecewise arithmetic progression sequences (PAPS); the singular points of those PAPS define the possible degrees of the elementary divisors (e.d.) at α, whereas the deviations, or gaps from the arithmetic progression sequence, define the multiplicity of the corresponding degree e.d.s. The links between the algebraic notion of the Segré characteristic and the geometric notion of the Weyr characteristic are established for such families of pencils. Two different procedures for the computation of the Segré characteristic are given; the first is based on the analysis of singular points of PAPS and the second is based on the conjugate partitioning property of the algebraic multiplicity of a zero of sF - G at α by the Weyr and Segré characteristics at a. In the latter case a Ferrers-type diagram is suggested as the tool for the computation of the Segré characteristic from the Weyr characteristic. The results provide a procedure for the computation of the Weierstrass part of the Kronecker form of right (left) regular pencils by rank tests on α-Toeplitz matrices.