An algorithm for computing Jordan chains and inverting analytic matrix functions

We present an efficient algorithm for obtaining a canonical system of Jordan chains for an n × n regular analytic matrix function A(λ) that is singular at the origin. For any analytic vector function b(λ), we show that each term in the Laurent expansion of A(λ)⁻¹b(λ) may be obtained from the previous terms by solving an (n + d) × (n+d) linear system, where d is the order of the zero of det A(λ) at λ = 0. The matrix representing this linear system contains A(0) as a principal submatrix, which can be useful if A(0) is sparse. The last several iterations can be eliminated if left Jordan chains are computed in addition to right Jordan chains. The performance of the algorithm in floating point and exact (rational) arithmetic is reported for several test cases. The method is shown to be forward stable in floating point arithmetic.     

The existence of matrices with prescribed characteristic and permanental polynomials

Let d(λ) and p(λ) be monic polynomials of degree n?2 with coefficients in F, an algebraically closed field or the field of all real numbers. Necessary and sufficient conditions for the existence of an n-square matrix A over F such that det(λI?A)=d(λ) and per(λI?A=p(λ) are given in terms of the coefficients of d(λ) and p(λ).     

A companion matrix analogue for orthogonal polynomials

Given a set of orthogonal polynomials {P_i(x)}, it is shown that associated with a polynomial a(x)=∑a_ip_i(x) there is a matrix A which possesses several of the properties of the usual companion form matrix C. An alternative and possibly preferable form A' is also suggested. A similarity transformation between A [orA'] and C is given. If b(x) is another polynomial then the matrix b(A) [or b(A')] has properties like those of b(C), relating to the greatest common divisor of a(x) and b(x).     

Lambert or Saccheri quadrilaterals as single primitive notions for plane hyperbolic geometry

With the aim of revealing their purely geometric nature, we rephrase two theorems of S. Yang and A. Fang [S. Yang, A. Fang, A new characteristic of Möbius transformations in hyperbolic geometry, J. Math. Anal. Appl. 319 (2006) 660-664] characterizing Möbius transformations as definability results in elementary plane hyperbolic geometry. We show not only that elementary plane hyperbolic geometry can be axiomatized in terms of the quaternary predicates λ or σ, with λ(abcd) to be read as ‘abcd is a Lambert quadrilateral’ and σ(abcd) to be read as ‘abcd is a Saccheri quadrilateral’, but also that all elementary notions of hyperbolic geometry can be positively defined (i.e. by using only quantifiers (∀ and ∃) and the connectives ∨ and ∧ in the definiens) in terms of λ or σ.     

The structure of the inverse to the Sylvester resultant matrix

Given polynomials a(λ) of degree m and b(λ) of degree n, we represent the inverse to the Sylvester resultant matrix of a and b, if this inverse exists, as a canonical sum of m+n dyadic matrices each of which is a rational function of zeros of a and b.     

Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals

Let T⊂[a,b] be a time scale with a,b∈T. In this paper we study the asymptotic distribution of eigenvalues of the following linear problem −uΔΔ=λuσ, with mixed boundary conditions αu(a)+βuΔ(a)=0=γu(ρ(b))+δuΔ(ρ(b)). It is known that there exists a sequence of simple eigenvalues k{λk}; we consider the spectral counting function , and we seek for its asymptotic expansion as a power of λ. Let d be the Minkowski (or box) dimension of T, which gives the order of growth of the number K(T,ε) of intervals of length ε needed to cover T, namely K(T,ε)≈εd. We prove an upper bound of N(λ) which involves the Minkowski dimension, N(λ,T)?Cλd/2, where C is a positive constant depending only on the Minkowski content of T (roughly speaking, its d-volume, although the Minkowski content is not a measure). We also consider certain limiting cases (d=0, infinite Minkowski content), and we show a family of self similar fractal sets where N(λ,T) admits two-side estimates.     

Left division in the free left distributive algebra on one generator

Let A be the free algebra on one generator satisfying the left distributive law a(bc)=(ab)(ac). Using a division algorithm for elements of an extension P of A, we prove some facts about left division in A, one consequence of which is a conjecture of J. Moody: If a,b,c,d∈A,ab=cd,a and b have no common left divisors, and c and d have no common left divisors, then a=c and b=d.     

Efficient algorithms for decomposing graphs under degree constraints

Stiebitz [Decomposing graphs under degree constraints, J. Graph Theory 23 (1996) 321-324] proved that if every vertex v in a graph G has degree d(v)?a(v)+b(v)+1 (where a and b are arbitrarily given nonnegative integer-valued functions) then G has a nontrivial vertex partition (A,B) such that d_A(v)?a(v) for every v∈A and d_B(v)?b(v) for every v∈B. Kaneko [On decomposition of triangle-free graphs under degree constraints, J. Graph Theory 27 (1998) 7-9] and Diwan [Decomposing graphs with girth at least five under degree constraints, J. Graph Theory 33 (2000) 237-239] strengthened this result, proving that it suffices to assume d(v)?a+b (a,b?1) or just d(v)?a+b-1 (a,b?2) if G contains no cycles shorter than 4 or 5, respectively.The original proofs contain nonconstructive steps. In this paper we give polynomial-time algorithms that find such partitions. Constructive generalizations for k-partitions are also presented.     

Explicit eigenvalue estimates for transfer operators acting on spaces of holomorphic functions

We consider transfer operators acting on spaces of holomorphic functions, and provide explicit bounds for their eigenvalues. More precisely, if Ω is any open set in Cd, and L is a suitable transfer operator acting on Bergman space A²(Ω), its eigenvalue sequence {λn(L)} is bounded by |λn(L)|?Aexp(−an1/d), where a,A>0 are explicitly given.     

Imbedding conditions for λ-matrices

We say that A(λ) is λ-imbeddable in B(λ) whenever B(λ) is equivalent to a λ-matrix having A(λ) as a submatrix. In this paper we solve the problem of finding a necessary and sufficient condition for A(λ) to be λ-imbeddable in B(λ). The solution is given in terms of the invariant polynomials of A(λ) and B(λ). We also solve an analogous problem when A(λ) and B(λ) are required to be equivalent to regular λ-matrices. As a consequence we give a necessary and sufficient condition for the existence of a matrix B, over a field F, with prescribed similarity invariant polynomials and a prescribed principal submatrix A.     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

Schauder estimates for a degenerate second order elliptic operator on a cube

In the present article we are concerned with a class of degenerate second order differential operators LA,b defined on the cube d[0,1], with d?1. Under suitable assumptions on the coefficients A and b (among them the assumption of their Hölder regularity) we show that the operator LA,b defined on C²(d[0,1]) is closable and its closure is m-dissipative. In particular, its closure is the generator of a C₀-semigroup of contractions on C(d[0,1]) and C²(d[0,1]) is a core for it. The proof of such result is obtained by studying the solvability in Hölder spaces of functions of the elliptic problem λu(x)−LA,bu(x)=f(x), x∈d[0,1], for a sufficiently large class of functions f.     

The spectral bounds of the discrete Schrödinger operator

Let H(λ)=−Δ+λb be a discrete Schrödinger operator on ?²(Zd) with a potential b and a non-negative coupling constant λ. When b≡0, it is well known that σ(−Δ)=[0,4d]. When b?0, let and be the bounds of the spectrum of the Schrödinger operator. One of the aims of this paper is to study the influence of the potential b on the bounds 0 and 4d of the spectrum of −Δ. More precisely, we give a necessary and sufficient condition on the potential b such that s(−Δ+λb) is strictly positive for λ small enough. We obtain a similar necessary and sufficient condition on the potential b such that M(−Δ+λb) is lower than 4d for λ small enough. In dimensions d=1 and d=2, the situation is more precise. The following result was proved by Killip and Simon (2003) (for d=1) in [5], then by Damanik et al. (2003) (for d=1 and d=2) in [3]:

     

Regular abelian Banach algebras of linear maps of operator algebras

With H a complex Hilbert space we study regular abelian Banach subalgebras of the Banach algebra of bounded linear maps of B(H) into itself. If a ? b denotes the map x → axb, a, b, x ? B(H), it is shown that normalized positive maps in algebras of the form A ? A with A an abelian C¹-algebra, can be described by a generalized Bochner theorem.     

On connected components in the set of divisors of a monic matrix polynomial

Let A(λ) be a monic matrix polynomial. The topological properties (in particular, connectedness) of the set of all right monic divisors of A(λ) with the same determinant are investigated. It turns out that such set of divisors is not connected in general. Sufficient conditions (in terms of the elementary divisors of A(λ)) for its connectedness are provided.     

Schur-class multipliers on the Arveson space: De Branges-Rovnyak reproducing kernel spaces and commutative transfer-function realizations

An interesting and recently much studied generalization of the classical Schur class is the class of contractive operator-valued multipliers S(λ) for the reproducing kernel Hilbert space H(kd) on the unit ball Bd⊂Cd, where kd is the positive kernel kd(λ,ζ)=1/(1−〈λ,ζ〉) on Bd. The reproducing kernel space H(KS) associated with the positive kernel KS(λ,ζ)=(I−S(λ)S^∗(ζ))⋅kd(λ,ζ) is a natural multivariable generalization of the classical de Branges-Rovnyak canonical model space. A special feature appearing in the multivariable case is that the space H(KS) in general may not be invariant under the adjoints of the multiplication operators on H(kd). We show that invariance of H(KS) under for each j=1,…,d is equivalent to the existence of a realization for S(λ) of the form S(λ)=D+C⁻¹(I−λ₁A₁−?−λdAd)(λ₁B₁+?+λdBd) such that connecting operator has adjoint U^∗ which is isometric on a certain natural subspace (U is “weakly coisometric”) and has the additional property that the state operators A₁,…,Ad pairwise commute; in this case one can take the state space to be the functional-model space H(KS) and the state operators A₁,…,Ad to be given by (a de Branges-Rovnyak functional-model realization). We show that this special situation always occurs for the case of inner functions S (where the associated multiplication operator MS is a partial isometry), and that inner multipliers are characterized by the existence of such a realization such that the state operators A₁,…,Ad satisfy an additional stability property.     

A conjecture involving generalized matrix functions

This paper resolves the following conjecture of R. Merris: Let d^G_λ be the generalized matrix function determined by a subgroup G of the symmetric group S_m and an irreducible complex character λ of G. If A, B, and A?B are m-square positive semidefinite hermitian m-square matrices and d^G_λ(A)=d^G_λ(B)≠0, then A=B.     

The algebraic theory of latent projectors in lambda matrices

Multivariable systems and controls are often formulated in terms of differential equations, which give rise to lambda matrices of the form A(λ) = A₀λⁿ + A₁λ^n-1 + ? + A_n. The inverses of regular lambda matrices can be represented by the latent projectors or matrix residues that have very specific properties. This paper describes the general theory of latent roots, latent vectors, and latent projectors and gives the relationships to eigenvalues, eigenvectors, and eigenprojectors of the companion form.     

Completely strong path-connected tournaments

Let T = (V, A) be a tournament with p vertices. T is called completely strong path-connected if for each arc (a, b) ∈ A and k (k = 2, 3,…, p), there is a path from b to a of length k (denoted by P_k(a, b)) and a path from a to b of length k (denoted by P′_k(a, b)). In this paper, we prove that T is completely strong path-connected if and only if for each arc (a, b) ∈ A, there exist P₂(a, b), P′₂(a, b) in T, and T satisfies one of the following conditions: (a) T/T₀-type graph, (b) T is 2-connected, (c) for each arc (a, b) ∈ A, there exists a P′_p?1(a, b) in T.     

Quadratic matrix polynomials with a parameter

In this paper we examine matrix polynomials of the form L(λ) = Aλ² + εBλ + C in which ε is a parameter and A, B, C are positive definite. This arises in a natural way in the study of damped vibrating systems. The main results are concerned with the generic case in which det L(λ) has at least 2n − 1 distinct zeros for all ε ϵ [0, ∞). The values of ε at which there is a multiple zero of det L(λ) are of major interest in this analysis. The dependence of first degree factors of L(λ) on ε is also discussed.