POWERS OF REAL SYMMETRIC DIFFERENTIAL EXPRESSIONS WITHOUT SMOOTHNESS ASSUMPTIONS

Abstract

In recent years many authors have studied properties of powers of symmetric (formally self-adjoint) ordinary linear differential expressions. So that these powers can be formed in the classical way these authors have placed heavy smoothness assumptions on the coefficients. Here we show that no differentiability conditions whatsoever are needed on the coefficients in order to form powers of a given expression-provided these powers are formed in the quasi-differential sense rather than the classical one.     

ON J-SYMMETRIC QUASI-DIFFERENTIAL EXPRESSIONS WITH MATRIX-VALUED COEFFICIENTS

Abstract

A very general class of quasi-differential expressions which generate J-symmetric operators in Hilbert space is defined with the aid of Green's formula and characterized in terms of the coefficients. Furthermore, the limit-point condition is discussed for not necessarily symmetric expressions.     

Boundary conditions for quasi-differential operators in L p spaces

For a very general class of ordinary quasi-differential expressions M with matrix-valued coefficients and for p, q ∈ [1, ∞) or p, q ∈ (1, ∞] all operators T of a subspace of LP into Lq which satisfy T _M ⁰ ? T ? Tm and are Fredholm operators with index zero are characterized by suitable boundary conditions, where Tm and T _M ⁰ are the maximal and minimal operators associated to M. This generalizes a result of Evans and Ibrahim for p = q = 2.     

Computing the determinants of matrix Padé approximation

This paper is devoted to a computational problem of two special determinants which appear in the construction of generalized inverse matrix Padé approximants of type [n/2k] for the given power series with matrix coefficients. The main tools to be used are well-known Schur complement theorem and Arnoldi process for skew-symmetric systems.     

The Spectra of General Differential Operators in the Direct Sum Spaces

In this paper, the general ordinary quasi-differential expression M _pof n-th order with complex coefficients and its formal adjoint M _p ⁺ on any finite number of intervals I _p =(a _p ,b _p ),p= 1,...,N, are considered in the setting of the direct sums of L _wp ² (a _p ,b _p )-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations of those in a symmetric case in [1], [14], [15], [16], [17] and of a general case with one interval in [2], [11], [12], whilst others are new.     

Existence for a matrix equation arising in microelectronics

In this paper we rigorously show the existence of solutions of a matrix equation which arises in the design of micro electronical circuits. This equation was studied by Szidarovszky and Palusinsk [Appl. Math. Comput. 64, 115-119(1994)], who also presented an iterative algorithm for its solution. We show, via an example, that this algorithm could converge extremely slow in certian cases. The solution can then be used to minimize the reflection coefficients of the active signals.

     

The generalized Fibonomial matrix

The Fibonomial coefficients are known as interesting generalizations of binomial coefficients. In this paper, we derive a (k+1)th recurrence relation and generating matrix for the Fibonomial coefficients, which we call generalized Fibonomial matrix. We find a nice relationship between the eigenvalues of the Fibonomial matrix and the generalized right-adjusted Pascal matrix; that they have the same eigenvalues. We obtain generating functions, combinatorial representations, many new interesting identities and properties of the Fibonomial coefficients. Some applications are also given as examples.     

Pivoting strategy for rank-one modification ofLDM t-like factorization

The rank-one modification algorithm of theLDM ^t factorization was given by Bennett [1]. His method, however, could break down even when the matrix is nonsingular and well-conditioned. We introduce a pivoting strategy for avoiding possible break-down as well as for suppressing error growth in the modification process. The method is based on a symbolic formula of the rank-one modification of the factorization of a possibly singular nonsymmetric matrix. A new symbolic formula is also obtained for the inverses of the factor matrices. Repeated application of our method produces theLDM ^t-like product form factorization of a matrix. A numerical example is given to illustrate our pivoting method. An incomplete factorization algorithm is also introduced for updating positive definite matrix useful in quasi-Newton methods, in which the Fletcher and Powell algorithm [2] and the Gill, Murray and Saunders algorithm [4] are usually used.This paper is presented at the Japan SIAM Annual Meeting held at University of Tokyo, Japan, October 7–9, 1991.     

Estimating the spectral radius of a real matrix by discrete Lyapunov equation

For any real matrix A, this paper is concerned with the estimation of the spectral radius of A. The relationship between the weighted norm and the discrete Lyapunov equation of the matrix A is obtained. On the basis of the relationship, an iterative algorithm is presented to obtain the spectral radius of A and to estimate the solution of the corresponding linear discrete system. Several numerical examples are given to show that the iterative algorithm is effective.     

A Comparison Theorem for a Second Order Linear Quasi-Differential Equation

We are concerned with the non-oscillatory behaviour of a secondorder linear quasi-differential equation defined a.e. over ahalf-line. More particularly given that (py')' + qy = 0 is non-oscillatorywe show that by writing q in a form which is amenable to furtheranalysis we can generalise Sturm's comparison theorem.     

A formula for deficiency (L-andR-tests)

In this paper an analogue of the formulas [D. M. Chibisov,Teor. Veroyatn. Primen.,30, 269–288 (1985);Izv. Akad. Nauk UzSSR,6, 23–30 (1982)] for the difference between the power of a given asymptotically efficient test and that of the most powerful test is justified for one-sample L-and R-tests, i.e., tests based on linear combinations of order statistics and linear rank statistics. This formula directly yields the Hodges-Lehmann deficiency of corresponding tests. A general theorem is stated which is applied to L-and R-tests. The explicit expressions given by this formula for L- and R-tests are also presented. The expression related to R-tests agrees with the one obtained in [W. Albers, P. J. Bickel, and W. R. Van Zwet,Ann. Statist.,4, 108–156 (1976);6, 1170–1171 (1978)]. We present here a nontechnical (heuristic) proof of these results. Supported by the Russian Foundation for Fundamental Research (grant No. 93-011-1446). Proceedings of the XVI Seminar on Stability Problems for Stochastic Models, Part II, Eger, Hungary, 1994.     

The rationality of vector valued modular forms associated with the Weil representation

 In a recent paper [Duke Math. J., 97, 219–233], Borcherds asks whether or not the spaces of vector valued modular forms associated to the Weil representation have bases of modular forms whose Fourier expansions have only integer coefficients. We give an affirmative answer to Borcherds' question. This strengthens and simplifies Borcherds' main theorem which is a generalization of a theorem of Gross, Kohnen, and Zagier [Math. Ann., 278, 497–562]. Received: 27 September 2001 / Revised version: 22 July 2002 / Published online: 28 March 2003 Mathematics Subject Classification (1991): 11F30; 11F27     

On a Conjectured Formula for Quiver Varieties

In A.S. Buch and W. Fulton [Invent. Math. 135 (1999), 665–687] a formula for the cohomology class of a quiver variety is proved. This formula writes the cohomology class of a quiver variety as a linear combination of products of Schur polynomials. In the same paper it is conjectured that all of the coefficients in this linear combination are non-negative, and given by a generalized Littlewood-Richardson rule, which states that the coefficients count certain sequences of tableaux called factor sequences. In this paper I prove some special cases of this conjecture. I also prove that the general conjecture follows from a stronger but simpler statement, for which substantial computer evidence has been obtained. Finally I will prove a useful criterion for recognizing factor sequences.     

The existence of soliton metrics for nilpotent Lie groups

We show that a left-invariant metric g on a nilpotent Lie group N is a soliton metric if and only if a matrix U and vector v associated the manifold (N, g) satisfy the matrix equation U v = [1], where [1] is a vector with every entry a one. We associate a generalized Cartan matrix to the matrix U and use the theory of Kac–Moody algebras to analyze the solution spaces for such linear systems. An application to the existence of soliton metrics on certain filiform Lie groups is given.     

Quantum Cohomology of a Pfaffian Calabi–Yau Variety: Verifying Mirror Symmetry Predictions

We formulate a generalization of Givental–Kim's quantum hyperplane principle. This is applied to compute the quantum cohomology of a Calabi–Yau 3-fold defined as the rank 4 locus of a general skew-symmetric 7×7 matrix with coefficients in P ⁶. The computation verifies the mirror symmetry predictions of Rødland [25].     

On an inverse problem for nonnegative and eventually nonnegative matrices

Let σ= (λ₁,···λ_n)⊂C. We discuss conditions for which σ is the spectrum of a nonnegative or eventually nonnegative matrix. This brings us to study rational functions with nonnegative Maclaurin coefficients. A conjecture for special sets σ is stated and some evidence in support of this conjecture is given. Dedicated to my teacher Professor Menachem Schiffer on the occasion of his sixty-fifth birthday.     

Matrix polynomials,roots and spectral factors

This paper discusses some properties of matrix polynomials and a computational procedure for finding the matrix roots of such polynomials and their relationship to spectral factorization. Polynomials of order n with square matrix coefficients of order N are considered. The computational procedure is of interest in the analysis and design of multivariable control systems.     

Matrix Extension with Symmetry and Applications to Symmetric Orthonormal Complex M-wavelets

Matrix extension with symmetry is to find a unitary square matrix P of 2π-periodic trigonometric polynomials with symmetry such that the first row of P is a given row vector p of 2π-periodic trigonometric polynomials with symmetry satisfying p[`(p)]^T=1\mathbf {p}\overline{\mathbf{p}}^{T}=1 . Matrix extension plays a fundamental role in many areas such as electronic engineering, system sciences, wavelet analysis, and applied mathematics. In this paper, we shall solve matrix extension with symmetry by developing a step-by-step simple algorithm to derive a desired square matrix P from a given row vector p of 2π-periodic trigonometric polynomials with complex coefficients and symmetry. As an application of our algorithm for matrix extension with symmetry, for any dilation factor M, we shall present two families of compactly supported symmetric orthonormal complex M-wavelets with arbitrarily high vanishing moments. Wavelets in the first family have the shortest possible supports with respect to their orders of vanishing moments; their existence relies on the establishment of nonnegativity on the real line of certain associated polynomials. Wavelets in the second family have increasing orders of linear-phase moments and vanishing moments, which are desirable properties in numerical algorithms.     

Approximation of multivariate distribution functions

In the paper the unknown distribution function is approximated with a known distribution function by means of Taylor expansion. For this approximation a new matrix operation — matrix integral — is introduced and studied in [PIHLAK, M.: Matrix integral, Linear Algebra Appl. 388 (2004), 315–325]. The approximation is applied in the bivariate case when the unknown distribution function is approximated with normal distribution function. An example on simulated data is also given.      

Insensitive bounds for the moments of the sojourn time distribution in the M/G/1 processor-sharing queue

This paper studies the M/G/1 processor-sharing (PS) queue, in particular the sojourn time distribution conditioned on the initial job size. Although several expressions for the Laplace-Stieltjes transform (LST) are known, these expressions are not suitable for computational purposes. This paper derives readily applicable insensitive bounds for all moments of the conditional sojourn time distribution. The instantaneous sojourn time, i.e., the sojourn time of an infinitesimally small job, leads to insensitive upper bounds requiring only knowledge of the traffic intensity and the initial job size. Interestingly, the upper bounds involve polynomials with so-called Eulerian numbers as coefficients. In addition, stochastic ordering and moment ordering results for the sojourn time distribution are obtained. AMS Subject Classification: 60K25, 60E15 This work has been partially funded by the Dutch Ministry of Economic Affairs under the program ‘Technologische Samenwerking ICT-doorbraakprojecten’, project TSIT1025 BEYOND 3G.