Spectral factorization of rectangular rational matrix functions with application to discrete Wiener-Hopf equations

The properties of a discrete Wiener-Hopf equation are closely related to the factorization of the symbol of the equation. We give a necessary and sufficient condition for existence of a canonical Wiener-Hopf factorization of a possibly nonregular rational matrix function W relative to a contour which is a positively oriented boundary of a region in the finite complex plane. The condition involves decomposition of the state space in a minimal realization of W and, if it is satisfied, we give explicit formulas for the factors. The results are generalized by means of centered realizations to arbitrary rational matrix functions. The proposed approach can be used to solve discrete Wiener-Hopf equations whose symbols are rational matrix functions which admit canonical factorization relative to the unit circle.     

有限扩张环

证明了环的有限扩张性可以传递到矩阵环上;通过PP环,半遗传环以及有限余非奇异环刻划了有限扩张环,并推广了文献[2]的定理2.1; 对于FGF与CF猜测,给出了部分肯定的回答,即右有限扩张右CF环是右CEP的,从而是右aritian的,改进了文献[6]的定理3.7.     

A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator

This is a sequel of the article by Borichev, Golinskii and Kupin (2009) [1], where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk, which satisfy certain growth hypotheses. These results were applied to get Lieb–Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev et al. (2009) [1]. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb–Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb–Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or finite-band) Jacobi matrix.     

The diamond partial order in rings

In this paper we introduce a new partial order on a ring, namely the diamond partial order. This order is an extension of a partial order defined in a matrix setting in [J.K. Baksalary and J. Hauke, A further algebraic version of Cochran’s theorem and matrix partial orderings, Linear Algebra and its Applications, 127, 157–169, 1990]. We characterize the diamond partial order on rings and study its relationships with other partial orders known in the literature. We also analyse successors, predecessors and maximal elements under the diamond order.     

Lagrange基函数的复矩阵有理插值及连分式插值

1引言 矩阵有理插值问题与系统线性理论中的模型简化问题和部分实现问题有着紧密的联系~[1][2]，在矩阵外推方法中也常常涉及线性或有理矩阵插值问题~[3]。按照文~[1]的阐述。目前已经研究的矩阵有理插值问题包括矩阵幂级数和Newton-Pade逼近。Hade逼近，联立Pade逼近，M-Pade逼近，多点Pade逼近等。显然，上述各种形式的矩阵Pade逼上梁山近是矩     

Adjoints of rationally induced composition operators

We give an elementary proof of a formula recently obtained by Hammond, Moorhouse, and Robbins for the adjoint of a rationally induced composition operator on the Hardy space H² [Christopher Hammond, Jennifer Moorhouse, Marian E. Robbins, Adjoints of composition operators with rational symbol, J. Math. Anal. Appl. 341 (2008) 626-639]. We discuss some variants and implications of this formula, and use it to provide a sufficient condition for a rationally induced composition operator adjoint to be a compact perturbation of a weighted composition operator.     

A Note on Symbolic Integration with Polylogarithms

We generalize partially Liouville’s theorem on integration in finite terms to allow polylogarithms of any order to occur in the integral in addition to elementary functions. The result is a partial generalization of a theorem proved by the author for the dilogarithm. It is also a partial proof of a conjecture postulated by the author in 1994 [1]. The basic conclusion is that an associated function to the nth polylogarithm appears linearly with logarithms appearing possibly in a polynomial way with non-constant coefficients.     

MATRIX ALGORITHMS AND ERROR FORMULA FOR BIVARIATE THIELE-TYPE RECTANGULAR MATRIX VALUED RATIONAL INTERPOLATION

A new method for the construction of bivariate matrix valued rational interpolants (BGIRI) on a rectangular grid is presented in [6]. The rational interpolants are of Thiele-type continued fraction form with scalar denominator. The generalized inverse introduced by [3]is gen-eralized to rectangular matrix case in this paper. An exact error formula for interpolation is ob-tained, which is an extension in matrix form of bivariate scalar and vector valued rational interpola-tion discussed by Siemaszko[l2] and by Gu Chuangqing [7] respectively. By defining row and col-umn-transformation in the sense of the partial inverted differences for matrices, two type matrix algorithms are established to construct corresponding two different BGIRI, which hold for the vec-tor case and the scalar case.     

A fixed point theorem with applications to truncated lattices

We prove a fixed point theorem related to the set P2 of [17]. The result gives access to nontrivial infinite ordered sets with the fixed point property. We also show how the result can be used to provide an elementary proof of part of Baclawski and Björner’s results on truncated lattices.Dedicated to the memory of Ivan RivalReceived December 1, 2002; accepted in final form June 18, 2004.This revised version was published online in August 2005 with a corrected cover date.     

An improved lower bound for deflections of laminated cantilever beams including the effect of transverse shear deformation

A previous paper (reference [3]) has formulated a version of the well-known plane stress problem of the end-loaded cantilever beam of narrow rectangular cross section in a way which allowed the statement of upper and lower bounds for the deflection of the beam, through use of the principles of minimum potential and complementary energy. In extension of the work in reference [3] we here derive an improved lower bound, with the consequence that now the upper and the lower bound coincide for the limiting case of vanishing depth to span ratio of the beam. This means that we have a rigorous proof of the correctness of the ‘elementary strength of materials’ result for the case of a very slender beam, in addition to quantitatively reliable corrections to the elementary result for the case of beams which are not slender.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.     

Determination of the underground contour of a submerged apron with a region of constant velocity when there is a salt backwater

Within the framework of two-dimensional seepage theory, the underground contour of a submerged apron with a region of constant velocity in the case where there is a layer of stagnant salt water under the apron is constructed. The solution of the corresponding boundary-value problem is found by Polubarinova-Kochina's method [1] using the results obtained in [2]. The results of numerical calculations are given and the influence of the fundamental defining parameters of the model on the shape and size of the underground contour of the apron is analysed. Mention is made of special and limiting cases: a scheme with a water-confining stratum [3], an unsubmerged apron [2] and flow around a tongue [4,5].     

A testable version of the Pareto-stable CAPM

The Capital Asset Pricing Model (CAPM) under the Pareto-stable assumptions was first developed by Fama [1] and Ross [2]. Fama treated a restricted case and neither he nor Ross gave an expression of the coefficient beta that could lead to an empirical use of the stable CAPM. Using new developments on the multivariate structure of stable laws [3], we give a self-contained version of the Pareto-stable CAPM with several convenient expressions of beta [4].Some of these expressions provide us with straightforward estimators of beta. We also address the issue of the Best Linear Unbiased Estimator (BLUE). Unfortunately, the BLUE program is solved only in Fama's restricted case.We have used the backtesting procedure proposed by Black in [5] in order to test our various estimators of beta. Black's procedure tends to show that leverage effects linked to beta can be efficiently exploited on the French market. Therefore, a simple statistical test based on this procedure was devised to determine whether beta's predictive power could still be improved with the stable hypothesis.Although no improvement has been realized so far, the statistical estimation of the stable beta is still an open issue and the testing procedure may prove to be of use for other new estimators.     

Local Wiener–Hopf factorization and indices over arbitrary fields

In this paper, a generalization to arbitrary fields of the usual Wiener–Hopf equivalence of complex valued rational matrix functions is given and the left local Wiener–Hopf factorization indices defined in our previous work [A. Amparan, S. Marcaida, I. Zaballa, Local realizations and local polynomial matrix representations of systems, Linear Algebra Appl. 425 (2007) 757–775] are proved to form a complete system of invariants for this equivalence relation. For the case when the field is algebraically closed a reduced form of a controllable matrix pair under the feedback equivalence is presented for which the controllability indices can be written as sums of the local controllability indices [A. Amparan, S. Marcaida, I. Zaballa, On the existence of linear systems with prescribed invariants for system similarity, Linear Algebra Appl. 413 (2006) 510–533].     

On Feller processes with sample paths in Besov spaces

Under mild regularity assumptions on its domain the infinitesimal generator of a Feller process is known to be a pseudo-differential operator. We give a simple condition on the symbol of the generator in order to characterize the smoothness of the sample paths of real-valued Feller processes in terms of Besov spaces . Our result extends previous papers on the paths of Gaussian, symmetric -stable [6], [20], and Lévy processes [11]. Received: 31 May 1996 / Revised version: 10 December 1996     

Wiener–Hopf indices of unitary functions on the unit circle in terms of realizations and related results on Toeplitz operators

We provide new formulas for the Wiener–Hopf factorization indices of a rational matrix function $R$ which has neither poles nor zeros on the unit circle. In addition, we recover recent results on the Fredholm characteristics of the Toeplitz operator with symbol $R$ via the method of matricial coupling. Furthermore, we present an alternative formula for the index in terms of the Fourier coefficients of $R$.     

The class of infinite dimensional neat reducts of quasi‐polyadic algebras is not axiomatizable

SC, CA, QA and QEA denote the classes of Pinter's substitution algebras, Tarski's cylindric algebras, Halmos' quasi‐polyadic algebras and quasi‐polyadic equality algebras, respectively. Let ω ≤ α < β and let K ∈ {SC,CA,QA,QEA}. We show that the class of α ‐dimensional neat reducts of algebras in K_β is not elementary. This solves a problem in [3]. Also our result generalizes results proved in [2] and [3]. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The Linear Rational Pseudospectral Method for Boundary Value Problems

We consider the version of the pseudospectral method for solving boundary value problems which replaces the differential operator with a matrix constructed from the elementary differentiation matrices whose elements are the derivatives of the Lagrange fundamental polynomials at the collocation points. The iterative solution of the resulting system of equations then requires the recurrent application of that differentiation matrix. Since global polynomial interpolation on the interval only gives useful approximants for points which accumulate in the vicinity of the extremities, the matrix is ill-conditioned. To reduce this drawback, we use Kosloff and Tal-Ezer's suggestion to shift the collocation points closer to equidistant by a conformal map. However, instead of applying their change of variable setting, we extend to stationary equations the linear rational collocation method introduced in former work on partial differential equations. Numerically about as efficient, this does not require any new coding if one starts from an efficient program for the polynomial differentiation matrices.This revised version was published online in October 2005 with corrections to the Cover Date.     

A fraction-free unit-circle zero location test for a polynomial with any singularity profile

This paper presents a fraction-free (FF) version of the Bistritz test to determine the zero location (ZL) of a polynomial with respect to the unit circle. The test has the property that when it is invoked on a polynomial with Gaussian or real integer coefficients, it is an efficient integer algorithm completed without fractions over the respective integral domain. The test is not restricted to integers but remains integer preserving (IP) in all possible encounters of abnormalities and singularities. We define a symmetric subresultant polynomial sequence (SSPS) for the Sylvester matrix of two symmetric polynomials. We then show that the sequence of polynomials produced by the FF test coincides with the SSPS of its first two polynomials when the test is normal and the SSPS is strongly nonsingular, or else its polynomials match the non-singular subresultant polynomial and pass over intermediate gaps of singular subresultants in an IP and efficient manner. This relationship (interesting in its own right) is used to show that the test is IP and normally attains integers of minimal size.     

Orders in rings without identity

G. Lallement [5] proved that every idem potent congruence class of a regular semigroup contains an idem potent. P. Edwards [4] generalized this property of congruences to eventually regular semigroups. Using the natural partial order of the semigroup (see [6]) a weakened version of this result will be proved for the more general class of E-inversive semigroups. But for particular congruences the original result of Lallement still holds for every E-inversive semigroup. Finally, conditions for a congruence on a general semigroup (with E(S) a subsemigroup, resp.) are given, which ensure that Lallement's result holds.