LP formulations of the shortest path tree problem

Based on a pair of primal-dual LP formulations of the shortest path tree problem, the first algorithmic approach to reoptimizing the shortest paths subject to changes in the edge weights was proposed by S. Pallottino and M.G. Scutellá in 2003. We shall here focus solely on their introductory sections, propose some simplifications of the models considered, and finally relate the resulting models to the presentation of single-source shortest path problems in textbooks treating this subject with but limited or no reference to LP.Received: April 2004, Revised: August 2004, MSC classification: 90C05, 90C35, 90B10 Dedicated to the memory of Stefano Pallottino     

Taylor polynomials and non-homogeneous blow-ups

The graph of a function f is subjected to non-homogeneous dilatations around (x₀;f(x₀)), related to the Taylor expansion of f at x₀. Some natural questions about convergence are considered and answered. Finally, it is provided a counterexample to a statement which was presumed to be true in former literature.Mathematics Subject Classification (1991): Primary 41A10, 49Q15, 53A05, 54G20; Secondary 28A75, 28A78, 28A33, 54C20     

String Jacobi Matrices with Very Fast Decreasing Weights

In this work we investigate a class of string, compact Jacobi matrices with very rapidly decreasing weights. In particular, we have found Hellinger-Weyls function representation as ratio of two entire functions with known Taylor series representation. The asymptotical estimations of zeroes of this entire functions and of eigenvalues of a corresponding operator are found. The application of this theory to Rogers-Ramanujans functions is considered.Submitted: November 18, 2002 Revised: October 30, 2003     

Two-Wavelet Localization Operators on $$ L^p(\mathbb{R}^n) $$ for the Weyl-Heisenberg Group

We give results on the boundedness and compactness of localization operators with two admissible wavelets on for the Weyl-Heisenberg group.     

Existence theory for nonclassical entropy solutions of scalar conservation laws

We establish a general existence theory for the Cauchy problem associated with a scalar conservation law in one-space dimension. The flux-function is assumed to be nonconvex and we consider nonclassical entropy solutions selected by a kinetic relation. To solve the Cauchy problem, we construct a sequence of approximate solutions using a wave-front tracking scheme. The main difficulty is deriving a uniform estimate on the total variation of the approximate solutions. This is achieved here by introducing a generalized total variation functional, which is decreasing in time and, additionally, reduces to the standard total variation functional when the solutions contain only classical shocks. This functional seems sufficiently robust to be useful for systems as well.Received: June 3, 2002; revised: November 12, 2002     

The number of simple modules of a cellular algebra

Let n be a natural number, and let A be an indecomposable cellular algebra such that the spectrum of its Cartan matrix C is of the form {n, 1, …., 1}. In general, not every natural number could be the number of non-isomorphic simple modules over such a cellular algebra. Thus, two natural questions arise: (1) which numbers could be the number of non-isomorphic simple modules over such a cellular algebra A ? (2) Given such a number, is there a cellular algebra such that its Cartan matrix has the desired property ? In this paper, we shall completely answer the first question, and give a partial answer to the second question by constructing cellular algebras with the pre-described Cartan matrix.     

Adaptive quasi-likelihood estimate in generalized linear models

This paper gives a thorough theoretical treatment on the adaptive quasi-likelihood estimate of the parameters in the generalized linear models. The unknown covariance matrix of the response variable is estimated by the sample. It is shown that the adaptive estimator defined in this paper is asymptotically most efficient in the sense that it is asymptotic normal, and the covariance matrix of the limit distribution coincides with the one for the quasi-likelihood estimator for the case that the covariance matrix of the response variable is completely known.     

BGP reflection functors in root categories

We define the BGP-reflection functors in the derived categories and the root categories. By Ringel's Hall algebra approach, the BGP-reflection functor is applicable to obtain the classical Weyl group action on the Lie algebra.     

Bayes and empirical Bayes iteration estimators in two seemingly unrelated regression equations

For a system of two seemingly unrelated regression equations given by (?)(y_1 is an m×1 vector and y_2 is an n×1 vector,m≠n),employ- ing the covariance adjusted technique,we propose the parametric Bayes and empirical Bayes iteration estimator sequences for regression coefficients.We prove that both the covariance matrices converge monotonically and the Bayes iteration estimator squence is consistent as well.Based on the mean square error (MSE) criterion,we elaborate the su- periority of empirical Bayes iteration estimator over the Bayes estimator of single equation when the covariance matrix of errors is unknown.The results obtained in this paper further show the power of the covariance adiusted approach.     

On the finite sum representations of the Lauricella functions <Emphasis Type="Italic">F</Emphasis><Subscript>D</Subscript>

By using divided differences, we derive two different ways of representing the Lauricella function of n variables F_D⁽ⁿ⁾(a,b₁,b₂,...,b_n;c;x₁,x₂,...,x_n) as a finite sum, for b₁,b₂,...,b_n positive integers, and a,c both positive integers or both positive rational numbers with c–a a positive integer. AMS subject classification 33D45, 40B05, 40C99Jieqing Tan: Research supported by the National Natural Science Foundation of China under Grant No. 10171026 and Anhui Provincial Natural Science Foundation under Grant No. 03046102.Ping Zhou: Corresponding author. Research supported by NSERC of Canada.