On orthogonal polynomials with respect to the form -(x - c)S'

Summary A form (linear functional) $u$ is called regular if we can associate with it a sequence of monic orthogonal polynomials. On certain regularity conditions, the product of a non regular form by a polynomial can be regular. The purpose of this work is to establish regularity conditions of the form $-(x-c){\mathbf S}',$ where ${\mathbf S}$ is a classical (Bessel, Jacobi). We give the second-order recurrence relations and structure relations of its corresponding orthogonal polynomial sequence. We conclude with an example as an illustration.     

On the powers and the inverse of a tridiagonal matrix

In this paper, we present an eigendecomposition of a tridiagonal matrix. Tridiagonal matrix powers and inverse are derived. As consequence, we get some relations verified by the coefficients of the inverse and the powers of a tridiagonal matrix.     

Existence of (3, 1, 2)‐conjugate orthogonal diagonal latin squares

We shall refer to a diagonal Latin square which is orthogonal to its (3,1,2)‐conjugate, and the latter is also a diagonal Latin square, as a (3,1, 2)‐conjugate orthogonal diagonal Latin square, briefly CODLS. This article investigates the spectrum of CODLS and it is found that it contains all positive integers υ except 2, 3, 6, and possibly 10. © 2001 John Wiley & Sons, Inc. J Combin Designs 9: 297–308, 2001     

On the roots of the orthogonal polynomials and residual polynomials associated with a conjugate gradient method

In this paper we explore two sets of polynomials, the orthogonal polynomials and the residual polynomials, associated with a preconditioned conjugate gradient iteration for the solution of the linear system Ax = b. In the context of preconditioning by the matrix C, we show that the roots of the orthogonal polynomials, also known as generalized Ritz values, are the eigenvalues of an orthogonal section of the matrix C A while the roots of the residual polynomials, also known as pseudo-Ritz values (or roots of kernel polynomials), are the reciprocals of the eigenvalues of an orthogonal section of the matrix (C A)^?1. When C A is selfadjoint positive definite, this distinction is minimal, but for the indefinite or nonselfadjoint case this distinction becomes important. We use these two sets of roots to form possibly nonconvex regions in the complex plane that describe the spectrum of C A.     

An alternative proof of the Barker, Berman, Plemmons (BBP) result on diagonal stability and extensions

We revisit the theorem of Barker, Berman and Plemmons on the existence of a diagonal quadratic Lyapunov function for a stable linear time-invariant (LTI) dynamical system [G.P. Barker, A. Berman, R.J. Plemmons, Positive diagonal solutions to the Lyapunov equations, Linear and Multilinear Algebra 5(3) (1978) 249-256]. We use recently derived results to provide an alternative proof of this result and to derive extensions.     

关于正交设计与均匀设计的比较（Ⅱ）

本文通过对优化效率的比较，继续［１］说明在主要方面正交设计要比均匀设计更好。     

关于正交设计与均匀设计的比较（Ⅰ）

本文通过对优化效率的比较，说明在主要方面正交设计要比均匀设计更好。     

关于正交设计与均匀设计的比较（Ⅲ）

本文通过平均冒尖性（差值评分λ）和极大最小距离ｄ的比较，继续［１，２」说明正交设计的优化效率在主要方面要比均匀设计更好．另外，举反例表明“分散性带来冒尖性”并不绝对有序．     

均匀设计法在优化青霉素1~＃、2~＃菌种培养基配方中的应用

我们将均匀设计法与传统正交设计法相结合，通过实验成功地优化青霉素菌种配方，筛选培养基。（仅ｕ／ｍｌ这一项就比对照１＃配方高２０．７５％，比２＃配方高３１．８４％）。     

关于函数S(x)及S_*(x)的渐近性质

对于任意实数x∈(1,∞),定义S(x)=m in{m∈N∶x m!};x∈[1,∞),S*(x)=m ax{m∈N∶m!x}.主要目的是研究这两个函数的渐近性质,并给出了它们的渐近公式.     

On the residual finiteness and other properties of (relative) one-relator groups

A relative one-relator presentation has the form where is a set, is a group, and is a word on . We show that if the word on obtained from by deleting all the terms from has what we call the unique max-min property, then the group defined by is residually finite if and only if is residually finite (Theorem 1). We apply this to obtain new results concerning the residual finiteness of (ordinary) one-relator groups (Theorem 4). We also obtain results concerning the conjugacy problem for one-relator groups (Theorem 5), and results concerning the relative asphericity of presentations of the form (Theorem 6).

     

On WLI-ideal space and the properties of WLI-ideals in lattice implication algebra (LIA)

In classical logic and non-classical logic, Modus Ponens (MP rule) is very useful for a formal deduction. Firstly, we focus on the properties of WLI-ideal of lattice implication algebras based on the definition of ⊕ operation and ? operation, which are abstraction of MP rule. On the other hand, we consider the WLI-ideal topological space (L,T _W(L)) based on LIA, and its topological properties are investigated, it is an abstraction of MP rule-base. Finally, we consider the fuzzification of WLI-ideals in lattice implication algebras, and the properties of FWLI-ideals are investigated.