On formally real modules

The aim of this paper is to develop the study of modules in real algebra. By introducing two kinds of reality for modules over commutative rings, we establish some results on semireal modules and real modules. Our results involve real radicals of submodules and orderings on modules.     

On designs and formally self-dual codes

Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn2 (mol 8) [n 6 (mod 8)], then the words of a fixed weight inC C hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.The author thanks the University of Illinois at Chicago for their hospitality while this work was in progress.This work was supported in part by NSA Grant MDA 904-91-H-0003.     

Piecewise homotopy analysis method and convergence analysis for formally well-posed initial value problems

In this paper, we apply the Schwarz waveform relaxation (SWR) method to the one-dimensional Schrödinger equation with a general linear or a nonlinear potential. We propose a new algorithm for the Schrödinger equation with time-independent linear potential, which is robust and scalable up to 500 subdomains. It reduces significantly computation time compared with the classical algorithms. Concerning the case of time-dependent linear potential or the nonlinear potential, we use a preprocessed linear operator for the zero potential case as a preconditioner which leads to a preconditioned algorithm. This ensures high scalability. In addition, some newly constructed absorbing boundary conditions are used as the transmission conditions and compared numerically.     

On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T _max)/D(T _min) associated withM[.] in terms of the behaviour of the determinants det [[f _rg_s](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.     

On some Hasse Principles over formally real fields

Mathematische Zeitschrift -     

On the convergence of iterates

THIS PAPER points out that some of Thron's [5] results on the convergence of iterates of a real valued function of a real variable can be extended to general situations afforded, for instance, by metric and gauge spaces. A few examples are given at the end to illustrate the existence of the special type of operators dealt with in this paper.     

On the convergence of SUMT

This paper investigates some convergence properties of Fiacco and McCormick's SUMT algorithm. A convex programming problem is given for which the SUMT algorithm generates a unique unconstrained minimizing trajectory having an infinite number of accumulation points, each a global minimizer to the original convex programming problem.     

On formally real division algebras and quasifields of finite rank

For each natural number n with n2 we construct proper, formally real division algebras and proper, formally real quasifields of dimension n over their kernel.Dedicated to Professor Helmut Karzel on the occasion of his 70th birthday     

On the universal convergence sets

Summary In this paper we study various universal convergence sets, in connection with some problem arising from the Korovkin approximation theory. This analysis is made in the context of topological vector lattices, of commutative Banach algebras, of Banach algebras with an involution and of C ^*-algebras. Some applications and examples are given in specific function spaces and function algebras.Work performed out under the auspices of the G.N.A.F.A. (C.N.R.) and of Ministero della Pubblica Istruzione (60%) for the year 1982.     

On the convergence of cross decomposition

Cross decomposition is a recent method for mixed integer programming problems, exploiting simultaneously both the primal and the dual structure of the problem, thus combining the advantages of Dantzig—Wolfe decomposition and Benders decomposition. Finite convergence of the algorithm equipped with some simple convergence tests has been proved. Stronger convergence tests have been proposed, but not shown to yield finite convergence.In this paper cross decomposition is generalized and applied to linear programming problems, mixed integer programming problems and nonlinear programming problems (with and without linear parts). Using the stronger convergence tests finite exact convergence is shown in the first cases. Unbounded cases are discussed and also included in the convergence tests. The behaviour of the algorithm when parts of the constraint matrix are zero is also discussed. The cross decomposition procedure is generalized (by using generalized Benders decomposition) in order to enable the solution of nonlinear programming problems.     

On the convergence of uncertain sequences

Uncertain variables are measurable functions from uncertainty spaces to the set of real numbers. In this paper, a new kind of convergence, convergence uniformly almost surely (convergence uniformly a.s.), is presented. Then, relations between convergence uniformly almost surely and convergence almost surely (convergence a.s.), convergence in measure, convergence in mean, and convergence in distribution are discussed.