Variational approximations for couplers and Y junctions

A simple variational method is used to determine the modal properties of single-mode couplers and Y junctions when the cores are close together or coalesce. The results complement the well-known perturbation expressions for large core separation. Together the two approximation methods provide accurate approximations over the complete range of separation from infinite separation to complete coalescence.     

N2- and O2-broadening parameters in the ν3 band of 14N216O

Broadening parameters of lines belonging to the ν₃ band of N₂O in collision with N₂ or O₂ molecules have been measured with a vacuum high-resolution grating spectrometer. The measurements were carried out in the R branch, up to J = 73 at 297 K and up to J = 50 at 195 K. The experimental results are compared with values calculated from two semiclassical theories.     

A comparison of two methods for solving 0–1 integer programs using a general purpose simulated annealing algorithm

0–1 problems are often difficult to solve. Although special purpose algorithms (exact as well as heuristic) exist for solving particular problem classes or problem instances, there are few general purpose algorithms for solving practical-sized instances of 0–1 problems. This paper deals with a general purpose heuristic algorithm for 0–1 problems. In this paper, we compare two methods based on simulated annealing for solving general 0–1 integer programming problems. The two methods differe in the scheme used for neighbourhood transitions in the simulated annealing framework. We compare the performance of the two methods on the set partitioning problem.     

Study of the initial value problems appearing in a method of factorization of second-order elliptic boundary value problems

In [J. Henry, A.M. Ramos, Factorization of second order elliptic boundary value problems by dynamic programming, Nonlinear Analysis. Theory, Methods & Applications 59 (2004) 629–647] we presented a method for factorizing a second-order boundary value problem into a system of uncoupled first-order initial value problems, together with a nonlinear Riccati type equation for functional operators. A weak sense was given to that system but we did not perform a direct study of those equations. This factorization utilizes either the Neumann to Dirichlet (NtD) operator or the Dirichlet to Neumann (DtN) operator, which satisfy a Riccati equation. Here we consider the framework of Hilbert–Schmidt operators, which provides tools for a direct study of this Riccati type equation. Once we have solved the system of Cauchy problems, we show that its solution solves the original second-order boundary value problem. Finally, we indicate how this techniques can be used to find suitable transparent conditions.     

Kenneth O. May, 1915–1977 : His early life to 1946

Kenneth Ownsworth May graduated from the University of California at Berkeley in 1936 with highest honors in mathematics. The following year he received his Masters degree and became a fellow of the Institute of Current World Affairs, and during the next two years he traveled to England, Europe, and Russia. On his return to the United States he became active in the Communist Party, the consequences of which would plague him for years. He joined the United States Army in 1942, serving with distinction, and after the war returned to Berkeley, where he obtained his Ph.D. in 1946. He immediately accepted an assistant professorship at Carleton College in Northfield, Minnesota, later moving to the University of Toronto.This part of May's biography focuses on the events up to his accepting a position at Carleton College. In this early phase his openness, his emphasis on good communications in the process of education, and his interest in practical procedures emerge which later set the background for his successful career as a leading historian of mathematics and the founding editor of Historia Mathematica.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.