On the limiting behavior of Burger's equation

The limiting behavior as the viscosity goes to zero of the solution of the first boundary value problem for Burger's equation is considered. The method consists in identifying the solution of Burger's equation with the optimal control of an appropriate stochastic control problem.     

An algorithm for the quadratic assignment problem using Bender's decomposition

A well known method used for solving quadratic assignment problems proceeds by the construction of an equivalent much larger linear assignment problem with many side constraints. The disadvantage of this method lies in the weakness of the bounds obtained by solving the linear problem. An alternate linearization has been suggested using a general method of Glover. In this paper the mixed integer program obtained by Glover's method is discussed and a solution using Bender's decomposition is proposed.     

A generalization of Cooke's integral inversion formula with application to remote-sensing theory

The remote sensing of environmental particulate pollutants, particularly their size distribution, frequently leads to the solution of first-kind Fredholm integral equations. The corresponding physical kernel tends to smooth the behavior of the required function for all values of the dependent variable. Thus, the problem is ill posed and needs regularization by the introduction of constraints on the solution (closure condition). However, under physically realistic conditions, the original problem can be transformed so that it presents a unique and stable solution. One such condition is the so-called anomalous-diffraction approximation, for which we provide two alternate inversion formulae. We derive a new inversion formula (see our theorem) which generalizes that of Cooke and which also provides, as a special case, one of Titchmarsh's formulae. We propose a unifying viewpoint for a number of known integral inversion formulae, including those of Fox (his first theorem), Hardy, Hankel, Titchmarsh, Cooke, and our own, along with the mutual interrelationships that exist between them (Fig. 1 and Table 1). One solution to the particulate sounding problem is then obtained from a direct application of our formula [Eq. (25)]. An alternate solution is likewise obtained by applying Titchmarsh's formula (II) [Eq. (27)]. Both solutions can be independently recovered from Fox's first theorem, although under somewhat more restrictive conditions. They are shown to be identical, and to provide the unique solution to the remote sensing problem considered.     

The Laplace and poisson equations in Schwarzschild's space-time

The method of separation of variables is used to solve the Laplace equation in Schwarzschild's space-time. The solutions are given explicitly in series form and in terms of Legendre functions. Green's function is determined and remarks are made on the solution of Poisson's equation for a point source.     

Application of operations research in Israel's water sector

Israel's water sector has moved from a period of development, which ended in the mid-1960's, to an era of scarcity. Over 95% of the natural water potential is already being utilized, and there is severe competition for this scarce resource between economic sectors and geographic regions. Management of development, design and operation of the water systems is therefore an acute problem, with implications ranging from national policy to efficiency in daily operation. Operations research methodologies have been developed and applied quite extensively over the last 15 years in Israel's water sector, dealing with the full range of its problems. The paper is a survey of these applications, aimed at providing a realistic assessment of their value, from which water resources systems analysts in other countries may derive some guidelines for their own work.     

American attempts to solve Newton's pasturage problem

Frederick Emerson's North American Arithmetic contained a “pasturage problem” which baffled his compatriots. Actually, as the Americans discovered forty-two years later, this problem was taken from Isaac Newton's Arithmetica Universalis. The history of this problem illuminates the tradition of standard artificial exercises, the isolation of American mathematics, a chain of mathematical and historical plagiarisms, and changing patterns of arithmetical reasoning.     

A note on Sheppard's corrections for grouping and maximum likelihood estimation

Sheppard's corrections for grouping can, in the case of an underlying normal distribution, be interpreted as a first step to the solution of the maximum likelihood equations which incorporate the grouping problem. This result of Lindley (for the univariate) and Haitovsky (for the bivariate) is generalized to the multivariate normal distribution, making use of recent results in matrix algebra. Also, formulae concerning the efficiency lost in grouping are generalized to the multivariate case.     

A remark on Goldbach's problem

In the present note we show that the conjectured asymptotic formulas for Goldbach's problem and the allied twin-primes conjecture are available conditionally.     

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).     

Comments on practical implementation of Gomory's fractional algorithm

Gomory's fractional algorithm is widely used for solving integer linear programming problems. It is strongly influenced by round off errors and consequently has the disadvantage that in many cases the optimal solution is not reached. The present note describes a simple modification of this algorithm. The essential feature of this modification consists in conserving the integral structure of the problem. Thus the influence of round off errors is avoided. The power of our method is illustrated by a numerical example.     

A variant of Nitsche's method

We present a method that aims to reconcile Nitsche's method with the traditional finite element method ('weak' versus 'strong implementation' of essential boundary conditions). We retain the original idea of a variational formulation based on an extended energy, but replace the original boundary terms by domain terms involving weak derivatives. The solution of the proposed method coincides, for the Poisson problem, with the one of the traditional method, which in particular shows monotonicity under the standard angle condition for the Courant element. For more general second-order problems, it allows for the weighting of boundary terms inherent to Nitsche's method. This is of particular interest for singularly perturbed problems.     

Roth's removal rule revisited

The theory of companion matrices is used to give explicit representations for the matrices needed in Roth's removal rule. These are then used to give simple proofs for the cyclic decomposition theorem, as well as for Roth's similarity theorem for matrices over a field.     

Matrices of 0's and 1's with total support

We determine when there exists a matrix of 0's and 1's with total support (a pattern of a doubly stochastic matrix) having a specified row sum vector and column sum vector.     

The Ladies' Diary or Woman's Almanack, 1704–1841

The existence of the Ladies' Diary or the Woman's Almanack, an 18th century English magazine devoted largely to problems and puzzles in mathematics, indicates that stereotypes about the inability of women to understand and enjoy mathematics were less strongly believed in the 18th century than they are today. The beginning of the Ladies' Diary coincides with the popularization of mathematics and the growth of mathematical literacy. However, as mathematical literacy spread in response to developing technology's requirements for more mathematically sophisticated workers, women, not part of this need, were left behind. This effect is reflected in the decline in the number of women contributors over the life of the publication.     

Explicit solutions of Maxwell's equations on a space of constant curvature

An explicit solution is given to the Cauchy problem for the source-free Maxwell's equations in a vacuum on a space-time of the form $\text{R}$¹ X M₃, where M₃ is a 3-manifold of constant curvature. This solution satisfies Huyghens' Principle, that all electromagnetic radiation propagates at exactly the speed of light. The solution is obtained by harmonic analysis on M₃, and in the process a generating class of plane wave solutions is found. These solutions approximate the flat-space plane wave solutions in a neighbourhood of a point, but their global properties are somewhat different. The solutions obtained are easily transplanted to the Robertson-Walker models of General Relativity by re-scaling the time variable.     

The mathematical technique in Fermat's deduction of the law of refraction

This article attempts to explain Fermat's not quite obvious calculations connected with his deduction of the law of refraction in Analysis ad refractiones (1662), and to describe the development which led to these calculations. In 1657 Fermat tried to deduce a law of refraction based on the principle that light follows the quickest path between two given points. He did not succeed because he found that the calculations were too long and tedious. The calculations are indeed complicated, but if Fermat, in 1657, had been willing to accept Descartes' law of refraction he would probably also have seen that it solved his problem. However, Fermat was of the opinion that Descartes' law was wrong and, therefore, he did not expect that solution. Only in 1662, when he succeeded in reducing the calculations substantially, did he realize that they led to the sine law of Descartes.     

On Green's function in higher order stochastic differential equations

In Adomian's solution of linear or nonlinear, deterministic or stochastic, differential equations, the decomposition of the linear part into L+$\text{R}$ can be made so as to simplify Green's function and, consequently, the resulting computations.     

Wagner's theorem for Torus graphs

Wagner's theorem (any two maximal plane graphs having p vertices are equivalent under diagonal transformations) is extended to maximal torus graphs, graphs embedded in the torus with a maximal set of edges present. Thus any maximal torus graph having p vertices may be diagonally transformed into any other maximal torus graph having p vertices. As with Wagner's theorem, a normal form representing an intermediate stage in the above transformation is displayed. This result, along with Wagner's theorem, may make possible constructive characterizations of planar and toroidal graphs, through a wholly combinatorial definition of diagonal transformation.