The evolution of many allele frequencies can be understood through the use of difference equations. Let n represent (equallyspaced) discrete points in time or generations, and let A(n) represent the size of a population at time n. A first-order...
The last fifteen years or so have seen enormous research activity in the development of parabolic wave equations (PWEs). These were first discovered by a pair of Soviet scientists shortly after World War 11, during their investigations of different...
Geometric modeling is the representation and manipulation of geometic data in a computer. It is the mathematics that is central to Computer Aided Design and Computer Aided Manufacturing (CAD/CAM). Its central objective in this context is to balance the...
Current trigonometry texts begin with trigonometric ratios, continue through the solution of right and other triangles, and go on to applications to the familiar problems on the height of a tree or the distance across a lake. The presentation leads one...
Genetic algorithms (GAS) are iterative processes for finding good solutions to optimization problems. GAS operate on a population of individuals which represents a set of potential solutions to a given problem. We usually think of such a population at...
A recent publication, Toward a Lean and Lively Calculus [Douglas 19871, contains reports and articles on the teaching of calculus. In his review of the book, R.W. Hamming [I9881 maintains that the calculus texts currently in use are very much alike,...
The problem of the title, that of flipping a coin by telephone, was proposed and solved by Manuel Blum [19821. One cannot do better in sketching the problem than by quoting the first paragraph of his abstract: Alice and Bob want to flip a coin by...
Many problems in engineering, planning, and manufacturing can be modeled as that of minimizing or maximizing a cost function over a finite set of discrete variables. This class of combinatorial optimization problems has received much attention over the...
The lifeguard problem is a reincarnation of one of the first problems to which calculus was applied; and, as with many old chestnuts, it continues to draw attention. Whitley solved this problem and suggested an approximation strategy for his lifeguard...
Most physical phenomena, whether in the area of fluid dynamics, electricity, magnetism, heat flow and other domains can be described by PDEs. MG was first successfully applied to PDEs that have outcomes that change with respect to space but not time,...
The power and usefulness of neural networks is demonstrated by the success of the TNA (Thermal Neutron Analysis) project, developed by SAIC (Science Applications International Corp.). TNA is a neural-network-based machine used at airports that detects...
The subject of wavelets has been around for less than ten years. Yet it seems to be popping up everywhere. The major reason for this is its usefulness in applications. Wavelets reduce the data requirements in fields such as image analysis and signal...
There are many ways of describing the discovery of error-correcting codes, and I will place the emphasis on the mathematics involved. Their discovery is a good illustration of Pasteur's rule: Luck favors the prepared mind. In 1946-47, I was using a...
When you receive a letter that apparently was sent by your acquaintance John Smith, how do you know that he is indeed the sender? The answer is obvious: You verify that the signature at the bottom of the letter looks like his, has not been changed from...
Newton and Leibniz were independent originators of the calculus in the last half of the seventeenth century, and both thought in terms of infinitesimals, at least initially. Infinitesimals were somewhat problematic, and their use was severely...
Combinatorial game theory was axiomatized by Conway [I9761 and was developed by Berlekamp, Conway, and Guy [19821. It has provided insight into many mathematical games (including games on graphs, Nim variants, and Sprouts) as well as some more popular...
Whether it is downloading files from the Internet, having conversations between cell phones, or sending information from a laptop to a printer, we often want to transmit data in situations where we need to worry about interference from other signals...
The 2008 Democratic Primary season was one of the most riveting in recent history in part because of the lengthy contest between New York Senator Hillary Clinton and Illinois Senator Barack Obama. Of the eight candidates who mounted national campaigns,...
From the earliest days of computing, the development of efficient methods of calculation has played a key role in expanding the diversity of problems computers can solve. Computational complexity studies the measurement of resources, especially time...
A good mystery often begins with its main characters innocently pursuing a commonplace activity. In this instance, one of us (Goldman) was working on a study of trends in fertility and mortality in the Republic of Korea. To cope with some of the gaps...