Harold R. Parks

A Primer of Real Analytic Functions

Preface to the Second Edition * Preface to the First Edition * Elementary Properties * Multivariable Calculus of Real Analytic Functions * Classical Topics * Some Questions of Hard Analysis * Results Motivated by Partial Differential Equations * Topics in Geometry * Bibliography * Index

The Plateau Problem

Thanks to David Brewster’s 1824 translation of Legendre’s Elements de Geometrie, even children in school know that a straight line is the shortest distance between two points.

Soap-film-like minimal surfaces spanning knots

It is proved that any three-times continuously differentiable, nontrivial knot in three-dimensional euclidean space supports a surface that minimizes area among nearby surfaces but that does not touch all of the supporting knot. This provides a mathematical model of a physical phenomenon occurring in soap-films. The notion of a homotopically spanning surface is defined to determine an appropriate class of admissible surfaces, and it is shown that there is a lower bound on the area of admissible surfaces. The existence of an area minimizing admissible surface is then proved by the direct method based on earlier work of E. R. Reiffenberg.

Computing Least Area Hypersurfaces Spanning Arbitrary Boundaries

The numerical least area problem for oriented hypersurfaces seeks algorithms which approximate area-minimizing hypersurfaces spanning a given boundary in Euclidean n-dimensional space. A mathematical model and numerical implementation are presented for finding the solution to the general least area problem for oriented surfaces in Euclidean three-dimensional space. (The mathematical model is valid for hypersurfaces of arbitrary Euclidean n-dimensional spaces.) There are no a priori restrictions on either the topological complexity of the given boundary or the topological type of the surfaces considered. As an example which illustrates the power of the method, the algorithm is applied to a boundary consisting of a pair of square-shaped linked curves. The resulting numerical surface is compared with an actual physical area-minimizing spanning surface (soap film).

The volume of the unit n-Ball

Summary Two simple proofs are given for the fact that the volume of the unit ball in n-dimensional Euclidean space approaches 0 as n approaches ∞. (Some authors use the term “unit sphere” for what is here called the unit ball.) One argument involves covering the unit ball by simplices. The other argument involves covering the unit ball by rectangular solids.

The least-gradient method for computing area minimizing hypersurfaces spanning arbitrary boundaries

In this paper we give a brief presentation of the least-gradient method. The least-gradient method is used to compute an approxination to a globally area minimizing oriented hypersurface having a given boundary, without the necessity of providing any a priori topological information about the area minimizing surface. The method has been successfully implemented. The first results were obtained, and published, for the case in which the boundary curve is required to lie on the surface of a convex body. Subsequent work has dealt with the general problem in which the given boundary curve is essentially unrestricted.

Fermat’s Last Theorem

Sometime during 1637 Pierre de Fermat (1601–1665) wrote in the margin of a book the assertion that he had found a truly marvelous proof of the following result:

The Foundations of Mathematics

Mathematics has a long and distinguished history. Babylonian tablets as old as 1800 BCE show clear evidence of sophisticated mathematical thinking. More elementary mathematics for keeping track of commerce and land area dates back even further.

The P/NP Problem

Mathematicians are always interested in problems and their solutions. In this chapter we will be interested specifically in problems that can be solved using an algorithm, that is, by a step-by-step procedure.

Ricci Flow and the Poincaré Conjecture

Jules Henri Poincare (1854–1912) was one of the great geniuses of nineteenth and twentieth-century mathematics. Born into a distinguished family of academicians and public servants, he showed early talent in mathematics and science. Indeed, the entire country of France watched in awe as Poincare developed from a child prodigy to a major leader in mathematics and physics.