From: Osher Doctorow [EMAIL PROTECTED], Mon. Sept. 23, 2002 12:32PM I refer readers to http://www.superstringtheory.com/forum, especially to the String - M Theory - Duality subforum of their Forum section (membership is free, and archives are open to members, and many of my postings are in the archives), during the last few days, in which I have provided literature references and sites from very recent research mostly that puts Analysis via determinants and/or negative exponentials at the forefront of science - not merely the interesting Fredholm type determinants and the Slater determinants, but determinant maximization in general (with constraints). Fields crossed by these include quantum theory, general relativity, information theory, communications theory, experimental design, system identification, statistics as a whole, geometry, computer programming, entropy, experimental design, algorithms including path-finding algorithms for convex optimization, etc.
Let me very briefly recapitulate why determinants are Analysis-based rather than Algebra-based. The expression 1 + y - x, which can be generalized to c + y - x for arbitrary real constant c (or even to non-real expressions, but that is another story) or simply written y - x with incorporation of c into y or x, is continuous and CRITICAL to outgrowths of Analysis including probability-statistics (for Rare Event scenarios), fuzzy multivalued logics (see below for those who believe that logic is algebraic), proximity functions, geometry-topology based on proximity functions. Determinants generalize y - x to a finite alternating series. Alternating series in general generalize determinants. The same site (earlier postings) explains why fuzzy multivalued logics and logics in general are Analysis-based rather than Algebra-based, although many mathematical and non-mathematical logicians unfamiliar with Analysis have believed otherwise. Osher Doctorow