III. Mathematics

Mathematics probably ranks as the Greeks’ greatest achievement, in the eyes of many modern scientists. And amongst the general public it is fair to say that some Greek mathematicians are better known than any other figures from antiquity, with the possible exception of Alexander the Great. For example, Pythagoras, Euclid, and Archimedes are household names two thousand years after they lived and wrote their mathematics. The Greek alphabet, or some of it anyway, is widely known today because modern mathematics uses it by preference for symbols. For example, α, β, and θ for angles, the amazing π, Σ as the summation sign, and the χ-square test. The Greeks gave us ‘square’ and ‘cube’ numbers, e.g. 22 and 33. ‘Squaring the circle’ is modern English idiom for an impossible task – the original task, set by the Greeks, being the mathematical problem of constructing a square with the same area as a given circle. Aristophanes mentions this problem. Famously, Plato forbade the geometrically-challenged to enter his Academy. He also found mathematically interesting numbers for his ideal Republic. Greek mathematical texts are rich in mathematical concepts, methods, and results. Like Greek plays, they contain timeless insights and truths. Also like literary works they require analysis and interpretation.

The arithmetic of certain semigroups of positive operators

Some time ago, S. Bochner gave an interesting analysis of certain positive operators which are associated with the ultraspherical polynomials (1,2). Let {Pn(x)} denote these polynomials, which are orthogonal on [ − 1, 1 ] with respect to the measureand which are normalized by settigng Pn(1) = 1. (The fixed parameter γ will not be explicitly shown.) A sequence t = {tn} of real numbers is said to be ‘positive definite’, which we will indicate by writing , provided thatHere the coefficients an are real, and the prime on the summation sign means that only a finite number of terms are different from 0. This condition can be rephrased by considering the set of linear operators on the space of real polynomials which have diagonal matrices with respect to the basis {Pn(x)}, and noting that