By Otto Mutzbauer
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Subsequent work may suggest a very good reason, often about the symmetries of the problem that are embodied in these mathematical objects. Or, what was once seen as an idiosyncratic but effective technique, turns out to be generic for a class of problems of which this 14 INTRODUCTION one is an exemplar (technically, here about nonlinear integrable systems and their associated linear differential equations). ” The applications of classical analysis may still demand lengthy and intricate calculations, endemic to the enterprise.
Dyson and Lenard’s work shows an enormous amount of physical understanding, whatever the size of the proportionality constant. This is evident in the organization of the proof. Moreover, the aesthetics are more complicated than terms such as clean or messy might suggest. 31 ANALOGY AND SYZYGY . . members of any group of functions [U, V, W, . ], more than two in number, whose nullity is implied in the relation of double contact [namely, aU+bV+cW=0, a, b, c integers] , . . must be in syzygy. Thus PQ, PQR, and QR, must form a syzygy.
So Dirac (1927) might employ a first degree partial differential equation to describe relativistic electrons, and then say that the electrons are “four-component spinors”—spin up and down, positive and negative electrons. We might then justify our model by saying that the world is just this distinctive way, or at least it is very closely approximated by just this distinctive way. THE CENTRAL LIMIT THEOREM The Gaussian distribution or the normal distribution or the bell-shaped curve is among other things the asymptotic limit or limiting shape referred to in the Central Limit Theorem (“Central” here means important, playing a central role): sums of nice enough random variables end up (asymptotically) looking like a suitable Gaussian.