Chromosome Odds and Ends

Odds and Ends

We represent G2(n) ∼= SO(n)/O(2) × SO(n − 2). Then we may regard the dual object G2(n)̂ ∼= {(m1,m2);m1,m2 ≥ 0 and even}, as the highest weights (mi) of irreducible unitary representations of SO(n) which occur in G2(n). They satisfy mi = 0 for i ≥ 2 and m1,m2 are even. Given K on G2(n), we may then form the Fourier series of K. General theorems [8, 44.47] assure us of pointwise convergence except...

Dynamic Programming: Odds & Ends

Max-Cost Paths in DAGS Let G = (V,E) be a weighted directed graph. Dijkstra’s algorithm constructs a shortest path tree rooted at (any vertex) s in O(|E| log |V |) time (or O(|E|+ |V | log |V |) with more sophisticated data structures) as long as the algorithm does not contain any negative cycles. Surprisingly, the corresponding max-cost problem, i.e., finding a maximum cost simple path tree ro...

Dynamic Odds & Ends

Modal logic becomes action logic by adding programs as in propositional dynamic logic or the μ– calculus. Modal languages can be seen as decidable fragments of first-order logic that admit a natural bisimulation, and hence enjoy a good model theory. Recently, much stronger 'guarded fragments' of firstorder logic have been identified that enjoy the same pleasant features. The latter can serve as...

Bayesian Belief Networks: Odds and Ends

Predictive Medicine: The Story of Odds and Ends

Introduction Some years ago in a stimulating, and at times irritating, book entitled "Apes, Men and Morons," anthropologist Earnest A. Hooten made the comment that it is a very short-visioned medical science which works backward from the autopsy table, rather than forward from the cradle. He sees no bright future for scientific achievement in the healing arts by limiting its responsibility to r...