Let S2 be the 2-dimensional unit sphere and let Jα denote the nonlinear functional on the Sobolev space H1,2(S2) defined by Jα(u) = α 4 ∫

In (4) Young investigates the representation in terms of his exact seminormal units m^ of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive ...

[S1,Q15=MAC15(S1), P13=MAC13(S1,Q15)] [S2,Q25=MAC25(S2),P23=MAC23(S2,Q25)] [S4,Q46=MAC46(S4),P45=MAC45(S4,Q46)] Q36=MAC36(S1|S2|S3) P35=MAC35(S3, S1|S2, Q15, Q25, Q36) [S3, (S1|S2), Q15, Q25, Q36, P35] Q5B= MAC5B(S1|S2|S3|S4|S5) P56=MAC56(S5, (S1|S2|S3|S4),Q36,Q46,Q5B) [S5, (S1|S2|S3|S4),Q36, Q46, Q5B, P56] P6B=MAC6B(S6, ((S1|S2|S3|S4 |S5)| S7|S8),Q5B,Q7B,Q8B) [S6, ((S1|S2|S3|S4 |S5)| S7|S8),Q5...

For simplicity, we adopt the following rules: n denotes an element of N, X, X1 denote sets, r, p denote real numbers, s, x0, x1, x2 denote real numbers, S, T denote real normed spaces, f , f1, f2 denote partial functions from R to the carrier of S, s1 denotes a sequence of real numbers, and Y denotes a subset of R. The following propositions are true: (1) Let s2 be a sequence of real numbers an...