By Thomas M. Thompson
1. The foundation of error-correcting codes; an creation to coding; the paintings of Hamming; the Hamming-Holbrook patent; the Hamming codes are linear; the paintings of Golay; the concern controversy; 2. From coding to sphere packing; an advent to sphere packing; the Leech connection; the foundation of Leech's first packing in E24; the matrix for Leech's first packing; the Leech lattice; three. From sphere packing to new basic teams; is there an engaging crew in Leech's lattice?; the difficult promote of an easy workforce; twelve hours on Saturday, six on Wednesday; the constitution of zero; new basic teams; Appendix 1. Densest recognized sphere packings; Appendix 2. extra homes of the (12,24) Golay code and the similar Steiner procedure s (5,8,24); Appendix three. A calculation of the variety of spheres with facilities in LAMBDA2; Appendix four. The Mathieu team M24 and the order of M22; Appendix five. The evidence of lemma 3.3; Appendix 6. The sporadic uncomplicated teams; Bibliography; Index
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44. 58 [2 . 46) The decay p--, K°ei The decay p --* K°ei r(p---_ • . 003GeV z) ( 1301_eV f. )2 ( 101rGeV M_ )2 (1TeV" X X I [UQ(VTUL)'' . 70 12 . 48), and those are given, via UQ and U L, by the flavor structure of the underlying SUSY GUT. 37) h" = 2 hQ = symmetric, If we go to the basis of mass eigenstates h d = h LT = heT. 49) for up quarks (h" -- diag(A,,,Ac, At)) and charged leptons (h e = diag(Ae, A_,,A_)) , hQ also becomes diagonal with the same values as h_, up to two phases (phases cannot be absorbed by redefinition of Q fields: their phases are already fixed by requiring the diagonal h_ to be real), while hd = h LT = h • V_?
The smallnessand hierarchy of fermionmassesand mixingscan be restated in termsofapproximatechiral and flavor symmetrieswhichacton eachfermion type(-). 13) become approximate [s_sz31, etc. Hence these approximate sufficient to guarantee alone one finds that the equalities, chiral and flavor symmetries powerful means, for example by some new exact symmetry. 2). These results follow only if the 11, 13 and 31 entries of the Yukawa matrices are constrained . thus [s131 ans_tze, family symmetries by some more In many specific force these to vanish(,,,).
41) with winos produces which contribute the effective to nucleon decays: _ -- 1 a q n Mc2_r[U;kU_, - U,QU_](u,_)(uke,)[f(_,_,,) + f(fij,d_)] L=dd_ _ - 1 a rUq r rL M. 38), we generate five are the triangle diagram running through the triangle. 42) matrix (at the weak scale) and of left handed up and down quarks, respectively. factors, which are functions of scalar masses For simplicity we assume all the scalars to be of the same mass, and we set from now on all the f's equal. _, , ,.. 40).
From error-correcting codes through sphere packings to simple groups by Thomas M. Thompson