17 resultados para Lie
Resumo:
Let F = Ǫ(ζ + ζ –1) be the maximal real subfield of the cyclotomic field Ǫ(ζ) where ζ is a primitive qth root of unity and q is an odd rational prime. The numbers u1=-1, uk=(ζk-ζ-k)/(ζ-ζ-1), k=2,…,p, p=(q-1)/2, are units in F and are called the cyclotomic units. In this thesis the sign distribution of the conjugates in F of the cyclotomic units is studied.
Let G(F/Ǫ) denote the Galoi's group of F over Ǫ, and let V denote the units in F. For each σϵ G(F/Ǫ) and μϵV define a mapping sgnσ: V→GF(2) by sgnσ(μ) = 1 iff σ(μ) ˂ 0 and sgnσ(μ) = 0 iff σ(μ) ˃ 0. Let {σ1, ... , σp} be a fixed ordering of G(F/Ǫ). The matrix Mq=(sgnσj(vi) ) , i, j = 1, ... , p is called the matrix of cyclotomic signatures. The rank of this matrix determines the sign distribution of the conjugates of the cyclotomic units. The matrix of cyclotomic signatures is associated with an ideal in the ring GF(2) [x] / (xp+ 1) in such a way that the rank of the matrix equals the GF(2)-dimension of the ideal. It is shown that if p = (q-1)/ 2 is a prime and if 2 is a primitive root mod p, then Mq is non-singular. Also let p be arbitrary, let ℓ be a primitive root mod q and let L = {i | 0 ≤ i ≤ p-1, the least positive residue of defined by ℓi mod q is greater than p}. Let Hq(x) ϵ GF(2)[x] be defined by Hq(x) = g. c. d. ((Σ xi/I ϵ L) (x+1) + 1, xp + 1). It is shown that the rank of Mq equals the difference p - degree Hq(x).
Further results are obtained by using the reciprocity theorem of class field theory. The reciprocity maps for a certain abelian extension of F and for the infinite primes in F are associated with the signs of conjugates. The product formula for the reciprocity maps is used to associate the signs of conjugates with the reciprocity maps at the primes which lie above (2). The case when (2) is a prime in F is studied in detail. Let T denote the group of totally positive units in F. Let U be the group generated by the cyclotomic units. Assume that (2) is a prime in F and that p is odd. Let F(2) denote the completion of F at (2) and let V(2) denote the units in F(2). The following statements are shown to be equivalent. 1) The matrix of cyclotomic signatures is non-singular. 2) U∩T = U2. 3) U∩F2(2) = U2. 4) V(2)/ V(2)2 = ˂v1 V(2)2˃ ʘ…ʘ˂vp V(2)2˃ ʘ ˂3V(2)2˃.
The rank of Mq was computed for 5≤q≤929 and the results appear in tables. On the basis of these results and additional calculations the following conjecture is made: If q and p = (q -1)/ 2 are both primes, then Mq is non-singular.
Resumo:
Part I
Present experimental data on nucleon-antinucleon scattering allow a study of the possibility of a phase transition in a nucleon-antinucleon gas at high temperature. Estimates can be made of the general behavior of the elastic phase shifts without resorting to theoretical derivation. A phase transition which separates nucleons from antinucleons is found at about 280 MeV in the approximation of the second virial coefficient to the free energy of the gas.
Part II
The parton model is used to derive scaling laws for the hadrons observed in deep inelastic electron-nucleon scattering which lie in the fragmentation region of the virtual photon. Scaling relations are obtained in the Bjorken and Regge regions. It is proposed that the distribution functions become independent of both q2 and ν where the Bjorken and Regge regions overlap. The quark density functions are discussed in the limit x→1 for the nucleon octet and the pseudoscalar mesons. Under certain plausible assumptions it is found that only one or two quarks of the six types of quarks and antiquarks have an appreciable density function in the limit x→1. This has implications for the quark fragmentation functions near the large momentum boundary of their fragmentation region. These results are used to propose a method of measuring the proton and neutron quark density functions for all x by making measurements on inclusively produced hadrons in electroproduction only. Implications are also discussed for the hadrons produced in electron-positron annihilation.