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In terms of radians, 45-45 is p/4-p/4 and 30-60 is p/6-p/3. The first phase group describes leptons. The second describes quarks. Together they describe both matter and antimatter structures at the local infinitesimal domain of elementary particles. The phase group p-p/2 for bosons will be described separately. In the theory of complex analysis, the complex domain includes ¥ as a complex number with the following properties given an arbitrary complex number z: (1) z/¥=0 (z≠¥), (2) z±¥=¥ (z≠¥), (3) z/0=¥ (z≠0), (4) z·¥=¥ (z≠¥), (5) ¥/z=¥ (z≠¥). However, expressions such as ¥+¥, ¥-¥, and ¥/¥ are not defined. On the other hand if ¥ is replaced by 0 then two of the three expression can be defined in the real domain R. For unit circle in the complex plane, a complex number z becomes the imaginary exponent of the base of natural logarithm: e such that z=exp(iq) where q is the phase factor in radian measures. The n power of z is exp(inq). If q=p/4 then for n=4, z=-1 and for n=8, z=1. On the other hand, if q=p/3 then for n=3, z=-1 and for n=6, the 6th power of z is unity. However, if q=p/6 and for n=6 the 6th power of z becomes -1 and for n=12 the 12th power of z is 1. These show that the real basis for many power of z is (1,-1) and (-1,1) whose column-row matrix multiplications are Hadamard matrices.
Time independence: [∂E(g)]²=[∂F(a)×∂r(a)]·[∂F(b)×∂r(b)] and Mass independence: ¶a(t)·¶r(t)=c²
Re: 45-45 30-60 phase groups -
02-16-2008, 04:58 PM
Linear Mode
