research papers
Two methods are discussed in detail. In the first method the triplet relationship is treated using the first neighborhood, and the quartet relationship using its second neighborhood. For the triplet relationship it is found that the reliability φh +φk - φh+k ≃ 0 is enhanced when Rh ≃ Rk ≃ Rh+k and large. This conclusion is drawn from formula (16) giving the conditional probability of φh + φk - φh+k using an asymptotic development up to and including terms of order N-1/2. For the quartet relationship it is found that the reliability that φh + φk + φl - φh+k+l ≃ π given Rh+k ≃ Rh+l ≃ Rk+l ≃ 0 is diminished when Rh ≃ Rk ≃ Rl ≃ Rh+k+l and large. This conclusion is drawn from formula (19) using similar calculations for the triplet relationship. A heuristic theoretical discussion of this last result trying to explain this difference with the usual theories is given. In the second method the triplet relationship is treated using its first neighborhood. These calculations have been done using a 'normal' asymptotic development up to and including terms of order N-1/2. As a result a formula (28) is obtained that is (at least theoretically) able to predict negative cosine values. A third method that is proposed where one uses the ideas of Patterson superposition will be discussed in detail in a forthcoming paper.