This article will introduce the general formula of multiple Patterson function(MPF), and deduce therefrom phase structure invariants in reciprocal space that Hauptman and Karle introduced in a different way. By expand...This article will introduce the general formula of multiple Patterson function(MPF), and deduce therefrom phase structure invariants in reciprocal space that Hauptman and Karle introduced in a different way. By expanding the concept of the structure invariant, it may be found that a connection exists between the structure invariants in positive and reciprocal spaces. The phase question of MPF can be merged into the estimation question of phase structure invariants in the probability direct method.This theme will show the general group of Σ relationship without extra magnitudes. It might be one of the methods used in calculating MPF, and it primarily discusses the property of MPF.展开更多
The main purpose of this paper is to show that the Poincaré q-polynomials admit a representation in terms of the symmetric functions and the Patterson-Selberg (or Ruelle-type) spectral functions. We have shown th...The main purpose of this paper is to show that the Poincaré q-polynomials admit a representation in terms of the symmetric functions and the Patterson-Selberg (or Ruelle-type) spectral functions. We have shown that the q-series elliptic genera can be expressed in terms of q-analogs of the classical special functions, specially the equivalence between the spectral Patterson-Selberg and the Ruelle functions. The main result of this manuscript is to show that this representation can be used in theoretical physics and we analyze them in terms of the Patterson-Selberg spectral function R (s).展开更多
文摘This article will introduce the general formula of multiple Patterson function(MPF), and deduce therefrom phase structure invariants in reciprocal space that Hauptman and Karle introduced in a different way. By expanding the concept of the structure invariant, it may be found that a connection exists between the structure invariants in positive and reciprocal spaces. The phase question of MPF can be merged into the estimation question of phase structure invariants in the probability direct method.This theme will show the general group of Σ relationship without extra magnitudes. It might be one of the methods used in calculating MPF, and it primarily discusses the property of MPF.
文摘The main purpose of this paper is to show that the Poincaré q-polynomials admit a representation in terms of the symmetric functions and the Patterson-Selberg (or Ruelle-type) spectral functions. We have shown that the q-series elliptic genera can be expressed in terms of q-analogs of the classical special functions, specially the equivalence between the spectral Patterson-Selberg and the Ruelle functions. The main result of this manuscript is to show that this representation can be used in theoretical physics and we analyze them in terms of the Patterson-Selberg spectral function R (s).
