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Register a new userPfaffian Systems of A-Hypergeometric Systems II --- Holonomic Gradient Method
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Added by
Nobuki Takayama
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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We give two efficient methods to derive Pfaffian systems for A-hypergeometric systems for the application to the holonomic gradient method for statistics. We utilize the Hilbert driven Buchberger algorithm and Macaulay type matrices in the two methods.
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In this paper, we present an algorithm for computing a fundamental matrix of formal solutions of completely integrable Pfaffian systems with normal crossings in several variables. This algorithm is a generalization of a method developed for the bivariate case based on a combination of several reduction techniques and is implemented in the computer algebra system Maple.
This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Grobner deformations give bases. These bases also give those for twisted cohomology groups. For hypergeometric system associated to a class of order polytopes, these bases have a combinatorial description. The size of the bases associated to a subclass of the order polytopes have the growth rate of the polynomial order. Bases associated to two chain posets and bouquets are studied.
We give a new algorithm to find local maximum and minimum of a holonomic function and apply it for the Fisher-Bingham integral on the sphere $S^n$, which is used in the directional statistics. The method utilizes the theory and algorithms of holonomic systems.
We will introduce a modified system of A-hypergeometric system (GKZ system) by applying a change of variables for Groebner deformations and study its Groebner basis and the indicial polynomials along the exceptional hypersurface.
Coupled atom-cavity arrays, such as those described by the Jaynes-Cummings Hubbard model, have the potential to emulate a wide range of condensed matter phenomena. In particular, the strongly correlated states of the fractional quantum Hall effect can be realised. At some filling fractions, the fraction quantum Hall effect has been shown to possess ground states with non-abelian excitations. The most well studied of these states is the Pfaffian state of Moore and Read, which is the groundstate of a Hall Liquid with a 3-body interaction. In this paper we show how an effective 3-body interaction can be generated within the Cavity QED framework, and that a Pfaffian-like groundstate of these systems exists.
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