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Kenta Nishiyama

Publications and source records attributed to Kenta Nishiyama.

8 recordsLinked to original sources
arXiv2014

Pfaffian Systems of A-Hypergeometric Equations I: Bases of Twisted Cohomology Groups

This is the third revision. We study bases of Pfaffian systems for $A$-hypergeometric system. Gröbner 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.

math.CA↗
arXiv2013

Holonomic Gradient Descent for the Fisher-Bingham Distribution on the $d$-dimensional Sphere

We propose an accelerated version of the holonomic gradient descent and apply it to calculating the maximum likelihood estimate (MLE) of the Fisher-Bingham distribution on a $d$-dimensional sphere. We derive a Pfaffian system (an integrable connection) and a series expansion associated with the normalizing constant with an error estimation. These enable us to solve some MLE problems up to dimension $d=7$ with a specified accuracy.

math.ST↗
arXiv2013

Many toric ideals generated by quadratic binomials possess no quadratic Gröbner bases

Let $G$ be a finite connected simple graph and $I_{G}$ the toric ideal of the edge ring $K[G]$ of $G$. In the present paper, we study finite graphs $G$ with the property that $I_{G}$ is generated by quadratic binomials and $I_{G}$ possesses no quadratic Gröbner basis. First, we give a nontrivial infinite series of finite graphs with the above property. Second, we implement a combinatorial characterization for $I_{G}$ to be generated by quadratic binomials and, by means of the computer search, we classify the finite graphs $G$ with the above property, up to 8 vertices.

math.AC↗
arXiv2013

The Holonomic Rank of the Fisher-Bingham System of Differential Equations

The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the $n$-dimensional sphere $S^n$. The system is given in [Nakayama et al. (2011), Theorem 2] and it is shown that it is a holonomic system [Koyama]. We show that the holonomic rank of the system is equal to $2n+2$.

math.CA↗
arXiv2011

Incomplete A-Hypergeometric Systems

The incomplete beta function is an important special function in statistics. In modern theory of hypergeometric functions, we regard hypergeometric functions as pairings of twisted cycles and twisted cocycles. However, the incomplete beta function cannot be understood in this scheme; in other words, the domain of the integration is not cycle (incomplete). We will generalize the theory of A-hypergeometric systems for incomplete functions. We give a general study as well as a detailed study on an incomplete Gauss hypergeometric function.

math.CA↗