2013 Fiscal Year Final Research Report
Applications of the integration algorithm for D-modules
| Project/Area Number |
24740064
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| Research Category |
Grant-in-Aid for Young Scientists (B)
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| Allocation Type | Multi-year Fund |
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| Research Field |
General mathematics (including Probability theory/Statistical mathematics)
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| Research Institution | Kobe University |
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Principal Investigator |
NAKAYAMA Hiromasa 神戸大学, 理学(系)研究科(研究院), 研究員 (00595952)
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| Project Period (FY) |
2012-04-01 – 2014-03-31
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| Keywords | グレブナー基底 / 計算代数 / 数式処理 |
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| Research Abstract |
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. These enable us to solve some MLE problems up to dimension d=7 with a specified accuracy. The Fisher-Bingham system is a system of linear partial differential equations satisfied by the Fisher-Bingham integral for the n-dimensional sphere. We show that the holonomic rank of the system is equal to 2n+2.
We derive Groebner bases for Lauricella's hypergeometric differential equations with respect to a monomial order. By using these Groebner bases, we determine characteristic varieties and the singular loci of these differential equations.
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