距離空間内の勾配流とその応用

度量空间中的梯度流及其应用

• 批准号: 14F04320
• 负责人: 太田 慎一
• 金额: $ 0.96万
• 依托单位: Kyoto University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for JSPS Fellows
• 财政年份: 2014
• 资助国家: 日本
• 起止时间: 2014-04-25 至 2016-03-31
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-14F04320/
• 关键词: 凸関数; Loewner理論; 多変数関数; 作用素論; リーマン幾何; 曲率; 勾配流

In this academic year the researcher continued the work on extending Loewner’s theorem to several variables. The first result is an new LMI characterization of operator concave and monotone functions. This characterizes the functions as unique extremal solutions of linear matrix inequalities over some auxiliary Hilbert space. This result was presented at the ’Recent developments in operator algebras’ workshop in RIMS.Later using this LMI representation an exact solution formula was established which seems to be the key to obtain the analytic continuation part of Loewner’s theorem. This part is still being carried out at the moment. As a by-product an infinite dimensional version of the theory of matrix convex sets and non-commutative Hahn-Banach theorems were established as key tools.

在本学年中，研究人员继续将 Loewner 定理扩展到多个变量，第一个结果是算子凹函数和单调函数的新 LMI 表征，这将函数表征为某些辅助希尔伯特空间上线性矩阵不等式的唯一极值解。这个结果在 RIMS 的“算子代数的最新发展”研讨会上发表。后来使用这个 LMI 表示建立了精确的解公式，这似乎是获得洛纳定理的解析延拓部分目前仍在进行中，作为副产品，建立了矩阵凸集理论的无限维版本和非交换哈恩-巴纳赫定理作为关键工具。

项目成果

Discrete-time gradient flows and law of large numbers in Alexandrov spaces

• DOI: 10.1007/s00526-015-0837-y
• 发表时间: 2014-02
• 期刊: Calculus of Variations and Partial Differential Equations
• 影响因子: 2.1
• 作者: Shin-ichi Ohta;Miklós Pálfia

Gradient flows in K-convex and CAT(1)-spaces

K 凸和 CAT(1) 空间中的梯度流

• 发表时间: 2015
• 作者: Miklos Palfia;Denes Petz;Fumiharu Kobayashi and Yoshiko Kanada-En'yo;北澤直宏;Miklos Palfia
• 通讯作者: Miklos Palfia

Construction, characterization and approximation of operator means

算子平均值的构造、表征和近似

• 发表时间: 2015
• 作者: Miklos Palfia;Denes Petz;Fumiharu Kobayashi and Yoshiko Kanada-En'yo;北澤直宏;Miklos Palfia;Fumiharu Kobayashi and Yoshiko Kanada-En'yo;北澤直宏;Miklos Palfia
• 通讯作者: Miklos Palfia

Gradient flows and law of large numbers in Alexandrov spaces

Alexandrov 空间中的梯度流和大数定律

• 发表时间: 2014
• 作者: 武内房司・宮沢千尋（編集）;高津茂・北澤直宏（編集協力）;Miklos Palfia;Miklos Palfia;佐治 史;Miklos Palfia;Fumi SAJI;Miklos Palfia
• 通讯作者: Miklos Palfia

Operator means of probability measures and generalized Karcher equations

概率测量的算子平均值和广义 Karcher 方程

• DOI: 10.1016/j.aim.2015.11.019
• 发表时间: 2016
• 期刊: Advances in Mathematics
• 影响因子: 1.7
• 作者: C. Perfetti;G. Rius;X. Borrise and K. Abe.;Fumiharu Kobayashi;畠中省伍・高島義徳・橋爪章人・原田明;畠中省伍・高島義徳・原田明;Fumiharu Kobayashi;Miklos Palfia
• 通讯作者: Miklos Palfia