Weak convergence of vector measures with applications to real analysis

矢量测量与实际分析应用的收敛性较弱

• 批准号: 11640160
• 负责人: KAWABE Jun
• 金额: $ 1.28万
• 依托单位: SHINSHU UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-11640160/
• 关键词: weak convergence; injective tensor integral; fuzzy optimization; Fourier coefficients; nonlinear regulator; controller time-delay; martingale-like sequence; Riemannian submersion; weak convergence of vector measures; fuzzy set; generalized Norlu

We have studied weak convergence of vector measures with values in Banach spaces and nuclear spaces, and have applied it to several interesting problems in real analysis, probability theory, control theory, differential geometry and so on. Some of our important results are as follows :1. By an essential use of Bartle's bilinear integration theory, it is shown that the injective tensor product of positive vector measures in certain Banach lattices is jointly continuous with respect to the weak convergence of vector measures.2. The weak compactness of a set of control inputs is shown in the case that they are given by the gravity calculation of time dependent fuzzy membership functions. As an application, the existence of optimal solution is discussed in a fuzzy control for an open-loop system.3. We obtain a convergence theorem of compound probability measures on a uniform space for a net of uniformly equicontinuous transition probabilities. This result applies to Gaussian transition probabilities on a Hilbert spaces.4. We obtain a general theorem for the method (N,p_n, q_n)(C, 1) summability of the sequence {nB_n (x)}, which contains some theorems due to S.P.Khare, V.K.Tripathi and A.N.Singh and et al.5. It is shown that some central manifold exists in a neighborhood of a point of equilibrium.6. It is shown that a set of fuzzy membership functions in the NBV space is compact with respect to the weak^* topology. This result applies to the existence of fuzzy optimal control.7. A relation between two convergence theorems of maritingale in the limit, i.e., L^1-boundedness and integrability of stopped processes is studied.8. We define another almost complex structure (resp.almost contact structure) and an indefinite Kahlerian (resp. Sasakian) manifold with affine connection.

本文研究了取值于Banach空间和核空间的向量测度的弱收敛性，并将其应用于真实的分析、概率论、控制论、微分几何等领域中的一些有趣的问题，主要结果如下：1.利用Bartle的双线性积分理论，证明了某些Banach格上正向量测度的内射张量积关于向量测度的弱收敛是联合连续的.弱紧的一组控制输入的情况下，他们是由重力计算的时间依赖的模糊隶属函数。作为应用，讨论了开环系统模糊控制最优解的存在性.对于一致等度连续转移概率网，得到了一致空间上复合概率测度的一个收敛定理。这一结果适用于希尔伯特空间上的高斯转移概率。本文得到了序列{nB_n（x）}的方法（N，p_n，q_n）（C，1）可和性的一个一般定理，它包含了S. P. Khare，V. K. Tripathi和A. N. Singh等人的一些定理。证明了在平衡点的邻域内存在中心流形.证明了NBV空间中的模糊隶属函数集关于弱^* 拓扑是紧的。这一结果适用于模糊最优控制的存在性.证明了Maritingale的两个极限收敛定理之间的关系，研究了停止过程的L^1-有界性和可积性.我们定义了另一个几乎复结构（或几乎接触结构）和一个不定Kahlerian（或几乎接触结构）。Sasakian）流形。

项目成果

河邊淳: "Banach束に値をとる正値テンソル積測度の弱収束"京都大学数理解析研究所講究録. 1186(掲載予定). 1-14

Jun Kawabe：“正张量乘积的弱收敛取巴拿赫丛中的值”京都大学数学分析研究所研究记录1186（待出版）。

J.Kawabe: "Weak convergence of tensor products of vector measures with values in nuclear spaces"Bull.Austral.Math.Soc.. 59. 449-458 (1999)

J.Kawabe：“矢量测量的张量积与核空间中的值的弱收敛”Bull.Austral.Math.Soc.. 59. 449-458 (1999)

T.Mitsuishi, J.Kawabe and et al.: "Mamdani method and fuzzy optimal control (in Japanese)"Lecture Notes in RIMS, Kyoto University (Kokyuroku). 1100. 78-86 (1999)

T.Mitsuishi、J.Kawabe 等人：“Mamdani 方法和模糊最优控制（日语）”京都大学 RIMS 讲义（Kokyuroku）。

N.Endo, J.Kawabe and et al.: "Compactness of a set of time dependent fuzzy membership functions and fuzzy optimal control (in Japanese)"Lecture Notes in RIMS, Kyoto University (Kokyuroku). 1186 (to appear). 216-223

N.Endo、J.Kawabe 等人：“Compactness of a set of time dependent fuzzy隶属函数和模糊最优控制（日语）”京都大学 RIMS 讲义（Kokyuroku）。

河邊 淳: "Strassenの定理のベクトル測度への拡張"実解析学シンポジウム報告集. 204-211 (1999)

Jun Kawabe：“施特拉森定理对矢量测量的扩展”实分析研讨会报告 204-211 (1999)。