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Timothee Benard - One of the best experts on this subject based on the ideXlab platform.

  • transience of a symmetric random walk in infinite measure
    arXiv: Dynamical Systems, 2020
    Co-Authors: Timothee Benard
    Abstract:

    We consider a random walk on a second countable locally Compact Topological Space endowed with an invariant Radon measure. We show that if the walk is symmetric and if every subset which is invariant by the walk has zero or infinite measure, then one has transience in law for almost every starting point. We then deduce a converse to Eskin-Margulis recurrence theorem.

Shidama Yasunari - One of the best experts on this subject based on the ideXlab platform.

  • Functional Space Consisted by Continuous Functions on Topological Space
    'Walter de Gruyter GmbH', 2021
    Co-Authors: Yamazaki Hiroshi, Miyajima Keiichi, Shidama Yasunari
    Abstract:

    In this article, using the Mizar system [1], [2], first we give a definition of a functional Space which is constructed from all continuous functions defined on a Compact Topological Space [5]. We prove that this functional Space is a Banach Space [3]. Next, we give a definition of a function Space which is constructed from all continuous functions with bounded support. We also prove that this function Space is a normed Space

  • Functional Space Consisted by Continuous Functions on Topological Space
    'Walter de Gruyter GmbH', 2021
    Co-Authors: Yamazaki Hiroshi, Miyajima Keiichi, Shidama Yasunari
    Abstract:

    In this article, using the Mizar system [1], [2], first we give a definition of a functional Space which is constructed from all continuous functions defined on a Compact Topological Space [5]. We prove that this functional Space is a Banach Space [3]. Next, we give a definition of a function Space which is constructed from all continuous functions with bounded support. We also prove that this function Space is a normed Space.Hiroshi Yamazaki - Nagano Prefectural Institute of Technology, Nagano, JapanKeiichi Miyajima - Ibaraki University, Ibaraki, JapanYasunari Shidama - Karuizawa Hotch 244-1, Nagano, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3- 319-20614-1. doi:10.1007/978-3-319-20615-8 17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear Spaces. Formalized Mathematics, 12(3):269–275, 2004.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Yasumasa Suzuki. Banach Space of bounded real sequences. Formalized Mathematics, 12 (2):77–83, 2004.Hiroshi Yamazaki. Functional sequence in norm Space. Formalized Mathematics, 28(4): 263–268, 2020. doi:10.2478/forma-2020-0023.291496

William Moran - One of the best experts on this subject based on the ideXlab platform.

  • Topological feedback entropy and Nonlinear stabilization
    IEEE Transactions on Automatic Control, 2004
    Co-Authors: Girish N. Nair, Robin J. Evans, Iven Mareels, William Moran
    Abstract:

    It is well known in the field of dynamical systems that entropy can be defined rigorously for completely deterministic open-loop systems. However, such definitions have found limited application in engineering, unlike Shannon's statistical entropy. In this paper, it is shown that the problem of communication-limited stabilization is related to the concept of Topological entropy, introduced by Adler et al. as a measure of the information rate of a continuous map on a Compact Topological Space. Using similar open cover techniques, the notion of Topological feedback entropy (TFE) is defined in this paper and proposed as a measure of the inherent rate at which a map on a nonCompact Topological Space with inputs generates stability information. It is then proven that a Topological dynamical plant can be stabilized into a Compact set if and only if the data rate in the feedback loop exceeds the TFE of the plant on the set. By taking appropriate limits in a metric Space, the concept of local TFE (LTFE) is defined at fixed points of the plant, and it is shown that the plant is locally uniformly asymptotically stabilizable to a fixed point if and only if the data rate exceeds the plant LTFE at the fixed point. For continuously differentiable plants in Euclidean Space, real Jordan forms and volume partitioning arguments are then used to derive an expression for LTFE in terms of the unstable eigenvalues of the fixed point Jacobian.

Yamazaki Hiroshi - One of the best experts on this subject based on the ideXlab platform.

  • Functional Space Consisted by Continuous Functions on Topological Space
    'Walter de Gruyter GmbH', 2021
    Co-Authors: Yamazaki Hiroshi, Miyajima Keiichi, Shidama Yasunari
    Abstract:

    In this article, using the Mizar system [1], [2], first we give a definition of a functional Space which is constructed from all continuous functions defined on a Compact Topological Space [5]. We prove that this functional Space is a Banach Space [3]. Next, we give a definition of a function Space which is constructed from all continuous functions with bounded support. We also prove that this function Space is a normed Space

  • Functional Space Consisted by Continuous Functions on Topological Space
    'Walter de Gruyter GmbH', 2021
    Co-Authors: Yamazaki Hiroshi, Miyajima Keiichi, Shidama Yasunari
    Abstract:

    In this article, using the Mizar system [1], [2], first we give a definition of a functional Space which is constructed from all continuous functions defined on a Compact Topological Space [5]. We prove that this functional Space is a Banach Space [3]. Next, we give a definition of a function Space which is constructed from all continuous functions with bounded support. We also prove that this function Space is a normed Space.Hiroshi Yamazaki - Nagano Prefectural Institute of Technology, Nagano, JapanKeiichi Miyajima - Ibaraki University, Ibaraki, JapanYasunari Shidama - Karuizawa Hotch 244-1, Nagano, JapanGrzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, Karol Pak, and Josef Urban. Mizar: State-of-the-art and beyond. In Manfred Kerber, Jacques Carette, Cezary Kaliszyk, Florian Rabe, and Volker Sorge, editors, Intelligent Computer Mathematics, volume 9150 of Lecture Notes in Computer Science, pages 261–279. Springer International Publishing, 2015. ISBN 978-3- 319-20614-1. doi:10.1007/978-3-319-20615-8 17.Grzegorz Bancerek, Czesław Bylinski, Adam Grabowski, Artur Korniłowicz, Roman Matuszewski, Adam Naumowicz, and Karol Pak. The role of the Mizar Mathematical Library for interactive proof development in Mizar. Journal of Automated Reasoning, 61(1):9–32, 2018. doi:10.1007/s10817-017-9440-6.Bruce K. Driver. Analysis Tools with Applications. Springer, Berlin, 2003.Takaya Nishiyama, Keiji Ohkubo, and Yasunari Shidama. The continuous functions on normed linear Spaces. Formalized Mathematics, 12(3):269–275, 2004.Laurent Schwartz. Théorie des ensembles et topologie, tome 1. Analyse. Hermann, 1997.Yasumasa Suzuki. Banach Space of bounded real sequences. Formalized Mathematics, 12 (2):77–83, 2004.Hiroshi Yamazaki. Functional sequence in norm Space. Formalized Mathematics, 28(4): 263–268, 2020. doi:10.2478/forma-2020-0023.291496

Tomas Dominguez Benavides - One of the best experts on this subject based on the ideXlab platform.

  • the szlenk index and the fixed point property under renorming
    Fixed Point Theory and Applications, 2010
    Co-Authors: Tomas Dominguez Benavides
    Abstract:

    Assume that is a Banach Space such that its Szlenk index is less than or equal to the first infinite ordinal . We prove that can be renormed in such a way that with the resultant norm satisfies , where is the Garcia-Falset coefficient. This leads us to prove that if is a Banach Space which can be continuously embedded in a Banach Space with , then, can be renormed to satisfy the w-FPP. This result can be applied to Banach Spaces which can be embedded in , where is a scattered Compact Topological Space such that . Furthermore, for a Banach Space , we consider a distance in the Space of all norms in which are equivalent to (for which becomes a Baire Space). If , we show that for almost all norms (in the sense of porosity) in , satisfies the w-FPP. For general reflexive Spaces (independently of the Szlenk index), we prove another strong generic result in the sense of Baire category.