Introduction to Koopman Operator Theory V
This is the fifth article in a series on Koopman Operator Theory. Koopman Operator Theory (KOT) is a theory and application of nonlinear dynamics based on the elegant theorem developed by Koopman and Von Neumann in the 1930’s as a precursor to the Feynman-Kac formula path integral.
It is known as the KvN Integral crucial to the development of Ergodic Theory and later Dynamic and Chaotic Systems Theory by M′ezic [3][5][7][15].
The linear Koopman Operator Theory includes a state space of infinite dimensions to control a finite dimensional nonlinear dynamic system using a dynamic mode decomposition of data snapshot eigenvalues and mode functions. This data can be stochastic non-deterministic [17][19][23].
The KOT has been used to compute critical eigenvector energy velocity ratios for 2D Ising and Potts matrix models [20][21]. Using the magnetic coupling constant to derivative of alpha times beta + derivative of beta times alpha.
The system can be approximated using this parameter and does not require a more complex analysis. The results of this paper were compared to a Monte Carlos simulation output with favorable results [22][23].? Even number Koopman = RG in 2D Ising Model finite section for magnetic coupling.
This is a specific form of a spin glass; spin glass is special lattice where each particle has unique spin. One of the primary open problems in physics. These models are crucial to the understanding of string theory and high energy physics in the definition of energy spin states in nonlinear matrices. This is one of the problems in physics which is important to the unification of quantum mechanics and general relativity. Otherwise known as quantum gravity.
This 2D Ising Model is very popular with strings, defines EM field lattice matrix for string manifolds. I used it for string boundaries in previous papers [17].
Please see figure for 2D Ising Model
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References
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[3] Petar Bevanda, Stefan Sosnowski, Sandra Hirche. Koopman Operator Dynamical Models: Learning, Analysis and Control. arXiv:2102.02522v2 [eess.SY] 22 Dec 2021. https://www.sciencedirect.com/science/article/abs/pii/S1367578821000729
[4] Brodsky, Stanley. (2008). Novel LHC Phenomena. 002. 10.22323/1.045.0002.
[5] Steven L. Brunton and Matthew J. Colbrook. 23/01/2023, 22:13 Resilient Data-driven Dynamical Systems with Koopman: An Infinite-dimensional Numerical Analysis Perspective https://www.damtp.cam.ac.uk/user/mjc249/pdfs/Koopman_NA.pdf
[6] Brunton, Steven & Kutz, J. & Fu, Xing & Grosek, Jacob. (2016). Dynamic Mode Decomposition for Robust PCA with Applications to Foreground/Background Subtraction in Video Streams and Multi-Resolution Analysis. https://www.researchgate.net/publication/301627453_Dynamic_Mode_Decomposition_for_Robust_PCA_with_Applications_to_ForegroundBackground_Subtraction_in_Video_Streams_and_Multi-Resolution_Analysis
[7] P. Carta, E. Gozzi, D. Mauro. Koopman-von Neumann Formulation of Classical Yang-Mills Theories: I. arXiv:hep-th/0508244v1 31 Aug 2005. https://arxiv.org/abs/hep-th/0508244
[8] B. Duplantier et al. Schramm Loewner Evolution and Liouville Quantum Gravity. Phys.Rev.Lett. 107 (2011) 131305 arXiv:1012.4800 [math-ph].
[9] Razvan Ciuca, Oscar F. Hern′andez and Michael Wolman. A Convolutional Neural Network For Cosmic String Detection in CMB Temperature Maps. 14 Mar 2019.arXiv:1708.08878v3 [astro-ph.CO]
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[10] Et. Al. Constraints on cosmic strings using data from the ?rst Advanced LIGO observing run. [Submitted on 11 Sep 2008]https://arxiv.org/abs/1712.01168v2
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[12] Isabel Fernandez-Nu~nez and Oleg Bulashenko. Wave propagation in metamaterials mimicking the topology of a cosmic string. 6 Mar 2018. arXiv:1711.02420v2 [physics.optics].
[13] Junkee Jeon and Ji-Hun Yoon. Discount Barrier Option Pricing with a Stochastic Interest Rate: Mellin Transform Techniques and Method of Images. Commun. Korean Math. Soc. 33 (2018), No. 1, pp. 345–360. https://doi.org/10.4134/CKMS.c170060. pISSN: 1225-1763 / eISSN: 2234-3024.
[14] E. Kiritsis. String Theory in a Nutshell. Princeton University Press. ISBN: 10:-0-691-12230-X. 19 March 2007.
[15] Milan Korda, Yoshihiko Susuki, Igor Mezi?, Power grid transient stabilization using Koopman model predictive control. IFAC-PapersOnLine, Volume 51, Issue 28, 2018,
Pages 297-302, ISSN 2405-8963, https://doi.org/10.1016/j.ifacol.2018.11.718
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[16] Vihar Kurama. Beginner's Guide to Boltzmann Machines in PyTorch. May 2021. https://blog.paperspace.com/beginners-guide-to-boltzmann-machines-pytorch/
[17] S. Little. AdS-CFT Stochastic Feynman-Kac Mellin Transform with Chaotic Boundaries. Academia.edu. December 28, 2021. https://www.academia.edu/66244508/AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform_with_Chaotic_Boundaries?source=swp_share.
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[18] Jorge Mallo. An introduction to Koopman operator theory and its applications. Universidad de Deusto. October 2019. https://cmc.deusto.eus/wp-content/uploads/2019/10/07-koopman-jorgeMallo.pdf
[19] Gerard McCaul and Denys. I. Bondar. How to Win Friends and In?uence Functionals: Deducing Stochasticity From Deterministic Dynamics. arXiv:1904.04918v2 [cond-mat.stat-mech] 1 Dec 2020. https://arxiv.org/abs/1904.04918
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[20] William T. Redman 9 June 2020. Renormalization Group as a Koopman Operator. arXiv:1912.13010v3 [cond-mat.stat-mech] [1912.13010v3] Renormalization Group as a Koopman Operator (arxiv.org)
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[21] Reinhold Schneider, L. Sallandt, M. Oster and P. Trunschke. (TU Berlin) (Deep Neural Networks: Niklas Nüsken U Potsdam, Lorenz. Richter ZIP Berlin - U Cottbus. FBSDE (forward-backward stochastic di?erential equations) and Hierarchical Tensors.
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[22] El-Showk, Sheer; Paulos, Miguel F.; Poland, David; Rychkov, Slava; Simmons-Duffin, David; Vichi, Alessandro (2014). "Solving the 3d Ising Model with the Conformal Bootstrap II. C -Minimization and Precise Critical Exponents" (PDF). Journal of Statistical Physics. 157 (4–5): 869–914. arXiv:1403.4545. https://inspirehep.net/literature/1286327
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[23] Favio Vázquez. Deep Learning made easy with Deep Cognition. Dec 21, 2017. https://becominghuman.ai/deep-learning-made-easy-with-deep-cognition-403fbe445351
[24] Williams, M.O., Kevrekidis, I.G. & Rowley, C.W. A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition. J Nonlinear Sci 25, 1307–1346 (2015). https://doi.org/10.1007/s00332-015-9258-5
[25] Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.