Koopman Tori, Knots and Elliptic Curves Basic Theorems
This article will focus on the continuing work on Koopman operators and the incorporation into the Feynman-Kac quantum fields and AdS space, with Boltzmann Machine correlations for machine learning and AI.
We will demonstrate the proof and relationship for the Koopman Feynman-Kac Mellin Space with a D2-brane in the DBI action and compacted onto a chaotic KAM Torus.
The work of Asakawa et. al focuses on the D2-brane fermionic string as a basis for the qubit, the fundamental unit for quantum computing.
Our approach is inclusive with possible applications to quantum gravity, gravitational waves-cosmic strings, fluid dynamics, plasma fusion energy, ion propulsion, fluid neural networks, metamaterial cloaking, electromagnetic singularities, robotic systems, econometrics, stock pricing, and many others.
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Using the parameters of the non-integrable KAM torus and incorporating the chaotic SLE6 boundary conditions, the Stochastic Feynman-Kac DBI AdS/CFT exhibits chaotic behavior. This is leading to the conjecture of the simplex shape of VSHE as the Schr?dinger Equation in triangular quantum well. The revised DBI Action is used.
This was used to define the simplex Wilson Loop boundary Causal Dynamical Triangulations (CDT).
Figure 5: courtesy of Scott Little
With initial KvN conditions, which will be used for the D-branes wrapped on the KAM Torus
Theorem: proof and relationship for the Koopman Feynman-Kac Mellin Space with a D2-brane in the DBI action and compacted onto a chaotic KAM Torus.
The D2-brane has been defined and the previous theorems on the Koopman operator with the Feynman-Kac AdS holographic Boltzmann Machine are included for continuity.
The Boltzmann Machine as a renormalization group was defined as the 2D representation of the Koopman operator. This is a natural evolution to the present existence and uniqueness of this theorem.
One method to work with D-branes is to measure them within a target space M and not a world sheet or volume. This is the method used by other researchers to create infinite copies of the D-brane submanifold called a foliation, with each singular D-brane referred to as a leaf.
Using the parameters of the non-integrable KAM torus and incorporating the chaotic SLE6 boundary conditions, the Stochastic Feynman-Kac DBI AdS/CFT exhibits chaotic behavior.
The DBI action is the initial point for the D2-brane on an electromagnetic field with BPS, non-chaotic and three dimensions. The modified Koopman D2-brane DBI action on the chaotic perturbative KAM Kolmogorov–Arnold–Moser (KAM) torus is a dilaton singular field.
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To determine the dilaton factor we will equate it to the Koopman operator that represents the extra dimension functional including the electromagnetic field needed for the DBI action.
The Kolmogorov-Arnold-Moser (KAM) Theorem describes the KAM tori with irrational winding numbers that are non-integrable and disintegrate into elliptical and hyperbolic points and the 2D section is a Poincare circle. The next article will discuss elliptic curves in detail.
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If allowed to continue outside of the tori, the non-integrability will continue to increase, raising the value of the Lyapunov Index and non-harmonic oscillations. The index will equal zero if the oscillations are completely harmonic. This is analogous to the ranking of the Birch-Swinner-Dyer (BSD) conjecture described in the next article.
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One area of study for the non-integrable KAM Tori is the Liouville Schramm-Loewner Evolution (SLE) on a chaotic quantum boundary condition.
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The quantum well exists inside the sides of the triangle. Sides are UV cut-off distance. CDT is used to define lattice quantum gravity. The Wilson Loop is the path in spacetime for quark–antiquark pairs from one point and annihilated at another point.
Wilson Loops confine quark triangular quantum wells, and loops in Loop Quantum Gravity. Previous theorems are:
Theorem 1. There exists a solution of the oscillatory Airy Ai Schr?dinger equation (AiS) defined within a causal dynamical triangular (CDT) quantum well. The function is non-integrable.
Theorem 2. The triangular quantum wells can be summed together to form a CDT mesh slice of a toroid that includes Ising spin lattice characteristics.
We will now define the Koopman D2-brane DBI action on a fuzzy KAM torus. This is one of the more flexible models that can be applied in other areas besides string-M theory and quantum gravity.
These applications include turbulent flows of plasma inside a tokamak fusion reactor, chaotic flows within ion propulsion and various forms of energy transfer, electromagnetic singularities-cloaking, AdS holographic neural networks with fuzzy multiple dimensions i.e. Boltzmann Machines.
Stock pricing behavior models have utilized D2-brane string interactions to model pricing fluctuations.
For the KAM (Kolmogorov–Arnold–Moser) theorem the torus will remain stable if perturbed along the x-axis initially. The Lyapunov frequencies will increase with the time axis continuation, and the quantum fluctuations correspond to the zero mode Jacobian matrix of the vector values.
The knots on the torus are defined as diophantine equations with rational roots and frequency varied continuously on the Lyapunov family.
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I have deliberately left out mathematical equations, theorems, and proofs in order to make the material shorter and more accessible. These are available in various places including papers published in Academia.edu , Google Scholar, and in my book KKOGFAB Theory available on Amazon.com KKOGFAB .
Here are some links:
Papers on Academia.edu and Google Scholar?Papers ?|?Outline ?|?Primer
References are listed at the end of each article.
References for main body of work
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1.????? T. Asakawa, S. Sasa, and S. Watamura. D-branes in Generalized Geometry and Dirac-Born-Infeld Action. Particle Theory and Cosmology Group, Tohoku University. arXiv:1206.6964v2 [hep-th] 19 Jul 2012
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2.????? Eugeny Babichev, Philippe Brax,Chiara Caprini, J′er?ome Martin, Dani`ele A. Steer. Dirac Born Infeld (DBI) Cosmic Strings. 11 Sep 2008. arXiv:0809.2013v1 [hep-th].
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3.????? Erik Bartoˇs and Richard Pinˇc′ak.? Identification of market trends with string and D2-brane maps. arXiv:1607.05608v1 [q-fin.ST ] 18 Jul 2016. https://arxiv.org/pdf/1607.05608.pdf
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4.????? K. Becker, M. Becker, J. & Schwartz. String Theory and M-Theory: An Introduction. Cambridge University Press, New York. ISBN: 10-521-86069-5. 2007.
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5.????? N. Beisert et al., \Review of AdS/CFT Integrability: An Overview", Lett. Math. Phys. 99, 3 (2012), arXiv:112.3982.
Birch, Bryan ;?Swinnerton-Dyer, Peter ?(1965). "Notes on Elliptic Curves (II)".?J. Reine Angew. Math. ?165?(218): 79–108.?doi :10.1515/crll.1965.218.79 .?S2CID ?122531425 .
I. Blake; G. Seroussi; N. Smart (2000).?Elliptic Curves in Cryptography. LMS Lecture Notes. Cambridge University Press.?ISBN ?0-521-65374-6 .
Brown, Ezra (2000), "Three Fermat Trails to Elliptic Curves",?The College Mathematics Journal,?31?(3): 162–172,?doi :10.1080/07468342.2000.11974137 ,?S2CID ?5591395 , winner of the MAA writing prize the?George Pólya Award
9.????? Brodsky, Stanley. (2008). Novel LHC Phenomena. 002. 10.22323/1.045.0002.
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10.?? 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.
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11.?? B. Duplantier et al. Schramm Loewner Evolution and Liouville Quantum Gravity. Phys.Rev.Lett. 107 (2011) 131305 arXiv:1012.4800 [math-ph].
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12.?? Béatrice I. Chetard Last update: October 3, 2017 Elliptic curves as complex tori. https://bchetard.wordpress.com/wp-content/uploads/2017/10/main.pdf
领英推荐
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13.?? 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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14.?? Joshua Erlich . Stochastic Emergent Quantum Gravity. v1] Wed, 18 Jul 2018. arXiv:1807.07083 ?
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15.?? 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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16.?? 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].
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17.?? O. Gonz′alez-Gaxiola and J. A. Santiago. Symmetries, Mellin Transform and the Black-Scholes Equation (A Nonlinear Case). Int. J. Contemp. Math. Sciences, Vol. 9, 2014, no. 10, 469 – 478. HIKARI Ltd, www.m-hikari.com . https://dx.doi.org/10.12988/ijcms.2014.4673
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18.?? Koji Hashimoto. AdS/CFT as a deep Boltzmann machine. Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan. (Dated: March 13, 2019). https://arxiv.org/abs/1903.04951
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19.?? 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.
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20.?? E. Kiritsis. String Theory in a Nutshell. Princeton University Press. ISBN: 10:-0-691-12230-X. 19 March 2007.
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21.?? Stefan Klus etal2022. Koopman analysis of quantum systems. J.Phys.A:Math.Theor.55 314002. https://iopscience.iop.org/article/10.1088/1751-8121/ac7d22/meta
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22.?? Vihar Kurama. Beginner's Guide to Boltzmann Machines in PyTorch. May 2021. https://blog.paperspace.com/beginners-guide-to-boltzmann-machines-pytorch/
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23.?? Sangmin Lee and L′arus Thorlacius. Strings and D-Branes at High Temperature. Joseph Henry Labora, toriesPrinceton UniversityarXiv:hep-th/9707167v1 18 Jul 1997.
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24.?? ?M. Li, R. Miao & R. Zheng. Meta-Materials Mimicking Dynamic Spacetime D-Brane and Non-Commutativity in String Theory. 03 February 2011. arXiv: 1005.5585v2.
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25.?? 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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26.?? ?S. Little. Chaotic Boundaries of AdS/CFT Stochastic Feynman-Kac Mellin Transform. Academia.edu . December 28, 2021. https://www.academia.edu/66245245/Chaotic_Boundaries_of_AdS_CFT_Stochastic_Feynman_Kac_Mellin_Transform?source=swp_share .
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27.?? S. Little. Feynman-Kac Formulation of Stochastic String DBI Helmholtz Action. Academia.edu . July 13, 2021. https://www.academia.edu/49860679/Feynman_Kac_Formulation_of_Stochastic_String_DBI_Helmholtz_Action .
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28.?? S. Little. Liouville SLE Boundaries on CFT Torus Defined with Stochastic Schr?dinger Equation. SIAM Conference on Analysis of Partial Differential Equations (PD11) December 7-10, 2015.?https://www.siam.org/meetings/pd15/ . Session:?https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=2189311
29.?? S. Little. Session Chair and Contributed Speaker (CP10): Stochastic Helmholtz Finite Volume Method for DBI String-Brane Theory Simulations. SIAM Conference on Analysis of Partial Differential Equations (PD19),?December 11 – 14, 2019, La Quinta Resort & Club, La Quinta, California, Session:?https://meetings.siam.org/sess/dsp_programsess.cfm?SESSIONCODE=67694
30.?? David A. Lowe. Mellin transforming the minimal model CFTs: AdS/CFT at strong curvature. 17 Feb 2016. arXiv:1602.05613v1 [hep-th]
31.?? J. M. Maldacena, “The large N limit of superconformal ?eld theories and super-gravity,” Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200].
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32.?? Pallab Basu, Diptarka Das, Leopoldo A. Pando-Zayas, Dori Reichmann. Chaos in String Theory.[arXiv:1103.4101, 1105.2540, 1201.5634, and ongoing work.]
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33.?? Pallab Basu, Leopoldo A. Pando Zayas, Phys.Lett.B 11 (2011) 00418, [arXiv:1103.4107]. Pallab Basu, Diptarka Das, Archisman Ghosh, Phys.Lett.B 11 (2011) 00417, [arXiv:1103.4101].
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34.?? William T. Redman 9 June 2020. arXiv:1912.13010v3 [cond-mat.stat-mech]
35.?? Paul Schreivogl and Harold Steinacker. Generalized Fuzzy Torus and its Modular Properties. Faculty of Physics, University of Vienna. Published online October 17, 2013
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37.?? S. Sheffield. Conformal weldings of random surfaces: SLE and the quantum gravity zipper. 23 Sep 2015. arXiv:1012.4797v2 [math.PR ]
38.?? UCLA Ozcan Research Group. UCLA engineers use deep learning to reconstruct holograms and improve optical microscopy. November 20, 2017. https://phys.org/news/2017-11-ucla-deep-reconstruct-holograms-optical.html
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39.?? Favio Vázquez. Deep Learning made easy with Deep Cognition. Dec 21, 2017. https://becominghuman.ai/deep-learning-made-easy-with-deep-cognition-403fbe445351
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40.?? David Viennot and Lucile Aubourg. Chaos, decoherence and emergent extra dimensions in D-brane dynamics with fluctuations. Class. Quantum Grav. 35 (2018) 135007 (18pp) https://doi.org/10.1088/1361-6382/aac603
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Wiles, Andrew ?(2006).?"The Birch and Swinnerton-Dyer conjecture" ?(PDF). In Carlson, James;?Jaffe, Arthur ;?Wiles, Andrew ?(eds.).?The Millennium prize problems. American Mathematical Society. pp.?31–44.?ISBN ?978-0-8218-3679-8 .?MR ?2238272 . Archived from?the original ?(PDF)?on 29 March 2018. Retrieved?16 December?2013.
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42.?? 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
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43.?? Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.
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Other references
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.
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INTERNATIONAL COLLABORATIONS:
E.Deotto (MIT, USA); M.Reuter (Mainz University, Germany); A.A.Abrikosov (jr) (ITEP, Moscow); I.Fiziev (Sofia University, Bulgaria).