Koopman Tori, Knots and Elliptic Curves, Knots and Elliptic Curves Intro
Scott Little PhD
Innovative senior data scientist, professor, engineering coach-thought leader, entrepreneur, author
The work by Hirosi Ooguri and Cumrun Vafa demonstrate the Loop Quantum Gravity (LQG) Wilson Loop path of a topological string is the S³ simple knot invariant to the string orientation spectrum. The spectrum is the equivalent of an M2 brane embedded into an M5 brane with conifold symmetry. The Wilson Loop knot is also correlated with N=1 supersymmetric 4D world sheet instantons.
The theorem by Lickorish–Wallace in the 1960s states that a 3D manifold can be created from a regular space by cutting and re-connecting a series of hyper dimensional (d-1) knots within that space known as surgical alterations.
The knot concordance group is a mapping on a solid hypersurface which is injective or one to one on a torus and is fractal if the group is a satellite or peripheral to the main knot. The knots on the torus are defined as diophantine equations with rational roots and frequency varied continuously on the Lyapunov family.
The satellite knot in this figure is a trefoil, one of the more common and versatile in the knot family. The trefoil has the following topological properties:
The trefoil knot has been used in art and religion over the years, representing the infinite nature of spiritual dimensions. Knots are associated with magical properties such as binding, with the Egyptian “Tat” Knot of Isis and Celtic knots used in wiccan ceremonies according to Livio, there are folkloric tales of knots being used by magic practitioners to control the weather and bind the winds for sailors.
Knots and tori also feature in religious symbolism such as the trinity of the Father, Son, Holt Spirit in Christianity and the Kabbala in Jewish mysticism. The mandala is a geometric toroidal shaped religious symbol in Hindu and Buddhist Tantrism with a knot node that represents sacred space and infinity. A meditation bowl is shaped like a torus and when struck, vibrates at a specific frequency used for meditation.
In Ancient Greece Alexander the Great sliced through the impossible Gordian knot. The artist E.M. Escher used knots in many of his multi-dimensional artworks. The labyrinth is a holy maze used in churches as a walking meditation since the Middle Ages. Knots in the maze path function as points of spiritual inflection. The knot singularity or node might be considered in some belief systems to be a portal or dimensional vortex.
Normally string theory or loop quantum gravity (LQG) is integrable. In special cases such as the Kolmogorov-Arnold-Moser (KAM) Theorem, it is non-integrable. This is where space is compacted on a torus with an irrational winding number, or how many times the string is wrapped around the torus. In 4-dimensional phase space, the 2-torus (Poincare Circle) with irrational winding number will survive, while the rational or resonant tori will decompose into elliptical and hyperbolic points.
The elliptical points are defined on an elliptic curve. An elliptic curve is a smooth curve with a cubic third order polynomial described on a plane including a singular infinite point (singularity pole, knot). Normally these are on a complex space and embedded into a simple circular. In our case, we are using a complex space and the KAM torus is the choice. The Weierstrass version has a meromorphic singularity pole in the 2d space.
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
References for main body of work
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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
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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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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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
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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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].
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. http://dx.doi.org/10.12988/ijcms.2014.4673
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
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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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
22. Vihar Kurama. Beginner's Guide to Boltzmann Machines in PyTorch. May 2021. https://blog.paperspace.com/beginners-guide-to-boltzmann-machines-pytorch/
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.
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.
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.
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.
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.
28. S. Little. Liouville SLE Boundaries on CFT Torus Defined with Stochastic Schrödinger Equation. SIAM Conference on Analysis of Partial Differential Equations (PD11) December 7-10, 2015. http://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
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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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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
39. Favio Vázquez. Deep Learning made easy with Deep Cognition. Dec 21, 2017. https://becominghuman.ai/deep-learning-made-easy-with-deep-cognition-403fbe445351
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
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
43. Git Hub Python boltzmannclean https://github.com/facultyai/boltzmannclean.
Other references
Milan Korda, Yoshihiko Susuki, Igor Mezić,. Power grid transient stabilization using Koopman model predictive control. IFAC-PapersOnLine, Volume 51, Issue 28,2018, Pages 297-302, ISSN 2405-8963.
INTERNATIONAL COLLABORATIONS:
E.Deotto (MIT, USA); M.Reuter (Mainz University, Germany); A.A.Abrikosov (jr) (ITEP, Moscow); I.Fiziev (Sofia University, Bulgaria).