Koopman Tori, Knots, Elliptic Curves and Gamma Ray Bursts
This is an interim article focused on one of the many applications of Koopman Operator Theory, Tori, Knots, and Elliptic Curves (KKOGFAB). This particular application is the mapping of toroidal spaces to gamma ray bursts based on black hole accretion disk collapse. This is related to my work on gravitational waves-cosmic strings, electromagnetic pulse (EMP)-solar storms and seismicity.
Other applications include PDEs, fluid dynamics, string brane M-theory, AdS/CFT, quantum gravity field theory, knot theory, complexity chaos, artificial intelligence (AI), neural networks, quantum computing, cryptography, metamaterial cloaking, electromagnetic singularities, plasma fusion ion energy propulsion, robotics, economics stocks, and biology genetics.
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This series of articles 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.
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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 simplex Wilson Loop boundary Causal Dynamical Triangulation (CDT). Here is a simple diagram of a KAM Torus:
Gamma Ray Burst
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Since Wikipedia has a great site on gamma ray burst, I show a brief introduction here with links, no reason to re-invent the wheel. Please access this site for a more detailed introduction.
“In?gamma-ray astronomy,?gamma-ray bursts?(GRBs) are immensely energetic events occurring in distant?galaxies?which represent the brightest and "most powerful class of explosion in the universe."[1][2][3][4]?These extreme?electromagnetic events?are second only to the?Big Bang?as the most energetic and luminous phenomenon ever known.[5][6]?Gamma-ray bursts can last from ten milliseconds to several hours.[7][8]?After the initial flash of?gamma rays, a longer-lived?§?afterglow?is emitted, usually in the longer wavelengths of?X-ray,?ultraviolet,?optical,?infrared,?microwave?or?radio?frequencies.[9].
The intense radiation of most observed GRBs is thought to be released during a?supernova?or?superluminous supernova?as a high-mass?star?implodes to form a?neutron star?or a?black hole. From?gravitational wave?observations,?§?short-duration?(sGRB) events describe a subclass of GRB signals that are now known to originate from the cataclysmic?merger of binary neutron stars.[10]” https://en.wikipedia.org/wiki/Gamma-ray_burst
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Here is a diagram from the Wikipedia site:
?Here is an image from Phys.org article “Simulations reveal black holes inherit magnetic fields from parent stars” on black hole burst origins that show the toroidal shape:
Finally, here is a Google AI summary on ‘Black Hole Gamma Ray Burst’.
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Black holes can create bursts of radiation when they form, or when matter swirls into them:
??????????? Gamma-ray bursts
When a massive star collapses into a black hole, matter swirling into the black hole can create two jets that shoot outward at nearly the speed of light. This produces a gamma-ray burst, which is a powerful explosion that releases a huge amount of energy in a short period of time. The afterglow of a gamma-ray burst can be visible for months or years.
??????????? Supernova
When a massive star runs out of fuel, its core collapses and forms a black hole in a process called a supernova. The supernova blasts part of the star into space.
??????????? Direct collapse
A star can also collapse into a black hole without a supernova in a process called direct collapse. This process is gentler and doesn't disturb nearby objects.
Here are some other things to know about black holes:
??????????? Most galaxies have a supermassive black hole at their center.
??????????? The Event Horizon Telescope took the first picture of a black hole in April 2019.
??????????? The sources of most gamma-ray bursts are billions of light years away from Earth.
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Here is another Google AI summary on ‘Gamma Ray Burst Torus’.
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A "gamma ray burst torus" refers to a theoretical ring of dense material, often described as a torus shape, that is thought to form around a newly formed black hole following a massive stellar collapse, which is the primary source of a gamma ray burst (GRB); this torus plays a crucial role in the powerful, highly collimated jets of radiation that characterize GRBs.
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Key points about gamma ray burst torus:
??????????? Origin:
When a massive star collapses to form a black hole, the remaining stellar material can be ejected outwards, forming a torus-like structure around the black hole.
??????????? Jet formation:
The powerful magnetic fields near the black hole are believed to channel the energy from the collapsing star into focused jets, which are thought to be launched from the torus's inner region.
??????????? Radiation emission:
The interaction between the jets and the torus material is considered a key mechanism for producing the intense gamma rays observed in GRBs.
Important aspects of the torus model:
??????????? Collimation:
The torus acts as a funnel, helping to collimate the gamma-ray jets into narrow beams, which explains why GRBs are observed only from specific directions.
??????????? Variability:
The interaction between the jets and the torus can lead to variations in the observed gamma-ray emission, including rapid fluctuations and multiple pulses.
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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.
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领英推荐
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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
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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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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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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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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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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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