67 research outputs found
This Week's Finds in Mathematical Physics (1-50)
These are the first 50 issues of This Week's Finds of Mathematical Physics,
from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity,
topological quantum field theory, knot theory, and applications of
-categories to these subjects. However, there are also digressions into Lie
algebras, elliptic curves, linear logic and other subjects. They were typeset
in 2020 by Tim Hosgood. If you see typos or other problems please report them.
(I already know the cover page looks weird).Comment: 242 page
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
Statistical Mechanics of 2+1 Gravity From Riemann Zeta Function and Alexander Polynomial:Exact Results
In the recent publication (Journal of Geometry and Physics,33(2000)23-102) we
demonstrated that dynamics of 2+1 gravity can be described in terms of train
tracks. Train tracks were introduced by Thurston in connection with description
of dynamics of surface automorphisms. In this work we provide an example of
utilization of general formalism developed earlier. The complete exact solution
of the model problem describing equilibrium dynamics of train tracks on the
punctured torus is obtained. Being guided by similarities between the dynamics
of 2d liquid crystals and 2+1 gravity the partition function for gravity is
mapped into that for the Farey spin chain. The Farey spin chain partition
function, fortunately, is known exactly and has been thoroughly investigated
recently. Accordingly, the transition between the pseudo-Anosov and the
periodic dynamic regime (in Thurston's terminology) in the case of gravity is
being reinterpreted in terms of phase transitions in the Farey spin chain whose
partition function is just a ratio of two Riemann zeta functions. The mapping
into the spin chain is facilitated by recognition of a special role of the
Alexander polynomial for knots/links in study of dynamics of self
homeomorphisms of surfaces. At the end of paper, using some facts from the
theory of arithmetic hyperbolic 3-manifolds (initiated by Bianchi in 1892), we
develop systematic extension of the obtained results to noncompact Riemannian
surfaces of higher genus. Some of the obtained results are also useful for 3+1
gravity. In particular, using the theorem of Margulis, we provide new reasons
for the black hole existence in the Universe: black holes make our Universe
arithmetic. That is the discrete Lie groups of motion are arithmetic.Comment: 69 pages,11 figures. Journal of Geometry and Physics (in press
Entangled graphs on surfaces in space
In the chemical world, as well as the physical, strands get tangled. When those strands form loops, the mathematical discipline of ‘knot theory’ can be used to analyse and describe the resultant tangles. However less has been studied about the situation when the strands branch and form entangled loops in either finite structures or infinite periodic structures. The branches and loops within the structure form a ‘graph’, and can be described by mathematical ‘graph theory’, but when graph theory concerns itself with the way that a graph can fit in space, it typically focuses on the simplest ways of doing so. Graph theory thus provides few tools for understanding graphs that are entangled beyond their simplest spatial configurations.
This thesis explores this gap between knot theory and graph theory. It is focussed on the introduction of small amounts of entanglement into finite graphs embedded in space. These graphs are located on surfaces in space, and the surface is chosen to allow a limited amount of complexity. As well as limiting the types of entanglement possible, the surface simplifies the analysis of the problem – reducing a three-dimensional problem to a two-dimensional one.
Through much of this thesis, the embedding surface is a torus (the surface of a doughnut) and the graph embedded on the surface is the graph of a polyhedron. Polyhedral graphs can be embedded on a sphere, but the addition of the central hole of the torus allows a certain amount of freedom for the entanglement of the edges of the graph. Entanglements of the five Platonic polyhedra (tetrahedron, octahedron, cube, dodecahedron, icosahedron) are studied in depth through their embeddings on the torus. The structures that are produced in this way are analysed in terms of their component knots and links, as well as their symmetry and energy.
It is then shown that all toroidally embedded tangled polyhedral graphs are necessarily chiral, which is an important property in biochemical and other systems. These finite tangled structures can also be used to make tangled infinite periodic nets; planar repeating subgraphs within the net can be systematically replaced with a tangled version, introducing a controlled level of entanglement into the net.
Finally, the analysis of entangled structures simply in terms of knots and links is shown to be deficient, as a novel form of tangling can exist which involves neither knots nor links. This new form of entanglement is known as a ravel. Different types of ravels can be localised to the immediate vicinity of a vertex, or can be spread over an arbitrarily large scope within a finite graph or periodic net. These different forms of entanglement are relevant to chemical and biochemical self-assembly, including DNA nanotechnology and metal-ligand complex crystallisation
Bifurcation analysis of the Topp model
In this paper, we study the 3-dimensional Topp model for the dynamicsof diabetes. We show that for suitable parameter values an equilibrium of this modelbifurcates through a Hopf-saddle-node bifurcation. Numerical analysis suggests thatnear this point Shilnikov homoclinic orbits exist. In addition, chaotic attractors arisethrough period doubling cascades of limit cycles.Keywords Dynamics of diabetes · Topp model · Reduced planar quartic Toppsystem · Singular point · Limit cycle · Hopf-saddle-node bifurcation · Perioddoubling bifurcation · Shilnikov homoclinic orbit · Chao
Hadron models and related New Energy issues
The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation
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