A running commentary is being developed within my LinkedIn blogging area. Besides editing the overall document, the end notes will be using some of these reference materials below.
Isoperimetric Quotient for Fullerenes and Other Polyhedral Cages Tomaž Pisanski ,† Matjaž Kaufman ,*† Drago Bokal ,† Edward C. Kirby ,‡ Ante Graovac § Inštitut za matematiko, fiziko in mehaniko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenia, Resource Use Institute, 14 Lower Oakfield, Pitlochry, Perthshire PH16 5DS, Scotland, UK The Rugjer Bošković Institute, Bijenička c. 54, HR-10001 Zagreb, POB. 1016, Croatia J. Chem. Inf. Comput. Sci., 1997, 37 (6), pp 1028–1032 DOI: 10.1021/ci970228e Publication Date (Web): November 24, 1997 b Copyright © 1997 American Chemical Society Abstract: The notion of Isoperimetric Quotient (IQ) of a polyhedron has been already introduced by Polya. It is a measure that tells us how spherical is a given polyhedron. If we are given a polyhedral graph it can be drawn in a variety of ways in 3D space. As the coordinates of vertices belonging to the same face may not be coplanar the usual definition of IQ fails. Therefore, a method based on a proper triangulation (obtained from omni-capping) is developed that enables one to extend the definition of IQ and compute it for any 3D drawing. The IQs of fullerenes and other polyhedral cages are computed and compared for their NiceGraph and standard Laplacian 3D drawings. It is shown that the drawings with the maximal IQ values reproduce well the molecular mechanics geometries in the case of fullerenes and exact geometries for Platonic and Archimedean polyhedra.
In the equations of general relativity, G is often multiplied by 8π. Hence writings in particle physics and physical cosmology often normalize 8πG to 1. This normalization results in the reduced Planck energy, defined as:
- natural number after zero.
- e, approximately equal to 2.718281828459045235360287…
- i, the imaginary unit such that i2 = -1.
- (square root of 2), the length of the diagonal of a square with unit sides, approximately equal to 1.414213562373095048801688.
- Giunti M. and Mazzola C. (2012), “Dynamical systems on monoids: Toward a general theory of deterministic systems and motion“. In Minati G., Abram M., Pessa E. (eds.), Methods, models, simulations and approaches towards a general theory of change, pp. 173-185, Singapore: World Scientific. ISBN 978-981-4383-32-5.
- Vladimir Igorevic Arnol’d “Ordinary differential equations“, various editions from MIT Press and from Springer Verlag, chapter 1 “Fundamental concepts“.
- I. D. Chueshov “Introduction to the Theory of Infinite-Dimensional Dissipative Systems” online version of first edition on the EMIS site .
- Roger Temam “Infinite-Dimensional Dynamical Systems in Mechanics and Physics” Springer Verlag 1988, 1997.
THEORY OF DYNAMICAL SYSTEMS AND GENERAL TRANSFORMATION. GROUPS WITH INVARIANT MEASURE. A. B. Katok, Ya. G. Sinai, and A. M. Stepin.
page 1= Printable PDF of this page only
page2 = Printable PDF of this page only
March 20 George Polya can rightly be called the father of problem solving in mathematics education. https://en.wikipedia.org/wiki/George_P%C3%B3lya
Notes: March 5, 2017
1. Virtual particles: https://en.wikipedia.org/wiki/Virtual_particle
2. Pertubation theory: https://en.wikipedia.org/wiki/Perturbation_theory_(quantum_mechanics)
3. Hamiltonian: https://en.wikipedia.org/wiki/Hamiltonian_(quantum_mechanics)
http://www.pnas.org/content/103/28/10612.full.pdf Packing, tiling, and covering with tetrahedra J. H. Conway* and S. Torquato
http://doye.chem.ox.ac.uk/research/cluster_structure.html Jonathan Doyle, Cambridge
On inflation, cosmological constant, and SUSY breaking Andrei Linde
On the problem of scale: a general theory of morphogenesis and normative policy signals for economic evolution
1.54 steradians = or ≠ a gap of 7.36∗
http://plato.stanford.edu/entries/spacetime-theories/ Absolute and Relational Theories of Space and Motion Nick Huggett <email@example.com> Carl Hoefer <firstname.lastname@example.org>
Dr. Claus Kiefer. Institut für Theoretische Physik Universität zu Köln … 0221 470-4300 (secretary). Fax: 0221 470-2189. E-mail: email@example.com
- Why Trust A Theory? from a conference, Reconsidering Scientific Methodology in Light of Modern Physics, 7-9 December, 2015
- Study algebraic phenomena inhabiting the murky boundary between finite and infinite with Alexandre Borovik
- Study the tension between intuitive infinitesimals and formal mathematical analysis with Mikhail G. Katz (and David Tall)
- Crossroads in the History of Mathematics and Mathematics Education edited by Bharath Sriraman, University of Montana
- Wendy Freedman, University of Chicago, once director of the Carnegie Observatories in Pasadena, California, and an expert on the Cepheid variable whereby a star pulsates radially, varying in both diameter and temperature and producing changes in brightness with a well-defined stable period and amplitude.
- Lewis Carroll‘s The Mad Gardener’s Song includes the lines “He thought he saw a Garden-Door / That opened with a key: / He looked again, and found it was / A double Rule of Three“
- Quantum Electrodynamics and Planck-Scale, Rainer Collier, 28 Sep 2017 arXiv:1710.00618v Institute of Theoretical Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
- A smooth function which is nowhere real analytic
- Peter Mohr <WidmaierMohr@t-online.de> Cross sections at sub-Coulomb energies
- homotopy group of SU(2)
- topological vacua
- abelian gauge group
- Yang–Mills theory
- BPST instanton after its discoverers
- Alexander Belavin
- Alexander Polyakov
- Albert S. Schwarz
- Yu. S. Tyupkin
- pure gauge at spatial infinity
- nonrenormalization theorems
- perturbation theory
- Equations of motion are grouped under three main types of motion:
- translations, rotations, oscillations
- Robin Hartshorne
- The idea of T-duality was first noted by Bala Sathiapalan in an obscure paper in 1987.
- Newtonian constant of gravitation
- Newtonian constant of gravitation over h-bar c
- Planck constant
- Planck constant in eV/Hz
- Planck length
- Planck mass On 20 May 2019, World Metrology Day, the world said goodbye to the original kilogram as the redefinition of the SI base units come into force.
- Planck mass energy equivalent in GeV
- Planck temperature
- Planck time
- reduced Planck constant
- reduced Planck constant in eV s
- reduced Planck constant times c in MeV fm
- speed of light in vacuum
- vacuum electric permittivity
- vacuum magnetic permeabilityy
Period doubling bifurcation
- …a fine string is maintained in transverse vibration by connecting one of its extremities with the vibrating prong of a massive tuning-fork, the direction of motion of the point of attachment being parallel to the length of the string …the string may settle down into a state of permanent and vigorous vibration whose period is double that of the point of attachment.
- Nonlinear dynamics: chaos are period doubling… intermittency, horseshoes and homoclinic orbits.
- chaos is always regarded as intrinsic randomicity of determinate dynamical systems.
- …attractor undergoes a period–doubling bifurcation which converts it from a period-1 to a period-2 attractor. This bifurcation is indicated by the forking of the curve
- The -coordinate of the Poincaré section of a time-asymptotic orbit plotted against the quality-factor .
- Poincairé (1) bifurcation theory (1885), The Future of Mathematics (PDF) (Weil) (Wikipedia), Sphere, Homology Sphere (Evelyn Lamb, 2017)
- On the scaling structure for period doubling (PDF), Garrett Birkhoff, Marco Martens & Charles Tresser, Astérisque, Société mathématique de France, 286 (2003), p. 167-186
- Connecting period-doubling cascades to chaos, Evelyn Sander, James A. Yorke 17 Feb 2010
- ventricular fibrillation (VF) is an application of period doubling
Fourier transform transforms
- Fourier, Quantum Electrodynamics and Planck-Scale,Rainer Collier, 28 Sep 2017 arXiv:1710.00618vInstitute of Theoretical Physics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany
- Euler, https://www.youtube.com/watch?v=IUTGFQpKaPU Yet, we ask, what are these people thinking when they voted for these equations:
- The Dirac equation. 27,061 votes. 34%
- Euler’s identity. 13,745 votes. 17%
- Pi. 9,937 votes. 13%
- The wave equation. 3,761 votes. 5%
- Riemann’s formula. 4,271 votes. 5%
- The Euler-Lagrange equation. 3,176 votes. 4%
- The Yang-Baxter equation. 1,592 votes. 2%
- Bayes’s theorem. 2,958 votes. 4%
The logic of quantum indeterminacy
- Doubly Special Relativity theory (DSR), loop quantum gravitation, the introduction of non-commutative geometries, the use of specially deformed Lorentz algebras, as well as several generalized uncertainty principles (GUP) in which the Planck momentum occurs.