User:Gandalf61/Sandbox

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Sandbox stuff

1 Early universe
- 1.1 Quark epoch
2 modular group
3 List of chaotic maps
4 Scraps
- 4.1 Plastic number
- 4.2 Georgy Fedoseevich Voronoy

[edit] Early universe

Planck epoch : earlier than 10^-43s

10^-43s : Gravity separates from other forces

Grand unification epoch - 10^-43s to 10^-35s

10^-35s : Strong force separates from other forces; X and Y bosons no longer created

Inflationary epoch : 10^-35s to 10^-32s

10^-32s : Inflation ends; quark-gluon plasma created by reheating.

Electroweak epoch : 10^-32s to 10^-12s

10^-12s : Weak force separates from other forces; W and Z bosons no longer created; weak force becomes short range force

Quark epoch : 10^-12s to 10^-6s - universe is quark-gluon plasma

10^-6s : Quarks become confined in hadrons

Hadron epoch : 10^-6s to 1s - as temperature falls, new hadron-antihadron pairs no longer created; general annihilation of hadron-antihadron pairs; remaining heavier hadrons decay leaving only protons and neutrons

1s : Neutrions cease to interact with other particles due to falling density

Lepton epoch : 1s to 3s

3s : Electron-position pairs no longer created; general annihilation of electron-positron pairs leaving one electron per proton; universe now dominated by photons

Photon epoch : 3s to 300,000 years - Big Bang nucleosynthesis occurs between 100s and 300s

300,000 years : atoms form in recombination; photons no longer react significantly with matter; universe is transparent to photons; origin of cosmic microwave background

[edit] Quark epoch

In physical cosmology the quark epoch was the period in the evolution of the early universe when the fundamental interactions of gravitation, electromagnetism, the strong interaction and the weak interaction had taken their present forms, but the temperature of the universe was still too high to allow quarks to bind together to form hadrons. The quark epoch began approximately 10^-12 seconds after the Big Bang, when the preceding electroweak epoch ended as the electroweak interaction separated into the weak interaction and electromagnetism. During the quark epoch the universe was filled with a dense, hot quark-gluon plasma, containing quarks, leptons and their antiparticles. Collisions between particles were too energetic to allow quarks to combine into mesons or baryons. The quark epoch ended when the universe was about 10^-6 seconds old, when the average energy of particle interactions had fallen below the binding energy of hadrons. The following period, when quarks were confined within hadrons, is known as the hadron epoch.

Allday, Jonathan (2002). Quarks, Leptons and the Big Bang. Second Edition. ISBN 978-0750308069.
Physics 175: Stars and Galazies - The Big Bang, Matter and Energy; Ithaca College, New York

[edit] modular group

[edit] Applications to Number Theory

[...include here at least some mention of quadratic forms, the fundamental domain (modular curve) and modular forms...]

[edit] Relationship to Lattices

The lattice Δ_τ is the set of points in the complex plane generated by the values 1 and τ (where τ is not real). So

$\Delta_{\tau}=\{m+n\tau : m,n \in Z\}$

Clearly τ and -τ generate the same lattice i.e. Δ_τ=Δ_-τ. In addition, τ and τ' generate the same lattice if they are related by a fractional linear transformation that is a member of the modular group.

[edit] Relationship to Quadratic Forms

The set of values taken by the positive definite binary quadratic form am²+bmn+cn² is related to the lattice Δ_τ as follows :-

$\{ am^2+bmn+cn^2:m,n \in Z \} = \{a|z|^2 : z \in \Delta_\tau \}$

where τ is chosen such that Re(τ)=b/2a and |τ|² = c/a.

[edit] Congruence Subgroups

[...brief mention and definition of congruence subgroup, this really deserves its own article independent of Γ]

[edit] List of chaotic maps

Name	Time	Space	Dimension
2x mod 1 map	Discrete	Real	1
Arnold's Cat map	Discrete	Real	2
Baker's map	Discrete	Real	2
Boundary map
Bogdanov map	Discrete	Real	2
Chossat-Golubitsky symmetry map
Cantor set	Discrete	Discrete	1
Cellular automata	Discrete	Discrete	1 or 2
Circle map
Cob Web map
Complex map
Complex Cubic map
Degenerate Double Rotor map
Double Rotor map
Duffing map	Discrete	Real	2
Duffing equation	Continuous	Real	2
Gauss map
Generalized Baker map
Gingerbreadman map	Discrete	Real	2
Gumowski/Mira map
Harmonic map
Hénon map	Discrete	Real	2
Hénon with 5th order polynomial
Hitzl-Zele map
Horseshoe map	Discrete	Real	2
Hyperbolic map
Ikeda map
Inclusion map
Julia map	Discrete	Complex	1
Koch curve	Discrete	Discrete	2
Kaplan-Yorke map	Discrete	Real	2
Langton's ant	Discrete	Discrete	2
linear map on unit square
Logistic map	Discrete	Real	1
Lorenz attractor	Continuous	Real	3
Lorenz system's Poincare Return map
Lozi map	Discrete	Real	2
Lyapunov fractal
Mandelbrot map	Discrete	Complex	1
Menger sponge	Discrete	Discrete	2
Mitchell-Green gravity set	Discrete	Real	2
Nordmark truncated map
Piecewise Linear map
Pullback map
Pulsed Rotor & standard map
Quadratic map
Quasiperiodicity map
Rabinovich-Fabrikant equations	Continuous	Real	3
Random Rotate map
Rössler map	Continuous	Real	3
Sierpinski carpet	Discrete	Discrete	2
Symplectic map
Tangent map
Tent map	Discrete	Real	1
Tinkerbell map
Triangle map
Van der Pol map
Zaslavskii map	Discrete	Real	2
Zaslavskii rotation map

[edit] Scraps

[edit] Plastic number

$\sqrt[3]{\frac{1}{2}+\frac{1}{6}\sqrt{\frac{23}{3}}}+\sqrt[3]{\frac{1}{2}-\frac{1}{6}\sqrt{\frac{23}{3}}}$

THE PLASTIC NUMBER AND CONCLUSIONS I first came across Padovan, Hans Van der Laan and the Plastic Number after reading Ian Stewart's column in the Scientific American, which was a successor to Martin Gardner's one [Stewart 1996]. I tried to gain some insight from Padovan's earlier book on Van der Laan's work [Padovan 1994]. Van der Laan found that he could not make the golden section work in three dimensions and worked from first principles on the mathematics to achieve the effect he wanted.

[edit] Georgy Fedoseevich Voronoy

Born: 28 April 1868 in Zhuravka, Poltava guberniya, Russia (now Ukraine) Died: 20 Nov 1908 in Warsaw, Poland

Georgy Voronoy studied at the gymnasium in Priluki graduating in 1885. He then entered the University of St Petersburg, joining the Faculty of Physics and Mathematics.

After graduating from St Petersburg in 1889, Voronoy decided to remain there and work for his teaching qualification. He was awarded a Master's Degree in 1894 for a dissertation on the algebraic integers associated with the roots of an irreducible cubic equation.

Voronoy lectured at Warsaw University, being appointed professor of pure mathematics there. He wrote his doctoral thesis on algorithms for continued fractions which he submitted to the University of St Petersburg. In fact both Voronoy's master's thesis and his doctoral thesis were of such high quality that they were awarded the Bunyakovsky prize by the St Petersburg Academy of Sciences.

Later Voronoy worked on the theory of numbers, in particular he worked on algebraic numbers and the geometry of numbers. He extended work by Zolotarev and his work was the starting point for Vinogradov's investigations. His methods were also used by Hardy and Littlewood.

In 1904 Voronoy attended the Third International Congress of Mathematicians at Heidelberg. There he met Minkowski and they discovered that they were each working on similar topics.

The concept of Voronoi diagrams first appeared in works of Descartes as early as 1644. Descartes used Voronoi-like diagrams to show the disposition of matter in the solar system and its environs.

The first man who studied the Voronoi diagram as a concept was a German mathematician G. L. Dirichlet. He studied the two- and three dimensional case and that is why this concept is also known as Dirichlet tessellation. However it is much better known as a Voronoi diagram because another German mathematician M. G. Voronoi in 1908 studied the concept and defined it for a more general n-dimensional case.

Very soon after it was defined by Voronoi it was developed independently in other areas like meteorology and crystalography. Thiessen developed it in meteorology in 1911 as an aid to computing more accurate estimates of regional rainfall averages. In the field of crystalography German researchers dominated and Niggli in 1927 introduced the term Wirkungsbereich (area of influence) as a reference to a Voronoi diagram.

http://www.comp.lancs.ac.uk/~kristof/research/notes/voronoi/voronoi.pdf

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