User:Gandalf61/sandbox

test

$a^{b}$ $a^{b}a^{b}$ $a^{b}=a^{b}$ $a^{b}\to a^{b}$ $a^{b}\rightarrow a^{b}$ $a^{b}c^{d}$ $a^{b}=c^{d}$ $a^{b}\to c^{d}$ $a^{b}\rightarrow c^{d}$

Mathematics good article nominations

My nominations:

Article (Review)	Nominated on	Review started on	Review started by	Review completed on	Result	Reasons
Complete icosahedron (Review)	5 August 2009	9 October 2009	Edge3	12 October 2009	Pass
Jeep problem (Review)	5 August 2009	13 August 2009	Protonk	7 September 2009	Fail	3 major issues including lack of details of proof that solution described in article is optimal. "requires a substantial rewrite"
Mathematics and art (Review)	5 August 2009	13 August 2009	Protonk	7 September 2009	Fail	4 major issues including lack of comprehensiveness. "has a way to go before reaching GA standards of comprehensiveness & accuracy"
Penrose tiling (Review)	5 August 2009	13 Nov 2009	Wizardman	27 Nov 2009	Pass
Proof without words (Review)	5 August 2009	10 August 2009	Edge3	21 August 2009	Fail	Too short; insufficiently broad in its coverage.

Other recent mathematics GA nominations:

Article (Review)	Nominated on	Nominated by	Review started on	Review started by	Review completed on	Result	Remarks
Hilbert space (Review)	9 September 2009	Sławomir Biały	10 September 2009	Jakob.scholbach	14 September 2009	Pass
Area of a disk	6 October 2009	Classicalecon	3 December 2009	Jimfbleak	9 December 2009	Fail	"... it needs: Work to get the language to a better form, change to inline references, closure of the proofs where presented and needs to be written to be more comprehensible to the target audience"
Four color theorem (Review)	6 October 2009	Classicalecon	7 October 2009	Pyrotec	29 October 2009	Fail	Closing remarks "It needs, in my opinion, only page numbers for book citations to make GA"

van Eck's sequence

In recreational mathematics van Eck's sequence is an integer sequence that starts with a₀=0. For n>=0, if there exists an m < n such that a_m = a_n, take the largest such m and set a_n+1 = n-m; otherwise a_n+1 = 0. Thus the first occurrence of an integer in the sequence is followed by a 0, and the second and subsequence occurrences are followed by the size of the gap between the most recent two occurrences.

The first few terms of the sequence are:

0, 0, 1, 0, 2, 0, 2, 2, 1, 6, 0, 5, 0, 2, 6, 5, 4, 0, 5 ...

The sequence was named by Neil Sloane after Jan Ritsema van Eck, who contributed it to the On-Line Encyclopedia of Integer Sequences in 2010.

Properties

It is know that the sequence contains infinitely many zeros and that it is unbounded.

It is conjectured, but not proved, that the sequence contains every positive integer, and that every pair of non-negative integers apart from (1,1) appears as consecutive terms in the sequence.

References

[[Category:Integer sequences]] [[Category:Unsolved problems in mathematics]] {{math-stub}}

Zapotec civilization - Zapotec writing

The Zapotecs developed a calendar and a logosyllabic system of writing that used a separate glyph to represent each of the syllables of the language. This writing system is one of several candidates thought to have been the first writings system of Mesoamerica and the predecessor of the writing systems developed by the Maya, Mixtec, and Aztec civilizations.

The earliest examples of Zapotec writing are on stone monuments carved with images of slain and mutilated captives and brief inscriptions. The majority of these monuments were found in Monte Albán, but one was found in nearby San José Mogote. The San José Mogote monument shows what appears to be the stylised image of a disembowelled captive above the inscription "1 Earthquake". This is a date in the Zapotec calendar, but as birth dates were used as names in Zapotec culture, it may also be the name of the captive. This inscription was thought to be dated to 500 BC, but there is now some disagreement about this date. Other inscribed monuments have been reliably dated to the period known as Monte Albán I (400 to 200 BCE), which still makes them the some of the earliest texts in Mesoamerica.

Zapotec script is read in columns from top to bottom. Its execution is somewhat cruder than that of the later Classic Maya. Zapotec used a vigesimal bar-and-dot numeral system and a calendar which combined the 365-day solar calendar (called yza) and the 260-day sacred calendar (called piye); these are similar to other Mesoamerican numeral systems and Mesoamerican calendars. This allows some Zapotec inscriptions to be recognized as dates and numbers, but the majority of Zapotec writing is still poorly understood. There are a relatively small number of texts available to researchers (how many ?) and most of these texts are no more than 10 glyphs in length. In addition, the underlying language itself is not certain, because the first European record of the Zapotec language dates from no earlier than the 16th century, and the ancient form of Zapotec from a thousand years earlier is not documented at all.

Mesoamerican writing systems:

Another candidate for earliest writing system in Mesoamerica is the writing system of the Zapotec culture. Rising in the late Pre-Classic era after the decline of the Olmec civilization, the Zapotecs of present day Oaxaca built an empire around Monte Alban. On a few monuments at this archaeological site, archaeologists have found extended text in a glyphic script. Some signs can be recognized as calendric information but the script as such remains undeciphered. Read in columns from top to bottom, its execution is somewhat cruder than that of the later Classic Maya and this has led epigraphers to believe that the script was also less phonetic than the largely syllabic Mayan script. These are, however, speculations.

The earliest known monument with Zapotec writing is a "Danzante" stone, officially known as Monument 3, found in San Jose Mogote, Oaxaca. It has a relief of what appears to be dead and bloodied captive with two glyphic signs between his legs, probably his name. First dated to 500–600 BCE, this was earlier considered the earliest writing in Mesoamerica. However doubts have been expressed as to this dating and the monument may have been reused. The Zapotec script went out of use only in the late Classic period.

Zapotec, ancientscripts.com:

The Zapotec script has great antiquity, being one of the earliest writing systems in Mesoamerica. The first examples of Zapotec writing are in the form of danzante slabs, stone monuments carved with the image of slain and mutilated captives and a brief inscriptions. The majority of danzantes are found in Monte Albán, but one is found in the nearby town of San José Mogote. While once the San José Mogote danzante was thought to be the most ancient Zapotec inscription (dated to 500 BC), there is now considerable argument against this date. However, regardless of the status of the San José Mogote slab, danzantes are generally dated to the period known as Monte Albán I (400 to 200 BCE), still making them the some of the earliest texts in Mesoamerica

Compared to other major Mesoamerican writing systems, Zapotec is still poorly understood. First of all, the underlying language itself presents a problem. The first European record of the Zapotec language dates from no earlier than the 16th century, and the ancient form of Zapotec from a thousand years earlier is not documented at all. Efforts are underway to reconstruct this "proto-Zapotec" language from the modern Zapotec languages, but such tasks take years to refine. Another factor contributing to the lack of progress in Zapotec decipherment is the relatively meager number of texts available to researchers. On top of this problem, the known texts are usually very short, no more than 10 glyphs, making it difficult to discover grammatical structures like sentences.

What we do know about Zapotec derive mostly from comparing with similar features in other Mesoamerican writing systems. Like other Mesoamerican scripts, Zapotec used the bar-and-dot notation to represent numbers. In terms of time-keeping, the Zapotecs employed the 365-day solar calendar (called yza) and the 260-day sacred calendar (called piye).

Unlike later Mixtec and Aztec scripts, Zapotec was much more textual, possibly capable of representing sentences. Zapotec very well could be a logophonetic writing system, but likely not as extensively phonetic as epi-Olmec or Maya. For instance, the number of non-calendrical glyphs range between 80 to 90, making it possible that Zapotec contained a mixture of logograms and phonograms. Also, the way signs are joined into compounds might indicate affixes, possibly spelled out phonetically, attached to a root logogram to form a noun or verb phrase. Some tantalizing clues come from Javier Urcid, whose studies have shown that possible instances of homophonic principle, or "rebus writing", are used in naming personages. He also demonstrated that grammatical constructs (like short sentences) might be present in some monuments (which you will see below).

The Zapotec system very likely was the source of the Mixtec system, which is characterized by a highly pictorial and minimal set of logograms, and by the use of the rebus principle for rough phonetic spelling of names. In fact, the Zapotec writing system started to be replaced by the early form of Mixtec script by the 10th century CE. When the Spanish conquistadores arrived in Oaxaca in the 16th century CE, the Zapotec script has long been forgotten, although the Zapotec language continues to be spoken to this date.

modular group

Applications to Number Theory

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

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.

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.

Congruence Subgroups

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

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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

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Scraps

Penrose tiling: Construction by L-system

The rhombus (P3) Penrose tiling can be drawn using the following L-system:

Evolution of L-system for n=1,2,3,4,5 and 6.

variables:	1 6 7 8 9 [ ]
constants:	+ −
start:	[7]++[7]++[7]++[7]++[7]
rules:	6 → 81++91−−−−71[−81−−−−61]++
	7 → +81−−91[−−−61−−71]+
	8 → −61++71[+++81++91]−
	9 → −−81++++61[+91++++71]−−71
	1 → (eliminated at each iteration)
angle:	36º

Here 1 means "draw forward", + means "turn left by angle", and − means "turn right by angle" (see turtle graphics). The [ means save the present position and direction to restore them when corresponding ] is executed. The symbols 6, 7, 8 and 9 do not correspond to any action; they are there only to produce the correct curve evolution.

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