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[edit] Example of spigot algorithm

[edit] Example

This example illustrates the working of a spigot algorithm by calculating the binary digits of the natural logarithm of 2 (sequence A068426 in OEIS) using the identity

\ln(2)=\sum_{k=1}^{\infty}\frac{1}{k2^k}\, .

To start calculating binary digits from, say, the 8th place we multiply this identity by 27:

2^7\ln(2) =2^7\sum_{k=1}^{\infty}\frac{1}{k2^k}\, .

We then divide the infinite sum into a "head", in which the exponents of 2 are greater than or equal to zero, and a "tail", in which the exponents of 2 are negative:

2^7\ln(2) =\sum_{k=1}^{7}\frac{2^{7-k}}{k}+\sum_{k=8}^{\infty}\frac{1}{k2^{k-7}}\, .

We are only interested in the fractional part of this value, so we can replace each of the terms in the "head" by

\frac{2^{7-k} \mod k}{k}\, .

Calculating each of these terms and adding them to a running total where we again only keep the fractional part, we have:

k A = 27-k B = A mod k C = B / k Sum of C mod 1
1 64 0 0 0
2 32 0 0 0
3 16 1 1/3 1/3
4 8 0 0 1/3
5 4 4 4/5 2/15
6 2 2 1/3 7/15
7 1 1 1/7 64/105

We add a few terms in the "tail", noting that the error introduced by truncating the sum is less than the final term:

k D = 1/k2k-7 Sum of D Maximum error
8 1/16 1/16 1/16
9 1/36 13/144 1/36
10 1/80 37/360 1/80

Adding the "head" and the first few terms of the "tail" together we get:

2^7\ln(2)\mod{1} \approx \frac{64}{105}+\frac{37}{360}=0.10011100 \cdots_{2} + 0.00011010 \cdots_{2} = 0.1011 \cdots_{2}\, ,

so the 8th to 11th binary digits in the binary expansion of ln(2) are 1, 0, 1, 1.

The same approach can be used to calculate digits of the binary expansion of ln(2) starting from an arbitary nth position. The number of terms in the "head" sum increases linearly with n, but the complexity of each term only increases with the logarithm of n if an efficient method of modular exponentiation is used. The precision of calculations and intermediate results and the number of terms taken from the "tail" sum are all independent of n, and only depend on the number of binary digits that are being calculated - single precision arithmetic can be used to calculate around 12 binary digits, regardless of the starting position.

[edit] Timeline of the future in forecasts - sorted by date

    • 2010 10 petaFLOPS supercomputer - NEC, Tokyo Institute of Technology[3]
    • 2012 10 petaFLOPS supercomputer- Riken[4]
    • 2013 - voice control replace keyboard/mouse interface for 30% of routine tasks - TechCast[5]
    • 2014 All communications are IP-based - Paul Mockapetris, inventor of the DNS system, 2004[6]
    • 2015 - one third of US fighting strength will be composed of robots - US Department of Defense, 2006[7]
    • 2018 - robots will routinely carry out surgery, South Korea government 2007[8]
    • 2019 - $1,000 computer will match the processing power of the human brain - Ray Kurzweil[9]
    • 2030 - robots capable of performing at human level at most manual jobs Marshall Brain[13]
    • 2034 - robots (home automation systems) performing most household tasks, Helen Greiner, Chairman of iRobot[14]
    • 2035 - first completely autonomous robot soldiers in operation - US Department of Defense, 2006[7]
    • 2038 - first completely autonomous robot flying car in operation - US Department of Technology, 2007[7]
    • 2050 - computer costing a few hundred pounds will have the capacity of the human mind - Hans Moravec[16]
    • 2055 - $1,000 computer will match the processing power of all human brains on Earth - Ray Kurzweil[9]
  1. ^ Robots get bookish in libraries, BBC News
  2. ^ Trials for 'bionic' eye implants, BBC News
  3. ^ NEC claims 10-Petaflop supercomputing breakthrough
  4. ^ Taking on the Challenge of a 10-Petaflop Computer, Riken News April 2006
  5. ^ a b Latest Forecast Results, TechCast
  6. ^ Net pioneer predicts web future, BBC News
  7. ^ a b c Launching a new kind of warfare, Guardian Online
  8. ^ Robotic age poses ethical dilemma, BBC News
  9. ^ a b The Coming Merging of Mind and Machine, Ray Kurzweil
  10. ^ The Technology Lane on the Road to a Zettaflops
  11. ^ a b Interview with Arthur C. Clarke, November 30, 2001
  12. ^ Cory Doctorow, "Thought Experiments: When the Singularity is More Than a Literary Device: An Interview with Futurist-Inventor Ray Kurzweil"
  13. ^ 2003 Robotic Nation, Marshall Brain
  14. ^ Interview: Helen Greiner, Chairman and Cofounder of iRobot, Corp
  15. ^ *Kurzweil, Raymond (2005), The Singularity Is Near, New York: Viking, ISBN 0-670-03384-7 
  16. ^ Robots rule OK?, BBS News

Not yet sorted...

[edit] Culture and leisure

[edit] Demographics

[edit] Energy

[edit] Environment

[edit] Nanotechnology

[edit] Politics and economics

  • World economic growth
    • 2025 - one billion dollar-millionaires worldwide - James Canton, The Extreme Future[17]
    • 2050 - China's GDP exceeds that of US - Goldman Sachs,[18] Price Waterhouse Coopers[19]

[edit] Transportation

[edit] Space

[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 Δτ 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. Δτ=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 Δτ 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