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Power law

Power law
An example power-law graph, being used to demonstrate ranking of popularity. To the right is the long tail, and to the left are the few that dominate (also known as the 80–20 rule). In statistics, a power law is a functional relationship between two quantities, where a relative change in one quantity results in a proportional relative change in the other quantity, independent of the initial size of those quantities: one quantity varies as a power of another. Empirical examples of power laws[edit] Properties of power laws[edit] Scale invariance[edit] One attribute of power laws is their scale invariance. , scaling the argument by a constant factor causes only a proportionate scaling of the function itself. That is, scaling by a constant simply multiplies the original power-law relation by the constant . and , and the straight-line on the log-log plot is often called the signature of a power law. Lack of well-defined average value[edit] A power-law has a well-defined mean over only if for where , and .

Table of mathematical symbols When reading the list, it is important to recognize that a mathematical concept is independent of the symbol chosen to represent it. For many of the symbols below, the symbol is usually synonymous with the corresponding concept (ultimately an arbitrary choice made as a result of the cumulative history of mathematics), but in some situations a different convention may be used. For example, depending on context, the triple bar "≡" may represent congruence or a definition. Each symbol is shown both in HTML, whose display depends on the browser's access to an appropriate font installed on the particular device, and in TeX, as an image. Guide[edit] This list is organized by symbol type and is intended to facilitate finding an unfamiliar symbol by its visual appearance. Basic symbols: Symbols widely used in mathematics, roughly through first-year calculus. Basic symbols[edit] Symbols based on equality sign[edit] Symbols that point left or right[edit] Brackets[edit] Other non-letter symbols[edit]

The Small-World Phenomenon: An Algorithmic Perspective 1 Jon Kleinberg 2 Abstract: Long a matter of folklore, the ``small-world phenomenon'' -- the principle that we are all linked by short chains of acquaintances -- was inaugurated as an area of experimental study in the social sciences through the pioneering work of Stanley Milgram in the 1960's. This work was among the first to make the phenomenon quantitative, allowing people to speak of the ``six degrees of separation'' between any two people in the United States. Since then, a number of network models have been proposed as frameworks in which to study the problem analytically. But existing models are insufficient to explain the striking algorithmic component of Milgram's original findings: that individuals using local information are collectively very effective at actually constructing short paths between two points in a social network. The Small-World Phenomenon. Milgram's basic small-world experiment remains one of the most compelling ways to think about the problem. The Present Work. .

THE LAST DAYS OF THE POLYMATH People who know a lot about a lot have long been an exclusive club, but now they are an endangered species. Edward Carr tracks some down ... From INTELLIGENT LIFE Magazine, Autumn 2009 CARL DJERASSI can remember the moment when he became a writer. His wife, Diane Middlebrook, thought it was a ridiculous idea. Even at 85, slight and snowy-haired, Djerassi is a det­ermined man. Eventually Djerassi got the bound galleys of his book. Diane Middlebrook died of cancer in 2007 and, as Djerassi speaks, her presence grows stronger. Carl Djerassi is a polymath. His latest book, “Four Jews on Parnassus”, is an ima­gined series of debates between Theodor Adorno, Arnold Schönberg, Walter Benjamin and Gershom Scholem, which touches on art, music, philosophy and Jewish identity. The word “polymath” teeters somewhere between Leo­nardo da Vinci and Stephen Fry. “To me, promiscuity is a way of flitting around. Djerassi is right to be suspicious of flitting. Young’s achievements are staggering.

Clustering coefficient In graph theory, a clustering coefficient is a measure of the degree to which nodes in a graph tend to cluster together. Evidence suggests that in most real-world networks, and in particular social networks, nodes tend to create tightly knit groups characterised by a relatively high density of ties; this likelihood tends to be greater than the average probability of a tie randomly established between two nodes (Holland and Leinhardt, 1971;[1] Watts and Strogatz, 1998[2]). Two versions of this measure exist: the global and the local. The global version was designed to give an overall indication of the clustering in the network, whereas the local gives an indication of the embeddedness of single nodes. Global clustering coefficient[edit] The global clustering coefficient is based on triplets of nodes. Watts and Strogatz defined the clustering coefficient as follows, "Suppose that a vertex has neighbours; then at most edges can exist between them (this occurs when every neighbour of ). over all

An Introduction to Wavelets: What Do Some Wavelets Look Like? W hat do S ome W avelets L ook L ike? Wavelet transforms comprise an infinite set. The different wavelet families make different trade-offs between how compactly the basis functions are localized in space and how smooth they are. Some of the wavelet bases have fractal structure. This figure was generated using the WaveLab command: wave=MakeWavelet(2, -4, 'Daubechies', 4, 'Mother', 2048). Within each family of wavelets (such as the Daubechies family) are wavelet subclasses distinguished by the number of coefficients and by the level of iteration. The number next to the wavelet name represents the number of vanishing moments (A stringent mathematical definition related to the number of wavelet coefficients) for the subclass of wavelet. wave = MakeWavelet(2,-4,'Daubechies',6,'Mother', 2048); wave = MakeWavelet(2,-4,'Coiflet',3,'Mother', 2048); wave = MakeWavelet(0,0,'Haar',4,'Mother', 512); wave = MakeWavelet(2,-4,'Symmlet',6,'Mother', 2048); Or click HERE to download a PDF version (360 Kbytes)

Exclusive: How Google's Algorithm Rules the Web | Wired Magazine Want to know how Google is about to change your life? Stop by the Ouagadougou conference room on a Thursday morning. It is here, at the Mountain View, California, headquarters of the world’s most powerful Internet company, that a room filled with three dozen engineers, product managers, and executives figure out how to make their search engine even smarter. This year, Google will introduce 550 or so improvements to its fabled algorithm, and each will be determined at a gathering just like this one. You might think that after a solid decade of search-market dominance, Google could relax. Still, the biggest threat to Google can be found 850 miles to the north: Bing. Team Bing has been focusing on unique instances where Google’s algorithms don’t always satisfy. Even the Bingers confess that, when it comes to the simple task of taking a search term and returning relevant results, Google is still miles ahead. Google’s response can be summed up in four words: mike siwek lawyer mi.

Primality Proving 2.1: Finding very small primes For finding all the small primes, say all those less than 10,000,000,000; one of the most efficient ways is by using the Sieve of Eratosthenes (ca 240 BC): Make a list of all the integers less than or equal to n (greater than one) and strike out the multiples of all primes less than or equal to the square root of n, then the numbers that are left are the primes. (See also our glossary page.) For example, to find all the odd primes less than or equal to 100 we first list the odd numbers from 3 to 100 (why even list the evens?) The first number is 3 so it is the first odd prime--cross out all of its multiples. Now the first number left is 5, the second odd prime--cross out all of its multiples. This method is so fast that there is no reason to store a large list of primes on a computer--an efficient implementation can find them faster than a computer can read from a disk. To find individual small primes trial division works well.

PageRank Algorithm used by Google Search to rank web pages PageRank (PR) is an algorithm used by Google Search to rank web pages in their search engine results. It is named after both the term "web page" and co-founder Larry Page. PageRank works by counting the number and quality of links to a page to determine a rough estimate of how important the website is. Currently, PageRank is not the only algorithm used by Google to order search results, but it is the first algorithm that was used by the company, and it is the best known.[2][3] As of September 24, 2019, all patents associated with PageRank have expired.[4] Description[edit] A PageRank results from a mathematical algorithm based on the webgraph, created by all World Wide Web pages as nodes and hyperlinks as edges, taking into consideration authority hubs such as cnn.com or mayoclinic.org. History[edit] The eigenvalue problem behind PageRank's algorithm was independently rediscovered and reused in many scoring problems. Algorithm[edit] where At

Numbers: Facts, Figures & Fiction Click on cover for larger image Numbers: Facts, Figures & Fiction by Richard Phillips. Published by Badsey Publications. See sample pages: 24, 82, 103. Order the book direct from Badsey Publications price £12. In Australia it is sold by AAMT and in the US by Parkwest. For those who need a hardback copy, a limited number of the old 1994 hardback edition are still available. Have you ever wondered how Room 101 got its name, or what you measure in oktas? This new edition has been updated with dozens of new articles, illustrations and photographs. Some press comments – "This entertaining and accessible book is even more attractive in its second edition..." – Jennie Golding in The Mathematical Gazette "...tangential flights into maths, myth and mystery..." – Vivienne Greig in New Scientist ... and on the first edition – Contents –

Nash equilibrium In game theory, the Nash equilibrium is a solution concept of a non-cooperative game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only their own strategy.[1] If each player has chosen a strategy and no player can benefit by changing strategies while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitutes a Nash equilibrium. The reality of the Nash equilibrium of a game can be tested using experimental economics method. Stated simply, Amy and Will are in Nash equilibrium if Amy is making the best decision she can, taking into account Will's decision while Will's decision remains unchanged, and Will is making the best decision he can, taking into account Amy's decision while Amy's decision remains unchanged. Applications[edit] History[edit] The Nash equilibrium was named after John Forbes Nash, Jr. Let .

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