 # P versus NP problem Diagram of complexity classes provided that P≠NP. The existence of problems within NP but outside both P and NP-complete, under that assumption, was established by Ladner's theorem. The P versus NP problem is a major unsolved problem in computer science. Informally, it asks whether every problem whose solution can be quickly verified by a computer can also be quickly solved by a computer. It was essentially first mentioned in a 1956 letter written by Kurt Gödel to John von Neumann. Gödel asked whether a certain NP complete problem could be solved in quadratic or linear time. The precise statement of the P=NP problem was introduced in 1971 by Stephen Cook in his seminal paper "The complexity of theorem proving procedures" and is considered by many to be the most important open problem in the field. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US\$1,000,000 prize for the first correct solution. Context

Time complexity Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, where an elementary operation takes a fixed amount of time to perform. Thus the amount of time taken and the number of elementary operations performed by the algorithm differ by at most a constant factor. Since an algorithm's performance time may vary with different inputs of the same size, one commonly uses the worst-case time complexity of an algorithm, denoted as T(n), which is defined as the maximum amount of time taken on any input of size n. Time complexities are classified by the nature of the function T(n). For instance, an algorithm with T(n) = O(n) is called a linear time algorithm, and an algorithm with T(n) = O(2n) is said to be an exponential time algorithm. Table of common time complexities The following table summarizes some classes of commonly encountered time complexities. Constant time Logarithmic time Polylogarithmic time and .

Computational complexity theory Computational complexity theory is a branch of the theory of computation in theoretical computer science and mathematics that focuses on classifying computational problems according to their inherent difficulty, and relating those classes to each other. A computational problem is understood to be a task that is in principle amenable to being solved by a computer, which is equivalent to stating that the problem may be solved by mechanical application of mathematical steps, such as an algorithm. A problem is regarded as inherently difficult if its solution requires significant resources, whatever the algorithm used. The theory formalizes this intuition, by introducing mathematical models of computation to study these problems and quantifying the amount of resources needed to solve them, such as time and storage. Closely related fields in theoretical computer science are analysis of algorithms and computability theory. Computational problems Problem instances Turing machine

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 –

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.

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.

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 Properties of power laws Scale invariance 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 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 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 Symbols based on equality sign Symbols that point left or right Brackets Other non-letter symbols

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)

Specifications for ACME Klein Bottles Specifications for ACME Klein Bottles No two Acme Klein Bottles are alike! Since Acme glassblowers individually hand craft each one, dimensions will vary and you may find occasional bubbles or streaks in the glass. WARNING! Although originating in the 4th dimension, Acme Klein Bottles are immersed (not embedded) in 3 dimensions, using special techniques known to students of advanced topology. We construct our glass Acme Klein Bottles from materials which resist both oxidation and reduction. What with the decay of protons, Acme recognizes that our products have a finite lifetime and may not outlast your physical universe. A Klein Bottle has zero volume, so we suggest that you do not use it as a personal flotation device. ACME KLEIN BOTTLE - Where there's one side to every problem

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