PlanetMath - PlanetMath PlanetMath is a free, collaborative, online mathematics encyclopedia. The emphasis is on peer review, rigour, openness, pedagogy, real-time content, interlinked content, and community-drivenness. PlanetMath is intended to be a comprehensive online encyclopedia of mathematics. The project is located at the Digital Library Research Lab at Virginia Tech. PlanetMath was started when a popular free online math encyclopedia mathworld.com was taken offline by an court injunction as a result of the CRC Press lawsuit against the company Wolfram Research Inc and its author and employee Eric Weisstein. PlanetMath uses the same copyleft GFDL license used by Wikipedia. An author who starts a new article becomes the owner of that article; he or she may then choose to grant editing rights to other individuals or groups. All.
Jacobson radical - the main diagonal. If K is a field and R = K[[X1,...,Xn]] is a ring of formal power series, then J(R) consists of those power series whose constant term is zero. More generally: the Jacobson radical of every local ring consists precisely of the ring's non-units. Start with a finite quiver Γ and a field K and consider the quiver algebra KΓ (as described in the quiver article). The Jacobson radical of this ring is generated by all the paths in Γ of length ≥ 1. some more examples of non-trivial Jacobson radicals would be nice. Rings of continuous functions? Endomorphism rings? Properties Unless R is the trivial ring {0}, the Jacobson radical is always a proper ideal in R. If R is commutative and finitely generated, then J(R) is equal.
Initial object - the category of all small categories with functors as morphisms has the empty category as initial object and the one-object-one-morphism category as terminal object. Any topological space X can be viewed as a category by taking the open sets as objects, and a single morphism between two open sets U and V if and only if U ⊂ V. The empty set is the initial object of this category, and X is the terminal object. If X is a topological space (viewed as a category as above) and C is some small category, we can form the category of all contravariant functors from X to C, using natural transformations as morphisms. This category is called the category of presheaves on X with values in C. If C has an initial object.
Intermediate treatment of tensors - contents showTocToggle("show","hide") 1 Definition 2 Transformation rules 3 Further reading Definition The formal definition of a tensor quantity begins with a finite-dimensional vector space , which furnishes the uniform "building blocks" for tensors of all valences. In typical applications, is the tangent space at a point of a manifold; the elements of typically represent physical quantities such as velocities and forces. The space of -valent tensors, denoted here by is obtained by taking the tensor product of copies of and copies of the dual vector space . To wit, In order to represent a tensor by a concrete array of numbers, we require a frame of reference, which is essentially a basis of , say Every vector in can be "measured" relative to this basis, meaning that for every there exist.
Entropy encoding - of the most common entropy encoding techniques are Huffman coding, Range encoder and arithmetic encoding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code such as Unary coding, Elias Gamma coding, Fibonacci coding, Golomb coding, or Rice coding may be useful. An earlier version of the above article was posted on PlanetMath. This article is open content. See also: entropy, Universal code.
Divergence theorem - surface of a small sphere relative to the surface area of the sphere. To wit, where S denotes the sphere of radius r about a point p in R3, and the integral is a surface integral taken with respect to N, the normal to that sphere. The non-infinitesimal interpretation of divergence is given by Gauss's Theorem. This theorem is a conservation law, stating that the volume total of all sinks and sources, i.e. the volume integral of the divergence, is equal to the net flow across the volume's boundary. In symbols, ∫V div F ∂V = ∫S (F·N) ∂S where V, a subset of R3, is a compact region with a smooth boundary, and S = ∂V is that boundary oriented by outward-pointing normals. We note that Gauss's theorem follows from.
Arithmetic coding - . Therefore the size of Ii is PiR. If the next symbol is xi, then restrict the output to the new interval Ii. Note that at each stage, all the possible intervals are pairwise disjoint. Therefore a specific sequence of symbols produces exactly one unique output range, and the process can be reversed. Since arithmetic encoders are typically implemented on binary computers, the actual output of the encoder is generally the shortest sequence of bits representing the fractional part of a rational number in the final interval. Suppose our entire input string contains M symbols: then xi appears exactly PiM times in the input. Therefore, the size of the final interval will be By Shannon's theorem, this is the total entropy of the original message. Therefore arithmetic encoding is near-optimal entropy.
Church integer - particular Church integer might be church 0 = \\ f x -> x church n = c where c f x = c' f (f x) where c' = church (n - 1) The transformation from a Church integer to an integer might be unchurch n = n (+1) 0 Thus the (+1) function would be applied to an initial value of 0 n times, yielding the ordinary integer n. See lambda calculus for another expression of the same idea. An earlier version of the above article was posted on PlanetMath. This article is open-content..
Splitting lemma - a in A such that q(a) = b, and t(b) = 0, then 0 = tq(a) = a; and therefore, b = 0. (By exactness, im q = ker r, so im q is a normal subgroup of B in the case of groups.) This proves that B is the direct sum (alternatively, a semidirect product) of im q and ker t. So, for all b in B, b can be uniquely identified by some a in A, k in ker t, such that b = q(a) + k. By exactness, ker rq = A, and so ker r = im q. The subsequence B → C → 0 implies that f is onto; therefore for any c in C there exists some b = q(a) + k such that c.
Pi - together with Peter Borwein and Simon Plouffe, discovered a new formula for π as an infinite series: This formula permits one to easily compute the nth binary or hexadecimal digit of π, without having to compute the preceeding n-1 digits. Bailey's website contains the derivation as well as implementations in various programming languages. Other formulas that have been used to compute pi include: (Ramanujan) (David Chudnovsky and Gregory Chudnovsky) Open questions The most pressing open question about π is whether it is normal, i.e. whether any digit block occurs in the expansion of π just as often as one would statistically expect if the digits had been produced completely randomly. This should be true in any base, not just in base 10. It isn't even known which of the digits 0,...,9.
Open content - record label Magnatune [1] - open content record label Nupedia [1] - peer-reviewed encyclopedia Opencode [1] - consortium for open research and content OpenContent [1] - open source licensing scheme for information content Open Content for Education [1] Open-education.org [1] - Portal and advocacy-site for collaborative creation of Open Content Educational materials. Open Gaming Center - an open content experiment to create a games and gaming encyclopedia Openlaw [1] - Experiment in the open crafting of legal arguments Opsound [1] - Open sound pool, a record label. Open Directory Project [1] - web directory like Yahoo. Open Music Registry [1] - Open sharing of music using an Open Audio License Open Photo [1] - stock photos OYEZ [1] - US Supreme Court multimedia Prelinger Archives [1] - government and advertising films.
Matroid - element from a circuit yields a basis, so all circuits have the same number of elements too, one more than the rank. In the first example matroid above, a basis is a basis in the sense of linear algebra of the subspace spanned by M. In the second example, a basis is the same as a spanning tree of the graph G. In the third example, a basis is any subsets of M with k elements. If N is a subset of the matroid M, then N becomes a matroid by considering a subset of N independent if and only if it is independent in M. This allows to talk about the rank of any subset of M. The rank function r assigns a natural number to every subset of M.
MathWorld - content was to remain in print only. The site was taken down by a court injunction. The case was later settled out of court, with WRI paying an unspecified amount and other stipulations. The site became available free to the public again afterwards. This case made a wave of headlines in the online publishing circle. Most people accused CRC Press of corporate greed, and demanded a free online encyclopedia. Many people depended on having a free mathematics encyclopedic resource available on the internet. Because of the temporary closure of Mathworld, a number of individuals have tried to start a free online math encyclopedia from scratch. Most notable are PlanetMath and this Wikipedia which carries a sizable mathematics reference accompanying its other reference material..
List of mathematical topics (P-R) - -- Pentomino -- Percentage -- Perfect crystal -- Perfect matching -- Perfect number -- Perfect square -- Perimeter -- Period -- Perelman, Grigori -- Permanent -- Permutation -- Permutation group -- Permutation matrix -- Perrin pseudoprime -- Perron Integral -- Perspective -- Perspective distortion -- Perturbation theory -- Pervouchine -- Pervushin -- Pervushin, Ivan Mikheevich -- Petersen graph -- Petersen, Julius -- Peter-Weyl theorem -- Phase diagram -- Phase velocity -- Phase (waves) -- Phasor -- Phenomenology -- Philolaus -- Philosophy of mathematics -- Photon -- Physical constants -- Physical geodesy -- Physics -- Pi -- Pi Approximation Day -- Pi Day -- Picard -- Picard, Charles Emile -- Pick's theorem Talk:Pick's theorem - PID controller -- Pigeonhole principle -- Pincherle derivative -- Pisanski, Tomas -- Pitman-Koopman-Darmois theorem -- Planar graph.
