|
Classification Theorem
In mathematics, a classification theorem answers the classification problem "What are the objects of a given type, up to some equivalence?". It gives a non-redundant enumeration: each object is equivalent to exactly one class. A few issues related to classification are the following. *The equivalence problem is "given two objects, determine if they are equivalent". *A complete set of invariants, together with which invariants are solves the classification problem, and is often a step in solving it. *A (together with which invariants are realizable) solves both the classification problem and the equivalence problem. * A canonical form solves the classification problem, and is more data: it not only classifies every class, but provides a distinguished (canonical) element of each class. There exist many classification theorems in mathematics, as described below. Geometry * Classification of Euclidean plane isometries * Classification theorems of surfaces ** Classification of two ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank 3 Permutation Group
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as: Level or position in a hierarchical organization * Academic rank * Diplomatic rank * Hierarchy * Hierarchy of the Catholic Church * Military rank * Police ranks of the United States * Ranking member, S politicsthe most senior member of a committee from the minority party, and thus second-most senior member of a committee * Imperial, royal and noble ranks Level or position in society *Social class *Social position *Social status Places * Rank, Iran, a village * Rank, Nepal, a village development committee People * Rank (surname), a list of people with the name Arts, entertainment, and media Music * ''Rank'' (album), a live album by the Smiths * "Rank", a song by Artwork from '' A Bugged Out Mix'' Other arts, entertainment, and media * Rank (chess), a row of the chessboard * ''Rank'' (film), a short film directed by David Yates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Fatou Components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou. Rational case If f is a rational function :f = \frac defined in the extended complex plane, and if it is a nonlinear function (degree > 1) : d(f) = \max(\deg(P),\, \deg(Q))\geq 2, then for a periodic component U of the Fatou set, exactly one of the following holds: # U contains an attracting periodic point # U is parabolic # U is a Siegel disc: a simply connected Fatou component on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. # U is a Herman ring: a double connected Fatou component (an annulus) on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle. File:Julia-set_N_z3-1.png, Julia set (white) and Fatou set (dark red/green/blue) for f: z\mapsto z-\frac(z) with g: z \mapsto z^3-1 in the complex plane. Basilica Juli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classification Of Discontinuities
Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values. The oscillation of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * in an essential discontinuity, oscillation measures the failure of a limit to exist; the limit is constant. A special case is if the fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sylvester's Law Of Inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadratic form, and ''S'' is any invertible matrix such that ''D'' = ''SAS''T is diagonal, then the number of negative elements in the diagonal of ''D'' is always the same, for all such ''S''; and the same goes for the number of positive elements. This property is named after James Joseph Sylvester who published its proof in 1852. Statement Let ''A'' be a symmetric square matrix of order ''n'' with real number, real entries. Any non-singular matrix ''S'' of the same size is said to transform ''A'' into another symmetric matrix , also of order ''n'', where ''S''T is the transpose of ''S''. It is also said that matrices ''A'' and ''B'' are Matrix congruence, congruent. If ''A'' is the coeffi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
