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User:Richard Pinch/Articles

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Revision as of 18:36, 1 January 2009

Articles I have started

  • Stub Absorbing element: An element whose behaviour with respect to an algebraic binary operation is like that of zero with respect to multiplication. [e]
  • Stub Albert algebra: An exceptional Jordan algebra, consisting of 3×3 self-adjoint matrices over the octonions. [e]
  • Stub Algebra over a field: A ring containing an isomorphic copy of a given field in its centre. [e]
  • Stub Algebraic independence: The property of elements of an extension field which satisfy only the trivial polynomial relation. [e]
  • Developing Article Algebraic number field: A field extension of the rational numbers of finite degree; a principal object of study in algebraic number theory. [e]
  • Stub Alternant code: A class of parameterised error-correcting codes which generalise the BCH codes. [e]
  • Developing Article Altitude (geometry): In a triangle, a line from a vertex perpendicular to the opposite side. [e]
  • Stub Arithmetic function: A function defined on the set of positive integers, usually with integer, real or complex values, studied in number theory. [e]
  • Stub Artin-Schreier polynomial‎: A type of polynomial whose roots generate extensions of degree p in characteristic p. [e]
  • Stub Associativity: A property of an algebraic operation such as multiplication: a(bc) = (ab)c. [e]
  • Developing Article Automorphism: An isomorphism of an algebraic structure with itself: a permutation of the underlying set which respects all algebraic operations. [e]
  • Stub Average order of an arithmetic function‎: A simple or well-known function, usually continuous and montonic, which on average takes the same or closely approximate values as a given arithmetic function. [e]
  • Developing Article Baer-Specker group‎: An example of an infinite Abelian group which is a building block in the structure theory of such groups. [e]
  • Stub Barycentre: The centre of mass of a body or system of particles, a weighted average where certain forces may be taken to act. [e]
  • Developing Article Barycentric coordinates‎: The weights that would have to be assigned to a system of reference points to yield a given position as barycentre are used as coordinates. [e]
  • Stub Binary operation: A function of two elements within a set, which assigns another value from among the elements of the set. [e]
  • Developing Article Brun-Titchmarsh theorem: An upper bound on the distribution on primes in an arithmetic progression. [e]
  • Stub Cameron-Erdos conjecture‎: Add brief definition or description
  • Stub Cartesian product: The set of ordered pairs whose elements come from two given sets. [e]
  • Stub Centraliser: The set of all group elements which commute with every element of a given subset. [e]
  • Stub Centre of a group: The subgroup of a group consisting of all elements which commute with every element of the group. [e]
  • Stub Centre of a ring: The subring of a ring consisting of all elements which commute with every element of the ring. [e]
  • Developing Article Cevian line: A line from the vertex of a triangle to some point on the opposite edge. [e]
  • Developing Article Chain rule: A rule in calculus for differentiating a function of a function. [e]
  • Stub Characteristic function: A function on a set which takes the value 1 on a given subset and 0 on its complement. [e]
  • Stub Characteristic polynomial: The polynomial attached to a square matrix or endomorphism det(A-XI)=0. [e]
  • Circumcentre: The centre of the circle that goes through the vertices of a triangle or a cyclic polygon. [e]
  • Developing Article Closure operator: An idempotent unary operator on subsets of a given set, mapping a set to a larger set with a particular property. [e]
  • Stub Cocountable topology: The topology on a space in which the open sets are those with countable complements, or the empty set. [e]
  • Stub Cofactor (mathematics): A component of a matrix computation of the determinant; a signed determinant of a matrix minor. [e]
  • Stub Cofinite topology: The topology on a space in which the open sets are those with finite complement, or the empty set. [e]
  • Developing Article Commutativity: A property of a binary operation (such as addition or multiplication), that the two operands may be interchanged without affecting the result. [e]
  • Stub Commutator: A measure of how close two elements of a group are to commuting. [e]
  • Stub Compactification: A compact space in which a given topological space can be embedded as a dense subset. [e]
  • Stub Compactness axioms: Properties of a toplogical space related to compactness. [e]
  • Stub Complement (linear algebra)‎: A pair of subspaces which form an (internal) direct sum. [e]
  • Stub Complement (set theory)‎: The set containing those elements of a set (or "universal" set) which are not contained in a given set. [e]
  • Stub Complex conjugation: The operation on complex numbers which changes the sign of the imaginary part, x+iyx-iy [e]
  • Developing Article Conductor of an abelian variety: A measure of the nature of the bad reduction at some prime. [e]
  • Developing Article Congruent triangles: In Euclidean geometry, triangles which can be superposed by a rigid motion. [e]
  • Stub Conjugation (group theory)‎: The elements of any group that may be partitioned into conjugacy classes. [e]
  • Stub Connected space: A topological space in which there is no non-trivial subset which is both open and closed. [e]
  • Stub Content (algebra): The highest common factor of the coefficients of a polynomial. [e]
  • Developing Article Continuant (mathematics): An algebraic expression which has applications in generalized continued fractions and as the determinant of a tridiagonal matrix. [e]
  • Convolution: Add brief definition or description
  • Stub Coprime: Integers, or more generally elements of a ring, which have no non-trivial common factor. [e]
  • Stub Countability axioms in topology: Properties that a topological space may satisfy which refer to the countability of certain structures within the space. [e]
  • Developing Article Cubic reciprocity: Various results connecting the solvability of two related cubic equations in modular arithmetic, generalising the concept of quadratic reciprocity. [e]
  • Stub Cyclic group: A group consisting of the powers of a single element. [e]
  • Developing Article Cyclic polygon: A polygon whose vertices lie on a single circle. [e]
  • Stub Cyclotomic field: An algebraic number field generated over the rational numbers by roots of unity. [e]
  • Stub Cyclotomic polynomial: A polynomial whose roots are primitive roots of unity. [e]
  • Stub Delta form: A modular form arising from the discriminant of an elliptic curve: a cusp form of weight 12 and level 1 for the full modular group and a Hecke eigenform. [e]
  • Stub Derivation (mathematics): A map defined on a ring which behaves formally like differentiation: D(x.y)=D(x).y+x.D(y). [e]
  • Stub Diagonal matrix: A square matrix which has zero entries off the main diagonal. [e]
  • Developing Article Different ideal: An invariant attached to an extension of algebraic number fields which encodes ramification data. [e]
  • Stub Differential ring: A ring with added structure which generalises the concept of derivative. [e]
  • Developing Article Dirichlet series: An infinite series whose terms involve successive positive integers raised to powers of a variable, typically with integer, real or complex coefficients. [e]
  • Stub Discrete metric: The metric on a space which assigns distance one to any distinct points, inducing the discrete topology. [e]
  • Developing Article Discrete space: A topological space with the discrete topology, in which every subset is open (and also closed). [e]
  • Developing Article Discriminant of a polynomial: An invariant of a polynomial which vanishes if it has a repeated root: the product of the differences between the roots. [e]
  • Developing Article Discriminant of an algebraic number field: An invariant attached to an extension of algebraic number fields which describes the geometric structure of the ring of integers and encodes ramification data. [e]
  • Developing Article Disjoint union: A set containing a copy of each of a family of two or more sets, so that the copies do not overlap. [e]
  • Stub Distributivity: A relation between two binary operations on a set generalising that of multiplication to addition: a(b+c)=ab+ac. [e]
  • Stub Division (arithmetic): The process of determing how many copies of one quantity are required to make up another; repeated subtraction; the inverse operation to multiplication. [e]
  • Stub Division ring: (or skew field), In algebra it is a ring in which every non-zero element is invertible. [e]
  • Stub Divisor (algebraic geometry): A formal sum of subvarieties of an algebraic variety. [e]
  • Stub Door space: A topological space in which each subset is open or closed. [e]
  • Stub Dowker space: A topological space that is T4 but not countably paracompact. [e]
  • Empty set: In set theory, this is a set without elements, usually denoted \{~\} or \empty. The empty set is a subset of any set. [e]
  • Stub End (topology): For a topological space this generalises the notion of "point at infinity" of the real line or plane. [e]
  • Developed Article Equivalence relation: A reflexive symmetric transitive binary relation on a set. [e]
  • Stub Erdos-Fuchs theorem: A statement about the number of ways that numbers can be represented as a sum of two elements of a given set. [e]
  • Stub Error function: A function associated with the cumulative distribution function of the normal distribution. [e]
  • Stub Essential subgroup: A subgroup of a group which has non-trivial intersection with every other non-trivial subgroup. [e]
  • Stub Exact sequence: A sequence of algebraic objects and morphisms which is used to describe or analyse algebraic structure. [e]
  • Developed Article Factorial: The number of ways of arranging n labeled objects in order; the product of the first n integers. [e]
  • Stub Field automorphism: An invertible function from a field onto itself which respects the field operations of addition and multiplication. [e]
  • Stub Filter (mathematics)‎: A family of subsets of a given set which has properties generalising the notion of "almost all natural numbers". [e]
  • Stub Frattini subgroup: The intersection of all maximal subgroups of a group. [e]
  • Stub Free group: A group in which there is a generating set such that every element of the group can be written uniquely as the product of generators. [e]
  • Stub Frobenius map: The p-th power map considered as acting on commutative algebras or fields of prime characteristic p. [e]
  • Stub Function composition: The successive application of two functions. [e]
  • Stub Functional equation: A relation between the values of a function at different points, such as periodicity or symmetry. [e]
  • Developing Article Generic point: A point of a topological space which is not contained in any proper closed subset; a point satisfying no special properties. [e]
  • Stub Genus field: The maximal absolutely abelian unramified extension of a number field. [e]
  • Developing Article Group action: A way of describing symmetries of objects using groups. [e]
  • Developing Article Group homomorphism: A map between group which preserves the group structure. [e]
  • Stub Group isomorphism problem: The decision problem of determining whether two group presentations present isomorphic groups. [e]
  • Developing Article Hall polynomial: The structure constants of Hall algebra. [e]
  • Stub Hall-Littlewood polynomial‎: Symmetric functions depending on a parameter t and a partition λ. [e]
  • Stub Heine–Borel theorem: In Euclidean space of finite dimension with the usual topology, a subset is compact if and only if it is closed and bounded. [e]
  • Developing Article Hutchinson operator: A collection of functions on an underlying space. [e]
  • Developing Article Idempotence: The property of an operation that repeated application has no effect. [e]
  • Stub Idempotent element: An element or operator for which repeated application has no further effect. [e]
  • Stub Identity element: An element whose behaviour with respect to a binary operation generalises that of zero for addition or one for multiplication. [e]
  • Stub Identity function: The function from a set to itself which maps each element to itself. [e]
  • Stub Identity matrix: A square matrix with ones on the main diagonal and zeroes elsewhere: the identity element for matrix multiplication. [e]
  • Stub Incentre: The centre of the incircle, a circle which is within a triangle and tangent to its three sides. [e]
  • Stub Injective function: A function which has different output values on different input values. [e]
  • Developing Article Integral closure: The ring of elements of an extension of a ring which satisfy a monic polynomial over the base ring. [e]
  • Developing Article Interior (topology): The union of all open sets contained within a given subset of a topological space. [e]
  • Stub Intersection: The set of elements that are contained in all of a given family of two or more sets. [e]
  • Stub Isolated singularity: A point at which function of a complex variable is not holomorphic, but which has a neighbourhood on which the function is holomorphic. [e]
  • Developing Article Jordan's totient function: A generalisation of Euler's totient function. [e]
  • Stub Justesen code: A class of error-correcting codes which are derived from Reed-Solomon codes and have good error-control properties. [e]
  • Stub KANT: A computer algebra system for mathematicians interested in algebraic number theory. [e]
  • Developing Article Kernel of a function: The equivalence relation on the domain of a function defined by elements having the same function value: the partition of the domain into fibres of a function. [e]
  • Stub Kronecker delta: A quantity depending on two subscripts which is equal to one when they are equal and zero when they are unequal. [e]
  • Stub Krull dimension: In a ring, one less than the length of a maximal ascending chain of prime ideals. [e]
  • Stub Lambda function: The exponent of the multiplicative group modulo an integer. [e]
  • Stub Lattice (geometry): A discrete subgroup of a real vector space. [e]
  • Stub Limit point: A point which cannot be separated from a given subset of a topological space; all neighbourhoods of the points intersect the set. [e]
  • Developing Article Littlewood polynomial: A polynomial all of whose coefficients are plus or minus 1. [e]
  • Stub Manin obstruction: A measure of the failure of the Hasse principle for geometric objects. [e]
  • Developing Article Median algebra: A set with a ternary operation satisfying a set of axioms which generalise the notion of median or majority function, as a Boolean function. [e]
  • Developing Article Minimal polynomial: The monic polynomial of least degree which a square matrix or endomorphism satisfies. [e]
  • Developing Article Möbius function‎: Arithmetic function which takes the values -1, 0 or +1 depending on the prime factorisation of its input n. [e]
  • Developing Article Modulus (algebraic number theory)‎: A formal product of places of an algebraic number field, used to encode ramification data for abelian extensions of a number field. [e]
  • Developing Article Monogenic field: An algebraic number field for which the ring of integers is a polynomial ring. [e]
  • Developing Article Monoid: An algebraic structure with an associative binary operation and an identity element. [e]
  • Developing Article Monotonic function: A function on an ordered set which preserves the order. [e]
  • Stub Moore determinant: A determinant defined over a finite field which has successive powers of the Frobenius automorphism applied to the first column. [e]
  • Stub Morita conjectures: Three conjectures in topology relating to normal spaces, now proved. [e]
  • Stub Neighbourhood: Add brief definition or description
  • Nine-point centre: Add brief definition or description
  • Developing Article Noetherian module: Module in which every ascending sequence of submodules has only a finite number of distinct members. [e]
  • Stub Normal extension: A field extension which contains all the roots of an irreducible polynomial if it contains one such root. [e]
  • Stub Normal number: A real number whose digits in some particular base occur equally often in the long run. [e]
  • Stub Normal order of an arithmetic function‎: A simple or well-known function, usually continuous and montonic, which "usually" takes the same or closely approximate values as a given arithmetic function. [e]
  • Stub Normaliser: The elements of a group which map a given subgroup to itself by conjugation. [e]
  • Stub Nowhere dense set: A set in a topological space whose closure has empty interior. [e]
  • Null set: Add brief definition or description
  • Stub Number of divisors function: The number of positive integer divisors of a given number. [e]
  • Stub Number Theory Foundation: A non-profit organisation based in the United States which supports research and conferences in the field of number theory. [e]
  • Developing Article Order (group theory): For a group, its cardinality; for an element of a group, the least positive integer (if one exists) such that raising the element to that power gives the identity. [e]
  • Stub Order (relation): An irreflexive antisymmetric transitive binary relation on a set. [e]
  • Stub Order (ring theory): A ring which is finitely generated as a Z-module. [e]
  • Stub Ordered field: A field with a total order which is compatible with the algebraic operations. [e]
  • Stub Ordered pair: Two objects in which order is important. [e]
  • Stub p-adic metric: A metric on the rationals in which numbers are close to zero if they are divisible by a large power of a given prime p. [e]
  • Developing Article Partition (mathematics): Concepts in mathematics which refer either to a partition of a set or an ordered partition of a set, or a partition of an integer, or a partition of an interval. [e]
  • Stub Partition function (number theory): The number of additive partitions of a positive integer. [e]
  • Developing Article Pedal triangle: Triangle whose vertices are located at the feet of the perpendiculars from some given point to the sides of a specified triangle. [e]
  • Stub Pointwise operation: Method of extending an operation defined on an algebraic struture to a set of functions taking values in that structure. [e]
  • Stub Pole (complex analysis): A type of singularity of a function of a complex variable where it behaves like a negative power. [e]
  • Stub Power set: The set of all subsets of a given set. [e]
  • Stub Preparata code: A class of non-linear double-error-correcting codes. [e]
  • Developing Article Primitive root: A generator of the multiplicative group in modular arithmetic when that group is cyclic. [e]
  • Developing Article Product topology: Topology on a product of topological spaces whose open sets are constructed from cartesian products of open sets from the individual spaces. [e]
  • Developing Article Quadratic field: A field which is an extension of its prime field of degree two. [e]
  • Stub Quadratic residue: A number which is the residue of a square integer with respect to a given modulus. [e]
  • Stub Quotient topology: The finest topology on the image set that makes a surjective map from a topological space continuous. [e]
  • Developing Article Relation (mathematics): A property which holds between certain elements of some set or sets. [e]
  • Stub Relation composition: Formation of a new relation S o R from two given relations R and S, having as its most well-known special case the composition of functions. [e]
  • Stub Removable singularity: A singularity of a complex function which can be removed by redefining the function value at that point. [e]
  • Stub Residual property (mathematics): A concept in group theory on recovered element properties. [e]
  • Stub Resolution (algebra): An exact sequence which is used to describe the structure of a module. [e]
  • Stub Resultant (algebra): An invariant which determines whether or not two polynomials have a factor in common. [e]
  • Developing Article Resultant (statics): A single force having the same effect as a system of forces acting at different points. [e]
  • Developing Article Rigid motion: A transformation which preserves the geometrical properties of the Euclidean spacea distance-preserving mapping or isometry. [e]
  • Developing Article Ring homomorphism: Function between two rings which respects the operations of addition and multiplication. [e]
  • Developing Article Root of unity: An algebraic quantity some power of which is equal to one. [e]
  • Stub S-unit: An element of an algebraic number field which has a denominator confined to primes in some fixed set. [e]
  • Developing Article Selberg sieve: A technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. [e]
  • Stub Semigroup: An algebraic structure with an associative binary operation. [e]
  • Developing Article Separation axioms: Axioms for a topological space which specify how well separated points and closed sets are by open sets. [e]
  • Stub Series (group theory): A chain of subgroups of a group linearly ordered by subset inclusion. [e]
  • Stub Singleton set: A set with exactly one element. [e]
  • Developing Article Sober space: A topological space in which every irreducible closed set has a unique generic point. [e]
  • Stub Srivastava code: A class of parameterised error-correcting codes which are a special case of alternant codes. [e]
  • Stub Stably free module: A module which is close to being free: the direct sum with some free module is free. [e]
  • Developing Article Stirling number: Coefficients which occur in the Stirling interpolation formula for a difference operator. [e]
  • Stub Subgroup: A subset of a group which is itself a group with respect to the group operations. [e]
  • Developing Article Subspace topology: An assignment of open sets to a subset of a topological space. [e]
  • Stub Sum-of-divisors function: The function whose value is the sum of all positive divisors of a given positive integer. [e]
  • Stub Surjective function: A function for which every possible output value occurs for one or more input values: the image is the whole of the codomain. [e]
  • Stub Sylow subgroup: A subgroup of a finite group whose order is the largest possible power of one of the primes factors of the group order. [e]
  • Stub Symmetric difference: The set of elements that lie in exactly one of two sets. [e]
  • Developing Article Szpiro's conjecture: A relationship between the conductor and the discriminant of an elliptic curve. [e]
  • Stub Tau function: An arithmetic function studied by Ramanjuan, the coefficients of the q-series expansion of the modular form Delta. [e]
  • Stub Theta function: An analytic function which is a modular form of weight one-half; more generally, the generating function for a quadratic form. [e]
  • Stub Totient function: The number of integers less than or equal to and coprime to a given integer. [e]
  • Developing Article Transitive relation: A relation with the property that if x→y and y→z then x→z. [e]
  • Developing Article Turan sieve: A technique for estimating the size of "sifted sets" of positive integers which satisfy a set of conditions which are expressed by congruences. [e]
  • Stub Tutte matrix: A matrix used to determine the existence of a perfect matching in a graph: that is, a set of edges which is incident with each vertex exactly once. [e]
  • Stub Weierstrass preparation theorem: A description of a canonical form for formal power series over a complete local ring. [e]
  • Developing Article Zero matrix: A matrix consisting entirely of zero entries. [e]
  • Developing Article Zipf distribution: Observation that states that, in a population consisting of many different types, the proportion belonging to the nth most common type is approximately proportional to 1/n. [e]
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