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# User:Richard Pinch/Articles

(Difference between revisions)
 Revision as of 12:17, 1 January 2009 (view source) (→Articles I have started: added Order (ring theory))← Older edit Revision as of 18:36, 1 January 2009 (view source) (→Articles I have started: added Integral closure)Newer edit → Line 100: Line 100: {{rpl|Incentre}} {{rpl|Incentre}} {{rpl|Injective function}} {{rpl|Injective function}} + {{rpl|Integral closure}} {{rpl|Interior (topology)}} {{rpl|Interior (topology)}} {{rpl|Intersection}} {{rpl|Intersection}}

## Articles I have started

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