Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free |

CZ thanks . AUGUST 2014 donors; special to Darren Duncan. SEPTEMBER 2014 donations open; need minimum total $100. Let's exceed that. Donate hereTreasurer's Financial Report -- Thanks to August content contributors. -- |

# User:Richard Pinch/Articles

### From Citizendium, the Citizens' Compendium

(Difference between revisions)

(→Articles I have started: added Order (ring theory)) |
(→Articles I have started: added Integral closure) |
||

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}} |

## Revision as of 18:36, 1 January 2009

## 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*+i*y*→*x*-i*y*^{[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 . 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]}