Encyclopedia of Mathematics
The Encyclopedia of Mathematics is a large reference work in mathematics. It is available in book form and on CD-ROM; the 2002 version contains more than 8,000 entries covering most areas of mathematics at a graduate level, the presentation is technical in nature. The encyclopedia is edited by Michiel Hazewinkel and was published by Kluwer Academic Publishers until 2003, when Kluwer became part of Springer; the CD-ROM contains three-dimensional objects. The encyclopedia has been translated from the Soviet Matematicheskaya entsiklopediya edited by Ivan Matveevich Vinogradov and extended with comments and three supplements adding several thousand articles; until November 29, 2011, a static version of the encyclopedia could be browsed online free of charge online. This URL now redirects to the new wiki incarnation of the EOM. A new dynamic version of the encyclopedia is now available as a public wiki online; this new wiki is a collaboration between the European Mathematical Society. This new version of the encyclopedia includes the entire contents of the previous online version, but all entries can now be publicly updated to include the newest advancements in mathematics.
All entries will be monitored for content accuracy by members of an editorial board selected by the European Mathematical Society. Vinogradov, I. M. Matematicheskaya entsiklopediya, Sov. Entsiklopediya, 1977. Hazewinkel, M. Encyclopaedia of Mathematics, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 1, Kluwer, 1987. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 2, Kluwer, 1988. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 3, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 4, Kluwer, 1989. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 5, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 6, Kluwer, 1990. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 7, Kluwer, 1991. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 8, Kluwer, 1992. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 9, Kluwer, 1993. Hazewinkel, M. Encyclopaedia of Mathematics, Vol. 10, Kluwer, 1994. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement I, Kluwer, 1997. Hazewinkel, M. Encyclopaedia of Mathematics, Supplement II, Kluwer, 2000.
Hazewinkel, M. Encyclopaedia of Mathematics, Supplement III, Kluwer, 2002. Hazewinkel, M. Encyclopaedia of Mathematics on CD-ROM, Kluwer, 1998. Encyclopedia of Mathematics, public wiki monitored by an editorial board under the management of the European Mathematical Society. List of online encyclopedias Official website Publications by M. Hazewinkel, at ResearchGate
Free abelian group
In abstract algebra, a free abelian group or free Z-module is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation, associative and invertible. A basis is a subset such that every element of the group can be found by adding or subtracting basis elements, such that every element's expression as a linear combination of basis elements is unique. For instance, the integers under addition form a free abelian group with basis. Addition of integers is commutative and has subtraction as its inverse operation, each integer is the sum or difference of some number of copies of the number 1, each integer has a unique representation as an integer multiple of the number 1. Free abelian groups have properties, they have applications in algebraic topology, where they are used to define chain groups, in algebraic geometry, where they are used to define divisors. Integer lattices form examples of free abelian groups, lattice theory studies free abelian subgroups of real vector spaces.
The elements of a free abelian group with basis B may be described in several equivalent ways. These include formal sums over B, which are expressions of the form ∑ a i b i where each coefficient ai is a nonzero integer, each factor bi is a distinct basis element, the sum has finitely many terms. Alternatively, the elements of a free abelian group may be thought of as signed multisets containing finitely many elements of B, with the multiplicity of an element in the multiset equal to its coefficient in the formal sum. Another way to represent an element of a free abelian group is as a function from B to the integers with finitely many nonzero values; every set B has a free abelian group with B as its basis. This group is unique in the sense that every two free abelian groups with the same basis are isomorphic. Instead of constructing it by describing its individual elements, a free group with basis B may be constructed as a direct sum of copies of the additive group of the integers, with one copy per member of B.
Alternatively, the free abelian group with basis B may be described by a presentation with the elements of B as its generators and with the commutators of pairs of members as its relators. The rank of a free abelian group is the cardinality of a basis; every subgroup of a free abelian group is itself free abelian. The integers, under the addition operation, form a free abelian group with the basis; every integer n is a linear combination of basis elements with integer coefficients: namely, n = n × 1, with the coefficient n. The two-dimensional integer lattice, consisting of the points in the plane with integer Cartesian coordinates, forms a free abelian group under vector addition with the basis. Letting these basis vectors be denoted e 1 = and e 2 =, the element can be written = 4 e 1 + 3 e 2 where'multiplication' is defined so that 4 e 1:= e 1 + e 1 + e 1 + e 1. In this basis, there is no other way to write. However, with a different basis such as, where f 1 = and f 2 =, it can be written as = f 1 + 3 f 2.
More every lattice forms a finitely-generated free abelian group. The d-dimensional integer lattice has a natural basis consisting of the positive integer unit vectors, but it has many other bases as well: if M is a d × d integer matrix with determinant ±1 the rows of M form a basis, conversely every basis of the integer lattice has this form. For more on the two-dimensional case, see fundamental pair of periods; the direct product of two free abelian groups is itself free abelian, with basis the disjoint union of the bases of the two groups. More the direct product of any finite number of free abelian groups is free abelian; the d-dimensional integer lattice, for instance, is isomorphic to the direct product of d copies of the integer group Z. The trivial group is considered to be free abelian, with basis the empty set, it may be interpreted as a direct product of zero copies of Z. For infinite families of free abelian groups, the direct product is not free abelian. For instance the Baer–Specker group Z N
In mathematics, the modular group is the projective special linear group PSL of 2 x 2 matrices with integer coefficients and unit determinant. The matrices A and -A are identified; the modular group acts on the upper-half of the complex plane by fractional linear transformations, the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. The modular group Γ is the group of linear fractional transformations of the upper half of the complex plane which have the form z ↦ a z + b c z + d where a, b, c, d are integers, ad − bc = 1; the group operation is function composition. This group of transformations is isomorphic to the projective special linear group PSL, the quotient of the 2-dimensional special linear group SL over the integers by its center. In other words, PSL consists of all matrices where a, b, c, d are integers, ad − bc = 1, pairs of matrices A and −A are considered to be identical; the group operation is the usual multiplication of matrices. Some authors define the modular group to be PSL, still others define the modular group to be the larger group SL.
Some mathematical relations require the consideration of the group GL of matrices with determinant plus or minus one. PGL is the quotient group GL/. A 2 × 2 matrix with unit determinant is a symplectic matrix, thus SL = Sp, the symplectic group of 2x2 matrices; the unit determinant of implies that the fractions a/b, a/c, c/d and b/d are all irreducible, having no common factors. More if p/q is an irreducible fraction a p + b q c p + d q is irreducible. Any pair of irreducible fractions can be connected in this way, i.e.: for any pair p/q and r/s of irreducible fractions, there exist elements ∈ SL such that r = a p + b q and s = c p + d q. Elements of the modular group provide a symmetry on the two-dimensional lattice. Let ω 1 and ω 2 be two complex numbers whose ratio is not real; the set of points Λ = is a lattice of parallelograms on the plane. A different pair of vectors α 1 and α 2 will generate the same lattice if and only if = for some matrix in GL, it is for this reason that doubly periodic functions, such as elliptic functions, possess a modular group symmetry.
The action of the modular group on the rational numbers can most be understood by envisioning a square grid, with grid point corresponding to the fraction p/q. An irreducible fraction is one, visible from the origin. Note that any member of the modular group maps the projectively extended real line one-to-one to itself, furthermore bijectively maps the projectively extended rational line to itself, the irrationals to the irrationals, the transcendental numbers to the transcendental numbers, the non-real numbers to the non-real numbers, the upper half-plane to the upper half-plane, et cetera. If p n − 1 / q n − 1 and p n / q n are two successive convergents of a continued fraction the matrix (
Michiel Hazewinkel is a Dutch mathematician, Emeritus Professor of Mathematics at the Centre for Mathematics and Computer and the University of Amsterdam known for his 1978 book Formal groups and applications and as editor of the Encyclopedia of Mathematics. Born in Amsterdam to Jan Hazewinkel and Geertrude Hendrika Werner, Hazewinkel studied at the University of Amsterdam, he received his BA in Mathematics and Physics in 1963, his MA in Mathematics with a minor in Philosophy in 1965 and his PhD in 1969 under supervision of Frans Oort and Albert Menalda for the thesis "Maximal Abelian Extensions of Local Fields". After graduation Hazewinkel started his academic career as Assistant Professor at the University of Amsterdam in 1969. In 1970 he became Associate Professor at the Erasmus University Rotterdam, where in 1972 he was appointed Professor of Mathematics at the Econometric Institute. Here he was thesis advisor of Roelof Stroeker, M. van de Vel, Jo Ritzen, Gerard van der Hoek. From 1973 to 1975 he was Professor at the Universitaire Instelling Antwerpen, where Marcel van de Vel was his PhD student.
From 1982 to 1985 he was appointed part-time Professor Extraordinarius in Mathematics at the Erasmus Universiteit Rotterdam, part-time Head of the Department of Pure Mathematics at the Centre for Mathematics and Computer in Amsterdam. In 1985 he was appointed Professor Extraordinarius in Mathematics at the University of Utrecht, where he supervised the promotion of Frank Kouwenhoven, Huib-Jan Imbens, J. Scholma and F. Wainschtein. At the Centre for Mathematics and Computer CWI in Amsterdam in 1988 he became Professor of Mathematics and head of the Department of Algebra and Geometry until his retirement in 2008. Hazewinkel has been managing editor for journals as Nieuw Archief voor Wiskunde since 1977, he was managing editor for the book series Mathematics and Its Applications for Kluwer Academic Publishers in 1977. Hazewinkel was member of 15 professional societies in the field of Mathematics, participated in numerous administrative tasks in institutes, Program Committee, Steering Committee, Consortiums and Boards.
In 1994 Hazewinkel was elected member of the International Academy of Computer Sciences and Systems. Hazewinkel has authored and edited several books, numerous articles. Books, selection: 1970. Géométrie algébrique-généralités-groupes commutatifs. With Michel Demazure and Pierre Gabriel. Masson & Cie. 1976. On invariants, canonical forms and moduli for linear, finite dimensional, dynamical systems. With Rudolf E. Kalman. Springer Berlin Heidelberg. 1978. Formal groups and applications. Vol. 78. Elsevier. 1993. Encyclopaedia of Mathematics. Ed. Vol. 9. Springer. Articles, a selection: Hazewinkel, Michiel. "Moduli and canonical forms for linear dynamical systems II: The topological case". Mathematical Systems Theory. 10: 363–385. Doi:10.1007/BF01683285. Archived from the original on 12 December 2013. Hazewinkel, Michiel. "On Lie algebras and finite dimensional filtering". Stochastics. 7: 29–62. Doi:10.1080/17442508208833212. Archived from the original on 12 December 2013. Hazewinkel, M.. J.. "Nonexistence of finite-dimensional filters for conditional statistics of the cubic sensor problem".
Systems & Control Letters. 3: 331–340. Doi:10.1016/0167-691190074-9. Hazewinkel, Michiel. "The algebra of quasi-symmetric functions is free over the integers". Advances in Mathematics. 164: 283–300. Doi:10.1006/aima.2001.2017. Homepage
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert; the development of homological algebra was intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied through both their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, topological spaces, other'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences. From its origins, homological algebra has played an enormous role in algebraic topology.
Its influence has expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes. Homological algebra began to be studied in its most basic form in the 1800s as a branch of topology, but it wasn't until the 1940s that it became an independent subject with the study of objects such as the ext functor and the tor functor, among others; the notion of chain complex is central in homological algebra. An abstract chain complex is a sequence of abelian groups and group homomorphisms, with the property that the composition of any two consecutive maps is zero: C ∙: ⋯ ⟶ C n + 1 ⟶ d n + 1 C n ⟶ d n C n − 1 ⟶ d n − 1 ⋯, d n ∘ d n + 1 = 0; the elements of Cn are called n-chains and the homomorphisms dn are called the boundary maps or differentials.
The chain groups Cn may be endowed with extra structure. The differentials must preserve the extra structure. For notational convenience, restrict attention to abelian groups; every chain complex defines two further sequences of abelian groups, the cycles Zn = Ker dn and the boundaries Bn = Im dn+1, where Ker d and Im d denote the kernel and the image of d. Since the composition of two consecutive boundary maps is zero, these groups are embedded into each other as B n ⊆ Z n ⊆ C n. Subgroups of abelian groups are automatically normal. A chain complex is called an exact sequence if all its homology groups are zero. Chain complexes arise in abundance in algebra and algebraic topology. For example, if X is a topological space the singular chains Cn are formal linear combinations of continuous maps from the standard n-simplex into X. In all these cases, there are natural differentials dn making Cn into a chain complex, whose homology reflects the structure of the topological space X, the simplicial complex K, or the abelian group A.
In the case of topological spaces, we arrive at the notion of singular homology, which plays a fundamental role in investigating the properties of such spaces, for example, manifolds. On a philosophical level, homological algebra teaches us that certain chain complexes associated with algebraic or geometric objects contain a lot of valuable algebraic information about them, with the homology being only the most available part. On a technical level, homological
In linear algebra, the Cayley–Hamilton theorem states that every square matrix over a commutative ring satisfies its own characteristic equation. If A is a given n×n matrix and In is the n×n identity matrix the characteristic polynomial of A is defined as p = det, where det is the determinant operation and λ is a scalar element of the base ring. Since the entries of the matrix are polynomials in λ, the determinant is an n-th order monic polynomial in λ; the Cayley–Hamilton theorem states that substituting the matrix A for λ in this polynomial results in the zero matrix, p = 0. The powers of A, obtained by substitution from powers of λ, are defined by repeated matrix multiplication; the theorem allows An to be expressed as a linear combination of the lower matrix powers of A. When the ring is a field, the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial; the theorem was first proved in 1853 in terms of inverses of linear functions of quaternions, a non-commutative ring, by Hamilton.
This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. The theorem holds for general quaternionic matrices. Cayley in 1858 stated it for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case; the general case was first proved by Frobenius in 1878. For a 1×1 matrix A =, the characteristic polynomial is given by p = λ − a, so p = − a1,1 = 0 is obvious; as a concrete example, let A =. Its characteristic polynomial is given by p = det = det = − = λ 2 − 5 λ − 2; the Cayley–Hamilton theorem claims that, if we define p = X 2 − 5 X − 2 I 2 p = A 2 − 5 A − 2 I 2 =. We can verify by computation that indeed, A 2 − 5 A − 2 I 2 = − − =. For a generic 2×2 matrix, A =, the characteristic polynomial is given by p = λ2 − λ +, so the Cayley–Hamilton theorem states that p = A 2 − A + I 2 =. For a general n×n invertible matrix A, i.e. one with nonzero determinant, A−1 can thus be written as an -th order polynomial expression in A: As indicated, the Cayley–Hamilton theorem amounts to the identity The coefficients ci are given by the elementary symmetric polynomials of the eigenvalues of A.
Using Newton identities, the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenval
In mathematics, a ring is one of the fundamental algebraic structures used in abstract algebra. It consists of a set equipped with two binary operations that generalize the arithmetic operations of addition and multiplication. Through this generalization, theorems from arithmetic are extended to non-numerical objects such as polynomials, series and functions. A ring is an abelian group with a second binary operation, associative, is distributive over the abelian group operation, has an identity element. By extension from the integers, the abelian group operation is called addition and the second binary operation is called multiplication. Whether a ring is commutative or not has profound implications on its behavior as an abstract object; as a result, commutative ring theory known as commutative algebra, is a key topic in ring theory. Its development has been influenced by problems and ideas occurring in algebraic number theory and algebraic geometry. Examples of commutative rings include the set of integers equipped with the addition and multiplication operations, the set of polynomials equipped with their addition and multiplication, the coordinate ring of an affine algebraic variety, the ring of integers of a number field.
Examples of noncommutative rings include the ring of n × n real square matrices with n ≥ 2, group rings in representation theory, operator algebras in functional analysis, rings of differential operators in the theory of differential operators, the cohomology ring of a topological space in topology. The conceptualization of rings was completed in the 1920s. Key contributors include Dedekind, Hilbert and Noether. Rings were first formalized as a generalization of Dedekind domains that occur in number theory, of polynomial rings and rings of invariants that occur in algebraic geometry and invariant theory. Afterward, they proved to be useful in other branches of mathematics such as geometry and mathematical analysis; the most familiar example of a ring is the set of all integers, Z, consisting of the numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …The familiar properties for addition and multiplication of integers serve as a model for the axioms for rings. A ring is a set R equipped with two binary operations + and · satisfying the following three sets of axioms, called the ring axioms R is an abelian group under addition, meaning that: + c = a + for all a, b, c in R. a + b = b + a for all a, b in R.
There is an element 0 in R such that a + 0 = a for all a in R. For each a in R there exists −a in R such that a + = 0. R is a monoid under multiplication, meaning that: · c = a · for all a, b, c in R. There is an element 1 in R such that a · 1 = a and 1 · a = a for all a in R. Multiplication is distributive with respect to addition, meaning that: a ⋅ = + for all a, b, c in R. · a = + for all a, b, c in R. As explained in § History below, many authors follow an alternative convention in which a ring is not defined to have a multiplicative identity; this article adopts the convention that, unless otherwise stated, a ring is assumed to have such an identity. A structure satisfying all the axioms except the requirement that there exists a multiplicative identity element is called a rng. For example, the set of integers with the usual + and ⋅ is a rng, but not a ring; the operations + and ⋅ are called multiplication, respectively. The multiplication symbol ⋅ is omitted, so the juxtaposition of ring elements is interpreted as multiplication.
For example, xy means x ⋅ y. Although ring addition is commutative, ring multiplication is not required to be commutative: ab need not equal ba. Rings that satisfy commutativity for multiplication are called commutative rings. Books on commutative algebra or algebraic geometry adopt the convention that ring means commutative ring, to simplify terminology. In a ring, multiplication does not have to have an inverse. A commutative ring such; the additive group of a ring is the ring equipped just with the structure of addition. Although the definition assumes that the additive group is abelian, this can be inferred from the other ring axioms; some basic properties of a ring follow from the axioms: The additive identity, the additive inverse of each element, the multiplicative identity are unique. For any element x in a ring R, one has x0 = 0 = 0x and x = –x. If 0 = 1 in a ring R R has only one element, is called the zero ring; the binomial formula holds for any commuting pair of elements. Equip the set Z 4 = with the following operat