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List Of Things Named After Richard Dedekind
This is a list of things named after Richard Dedekind. Richard Dedekind (1831–1916), a mathematician, is the eponym of all of the things (and topics) listed below. * 19293 Dedekind *Cantor–Dedekind axiom *Dedekind completeness *Dedekind cut *Dedekind discriminant theorem *Dedekind domain *Dedekind eta function * Dedekind function *Dedekind group *Dedekind number ** Dedekind's problem * Dedekind–Peano axioms *Dedekind psi function *Dedekind ring *Dedekind sum *Dedekind valuation *Dedekind zeta function *Dedekind–Hasse norm *Dedekind-infinite set *Dedekind–MacNeille completion * Dedekind's axiom *Dedekind's complementary module * Dedekind lattice *Jordan–Dedekind lattice A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bou ... * Dedekind's theorem on ellipsoids of equilibrium {{ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Dedekind
Julius Wilhelm Richard Dedekind (6 October 1831 – 12 February 1916) was a German mathematician who made important contributions to number theory, abstract algebra (particularly ring theory), and the axiomatic foundations of arithmetic. His best known contribution is the definition of real numbers through the notion of Dedekind cut. He is also considered a pioneer in the development of modern set theory and of the philosophy of mathematics known as '' Logicism''. Life Dedekind's father was Julius Levin Ulrich Dedekind, an administrator of Collegium Carolinum in Braunschweig. His mother was Caroline Henriette Dedekind (née Emperius), the daughter of a professor at the Collegium. Richard Dedekind had three older siblings. As an adult, he never used the names Julius Wilhelm. He was born in Braunschweig (often called "Brunswick" in English), which is where he lived most of his life and died. He first attended the Collegium Carolinum in 1848 before transferring to the Univ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Ring
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors. There are at least three other characterizations of Dedekind domains that are sometimes taken as the definition: see below. A field is a commutative ring in which there are no nontrivial proper ideals, so that any field is a Dedekind domain, however in a rather vacuous way. Some authors add the requirement that a Dedekind domain not be a field. Many more authors state theorems for Dedekind domains with the implicit proviso that they may require trivial modifications for the case of fields. An immediate consequence of the definition is that every principal ideal domain (PID) is a Dedekind domain. In fact a Dedekind domain is a unique factorization domain (UFD) if and only if it is a PID. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jordan–Dedekind Lattice
A lattice is an abstract structure studied in the mathematical subdisciplines of order theory and abstract algebra. It consists of a partially ordered set in which every pair of elements has a unique supremum (also called a least upper bound or join) and a unique infimum (also called a greatest lower bound or meet). An example is given by the power set of a set, partially ordered by inclusion, for which the supremum is the union and the infimum is the intersection. Another example is given by the natural numbers, partially ordered by divisibility, for which the supremum is the least common multiple and the infimum is the greatest common divisor. Lattices can also be characterized as algebraic structures satisfying certain axiomatic identities. Since the two definitions are equivalent, lattice theory draws on both order theory and universal algebra. Semilattices include lattices, which in turn include Heyting and Boolean algebras. These ''lattice-like'' st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Lattice
In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition, ;Modular law: implies where are arbitrary elements in the lattice, ≤ is the partial order, and ∨ and ∧ (called join and meet respectively) are the operations of the lattice. This phrasing emphasizes an interpretation in terms of projection onto the sublattice , a fact known as the diamond isomorphism theorem. An alternative but equivalent condition stated as an equation (see below) emphasizes that modular lattices form a variety in the sense of universal algebra. Modular lattices arise naturally in algebra and in many other areas of mathematics. In these scenarios, modularity is an abstraction of the 2nd Isomorphism Theorem. For example, the subspaces of a vector space (and more generally the submodules of a module over a ring) form a modular lattice. In a not necessarily modular lattice, there may stil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind's Complementary Module
In algebraic number theory, the different ideal (sometimes simply the different) is defined to measure the (possible) lack of duality in the ring of integers of an algebraic number field ''K'', with respect to the field trace. It then encodes the ramification data for prime ideals of the ring of integers. It was introduced by Richard Dedekind in 1882. Definition If ''O''''K'' is the ring of integers of ''K'', and ''tr'' denotes the field trace from ''K'' to the rational number field Q, then : x \mapsto \mathrm~x^2 is an integral quadratic form on ''O''''K''. Its discriminant as quadratic form need not be +1 (in fact this happens only for the case ''K'' = Q). Define the ''inverse different'' or ''codifferent'' or ''Dedekind's complementary module'' as the set ''I'' of ''x'' ∈ ''K'' such that tr(''xy'') is an integer for all ''y'' in ''O''''K'', then ''I'' is a fractional ideal of ''K'' containing ''O''''K''. By definition, the different ideal δ''K'' is the inverse fraction ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Geometry
Ordered geometry is a form of geometry featuring the concept of intermediacy (or "betweenness") but, like projective geometry, omitting the basic notion of measurement. Ordered geometry is a fundamental geometry forming a common framework for affine, Euclidean, absolute, and hyperbolic geometry (but not for projective geometry). History Moritz Pasch first defined a geometry without reference to measurement in 1882. His axioms were improved upon by Peano (1889), Hilbert (1899), and Veblen (1904). Euclid anticipated Pasch's approach in definition 4 of ''The Elements'': "a straight line is a line which lies evenly with the points on itself". Primitive concepts The only primitive notions in ordered geometry are points ''A'', ''B'', ''C'', ... and the ternary relation of intermediacy 'ABC''which can be read as "''B'' is between ''A'' and ''C''". Definitions The ''segment'' ''AB'' is the set of points ''P'' such that 'APB'' The ''interval'' ''AB'' is the segment ''AB'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind–MacNeille Completion
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dedekind cuts used by Dedekind to construct the real numbers from the rational numbers. It is also called the completion by cuts or normal completion. Order embeddings and lattice completions A partially ordered set (poset) consists of a set of elements together with a binary relation on pairs of elements that is reflexive ( for every ''x''), transitive (if and then ), and antisymmetric (if both and hold, then ). The usual numeric orderings on the integers or real numbers satisfy these properties; however, unlike the orderings on the numbers, a partial order may have two elements that are ''incomparable'': neither nor holds. Another familiar example of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind-infinite Set
In mathematics, a set ''A'' is Dedekind-infinite (named after the German mathematician Richard Dedekind) if some proper subset ''B'' of ''A'' is equinumerous to ''A''. Explicitly, this means that there exists a bijective function from ''A'' onto some proper subset ''B'' of ''A''. A set is Dedekind-finite if it is not Dedekind-infinite (i.e., no such bijection exists). Proposed by Dedekind in 1888, Dedekind-infiniteness was the first definition of "infinite" that did not rely on the definition of the natural numbers. A simple example is \mathbb, the set of natural numbers. From Galileo's paradox, there exists a bijection that maps every natural number ''n'' to its square ''n''2. Since the set of squares is a proper subset of \mathbb, \mathbb is Dedekind-infinite. Until the foundational crisis of mathematics showed the need for a more careful treatment of set theory, most mathematicians assumed that a set is infinite if and only if it is Dedekind-infinite. In the early twentieth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind–Hasse Norm
In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function (mathematics), function on an integral domain that generalises the notion of a Euclidean domain#Definition, Euclidean function on Euclidean domains. Definition Let ''R'' be an integral domain and ''g'' : ''R'' → Z≥0 be a function from ''R'' to the non-negative integers. Denote by 0''R'' the additive identity of ''R''. The function ''g'' is called a ''Dedekind–Hasse norm'' on ''R'' if the following three conditions are satisfied: * ''g''(''a'') = 0 if and only if ''a'' = 0''R'', * for any nonzero elements ''a'' and ''b'' in ''R'' either: ** ''b'' divisibility (ring theory), divides ''a'' in ''R'', or ** there exist elements ''x'' and ''y'' in ''R'' such that 0 < ''g''(''xa'' − ''yb'') < ''g''(''b''). The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Zeta Function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at ''s'' = 1, and its values encode arithmetic data of ''K''. The extended Riemann hypothesis states that if ''ζ''''K''(''s'') = 0 and 0 1. In the case ''K'' = Q, this definition reduces to that of the Riemann zeta function. Euler product The Dedekind zeta function of K has an Euler product which is a product over all the prime ideals \mathfrak of \mathcal_K :\zeta_K (s) = \prod_ \frac,\text(s)>1. This is the expression in analytic terms of the uniqueness of prime factorization of id ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Valuation
In algebra (in particular in algebraic geometry or algebraic number theory), a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field. Definition One starts with the following objects: *a field and its multiplicative group ''K''×, *an abelian totally ordered group . The ordering and group law on are extended to the set by the rules * for all ∈ , * for all ∈ . Then a valuation of is any map : which satisfies the following properties for all ''a'', ''b'' in ''K'': * if and only if , *, *, with equality if ''v''('' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Sum
In mathematics, Dedekind sums are certain sums of products of a sawtooth function, and are given by a function ''D'' of three integer variables. Dedekind introduced them to express the functional equation of the Dedekind eta function. They have subsequently been much studied in number theory, and have occurred in some problems of topology. Dedekind sums have a large number of functional equations; this article lists only a small fraction of these. Dedekind sums were introduced by Richard Dedekind in a commentary on fragment XXVIII of Bernhard Riemann's collected papers. Definition Define the sawtooth function (\!( \, )\!) : \mathbb \rightarrow \mathbb as :(\!(x)\!)=\begin x-\lfloor x\rfloor - 1/2, &\mboxx\in\mathbb\setminus\mathbb;\\ 0,&\mboxx\in\mathbb. \end We then let :D: \mathbb^2\times (\mathbb-\)\to \mathbb be defined by :D(a,b;c)=\sum_ \left(\!\!\left( \frac \right)\!\!\right) \! \left(\!\!\left( \frac \right)\!\!\right), the terms on the right being the Dedekin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
