database MathematicsDomain ring theory , a nontrivial ring without left or right zero divisors Integral ... query language for the relational data model Domain theory , a branch of mathematics that studies special ...NOTOC Wiktionary domainDomain may refer to Domain can be used for a name and science General Territory ... government Public domain , a body of works and knowledge without proprietary interest Eminent domain , the power of government to confiscate private property for public use Steve Alten Domain trilogy Domain trilogy is a trilogy of books regarding the Mayanism December 21.2C 2012 Mayan 2012 myths , written by Steve Alten Domain board game Domain , a game published by Parker Brothers in 1983 Sciences Domain biology , a subdivision even larger than a kingdom Domain knowledge , a specific expert knowledge valid for a pre selected area of activity, such as surgery Domain specificity , a theoretical ... Domain wall , a term used in physics which can have one of two distinct but similar meanings in either magnetism or string theory Magnetic domain , a region within a magnetic material which has uniform magnetization Protein domain , a part of a protein that can exist independently of the rest of the protein chain Information technology Administrative domain , a service provider holding a security repository permitting to easily authenticate and authorize clients with credentials Application domain , the kinds of purposes for which users use a software system Broadcast domain , in computer networking, a group of special purpose addresses to receive network announcements Clock domain crossing , when a signal crosses from one clock domain into another CLR application domain , a mechanism for separating executed applications similar to a process Collision domain , a physical network segment that is a shared medium where data packets can collide with one another Data domain , in database theory, a set of all permitted values Domain software engineering , a field of study that defines a set ... more details
uses see Mathematics disambiguation and Math disambiguation . File Euclid.jpg thumb Euclid , Greek ... . ref Mathematics from Greek language Greek m th ma , knowledge, study, learning is the study ... reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement ... mathematics has been a human activity for as far back as History of Mathematics written records exist. Logic Rigorous arguments first appeared in Greek mathematics , most notably in Euclid Euclid s Euclid s Elements Elements . Mathematics developed at a relatively slow pace until the Renaissance , when ... of Mathematics 1. Newton and Leibniz , BBC Radio 4 , 27 09 2010. ref Carl Friedrich Gauss 1777 1855 referred to mathematics as the Queen of the Sciences . ref name Waltershausen Waltershausen ref Benjamin Peirce 1809 1880 called mathematics the science that draws necessary conclusions . ref Peirce, p. 97. ref David Hilbert said of mathematics We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules. Rather, it is a conceptual ... , Basel, Birkh user 1992 . ref Albert Einstein 1879 1955 stated that as far as the laws of mathematics ... . ref name certain Mathematics is used throughout the world as an essential tool in many fields, including natural science , engineering , medicine , and the social sciences . Applied mathematics , the branch of mathematics concerned with application of mathematical knowledge to other fields ... in pure mathematics , or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, and practical applications for what began as pure mathematics are often discovered. ref Peterson ref Etymology The word mathematics comes from ... of which mean to learn . The word mathematics in Greek came to have the narrower and more technical ... more details
In mathematics , a tube may refer to A cylinder geometry cylinder from elementary geometry A tube domain in several complex variables A tubular neighborhood in differential geometry mathdab ... more details
Notability date October 2011 A programming domain defines a specific kind of use for a programming language . Some examples of programming domains are Application software General purpose applications Rapid software prototyping Financial time series analysis Natural language processing Artificial intelligence reasoning Expert systems Relational database querying Theorem proving Systems design and implementation Application scripting Domain specific applications Programming education Internet Symbolic mathematics Numerical mathematics Statistical applications Text processing Matrix algorithms See also Domain specific language Unreferenced date June 2007 compu lang stub Category Programming language topics Domain Category Computer languages ... more details
Unreferenced date December 2009 Domain knowledge is that valid knowledge used to refer to an area of human endeavour, an autonomous computer activity, or other specialized discipline. Domain expert Specialists and experts use and develop their own domain knowledge. If the concept domain knowledge or domain expert is used, we emphasize a specific domain which is an object of the discourse interest problem. Knowledge capture More particular, in software engineering , domain knowledge is knowledge about the environment in which the target system operates, for example, software agent s. Domain knowledge is important, because it usually must be learned from software users in the domain as domain specialists experts , rather than from software developers. Expert s domain knowledge frequently informal and ill structured is transformed in computer programs and active data, for example in a set of rules in knowledge bases, by knowledge engineer s. Communicating between end users and software developers is often difficult. They must find a common language to communicate in. Developing enough shared vocabulary to communicate can often take a while. The same knowledge can be included in different domain knowledge. Knowledge which may be efficient in every domain is called domain independent knowledge, for example logic s and mathematics. Operations on domain knowledge are performed by meta knowledge . Literature Hj rland, B. & Albrechtsen, H. 1995 . Toward A New Horizon in Information Science Domain Analysis. Journal of the American Society for Information Science, 1995, 46 6 , 400 425. See also Domain engineering Knowledge engineering Problem domain Artificial Intelligence DEFAULTSORT Domain Knowledge Category Knowledge az Elm sah si de Wissensgebiet ... more details
Unreferenced stub auto yes date December 2009 In the formal sciences , the domain of discourse , also called the universe of discourse or simply universe , is the set mathematics set of entities over which certain variable mathematics variable s of interest in some formal treatment may range. The domain of discourse is usually identified in the preliminaries, so that there is no need in the further treatment to specify each time the range of the relevant variables. For example, in an interpretation logic interpretation of first order logic , the domain of discourse is the set of individuals that the quantifier s range over. In one interpretation, the domain of discourse could be the set of real number s in another interpretation, it could be the set of natural number s. If no domain of discourse has been identified, a proposition such as math x x sup 2 sup 2 is ambiguous. If the domain of discourse is the set of real numbers, the proposition is false, with math 1 x 2 as counterexample if the domain is the set of naturals, the proposition is true, since 2 is not the square of any natural number. The term universe of discourse generally refers to the collection of objects being discussed in a specific discourse. In model theoretical semantics, a universe of discourse is the set of entities that a model is based on. The term universe of discourse is generally attributed to Augustus De Morgan 1846 and was also used by George Boole 1854 in his The Laws of Thought Laws of Thought . A database is a model of some aspect of the reality of an organisation. It is conventional to call this reality the universe of discourse or domain of discourse . citation needed date February 2011 See also Wiktionary Universe mathematics Term algebra DomainmathematicsDomain theory Interpretation logic DEFAULTSORT Domain Of Discourse Category Semantics Category Predicate logic Logic stub ca Domini de discurs de Diskursuniversum es Dominio de discurso fr Univers du discours ja pt Universo ... more details
Unreferenced date December 2009 Expert subject Mathematics date November 2008 In computing, the attribute domain is the set of Value computer science value s allowed in an Attribute computing attribute . For example Rooms in hotel 1 300 Age 1 99 Married yes or no Nationality Sri Lankan, Indian, American, or British For the relational model it is a requirement that each part of a tuple be atomic. The consequence is that each value in the tuple must be of some basic type, like a String computer science string or an integer . For the elementary type to be atomic it cannot be broken into more pieces. Alas, the domain is an elementary type, and attribute domain the domain a given attribute belongs to an abstraction belonging to or characteristic of an entity. DEFAULTSORT Attribute Domain Category Type theory Category Database theory ... more details
In mathematics, a GCD domain is an integral domain R with the property that any two non zero elements ... Ring Theory publisher Springer date 2000 series Mathematics and Its Applications isbn 0792364929 language English page 479 ref A GCD domain generalizes a unique factorization domain to the non Noetherian setting in the following sense an integral domain is a UFD if and only if it is a GCD domain ... . Properties Every irreducible element of a GCD domain is prime however irreducible elements need not exist, even if the GCD domain is not a field . A GCD domain is integrally closed , and every nonzero ... proof ref In other words, every GCD domain is a Schreier domain . For every pair of elements x , y of a GCD domain R , a GCD d of x and y and a LCM m of x and y can be chosen such that nowrap ... denotes the equivalence relation of being associate elements . If R is a GCD domain, then the polynomial ring R X sub 1 sub ,..., X sub n sub is also a GCD domain, and more generally, the group ring R G is a GCD domain for any torsion free commutative group G . ref Robert W. Gilmer, Commutative semigroup rings , University of Chicago Press, 1984, p. 172. ref For a polynomial in X over a GCD domain ... , which is valid over GCD domains. Examples A unique factorization domain is a GCD domain. Among the GCD domains, the unique factorization domains are precisely those that are also atomic domain s which ... . A B zout domain i.e., an integral domain where every finitely generated ideal is principal is a GCD domain. Unlike principal ideal domain s where every ideal is principal , a B zout domain need not be a unique factorization domain for instance the ring of entire function s is a non atomic B zout domain, and there are many other examples. An integral domain is a Pr fer domain Pr fer GCD domain if and only if it is a B zout domain. Fact date April 2009 If R is a non atomic GCD domain, then R X is an example of a GCD domain that is neither a unique factorization domain since it is non atomic ... more details
internalize previously excluded areas of interest within a problem domain. In mathematics, the term defines a Domainmathematicsdomain where the parameter s defining the boundaries of the domain and sufficient map pings into a set mathematics set of range s including itself are not well enough understood to provide a systematic description of the domain. This would be a target space of meta tools designed to explore a search space . Alternatively, a domain specifically defined by some extrinsic problem system to differentiate it from the set of all domains. See domain theory for the mathematical ...Merge Application domain date February 2010 A problem domain is the area of expertise or application that needs to be examined to solve a problem . A problem domain is simply looking at only the topics you are interested in, and excluding everything else. For example, if you were developing a system trying to measure good practice in medicine, you wouldn t include carpet drawings at hospitals in your problem domain. In this example the domain refers to relevant topics solely within your interest medicine. This points to one of the limitations of overly specific and bounded problem domains, one may think they are interested in medicine and not interior design, but a better solution exists outside of the problem domain as it was initially conceived. For example, when IDEO researchers noticed ... Although not originally within the bounded problem domain of measuring good practices in medicine, this non intuitive finding could then be added to the domain space. Arational, problem seeking and non ... a domain as a minimal set of sources for mappings relative to the problem a specific instance of applying Occam s Razor . Having defined a specific problem domain with sufficient parameters and mappings ... problem domain, and its immediate mappings should not be included within the problem domain, but should ... domain analysis Domain model References Reflist Category Systems engineering Category Data modeling ... more details
for Industrial and Applied Mathematics volume 1 year 1953 pages 35 51 ref See Time domain Origin of term ...In electronics , control systems engineering , and statistics , frequency domain is a term used to describe the domain for analysis of mathematical function s or Signal information theory signals with respect ... domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. A frequency domain ... of mathematical Operator mathematics operator s called a Transform mathematics transform . An example ... of sine wave frequency components. The spectrum of frequency components is the frequency domain representation of the signal. The inverse Fourier transform converts the frequency domain function ... in the frequency domain. Note that recent advances in the field of signal processing have also allowed to define representations or transforms that result in a joint time frequency domain, with the instantaneous frequency being a key link between the time domain and the frequency domain. Magnitude ..., the frequency spectrum is complex, describing the Magnitude mathematics magnitude and phase ... the information in a frequency domain representation to generate a frequency spectrum or spectral ... is a frequency domain description that can be applied to a large class of signals that are neither ... of a wide sense stationary random process. Different frequency domains Although the frequency domain ... to analyze time functions and are referred to as frequency domain methods. These are the most ... of the visible anchor transform domain with respect to any transform. The above transforms can be interpreted as capturing some form of frequency, and hence the transform domain is referred to as a frequency domain. Discrete frequency domain The Fourier transform of a periodic signal only has energy ... using a discrete frequency domain . Dually, a discrete time signal gives rise to a periodic ... more details
Unreferenced date December 2009 Other uses Restriction disambiguation In mathematics , the notion of restriction of a function is defined as follows If f E F is a function mathematics function from E to F , and A is a subset of E , then the restriction of f to A is the partial function math f A A to F math having the graph math G f A x,y in G f mid x in A math . In rough words, it is the same function , but only defined on math A cap mathrm dom , f math . More generally, the restriction or domain restriction or left restriction A R of a binary relation R between E and F may be defined as a relation having domain A , codomain F and graph G A R x , y G R x A . Similarly, one can define a right restriction or range restriction R B . Indeed, one could define a restriction to a subset of E x F , and the same applies to n ary Relation mathematics relations . These cases do not fit into the scheme of sheaf mathematics sheaves . The domain anti restriction or domain subtraction of a function or binary relation R with domain E and codomain F by a set A may be defined as E A R it removes all elements of A from the domain E . It is sometimes denoted A R . Similarly, the range anti restriction or range subtraction of a function or binary relation R by a set B is defined as R F B it removes all elements of B from the codomain F . It is sometimes denoted R B . Examples The restriction of the non injective function math f mathbb R to mathbb R x mapsto x 2 math to math mathbb R 0, infty math is the injection math f mathbb R to mathbb R x mapsto x 2 math . The inclusion map of a set A into a superset E of A is the restriction of the identity function on E to A . See also Deformation retract Function mathematics Restrictions and extensions Binary relation Restriction DEFAULTSORT Restriction Mathematics Category Sheaf theory ca Restricci matem tiques cs Restrikce zobrazen de Einschr nkung es Restricci n matem ticas it Restrizione di una funzione ru fi Rajoittuma ... more details
Image Star shaped.png right thumb A star domain equivalently, a star convex or star shaped set is not necessarily convex set convex in the ordinary sense. Image Not star shaped.svg right thumb An annulus mathematics annulus is not a star domain. In mathematics , a Set mathematics set math S math in the Euclidean space R sup n sup is called a star domain or star convex set , star shaped or radially convex set if there exists x sub 0 sub in S such that for all x in S the line segment from x sub 0 sub to x is in S . This definition is immediately generalizable to any real number real or complex number complex vector space . Intuitively, if one thinks of S as of a region surrounded by a wall, S is a star domain if one can find a vantage point x sub 0 sub in S from which any point x in S is within line of sight. Examples Any line or plane in R sup n sup is a star domain. A line or a plane with a single point removed is not a star domain. If A is a set in R sup n sup , the set math B ta a in A, t in 0,1 math obtained by connecting any point in A to the origin is a star domain. Any non empty convex set is a star domain. A set is convex if and only if it is a star domain with respect to any point in that set. A cross shaped figure is a star domain but is not convex. Properties The closure topology closure of a star domain is a star domain, but the interior topology interior of a star domain is not necessarily a star domain. Any star domain is a contractible space contractible set, via a straight line homotopy . In particular, any star domain is a simply connected set . The union and intersection of two star domains is not necessarily a star domain. A nonempty open star domain S in R sup n sup is diffeomorphism diffeomorphic to R sup n sup . See also Art gallery problem Star polygon &mdash an unrelated term Star shaped polygon Balanced set References Ian Stewart, David Tall, Complex Analysis . Cambridge University Press, 1983, ISBN 0 521 28763 4, mr 0698076 C.R. Smith, A characterization ... more details
Time domain is a term used to describe the analysis of mathematical function mathematics function s, physical signal information theory signal s or time series of economics economic or environmental statistics environmental data, with respect to time . In the time domain, the signal or function s value is known for all real number s, for the case of continuous time , or at various separate instants in the case of discrete time . An oscilloscope is a tool commonly used to visualize real world signals in the time domain. Speaking non technically, a time domain graph shows how a signal changes over time, whereas a frequency domain graph shows how much of the signal lies within each given frequency band over a range of frequencies. Origin of term The use of the contrasting terms time domain and frequency domain developed in US communication engineering in the 1950s and early 1960s, with the terms appearing together in 1961. ref http jeff560.tripod.com t.html Earliest Known Uses of Some of the Words of Mathematics T , Jeff Miller, March 25, 2009 ref ref citation first W. F. last Trench title A General Class of Discrete Time Invariant Filters journal Journal of the Society for Industrial and Applied Mathematics volume 9 year 1961 pages 405 421 ref See also Frequency domain References reflist Category Time domain analysis math stub statistics stub ca Domini temporal de Zeitbereich es Dominio del tiempo fr Domaine temporel it Dominio del tempo nl Tijddomein ja pt Dom nio do tempo ro Domeniu temporal zh ... more details
A plateau of a function mathematics function is a part of its domainmathematicsdomain where the function has constant value. More formally, let U , V be topological space s. A plateau for a function f U V is a path connected set of points P of U such that for some y we have f p y for all p in P . See also Level set Contour line PlanetMath attribution id 3374 title plateau Category Topology topology stub es Meseta matem ticas eo Altebena o matematiko ... more details
Orphan date August 2009 In mathematics , a Goldman domain A is an integral domain whose field of fractions is a finitely generated A algebra. ref name Ref Goldman domains ideals are called G domains ideals in Kaplansky 1974 . ref They are named after Oscar Goldman mathematician Oscar Goldman . An overring i.e., an intermediate ring lying between the ring and its field of fractions of a Goldman domain is again a Goldman domain. There exists a Goldman domain where all nonzero prime ideals are maximal although there are infinitely many prime ideals. ref name Ref a Kaplansky, pp. 13 ref An ideal ring theory ideal I in a commutative ring A is called a Goldman ideal if the quotient ring quotient A I is a Goldman domain. A Goldman ideal is thus prime ideal prime , but not necessarily maximal ideal maximal . In fact, a commutative ring is a Jacobson ring if and only if every Goldman ideal in it is maximal. The notion of a Goldman ideal can be used to give a slightly sharpened characterization of a radical of an ideal the radical of an ideal I is the intersection of all Goldman ideals containing I . Notes reflist References Citation last1 Kaplansky first1 Irving author1 link Irving Kaplansky title Commutative rings publisher University of Chicago Press edition Revised id MathSciNet id 0345945 isbn 0226424545 year 1974 DEFAULTSORT Goldman Domain Category Ring theory Abstract algebra stub ... more details
In mathematics, value commonly refers to the output of a Function mathematics function . In the most basic case, that of Unary function unary , single valued functions, there is one input the argument of a function argument and one output the value of the function . A real valued function is a Function mathematics function that associates to every element mathematics element of the domainmathematicsdomain a real number in the image mathematics image . Example If the function math f math is defined by prescribing that math f x 2x 2 3x 1 math for each real number math x math , then the input 3 will yield the function value 10 since indeed nowrap 1 2 3 sup 2 sup 3 3 1 10 . The function math f math is real valued, since each and every possible function value is real. On the other hand, it is not injective , since different inputs may yield the same value e.g., math f 1.5 10 math , too. In some contexts, for convenience, functions may be considered to have arity several arguments and or Multivalued function several values also cf. the discussion in the article Function mathematics Functions with multiple inputs and outputs function . However, strictly seen, this is not an extension, since such functions may be considered as having single families and or sets as input or output. Value is also used in other senses, e.g., to specify a certain instance of a Variable mathematics variable . Example math f x 0 math for two separate values of math x math , namely, for math x 0.5 math and for math x 1 math . See also Absolute value Function of a real variable Truth value Category Elementary mathematics math stub ar ca Funci real he nl Re el waardige functie sv V rde matematik ... more details
form a set called its domainmathematicsdomain . The set which contains the values produced is called the codomain , but the set of actual values attained by the operation is its range mathematics ... operation Binary operation Related topics col begin col break Arity Binary relation DomainmathematicsDomain col break Function mathematics Function Multigrade operator Operator mathematics col break ... hand, take two values, and include addition , subtraction , multiplication , division mathematics ... on Set mathematics sets include the binary operations union mathematics union and intersection mathematics intersection and the unary operation of complementation mathematics complementation . Operations on function mathematics function s include Function composition composition and convolution . Operations ... dissimilar objects. A vector can be multiplied by a scalar mathematics scalar to form another vector .... An operation is like an Operator mathematics operator , but the point of view is different. For instance ... mathematics function of the form V Y , where V X sub 1 sub X sub k sub . The sets X sub ... set indexing the arguments. Often, use of the term operation implies that the domain of the function ... be 1, in the most general sense given here, operation is synonymous with function mathematics function , mapping mathematics map and mapping mathematics mapping , that is, a relation mathematics relation , for which each element of the domain input set is associated with exactly one element of the codomain ... DEFAULTSORT Operation Mathematics Category Elementary mathematics Category Abstract algebra ar ... mathematics sk Matematick oper cia sl Matemati na operacija sr sh Operacija ... more details
Image Codomain2.SVG right thumb 250px Illustration showing f , a function from domain X to codomain Y . The smaller oval inside Y is the Image mathematics image of f , sometimes called the range mathematics range of f . In mathematics , the domain of definition or simply the domain of a function mathematics ... is defined. That is, the function provides an output or Value mathematics value for each member of the domain ..., the domain of cosine is the set of all real numbers , while the domain of the square root consists ... whose domain is a subset of the real numbers , when the function is represented in an xy Cartesian coordinate system , the domain is represented on the x axis. The image and caption below are problematic ... text decoration overline x span has domain that consists of all real numbers between 0 and positive infinity Formal definition This section is linked from Complex analysis Given a Function mathematics function f X Y , the set X is the domain of f the set Y is the codomain of f . In the expression f ..., and the value as the output. The image mathematics image sometimes called the range mathematics range ... . A well defined function must carry every element of its domain to an element of its codomain. For example ... s, math mathbb R math , cannot be its domain. In cases like this, the function is either defined on math ... of f to f x 1 x , for x 0, f 0 0, then f is defined for all real numbers, and its domain is math mathbb R math . Any function can be restricted to a subset of its domain. The restriction of g ... domain The natural domain of a formula is the set of values for which it is defined, typically within the reals but sometimes among the integers or complex numbers. For instance the natural domain of square ... a natural domain the set of possible values of the function is typically called its range. ref cite ... last2 Johnson page 60 year 1984 isbn 0 521 25012 9 publisher Cambridge University Pressd ref Domain of a partial function further2 Partial function Domain of a partial function There are two distinct ... more details
Other uses Null disambiguation In mathematics , the word null from German language German null , which is from Latin nullus , both meaning zero , or none ref name oed cite journal title null journal The Oxford English Dictionary , Draft Revision March 2004 url http dictionary.oed.com year 2004 accessdate 2007 04 05 ref means of or related to having Empty set zero members in a set or a value of zero . Sometimes the symbol is used to distinguish null from 0 . In a norm mathematics normed vector space the null vector vector space null vector is the zero vector in a norm mathematics seminormed vector space such as Minkowski space Causal structure Minkowski space , null vectors are, in general, non zero. In set theory , the empty set null set is the set with zero elements and in measure theory , a null set is a set with zero measure. A function mathematics mathematical mapping is said to be null potent or nilpotent if repeated application can map the whole domain into the null element. A null space of a mapping is the part of the domain that is mapped into the null element of the image the inverse image of the null element . In statistics , a null hypothesis is a proposition presumed true unless statistical evidence indicates otherwise. References references Category Mathematical objects Category Nothing ... more details
In mathematics , a B zout domain is an integral domain in which the sum of two principal ideal s is again ... finitely generated module finitely generated ideal is principal. Any principal ideal domain PID is a B zout domain, but a B zout domain need not be Noetherian ring , so it could have non finitely generated ideals which obviously excludes being a PID if so, it is not a unique factorization domain UFD , but still is a GCD domain . The theory of B zout domains retains many of the properties of PIDs ..., and so certainly not a PID. The following general construction produces a B zout domain S that is not a UFD from any B zout domain R that is not a field, for instance from a PID the case nowrap R Z ... generated and so X has no factorization in S . One shows as follows that S is a B zout domain ... constant polynomial r in R lies in nowrap aS bS . Also, since R is a B zout domain, the gcd ... as well, which completes the proof. Properties A ring is a B zout domain if and only if it is an integral domain in which any two elements have a greatest common divisor that is a linear combination ... than the mere existence of a gcd. An integral domain where a gcd exists for any two elements is called a GCD domain and thus B zout domains are GCD domains. In particular, in a B zout domain ..., they need not exist . For a B zout domain R , the following conditions are all equivalent R is a principal ideal domain. R is Noetherian. R is a unique factorization domain UFD . R satisfies the Ascending ... nonzero nonunit in R factors into a product of irreducibles R is an atomic domain . The equivalence of 1 and 2 was noted above. Since a B zout domain is a GCD domain, it follows immediately that 3 ... chain of finitely generated ideals, so in a B zout domain an infinite ascending chain of principal ideals. 4 and 2 are thus equivalent. A B zout domain is a Pr fer domain , i.e., a domain in which each ... domain. Roughly speaking, one may view the implications B zout domain implies Pr fer domain ... more details
In convex analysis , a branch of mathematics, the effective domain is an extension of the domain of a function . Given a vector space X then a convex function mapping to the extended reals , math f X to mathbb R cup pm infty math , has an effective domain defined by math operatorname dom x in X f x infty . , math ref name AB cite book last1 Aliprantis first1 C.D. last2 Border first2 K.C. title Infinite Dimensional Analysis A Hitchhiker s Guide edition 3 publisher Springer year 2007 isbn 978 3 540 32696 0 doi 10.1007 3 540 29587 9 page 254 ref ref cite book first1 Hans last1 F llmer first2 Alexander last2 Schied title Stochastic finance an introduction in discrete time publisher Walter de Gruyter year 2004 edition 2 isbn 978 3 11 018346 7 page 400 ref If the function is concave function concave , then the effective domain is math operatorname dom f x in X f x infty . , math ref name AB The effective domain is equivalent to the projection of the epigraph of a function math f X to mathbb R cup pm infty math onto X . That is math operatorname dom f x in X exists y in mathbb R x,y in operatorname epi f . , math ref name Rockafellar cite book author Rockafellar, R. Tyrrell title Convex Analysis publisher Princeton University Press location Princeton, NJ year 1997 origyear 1970 isbn 978 0 691 01586 6 page 23 ref Note that if a convex function is mapping to the normal real number line given by math f X to mathbb R math then the effective domain is the same as the normal definition of the domain. A function math f X to mathbb R cup pm infty math is a proper convex function if and only if the effective domain of f is nonempty and math f x infty math for every math x in X math . ref name Rockafellar References Reflist math stub Category Convex analysis Category Functions and mappings ... more details
Other uses Data domain disambiguation In data management and database analysis, a data domain refers to all the unique values which a data element may contain. The rule for determining the domain boundary may be as simple as a data type with an enumeration enumerated list of values. ref cite web url http books.google.co.uk books?id 3BXTfCtR8zsC&pg PA302 title Enterprise knowledge management the data quality approach first David last Loshin work books.google.co.uk year 2001 ISBN 978 0124558403 accessdate 19 August 2011 ref For example, a database table database table that has information about people, with one record per person, might have a gender column . This gender column might be declared as a Data type Strings string data type , and allowed to have one of two known Code metadata code values M for male, F for female and Null SQL NULL for records where gender is unknown or not applicable or arguably U for unknown as a sentinel value . The data domain for the gender column is M , F . In a database normalization normalized data model , the Master data management reference domain is typically specified in a reference table . Following the previous example, a Gender reference table would have exactly two records, one per allowed value excluding NULL. Reference tables are formally related to other tables in a database by the use of foreign key s. Less simple domain boundary rules, if database enforced, may be implemented through a check constraint or, in more complex cases, in a database ... declaring that the values must be greater than zero. This definition combines the concepts of domain as an area over which control is exercised and the mathematical idea of a set mathematics set of values of an independent variable for which a function mathematics function is defined. See domainmathematics . References reflist See also Data modeling Foreign key ISO IEC 11179 Metadata standards Master data management Database normalization Primary key Relational database DEFAULTSORT Data Domain ... more details
example is the ring Z of all integer s. Every field mathematics field is an integral domain. Conversely, every artinian ring Artinian integral domain is a field. In particular, all finite integral domains are finite field s more generally, by Wedderburn s little theorem , finite Domain ring theory ...Merge to Domain ring theory date February 2012 In abstract algebra , an integral domain is a commutative ... of the integer s and provide a natural setting for studying divisibility. An integral domain is a commutative domain ring theory domain with identity. ref Rowen 1994 , Google books EmO9ejuMHNUC p. 99 page 99 . ref The above is how integral domain is almost universally defined, but there is some ... and J.C. Robson Noncommutative Noetherian Rings Graduate studies in Mathematics Vol. 30, AMS ref However, we follow the much more usual convention of reserving the term integral domain for the commutative case and use domain ring theory domain for the noncommutative case curiously, the adjective ... entire ring for integral domain. ref Pages 91 92 of Lang Algebra edition 3 ref Some specific kinds ... ring s integral domains integrally closed domain s unique factorization domain s principal ideal domain s Euclidean domain s field mathematics field s The absence of zero divisor s means that in an integral domain the cancellation property holds for multiplication by any nonzero element a an equality nowrap ab ac implies nowrap b c . Algebraic structures cTopic ring mathematics Ring like structures Definitions There are a number of equivalent definitions of integral domain An integral domain ... to zero. An integral domain is a commutative ring with identity in which the zero ideal ring theory ideal 0 is a prime ideal . An integral domain is a ring with identity that is a subring of a field. This means it is also a commutative ring with identity. An integral domain is a commutative ring ... integral domain that is not a field, possessing infinite descending sequences of ideals such as math ... more details
, and a total map is a total function . Related terms like Domainmathematicsdomain , codomain ...Unreferenced date November 2009 otheruses Map disambiguation In most of mathematics and in some related technical fields, the term mapping , usually shortened to map , is either a synonym for Function mathematics function , or denotes a particular kind of function which is important in that branch, or denotes something conceptually similar to a function. In graph theory , a map is a drawing of a graph mathematics graph on a surface without overlapping edges a planar graph , similar to a political map . Maps as functions In many branches of mathematics, the term is used to mean a function with a specific property of particular importance to that branch. For instance, a map is a continuous function in topology , a linear map linear transformation in linear algebra , etc. In Wikipedia, we always include a relevant adjective like continuous or smooth to avoid confusion . In contrast, in category theory , map is often used as a synonym for morphism or arrow, thus for something more general than a function. Some authors, such as Serge Lang , use map as a general term for an association of an element in the range with each element in the domain, and function only to refer to maps in which the range is a Field mathematics field . Sets of maps of special kinds are the subjects of many important theories see for instance Lie group , mapping class group , permutation group . many more to add here In formal logic , the term is sometimes used for a functional predicate , whereas a function is a model logic model of such a Predicate logic predicate in set theory . In the theory of dynamical system s, a map denotes an Discrete time dynamical system evolution function used to create Dynamical ... Correspondence mathematics Homeomorphism Homomorphism List of chaotic maps Mapping class group Morphism Projection mathematics Topology DEFAULTSORT Map Mathematics Category Functions and mappings ... more details
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