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 ... system s, a map denotes an Discrete time dynamical system evolution function used to create Dynamical system Maps discrete dynamical systems . See also Poincar map . A partial map is a partial function , and a total map is a total function . Related terms like Domain mathematics domain , codomain ... Correspondence mathematics Homeomorphism Homomorphism List of chaotic maps Mapping class group Morphism Projection mathematics Topology DEFAULTSORT MapMathematics Category Functions and mappings ... 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 ..., then mathematical reasoning often provides insight or predictions. Through the use of abstraction mathematics abstraction and logic al reasoning , mathematics developed from counting , calculation , measurement .... Practical 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 ... History 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 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 ..., 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, inspires ... 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 the ancient ... mean to learn . The word mathematics in Greek came to have the narrower and more technical meaning ... until around 1700, the term mathematics more commonly meant astrology or sometimes astronomy ... more details
The Interface for Metadata Access Points IF MAP is an open standard client server protocol developed by the Trusted Computing Group TCG as one of the core protocols of the Trusted Network Connect TNC open architecture. IF MAP provides a common interface between the Metadata Access Point MAP , a database server acting as a clearinghouse for information about security events and objects, and other elements of the TNC architecture. http www.computer.org portal web computingnow archive news065 The IF MAP protocol defines a publish subscribe search mechanism with a set of identifiers and data types. History The IF MAP protocol was first published by the TCG on April 28, 2008. Originally, the IF MAP specification was developed to support data sharing across various vendor s devices and applications ... of the IF MAP spec was published on September 13, 2010. The 2.0 version separated the base protocol ... 2010 02 22 if map based intercloud testbed Industrial Control Systems , http www.automation.com ... within the MAP framework. https www.networkworld.com news 2010 091310 trusted computing group cloud security.html IF MAP Community IF MAP.com is the meeting place for IF MAP Community. Launched in December ... and contributing to the IF MAP world. IF MAP Adoption IF MAP is supported by a variety of vendors ... and Orchestration IF MAP Server http www.insightix.com Insightix BSA Business Security Assurance ... projects strongswan wiki IfMap strongSwan Open Source IPsec VPN Gateway with IF MAP ... if map based intercloud testbed IF MAP Based Intercloud Testbed In Planning http www.automation.com ... www.if map.org IF MAP Web site http www.trustedcomputinggroup.org Trusted Computing Group http www.trustedcomputinggroup.org resources tnc ifmap binding for soap specification TNC IF MAP Binding for SOAP Specification http ifmapdev.com IF MAP Developer Resources http code.google.com p omapd omapd Opensource IF MAP Server Category Computer network security Category Trusted computing Category Network ... more details
Other uses pp move indef A map is a visual representation of an area a symbolic depiction highlighting ... any space , real or imagined, without regard to context language use context or scale map scale e.g. ... thumb 200 px A celestial map from the 17th century, by the Dutch cartographer Frederik de Wit . Cartography or map making is the study and practice of crafting representations of the Earth upon ... maps. In terms of quantity, the largest number of drawn map sheets is probably made up by local surveys ... Mappa Mundi , about 1300, Hereford Cathedral , England. A classic T O map with Jerusalem at centre, east toward the top, Europe the bottom left and Africa on the right. The orientation of a map is the relationship between the directions on the map and the corresponding compass direction s in reality ... the T and O map s, were drawn with East at the top meaning that the direction up on the map ... is that North is at the top of a map. Several kinds of maps are often traditionally not oriented ... of Edo show the Tokyo Imperial Palace Japanese imperial palace as the top , but also at the centre, of the map. Labels on the map are oriented in such a way that you cannot read them properly unless you ... European T and O map s such as the Hereford Mappa Mundi were centred on Jerusalem with East at the top ... equidistant projection Polar map s of the Arctic or Antarctica Antarctic regions are conventionally centred on the pole the direction North would be towards or away from the centre of the map ... have the 0 meridian towards the top of the page. Reversed map s, also known as Upside Down ... Fuller s Dymaxion map s are based on a projection of the Earth s sphere onto an icosahedron . The resulting ... typically project north at the top of the map, but use math degrees 0 is east, degrees increase ... from 450. Scale and accuracy File Blank globe.svg thumb right 175px A global view map of Europe, Western Asia and Africa. Many, but not all, maps are drawn to a Scale map scale , expressed as a ratio ... more details
In mathematics , a connector is a map which can be defined for a linear connection and used to define the covariant derivative on a vector bundle from the linear connection. Category Connection mathematics differential geometry stub ... more details
saved book title Mathematics subtitle An overview cover image Math.svg cover color Mathematics Main article Mathematics Supporting articles History of mathematics Mathematical beauty Mathematical notation Category Wikipedia books on mathematicsMathematics ... more details
Wiktionarypar mathematicsMathematics is the body of knowledge justified by deductive reasoning about abstract structures, starting from axioms and definitions. Mathematics may also refer to Mathematics producer , a hip hop producer Mathematics album Mathematics album , an album by the band The Servant Mathematics song Mathematics song , a song by Mos Def Mathematics Little Boots song Mathematics Little Boots song , a song by Little Boots Mathematics Cherry Ghost song Mathematics Cherry Ghost song , a song by Cherry Ghost Mathematics Magazine , a publication of the Mathematical Association of America See also Category Mathematics Portal Mathematics Math disambiguation Mathematica disambiguation disambig fr Math homonymie it Mathematics lv Mathematics ... more details
be 1, in the most general sense given here, operation is synonymous with function mathematics function , mapping mathematicsmap and mapping mathematics mapping , that is, a relation mathematics relation ... 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 ... form a set called its domain mathematics domain . 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 ... 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 ... operation Binary operation Related topics col begin col break Arity Binary relation Domain mathematics Domain col break Function mathematics Function Multigrade operator Operator mathematics col break ... DEFAULTSORT Operation Mathematics Category Elementary mathematics Category Abstract algebra ar ... matem tica ru scn Upirazzioni matim tica simple Operation mathematics sk Matematick ... 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 economics , a correspondence between two sets A and B is a mapmathematicsmap f A P B from the elements ...unreferenced date August 2011 In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings. In general mathematics , correspondence is an alternative term for a Relation mathematics relation between two Set mathematics sets . Hence a correspondence of sets X and Y is any subset of the Cartesian product X × Y of the sets. anchor algebraic geometry In algebraic geometry , a correspondence between algebraic variety algebraic varieties V and W is in the same fashion a subset R of V × W , which is in addition required to be closed in the Zariski topology . It therefore means any relation that is defined by algebraic equations. There are some important examples, even when V and W are algebraic curve s for example the Hecke operator s of modular form theory may be considered as correspondences of modular curve s. However, the definition of a correspondence in algebraic geometry is not completely standard. For instance, Fulton, in his book on Intersection theory , uses the definition above. In literature, however, a correspondence from a variety X to a variety Y is often taken to be a subset Z of X × Y such that Z is finite and surjective over each component of X . Note the asymmetry in this latter definition which talks about a correspondence from X to Y rather than a correspondence between X and Y . The typical example of the latter kind of correspondence is the graph of a function f X &rarr Y In category theory , a correspondence from math C math to math D math is a functor math C op times D to mathbf Set math . It is the opposite of a profunctor . One to one correspondence is an alternate name for a bijection ... mathematics i.e., a Relation mathematics relation , except that the range is over sets instead of elements ... is thought of as the generalization of a function mathematics function , rather than as a special case ... 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 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 . The range anti restriction R B is defined by R F 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
ideas from discrete mathematics to real world problems, such as in operations research . Although the main objects of study in discrete mathematics are discrete objects, analytic methods from continuous mathematics are often employed as well. Grand challenges, past and present File Four Colour Map ...For the mathematics journal Discrete Mathematics journal File 6n graf.svg thumb 250px Graph mathematics Graphs like this are among the objects studied by discrete mathematics, for their interesting graph ... in developing computer algorithm s. Discrete mathematics is the study of Mathematics mathematical ... , the objects studied in discrete mathematics such as integer s, Graph mathematics graphs , and statements in Mathematical logic logic ref Richard Johnsonbaugh, Discrete Mathematics , Prentice Hall ... title Discrete mathematics urlname DiscreteMathematics ref Discrete mathematics therefore excludes topics in continuous mathematics such as calculus and Mathematical analysis analysis . Discrete objects can often be enumeration enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable set s ref Norman L. Biggs , Discrete mathematics , Oxford University Press, 2002. ref sets that have the same cardinality as subsets of the natural ... agreed, definition of the term discrete mathematics. ref Brian Hopkins, Resources for Teaching Discrete Mathematics , Mathematical Association of America, 2008. ref Indeed, discrete mathematics ... notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics ... discrete mathematics are useful in studying and describing objects and problems in branches of computer ... Penguin Books year 2002 isbn 0 691 11533 8 ref The history of discrete mathematics has involved a number ... more details
mapmathematicsmap from a set E into itself thus p p Id sub E sub and F p E be the image of p . If we denote by the map p viewed as a map from E onto F and by i the Injective function ... P . Various other projections, called projection map s have been defined for the need of cartography ... sup th sup projection set theory projection map , written proj sub j sub , that takes an element ... sub . This map is always surjective . A mapping that takes an element to its equivalence class under a given equivalence relation is known as the canonical projection. The evaluation map sends a function ... product math prod i in X Y i math , and the evaluation map is a projection map from ... to arbitrary category mathematics categories . The product category theory product of some ... categories. The projection from the Cartesian product of set mathematics sets , the product topology of topological space s which is always surjective and open map open , or from the direct product of groups direct product of group mathematics groups , etc. Although these morphisms are often ... bundle includes a projection map as part of its definition. Locally at least this map looks like a projection map in the sense of the product topology, and is therefore open and surjective. In topology , a retract is a continuous map r X X which restricts to the identity map on its image. This satisfies ... map. A retract which is homotopic to the identity is known as a deformation retract . This term ... projection or resolute of one vector geometric vector onto another. DEFAULTSORT Projection Mathematics ... more details
In category theory , a branch of abstract mathematics, a tower is defined as follows. Let math mathcal I math be the poset math cdots rightarrow 2 rightarrow 1 rightarrow 0 math of whole numbers in reverse order, regarded as a category. A countable tower of objects in a category math mathcal A math is a functor from math mathcal I math to math mathcal A math . In other words, a tower of math mathcal A math is a family of objects math A i i geq 0 math in math mathcal A math where there exists a map math A i rightarrow A j math iff math i j math and the composition math A i rightarrow A j rightarrow A k math is the map math A i rightarrow A k math Example Let math M i M math for some math R math module math M math . Let math M i rightarrow M j math be the identity map for math i j math . Then math M i math forms a tower of modules. Reference Section 3.5 of Citation last Weibel first Charles A. title An Introduction to Homological Algebra publisher Cambridge University Press series Cambridge Studies in Advanced Mathematics volume 38 year 1994 isbn 978 0 521 55987 4 Category Category theory ... more details
Folk mathematics can mean The mathematical folklore that circulates among mathematicians The informal mathematics used in everyday life, as studied in ethno cultural studies of mathematics. disambig Category Mathematical disambiguation ... more details
Unreferenced date December 2009 A mathematics journal is a scientific journal which publishes exclusively or almost exclusively mathematics papers. A practical definition of the current state of mathematics , as a research field, is that it consists of theorem s with proofs published in a reputable mathematics journal, and which usually have passed through the process of peer review . In some exceptional cases, the statement of a conjecture , or the introduction of some new method or definition might assume relevance. A relatively small proportion of mathematics papers concerned with pure mathematics are published through more general, science based learned journals. Applied mathematics may be published in publications more oriented towards engineering , but sometimes also biology and other sciences. Hundreds of such journals exist. Some of the most prestigious journals in pure mathematics are Annals of Mathematics , Publications Math matiques de l IH S , Acta Mathematica , and Inventiones Mathematicae . See also List of mathematics journals DEFAULTSORT Mathematical Journal Category Mathematics journals nl Wiskundig tijdschrift ... more details
Expert subject Mathematics date November 2008 unreferenced date July 2010 In the philosophy of mathematics , ordinary mathematics is an inexact term, used to distinguish the body of most mathematical work from that of, for example, constructivism mathematics constructivist , intuitionism intuitionist , or finitism finitist mathematics. Ordinary mathematics is usually studied within the universe mathematics universe Universe mathematics In ordinary mathematics SN , or sometimes V sub sub see Von Neumann universe . Contrast with finitist mathematics, which limits to the study of V sub sub see hereditarily finite set s , or with metamathematics and the study of large cardinal s, which study objects contained in a larger universe. Ordinary mathematicians generally assume the axiom of choice at least, because it makes their work easier , whereas constructivists reject it on the grounds that it is non constructive , and also reject the law of excluded middle , which can be derived from it. Category Philosophy of mathematics math stub ... more details
Italic title Mathematics of Computation ref http www.ams.org mcom aboutmcom.html Mathematics of Computation Journal overview , retrieved April 2007 ref is a quarterly mathematics journal focused on computational mathematics that is published by the American Mathematical Society . It was established in 1943. The articles in all volumes older than five years are available electronically free of charge. ref http www.ams.org jourcgi jrnl toolbar nav mcom all Mathematics of Computation Archive ref References reflist Category Mathematics journals Category Quarterly journals sci journal stub ... more details
Vect sub K sub vector space s over the field mathematics field K K linear map s Fiber bundle s with bundle ...In mathematics , a category is an algebraic structure that comprises objects that are linked by arrows ... are set mathematics sets and whose arrows are function mathematics functions . On the other hand, any ... idea of category theory , a branch of mathematics which seeks to generalize all of mathematics in terms ... branch of modern mathematics can be described in terms of categories, and doing so often reveals deep insights and similarities between seemingly different areas of mathematics. For more extensive motivational ... in bold or italics examples include category of sets Set , the category of set mathematics sets and function mathematics set functions category of rings Ring , the category of ring mathematics rings and ring ... map s. All of the preceding categories have the identity function identity map as identity ... g o f h o g o f , and identity mathematics identity for every object x , there exists a morphism 1 sub ... if both ob C and hom C are actually Set mathematics sets and not proper class es, and large otherwise ... a , b is a set, called a homset . Many important categories in mathematics such as the category of sets ... together with all function mathematics function s between sets, where composition is the usual function ... commonly used category in mathematics. The category category of relations Rel consists of all Set mathematics sets , with binary relation s as morphisms. Abstracting from Relation mathematics relations ... discrete category discrete . For any given Set mathematics set I , the discrete category on I is the small ... for categories. Any group mathematics group can be seen as a category with a single object in which ... continuously differentiable maps category of metric spaces Met metric space s short map s category of modules R Mod R Modules, where R is a Ring module homomorphisms category of rings Ring ring mathematics ring s ring homomorphism s category of sets Set Set mathematics set s Function mathematics ... more details
Refimprove date December 2010 The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics . The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of mathematics in people s lives. The logical and structural nature of mathematics ... of a Proposition mathematics mathematical proposition ? What is the relation between logic and mathematics? What is the role of hermeneutics in mathematics? What kinds of inquiry play a role in mathematics? What are the objectives of mathematical inquiry? What gives mathematics its hold on experience ? What are the human trait s behind mathematics? What is mathematical beauty ? What is the source and nature of mathematical truth? What is the relationship between the abstract world of mathematics and the material universe? The terms philosophy of mathematics and mathematical philosophy are frequently ... of Mathematics Book Review journal Philosophy of Science volume 36 issue 3 page 325 year 1969 . For example, when Edward Maziars proposes in a 1969 book review to distinguish philosophical mathematics ... with philosophy of mathematics . ref LONG REF ENDS HERE The latter, however, may be used to refer ... The Principles of Mathematics and Introduction to Mathematical Philosophy . History The origin of mathematics is subject to argument. Whether the birth of mathematics was a random happening or induced .... Many thinkers have contributed their ideas concerning the nature of mathematics. Today, some philosophers of mathematics aim to give accounts of this form of inquiry and its products as they stand ... philosophy and Eastern philosophy . Western philosophies of mathematics go as far back as Plato , who ... and issues related to infinity actual versus potential . Ancient Greece Greek philosophy on mathematics ... re evaluation of Greek philosophy of mathematics. According to legend, fellow Pythagoreans were ... more details
true The area of study known as the history of mathematics is primarily an investigation into the origin of discoveries in mathematics and, to a lesser extent, an investigation into the mathematical ... mathematical texts available are Plimpton 322 Babylonian mathematics c. 1900 BC , ref J. Friberg, Methods and traditions of Babylonian mathematics. Plimpton 322, Pythagorean triples, and the Babylonian ... Papyrus Egyptian mathematics c. 2000 1800 BC ref Cite book edition 2 publisher Dover Publications ... Egyptian Mathematics and Astronomy , pp. 71 96. ref and the Moscow Mathematical Papyrus Egyptian mathematics ... of mathematics as a subject in its own right begins in the 6th century BC with the Pythagoreans , who coined the term mathematics from the ancient Greek mathema , meaning subject of instruction . ref cite book author Heath title A Manual of Greek Mathematics page 5 ref Greek mathematics ... rigor in mathematical proof proofs and expanded the subject matter of mathematics. ref Sir Thomas L. Heath, A Manual of Greek Mathematics , Dover, 1963, p. 1 In the case of mathematics, it is the Greek contribution which it is most essential to know, for it was the Greeks who first made mathematics a science. ref Counting rods Chinese mathematics made early contributions, including a place value system . ref George Gheverghese Joseph, The Crest of the Peacock Non European Roots of Mathematics ... millennium AD in Indian mathematics India and was transmitted to the west via Islamic mathematics ... and.ac.uk HistTopics Indian numerals.html ref Islamic mathematics , in turn, developed and expanded the mathematics known to these civilizations. ref Adolf Yushkevich A.P. Juschkewitsch , Geschichte der Mathematik im Mittelalter , Teubner, Leipzig, 1964 ref Many Greek and Arabic texts on mathematics ... development of mathematics in Middle Ages medieval Europe . From ancient times through the Middle ... day. Prehistoric mathematics The origins of mathematical thought lie in the concepts of number ... more details
Italic title The Mathematics Enthusiast ISSN 1551 3440 is a triannual Peer review peer reviewed academic journal covering mathematics education , including historical, philosophical, and cross cultural perspectives on mathematics. It is published by Information Age Publishing and hosted by the department of mathematical sciences at The University of Montana . Its founder and editor in chief is Bharath Sriraman . The journal also includes a monograph series called the The Montana Mathematics Enthusiast Monographs in Mathematics Education . Abstracting and indexing The journal is abstracted and indexed in EBSCO Industries Academic Search Complete , PsycINFO , and Journals in Higher Education . External links Official website 1 http www.infoagepub.com index.php?id 43 http www.math.umt.edu tmme Journal page at University of Montana http www.infoagepub.com series The Montana Mathematics Enthusiast Series at Information Age Publishing DEFAULTSORT Mathematics Enthusiast Category Mathematics journals Category Education journals Category Triannual journals Category English language journals Category Mathematics education ... more details
Image Lifting diagram.png right thumb 100px Lift of f commutative diagram In the branch of mathematics called category theory , given a morphism f from an object X to an object Y , and a morphism g from an object Z to Y , a lift or lifting of f to Z is a morphism h from X to Z such that gh f . A basic example in topology is lifting a path topology path in one space to a path in a covering space . Consider, for instance, mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane . A path in the projective plane is a continuous map from the unit interval, 0,1 . We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have math begin align f colon& 0,1 to mathbb RP 2 , & qquad& text projective plane path g colon& S 2 to mathbb RP 2 , & qquad& text covering map h colon& 0,1 to S 2 . & qquad& text sphere path end align math Lifts are ubiquitous for example, the definition of fibration s see homotopy lifting property and the valuative criteria of separated morphism separated and proper map s of scheme mathematics schemes are formulated in terms of existence and in the last case unicity of certain lifts. Category Category theory categorytheory stub ... more details
In mathematics , the word kernel has several meanings. Kernel may mean a subset associated with a Mapmathematics mapping The kernel of a mapping is the set of elements that map to the Additive identity zero element such as zero or Null vector zero vector , as in kernel linear operator kernel of a linear operator and kernel matrix kernel of a matrix . In this context, kernel is often called nullspace . More generally, the kernel algebra kernel in algebra is the set of elements that map to the neutral element . Here, the mapping is assumed to be a homomorphism , that is, it preserves algebraic operations, and, in particular, maps neutral element to neutral element. The kernel is then the set of all elements that the mapping cannot distinguish from the neutral element. The Kernel category theory kernel in category theory is a generalization of this concept to morphism s rather than mappings between sets. In set theory, the Kernel set theory kernel of a function is the set of all pairs of elements that the function cannot distinguish, that is, they map to the same value. This is a generalization of the kernel concept above to the case when there is no neutral element. In set theory, the equalizer mathematics difference kernel or binary equalizer is the set of all elements where the values of two functions coincide. Kernel may also mean a function of two variables, which is used to define a mapping In integral calculus , the kernel also called integral kernel or kernel function is a function of two variables that defines an integral transform , such as the function k in math T f x int X k x, x f x , dx . math In partial differential equation s, when the solution of the equation for the right hand side f can be written as Tf above, the kernel becomes the Green s function . The heat kernel is the Green s function of the heat equation. In the case when the integral kernel depends only on the difference between its arguments, it becomes a convolution kernel , as in math T f ... more details
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