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
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 ... 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 Interface ... Table , Security Magazine , BNP Media. http opencloudconsortium.org 2010 02 22 if map based intercloud testbed IF MAP Based Intercloud Testbed In Planning http www.automation.com content ... 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 protocols ... 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 ... 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 word ... the T and O map s, were drawn with East at the top meaning that the direction up on the map corresponds ... is that North is at the top of a map. Several kinds of maps are often traditionally not oriented with North ... 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 put ... T and O map s such as the Hereford Mappa Mundi were centred on Jerusalem with East at the top. Indeed ... 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, respectively ... have the 0 meridian towards the top of the page. Reversed map s, also known as Upside Down maps ... 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 counter ... 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 such as 1 10,000, meaning that 1 of any unit of measurement ... 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
The flat map in differential geometry is a name for the mapping that converts coordinate basis vectors into corresponding coordinate 1 forms. A flat map in ring theory is a homomorphism f from a ring mathematics ring R to a ring S such that S is a flat module flat R module, where the action of R on S is given by f . See also Sharp map Category Ring theory Category Differential geometry Category Differential topology differential geometry stub Abstract algebra stub ... more details
Unreferenced date December 2009 For equivariance in estimation theory Invariant estimator In mathematics , an equivariant map is a function mathematics function between two Set mathematics sets that commutes with the group action action of a group . Specifically, let G be a group mathematics group and let X and Y be two associated group action G sets . A function f X Y is said to be equivariant if f g · x g · f x for all g G and all x in X . Note that if one or both of the actions are right actions the equivariance condition must be suitably modified f x · g f x · g right right f x · g g sup &minus 1 sup · f x right left f g · x f x · g sup &minus 1 sup left right Equivariant maps are homomorphism s in the Category mathematics category of G sets ... commutative diagram . Note that math g cdot math denotes the map that takes an element math z ..., if X and Y are the representation spaces of two linear representations of G then a linear map ... is an equivariant map in the special case of two linear representations actions. Alternatively, an intertwiner for representations of G over a field mathematics field K is the same thing as a module mathematics module homomorphism of K G module mathematics modules , where K G is the group ... other than the zero map only exists if the two representations are equivalent that is, are isomorphic as module mathematics modules . That intertwiner is then unique up to a multiplicative factor a non zero scalar mathematics scalar from K . These properties hold when the image of K G is a simple ... the same. Categorical description Equivariant maps can be generalized to arbitrary category mathematics ... in C , an equivariant map between those representations is simply a natural transformation from ... continuous function continuously . An equivariant map is then a continuous map f X Y between representations which commutes with the action of G . DEFAULTSORT Equivariant Map Category Group actions Category ... more details
Image DuffingMap.png right thumb 300px Plot of the Duffing map showing chaotic behavior, where a 2.75 and b 0.15. Image Tw duffing.png right thumb 300px Phase portrait of a two well Duffing oscillator a differential equation, rather than a map showing chaotic behavior. The Duffing map is a discrete time dynamical system . It is an example of a dynamical system that exhibits chaos theory chaotic behavior . The Duffing function mathematicsmap takes a point x sub n sub , y sub n sub in the plane mathematics plane and maps it to a new point given by math x n 1 y n , math math y n 1 bx n ay n y n 3. , math The map depends on the two constant mathematics constant s a and b . These are usually set to a 2.75 and b 0.2 to produce chaotic behaviour. It is a discrete version of the Duffing equation . External links http scholarpedia.org article Duffing oscillator Duffing oscillator on Scholarpedia Mathapplied stub Category Chaotic maps ... more details
In the mathematics mathematical theory of metric space s, a metric map is a Function mathematics function between metric spaces that does not increase any distance such functions are always continuous function continuous . These maps are the morphism s in the category of metric spaces , Met Isbell 1964 . They are also called Lipschitz continuity Lipschitz functions with Lipschitz constant 1, nonexpansive maps , nonexpanding maps , weak contractions , or short maps . Specifically, suppose that X and Y are metric spaces and is a function mathematics function from X to Y . Thus we have a metric map when, for any points x and y in X , math d Y f x ,f y leq d X x,y . math Here d sub X sub and d sub Y sub denote the metrics on X and Y respectively. A map between metric spaces is an isometry if and only if 1 it is metric, 2 it is a bijection , and 3 its inverse functions inverse is also metric. The composite function composite of metric maps is also metric. Thus metric spaces and metric maps form a category theory category Category of metric spaces Met Met is a subcategory of the category of metric spaces and Lipschitz functions, and the isomorphism s in Met are the isometries. One can say that is strictly metric if the inequality mathematics inequality is strict for every two different points. Thus a contraction mapping is strictly metric, but not necessarily the other way around. Note that an isometry is never strictly metric, except in the degeneracy mathematics degenerate case of the empty set empty space or a single point space. References cite journal author Isbell, J. R. authorlink John R. Isbell title Six theorems about injective metric spaces journal Comment. Math. Helv. volume 39 year 1964 pages 65 76 url http www.digizeitschriften.de resolveppn GDZPPN002058340 doi 10.1007 BF02566944 Category Metric geometry Category Lipschitz maps Geometry stub es Funci n corta fr Application non expansive it Funzione non espansiva pl Odwzorowanie nierozszerzaj ce pt Fun o ... more details
the same base field mathematics field F . A bilinear map is a function mathematics function B V × W &rarr X such that for any w in W the map v mapsto B v , w is a linear map from V to X , and for any v in V the map w mapsto B v , w is a linear map from W to X . In other words, if we hold the first entry of the bilinear map fixed, while letting the second entry vary, the result is a linear operator ... map B M × N T , where T is an abelian group mathematics group , such that for any n in N , m ... X . Examples This section is linked from Multilinear map matrix mathematics Matrix multiplication is a bilinear map M m , n × M n , p M m , p . If a vector space V over the real number s R carries an inner product space inner product , then the inner product is a bilinear map V × V R . In general, for a vector space V over a field F , a bilinear form on V is the same as a bilinear map V ...In mathematics , a bilinear operator is a function mathematics function combining elements of two vector space s to yield an element of a third vector space that is Linear map linear in each of its arguments ... use module mathematics modules over a commutative ring R . It also can be easily generalized to n ary functions, where the proper term is Multilinear map multilinear . For the case of a non commutative ... L V,W X of all bilinear maps is a linear subspace of the space viz. vector space , module mathematics ... a bilinear map into the real by means of a real bilinear form math scriptstyle v,w mapsto ... and W then each bilinear map can be uniquely represented by the matrix math B e i,f j math , and vice ... is a bilinear map from V × V to the base field. Let V and W be vector spaces over the same base field F . If f is a member of V and g a member of W , then b v , w f v g w defines a bilinear map V × W F . The cross product in R sup 3 sup is a bilinear map R sup 3 sup × R sup 3 sup R sup 3 sup . Let B V × W X be a bilinear map, and L U W be a linear map , then v , u B v , Lu is a bilinear ... 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
In the mathematical discipline of simplicial homology theory, a simplicial map is a Mapmathematicsmap between simplicial complex es with the property that the images of the vertices of a simplex always span a simplex. Note that this implies that vertices have vertices for images. Simplicial maps are thus determined by their effects on vertices. In particular, there are a finite number of simplicial maps between two given finite simplicial complexes. Simplicial maps induce continuous maps between the underlying polyhedra of the simplicial complexes one simply extends linearly using Barycentric coordinate system mathematics barycentric coordinates . Simplicial maps which are bijection bijective are called simplicial isomorphism isomorphisms . Simplicial approximation Let math f K rightarrow L math be a continuous map between the underlying polyhedra of simplicial complexes and let us write math text st v math for the simplicial complex star of a vertex. A simplicial map math f triangle K rightarrow L math such that math f text st v subseteq text st f triangle v math , is called a simplicial approximation to math f math . A simplicial approximation is homotopy homotopic to the map it approximates. References Munkres, James R. Elements of Algebraic Topology , Westview Press, 1995. ISBN 978 0201627282. See also Simplicial approximation theorem Category Algebraic topology ... more details
ref Mathematics In mathematics there is a very strong link between map coloring and graph coloring , since every map showing different areas has a corresponding graph. By far the most famous result in this area is the four color theorem , which states that any planar map can be colored with at most ...Map coloring is the act of assigning different colors to different features on a map. There are two very different uses of this term. The first is in cartography , choosing the colors to be used when producing a map. The second is in mathematics , where the problem is to determine the minimum number of colors needed to color a map so that no two adjacent features have the same color. Cartography Image Map of USA with state names.svg thumb left A map of the United States using colors to show political divisions using the Four color theorem . Image 2004US election map.svg thumb right The United States presidential election, 2004 US Presidential Election of 2004 , visualised using a choropleth map Image Easter Island map en.svg thumb left Topographic map of Easter Island , using colors to show elevations. Color is a very useful attribute to depict different features on a map. ref cite web url http geography.about.com od understandmaps a mapcolors.htm title Map Colors The Role of Colors on Maps author Matt Rosenberg ref Typical uses of color include displaying different political divisions, different elevations, or different kinds of roads. A choropleth map is a thematic map in which areas are colored differently to show the measurement of a statistical variable being displayed on the map. The choropleth map provides an easy way to visualize how a measurement varies across a geographic area or it shows the level of variability within a region. Displaying the data in different hues can greatly affect the understanding or feel of the map. ref cite article title Psychological aspects ... of the human visual system to make the map look three dimensional. ref cite journal title Color ... more details
Image Venn A subset B.svg 150px thumb right A is a subset of B , and B is a superset of A . In mathematics , if math A math is a subset of math B math , then the inclusion map also inclusion function , insertion , or canonical injection ref citation first1 S. last1 Mac Lane first2 G. last2 Birkhoff year 1967 , page 5 ref is the function mathematics function math i math that sends each element, math x math of math A math to math x math , treated as an element of math B math math i A rightarrow B, qquad i x x. math A hooked arrow math hookrightarrow math is sometimes used in place of the function arrow above to denote an inclusion map. This and other analogous injective functions ref citation first C. last Chevalley year 1956 , page 1 ref from substructure s are sometimes called natural injections . Given any morphism between object category theory objects X and Y , if there is an inclusion map into the domain mathematics domain math i A rightarrow X math , then one can form the Function mathematics Restrictions and extensions restriction fi of f . In many instances, one can also construct a canonical inclusion into the codomain R Y known as the range mathematics range of f . Applications of inclusion maps Inclusion maps tend to be homomorphism s of algebraic structure s thus, such inclusion maps are embedding s. More precisely, given a sub structure closed under some operations, the inclusion map will be an embedding for tautological reasons, given the very definition by restriction of what one checks. For example, for a binary operation math star math , to require that math i x star y i x star i y math is simply to say that math star math is consistently computed in the sub ... at nullary operations, which pick out a constant element. Here the point is that closure mathematics ... in algebraic topology where if A is a strong deformation retract of X , the inclusion map yields an isomorphism ... Map Category Functions and mappings Category Basic concepts in set theory de Inklusionsabbildung ... more details
Image Contour vs Surface Map.png thumb right 250px A 3D surface map of Mount St. Helens Mt. St. Helens with a 2D Contour line contour map above for comparison. In mathematics , geology , and cartography , a surface map is a dimension 2D Perspective graphical perspective representation of a Three dimensional space 3 dimensional surface . Surface maps usually represent real world entities such as landforms or the surfaces of objects. They can, however, serve as an abstraction where the third, or even all of the dimensions correspond to non spatial data. In this capacity they act more as Plot graphics graphs than maps. Category Maps Topology stub ... more details
In mathematics , particularly topology , a perfect map is a particular kind of continuous function between .... Formal definition Let X and Y be topological space s and let p be a map from X to Y that is continuous, closed map closed , surjective and such that p sup &minus 1 sup y is compact relative to X for each y in Y . Then p is known as a perfect map. Examples and properties 1. If p X Y is a perfect map and Y is compact space compact , then X is compact. 2. If p X Y is a perfect map and X is regular ... topology. 3. If p X Y is a perfect map and if X is locally compact , then Y is locally compact. 4. If p X Y is a perfect map and if X is second countable, then Y is second countable . 5. Every injective perfect map is a homeomorphism . This follows from the fact that a bijective closed map has a continuous inverse. 6. If p X Y is a perfect map and if Y is Connected space connected , then X need not be connected. For example, the constant map from a compact disconnected space to a singleton space is a perfect map. 7. A perfect map need not be open, as the following map shows p x x if x belongs to 1, 2 p x x &minus 1 if x belongs to 3, 4 This map is closed, continuous by the pasting lemma , and surjective and therefore is a perfect map the other condition is trivially satisfied . However ... to nowrap 1, 3 the range of p . Note that this map is a Quotient space quotient map and the quotient ... connected space local connectedness , second countability, local compactness etc we require that the map be not only continuous but also open. A perfect map need not be open see previous example , but these properties are still preserved under perfect maps. 9. Every homeomorphism is a perfect map. This follows from the fact that a bijective open map is closed and that since a homeomorphism is injective ... have precisely one element . 10. Every perfect map is a quotient map. This follows from the fact that a closed, continuous surjective map is always a quotient map. 11. Let G be a compact topological ... more details
In mathematics , a mapmathematics mapping math f V to W math from a complex vector space to another is said to be antilinear or conjugate linear or semilinear , though the latter term is more general if math f ax by bar a f x bar b f y math for all a , b in C and all x , y in V , where math bar a math and math bar b math are the complex conjugate s of a and b respectively. The composition mathematics composition of two antilinear maps is complex linear . An antilinear map math f V to W math may be equivalently described in terms of the linear map math bar f V to bar W math from math V math to the complex conjugate vector space math bar W math . Antilinear maps occur in quantum mechanics in the study of time reversal and in spinor calculus , where it is customary to replace the bars over the basis vectors and the components of geometric objects by dots put above the indices. References Horn and Johnson, Matrix Analysis, Cambridge University Press, 1985. ISBN 0 521 38632 2. antilinear maps are discussed in section 4.6 . Budinich, P. and Trautman, A. The Spinorial Chessboard . Spinger Verlag, 1988. ISBN 0 387 19078 3. antilinear maps are discussed in section 3.3 . See also Linear map Complex conjugate Sesquilinear form consimilar matrix Time reversal Category Functions and mappings Category Linear algebra eo Konjuglineara bildigo fa ia Mappamento antilinear it Trasformazione antilineare pl Przekszta cenie antyliniowe zh ... more details
In differential geometry , the sharp map is the mapping mathematics mapping that converts coordinate 1 form s into corresponding coordinate basis vector s. Definition Let math M math be a manifold and math , Gamma TM math denote the space of all sections of its tangent bundle . Fix a nondegenerate 0,2 tensor field math g in Gamma T M otimes 2 math , i.e., a metric tensor or a symplectic form . The definition math X flat i X g g X,. math yields a linear map sometimes called the flat map math flat Gamma TM to Gamma T M math which is an isomorphism , since math g math is non degenerate. Its inverse math sharp flat 1 Gamma T M to Gamma TM math is called the sharp map. Category Differential topology Category Differential geometry differential geometry stub ... more details
Orphan date August 2011 In mathematics, a Latt s map is a rational map f L sup &minus 1 sup from the complex sphere to itself such that is a holomorphic map from a complex torus to the complex sphere and L is an affine map z az b from the complex torus to itself. References Citation last1 Latt s first1 S title Sur l it ration des substitutions rationnelles et les fonctions de Poincar year 1918 journal Les Comptes rendus de l Acad mie des sciences volume 166 pages 26 28 Citation last1 Milnor first1 John Willard author1 link John Milnor title Dynamics on the Riemann sphere, publisher Eur. Math. Soc. year 2006 chapter On Latt s maps pages 9 43 mr 2348953 Category Dynamical systems ... more details
Orphan date November 2010 In Combinatorics combinatorial mathematics the map folding problem is the question of how many ways there are to fold a rectangular map along its creases. A related problem called the stamp folding problem is how many ways there are to fold a strip of stamps. ref MathWorld title Map Folding urlname MapFolding ref For example, there are six ways to fold a strip of three different stamps Image Stampfoldings1x3.png And there are eight ways to fold a 2× 2 map along its creases Image MapFoldings 2x2.png 350 px The problem is related to a problem in the mathematics of origami of whether a square with a crease pattern can be folded to a flat figure. Some simple extensions to the problem of folding a map are NP complete . ref cite journal url http erikdemaine.org papers MapFolding paper.pdf title When Can You Fold a Map? authors Esther M. Arkin, Michael A. Bender, Erik D. Demaine, Martin L. Demaine, Joseph S. B. Mitchell, Saurabh Sethia, Steven S. Skiena journal Computational Geometry Theory and Applications volume 29 issue 1 date September 2004 pages pp. 23 46 ref References references See also Martin Gardner, The Combinatorics of Paper Folding, Wheels, Life and Other Mathematical Amusements , New York W. H. Freeman, 1983 pp. 60&ndash 61. Folding a Strip of Labeled Stamps from The Wolfram Demonstrations Project http demonstrations.wolfram.com FoldingAStripOfLabeledStamps DEFAULTSORT Map Folding Category Paper folding Combin stub ... more details
The Zaslavskii map is a discrete time dynamical system . It is an example of a dynamical system that exhibits chaos theory chaotic behavior . The Zaslavskii map takes a point math x n,y n math in the plane mathematics plane and function mathematics maps it to a new point math x n 1 x n nu 1 mu y n epsilon nu mu cos 2 pi x n , textrm mod ,1 math math y n 1 e r y n epsilon cos 2 pi x n , math where mod is the modulo operation modulo operator with real arguments. The map depends on four Constant mathematics constant s &nu , &mu , &epsilon and r . Russel 1980 gives a Hausdorff dimension of 1.39 but Peter Grassberger Grassberger 1983 questions this value based on their difficulties measuring the correlation dimension . References cite journal author G.M. Zaslavskii title The Simplest case of a strange attractor journal Phys. Lett. A year 1978 volume 69 issue 3 pages 145 147 doi 10.1016 0375 9601 78 90195 0 http www.sciencedirect.com science? ob MImg& imagekey B6TVM 46S32M9 11N 1& cdi 5538& user 10& orig browse& coverDate 12 2F11 2F1978& sk 999309996&view c&wchp dGLbVtb zSkWA&md5 381ecc59b5847c0a67dbe457cae92c46&ie sdarticle.pdf LINK cite journal author D.A. Russel, J.D. Hanson, and E. Ott title Dimension of strange attractors journal Phys. Rev. year 1980 volume 45 issue 14 pages 1175 doi 10.1103 PhysRevLett.45.1175 bibcode 1980PhRvL..45.1175R http prola.aps.org abstract PRL v45 i14 p1175 1 LINK cite journal author Peter Grassberger P. Grassberger and I. Procaccia title Measuring the strangeness of strange attractors journal Physica year 1983 volume 9D pages 189 208 doi 10.1016 0167 2789 83 90298 1 bibcode 1983PhyD....9..189G http adsabs.harvard.edu cgi bin nph bib query?bibcode 1983PhyD....9..189G& db key PHY LINK See also List of chaotic maps Mathapplied stub Category Chaotic maps eo Mapo de Zaslavskii ... more details
Unreferenced date July 2009 Image Unintentional Humor, Way In, No Entry.jpg 250px right thumb Topological tube map of the London Underground In cartography and geology , a topology topological map is one that has been simplified so that only vital information remains and unnecessary detail has been removed. These maps lack scale, and distance and direction are subject to change and variation, but the relationship between points is maintained. A good example is the tube map of the London Underground . The name topological map is derived from topology , the branch of mathematics that studies the properties of objects that do not change as the object is deformed, much as the tube map retains useful information despite bearing little resemblance to the actual layout of the underground system. Not to be confused with a topographic map . See also Portal Atlas Main Outline of cartography Multicol 800 Aerial photography Animated mapping British Cartographic Society Cartogram Cartographic relief depiction Cartographic generalization Contour line Critical cartography Digital Cadastral DataBase Fantasy map Figure ground in map design Multicol break Four color theorem Gazetteer Geocode Geographic information system Geographic Information System GIS Geovisualization Here be dragons Isostasy Japanese map symbols List of cartographers Multicol break Locator map Map projection National Geospatial Intelligence Agency OpenStreetMap , a free project mapping the world s roads using Global Positioning System GPS Orthophoto Pictorial maps Planetary cartography Point of Beginning Sea level Terra incognita Multicol end Category Map types Cartography stub el he ... more details
Multiple issues context October 2009 refimprove June 2007 notability June 2010 In coalgebra theory, the notion of colinear map is dual to the notion for linear map of vector space , or more generally, for morphism between R module . Specifically, let R be a Ring mathematics ring , M,N,C be R modules, and math rho M M rightarrow M otimes C, rho N N rightarrow N otimes C math be right C comodule s. Then an R linear map math f M rightarrow N math is called a right comodule morphism , or right C colinear , if math rho N circ f f otimes 1 circ rho M math References Khaled AL Takhman, Equivalences of Comodule Categories for Coalgebras over Rings , J. Pure Appl. Algebra,.V. 173, Issue 3, September 7, 2002, pp. 245 271 Category Coalgebras algebra stub ... more details
Unreferenced date December 2009 In mathematics , a bundle map or bundle morphism is a morphism in the category mathematics category of fiber bundle s. There are two distinct, but closely related, notions of bundle map, depending on whether the fiber bundles in question have a common fiber bundle base space . There are also several variations on the basic theme, depending on precisely which category of fiber bundles is under consideration. In the first three sections, we will consider general fiber bundles in the category of topological spaces . Then in the fourth section, some other examples will be given. Bundle maps over a common base Let sub E sub E M and sub F sub F M be fiber bundles over a space M . Then a bundle map from E to F over M is a continuous map E F such that math pi F ... respectively. Then a continuous map E F is called a bundle map from E to F if there is a continuous map f M N such that the diagram Image BundleMorphism 04.svg 150px center commutes, that is, math pi F circ varphi f circ pi E math . In other words, is fiber preserving , and f is the induced map ... f , such a bundle map is said to be a bundle map covering f . Relation between the two notions It follows immediately from the definitions that a bundle map over M in the first sense is the same thing as a bundle map covering the identity map of M . Conversely, general bundle maps can be reduced ... N is a fiber bundle over N and f M N is a continuous map, then the pullback of F by f is a fiber bundle ... that a bundle map from E to F covering f is the same thing as a bundle map from E to f sup ... of a bundle map. First, one can consider fiber bundles in a different category of spaces. This leads, for example, to the notion of a smooth bundle map between smooth fiber bundles over a smooth manifold ... map is required to be a linear map on each fiber. In this case, such a bundle map covering f may ... sub f x sub of linear map s from E sub x sub to F sub f x sub . DEFAULTSORT Bundle Map Category Fiber ... more details
A bivariate map displays two Variable mathematics variables on a single map by combining two different sets of graphic symbols or colors. Bivariate mapping is an important technique in cartography . Given a set of geographic features, a bivariate map displays two Variable mathematics variables on a single map by combining two different sets of graphic symbols. It is a variation of simple choropleth map that portrays two separate phenomena simultaneously. The main objective of a bivariate map is to find a simple method for accurately and graphically illustrating the Correlation and dependence relationship between two spatially distributed variables. A bivariate map has potential to reveal relationships between variables more effectively than a side by side comparison of the corresponding univariate maps. A bivariate map is recent graphical method which is intended to convey the spatial distribution of two variables and the geographical concentration of their relationship. A bivariate choropleth map uses color to solve a problem of representation in four dimensions two spatial dimensions longitude and latitude and two statistical variables. Data classification and graphic representation of the classified data are two important processes involved in constructing a bivariate map. The number of classes should be possible to deal with by the reader. A rectangular legend box is divided into smaller boxes where each box represents a unique relationship of the variables. In general, bivariate maps are one of the alternatives to the simple univariate choropleth map s, although they are sometimes extremely difficult to understand the distribution of a single variable. Because conventional bivariate maps use two arbitrarily assigned color schemes and generate random color combinations ... between the two variables could be shown on the bivariate map. References Dunn R., 1989 ... Category Map types ... more details
distinguish Rotation mathematics In mathematics , a rotation map is a function that represents an undirected edge labeled graph , where each vertex enumerates its outgoing neighbors. Rotation maps were first introduced by Reingold, Vadhan and Wigderson Entropy waves, the zig zag graph product, and new constant degree expanders , 2002 in order to conveniently define the zig zag product and prove its properties. Given a vertex math v math and an edge label math i math , the rotation map returns the math i math th neighbor of math v math and the edge label that would lead back to math v math . Definition For a D regular graph G , the rotation map math mathrm Rot G N times D rightarrow N times D math is defined as follows math mathrm Rot G v,i w,j math if the i th edge leaving v leads to w , and the j th edge leaving w leads to v . Basic properties From the definition we see that math mathrm Rot G math is a permutation, and moreover math mathrm Rot G circ mathrm Rot G math is the identity map math mathrm Rot G math is an involution mathematics involution . Special cases and properties A rotation map is consistently labeled if all of the edges leaving each vertex are labeled in such a way that at each vertex, the labels of the incoming edges are all distinct. Every regular graph has some consistent labeling. A rotation map is math pi math consistent if math forall v mathrm Rot G v,i v i , pi i math . From the definition, a math pi math consistent rotation map is consistently labeled. See also Zig zag product Rotation system References Refbegin Citation first1 O. last1 Reingold first2 S. last2 Vadhan first3 A. last3 Widgerson title Entropy waves, the zig zag graph product, and new constant degree expanders and extractors journal 41st Annual Symposium on Foundations of Computer Science year 2000 doi 10.1109 SFCS.2000.892006 pages 3 13 Citation first O last Reingold title Undirected connectivity in log space journal Journal of the ACM year 2008 volume 55 issue 4 pages Article ... more details
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