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 topologicalmap refers to a map that has been topology 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 of a topologicalmap is the tube map of the London Underground . The name topologicalmap 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 mapMap 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 The article titled great circle distance explains how to find that quantity if one knows the two latitudes and longitudes. Multicol end Category Map types Cartography stub el ... more details
In mathematics , topological dynamics is a branch of the theory of dynamical systems in which qualitative, asymptotic properties of dynamical systems are studied from the viewpoint of general topology . Scope The central object of study in topological dynamics is a topological dynamical system , i.e. a topological space , together with a continuous map topology continuous transformation , a continuous flow, or more generally, a transformation semigroup semigroup of continuous transformations of that space. The origins of topological dynamics lie in the study of asymptotical properties of trajectories of systems of autonomous ordinary differential equation s, in particular, the behavior of limit sets and various manifestations of repetitiveness of the motion, such as periodic trajectories, recurrence and minimality, stability, non wandering point s. George Birkhoff is considered to be the founder of the field. A structure theorem for minimal distal flows proved by Hillel Furstenberg in the early ... and 1980s was devoted to topological dynamics of one dimensional maps, in particular, piecewise linear ... in topological dynamics are general metric spaces usually, compact space compact . This necessitates ... can be considered on an equal footing with more geometric actions. Topological dynamics has ... have topological analogues cf Kolmogorov Sinai entropy and topological entropy . See also Poincar Bendixson theorem Symbolic dynamics Topological conjugacy References eom author D.V.Anosov id T t093030 Scholarpedia title Topological dynamics urlname Topological dynamics curator Joseph Auslander Robert Ellis, Lectures on topological dynamics . W. A. Benjamin, Inc., New York 1969 Walter Gottschalk, Gustav A. Hedlund Gustav Hedlund , Topological dynamics . American Mathematical Society Colloquium ... of topological dynamics . Mathematics and its Applications, 257. Kluwer Academic Publishers ... Bookstore, 2010, ISBN 9780821849323 Category Topological dynamics ... more details
Basic notions in group theory In mathematics , a topological group is a group mathematics group G together with a topological space topology on G such that the group s binary operation and the group s inverse .... A topological group is a mathematical object with both an algebraic structure and a topological structure ... about continuous functions, because of the topology. Topological groups, along with continuous group ... Symmetry physics in physics . Formal definition A topological group G is a topological space and group ... function s. Here, G × G is viewed as a topological space by using the product topology . Although ..., are discussed below. In the end, this is not a serious restriction&mdash any topological group can be made Hausdorff in a canonical fashion. In the language of category theory , topological groups can be defined concisely as group object s in the category of topological spaces , in the same way that ordinary ... topological groups G and H is just a continuous group homomorphism G math to math H . An isomorphism of topological groups is a group isomorphism which is also a homeomorphism of the underlying topological ... must also be continuous. There are examples of topological groups which are isomorphic as ordinary groups but not as topological groups. Indeed, any nondiscrete topological group is also a topological group when considered with the discrete topology. The underlying groups are the same, but as topological groups there is not an isomorphism. Topological groups, together with their homomorphisms, form a category theory category . Examples Every group can be trivially made into a topological group ..., the theory of topological groups subsumes that of ordinary groups. The real number s R , together with addition as operation and its usual topology, form a topological group. More generally, Euclidean space Euclidean n space R sup n sup with addition and standard topology is a topological group. More generally yet, the additive groups of all topological vector space s, such as Banach space s or Hilbert ... more details
variational principle relates the notions of topological and measure theoretic entropy. Definition A topological dynamical system consists of a Hausdorff topological space X usually assumed to be compact space compact and a continuous function topology continuous self map f . Its topological ...In mathematics , the topological entropy of a topological dynamical system is a nonnegative real number that measures the complexity of the system. Topological entropy was first introduced in 1965 by Adler ... of the topological entropy for a system given by an iterated function , the topological entropy .... Definition of Adler, Konheim, and McAndrew Let X be a compact Hausdorff topological space ... continuous map f X &rarr X , the following limit exists math H C,f lim n to infty frac 1 n H C vee f 1 C vee ldots vee f n 1 C . math Then the topological entropy of f , denoted h f ... of these iterates, as seen by the partition C . Thus the topological entropy is the average per iteration amount of information needed to describe long iterations of the map f . Definition of Bowen ... map . For each natural number n , a new metric d sub n sub is defined on X by the formula math ... . Denote by N n , &epsilon the maximum cardinality of an n , &epsilon separated set. The topological entropy of the map f is defined by math h f lim epsilon to 0 left limsup n to infty frac 1 n log N ..., it measures complexity of the topological dynamical system X , f . Rufus Bowen extended this definition of topological entropy in a way which permits X to be noncompact. Notes references See also Milnor Thurston kneading theory For the measure of correlations in systems with topological order see Topological entanglement entropy References R. L. Adler, A. G. Konheim, M. H. McAndrew, 1965 , http links.jstor.org sici?sici 0002 9947 196502 114 3A2 3C309 3ATE 3E2.0.CO 3B2 N Topological Entropy , Transactions ... article Topological entropy Topological entropy at Scholarpedia planetmath id 6068 title Topological ... more details
a homeomorphism that will conjugate the one into the other. Topological conjugacy is important in the study ..., math circ math denotes function composition . Definition Let math X math and math Y math be topological ... call math h math a topological conjugation between math f math and math g math . Similarly, a flow ... map and the tent map are topologically conjugate. Citation needed date November 2010 the logistic map of unit height and the Bernoulli map are topologically conjugate. Citation needed date November 2010 the logistic map of unit height and the tent map of unit height are topologically conjugate. Discussion Topological conjugation defines an equivalence relation in the space of all continuous surjections of a topological space to itself, by declaring math f math and math g math to be related if they are topologically ... system s, since each class contains all functions which share the same dynamics from the topological ... math g n h 1 circ f n circ h math . Speaking informally, topological conjugation is a change of coordinates in the topological sense. However, the analogous definition for flows is somewhat restrictive ... math be mapped to orbits of math psi math homeomorphically. This motivates the definition of topological ... sharing the same dynamics, again from the topological viewpoint. Topological equivalence We say that math ... s math is such that math varphi h y ,s h psi y,t math , then math s 0 math . Overall, topological equivalence is a weaker equivalence criterion than topological conjugacy, as it does not require ... matched, thus failing to satisfy the topological conjugacy criterion while satisfying the topological equivalence criterion. Generalizations of dynamic topological conjugacy There are two reported extensions of the concept of dynamic topological conjugacy 1. Analogous systems defined as isomorphic ..., Topological Conjugacy and Adjoint Systems ref . Cited References references See also Commutative diagram planetmath id 4353 title topological conjugation Category Topological dynamics Category Homeomorphisms ... more details
A topological game is an infinite positional game of perfect information played between two players on a topological space . Players choose objects with topological properties such as points, open sets ... notions like topological closure and wikt convergence convergence . It turns out that some fundamental topological constructions have a natural counterpart in topological games examples of these are the Baire ... time, some topological properties that arise naturally in topological games can be generalized beyond a game theoretic context by virtue of this duality, topological games have been widely used to describe new properties of topological spaces, and to put known properties under a different light. The term topological game was first introduced by Berge, ref C. Berge, Topological games with perfect ..., M m. des Sc. Mat., Gauthier Villars, Paris 1957. ref ref A. R. Pears, On topological games ... with topological groups. A different meaning for topological game , the concept of topological ..., On topological properties defined by games, Topics in Topology Proc. Colloq. Keszthely 1972 , Colloq ... by topological games ref R. Telg rsky, Spaces defined by topological games, Fund. Math. 88 1975 ... games, and defines and studies topological games within topology. After more than 35 years, the term topological game became widespread, and appeared in several hundreds of publications. The survey paper of Telg rsky ref name Telgarsky 1987 R. Telg rsky, Topological Games On the 50th Anniversary ... RMJM Telgarsky Topological Games.pdf 3.2MB PDF ref emphasizes the origin of topological games from the Banach Mazur game . There are two other meanings of topological games, but these are used less frequently. The term topological game introduced by Leon Petrosjan ref L. A. Petrosjan, Topological games ... of antagonistic pursuit evasion games. The trajectories in these topological games are continuous ... plane games , and Gale s games Bridg It games were called topological games by David Gale in his invited ... more details
Refimprove date November 2009 Expert verify date November 2009 Topological computing is the designing ... of the electromagnetic field was studied in detail and a topological theory of guided waves ..., 1988 ref ref name Kouzaev91 G.A. Kouzaev, Mathematical fundamentals of topological electrodynamics .... ref The topological theory, nonlocal by its nature, describes the electric and magnetic fields by their topological schemes or skeletons Clarify date November 2009 composed of the field force map separatrices and field equilibrium manifolds. These skeletons are coupled to each other through the topological ... aspects of topological theory of electromagnetic field and applications of topology in physics and electromagnetism can be found from ref name Ranada F. Ranada , A topological theory of the electromagnetic field, Lett Math. Phys. , Vol. 18, pp. 97 106, 1989. ref ref name Barret T. W. Barret, Topological ..., Electromagnetic Theory and Computations A Topological Approach, Cambridge University Press, 2004 .... ref ref name Afanasief G. W. Afanasief, Topological Effects in Quantum Mechanics, Kluwer Acad ..., 31 Dec. 1997. ref ref name Boi L. Boi, Geometrical and topological foundations of theoretical ... Federation, 2054794, 05.26.1992 . ref They excel in increased noise immunity due to their topological ... of topological modulation of electromagnetic field , Russian Physics Doklady , Vol. 38, pp. 512 514 ... and G. A. Kouzaev, Topological computer , Computers and People , 1, pp. 2 5, 1992. ref Topological ... impulses is a topological processor proposed in 1991 1992. ref name Gvozdev92a ref name Gvozdev92c This processor united with other necessary conventional digital units makes up the topological computer. The first predicate logic processor based on the idea of topological computing was designed and tested by FPGA modeling in 2007. ref name Kouzaev08a ref name Kouzaev08b The first quantum topological ..., Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc. , 40, 31 2003 , Topological ... more details
. In mathematics and physics , a topological soliton or a topological defect is a solution of a system ... by their homotopy class . Topological defects are not only stable against small wiktionary ... transformation that will map them homotopically to a uniform or trivial solution ... Zumino Witten model in quantum field theory. Topological defects are believed to drive phase transition s in condensed matter physics. Notable examples of topological defects are observed in Lambda transition ... predict topological defects to have formed in the early universe . According to the Big Bang theory ... like what happens in condensed matter systems. In physical cosmology , a topological defect is an often ... known topological defects are magnetic monopole s, cosmic string s, domain wall s, Skyrmion s and Texture ... began breaking down in regions that spread at the speed of light topological defects occur where .... Types of topological defects Various different types of topological defects are possible, with the type ... . Observation Topological defects, of the cosmological type, are extremely high energy phenomena and are likely impossible to produce in artificial Earth bound physics experiments, but topological defects that formed during the universe s formation could theoretically be observed. No topological .... year 1979 title The topological theory of defects in ordered media journal Reviews of Modern Physics volume 51 issue 3 ref Topological methods have been used in several problems of condensed matter theory. Po naru and Toulouse used topological methods to obtain a condition for line string defects in liquid ... the existence, stability and classifications of topological defects in that medium. ref name mermin ... conjugacy classes of math pi 1 R math Stable defects Unlike in cosmology and field theory, topological ... research gr public cs top.html title Topological defects publisher Cambridge cosmology ... Topological entropy in physics Topological order Topological quantum field theory Topological quantum ... more details
In mathematics , a topological manifold is a topological space can even be a Hausdorff space separated space which looks locally like Euclidean space in a sense defined below. Topological manifolds form an important class of topological spaces with applications throughout mathematics. A manifold can mean a topological manifold, or more frequently, a topological manifold together with some additional structure. Differentiable manifold s, for example, are topological manifolds equipped with a differential structure . Every manifold has an underlying topological manifold, obtained simply by forgetting ... focuses purely on the topological aspects of manifolds. Formal definition A topological space ... n sup . A topological manifold is a locally Euclidean Hausdorff space . It is common to place additional requirements on topological manifolds. In particular, many authors define them to be paracompact ... of this article a manifold will mean a topological manifold. An n manifold will mean a topological ..., being locally Euclidean is a topological property . Manifolds inherit many of the local properties ... property for a topological space, it is common to add paracompactness to the definition of a manifold ... manifolds have all the topological properties of metric spaces. In particular, they are perfectly .... Dimensionality The dimension of a manifold is a topological property , meaning that any manifold ... to the domain or range of such a map . A space M is locally Euclidean if and only if it can be cover ... &psi &phi sup &minus 1 sup &phi U &cap V &rarr &psi U &cap V . Such a map is a homeomorphism ... of dimension 5. Manifolds with boundary A slightly more general concept is sometimes useful. A topological ... R n x n ge 0 . math The terminology is somewhat confusing every topological manifold is a topological ... cite journal last Gauld first D. B. year 1974 title Topological Properties of Manifolds journal ... Robion Kirby coauthors Siebenmann, Laurence C. title Foundational Essays on Topological Manifolds ... more details
In mathematics , a topological module is a module algebra module over a topological ring such that scalar multiplication and addition are continuous function topology continuous . Examples A topological vector space is a topological module over a topological field . An abelian group abelian topological group can be considered as a topological module over Z , where Z is the ring of integers with the discrete topology . A topological ring is a topological module over each of its subring s. A more complicated example is the I adic topology on a ring and its modules. Let I be an ideal ring theory ideal of a ring R . The sets of the form nowrap x I sup n sup , for all x in R and all positive integers n , form a base topology base for a topology on R that makes R into a topological ring. Then for any left R module M , the sets of the form nowrap x I sup n sup M , for all x in M and all positive integers n , form a base for a topology on M that makes M into a topological module over the topological ring R . Category Algebra Category Topology topology stub algebra stub ... more details
relation is defined on the topological space X . The map f is then the natural projection ...Image Topological space examples.svg frame right 300px Four examples and two non examples of topologies ... and 2,3 i.e. 2 , is missing. Topological spaces are mathematical structures that allow the formal definition ... and are a central unifying notion. The branch of mathematics that studies topological spaces in their own right is called topology . Definition A topological space is a set mathematics set X together ... of X are usually called points , though they can be any mathematical objects. A topological space ... main Characterizations of the category of topological spaces There are many other equivalent ways to define a topological space. In other words, each of the following defines a category theory category equivalent to the category of topological spaces above. For example, using de Morgan s laws ... sets is also closed. Using these axioms, another way to define a topological space is as a set X ... are the open sets. Another way to define a topological space is by using the Kuratowski closure ... main Comparison of topologies A variety of topologies can be placed on a set to form a topological ... every member of F . Continuous functions A function mathematics function between topological spaces ... of topological spaces with topological spaces as object category theory objects and continuous ... mathematics homology theory , and K theory , to name just a few. Examples of topological spaces ... as a different topological space. Any set can be given the discrete space discrete topology in which ... converges to every point of the space. This example shows that in general topological spaces, limits of sequences need not be unique. However, often topological spaces are required to be Hausdorff ... and edges. The Sierpi ski space is the simplest non discrete topological space. It has important ... finite set . Such spaces are called finite topological space s. Finite spaces are sometimes used ... more details
orphan date November 2009 Topological excitations are certain features of classical solutions of gauge field theory gauge field theories . Namely, a gauge field theory on a manifold math M math with a gauge group math G math may possess classical solutions with a quantized topology topological invariant called topological charge . The term topological excitation especially refers to a situation when the topological charge is an integral of a localized quantity. Examples ref F. A. Bais, Topological excitations in gauge theories An introduction from the physical point of view. Springer Lecture Notes in Mathematics, vol. 926 1982 ref 1 math M R 2 math , math G U 1 math , the topological charge is called magnetic flux . 2 math M R 3 math , math G SO 3 U 1 math , the topological charge is called magnetic charge . The concept of a topological excitation is almost synonymous with that of a topological defect . References See Wikipedia Footnotes on how to create references using ref ref tags which will then appear here automatically Reflist DEFAULTSORT Topological Excitations Category Theoretical physics ... more details
In mathematics , a topological semigroup is a semigroup which is simultaneously a topological space , and whose semigroup operation is continuous function continuous . ref Artur Hideyuki Tomita. http tatra.mat.savba.sk Full 14 10tomita.ps On sequentially compact both sides cancellative semigroups with sequentially continuous addition. ref A topological group is a topological semigroup. TODO Cite a more topical source instead. See also Strongly continuous semigroup Analytic semigroup Ellis Nakamura lemma References references Category Topology algebra stub topology stub ... more details
Noref date November 2009 In mathematics , a topological algebra A over a topological field K is a topological vector space together with a continuous multiplication math cdot A times A longrightarrow A math math a,b longmapsto a cdot b math that makes it an algebra over a field algebra over K . A unital associative algebra associative topological algebra is a topological ring . An example of a topological algebra is the algebra C 0,1 of continuous real valued functions on the closed unit interval 0,1 , or more generally any Banach algebra . The term was coined by David van Dantzig it appears in the title of his Thesis doctoral dissertation 1931 . The natural notion of subspace in a topological algebra is that of a topologically closed subalgebra . A topological algebra A is said to be generated by a subset S if A itself is the smallest closed subalgebra of A that contains S . For example by the Stone Weierstrass theorem , the set id sub 0,1 sub consisting only of the identity function id sub 0,1 sub is a generating set of the Banach algebra C 0,1 . Category Topological vector spaces Category Topological algebra Category Algebras topology stub pl Algebra topologiczna uk ... more details
Unreferenced date November 2009 In topology , two points of a topological space X are topologically indistinguishable if they have exactly the same neighborhood topology neighborhood s. That is, if x and y are points in X , and A is the set of all neighborhoods that contain x , and B is the set of all ... set can then be used to distinguish between the two points. A T0 space T sub 0 sub space is a topological ... of the separation axiom s. Topological indistinguishability defines an equivalence relation on any topological space X . If x and y are points of X we write x y for x and y are topologically ... 0 sub space s in particular, for Hausdorff space s the notion of topological indistinguishability ... if they are equal almost everywhere . In a topological group , x y if and only if x sup &minus ... pseudometric spaces and topological groups. In a uniform space, x y if and only if the pair x , y belongs ... relation on X which is just that of topological indistinguishability. Let X have the initial topology ... X there is a topology on X for which the notion of topological indistinguishability agrees with the given .... This is called the partition topology on X . Specialization preorder The topological ... by is precisely that of topological indistinguishability x &equiv y if and only if x &le y and y &le x . A topological space is said to be R0 space symmetric or R sub 0 sub if the specialization .... Topological indistinguishability is better behaved in these spaces and easier to understand ... are topologically indistinguishable. Let X be a topological space and let x and y be points of X . Denote ... if each of their components are topologically indistinguishable. Kolmogorov quotient Since topological indistinguishability is an equivalence relation on any topological space X , we can form ..., by the characteristic property of the quotient map any continuous map f X Y from X to a T sub 0 sub space factors through the quotient map q X KX . Although the quotient map q is generally not a homeomorphism ... more details
In the field of shape optimization , a topological derivative is, conceptually, a derivative of a function of a region with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. The topological derivative is often called a topological gradient , it comes from the second term of the topological asymptotic expansion, dealing only with infinitesimal perturbations. Shape optimization concerns itself with finding an optimal shape. That is, find math Omega math to minimize some scalar valued objective function , math J Omega math . Neglecting changes in topology, an initial guess can be improved by perturbing the shape of math Omega math by methods of calculus of variations and functional analysis . Applications Topological optimization Image processing Inverse problems References Reflist External links Allaire and al. http www.cmap.polytechnique.fr jouve papers toplev.pdf Structural optimization using topological and shape sensitivity via a level set method S. Amtutz, I. Horchani and M. Masmoudi http www.univ avignon.fr fileadmin documents Users Fiches X P crack.pdf Crack detection by the topological gradient method J. Soko owski and A. Zochowski http hal.inria.fr docs 00 07 35 18 PDF RR 3170.pdf On topological derivative in shape optimization Category Mathematical optimization math stub ... more details
In mathematics , a topological ring is a ring algebra ring R which is also a topological space such that both ... , where R × R carries the product topology . General comments The group of units of R may not be a topological .... Its unit group, called the idele group , is not a topological group in the subspace topology. Embedding the unit group of R into the product R × R as x , x sup 1 sup does make the unit group a topological ..., or equivalently, to define the topological ring as a ring which is a topological group for in which multiplication is continuous, too. Examples Topological rings occur in mathematical analysis , for examples as rings of continuous real valued function mathematics function s on some topological space ... s on some normed vector space all Banach algebra s are topological rings. The rational number rational , real number real , complex number complex and p adic number p adic numbers are also topological rings even topological fields, see below with their standard topologies. In the plane, split complex number s and dual numbers form alternative topological rings. See hypercomplex numbers for other ... I sup n sup U . This turns R into a topological ring. The I adic topology is Hausdorff space ... main Completion ring theory Every topological ring is a topological group with respect to addition and hence a uniform space in a natural manner. One can thus ask whether a given topological ring ... unique complete topological ring S which contains R as a dense topology dense subring such that the given ... series and the p adic number p adic integers are most naturally defined as completions of certain topological rings carrying I adic topologies. Topological fields Some of the most important examples are also field mathematics field s F . To have a topological field we should also specify that multiplicative ... examples. References springer id T t093110 title Topological ring author L. V. Kuzmin springer id T t093060 title Topological field author D. B. Shakhmatov Seth Warner Topological Rings . North ... more details
The topological censorship theorem states that general relativity does not allow an observer to probe the topology of spacetime any topological structure collapses too quickly to allow light to traverse it. More precisely, in a globally hyperbolic , asymptotically flat spacetime satisfying the null energy condition , every causal curve from past null infinity to future null infinity is fixed endpoint homotopic to a curve in a topologically trivial neighbourhood of infinity. References cite journal author John L. Friedman , Kristin Schleich , and Donald M. Witt year 1993 title Topological Censorship journal Phys.Rev.Lett. volume 71 pages 1486 1489 doi 10.1103 PhysRevLett.71.1486 pmid 10054420 issue 10 Category Lorentzian manifolds ... more details
p from x to y , i.e., a continuous map p 0,1 X with p 0 x and p 1 y . Path connected ... connected and every continuous map f S sup 1 sup X is homotopic to a constant map. Locally simply connected . A space X is locally simply connected space locally simply connected ... cover . Contractible . A space X is contractible if the identity function identity map on X is homotopic to a constant map. Contractible spaces are always simply connected. Hyper connected ... at every point. All topological group s are homogeneous. Finitely generated or Alexandrov . A space ... members of the category of topological spaces and continuous maps. Zero dimensional . A space is zero ... Winding number Knot invariant Linking number Fixed point property Topological quantum number Homotopy ... of topological spaces Category Homeomorphisms it Invariante topologico nl Topologische eigenschap ... more details
File Topological insulator band structure.svg thumb A idealized electronic band structure band structure for a topological insulator. The Fermi level falls within the bulk band gap which is traversed by topologically protected surface states. A topological insulator is a material that behaves as an Insulator .... In the bulk of a topological insulator the electronic band structure resembles an ordinary insulator, with the Fermi level falling between the conduction and valence bands. On the surface of a topological ... spin momentum locking or topological order . At a given energy the only other available electronic ... genus in topology, and are an example of topological order topologically ordered states. ref Cite journal last Kane first C. L. coauthors Mele, E. J. date 30. September 2005 title Z sub 2 sub Topological ... coauthors Taylor L. Hughes, Shou Cheng Zhang title Quantum Spin Hall Effect and Topological Phase ... issue 4 pages 045302 last Fu first Liang coauthors C. L. Kane title Topological insulators with inversion ... phases in 3D emergence of a topological gapless phase journal New Journal of Physics accessdate ... bulk solids of binary compounds involving bismuth . The first experimentally realized 3D topological ... A 3D Topological Dirac insulator in a quantum spin Hall phase journal Nature volume 452 issue 9 pages ... pmid 18432240 ref . A method to measure Z2 topological invariants was also demonstrated ref ... and M. Z. Hasan title Observation of Unconventional Quantum Spin Textures in Topological Insulators ... first M.Z. coauthors Kane, C.L. title Topological Insulators journal Review of Modern Physics accessdate ... systems are now believed to include topological insulators. ref Cite journal doi 10.1038 nmat2771 ... Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena ..., R. J. Cava, M. Z. Hasan title Observation of Time Reversal Protected Single Dirac Cone Topological ... 0295 5075 81 5 57006 ref Topological order is encoded in the existence of a gas of helical Dirac fermions ... more details
About quantum physics the graph theoretical concept topological sort In physics , topological order ref Xiao Gang Wen , http dao.mit.edu wen pub topo.pdf Topological Orders in Rigid States. Int. J. Mod ... transition order parameters and Long range order long range correlations . However, topological ... fractional statistics , edge state s, topological entropy , etc. Roughly speaking, topological order is a pattern of long range quantum entanglement in quantum states. States with different topological ... of topological order However, since late 1980s, it has become gradually apparent that Landau ... topological order ref Xiao Gang Wen , Intl. J. Mod. Phys ., B4 , 239 1990 , http dao.mit.edu wen pub topo.pdf Topological Orders in Rigid States ref . The name topological order is motivated by the low energy effective theory of the chiral spin states which is a topological quantum field theory TQFT ref Atiyah, Michael 1988 , Topological quantum field theories , Publications Mathe matiques ... ref Witten, Edward 1988 , Topological quantum field theory , Communications in Mathematical Physics .... Phys ., B4 , 239 1990 , Topological Orders in Rigid States ref . were introduced to characterize the different topological orders in chiral spin states. Recently, it was shown that topological orders can also be characterized by topological entropy in physics topological entropy ref Alexei Kitaev and John Preskill, Phys. Rev. Lett. 96 , 110404 2006 , Topological Entanglement Entropy ref ref Levin M. and Wen X G., http link.aps.org doi 10.1103 PhysRevLett.96.110405 Detecting topological order in a ground ... that chiral spin states do not describe high temperature superconductors, and the theory of topological ... chiral spin states and quantum Hall effect quantum Hall states allows one to use the theory of topological ... by topological orders, so the topological order does have experimental realizations. quantum Hall ... of the concept of topological order. But FQH states are not the first experiemntally discovered ... more details
For topological index in mathematics, see Atiyah Singer index theorem . In the fields of chemical graph theory , topology chemistry molecular topology , and mathematical chemistry , a topological index ... origyear pages quote isbn 3 527 29913 0 oclc doi url accessdate ref Topological indices are numerical ... . Topological indices are used for example in the development of quantitative structure activity ... isbn 0 12 406560 0 oclc doi url accessdate ref Calculation Topological descriptors are derived from ... by edges. The connections between the atoms can be described by various types of topological matrices ... number, usually known as graph invariant, graph theoretical index or topological index. ref name ... chemistry and bioinformatics current trends in drugs discovery with networks topological indices journal ... url issn ref As a result, the topological index can be defined as two dimensional descriptors that can ... or labeled and no need of energy minimization of the chemical structure. Types The simplest topological ... pages quote isbn 0 444 42244 7 oclc doi url accessdate ref More sophisticated topological indices ... in the molecule. Hundreds of indices were introduced. The Hosoya index is the first topological index recognized in chemical graph theory, and it is often referred to as the topological index. ref Cite journal last Hosoya first Haruo title Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons journal Bulletin of the Chemical ... name urlDescriptors Topological Descriptors cite web url http www.codessa pro.com descriptors topo index.htm title Topological Descriptors author Katritzky AR, Karelson M, Petrukhin R authorlink coauthors ... Chemistry pages 327 332 volume 8 issue 1 ref Discrimination capability and superindices A topological ... of topological index. To increase the discrimination capability a few topological indices may be combined to superindex . ref Cite journal title Isomer discrimination by topological information ... more details
analog in discrete Morse theory . See also Sperner s lemma Discrete exterior calculus Topological ... contribution 25 years proof of the Kneser conjecture The advent of topological combinatorics title ... 0507390 Trends in Topological Combinatorics http edocs.tu berlin.de diss 2004 lange carsten.pdf Combinatorial Curvatures, Group Actions, and Colourings Aspects of Topological Combinatorics cite book ... on Topological Methods in Combinatorics and Geometry publisher Springer year 2003 isbn 978 3540003625 ... Topological Combinatorics Category Combinatorics Category Topology de Topologische Kombinatorik ... more details
In mathematics, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory harv Lurie 2009 . See also Infinity category Simplicial category References Citation last1 Lurie first1 Jacob title Higher topos theory arxiv math.CT 0608040 publisher Princeton University Press series Annals of Mathematics Studies isbn 978 0 691 14049 0 978 0 691 14049 0 mr 2522659 year 2009 volume 170 Category Category theory ... more details
Unreferenced date August 2008 Merge pair of spaces date August 2008 In mathematics , more specifically algebraic topology , a pair math X,A math is short hand for an inclusion of topological spaces math i colon A hookrightarrow X math . Sometimes math i math is assumed to be a cofibration . A morphism from math X,A math to math X ,A math is given by two maps math f colon X rightarrow X math and math g colon A rightarrow A math such that math i circ g f circ i math . Pairs come up mainly in homology theory and cohomology theory , where chains in math A math are made equivalent to 0, when considered as chains in math X math . Heuristically, one often thinks of a pair math X,A math as being akin to the quotient space math X A math . There is a functor from spaces to pairs, which sends a space math X math to the pair math X, varnothing math . References reflist Category Algebraic topology fr Paire d espaces ... more details
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