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Bundle Gerbe
In mathematics, a bundle gerbe is a geometrical model of certain 1-gerbes with connection, or equivalently of a 2-class in Deligne cohomology. Topology U(1)-principal bundles over a space M (see circle bundle) are geometrical realizations of 1-classes in Deligne cohomology which consist of 1-form connections and 2-form curvatures. The topology of a U(1) bundle is classified by its Chern class, which is an element of H^2(M, \mathbb), the second integral cohomology of M. Gerbes, or more precisely 1-gerbes, are abstract descriptions of Deligne 2-classes, which each define an element of H^3(M, \mathbb), the third integral cohomology of ''M''. As a cohomology class in Deligne cohomology Recall for a smooth manifold M the p-th Deligne cohomology groups are defined by the hypercohomology of the complex \mathbb(q)_D^\infty = \underline(q) \to \mathcal_^0 \xrightarrow \mathcal_^1 \xrightarrow \cdots \xrightarrow \mathcal_^ called the weight q Deligne complex, where \mathcal^k_ is the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twist (mathematics)
In differential geometry Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ..., the twist of a ''Ribbon (mathematics), ribbon'' is its rate of change (mathematics), rate of axial rotation. Let a ribbon (X,U) be composted of space curve X=X(s), where s is the arc length of X, and U=U(s) the a unit normal vector, perpendicular at each point to X. Since the ribbon (X,U) has edges X and X'=X+\varepsilon U, the twist (or ''total twist number'') Tw measures the average winding number, winding of the edge curve X' around and along the axial curve X. According to Love (1944) twist is defined by : Tw = \dfrac \int \left( U \times \dfrac \right) \cdot \dfrac ds \; , where dX/ds is the unit tangent vector to X. The total twist number Tw can be decomposed (Moffatt & Ricca 1992) into ''normalized ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Exterior Derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree. The exterior derivative was first described in its current form by Élie Cartan in 1899. The resulting calculus, known as exterior calculus, allows for a natural, metric-independent generalization of Stokes' theorem, Gauss's theorem, and Green's theorem from vector calculus. If a differential -form is thought of as measuring the flux through an infinitesimal - parallelotope at each point of the manifold, then its exterior derivative can be thought of as measuring the net flux through the boundary of a -parallelotope at each point. Definition The exterior derivative of a differential form of degree (also differential -form, or just -form for brevity here) is a differential form of degree . If is a smooth function (a -form), then the exterior derivative of is the differential of . That is, is the unique -form such that for e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term was introduced by Benjamin Peirce in the context of his work on the classification of algebras. Examples *This definition can be applied in particular to square matrices. The matrix :: A = \begin 0 & 1 & 0\\ 0 & 0 & 1\\ 0 & 0 & 0 \end :is nilpotent because A^3=0. See nilpotent matrix for more. * In the factor ring \Z/9\Z, the equivalence class of 3 is nilpotent because 32 is congruent to 0 modulo 9. * Assume that two elements a and b in a ring R satisfy ab=0. Then the element c=ba is nilpotent as \beginc^2&=(ba)^2\\ &=b(ab)a\\ &=0.\\ \end An example with matrices (for ''a'', ''b''):A = \begin 0 & 1\\ 0 & 1 \end, \;\; B =\begin 0 & 1\\ 0 & 0 \end. Here AB=0 and BA=B. *By definition, any element of a nilsemigroup is nilpotent. Properties No nilpotent element c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Cohomology
Twisted may refer to: Film and television * ''Twisted'' (1986 film), a horror film by Adam Holender starring Christian Slater * ''Twisted'' (1996 film), a modern retelling of ''Oliver Twist'' * ''Twisted'', a 2011 Singapore Chinese film directed by Chai Yee Wei * ''Twisted'' (2004 film), a thriller starring Ashley Judd and Andy Garcia * ''Twisted'', a parody musical by StarKid Productions * ''Twisted'' (TV series), 2013 * "Twisted" (''Star Trek: Voyager''), a television episode * ''Twisted'' (web series), an Indian erotic thriller web series Software and games * '' Twisted: The Game Show'', a 1994 3DO game * Twisted (software), an event-driven networking framework * '' WarioWare: Twisted!'', a 2005 game for the Game Boy Advance Books * ''Twisted'' (book), a short story collection by crime writer Jeffery Deaver ** ''More Twisted'', a second short story collection by Deaver * '' Twisted'', a novel by Laurie Halse Anderson * ''Twisted'', a ''Pretty Little Liars'' novel by Sar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression is an example of a -form, and can be integrated over an interval contained in the domain of : :\int_a^b f(x)\,dx. Similarly, the expression is a -form that can be integrated over a surface : :\int_S (f(x,y,z)\,dx\wedge dy + g(x,y,z)\,dz\wedge dx + h(x,y,z)\,dy\wedge dz). The symbol denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form represents a volume element that can be integrated over a region of space. In general, a -form is an object that may be integrated over a -dimensional manifold, and is homogeneous of degree in the coordinate differentials dx, dy, \ldots. On an -dimensional manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic Class
In mathematics, a characteristic class is a way of associating to each principal bundle of ''X'' a cohomology class of ''X''. The cohomology class measures the extent the bundle is "twisted" and whether it possesses sections. Characteristic classes are global invariants that measure the deviation of a local product structure from a global product structure. They are one of the unifying geometric concepts in algebraic topology, differential geometry, and algebraic geometry. The notion of characteristic class arose in 1935 in the work of Eduard Stiefel and Hassler Whitney about vector fields on manifolds. Definition Let ''G'' be a topological group, and for a topological space X, write b_G(X) for the set of isomorphism classes of principal ''G''-bundles over X. This b_G is a contravariant functor from Top (the category of topological spaces and continuous functions) to Set (the category of sets and functions), sending a map f\colon X\to Y to the pullback operation f^*\colon b_G(Y) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Chern Character
Twisted may refer to: Film and television * ''Twisted'' (1986 film), a horror film by Adam Holender starring Christian Slater * ''Twisted'' (1996 film), a modern retelling of ''Oliver Twist'' * ''Twisted'', a 2011 Singapore Chinese film directed by Chai Yee Wei * ''Twisted'' (2004 film), a thriller starring Ashley Judd and Andy Garcia * ''Twisted'', a parody musical by StarKid Productions * ''Twisted'' (TV series), 2013 * "Twisted" (''Star Trek: Voyager''), a television episode * ''Twisted'' (web series), an Indian erotic thriller web series Software and games * '' Twisted: The Game Show'', a 1994 3DO game * Twisted (software), an event-driven networking framework * '' WarioWare: Twisted!'', a 2005 game for the Game Boy Advance Books * ''Twisted'' (book), a short story collection by crime writer Jeffery Deaver ** ''More Twisted'', a second short story collection by Deaver * '' Twisted'', a novel by Laurie Halse Anderson * ''Twisted'', a ''Pretty Little Liars'' novel by Sar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted K-theory
In mathematics, twisted K-theory (also called K-theory with local coefficients) is a variation on K-theory, a mathematical theory from the 1950s that spans algebraic topology, abstract algebra and operator theory. More specifically, twisted K-theory with twist ''H'' is a particular variant of K-theory, in which the twist is given by an integral 3-dimensional cohomology class. It is special among the various twists that K-theory admits for two reasons. First, it admits a geometric formulation. This was provided in two steps; the first one was done in 1970 (Publ. Math. de l' IHÉS) by Peter Donovan and Max Karoubi; the second one in 1988 by Jonathan Rosenberg iContinuous-Trace Algebras from the Bundle Theoretic Point of View In physics, it has been conjectured to classify D-branes, Ramond-Ramond field strengths and in some cases even spinors in type II string theory. For more information on twisted K-theory in string theory, see K-theory (physics). In the broader context of K-the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
K-theory
In mathematics, K-theory is, roughly speaking, the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is a cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It is also a fundamental tool in the field of operator algebras. It can be seen as the study of certain kinds of invariants of large matrices. K-theory involves the construction of families of ''K''-functors that map from topological spaces or schemes to associated rings; these rings reflect some aspects of the structure of the original spaces or schemes. As with functors to groups in algebraic topology, the reason for this functorial mapping is that it is easier to compute some topological properties from the mapped rings than from the original spaces or schemes. Examples of results gleaned from the K-theory approach include the Grothendieck–Riemann–Roch theorem, Bott periodicity, the Atiyahâ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
