|
Classifying Space For SO(n)
In mathematics, the classifying space \operatorname(n) for thspecial orthogonal group'' \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname(n). The isomorphism is given by pullback. Definition There is a canonical inclusion of real oriented Grassmannians given by \widetilde\operatorname_n(\mathbb^k)\hookrightarrow\widetilde\operatorname_n(\mathbb^), V\mapsto V\times\. Its colimit is:Milnor & Stasheff 74, section 12.2 The Oriented Universal Bundle on page 151 : \operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k). Since real oriented Grassmannians can be expressed as a homogeneous space by: : \widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GF(2)
(also denoted \mathbb F_2, or \mathbb Z/2\mathbb Z) is the finite field with two elements. is the Field (mathematics), field with the smallest possible number of elements, and is unique if the additive identity and the multiplicative identity are denoted respectively and , as usual. The elements of may be identified with the two possible values of a bit and to the Boolean domain, Boolean values ''true'' and ''false''. It follows that is fundamental and ubiquitous in computer science and its mathematical logic, logical foundations. Definition GF(2) is the unique field with two elements with its additive identity, additive and multiplicative identity, multiplicative identities respectively denoted and . Its addition is defined as the usual addition of integers but modulo 2 and corresponds to the table below: If the elements of GF(2) are seen as Boolean values, then the addition is the same as that of the logical XOR operation. Since each element equals its opposite (m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying+space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e., a topological space all of whose homotopy groups are trivial) by a proper free action of ''G''. It has the property that any ''G'' principal bundle over a paracompact manifold is isomorphic to a pullback of the principal bundle EG \to BG. As explained later, this means that classifying spaces represent a set-valued functor on the homotopy category of topological spaces. The term classifying space can also be used for spaces that represent a set-valued functor on the category of topological spaces, such as Sierpiński space. This notion is generalized by the notion of classifying topos. However, the rest of this article discusses the more commonly used notion of classifying space up to homotopy. For a discrete group ''G'', ''BG'' is a path-connected topological space ''X'' such that the fundamental group of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For SU(n)
In mathematics, the classifying space \operatorname(n) for the special unitary group \operatorname(n) is the base space of the universal \operatorname(n) principal bundle \operatorname(n)\rightarrow\operatorname(n). This means that \operatorname(n) principal bundles over a CW complex up to isomorphism are in bijection with homotopy classes of its continuous maps into \operatorname(n). The isomorphism is given by pullback. Definition There is a canonical inclusion of complex oriented Grassmannians given by \widetilde\operatorname_n(\mathbb^k)\hookrightarrow\widetilde\operatorname_n(\mathbb^), V\mapsto V\times\. Its colimit is: \operatorname(n) :=\widetilde\operatorname_n(\mathbb^\infty) :=\lim_\widetilde\operatorname_n(\mathbb^k). Since real oriented Grassmannians can be expressed as a homogeneous space by: : \widetilde\operatorname_n(\mathbb^k) =\operatorname(n+k)/(\operatorname(n)\times\operatorname(k)) the group structure carries over to \operatorname(n). Simplest classifyi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For U(n)
In mathematics, the classifying space for the unitary group U(''n'') is a space BU(''n'') together with a universal bundle EU(''n'') such that any hermitian bundle on a paracompact space ''X'' is the pull-back of EU(''n'') by a map ''X'' → BU(''n'') unique up to homotopy. This space with its universal fibration may be constructed as either # the Grassmannian of ''n''-planes in an infinite-dimensional complex Hilbert space; or, # the direct limit, with the induced topology, of Grassmannians of ''n'' planes. Both constructions are detailed here. Construction as an infinite Grassmannian The total space EU(''n'') of the universal bundle is given by :EU(n)=\left \. Here, ''H'' denotes an infinite-dimensional complex Hilbert space, the ''e''''i'' are vectors in ''H'', and \delta_ is the Kronecker delta. The symbol (\cdot,\cdot) is the inner product on ''H''. Thus, we have that EU(''n'') is the space of orthonormal ''n''-frames in ''H''. The Group action (mathematics), group action ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classifying Space For O(n)
In mathematics, the classifying space for the orthogonal group O(''n'') may be constructed as the Grassmannian of ''n''-planes in an infinite-dimensional real space \mathbb^\infty. Cohomology ring The cohomology ring of \operatorname(n) with coefficients in the field \mathbb_2 of two elements is generated by the Stiefel–Whitney classes:Hatcher 02, Theorem 4D.4. : H^*(\operatorname(n);\mathbb_2) =\mathbb_2 _1,\ldots,w_n Infinite classifying space The canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) induce canonical inclusions \operatorname(n)\hookrightarrow\operatorname(n+1) on their respective classifying spaces. Their respective colimits are denoted as: : \operatorname :=\lim_\operatorname(n); : \operatorname :=\lim_\operatorname(n). \operatorname is indeed the classifying space of \operatorname. See also * Classifying space for U(''n'') * Classifying space for SO(n) * Classifying space for SU(n) Literature * * * External links * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler Class
In mathematics, specifically in algebraic topology, the Euler class is a characteristic class of oriented, real vector bundles. Like other characteristic classes, it measures how "twisted" the vector bundle is. In the case of the tangent bundle of a smooth manifold, it generalizes the classical notion of Euler characteristic. It is named after Leonhard Euler because of this. Throughout this article E is an oriented, real vector bundle of rank r over a base space X. Formal definition The Euler class e(E) is an element of the integral cohomology group :H^r(X; \mathbf), constructed as follows. An orientation of E amounts to a continuous choice of generator of the cohomology :H^r(\mathbf^, \mathbf^ \setminus \; \mathbf)\cong \tilde^(S^;\mathbf)\cong \mathbf of each fiber \mathbf^ relative to the complement \mathbf^ \setminus \ of zero. From the Thom isomorphism, this induces an orientation class :u \in H^r(E, E \setminus E_0; \mathbf) in the cohomology of E relative to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pontryagin Class
In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. Definition Given a real vector bundle E over M, its k-th Pontryagin class p_k(E) is defined as :p_k(E) = p_k(E, \Z) = (-1)^k c_(E\otimes \Complex) \in H^(M, \Z), where: *c_(E\otimes \Complex) denotes the 2k-th Chern class of the complexification E\otimes \Complex = E\oplus iE of E, *H^(M, \Z) is the 4k-cohomology group of M with integer coefficients. The rational Pontryagin class p_k(E, \Q) is defined to be the image of p_k(E) in H^(M, \Q), the 4k-cohomology group of M with rational coefficients. Properties The total Pontryagin class :p(E)=1+p_1(E)+p_2(E)+\cdots\in H^*(M,\Z), is (modulo 2-torsion) multiplicative with respect to Whitney sum of vector bundles, i.e., :2p(E\oplus F)=2p(E)\smile p(F) for two vector bundles E and F over M. In terms of the individual Pon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all rational numbers is often referred to as "the rationals", and is closed under addition, subtraction, multiplication, and division by a nonzero rational number. It is a field under these operations and therefore also called the field of rationals or the field of rational numbers. It is usually denoted by boldface , or blackboard bold A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Characteristic (algebra)
In mathematics, the characteristic of a ring , often denoted , is defined to be the smallest positive number of copies of the ring's multiplicative identity () that will sum to the additive identity (). If no such number exists, the ring is said to have characteristic zero. That is, is the smallest positive number such that: : \underbrace_ = 0 if such a number exists, and otherwise. Motivation The special definition of the characteristic zero is motivated by the equivalent definitions characterized in the next section, where the characteristic zero is not required to be considered separately. The characteristic may also be taken to be the exponent of the ring's additive group, that is, the smallest positive integer such that: : \underbrace_ = 0 for every element of the ring (again, if exists; otherwise zero). This definition applies in the more general class of rngs (see '); for (unital) rings the two definitions are equivalent due to their distributive law. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
