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Quaternionic Projective Space
In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions \mathbb. Quaternionic projective space of dimension ''n'' is usually denoted by :\mathbb^n and is a closed manifold of (real) dimension 4''n''. It is a homogeneous space for a Lie group action, in more than one way. The quaternionic projective line \mathbb^1 is homeomorphic to the 4-sphere. In coordinates Its direct construction is as a special case of the projective space over a division algebra. The homogeneous coordinates of a point can be written : _0,q_1,\ldots,q_n/math> where the q_i are quaternions, not all zero. Two sets of coordinates represent the same point if they are 'proportional' by a left multiplication by a non-zero quaternion ''c''; that is, we identify all the : q_0,cq_1\ldots,cq_n/math>. In the language of group actions, \mathbb^n is the orbit space of \mathbb^\setminu ... [...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] |
K(Z,2)
K, or k, is the eleventh letter of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''kay'' (pronounced ), plural ''kays''. The letter K usually represents the voiceless velar plosive. History The letter K comes from the Greek letter Κ (kappa), which was taken from the Semitic kaph, the symbol for an open hand. This, in turn, was likely adapted by Semitic tribes who had lived in Egypt from the hieroglyph for "hand" representing /ḏ/ in the Egyptian word for hand, ⟨ ḏ-r-t⟩ (likely pronounced in Old Egyptian). The Semites evidently assigned it the sound value instead, because their word for hand started with that sound. K was brought into the Latin alphabet with the name ''ka'' /kaː/ to differentiate it from C, named ''ce'' (pronounced /keː/) and Q, named ''qu'' and pronounced /kuː/. In the earliest Latin inscriptions, the letters C, K and Q were all u ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphic
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definition Given two differentiable manifolds M and N, a continuously differentiable map f \colon M \rightarrow N is a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. Two C^r-differentiable manifolds are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line Over A Ring
In mathematics, the projective line over a ring is an extension of the concept of projective line over a field (mathematics), field. Given a ring (mathematics), ring ''A'' (with 1), the projective line P1(''A'') over ''A'' consists of points identified by projective coordinates. Let ''A''× be the group of units of ''A''; pairs and from are related when there is a ''u'' in ''A''× such that and . This relation is an equivalence relation. A typical equivalence class is written . , that is, is in the projective line if the right ideal, one-sided ideal generated by ''a'' and ''b'' is all of ''A''. The projective line P1(''A'') is equipped with a homography#Homography groups, group of homographies. The homographies are expressed through use of the matrix ring over ''A'' and its group of units ''V'' as follows: If ''c'' is in Z(''A''×), the center (group theory), center of ''A''×, then the Group action (mathematics), group action of matrix \left(\beginc & 0 \\ 0 & c \end\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set with two binary operations called ''addition'' and ''multiplication'', which obey the same basic laws as addition and multiplication of integers, except that multiplication in a ring does not need to be commutative. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. A ''ring'' may be defined as a set that is endowed with two binary operations called ''addition'' and ''multiplication'' such that the ring is an abelian group with respect to the addition operator, and the multiplication operator is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors apply the term ''ring'' to a further generalization, often called a '' rng'', that omits the requirement for a multiplicative identity, and instead call the structure defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Fractional Transformation
In mathematics, a linear fractional transformation is, roughly speaking, an inverse function, invertible transformation of the form : z \mapsto \frac . The precise definition depends on the nature of , and . In other words, a linear fractional transformation is a ''transformation (function), transformation'' that is represented by a ''fraction'' whose numerator and denominator are ''linear polynomial, linear''. In the most basic setting, , and are complex numbers (in which case the transformation is also called a Möbius transformation), or more generally elements of a field (mathematics), field. The invertibility condition is then . Over a field, a linear fractional transformation is the restriction (mathematics), restriction to the field of a projective transformation or homography of the projective line. When are integer, integers (or, more generally, belong to an integral domain), is supposed to be a rational number (or to belong to the field of fractions of the integral ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Möbius Group
Moebius, Mœbius, Möbius or Mobius may refer to: People * August Ferdinand Möbius (1790–1868), German mathematician and astronomer * Friedrich Möbius (art historian) (1928–2024), German art historian and architectural historian * Theodor Möbius (1821–1890), German philologist, son of August Ferdinand * Karl Möbius (1825–1908), German zoologist and ecologist * Paul Julius Möbius (1853–1907), German neurologist, grandson of August Ferdinand * Dieter Moebius (1944–2015), Swiss-born German musician * Mark Mobius (born 1936), emerging markets investments pioneer * Jean Giraud (1938–2012), French comics artist who used the pseudonym Mœbius Fictional characters * Mobius M. Mobius, a character in Marvel Comics * Mobius, also known as the Anti-Monitor, a supervillain in DC Comics * Johann Wilhelm Möbius, a character in the play '' The Physicists'' * Moebius, the main antagonistic faction in the video game '' Xenoblade Chronicles 3'' * Mobius, or Dr. Ignatio Mo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Projective Line
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane (also called the closed complex plane): the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical ex ... [...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] |
Stiefel–Whitney Class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to ''n'', where ''n'' is the rank of the vector bundle. If the Stiefel–Whitney class of index ''i'' is nonzero, then there cannot exist (n-i+1) everywhere linearly independent sections of the vector bundle. A nonzero ''n''th Stiefel–Whitney class indicates that every section of the bundle must vanish at some point. A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, S^1 \times\R, is zero. The Stiefel–Whitney class w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symplectic Group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic group and is also denoted by \mathrm(n). Many authors prefer slightly different notations, usually differing by factors of . The notation used here is consistent with the size of the most common matrices which represent the groups. In Cartan's classification of the simple Lie algebras, the Lie algebra of the complex group is denoted , and is the compact real form of . Note that when we refer to ''the'' (compact) symplectic group it is implied that we are talking about the collection of (compact) symplectic groups, indexed by their dimension . The name " symplectic group" was coined by Hermann Weyl as a replacement for the previous confusing names (line) complex group and Abelian linear group, and is the Greek analog of "complex". The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaternion-Kähler Symmetric Space
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symmetric space with positive Ricci curvature is compact and simply connected, and is a Riemannian product of quaternion-Kähler symmetric spaces associated to compact simple Lie groups. For any compact simple Lie group ''G'', there is a unique ''G''/''H'' obtained as a quotient of ''G'' by a subgroup : H = K \cdot \mathrm(1).\, Here, Sp(1) is the compact form of the SL(2)-triple associated with the highest root of ''G'', and ''K'' its centralizer in ''G''. These are classified as follows. The twistor spaces of quaternion-Kähler symmetric spaces are the homogeneous holomorphic contact manifolds, classified by Boothby: they are the adjoint varieties of the complex semisimple Lie groups. These spaces can be obtained by taking a projectivization In mathematics, projectivizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
