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Segre Classification
The Segre classification is an algebraic classification of rank two symmetric tensors. It was proposed by the italian mathematician Corrado Segre in 1884. The resulting types are then known as Segre types. It is most commonly applied to the energy–momentum tensor (or the Ricci tensor) and primarily finds application in the classification of exact solutions in general relativity. See also *Corrado Segre *Jordan normal form \begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ ... * Petrov classification References * See ''section 5.1'' for the Segre classification. * Linear algebra Tensors Tensors in general relativity {{math-physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Symmetric Tensor
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: :T(v_1,v_2,\ldots,v_r) = T(v_,v_,\ldots,v_) for every permutation ''σ'' of the symbols Alternatively, a symmetric tensor of order ''r'' represented in coordinates as a quantity with ''r'' indices satisfies :T_ = T_. The space of symmetric tensors of order ''r'' on a finite-dimensional vector space ''V'' is naturally isomorphic to the dual of the space of homogeneous polynomials of degree ''r'' on ''V''. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on ''V''. A related concept is that of the antisymmetric tensor or alternating form. Symmetric tensors occur widely in engineering, physics and mathematics. Definition Let ''V'' be a vector space and :T\in V^ a tensor of order ''k''. Then ''T'' is a symmetric tensor if :\tau_\sigma T = T\, for the braiding ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Corrado Segre
Corrado Segre (20 August 1863 – 18 May 1924) was an Italian mathematician who is remembered today as a major contributor to the early development of algebraic geometry. Early life Corrado's parents were Abramo Segre and Estella De Benedetti. Career Segre developed his entire career at the University of Turin, first as a student of Enrico D'Ovidio. In 1883 he published a dissertation on quadrics in projective space and was named an assistant to professors in algebra and analytic geometry. In 1885 he also assisted in descriptive geometry. He began to instruct in projective geometry, as a stand-in for Giuseppe Bruno, from 1885 to 1888. Then for 36 years, he had the chair in higher geometry following D'Ovidio. Segre and Giuseppe Peano made Turin known in geometry, and their complementary instruction has been noted as follows: The Erlangen program of Felix Klein appealed early on to Segre, and he became a promulgator. First, in 1885 he published an article on conics i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Ricci Tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bili ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Exact Solutions In General Relativity
In general relativity, an exact solution is a (typically closed form) solution of the Einstein field equations whose derivation does not invoke simplifying approximations of the equations, though the starting point for that derivation may be an idealized case like a perfectly spherical shape of matter. Mathematically, finding an exact solution means finding a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical non-gravitational fields such as the electromagnetic field. Background and definition These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations). Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor T^. (A field is described by a Lagrangian, varying with respect to the field should give the field equations and varyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Jordan Normal Form
\begin \lambda_1 1\hphantom\hphantom\\ \hphantom\lambda_1 1\hphantom\\ \hphantom\lambda_1\hphantom\\ \hphantom\lambda_2 1\hphantom\hphantom\\ \hphantom\hphantom\lambda_2\hphantom\\ \hphantom\lambda_3\hphantom\\ \hphantom\ddots\hphantom\\ \hphantom\lambda_n 1\hphantom\\ \hphantom\hphantom\lambda_n \end Example of a matrix in Jordan normal form. All matrix entries not shown are zero. The outlined squares are known as "Jordan blocks". Each Jordan block contains one number ''λi'' on its main diagonal, and 1s directly above the main diagonal. The ''λi''s are the eigenvalues of the matrix; they need not be distinct. In linear algebra, a Jordan normal form, also known as a Jordan canonical form, is an upper triangular matrix of a particular form called a Jordan matrix representing a linear operator on a finite-dimensional vector space with respect to some basis. Such a matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Petrov Classification
In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold. It is most often applied in studying exact solutions of Einstein's field equations, but strictly speaking the classification is a theorem in pure mathematics applying to any Lorentzian manifold, independent of any physical interpretation. The classification was found in 1954 by A. Z. Petrov and independently by Felix Pirani in 1957. Classification theorem We can think of a fourth rank tensor such as the Weyl tensor, ''evaluated at some event'', as acting on the space of bivectors at that event like a linear operator acting on a vector space: : X^ \rightarrow \frac \, _ X^ Then, it is natural to consider the problem of finding eigenvalues \lambda and eigenvectors (which are now referred to as eigenbivectors) X^ such that :\frac \, _ \, X^ = \la ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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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] [Amazon] |
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Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathematics), matrices. Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as line (geometry), lines, plane (geometry), planes and rotation (mathematics), rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to Space of functions, function spaces. Linear algebra is also used in most sciences and fields of engineering because it allows mathematical model, modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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Tensors
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic tenso ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |