Unreferenced auto yes date December 2009 Expert subject Physics date November 2008 In general relativity , optical scalars are a set of scalar physics scalars that describe various properties of null geodesic congruence general relativity congruence s. The three optical scalars used in general relativity are expansion , shear and twist vorticity and were first defined and used by Rainer K. Sachs Sachs 1961 . Given a vector field math k math tangent to the null geodesic, the optical scalars are defined as follows. Expansion Expand section date June 2008 The expansion of the null ray is defined as math theta , frac 1 2 k a a math Shear Expand section date June 2008 The shear of the null ray is defined as math sigma 2 , frac 1 2 k a b k c d g ca g db theta 2 math Twist Expand section date June 2008 The twist of the null ray is defined as math omega 2 , k a b k c d g ca g db math DEFAULTSORT Optical Scalars Category Mathematical methods in general relativity Relativity stub ... more details
In abstract algebra , extension of scalars is a means of producing a module mathematics module over a ring mathematics ring math S math from a module over another ring math R math , given a ring homomorphism homomorphism math f R to S math between them. Intuitively, the new module admits multiplication by more scalars than the original one, hence the name extension . Definition In this definition the rings are assumed to be ring mathematics Notes on the definition associative , but not necessarily commutative ring commutative , or to have an identity element identity . Also, modules are assumed ... from math M math through extension of scalars . Informally, extension of scalars is the tensor .... Examples One of the simplest examples is complexification , which is extension of scalars from the real ... L, one can extend scalars from K to L. In the language of fields, as module over a field is called a vector space , and thus extension of scalars converts a vector space over K to a vector space over ... scalars on an R module, the resulting module can be thought of alternatively as an S module, or as an R ... a field, as in representation theory . Just as one can extend scalars on vector spaces, one can also extend scalars on group algebra s and also on modules over group algebras, i.e., group representation s. Particularly useful is relating how irreducible representation s change under extension of scalars ... by 90 , is an irreducible 2 dimensional real representation, but on extension of scalars to the complex ..., but 2 complex eigenvalues. Interpretation as a functor Extension of scalars can be interpreted ... u S SM to SN math defined by math u S text id S otimes u math . Connection with restriction of scalars ... of scalars , define math Fu SM to N math to be the function composition composition math SM ... , and so is functorial . In the language of category theory , the extension of scalars functor is left adjoint to the restriction of scalars functor. See also Tensor product of fields References unref ... more details
In mathematics , restriction of scalars also known as Weil restriction is a functor which, for any finite field extension extension of fields L k and any algebraic variety X over L , produces another variety Res sub L k sub X , defined over k . It is useful for reducing questions about varieties over large fields to questions about more complicated varieties over smaller fields. Definition Let L k be a finite extension of fields, and X a variety defined over L . The functor math mathrm Res L k X math from k scheme mathematics schemes sup op sup to sets is defined by math mathrm Res L k X S X S times k L math In particular, the k rational points of math mathrm Res L k X math are the L rational points of X . The variety that representable functor represents this functor is called the restriction of scalars, and is unique up to unique isomorphism if it exists. From the standpoint of sheaf mathematics sheaves of sets, restriction of scalars is just a pushforward along the morphism Spec L math to math Spec k and is right adjoint to fiber product , so the above definition can be rephrased in much more generality. In particular, one can replace the extension of fields by any morphism of ringed ... less control over the behavior of the restriction of scalars. Properties For any finite extension of fields, the restriction of scalars takes quasiprojective varieties to quasiprojective varieties ... space s yields a restriction of scalars functor that takes algebraic stack s to algebraic stacks ... and math f t g t,1 e 1 dots g t,s e s math . 3 Restriction of scalars over a finite extension of fields ... is again a commutative group variety. Aleksander Momot applied restriction of scalars on group varieties ... of scalars on abelian variety abelian varieties e.g. elliptic curve s yields abelian varieties ... vs. Greenberg transforms Restriction of scalars is similar to the Greenberg transform, but does not generalize ... of scalars , Duke Math J., 134 2006 , 139 164. http math.berkeley.edu molsson homstackfinal.pdf ... more details
In general relativity , the Carminati McLenaghan invariants or CM scalars are a set of 16 scalar curvature invariant s for the Riemann tensor . This set is usually supplemented with at least two additional invariants. Mathematical definition The CM invariants consist of 6 real scalars plus 5 complex scalars, making a total of 16 invariants. They are defined in terms of the Weyl tensor math C abcd math and its left or right dual math star C acdb math , the Ricci tensor math R ab math , and the trace free Ricci tensor math S ab R ab frac 1 4 , R , g ab math In the following, it may be helpful to note that if we regard math S a b math as a matrix, then math S a m , S m b math is the square of this matrix, so the trace of the square is math S a b , S b a math , and so forth. The real CM scalars are math R R m m math the trace of the Ricci tensor math R 1 frac 1 4 , S a b , S b a math math R 2 frac 1 8 , S a b , S b c , S c a math math R 3 frac 1 16 , S a b , S b c , S c d , S d a math math M 3 frac 1 16 , S bc , S ef left C abcd , C aefd star C abcd , star C aefd right math math M 4 frac 1 32 , S ag , S ef , S c d , left C ac db , C befg star C ac db , star C befg right math The complex CM scalars are math W 1 frac 1 8 , left C abcd i , star C abcd right , C abcd math math W 2 frac 1 16 , left C ab cd i , star C ab cd right , C cd ef , C ef ab math math M 1 frac 1 8 , S ab , S cd , left C acdb i , star C acdb right math math M 2 frac 1 16 , S bc , S ef , left C abcd , C aefd star C acdb , star C aefd right frac 1 8 , i , S bc , S ef , star C abcd , C aefd math math M 5 frac 1 32 , S cd , S ef , left C aghb i , star C aghb right , left C acdb , C gefh star C acdb , star C gefh right math The CM scalars have the following degree mathematics degree s math R math is linear, math ... fluid solution s, the CM scalars comprise a complete set. Additional invariants may be required for more ... a manual with definitions and discussions of the CM scalars. Category Tensors in general relativity ... more details
In mathematical physics , Kundt spacetimes are Lorentzian manifold s admitting a geodesic null congruence with vanishing optical scalars expansion, twist and shear . A well known member of Kundt class is pp wave spacetime pp wave . Ricci flat Kundt spacetimes in arbitrary dimension are Petrov Type algebraically special . In four dimensions Ricci flat Kundt metrics of Petrov type III and N are completely known. ref H. Stephani et. al, Exact solutions of Einstein s field equations, 2nd Edition, Cambridge Univ. Press 2003 Kundt spacetimes are studied in Chapter 31 ref references Category Lorentzian manifolds relativity stub differential geometry stub ... more details
on a d dimensional torus is a vector multiplet containing d real scalars. Similarly, in an 11 ..., and it contains no scalars. However again its dimensional reduction on a d torus to a maximal gravity multiplet does contain scalars. Category Supersymmetry physics stub it Supermultipletto ... more details
and properties Scalars of vector spaces A vector space is defined as a set of vectors, a set of scalars ... k x . The scalars can be taken from any field, including the rational number rational , algebraic number algebraic , real, and complex numbers, as well as finite field s. Scalars as vector components ... space of dimension vector space dimension n is isomorphic to n dimensional real space R sup n sup . Scalars ..., not every scalar product space is a normed vector space. Scalars in modules When the requirement that the set of scalars form a field is relaxed so that it need only form a ring mathematics ring so that, for example, the division of scalars need not be defined , the resulting more general algebraic structure is called a module mathematics module . In this case the scalars may be complicated ... with the n × n matrices with entries from R as the scalars. Another example comes from ... more details
In General Relativity , the Weyl scalars are a set of five complex Scalar physics scalar quantities , math Psi 0, ldots, Psi 4 math , describing the curvature of a four dimensional spacetime . They are the expression of the ten independent degrees of freedom of the Weyl tensor math C abcd math in the Newman Penrose Formalism for general relativity . Given a null tetrad math l a, n a, m a, bar m a math , the scalars are given up to an overall conventional sign by math Psi 0 C alpha beta gamma delta l alpha m beta l gamma m delta , math math Psi 1 C alpha beta gamma delta l alpha n beta l gamma m delta , math math Psi 2 C alpha beta gamma delta l alpha m beta bar m gamma n delta , math math Psi 3 C alpha beta gamma delta l alpha n beta bar m gamma n delta , math math Psi 4 C alpha beta gamma delta n alpha bar m beta n gamma bar m delta . math Physical Interpretation Szekeres 1965 ref cite journal author P. Szekeres title The Gravitational Compass journal Journal of Mathematical Physics year 1965 volume 6 issue 9 pages 1387 1391 doi 10.1063 1.1704788 bibcode 1965JMP.....6.1387S . ref gave an interpretation of the different Weyl scalars at large distances math Psi 2 math is a Coulomb term, representing the gravitational monopole of the source math Psi 1 math & math Psi 3 math are ingoing and outgoing longitudinal radiation terms math Psi 0 math & math Psi 4 math are ingoing and outgoing transverse radiation terms. For a general asymptotically flat spacetime containing radiation Petrov Type I , math Psi 1 math & math Psi 3 math can be transformed to zero by an appropriate choice of null tetrad. Thus these can be viewed as gauge quantities. A particularly important case is the Weyl scalar math Psi 4 math . It can be shown to describe outgoing gravitational radiation in an asymptotically flat spacetime as math Psi 4 frac 1 2 left ddot h hat theta hat theta ddot h hat phi hat phi right i ddot h hat theta hat phi ddot h i ddot h times . math Here, math h math and math h times ... more details
In mathematics , particularly linear algebra , an orthogonal basis for an inner product space math V is a basis linear algebra basis for math V whose vectors are mutually orthogonal . If the vectors of an orthogonal basis are normalize linear algebra normalized , the resulting basis is an orthonormal basis . In functional analysis , an orthogonal basis is any basis obtained from a orthonormal basis or Hilbert basis using multiplication by nonzero Scalar mathematics scalars . Any orthogonal basis can be used to define a system of orthogonal coordinates . A linear combination of orthogonal basis can be used to reach any point in the vector space. Citation needed date September 2011 References MathWorld title Orthogonal Basis urlname OrthogonalBasis reflist Category Linear algebra Linear algebra stub mathanalysis stub de Orthogonalbasis uk ... more details
unreferenced date April 2007 In physics , a sigma model is a physical system that is described by a Lagrangian density of the form math mathcal L phi 1, phi 2, ldots, phi n sum i 1 n sum j 1 n g ij mathrm d phi i wedge mathrm d phi j math Depending on the scalars g sub ij sub it is either a linear sigma model or a non linear sigma model . The fields math phi i math , in general, provide a map from a base manifold called the worldsheet and a target Riemannian manifold that is often understood to be the spacetime . A basic example is provided by quantum mechanics which is a quantum field theory in one dimension. It s a sigma model with a base manifold given by the real line parameterizing the time or an interval, or the circle, etc. and a target space that is the real line. quantum stub Category Quantum field theory Category Physical systems ... more details
In mathematics , a linear representation &rho of a group G is a monomial representation if there is a finite index subgroup H and a one dimensional linear representation &sigma of H , such that &rho is equivalent to the induced representation Ind sub H sub sup G sup &sigma . Alternatively, one may define it as a representation whose image is in the monomial matrices . Here for example G and H may be finite group s, so that induced representation has a classical sense. The monomial representation is only a little more complicated than the permutation representation of G on the coset s of H . It is necessary only to keep track of scalars coming from &sigma applied to elements of H . References Springer id m m064780 title Monomial representation Category Representation theory of groups Algebra stub it Rappresentazione monomiale ... more details
at the expense of configuration effort. br 3. scalars connection oriented, fixed size, uni directional, and FIFO streams. Scalars are intended to be the highest performance communications mode, albeit ... purely on embedded communications, and adds the ideas of messages, packets, and scalars connected ... more details
Expert subject Physics date November 2008 The Goldberg Sachs theorem is a result in Einstein s theory of general relativity about vacuum solutions of the Einstein field equations relating the existence of a certain type of congruence general relativity congruence with algebraic properties of the Weyl tensor . More precisely, the theorem states that a Vacuum solution general relativity vacuum solution of the Einstein field equations will admit a shear free null geodesic congruence if and only if the Weyl tensor is Petrov classification The classification theorem algebraically special . The theorem is often used when searching for algebraically special vacuum solutions. Linearised gravity It has been shown by Dain and Moreschi 2000 that a corresponding theorem will not hold in linearized gravity , that is, given a solution of the linearised Einstein field equations admitting a shear free null congruence, then this solution need not be algebraically special. See also Geodesic Optical scalars References cite web author Dain, Sergio and Moreschi, Osvaldo, M. title The Goldberg Sachs theorem in linearized gravity work arXiv eprint server url http scitation.aip.org getabs servlet GetabsServlet?prog normal&id JMAPAQ000041000009006296000001&idtype cvips&gifs yes accessdate February 5, 2005 Category Theorems in general relativity relativity stub ... more details
space scalars are described as 0 blades, vectors are 1 blades, and area elements are 2 blades known as pseudoscalar s, in that they are one dimensional objects distinct from regular scalars. In three dimensional space, 0 blades are again scalars and 1 blades are three dimensional vectors, but in three ... more details
Unreferenced date December 2009 In the mathematical field of projective geometry , a projective frame is an ordered collection of points in projective space which can be used as reference points to describe any other point in that space. For example Given three distinct points on a projective line , any other point can be described by its cross ratio with these three points. In a projective plane , a projective frame consists of four points, no three of which lie on a projective line. In general, let K P sup n sup denote n dimensional projective space over an arbitrary field K . This is the projectivization of the vector space K sup n 1 sup . Then a projective frame is an n 2 tuple of points in general position in K P sup n sup . Here general position means that no subset of n 1 of these points lies in a hyperplane a projective subspace of dimension n &minus 1 . Sometimes it is convenient to describe a projective frame by n 2 representative vectors v sub 0 sub , v sub 1 sub , ..., v sub n 1 sub in K sup n 1 sup . Such a tuple of vectors defines a projective frame if any subset of n 1 of these vectors is a basis for K sup n 1 sup . The full set of n 2 vectors must satisfy linear dependence relation math lambda 0 v 0 lambda 1 v 1 cdots lambda n v n lambda n 1 v n 1 0. math However, because the subsets of n 1 vectors are linearly independent, the scalars sub j sub must all be nonzero. It follows that the representative vectors can be rescaled so that sub j sub 1 for all j 0,1,..., n 1. This fixes the representative vectors up to an overall scalar multiple. Hence a projective frame is sometimes defined to be a n 2 tuple of vectors which span K sup n 1 sup and sum to zero. Using such a frame, any point p in K P sup n sup may be described by a projective version of barycentric coordinates mathematics barycentric coordinates a collection of n 2 scalars sub j sub which sum to zero, such that p is represented by the vector math mu 0 v 0 mu 1 v 1 cdots mu n v n mu n 1 v ... more details
, translations, and reflections form the Poincar group . The convariant quantities are four scalars ... scalars math ds math is the invariant interval 4 scalar math u a math is the 4 velocity 4 vector and math ... more details
are the scalars , with vector addition and scalar multiplication obtained from the corresponding field ... of scalars main Extension of scalars Given a field extension, one can Extension of scalars extend scalars on associated algebraic objects. For example, given a real vector space, one can produce a complex ... of scalars for associative algebra s over defined over the field, such as polynomials or group algebra s and the associated group representation s. Extension of scalars of polynomials is often used implicitly ... more formally. Extension of scalars has numerous applications, as discussed in extension of scalars Applications extension of scalars applications . See also Field theory mathematics Field ... more details
admits a non split torus given by Weil restriction restriction of scalars over a separable extension. Restriction of scalars over an inseparable field extension will yield a commutative group scheme ..., then this gives the restriction of scalars a permutation module structure. Tori whose weight lattices ... are finite products of restrictions of scalars. For a general base scheme S , weights and coweights .... Example Let S be the restriction of scalars of G sub m sub over the field extension C R . This is a real torus whose real points form the Lie group of nonzero complex numbers. Restriction of scalars ... of scalars to K . Projective triviality If T is a torus over K whose weight lattice is a projective ... more details
can be assembled into a single Majorana spinor Majorana supercharge . The only scalars in such a theory are the complex scalars of the chiral superfield s. He found that the vacuum manifold of allowed vacuum expectation values for these scalars is not only complex but also a K hler manifold . If gravity ... algebra contains two representation theory representation s with scalars, the vector superfield vector multiplet which contains a complex scalar and the hypermultiplet which contains two complex scalars ... more details
particle, can be constructed from two Lorentz scalars math v 1 2 mathbf v 1 cdot mathbf v 1 mathbf p 1 cdot mathbf p 1 c 6 over E 1 2 math . More complicated scalarsScalars may also be constructed ... more details
the scalar matrices, and leaves one degree of freedom any such map is determined by its value on scalars ... algebras math gl n to k math from operators to scalars , as the commutator of scalars is trivial it is an abelian ... space math n times n math matrices to the Lie algebra k of scalars as k is abelian the Lie bracket ... n sl n oplus k math of operators matrices into traceless operators matrices and scalars operators matrices ... the counit map with the unit map math k to gl n math of inclusion of scalar transformation scalars to obtain a map math gl n to gl n math mapping onto scalars, and multiplying by n . Dividing by n ... math textstyle frac 1 n math times scalars so math gl n sl n oplus k, math but the splitting of the determinant would be as the n th root times scalars, and this does not in general define a function ... of scalar transformation scalars , sending math 1 in F math to the identity matrix trace is dual to scalars . In the language of bialgebra s, scalars are the unit , while trace is the counit ... more details
physically . Scalars in relativity theory Main Lorentz scalar In the theory of relativity , one considers ... quantities that are scalars in Classical physics classical non relativistic physics need to be combined ... more details
In mathematics , more specifically modern algebra and ring theory , a noncommutative ring is a Ring mathematics ring whose multiplication is not commutative that is, if R is a noncommutative ring, there exists a and b in R with a b b a , and conversely. Noncommutative rings are ubiquitous in mathematics, and occur in numerous sciences. For instance, Matrix mathematics matrix multiplication is never commutative, except in trivial cases, despite the fact that matrices arise naturally as rings of linear transformation s of some vector space over a field. Furthermore, mathematical physics and more generally linear algebra exploit the concept of a matrix often. Noncommutative rings also arise naturally in the representation theory of groups. Algebra over a field Algebras , and more specifically Group ring group algebras , occur also in noncommutative ring theory. The study of noncommutative rings is a major area of modern algebra. Influential work by Richard Brauer , Nathan Jacobson , I. N. Herstein and P. M. Cohn and other mathematicians, has led to much of modern day ring theory. Basic but influential concepts in the field include the Jacobson radical , the Jacobson density theorem , the Artin Wedderburn theorem s and the Brauer group . Discussion Often noncommutative rings possess interesting invariants that commutative ring s do not. As an example, there exist rings which contain non trivial proper left or right ideal ring theory ideals , but are still Simple ring simple that is contain no non trivial proper two sided ideals. The theory of vector space s is one illustration of a special case of an object studied in noncommutative ring theory. In linear algebra , the scalars of a vector space are required to lie in a field mathematics field , that is, a commutative division ring . The concept of a module mathematics module , however, requires only that the scalars lie in an abstract ring. Neither commutativity nor the division ring assumption is required on the scalar ... more details
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