Unreferenced date December 2009 This is a glossary of tensortheory . For expositions of tensortheory from different points of view, see TensorTensor intrinsic definition Application of tensortheory in engineering science For some history of the abstract theory see also Multilinear algebra . Classical notation Tensor Formulation Tensor order A tensor written in component form is an indexed array. The order of a tensor is the number of indices required. The rank of tensor used to mean the order, but now it means something different Rank The rank of the tensor is the minimal number of rank one tensor that you need to sum up to obtain this higher rank tensor. Rank one tensors are given the generalization of outer product to m vectors where m is the order of the tensor. Dyadic tensor A dyadic tensor has order two, and may be represented as a square matrix math matrix . The conventions a sub ... Metric tensor Strain tensor Stress energy tensorTensor field theory Jacobian matrix Tensor field ... Of TensorTheory Category Glossaries on mathematics Tensortheory Category Tensors Category Multilinear ... t orthogonal see below . A dyad is a tensor such as a sub i sub b sup j sup , product component ... of vectors Covariant tensor , Covariance and contravariance of vectors Contravariant tensor The classical ... means all indices are upper. Mixed tensor This refers to any tensor with lower and upper indices. Cartesian tensor Cartesian tensors are widely used in various branches of continuum mechanics ... of interest, the metric tensor is the Kronecker delta . This means that there is no need ... tensors and Tensor density tensor densities . All Cartesian tensor Cartesian tensor indices are written as subscripts. Cartesian tensor s achieve considerable computational simplification at the cost of generality and of some theoretical insight. Tensor contraction Contraction of a tensor Raising and lowering indices Symmetric tensor Antisymmetric tensor Multiple cross products Algebraic notation ... more details
Unreferenced date December 2009 Tensor s are frequently used in engineering to describe measured physical quantity quantities . Common applications Measuring Deformation mechanics deformations finite deformation tensors and Strain materials science strain strain tensor in continuum mechanics Representing diffusion as a tensor in Diffusion tensor imaging Describing moment of inertia moments of inertia as the Moment of inertia Moment of inertia tensor inertia tensor See also Application of tensortheory in physics Mathematical physics Tensor DEFAULTSORT Application Of TensorTheory In Engineering Category Tensors ... more details
Tensor network theory is a theory of brain function specifically in the cerebellum by Llinas and Pellionisz which provides a mathematical model of transformation of sensory covariant space time coordinates into motor contravariant coordinates by cerebellar neuronal networks. ref name Neuroscience1980 Pellionisz Cite journal author Pellionisz, A., Llin s, R. year 1980 month title Tensorial Approach To The Geometry Of Brain Function Cerebellar Coordination Via A Metric Tensor journal Neuroscience volume 5 issue pages 1125&mdash 1136 id url http usa siliconvalley.com inst pellionisz 80 metric 80 metric.html doi 10.1016 0306 4522 80 90191 8 pmid 6967569 ref ref name Neuroscience1985 Pellionisz Cite journal author Pellionisz, A., Llin s, R. year 1985 month title Tensor Network Theory Of The Metaorganization Of Functional Geometries In The Central Nervous System journal Neuroscience volume 16 issue 2 pages 245 273 url http usa siliconvalley.com inst pellionisz 85 metaorganization 85 metaorganization.html doi 10.1016 0306 4522 85 90001 6 pmid 4080158 ref References Reflist Neuroscience stub Category Neuroscience ... more details
Expert subject Mathematics date November 2008 In theoretical physics , a scalar tensortheory is a theory that includes both a Scalar field theory scalar field and a tensor field to represent a certain interaction. For example, the Brans Dicke theory of gravitation uses both a scalar field and a tensor field to mediate the gravitational interaction. Tensor fields and field theory Modern physics tries ... equation from it , even within SM, so that Zee s idea was taken 1992 for a scalar tensortheory ... al. , Int. J. of Theor. Phys. 31 1 109 , 1992 Scalar tensortheory with Higgs field. C.H. Brans, arXiv ... title Comments on the scalar tensortheory journal Int. J. Theor. Phys. volume 1 pages 25 year 1968 ... TensorTheory Category Tensors fr Th orie tenseur scalaire ... . Higher order tensors are found for example in the deformation theory and in General Relativity. Gravity as field theory In physics, forces as vectorial quantities are given as the derivative gradient ... is called a scalar theory . The gravitational force is dependent of the distance r of the massive ... are unchangeable. Einstein s theory of gravity, the General Relativity is of another nature. It unifies ... in this context is by using both tensor of degree n 1 and scalar fields, i.e. so that gravitation is not only given through a scalar field nor through the metric. These are scalar tensor theories of gravitation ... with a square scalar field, field theories of scalar tensor theories of gravitation ... dependent of the scalar field. Higher dimensional relativity and scalar tensor theories After ... in 1917 a generalization in a 5 dimensional manifold Kaluza Klein theory . This theory possesses .... This theory was modified in 1955 by P. Jordan in his Projective Relativity theory, in which ... . Following the Conform Equivalence theory , multidimensional theories of gravity are conform equivalent ... case of this is given by Jordan s theory, which, without breaking energy conservation as it should ... more details
Unreferenced date December 2009 Field theory mathematics Field theory is the branch of mathematics in which field mathematics field s are studied. This is a glossary of some terms of the subject. See field theory physics for the unrelated field theories in physics. Definition of a field A field is a commutative ring F , , in which 0 1 and every nonzero element has a multiplicative inverse. In a field ... say that E is generated by S over F . Primitive element field theory Primitive element An element ... &ne 0, and that f is injective . Fields, together with these homomorphisms, form a category theory ... of rational numbers. Their detailed properties are studied in algebraic number theory . Quadratic ... fields. Galois theory Galois extension A normal, separable field extension. Galois group The automorphism ... s. Kummer theory The Galois theory of taking n th roots, given enough roots of unity . It includes the general theory of quadratic extension s. Artin Schreier theory Covers an exceptional case of Kummer theory, in characteristic p . Normal basis A basis in the vector space sense of L over K , on which the Galois group of L over K acts transitively. Tensor product of fields A different foundational piece of algebra, including the compositum operation join of fields . Extensions of Galois theory Inverse problem of Galois theory Given a group G , find an extension of the rational number or other field with G as Galois group. Differential Galois theory The subject in which symmetry groups of differential equation s are studied along the lines traditional in Galois theory. This is actually an old idea, and one of the motivations when Sophus Lie founded the theory of Lie group s. It has not, probably, reached definitive form. Grothendieck s Galois theory A very abstract approach from algebraic geometry , introduced to study the analogue of the fundamental group . DEFAULTSORT Glossary Of Field Theory Category Glossaries on mathematics Field theory Category Field theory it Glossario di teoria ... more details
tensor product See also Glossary of ring theory References cite book author John A. Beachy title ...Module theory is the branch of mathematics in which module mathematics modules are studied. This is a glossary of some terms of the subject. Basic definition left R module A left module math M math over the ring mathematics ring math R math is an abelian group math M, math with an operation math R times M to M math called scalar multipliction satisfies the following condition math forall r,s in R, m,n in M math , math r m n rm rn math math r s m r s m math math 1 R , m m math right R module A right module math M math over the ring math R math is an abelian group math M, math with an operation math M times R to M math satisfies the following condition math forall r,s in R, m,n in M math , math m n r m r n r math math m s r r s m math math m 1 R m math Or it can be defined as the left module math M math over math R textrm op math the opposite ring of math R math . bimodule If an abelian group math M math is both a left math S math module and right math R math module, it can be made to a math S,R math bimodule if math s mt sm t , forall s in S, r in R, m in M math . submodule Given math M math is a left math R math module, a subgroup math N math of math M math is a submodule if math RN subseteq N math . homomorphism of math R math modules For two left math R math modules math M 1, M 2 math , a group homomorphism math phi M 1 to M 2 math is called homomorphism of math R math modules ... m in M math . It is a left ideal ring theory ideal of math R math . The annihilator of an element ... math is called a flat module if the Tensor product of modules tensor product functor math otimes R F ... module Operations on modules Direct sum of modules Tensor product of modules Hom functor Ext ... year 1993 isbn 0 201 55540 9 Citation last1 Passman first1 Donald S. title A course in ring theory ... Category Glossaries on mathematics Module Category Module theory ... more details
main Systems theory A glossary of terms as relating to systems theory. ref Because systems language introduces many new terms essential to understanding how a system works, a glossary of many of the significant terms is developed. ref NOTOC compactTOC2 A Adaptive capacity An important part of the Resilience network resilience of systems in the face of a wiktionary perturbation perturbation , helping to minimise loss of structural functionalism function in individual human, and collective social and biological systems. Allopoiesis is the process whereby a system produces something other than the system itself. Allostasis is the process of achieving stability, or homeostasis , through physiological or behavioral change. Autopoiesis is the process by which a system regenerates itself through the self .... Enantiostasis is the ability of an open system systems theory open system , especially a living ... is that property of either an open system systems theory open system or a closed system , especially ... Open System Model.gif 200px thumb right Open System Model basics Open system systems theory Open ... of a system , normally an open system systems theory open system , increases in complexity without ... wholes with emergent properties. NOTOC compactTOC2 See also Glossary of Unified Modeling Language terms List of basic science topics List of glossaries List of types of systems theory References references External links Wiktionary Category Systems theory http pespmc1.vub.ac.be ASC INDEXASC.html ... foundations ASCGlossary.htm The ASC Glossary of Cybernetics by the American Society for Cybernetics ftp ftp.vub.ac.be pub projects Principia Cybernetica Nodes Cybernetics glossary.txt ASC Glossary on Cybernetics and Systems Theory by Stuart Umpleby ed. from the American Society for Cybernetics ... and Systems , edited by Charles Fran ois , 1997 M nchen K. G. Saur. Systems DEFAULTSORT Glossary Of Systems Theory Category Glossaries on mathematics Systems theory Category Systems theory Category ... more details
This is a glossary of scheme theory . For an introduction to the theory of schemes in algebraic geometry , see affine scheme , projective space , sheaf mathematics sheaf and scheme mathematics scheme . The concern here is to list the fundamental technical definitions and properties of scheme theory. See also list of algebraic geometry topics . Points A scheme math S math is a locally ringed space , so a fortiori a topological space , but the meanings of point of math S math are threefold a point math P math of the underlying topological space a math T math valued point of math S math is a morphism from math T math to math S math , for any scheme math T math a geometric point , where math S math is defined over is equipped with a morphism to math textrm Spec K math , where math K math is a field mathematics field , is a morphism from math textrm Spec overline K math to math S math where math overline K math is an algebraic closure of math K math . Geometric points are what in the most classical cases, for example algebraic varieties that are complex manifold s, would be the ordinary sense points. The points math P math of the underlying space include analogues of the generic point s in the sense of Zariski , not that of Andr Weil , which specialise to ordinary sense points. The math T math valued points are thought of, via Yoneda s lemma , as a way of identifying math S math with the representable functor math h S math it sets up. Historically there was a process by which projective ... the extension from affine scheme s, where it is just the tensor product of R algebras , to all ... schemes, 1 algebraic curves , 2 algebraic surfaces . catenary A scheme is Catenary ring theory ... in algebraic number theory . A morphism math f math is tale morphism tale if it is flat and unramified ... on X . Notes Reflist References EGA book I EGA book IV 1 Hartshorne AG DEFAULTSORT Glossary Of Scheme Theory Category Glossaries on mathematics Scheme theory Category Scheme theory zh ... more details
inverses. See also Glossary of field theoryGlossary of module theory Notes reflist References EGA book IV 1 DEFAULTSORT Glossary Of Ring Theory Category Glossaries on mathematics Ring theory Category Ring theory nl Glossarium van de ringtheorie ...refimprove date August 2010 Ring theory is the branch of mathematics in which ring mathematics rings are studied that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject. Definition of a ring Ring mathematics Ring A ring is a Set mathematics set R with two binary operation s, usually called addition and multiplication , such that R is an abelian group under addition, a monoid under multiplication, and such that multiplication is both left and right distributive over addition. Note that rings are assumed to have multiplicative identities unless otherwise noted. The additive identity is denoted by 0 and the multiplicative identity by 1. Subring A subset S of the ring R , , which remains a ring when and are restricted to S and contains the multiplicative identity 1 of R is called a subring of R . Types of elements Central element Central An element r of a ring R is central if xr rx for all x in R . The set of all central ... with ax b . Idempotent An element r of a ring is idempotent if r sup 2 sup r . Unit ring theory Unit ... a sub 1 sub ,..., a sub n sub such that I Ra sub 1 sub R ... Ra sub n sub R . ideal ring theory ... ring theory A domain is a ring without zero divisors and in which 1 0. This is the noncommutative ... division ring is a field, as is every finite integral domain. Field theory mathematics Field theory is in fact an older branch of mathematics than ring theory. Finitely presented algebra span id ... ring and every domain ring theory domain is a prime ring. Primitive ring A left primitive ring is a ring ... in which every non zero non Unit ring theory unit element can be written as a product of prime element ... more details
player s loss. Most classical board games e.g. chess , checkers are zero sum . Game theory DEFAULTSORT Glossary Of Game Theory Category Game theory Category Glossaries on mathematics Game theory ...Unreferenced date December 2009 Game theory is the branch of mathematics in which game mathematics games are studied that is, models describing human behaviour. This is a glossary of some terms of the subject. Definitions of a game Notational conventions Real numbers math mathbb R math . The set of players math mathrm N math . Strategy space math Sigma prod i in mathrm N Sigma i math , where Player i s strategy space math Sigma i math is the space of all possible ways in which player i can play the game. A strategy for player i math sigma i math is an element of math Sigma i math . complements math sigma i math an element of math Sigma i prod j in mathrm N , j ne i Sigma j math , is a tuple of strategies for all players other than i . Outcome space math Gamma math is in most textbooks identical to Payoffs math mathbb R mathrm N math , describing how much gain money, pleasure, etc. the players are allocated by the end of the game. Normal form game A game in normal form is a function math pi prod i in mathrm N Sigma i to mathbb R mathrm N math Given the tuple of strategies chosen by the players, one is given an allocation of payments given as real numbers . A further generalization can be achieved by splitting the game into a composition of two functions math pi prod i in mathrm N Sigma i to Gamma math the outcome function of the game some authors call this function the game form ... outcome of the game. Extensive form game This is given by a tree graph theory tree , where at each vertex graph theory vertex of the tree a different player has the choice of choosing an graph theory edge . The outcome set of an extensive form game is usually the set of tree leaves. Cooperative ... is the set of players. Glossary Acceptable game is a game form such that for every possible preference ... more details
This is a glossary of properties and concepts in category theory in mathematics . Categories A category mathematics category A is said to be small provided that the class of all morphisms is a Set mathematics set i.e., not a proper class otherwise large . locally small provided that the morphisms between every pair of objects A and B form a set. Some authors assume a foundation in which the collection of all classes forms a conglomerate , in which case a quasicategory is a category whose objects and morphisms merely form a conglomerate. ref cite book last Ad mek first Ji coauthors Herrlich, Horst, and Strecker, George E title Abstract and Concrete Categories The Joy of Cats origyear 1990 url http katmat.math.uni bremen.de acc format PDF year 2004 publisher Wiley & Sons location New York isbn 0 471 60922 6 page 40 ref NB other authors use the term quasicategory with a different meaning. ref cite journal doi 10.1016 S0022 4049 02 00135 4 last Joyal first A. title Quasi categories and Kan complexes journal Journal of Pure and Applied Algebra volume 175 year 2002 issue 1 3 pages 207 222 ref harv ref isomorphic to a category B provided that there is an isomorphism between them. equivalent to a category B provided that there is an equivalence of categories equivalence between them. concrete category concrete provided that there is a faithful functor from A to Category of sets Set e.g., category of vector spaces Vec , category of groups Grp and category of topological spaces Top . discrete category discrete provided that each morphism is an identity morphism of some object . thin ... DEFAULTSORT Glossary Of Category Theory Category Category theory Category Glossaries on mathematics Category theory es Anexo Glosario de teor a de categor as ... of sets Set . In other words, f is the dual of an epimorphism. a section category theory retraction if it has a right inverse. a section category theory coretraction if it has a left inverse. Functors ... more details
problems of group theory is the classification of groups up to isomorphism. Groups together with group ... group on G . The theory of finite groups is very rich. Lagrange s theorem group theory ... for p adic analysis , class field theory , and l adic cohomology is the ring of p adic integers and the profinite ... theory are designed to tackle non abelian groups. There are several notions designed to measure ... group , whose order group theory order is about 10 sup 54 sup . The finite simple groups ... title An introduction to the theory of groups location New York publisher Springer Verlag year 1994 ... 0 691 08017 8 id MathSciNet id 0347778 year 1972 ref harv postscript None DEFAULTSORT Glossary Of Group Theory Category Group theory Category Glossaries on mathematics Group theory de Gruppentheorie Glossar ... more details
wiktionary Appendix Glossary of graph theory Graph theory is a growing area in mathematical research .... Basics A Graph mathematics graph G consists of two types of elements, namely vertex graph theory vertices and Edge graph theory edges . Every edge has two endpoints in the set of vertices, and is said ..., i.e. E G . ref cite book last Harris first John M. title Combinatorics and Graph Theory year 2000 ... new 26 forthcoming titles 28default 29 book 978 0 387 79710 6 ref A Loop graph theory loop is an edge ... to H . A subgraph H is a spanning subgraph , or Factor graph theory factor , of a graph G if it has ..., a Path graph theory path referred to what is now usually known as an open walk . Nowadays ... has no repeated vertices or edges, is called a Cycle graph theory cycle . Like path , this term traditionally ... cycles of every possible length from 3 to the order of the graph . The Girth graph theory ... theory circumference , the length of a longest simple cycle. The girth and circumference of an acyclic ... tree with 6 vertices and 5 edges. A tree graph theory tree is a connected acyclic simple graph. For directed ... graph theory star is K sub 1, k sub . An induced star with 3 edges is a claw . A caterpillar tree caterpillar ... graph theory clique in a graph is a set of pairwise adjacent vertices. Since any subgraph induced ... are operations which collapse a path and all vertices on it into a single edge see Minor graph theory ... disjoint from all other such paths in math G 1 math . Adjacency and degree In graph theory, degree ... graph theory degree , or valency , d sub G sub v of a vertex v in a graph G is the number of edges ... graph theory neighborhood of v and denoted N sub G sub v . When v is also included, it is called .... Spectral graph theory studies relationships between the properties of the graph and its adjacency ... to be cubic , or trivalent . A Factor graph theory k factor is a k regular spanning subgraph ... as other nonadjacent pairs. Independence In graph theory, the word independent usually carries ... more details
This is a glossary of some terms used in various branches of mathematics that are related to the fields of order theory order , lattice order lattice , and domain theory . Note that there is a structured list of order topics available as well. Other helpful resources might be the following overview articles completeness order theory completeness properties of partial orders distributivity order theory distributivity laws of order theory limit preserving order theory preservation properties of functions between posets. In the following, partial orders will usually just be denoted by their carrier sets. As long as the intended meaning is clear from the context, &le will suffice to denote the corresponding ... in the article on completeness order theory , any poset for which either all suprema or all ... order theory Distributive . A lattice L is called distributive if, for all x , y , and z in L ... . domain theory Domain . Domain is a general term for objects like those that are studied in domain theory . If used, it requires further definition. Down set . See lower set . duality order theory Dual . For a poset P , &le , the dual order P sup d sup P , &ge is defined by setting x &ge ... of a drawing of its transitive reduction . I An ideal order theory ideal is a subset X of a poset ... an acyclic binary relation also called an acyclic digraph . limit preserving function order theory ... and idempotent under function composition . Projections play an important role in domain theory . Pseudo ... 0. Q Quasiorder . See preorder . R limit preserving order theory Reflecting . A function f between ... differently, a Scott continuous function is one that limit preserving function order theory preserves ... approximates y . See also domain theory . Weak order . A partial order &le on a set X is a weak ... Mathematical surveys and monographs volume 150 Category Glossaries on mathematics Order theory Category Order theory Category Article Feedback 5 ... more details
to this fact. See also Glossary of tensortheory Notation Voigt notation Mandel notation Penrose ... the middle of the nineteenth century, is itself a tensortheory, and highly geometric, but it was some time before it was seen, with the theory of differential form s, as naturally unified with tensor ... Covariant derivative Application of tensortheory in engineering Curvature Diffusion MRI Mathematical ...other uses Dablink Note that in common usage, the term tensor is also used to refer to a tensor field . File Components stress tensor.svg right thumb 300px Stress, a second order tensor. The tensor s components ... themselves are also tensors. A tensor can be represented as a Array data structure Multidimensional arrays multi dimensional array of numerical values. The order also degree or rank of a tensor is the dimensionality ... dimensional array, and therefore is a 2nd order tensor. A vector can be represented as a 1 dimensional array and is a 1st order tensor. Scalars are single numbers and are thus zeroth order tensors. Tensors ..., the Stress mechanics Cauchy s stress theorem stress tensor stress tensor T takes a direction ... system . Taking a coordinate basis of a vector space basis or frame of reference and applying the tensor to it results in an organized multidimensional array representing the tensor in that basis, or as it looks from that frame of reference. The coordinate independence of a tensor then takes the form ... in to the notion of a tensor in a geometrical or physical setting, and the precise form of the transformation law determines the type or valence of the tensor. Tensors are important in physics because ... curvature tensor . ref name Kline cite book title Mathematical thought from ancient to modern ... year 1972 ref History The concepts of later tensor analysis arose from the work of Carl Gauss in differential geometry, and the formulation was much influenced by the theory of algebraic form s and invariants ... books?id O6lixBzbc0gC ref The word tensor itself was introduced in 1846 by William Rowan ... more details
for Wikipedia s glossary Help Glossary seealso List of glossaries NOTOC A glossary , also known as an idioticon , vocabulary , or clavis , is an alphabetical list of Term language terms in a particular domain of knowledge with the definition s for those terms. Traditionally, a glossary appears at the end of a book and includes terms within that book which are either newly introduced, uncommon or specialized. A bilingual glossary is a list of terms in one language which are defined in a second language or glossed by synonym s or at least near synonyms in another language. In a general sense, a glossary contains explanations of concept s relevant to a certain field of study or action. In this sense, the term is related to the notion of ontology . Automatic methods have been also provided that transform a glossary into an ontology ref R. Navigli, P. Velardi. http www.dsi.uniroma1.it navigli pubs Navigli Velardi IOS 2008.pdf From Glossaries to Ontologies Extracting Semantic Structure from Textual Definitions , Ontology Learning and Population Bridging the Gap between Text and Knowledge P. Buitelaar and P. Cimiano, Eds. , Series information for Frontiers in Artificial Intelligence and Applications, IOS Press, 2008, pp. 71 87. ref or a computational lexicon. ref R. Navigli. http www.dsi.uniroma1.it ... 2009 , Athens, Greece, March 30 April 3rd, 2009, pp. 594 602. ref Core glossary A core glossary is a simple glossary or defining dictionary which enables definition of other concepts, especially for newcomers .... In computer science, a core glossary is a prerequisite to a core ontology . An example of this is seen ... to a glossary therefore providing access to a kind of compound glossary of glossary entries found ... of the Definder System for Fully Automatic Glossary Construction . In Proc. of American Medical Informatics ... relations. See also Terminology extraction References reflist External links Wiktionary glossary http ... glossml glossml.pdf Glossary Markup Language GlossML , an open XML vocabulary specially ... more details
A Killing tensor , named after Wilhelm Killing , is a tensor , known in the theory of general relativity , math K math that satisfies math nabla alpha K beta gamma 0 , math where the parentheses on the indices refer to the symmetric tensor symmetric part . This is a generalization of a Killing vector. While Killing vectors are associated with continuous symmetries more precisely, differentiable , and hence very common, the concept of Killing tensor arises much less frequently. The Kerr metric Kerr solution is the most famous example of a semi Riemannian manifold manifold possessing a Killing tensor. See also Killing form Killing vector field Wilhelm Killing Category Riemannian geometry ... more details
In differential geometry , the Weyl curvature tensor , named after Hermann Weyl , is a measure of the curvature ... tensor , the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic . The Weyl tensor differs from the Riemann curvature tensor in that it does not convey information on how ... force. The Ricci curvature , or trace linear algebra trace component of the Riemann tensor contains ... tensor is the traceless component of the Riemann tensor. It is a tensor that has the same symmetries as the Riemann tensor with the extra condition that it be trace free Tensor contraction Metric contraction ... tensor vanishes identically. In dimensions 4, the Weyl curvature is generally nonzero. If the Weyl tensor ... system in which the metric tensor is proportional to a constant tensor. This fact was a key component of Nordstr m s theory of gravitation , which was an earlier precursor of general relativity . Definition The Weyl tensor can be obtained from the full curvature tensor by subtracting out various traces. This is most easily done by writing the Riemann tensor as a 0,4 valence tensor by contracting with the metric . The 0,4 valence Weyl tensor is then Harv Petersen 2006 p 92 NumBlk math W R ... n is the dimension of the manifold, g is the metric, R is the Riemann tensor, Ric is the Ricci tensor , s is the scalar curvature , and h small O small k denotes the Kulkarni&ndash Nomizu product ... 1,3 valent Weyl tensor is then given by contracting the above with the inverse of the metric. The decomposition EquationNote 1 expresses the Riemann tensor as an orthogonal direct sum of vector bundles ... decomposition , expresses the Riemann curvature tensor into its irreducible representation irreducible ... tensor further decomposes into invariant factors for the action of the special orthogonal group , the self dual and antiself dual parts W sup sup and W sup &minus sup . The Weyl tensor can also be expressed using the Schouten tensor , which is a trace adjusted multiple of the Ricci tensor, math ... more details
encountered in field theory. It has extensive functionality for tensor polynomial simplification including ...External links date October 2009 Tensor software is a class of mathematical software designed for manipulation and calculation with tensor s. Standalone open source software http www.aei.mpg.de peekas ... Maxima is a free open source computer algebra system which can be used for tensor algebra calculations ... without defining all components of the tensor explicitly . It comes with three tensor packages itensor for abstract indicial tensor manipulation, ctensor for component defined tensors, and atensor for algebraic tensor manipulation. http maxima.sourceforge.net docs manual en maxima 27.html SEC90 ... is a tensor analysis system written for the Mathematica system. It provides more than 250 ... is a tensor package written for the Mathematica system. It provides many functions relevant for General ... lee Ricci Ricci is a system for Mathematica 2.x and later for doing basic tensor analysis, available for free. http baldufa.upc.es xjaen ttc index.htm TTC Tools of Tensor Calculus is a Mathematica package for doing tensor and exterior calculus on differentiable manifolds. http www.inp.demokritos.gr sbonano RGTC EDC and RGTC Exterior Differential Calculus and Riemannian Geometry & Tensor Calculus are free Mathematica packages for tensor calculus especially designed but not only for general relativity ... tensor calculus package for Mathematica. http www.xact.es xAct Efficient Tensor Computer Algebra for Mathematica. xAct is a collection of packages for fast manipulation of tensor expressions. http library.wolfram.com ... connection and the basic tensors of General Relativity from a given metric tensor. http ... Tensor Toolbox Multilinear algebra MATLAB software. http www.mathworks.com matlabcentral fileexchange ... analysis . Libraries http www.oonumerics.org FTensor FTensor is a high performance tensor library ... threaded tensor library implemented in C used in Dynare . The library allows for folded unfolded ... more details
In mathematics , the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor . It is therefore a tensor field of TensorTensor rank rank three. It vanishes for the case of Riemannian geometry . Category Differential geometry Geometry stub ... more details
energy tensor is important in physics and engineering. Both of these are related by Einstein s theory ... of tensor quantities this kind of descent category theory descent argument justifies abstractly the whole ...Unreferenced date August 2008 In mathematics , physics and engineering , a tensor field assigns a tensor to each point of a mathematical space typically a Euclidean space or manifold . Tensor fields are used ... physics stress and strain tensor strain in materials, and in numerous applications in the physical sciences and engineering. As a tensor is a generalization of a scalar physics scalar a pure number representing a value, like length and a Euclidean vector vector a geometrical arrow in space , a tensor ... tensor fields , that is, fields defined over a manifold which define a tensor at every point of the manifold. An example is the Riemann curvature tensor . Geometric introduction Intuitively, a vector ... velocity at each point of the Earth s surface. The general idea of tensor field combines the requirement ... tensor &mdash with the idea that we don t want our notion to depend on the particular method ... explanation The contemporary mathematical expression of the idea of tensor field breaks it down into a two ... the tensor product concept is independent of any choice of basis, taking the tensor product of two ... of tensor field , namely as a section fiber bundle section of some tensor bundle . There are vector bundles which are not tensor bundles the M bius band for instance. This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space math V otimes cdots otimes V otimes ... . See also tangent bundle and cotangent bundle . Given two tensor bundles E M and F M , a map A E F from the space of sections of E to sections of F can be considered itself as a tensor section ..., where f is a smooth function on M . Thus a tensor is not only a linear map on the vector space ... more details
to the theory. For example, the Ricci tensor is a non metric contraction of the Riemann curvature tensor ...In multilinear algebra , a tensor contraction is an operation on one or more tensor s that arises from ... vector space dual . In components, it is expressed as a sum of products of scalar components of the tensor ... to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices one a subscript, the other a superscript of the tensor are set equal to each other and summed ... tensor with TensorTensor rank rank or order reduced by 2. Tensor contraction can be seen as a generalization ... from the Component free treatment of tensors Definition Tensor Product of Vector Spaces tensor product of these two spaces to the field k math C V otimes V rightarrow k math corresponding ... the contraction operation on a tensor of type 1,1 , which is an element of math V otimes V math ... obtains a basis free definition of the trace linear algebra trace . In general, a tensor of type m , n ... Fulton and Joe Harris, Representation Theory A First Course , Graduate Texts in Mathematics GTM ... operation, which is a linear map which yields a tensor of type m &minus 1, n ... dual basis . Since a general mixed dyadic tensor is a linear combination of decomposable tensors ... T T i j mathbf e i e j math be a mixed dyadic tensor. Then its contraction is math T i j mathbf ... being implied by the summation convention . The resulting contracted tensor inherits the remaining indices of the original tensor. For example, contracting a tensor T of type 2,2 on the second and third indices to create a new tensor U of type 1,1 is written as math T ab bc sum b T ab bc T a1 ... dyadic tensor. This tensor does not contract if its base vectors are dotted the result is the contravariant metric mathematics metric tensor , math g ij mathbf e i cdot mathbf e j math , whose rank ... product also known as a Metric tensor metric g , such contractions are possible. One uses the metric ... more details
Expert subject Physics date November 2008 There are two different tensor s sometimes referred to as the Lanczos tensor both named after Cornelius Lanczos A tensor in the theory of quadratic Lagrangian s, which vanishes in Four dimensional space four dimensions . The potential tensor H for the Weyl tensor C , this can be expressed as math C abcd H abc d H abd c H cda b H cdb a , math math g ac H bd H db g ad H bc H cb g bd H ac H ca g bc H ad H da 2 , math math 2H ef e f g ac g bd g ad g bc 3, , math where the Lanczos tensor has the symmetries math H abc H bac 0, , math math H abc H bca H cab 0, , math and where math H bd math is defined by math H bd stackrel mathrm def H e b d e H e b e d . math Thus, the Lanczos potential tensor is a gravitational field analog of the vector potential A for the electromagnetic field . See also Introduction to 2 spinors in general relativity World Scientific, 2003 by Peter O Donnell for a more detailed discussion of the Lanczos tensor and spinor. External links http www.worldscibooks.com physics 5222.html Introduction to 2 spinors in general relativity http www.arXiv.org abs gr qc 9904006 gr qc 9904006 Category Tensors Category Differential geometry Category Tensors in general relativity relativity stub ... more details
Polder tensor is a tensor used in description of magnetic Permeability electromagnetism permeability of ferrite s. ref http www.nature.com nature journal v182 n4642 abs 1821080a0.html G. G. Robbrecht, J. L. Verhaeghe, Measurements of the Permeability Tensor for Ferroxcube , Letters to Nature, Nature 182, 1080 18 October 1958 , doi 10.1038 1821080a0 ref The tensor notation needs to be used because ferrimagnetic materials become anisotropy anisotropic in the presence of magnetizing field. The tensor is described mathematically as ref http books.google.co.uk books?id lqHsnZoa7wAC&pg PA93&dq polder tensor&cd 5 v onepage&q polder 20tensor&f false Ricardo Marqu s, Ferran Mart n, Mario Sorolla, Metamaterials with negative parameter theory, design, and microwave applications, Willey, New Jersey, USA, 2009, ISBN 978 0 471 7458 2, page 93 ref math B begin bmatrix mu & j kappa & 0 j kappa & mu & 0 0 & 0 & mu 0 end bmatrix H math where math mu mu 0 left 1 frac omega 0 omega m omega 0 2 omega 2 right math math kappa mu 0 frac omega omega m omega 0 2 omega 2 math math omega 0 gamma mu 0 H 0 math math omega m gamma mu 0 M math and math gamma 17.6 cdot g math kHz A m is a gyromagnetic ratio and g is a factor between 1.9 2.4 depending on ferrite material. Magnetizing frequency f is expressed as math omega 2 pi f math , H sub 0 sub is a bias field, M is magnetization and math mu 0 math is magnetic permeability of free space . References Reflist Use dmy dates date September 2010 DEFAULTSORT Polder Tensor Category Ferrites ... more details
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