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c_9zipnmf5jjhe | In mathematics, the successor function or successor operation sends a natural number to the next one. The successor function is denoted by S, so S(n) = n + 1. For example, S(1) = 2 and S(2) = 3. The successor function is one of the basic components used to build a primitive recursive function. Successor operations are ... | Successor function |
c_aoqhroq1v7wq | In mathematics, the sum of two cubes is a cubed number added to another cubed number. | Sum of two cubes |
c_xbi4keeloz9h | In mathematics, the super-logarithm is one of the two inverse functions of tetration. Just as exponentiation has two inverse functions, roots and logarithms, tetration has two inverse functions, super-roots and super-logarithms. There are several ways of interpreting super-logarithms: As the Abel function of exponentia... | Super-logarithm |
c_6nmgffi26r5b | However, this is not true for negative values and so cannot be considered a full definition. The precise definition of the super-logarithm depends on a precise definition of non-integer tetration (that is, y x {\displaystyle {^{y}x}} for y not an integer). There is no clear consensus on the definition of non-integer te... | Super-logarithm |
c_irdj7iwk2crv | In mathematics, the supernatural numbers, sometimes called generalized natural numbers or Steinitz numbers, are a generalization of the natural numbers. They were used by Ernst Steinitz: 249–251 in 1910 as a part of his work on field theory. A supernatural number ω {\displaystyle \omega } is a formal product: ω = ∏ p p... | Supernatural number |
c_8tuhalb4qfhf | If no n p = ∞ {\displaystyle n_{p}=\infty } and there are only a finite number of non-zero n p {\displaystyle n_{p}} then we recover the positive integers. Slightly less intuitively, if all n p {\displaystyle n_{p}} are ∞ {\displaystyle \infty } , we get zero. Supernatural numbers extend beyond natural numbers by allow... | Supernatural number |
c_x9zlhzxnddrs | There is no natural way to add supernatural numbers, but they can be multiplied, with ∏ p p n p ⋅ ∏ p p m p = ∏ p p n p + m p {\displaystyle \prod _{p}p^{n_{p}}\cdot \prod _{p}p^{m_{p}}=\prod _{p}p^{n_{p}+m_{p}}} . Similarly, the notion of divisibility extends to the supernaturals with ω 1 ∣ ω 2 {\displaystyle \omega _... | Supernatural number |
c_6qwrc13fyd1d | We can also extend the usual p {\displaystyle p} -adic order functions to supernatural numbers by defining v p ( ω ) = n p {\displaystyle v_{p}(\omega )=n_{p}} for each p {\displaystyle p} . Supernatural numbers are used to define orders and indices of profinite groups and subgroups, in which case many of the theorems ... | Supernatural number |
c_zj0vj2bmty4p | In mathematics, the superquadrics or super-quadrics (also superquadratics) are a family of geometric shapes defined by formulas that resemble those of ellipsoids and other quadrics, except that the squaring operations are replaced by arbitrary powers. They can be seen as the three-dimensional relatives of the superelli... | Superquadrics |
c_1o4p6334uo7y | The superquadrics include many shapes that resemble cubes, octahedra, cylinders, lozenges and spindles, with rounded or sharp corners. Because of their flexibility and relative simplicity, they are popular geometric modeling tools, especially in computer graphics. It becomes an important geometric primitive widely used... | Superquadrics |
c_anges3b4ovpr | In modern computer vision literatures, superquadrics and superellipsoids are used interchangeably, since superellipsoids are the most representative and widely utilized shape among all the superquadrics. Comprehensive coverage of geometrical properties of superquadrics and methods of their recovery from range images an... | Superquadrics |
c_no3vw6tkknqj | In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in elliptic-curve cryptography. Their vertices represent supersingular elliptic curves over finite fields and their edges represent isogenies between curves. | Supersingular isogeny graph |
c_kkofx1ykzkdl | In mathematics, the support (sometimes topological support or spectrum) of a measure μ {\displaystyle \mu } on a measurable topological space ( X , Borel ( X ) ) {\displaystyle (X,\operatorname {Borel} (X))} is a precise notion of where in the space X {\displaystyle X} the measure "lives". It is defined to be the lar... | Support (measure theory) |
c_jp4qj7ij1hwp | In mathematics, the support function hA of a non-empty closed convex set A in R n {\displaystyle \mathbb {R} ^{n}} describes the (signed) distances of supporting hyperplanes of A from the origin. The support function is a convex function on R n {\displaystyle \mathbb {R} ^{n}} . Any non-empty closed convex set A is uni... | Support function |
c_6jg440ci8lwm | In mathematics, the support of a real-valued function f {\displaystyle f} is the subset of the function domain containing the elements which are not mapped to zero. If the domain of f {\displaystyle f} is a topological space, then the support of f {\displaystyle f} is instead defined as the smallest closed set containi... | Singular support |
c_tkvsno4wu81u | In mathematics, the surface subgroup conjecture of Friedhelm Waldhausen states that the fundamental group of every closed, irreducible 3-manifold with infinite fundamental group has a surface subgroup. By "surface subgroup" we mean the fundamental group of a closed surface not the 2-sphere. This problem is listed as Pr... | Surface subgroup conjecture |
c_1fh6exlnlmoc | A proof of this case was announced in the summer of 2009 by Jeremy Kahn and Vladimir Markovic and outlined in a talk August 4, 2009 at the FRG (Focused Research Group) Conference hosted by the University of Utah. A preprint appeared in the arxiv.org server in October 2009. Their paper was published in the Annals of Mat... | Surface subgroup conjecture |
c_4t1vfz1hig02 | In mathematics, the surgery structure set S ( X ) {\displaystyle {\mathcal {S}}(X)} is the basic object in the study of manifolds which are homotopy equivalent to a closed manifold X. It is a concept which helps to answer the question whether two homotopy equivalent manifolds are diffeomorphic (or PL-homeomorphic or ho... | Surgery structure set |
c_xy1838kfrj3r | In mathematics, the surreal number system is a totally ordered proper class containing not only the real numbers but also infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. Research on the Go endgame by John Horton Conway led to the original definition an... | Surreal form |
c_fccj4w0bjogk | If formulated in von Neumann–Bernays–Gödel set theory, the surreal numbers are a universal ordered field in the sense that all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers (including the hyperreal numbers) can be realized as subfields of th... | Surreal form |
c_y7cb2ptjh4z3 | In mathematics, the symbol × has a number of uses, including Multiplication of two numbers, where it is read as "times" or "multiplied by" Cross product of two vectors, where it is usually read as "cross" Cartesian product of two sets, where it is usually read as "cross" Geometric dimension of an object, such as noting... | Multiplication sign |
c_0oyzvsdbyzuy | However, the communication of these hybrid names with a Latin letter "x" is common, when the actual "×" symbol is not readily available. The multiplication sign is also used by historians for an event between two dates. When employed between two dates – for example 1225 and 1232 – the expression "1225×1232" means "no e... | Multiplication sign |
c_gh7rk09vqy98 | In mathematics, the symbolic method in invariant theory is an algorithm developed by Arthur Cayley, Siegfried Heinrich Aronhold, Alfred Clebsch, and Paul Gordan in the 19th century for computing invariants of algebraic forms. It is based on treating the form as if it were a power of a degree one form, which corresponds... | Symbolic method of invariant theory |
c_1zreen4zcp08 | In mathematics, the symmetric algebra S(V) (also denoted Sym(V)) on a vector space V over a field K is a commutative algebra over K that contains V, and is, in some sense, minimal for this property. Here, "minimal" means that S(V) satisfies the following universal property: for every linear map f from V to a commutativ... | Symmetric square |
c_6d4e4dhu7ceb | In mathematics, the symmetric closure of a binary relation R {\displaystyle R} on a set X {\displaystyle X} is the smallest symmetric relation on X {\displaystyle X} that contains R . {\displaystyle R.} For example, if X {\displaystyle X} is a set of airports and x R y {\displaystyle xRy} means "there is a direct fligh... | Symmetric closure |
c_7eie5qtf8ng5 | In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. | Symmetric decreasing rearrangement |
c_zo3el13y4vbe | In mathematics, the symmetric derivative is an operation generalizing the ordinary derivative. It is defined as The expression under the limit is sometimes called the symmetric difference quotient. A function is said to be symmetrically differentiable at a point x if its symmetric derivative exists at that point. If a ... | Second symmetric derivative |
c_wj0suwqkjp05 | A well-known counterexample is the absolute value function f(x) = |x|, which is not differentiable at x = 0, but is symmetrically differentiable here with symmetric derivative 0. For differentiable functions, the symmetric difference quotient does provide a better numerical approximation of the derivative than the usua... | Second symmetric derivative |
c_jdjsdbxtgx86 | In mathematics, the symmetric difference of two sets, also known as the disjunctive union and set sum, is the set of elements which are in either of the sets, but not in their intersection. For example, the symmetric difference of the sets { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} and { 3 , 4 } {\displaystyle \{3,4\}} is... | Symmetric difference |
c_tkmh5p6bbqb1 | In mathematics, the symmetry of second derivatives (also called the equality of mixed partials) refers to the possibility of interchanging the order of taking partial derivatives of a function f ( x 1 , x 2 , … , x n ) {\displaystyle f\left(x_{1},\,x_{2},\,\ldots ,\,x_{n}\right)} of n variables without changing the res... | Schwarz's theorem |
c_7ac4h8cqa3i3 | In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it. | Symplectization |
c_7na74orzrayv | In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form 1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1}... | Hyperreal number |
c_ys46bq406cnx | The transfer principle for ultrapowers is a consequence of Łoś's theorem of 1955. Concerns about the soundness of arguments involving infinitesimals date back to ancient Greek mathematics, with Archimedes replacing such proofs with ones using other techniques such as the method of exhaustion. In the 1960s, Abraham Robi... | Hyperreal number |
c_kenoffkyk4q3 | This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules that Robinson delineated. The application of hyperreal numbers and in particular the transfer principle to problems of analysis is called nonstandard analysis. One immed... | Hyperreal number |
c_5bze2iebsa4y | In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation λ ( x 3 + y 3 + z 3 ) + μ x y z = 0. {\displaystyle \lambda (x^{3}+y^{3}+z^{3})+\mu xyz=0.} Each curve in the family is... | Syzygetic pencil |
c_9ir4s2yowhxu | Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equi... | Syzygetic pencil |
c_mlykz5s5mjuf | More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points. The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passe... | Syzygetic pencil |
c_owsirflnlbcx | In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a compact 3-manifold. The tameness theorem was conjectured by Marden (1974). It was proved by Agol (2004) and, independent... | Marden conjecture |
c_1ecqplmdnd6h | In mathematics, the tanc function is defined for z ≠ 0 {\displaystyle z\neq 0} as | Tanc function |
c_4bn55w3o2xfe | In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities f... | Tangent spaces |
c_cm19m8snjmmr | In mathematics, the tanhc function is defined for z ≠ 0 {\displaystyle z\neq 0} asThe tanhc function is the hyperbolic analogue of the tanc function. | Tanhc function |
c_fs0uk0x8ihjr | In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k {\displaystyle k} -dimensional subspaces of V {\displaystyle V} , given a point in the Grassmannian corresponding to a k {\displaystyle k} -dimensional vector subspace W ⊆ V {\... | Universal vector bundle |
c_81h7wghz01a3 | Tautological bundles are constructed both in algebraic topology and in algebraic geometry. In algebraic geometry, the tautological line bundle (as invertible sheaf) is O P n ( − 1 ) , {\displaystyle {\mathcal {O}}_{\mathbb {P} ^{n}}(-1),} the dual of the hyperplane bundle or Serre's twisting sheaf O P n ( 1 ) {\display... | Universal vector bundle |
c_v7krwvemu256 | The tautological line bundle and the hyperplane bundle are exactly the two generators of the Picard group of the projective space.In Michael Atiyah's "K-theory", the tautological line bundle over a complex projective space is called the standard line bundle. The sphere bundle of the standard bundle is usually called th... | Universal vector bundle |
c_ffutucvkn2t9 | (cf. Bott generator.) More generally, there are also tautological bundles on a projective bundle of a vector bundle as well as a Grassmann bundle. The older term canonical bundle has dropped out of favour, on the grounds that canonical is heavily overloaded as it is, in mathematical terminology, and (worse) confusion w... | Universal vector bundle |
c_05ugd34wzwgq | In mathematics, the tautological one-form is a special 1-form defined on the cotangent bundle T ∗ Q {\displaystyle T^{*}Q} of a manifold Q . {\displaystyle Q.} In physics, it is used to create a correspondence between the velocity of a point in a mechanical system and its momentum, thus providing a bridge between Lagra... | Canonical symplectic form |
c_6qfflmubxucf | The tautological one-form plays an important role in relating the formalism of Hamiltonian mechanics and Lagrangian mechanics. The tautological one-form is sometimes also called the Liouville one-form, the Poincaré one-form, the canonical one-form, or the symplectic potential. A similar object is the canonical vector f... | Canonical symplectic form |
c_qbu3q0ii4yni | To define the tautological one-form, select a coordinate chart U {\displaystyle U} on T ∗ Q {\displaystyle T^{*}Q} and a canonical coordinate system on U . {\displaystyle U.} Pick an arbitrary point m ∈ T ∗ Q . | Canonical symplectic form |
c_ss6lw0sv3re0 | {\displaystyle m\in T^{*}Q.} By definition of cotangent bundle, m = ( q , p ) , {\displaystyle m=(q,p),} where q ∈ Q {\displaystyle q\in Q} and p ∈ T q ∗ Q . {\displaystyle p\in T_{q}^{*}Q.} | Canonical symplectic form |
c_p5p5g3orn6v9 | The tautological one-form θ m: T m T ∗ Q → R {\displaystyle \theta _{m}:T_{m}T^{*}Q\to \mathbb {R} } is given by with n = dim Q {\displaystyle n=\mathop {\text{dim}} Q} and ( p 1 , … , p n ) ∈ U ⊆ R n {\displaystyle (p_{1},\ldots ,p_{n})\in U\subseteq \mathbb {R} ^{n}} being the coordinate representation of p . {\dis... | Canonical symplectic form |
c_ej12c3iy7lwo | The canonical symplectic form, also known as the Poincaré two-form, is given by The extension of this concept to general fibre bundles is known as the solder form. By convention, one uses the phrase "canonical form" whenever the form has a unique, canonical definition, and one uses the term "solder form", whenever an a... | Canonical symplectic form |
c_jpau3bwdzg86 | In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways n people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on n vertices, the number of permutations on n element... | Telephone number (mathematics) |
c_tc2n03nripsl | In mathematics, the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. It is the free algebra on V, in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" alge... | Tensor coalgebra |
c_sq5utrb0h3v2 | The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all alg... | Tensor coalgebra |
c_blzn793e5rg4 | In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold. To do calculus on the tensor bundle a connection is needed, except for the special case of the exterior derivative of antisymmetric tensors. | Tensor bundle |
c_yswz0ilosxdi | In mathematics, the tensor product (TP) model transformation was proposed by Baranyi and Yam as key concept for higher-order singular value decomposition of functions. It transforms a function (which can be given via closed formulas or neural networks, fuzzy logic, etc.) into TP function form if such a transformation i... | TP model transformation |
c_ni05srsjku7g | A key underpinning of the transformation is the higher-order singular value decomposition.Besides being a transformation of functions, the TP model transformation is also a new concept in qLPV based control which plays a central role in the providing a valuable means of bridging between identification and polytopic sys... | TP model transformation |
c_3wuwka97ku98 | Further details on the control theoretical aspects of the TP model transformation can be found here: TP model transformation in control theory. The TP model transformation motivated the definition of the "HOSVD canonical form of TP functions", on which further information can be found here. It has been proved that the ... | TP model transformation |
c_pvvd34csrzlt | Thus, the TP model transformation can be viewed as a numerical method to compute the HOSVD of functions, which provides exact results if the given function has a TP function structure and approximative results otherwise. The TP model transformation has recently been extended in order to derive various types of convex T... | TP model transformation |
c_q0f6wnxkcy9j | In mathematics, the tensor product V ⊗ W {\displaystyle V\otimes W} of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V × W → V ⊗ W {\displaystyle V\times W\to V\otimes W} that maps a pair ( v , w ) , v ∈ V , w ∈ W {\displaystyle (v,w),\ v\in V,w\in W} to an elem... | Tensor product representation |
c_avymgqd8zszz | The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense that every element of V ⊗ W {\displaystyle V\otimes W} is a sum of elementary tensors. If bases are given for V and W, a basis of V ⊗ W {\displaystyle V\otimes W} is formed by all tensor products of a basis element of V and a basis element of W. ... | Tensor product representation |
c_tr06c4ix3pe8 | In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a ... | Tensor product of abelian groups |
c_hgrhni8rd1g6 | In mathematics, the tensor product of quadratic forms is most easily understood when one views the quadratic forms as quadratic spaces. If R is a commutative ring where 2 is invertible (that is, R has characteristic char ( R ) ≠ 2 {\displaystyle {\text{char}}(R)\neq 2} ), and if ( V 1 , q 1 ) {\displaystyle (V_{1},q_{1... | Tensor product of quadratic forms |
c_tlffc6h12mj1 | . . , a n ⟩ {\displaystyle q_{1}\cong \langle a_{1},...,a_{n}\rangle } q 2 ≅ ⟨ b 1 , . | Tensor product of quadratic forms |
c_gs1s1pmkwzrl | . . , b m ⟩ {\displaystyle q_{2}\cong \langle b_{1},...,b_{m}\rangle } then the tensor product has diagonalization q 1 ⊗ q 2 ≅ ⟨ a 1 b 1 , a 1 b 2 , . | Tensor product of quadratic forms |
c_0cyg0lia2hk3 | . . a 1 b m , a 2 b 1 , . | Tensor product of quadratic forms |
c_hvi4gslf5urc | . . , a 2 b m , . | Tensor product of quadratic forms |
c_4u6pmys7ehfl | . . , a n b 1 , . | Tensor product of quadratic forms |
c_y6fh3uey66wb | . . a n b m ⟩ . {\displaystyle q_{1}\otimes q_{2}\cong \langle a_{1}b_{1},a_{1}b_{2},...a_{1}b_{m},a_{2}b_{1},...,a_{2}b_{m},...,a_{n}b_{1},...a_{n}b_{m}\rangle .} | Tensor product of quadratic forms |
c_eqv4q13jbvxh | In mathematics, the tensor product of representations is a tensor product of vector spaces underlying representations together with the factor-wise group action on the product. This construction, together with the Clebsch–Gordan procedure, can be used to generate additional irreducible representations if one already kn... | Hom representation |
c_pw4yqots2xoc | In mathematics, the tensor product of two algebras over a commutative ring R is also an R-algebra. This gives the tensor product of algebras. When the ring is a field, the most common application of such products is to describe the product of algebra representations. | Tensor product algebra |
c_99kxy0w87oiu | In mathematics, the tensor product of two fields is their tensor product as algebras over a common subfield. If no subfield is explicitly specified, the two fields must have the same characteristic and the common subfield is their prime subfield. The tensor product of two fields is sometimes a field, and often a direct... | Complex embedding |
c_mj33zx4t9x07 | In mathematics, the tensor representations of the general linear group are those that are obtained by taking finitely many tensor products of the fundamental representation and its dual. The irreducible factors of such a representation are also called tensor representations, and can be obtained by applying Schur functo... | Tensor representation |
c_9x43bho63vs3 | More generally, a matrix group is any subgroup of the general linear group. A tensor representation of a matrix group is any representation that is contained in a tensor representation of the general linear group. For example, the orthogonal group O(n) admits a tensor representation on the space of all trace-free symme... | Tensor representation |
c_no7ykvls1vsd | In mathematics, the tensor-hom adjunction is that the tensor product − ⊗ X {\displaystyle -\otimes X} and hom-functor Hom ( X , − ) {\displaystyle \operatorname {Hom} (X,-)} form an adjoint pair: Hom ( Y ⊗ X , Z ) ≅ Hom ( Y , Hom ( X , Z ) ) . {\displaystyle \operatorname {Hom} (Y\otimes X,Z)\cong \operatorname... | Tensor-hom adjunction |
c_gwkdjp5zvwvv | In mathematics, the tent map with parameter μ is the real-valued function fμ defined by f μ ( x ) := μ min { x , 1 − x } , {\displaystyle f_{\mu }(x):=\mu \min\{x,\,1-x\},} the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ maps the unit interval into its... | Tent map |
c_ex5ywdwukgqy | Choosing for instance the parameter μ = 2, the effect of the function fμ may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval to get again the interval . Iterating the procedure, any point x0 of the interval assumes new subsequent positions as describ... | Tent map |
c_g8hsja49noke | In mathematics, the term "almost all" means "all but a negligible quantity". More precisely, if X {\displaystyle X} is a set, "almost all elements of X {\displaystyle X} " means "all elements of X {\displaystyle X} but those in a negligible subset of X {\displaystyle X} ". The meaning of "negligible" depends on the mat... | Almost all |
c_e822nf282bbe | In mathematics, the term "characteristic function" can refer to any of several distinct concepts: The indicator function of a subset, that is the function 1 A: X → { 0 , 1 } , {\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},} which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.There ... | Characteristic function |
c_0v9631j4v02c | {\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}} In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question: φ X ( t ) = E ( e i t X ) ... | Characteristic function |
c_saf8b2egstu7 | The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant. | Characteristic function |
c_kipoipbaz5pf | The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics. == References == | Characteristic function |
c_yt554wgmnoo4 | In mathematics, the term "graded" has a number of meanings, mostly related: In abstract algebra, it refers to a family of concepts: An algebraic structure X {\displaystyle X} is said to be I {\displaystyle I} -graded for an index set I {\displaystyle I} if it has a gradation or grading, i.e. a decomposition into a dire... | Graded (mathematics) |
c_ha88n405obtb | An algebraic structure is said to be doubly graded if the index set is a direct product of sets; the pairs may be called "bidegrees" (e.g. see Spectral sequence). A I {\displaystyle I} -graded vector space or graded linear space is thus a vector space with a decomposition into a direct sum V = ⨁ i ∈ I V i {\textstyle V... | Graded (mathematics) |
c_yzxk9p7o50ff | A graded ring is a ring that is a direct sum of additive abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} , with i {\displaystyle i} taken from some monoid, usually N {\displaystyle \mathbb {N} } or Z {\displaystyle \mathbb {Z} } , or semigroup (for a ring... | Graded (mathematics) |
c_r48jnz0kfrbh | The associated graded module of an R {\displaystyle R} -module M {\displaystyle M} with respect to a proper ideal I {\displaystyle I} is gr I M = ⨁ n ∈ N I n M / I n + 1 M {\textstyle \operatorname {gr} _{I}M=\bigoplus _{n\in \mathbb {N} }I^{n}M/I^{n+1}M} . A differential graded module, differential graded Z {\displa... | Graded (mathematics) |
c_x4h3t5urdqs3 | The graded Leibniz rule for a map d: A → A {\displaystyle d\colon A\to A} on a graded algebra A {\displaystyle A} specifies that d ( a ⋅ b ) = ( d a ) ⋅ b + ( − 1 ) | a | a ⋅ ( d b ) {\displaystyle d(a\cdot b)=(da)\cdot b+(-1)^{|a|}a\cdot (db)} . A differential graded algebra, DG-algebra or DGAlgebra is a graded algebr... | Graded (mathematics) |
c_8xulpi0ob5tt | A DGA is an augmented DG-algebra, or differential graded augmented algebra, (see Differential graded algebra). A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. | Graded (mathematics) |
c_h143nthesixf | A graded-commutative superalgebra satisfies the "supercommutative" law y x = ( − 1 ) | x | | y | x y . {\displaystyle yx=(-1)^{|x||y|}xy.} | Graded (mathematics) |
c_str16ocwyizx | for homogeneous x,y, where | a | {\displaystyle |a|} represents the "parity" of a {\displaystyle a} , i.e. 0 or 1 depending on the component in which it lies. CDGA may refer to the category of augmented differential graded commutative algebras. A graded Lie algebra is a Lie algebra that is graded as a vector space by a... | Graded (mathematics) |
c_lufbrbecb3bd | A graded Lie superalgebra is a graded Lie algebra with the requirement for anticommutativity of its Lie bracket relaxed. A supergraded Lie superalgebra is a graded Lie superalgebra with an additional super Z 2 {\displaystyle \mathbb {Z} _{2}} -gradation. A differential graded Lie algebra is a graded vector space over a... | Graded (mathematics) |
c_tlrohaj4eklm | The Graded Brauer group is a synonym for the Brauer–Wall group B W ( F ) {\displaystyle BW(F)} classifying finite-dimensional graded central division algebras over the field F. An A {\displaystyle {\mathcal {A}}} -graded category for a category A {\displaystyle {\mathcal {A}}} is a category C {\displaystyle {\mathcal {... | Graded (mathematics) |
c_utzm9kj4puag | In mathematics, the term "trivial" is often used to refer to objects (e.g., groups, topological spaces) with a very simple structure. These include, among others: Empty set: the set containing no or null members Trivial group: the mathematical group containing only the identity element Trivial ring: a ring defined on a... | Trivial solution |
c_o93cha5d9kyj | The trivial solution is the zero function while a nontrivial solution is the exponential function The differential equation f ″ ( x ) = − λ f ( x ) {\displaystyle f''(x)=-\lambda f(x)} with boundary conditions f ( 0 ) = f ( L ) = 0 {\displaystyle f(0)=f(L)=0} is important in mathematics and physics, as it could be used... | Trivial solution |
c_0layal1sciu7 | In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. ... | Braket notation |
c_140on8635blh | To distinguish this type of vector from those described above, it is common and useful in physics to denote an element ϕ {\displaystyle \phi } of an abstract complex vector space as a ket | ϕ ⟩ {\displaystyle |\phi \rangle } , to refer to it as a "ket" rather than as a vector, and to pronounce it "ket- ϕ {\displaystyle... | Braket notation |
c_fw6uwky1gho3 | In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, |1⟩ + |2⟩ is not necessarily equal to |3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as... | Braket notation |
c_79mezvzwf1pd | In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling (1949), usually it is an algebra of periodic functions with Fourier series f ( x ) = ∑ a n e i n x {\displaystyle f(x)=\sum a_{n}e^{inx}} Example We may consider the algebra of those functions f where the majorants c k ... | Beurling algebra |
c_3lvlzsz7tw6l | In mathematics, the term Cartan matrix has three meanings. All of these are named after the French mathematician Élie Cartan. Amusingly, the Cartan matrices in the context of Lie algebras were first investigated by Wilhelm Killing, whereas the Killing form is due to Cartan. | Cartan matrices |
c_psqa6d7ev08o | In mathematics, the term Riemann–Hilbert correspondence refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this. The original setting appearing in Hilbert's twenty-first ... | Riemann-Hilbert correspondence |
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