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Successor function Summary Successor_function 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sum of two cubes Summary Sum_of_two_cubes In mathematics, the sum of two cubes is a cubed number added to another cubed number. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Super-logarithm Summary Super-logarithm 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-logari... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Super-logarithm Summary Super-logarithm 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 conse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supernatural number Summary Supernatural_number 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 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supernatural number Summary Supernatural_number 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. Supernatura... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supernatural number Summary Supernatural_number 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 supe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supernatural number Summary Supernatural_number 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superquadrics Summary Superquadrics 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-di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superquadrics Summary Superquadrics 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 impor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superquadrics Summary Superquadrics 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersingular isogeny graph Summary Supersingular_isogeny_graph 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 th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Support (measure theory) Summary Support_(measure_theory) 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 {\displa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Support function Summary Support_function 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}} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Singular support Summary Support_(mathematics) 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 inst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface subgroup conjecture Summary Surface_subgroup_conjecture 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface subgroup conjecture Summary Surface_subgroup_conjecture 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surgery structure set Summary Surgery_structure_set 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 m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surreal form Summary Surcomplex_number 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 C... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surreal form Summary Surcomplex_number 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 numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multiplication sign Uses Multiplication_sign > Uses 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 "cro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Multiplication sign Uses Multiplication_sign > Uses 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 12... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symbolic method of invariant theory Summary Symbolic_method_of_invariant_theory 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric square Summary Symmetric_square 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric closure Summary Symmetric_closure 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 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric decreasing rearrangement Summary Symmetric_decreasing_rearrangement 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second symmetric derivative Summary Second_symmetric_derivative 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Second symmetric derivative Summary Second_symmetric_derivative 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric difference Summary Symmetric_difference 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 \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Schwarz's theorem Summary Clairaut's_theorem_(calculus) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symplectization Summary Symplectization In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperreal number Summary Hyperreal_field 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperreal number Summary Hyperreal_field 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hyperreal number Summary Hyperreal_field 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Syzygetic pencil Summary Hesse_pencil 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})+\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Syzygetic pencil Summary Hesse_pencil 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 coordi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Syzygetic pencil Summary Hesse_pencil 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 cub... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Marden conjecture Summary Marden_conjecture 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). I... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tanc function Summary Tanc_function In mathematics, the tanc function is defined for z ≠ 0 {\displaystyle z\neq 0} as | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tangent spaces Summary Tangent_spaces 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tanhc function Summary Tanhc_function 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universal vector bundle Summary Tautological_subbundle 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universal vector bundle Summary Tautological_subbundle 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universal vector bundle Summary Tautological_subbundle 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 sp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universal vector bundle Summary Tautological_subbundle (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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form 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 sy... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form 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 symp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form 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 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form {\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.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form 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 \mathb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical symplectic form Summary Canonical_symplectic_form 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 de... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Telephone number (mathematics) Summary Telephone_number_(mathematics) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor coalgebra Summary Tensor_algebra 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 vect... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor coalgebra Summary Tensor_algebra 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 s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor bundle Summary Tensor_bundle 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 antisymmetri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
TP model transformation Summary TP_model_transformation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
TP model transformation Summary TP_model_transformation 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 me... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
TP model transformation Summary TP_model_transformation 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 inf... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
TP model transformation Summary TP_model_transformation 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 be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product representation Summary Tensor_product 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product representation Summary Tensor_product 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 produc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of abelian groups Summary Tensor_product_of_abelian_groups 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 prod... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms 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 {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . , a n ⟩ {\displaystyle q_{1}\cong \langle a_{1},...,a_{n}\rangle } q 2 ≅ ⟨ b 1 , . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . , 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 , . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . a 1 b m , a 2 b 1 , . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . , a 2 b m , . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . , a n b 1 , . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product of quadratic forms Summary Tensor_product_of_quadratic_forms . . 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 .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hom representation Summary Symmetric_and_alternating_squares 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 gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor product algebra Summary Tensor_product_of_rings 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 representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complex embedding Summary Real_embedding 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor representation Summary Tensor_representation 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 represent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor representation Summary Tensor_representation 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tensor-hom adjunction Summary Tensor-hom_adjunction 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 \... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tent map Summary Tent_map 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Tent map Summary Tent_map 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 subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Almost all Summary Almost_all 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 "... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic function Summary Characteristic_functions 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, h... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic function Summary Characteristic_functions {\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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic function Summary Characteristic_functions The characteristic polynomial in linear algebra. The characteristic state function in statistical mechanics. The Euler characteristic, a topological invariant. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Characteristic function Summary Characteristic_functions The receiver operating characteristic in statistical decision theory. The point characteristic function in statistics. == References == | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 grad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 decompositi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 differen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) A graded-commutative superalgebra satisfies the "supercommutative" law y x = ( − 1 ) | x | | y | x y . {\displaystyle yx=(-1)^{|x||y|}xy.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Graded (mathematics) Summary Graded_(mathematics) 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 {\mathca... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivial solution Trivial and nontrivial solutions Triviality_(mathematics) > Trivial and nontrivial solutions 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 member... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Trivial solution Trivial and nontrivial solutions Triviality_(mathematics) > Trivial and nontrivial solutions 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braket notation Vectors vs kets Ket_vector > Vector spaces > Vectors vs kets 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braket notation Vectors vs kets Ket_vector > Vector spaces > Vectors vs kets 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Braket notation Vectors vs kets Ket_vector > Vector spaces > Vectors vs kets 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Beurling algebra Summary Beurling_algebra 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cartan matrices Summary Cartan_matrices 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemann-Hilbert correspondence Summary Riemann–Hilbert_correspondence 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 generalizati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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