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Subbundle Summary Subbundle In mathematics, a subbundle U {\displaystyle U} of a vector bundle V {\displaystyle V} on a topological space X {\displaystyle X} is a collection of linear subspaces U x {\displaystyle U_{x}} of the fibers V x {\displaystyle V_{x}} of V {\displaystyle V} at x {\displaystyle x} in X , {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subcompact cardinal Summary Subcompact_cardinal In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number κ is subcompact if and only if for every A ⊂ H(κ+) there is a non-trivial elementary embedding j:(H(μ+), B) → (H(κ+), A) (where H(κ+) is the set of all sets of cardinality ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subcompact cardinal Summary Subcompact_cardinal The relationship is analogous to that of extendible versus supercompact cardinals. Quasicompactness may be viewed as a strengthened or "boldface" version of 1-extendibility. Existence of subcompact cardinals implies existence of many 1-extendible cardinals, and hence many... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subcompact cardinal Summary Subcompact_cardinal Existence of a 2κ-supercompact cardinal κ implies existence of many quasicompact cardinals. Subcompact cardinals are noteworthy as the least large cardinals implying a failure of the square principle. If κ is subcompact, then the square principle fails at κ. Canonical inn... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subcompact cardinal Summary Subcompact_cardinal (Existence of such models has not yet been proved, but in any case the square principle can be forced for weaker cardinals.) Quasicompactness is one of the strongest large cardinal properties that can be witnessed by current inner models that do not use long extenders. Fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Slice chart Summary Slice_chart In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types of submanifolds depending on exactly which properties are required. Different authors of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Submodular valuation Summary Submodular_valuation In mathematics, a submodular set function (also known as a submodular function) is a set function that, informally, describes the relationship between a set of inputs and an output, where adding more of one input has a decreasing additional benefit (diminishing returns)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subpaving Summary Subpaving In mathematics, a subpaving is a set of nonoverlapping boxes of R⁺. A subset X of Rⁿ can be approximated by two subpavings X⁻ and X⁺ such that X⁻ ⊂ X ⊂ X⁺. In R¹ the boxes are line segments, in R² rectangles and in Rⁿ hyperrectangles. A R² subpaving can be also a "non-regular tiling by recta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subpaving Summary Subpaving Boxes present the advantage of being very easily manipulated by computers, as they form the heart of interval analysis. Many interval algorithms naturally provide solutions that are regular subpavings.In computation, a well-known application of subpaving in R² is the Quadtree data structure.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Algebra of dual numbers Summary Algebra_of_dual_numbers In mathematics, a subring of R is a subset of a ring that is itself a ring when binary operations of addition and multiplication on R are restricted to the subset, and which shares the same multiplicative identity as R. For those who define rings without requiring... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsequence Summary Subsequence In mathematics, a subsequence of a given sequence is a sequence that can be derived from the given sequence by deleting some or no elements without changing the order of the remaining elements. For example, the sequence ⟨ A , B , D ⟩ {\displaystyle \langle A,B,D\rangle } is a subsequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsequence Summary Subsequence Subsequences can contain consecutive elements which were not consecutive in the original sequence. A subsequence which consists of a consecutive run of elements from the original sequence, such as ⟨ B , C , D ⟩ , {\displaystyle \langle B,C,D\rangle ,} from ⟨ A , B , C , D , E , F ⟩ , {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsequential limit Summary Subsequential_limit In mathematics, a subsequential limit of a sequence is the limit of some subsequence. Every subsequential limit is a cluster point, but not conversely. In first-countable spaces, the two concepts coincide. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsequential limit Summary Subsequential_limit In a topological space, if every subsequence has a subsequential limit to the same point, then the original sequence also converges to that limit. This need not hold in more generalized notions of convergence, such as the space of almost everywhere convergence. The suprem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Subsequential limit Summary Subsequential_limit Similarly, the infimum of such a set is called the limit inferior, or liminf. See limit superior and limit inferior.If ( X , d ) {\displaystyle (X,d)} is a metric space and there is a Cauchy sequence such that there is a subsequence converging to some x , {\displaystyle x... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Universally measurable set Summary Universally_measurable_set In mathematics, a subset A {\displaystyle A} of a Polish space X {\displaystyle X} is universally measurable if it is measurable with respect to every complete probability measure on X {\displaystyle X} that measures all Borel subsets of X {\displaystyle X} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Radial set Summary Radial_set In mathematics, a subset A ⊆ X {\displaystyle A\subseteq X} of a linear space X {\displaystyle X} is radial at a given point a 0 ∈ A {\displaystyle a_{0}\in A} if for every x ∈ X {\displaystyle x\in X} there exists a real t x > 0 {\displaystyle t_{x}>0} such that for every t ∈ , {\display... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Cofinal sequence Summary Final_function In mathematics, a subset B ⊆ A {\displaystyle B\subseteq A} of a preordered set ( A , ≤ ) {\displaystyle (A,\leq )} is said to be cofinal or frequent in A {\displaystyle A} if for every a ∈ A , {\displaystyle a\in A,} it is possible to find an element b {\displaystyle b} in B {\d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Absolutely convex set Summary Disked_hull In mathematics, a subset C of a real or complex vector space is said to be absolutely convex or disked if it is convex and balanced (some people use the term "circled" instead of "balanced"), in which case it is called a disk. The disked hull or the absolute convex hull of a se... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reduced residue system Summary Reduced_residue_system In mathematics, a subset R of the integers is called a reduced residue system modulo n if: gcd(r, n) = 1 for each r in R, R contains φ(n) elements, no two elements of R are congruent modulo n.Here φ denotes Euler's totient function. A reduced residue system modulo n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reflexive transitive symmetric closure Summary Reflexive_transitive_closure In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, bu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Reflexive transitive symmetric closure Summary Reflexive_transitive_closure The closure of a subset is the result of a closure operator applied to the subset. The closure of a subset under some operations is the smallest superset that is closed under these operations. It is often called the span (for example linear spa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Nowhere-dense set Summary Nowhere-dense_set In mathematics, a subset of a topological space is called nowhere dense or rare if its closure has empty interior. In a very loose sense, it is a set whose elements are not tightly clustered (as defined by the topology on the space) anywhere. For example, the integers are now... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Sum-free sequence Summary Sum-free_sequence In mathematics, a sum-free sequence is an increasing sequence of positive integers, a 1 , a 2 , a 3 , … , {\displaystyle a_{1},a_{2},a_{3},\ldots ,} such that no term a n {\displaystyle a_{n}} can be represented as a sum of any subset of the preceding elements of the sequence... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summability kernel Summary Summability_kernel In mathematics, a summability kernel is a family or sequence of periodic integrable functions satisfying a certain set of properties, listed below. Certain kernels, such as the Fejér kernel, are particularly useful in Fourier analysis. Summability kernels are related to app... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Summation equation Summary Summation_equation In mathematics, a summation equation or discrete integral equation is an equation in which an unknown function appears under a summation sign. The theories of summation equations and integral equations can be unified as integral equations on time scales using time scale cal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supervector space Summary Super_linear_algebra In mathematics, a super vector space is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded vector space, that is, a vector space over a field K {\displaystyle \mathbb {K} } with a given decomposition of subspaces of grade 0 {\displaystyle 0} and grade 1 {\displaystyle 1} . The... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superabundant number Summary Superabundant_number In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number n is called superabundant precisely when, for all m < n σ ( m ) m < σ ( n ) n {\displaystyle {\frac {\sigma (m)}{m}}<{\frac {\sigma (n)}{n}}} where... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superabundant number Summary Superabundant_number Superabundant numbers were defined by Leonidas Alaoglu and Paul Erdős (1944). Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed. Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superadditive set function Summary Superadditive_set_function In mathematics, a superadditive set function is a set function whose value when applied to the union of two disjoint sets is greater than or equal to the sum of values of the function applied to each of the sets separately. This definition is analogous to th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supercommutative algebra Summary Supercommutative_ring In mathematics, a supercommutative (associative) algebra is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous elements x, y we have y x = ( − 1 ) | x | | y | x y , {\displaystyle yx=(-1)^{|x||y|}xy,} where |x| denotes the grade of the elem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supercommutative algebra Summary Supercommutative_ring Grassmann algebras (also known as exterior algebras) are the most common examples of nontrivial supercommutative algebras. The supercenter of any superalgebra is the set of elements that supercommute with all elements, and is a supercommutative algebra. The even su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supercommutative algebra Summary Supercommutative_ring That is, even elements always commute. Odd elements, on the other hand, always anticommute. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supercommutative algebra Summary Supercommutative_ring That is, x y + y x = 0 {\displaystyle xy+yx=0\,} for odd x and y. In particular, the square of any odd element x vanishes whenever 2 is invertible: x 2 = 0. {\displaystyle x^{2}=0.} Thus a commutative superalgebra (with 2 invertible and nonzero degree one component... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superellipsoid Summary Superellipsoid In mathematics, a superellipsoid (or super-ellipsoid) is a solid whose horizontal sections are superellipses (Lamé curves) with the same squareness parameter ϵ 2 {\displaystyle \epsilon _{2}} , and whose vertical sections through the center are superellipses with the squareness par... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superellipsoid Summary Superellipsoid The main advantage of describing objects and envirionment with superellipsoids is its conciseness and expressiveness in shape. Furthermore, a closed-form expression of the Minkowski sum between two superellipsoids is available. This makes it a desirable geometric primitive for robo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superelliptic curve Summary Superelliptic_curve In mathematics, a superelliptic curve is an algebraic curve defined by an equation of the form y m = f ( x ) , {\displaystyle y^{m}=f(x),} where m ≥ 2 {\displaystyle m\geq 2} is an integer and f is a polynomial of degree d ≥ 3 {\displaystyle d\geq 3} with coefficients in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superintegrable Hamiltonian system Summary Superintegrable_Hamiltonian_system In mathematics, a superintegrable Hamiltonian system is a Hamiltonian system on a 2 n {\displaystyle 2n} -dimensional symplectic manifold for which the following conditions hold: (i) There exist k > n {\displaystyle k>n} independent integrals... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superintegrable Hamiltonian system Summary Superintegrable_Hamiltonian_system (iii) The matrix function s i j {\displaystyle s_{ij}} is of constant corank m = 2 n − k {\displaystyle m=2n-k} on N {\displaystyle N} . If k = n {\displaystyle k=n} , this is the case of a completely integrable Hamiltonian system. The Mishch... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superintegrable Hamiltonian system Summary Superintegrable_Hamiltonian_system Let invariant submanifolds of a superintegrable Hamiltonian system be connected compact and mutually diffeomorphic. Then the fibered manifold F {\displaystyle F} is a fiber bundle in tori T m {\displaystyle T^{m}} . There exists an open neigh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superintegrable Hamiltonian system Summary Superintegrable_Hamiltonian_system These coordinates are the Darboux coordinates on a symplectic manifold U {\displaystyle U} . A Hamiltonian of a superintegrable system depends only on the action variables I A {\displaystyle I_{A}} which are the Casimir functions of the coind... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superior highly composite number Summary Superior_highly_composite_number In mathematics, a superior highly composite number is a natural number which, in a particular rigorous sense, has many divisors. Particularly, it's defined by a ratio between the number of divisors an integer has and that integer raised to some p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superior highly composite number Summary Superior_highly_composite_number The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number n there exists a positive real number ε such that for all natural numbers k smaller than n we have and for all natural numbe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supermodule Summary Free_supermodule In mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework for studying the concept supersymmetry in theoretical physics. Supermodules over a commutative superalgebra can be viewe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supermodule Summary Free_supermodule These reason is that it is often necessary or useful to extend the field of scalars to include odd variables. In doing so one moves from fields to commutative superalgebras and from vector spaces to modules. In this article, all superalgebras are assumed be associative and unital un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superparticular number Summary Superparticular_number In mathematics, a superparticular ratio, also called a superparticular number or epimoric ratio, is the ratio of two consecutive integer numbers. More particularly, the ratio takes the form: n + 1 n = 1 + 1 n {\displaystyle {\frac {n+1}{n}}=1+{\frac {1}{n}}} where n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superparticular number Summary Superparticular_number When 3 and 4 are compared, they each contain a 3, and the 4 has another 1, which is a third part of 3. Again, when 5, and 4 are compared, they contain the number 4, and the 5 has another 1, which is the fourth part of the number 4, etc. Superparticular ratios were w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superpartient ratio Summary Superpartient_ratio In mathematics, a superpartient ratio, also called superpartient number or epimeric ratio, is a rational number that is greater than one and is not superparticular. The term has fallen out of use in modern pure mathematics, but continues to be used in music theory and in ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Superperfect number Summary Superperfect_number In mathematics, a superperfect number is a positive integer n that satisfies σ 2 ( n ) = σ ( σ ( n ) ) = 2 n , {\displaystyle \sigma ^{2}(n)=\sigma (\sigma (n))=2n\,,} where σ is the divisor summatory function. Superperfect numbers are not a generalization of perfect numb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersingular Abelian variety Summary Supersingular_variety In mathematics, a supersingular variety is (usually) a smooth projective variety in nonzero characteristic such that for all n the slopes of the Newton polygon of the nth crystalline cohomology are all n/2 (de Jong 2014). For special classes of varieties such ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersingular Abelian variety Summary Supersingular_variety Supersingular elliptic curves can also be characterized by the slopes of their crystalline cohomology, and the term "supersingular" was later extended to other varieties whose cohomology has similar properties. The terms "supersingular" or "singular" do not me... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersingular Abelian variety Summary Supersingular_variety Elliptic curves in non-zero characteristic with an unusually large ring of endomorphisms of rank 4. Supersingular Abelian variety Sometimes defined to be an abelian variety isogenous to a product of supersingular elliptic curves, and sometimes defined to be an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersingular Abelian variety Summary Supersingular_variety Certain K3 surfaces in non-zero characteristic. Supersingular Enriques surface. Certain Enriques surfaces in characteristic 2. A surface is called Shioda supersingular if the rank of its Néron–Severi group is equal to its second Betti number. A surface is call... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Supersolvable arrangement Summary Supersolvable_arrangement In mathematics, a supersolvable arrangement is a hyperplane arrangement which has a maximal flag with only modular elements. Equivalently, the intersection semilattice of the arrangement is a supersolvable lattice, in the sense of Richard P. Stanley. As shown ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface bundle Summary Surface_bundle In mathematics, a surface bundle is a bundle in which the fiber is a surface. When the base space is a circle the total space is three-dimensional and is often called a surface bundle over the circle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface bundle over the circle Summary Surface_bundle_over_the_circle In mathematics, a surface bundle over the circle is a fiber bundle with base space a circle, and with fiber space a surface. Therefore the total space has dimension 2 + 1 = 3. In general, fiber bundles over the circle are a special case of mapping to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface bundle over the circle Summary Surface_bundle_over_the_circle Here is the construction: take the Cartesian product of a surface with the unit interval. Glue the two copies of the surface, on the boundary, by some homeomorphism. This homeomorphism is called the monodromy of the surface bundle. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface bundle over the circle Summary Surface_bundle_over_the_circle It is possible to show that the homeomorphism type of the bundle obtained depends only on the conjugacy class, in the mapping class group, of the gluing homeomorphism chosen. This construction is an important source of examples both in the field of l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface bundle over the circle Summary Surface_bundle_over_the_circle This is the fibered part of William Thurston's geometrization theorem for Haken manifolds, whose proof requires the Nielsen–Thurston classification for surface homeomorphisms as well as deep results in the theory of Kleinian groups. In geometric grou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Topological surface In general Closed_surface > In general In mathematics, a surface is a geometrical shape that resembles a deformed plane. The most familiar examples arise as boundaries of solid objects in ordinary three-dimensional Euclidean space R3, such as spheres. The exact definition of a surface may depend on ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Topological surface In general Closed_surface > In general A surface is a two-dimensional space; this means that a moving point on a surface may move in two directions (it has two degrees of freedom). In other words, around almost every point, there is a coordinate patch on which a two-dimensional coordinate system is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface (geometry) Summary Surface_(mathematics) In mathematics, a surface is a mathematical model of the common concept of a surface. It is a generalization of a plane, but, unlike a plane, it may be curved; this is analogous to a curve generalizing a straight line. There are several more precise definitions, dependin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surface (geometry) Summary Surface_(mathematics) The exact definition of a surface may depend on the context. Typically, in algebraic geometry, a surface may cross itself (and may have other singularities), while, in topology and differential geometry, it may not. A surface is a topological space of dimension two; this... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surjective map Summary Surjective_function In mathematics, a surjective function (also known as surjection, or onto function ) is a function f such that every element y can be mapped from some element x such that f(x) = y. In other words, every element of the function's codomain is the image of at least one element of ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surjective map Summary Surjective_function Every surjective function has a right inverse assuming the axiom of choice, and every function with a right inverse is necessarily a surjection. The composition of surjective functions is always surjective. Any function can be decomposed into a surjection and an injection. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Surjunctive group Summary Surjunctive_group In mathematics, a surjunctive group is a group such that every injective cellular automaton with the group elements as its cells is also surjective. Surjunctive groups were introduced by Gottschalk (1973). It is unknown whether every group is surjunctive. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symbolic language (mathematics) Summary Symbolic_language_(mathematics) In mathematics, a symbolic language is a language that uses characters or symbols to represent concepts, such as mathematical operations, expressions, and statements, and the entities or operands on which the operations are performed. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric Boolean function Summary Symmetric_Boolean_function In mathematics, a symmetric Boolean function is a Boolean function whose value does not depend on the order of its input bits, i.e., it depends only on the number of ones (or zeros) in the input. For this reason they are also known as Boolean counting functi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric Boolean function Summary Symmetric_Boolean_function . . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric bilinear form Summary Symmetric_bilinear_form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric bilinear form Summary Symmetric_bilinear_form Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for V. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple ki... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices In mathematics, a symmetric matrix M {\displaystyle M} with real entries is positive-definite if the real number z T M z {\displaystyle z^{\textsf {T}}Mz} is positive for every nonzero real column vector z , {\displaystyle z,} where z T {\displaystyle z^... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite. A matrix is thus positive-definite if and only if it is the matrix o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices In other words, a matrix is positive-definite if and only if it defines an inner product. Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathemat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices A matrix M is positive-definite if and only if it satisfies any of the following equivalent conditions. M is congruent with a diagonal matrix with positive real entries. M is symmetric or Hermitian, and all its eigenvalues are real and positive. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices M is symmetric or Hermitian, and all its leading principal minors are positive. There exists an invertible matrix B {\displaystyle B} with conjugate transpose B ∗ {\displaystyle B^{*}} such that M = B ∗ B . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Positive semidefinite matrices Summary Positive-definite_matrices {\displaystyle M=B^{*}B.} A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed. Positive-definite an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Monomial symmetric polynomial Summary Symmetric_polynomial In mathematics, a symmetric polynomial is a polynomial P(X1, X2, …, Xn) in n variables, such that if any of the variables are interchanged, one obtains the same polynomial. Formally, P is a symmetric polynomial if for any permutation σ of the subscripts 1, 2, .... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Monomial symmetric polynomial Summary Symmetric_polynomial Indeed, a theorem called the fundamental theorem of symmetric polynomials states that any symmetric polynomial can be expressed in terms of elementary symmetric polynomials. This implies that every symmetric polynomial expression in the roots of a monic polynom... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Monomial symmetric polynomial Summary Symmetric_polynomial Symmetric polynomials also form an interesting structure by themselves, independently of any relation to the roots of a polynomial. In this context other collections of specific symmetric polynomials, such as complete homogeneous, power sum, and Schur polynomia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemann symmetric space Summary Riemann_symmetric_space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, leading to consequen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemann symmetric space Summary Riemann_symmetric_space More generally, a Riemannian manifold (M, g) is said to be symmetric if and only if, for each point p of M, there exists an isometry of M fixing p and acting on the tangent space T p M {\displaystyle T_{p}M} as minus the identity (every symmetric space is complete... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Riemann symmetric space Summary Riemann_symmetric_space Riemannian symmetric spaces arise in a wide variety of situations in both mathematics and physics. Their central role in the theory of holonomy was discovered by Marcel Berger. They are important objects of study in representation theory and harmonic analysis as w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetric tensor Summary Symmetric_tensor In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\ldots ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\ldots ,v_{\sigma r})} for every per... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetry in mathematics Symmetric tensors Symmetry_in_mathematics > Symmetry in abstract algebra > Symmetric tensors In mathematics, a symmetric tensor is tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T ( v σ 1 , v σ 2 , … , v σ r ) {\displaystyle T(v_{1},v_{2},\dots ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symmetrizable compact operator Summary Symmetrizable_compact_operator In mathematics, a symmetrizable compact operator is a compact operator on a Hilbert space that can be composed with a positive operator with trivial kernel to produce a self-adjoint operator. Such operators arose naturally in the work on integral ope... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symplectic integrator Summary Symplectic_integrator In mathematics, a symplectic integrator (SI) is a numerical integration scheme for Hamiltonian systems. Symplectic integrators form the subclass of geometric integrators which, by definition, are canonical transformations. They are widely used in nonlinear dynamics, m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Symplectic matrix Summary Symplectic_matrix In mathematics, a symplectic matrix is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix M {\displaystyle M} with real entries that satisfies the condition where M T {\displaystyle M^{\text{T}}} denotes the transpose of M {\displaystyle M} and Ω {\displaystyle \Omega } is a fixe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lagrangian subspace Summary Symplectic_product In mathematics, a symplectic vector space is a vector space V over a field F (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping ω: V × V → F that is Bilinear Linear in each argument separately; Alternating ω(v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lagrangian subspace Summary Symplectic_product Working in a fixed basis, ω can be represented by a matrix. The conditions above are equivalent to this matrix being skew-symmetric, nonsingular, and hollow (all diagonal entries are zero). This should not be confused with a symplectic matrix, which represents a symplectic... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Lagrangian subspace Summary Symplectic_product If V is finite-dimensional, then its dimension must necessarily be even since every skew-symmetric, hollow matrix of odd size has determinant zero. Notice that the condition that the matrix be hollow is not redundant if the characteristic of the field is 2. A symplectic fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Hamiltonian isotopy Summary Hamiltonian_isotopy In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds. In classical mechanics, a symplectomorphism represents a transformation of phase space that is volume-preserving and preserves the symplectic structure of phas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Syndetic set Summary Syndetic_set In mathematics, a syndetic set is a subset of the natural numbers having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural numbers is bounded. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
System of bilinear equations Summary System_of_bilinear_equations In mathematics, a system of bilinear equations is a special sort of system of polynomial equations, where each equation equates a bilinear form with a constant (possibly zero). More precisely, given two sets of variables represented as coordinate vectors... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
System of differential equations Summary Differential_system In mathematics, a system of differential equations is a finite set of differential equations. Such a system can be either linear or non-linear. Also, such a system can be either a system of ordinary differential equations or a system of partial differential e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Over-determined system Summary Over-determined_system In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns. An overdetermined system is almost always inconsistent (it has no solution) when constructed with random coefficients. However, an overdetermined system wil... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Over-determined system Summary Over-determined_system Each unknown can be seen as an available degree of freedom. Each equation introduced into the system can be viewed as a constraint that restricts one degree of freedom. Therefore, the critical case occurs when the number of equations and the number of free variables... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Over-determined system Summary Over-determined_system For every variable giving a degree of freedom, there exists a corresponding constraint. The overdetermined case occurs when the system has been overconstrained — that is, when the equations outnumber the unknowns. In contrast, the underdetermined case occurs when th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous equation Summary Homogeneous_system_of_linear_equations In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variables. For example, { 3 x + 2 y − z = 1 2 x − 2 y + 4 z = − 2 − x + 1 2 y − z = 0 {\displaystyle {\begin{cases}3x+2y-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Homogeneous equation Summary Homogeneous_system_of_linear_equations In mathematics, the theory of linear systems is the basis and a fundamental part of linear algebra, a subject used in most modern mathematics. Computational algorithms for finding the solutions are an important part of numerical linear algebra, and pla... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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