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wiki_3400_chunk_0 | Strictly convex space | In mathematics, a strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex space is one for which, given any two distinct points x and y on the unit sphere ∂B (i.e. the boundary of the unit ball B of X), the segment joining x a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3401_chunk_0 | Strong prime | In mathematics, a strong prime is a prime number with certain special properties. The definitions of strong primes are different in cryptography and number theory. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3402_chunk_0 | Strong topology | In mathematics, a strong topology is a topology which is stronger than some other "default" topology. This term is used to describe different topologies depending on context, and it may refer to: the final topology on the disjoint union the topology arising from a norm the strong operator topology the strong topology (... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3403_chunk_0 | Stuck unknot | In mathematics, a stuck unknot is a closed polygonal chain in three-dimensional space (a skew polygon) that is topologically equal to the unknot but cannot be deformed to a simple polygon when interpreted as a mechanical linkage, by rigid length-preserving and non-self-intersecting motions of its segments. Similarly a ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3404_chunk_0 | Stunted projective space | In this way the vector fields on spheres question was reduced to a question on stunted projective spaces: for RPn,m, is there a degree one mapping on the 'next cell up' (of the first dimension not collapsed in the 'stunting') that extends to the whole space? Frank Adams showed that this could not happen, completing the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3405_chunk_0 | Sub-Riemannian manifold | The Hausdorff dimension of such metric spaces is always an integer and larger than its topological dimension (unless it is actually a Riemannian manifold). Sub-Riemannian manifolds often occur in the study of constrained systems in classical mechanics, such as the motion of vehicles on a surface, the motion of robot ar... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3406_chunk_0 | Subadditive utility | In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function on each of the sets. This is thematically related to the subadditivity property of real-valued functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3407_chunk_0 | Subalgebra | In mathematics, a subalgebra is a subset of an algebra, closed under all its operations, and carrying the induced operations. "Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a comm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3408_chunk_0 | 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 , {\displaystyle X,} that make up a v... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3409_chunk_0 | 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. For current inner models, the elementary embedding... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3410_chunk_0 | 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). The natural diminishing returns property which m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3411_chunk_0 | 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. In image tracing context an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3412_chunk_0 | 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 the existence of a multiplicative identity, a subring o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3413_chunk_0 | 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 |
wiki_3414_chunk_0 | 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 supremum of the set of all subsequential limits of som... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3415_chunk_0 | 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,} then the sequence also converges to x . {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3416_chunk_0 | Absolutely convex set | 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 set is the intersection of all disks contain... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3417_chunk_0 | 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 can be formed from a complete residue system modulo n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3418_chunk_0 | 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 nowhere dense among the reals, whereas the inte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3419_chunk_0 | 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 calculus. A summation equation compares to a diff... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3420_chunk_0 | Supervector space | 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 study of super vector spaces and their general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3421_chunk_0 | 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 σ denotes the sum-of-divisors function (i.e., the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3422_chunk_0 | 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 the notion of superadditivity for real-valued functions. It is c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3423_chunk_0 | Supercommutative algebra | 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 element and is 0 or 1 (in Z2) according to whether the grad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3424_chunk_0 | Supercommutative algebra | 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 subalgebra of a supercommutative algebra is always a comm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3425_chunk_0 | Supercommutative algebra | That is, even elements always commute. Odd elements, on the other hand, always anticommute. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3426_chunk_0 | Supercommutative algebra | 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) always contains nilpotent elements. A Z-graded antico... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3427_chunk_0 | 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 parameter ϵ 1 {\displaystyle \epsilon _{1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3428_chunk_0 | 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 a field k {\displaystyle k} ; more precisely, it... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3429_chunk_0 | 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 Mishchenko-Fomenko theorem for superintegrable Hamiltonian systems generalizes the L... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3430_chunk_0 | 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 coinduced Poisson structure on F ( U ) {\displaystyle F(U)} . The Liouville-Arnold ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3431_chunk_0 | 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 numbers k larger than n we have where d(n), the divisor function, denotes the n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3432_chunk_0 | 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 viewed as generalizations of super vector ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3433_chunk_0 | 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 unless stated otherwise. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3434_chunk_0 | 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 numbers, but have a common generalization. The term ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3435_chunk_0 | Supersingular Abelian 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 mean that the variety has singularities. Examples include: Sup... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3436_chunk_0 | 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 by Hiroaki Terao, a complex hyperplane arrangement is supers... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3437_chunk_0 | 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 |
wiki_3438_chunk_0 | 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 tori. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3439_chunk_0 | 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 low-dimensional topology as well as in geometric group theory. In the f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3440_chunk_0 | 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 group theory the fundamental groups of such bundles give an important clas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3441_chunk_0 | Topological surface | 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 the context. Typically, in algebraic geometry, a surface ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3442_chunk_0 | Surface (geometry) | 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, depending on the context and the mathematical tools that ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3443_chunk_0 | Surface (geometry) | 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 means that a moving point on a surface may move ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3444_chunk_0 | Surjective map | 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 its domain. It is not required that x be un... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3445_chunk_0 | Surjective map | 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 |
wiki_3446_chunk_0 | 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 functions.There are 2n+1 symmetric n-ary Boolean functions. Instead ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3447_chunk_0 | 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 |
wiki_3448_chunk_0 | 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 {\displaystyle B} that maps every pair ( u , v ) {\displaysty... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3449_chunk_0 | 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 kind of basis known as an orthogonal basis (at least when ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3450_chunk_0 | Positive semidefinite 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^{\textsf {T}}} is the transpose of z {\displaystyle z} . More gene... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3451_chunk_0 | Positive semidefinite 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 of a positive-definite quadratic form or Hermitian form. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3452_chunk_0 | Positive semidefinite 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 mathematics. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3453_chunk_0 | Positive semidefinite 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 |
wiki_3454_chunk_0 | Positive semidefinite 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 |
wiki_3455_chunk_0 | Positive semidefinite 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 and positive-semidefinite real matrices are at the basis of convex o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3456_chunk_0 | Monomial 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, ..., n one has P(Xσ(1), Xσ(2), …, Xσ(n)) = P(X1, X2, …, Xn).... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3457_chunk_0 | Monomial 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 polynomial can alternatively be given as a polynomial expression i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3458_chunk_0 | Monomial 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 polynomials play important roles alongside the elementary ones. The ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3459_chunk_0 | 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 consequences in the theory of holonomy; or algebraically through ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3460_chunk_0 | 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 well as in differential geometry. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3461_chunk_0 | 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 permutation σ of the symbols {1, 2, ..., r}. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3462_chunk_0 | Symmetry in mathematics | 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 ,v_{r})=T(v_{\sigma 1},v_{\sigma 2},\dots ,v_{\sigma r})} for every permutation σ of the symbols {1,2,...,r}. Alterna... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3463_chunk_0 | 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 operators of Hilbert, Korn, Lichtenstein and Marty required to solve elli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3464_chunk_0 | 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, molecular dynamics, discrete element methods, acceler... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3465_chunk_0 | 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 fixed 2 n × 2 n {\displaystyle 2n\times 2n} nons... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3466_chunk_0 | Lagrangian subspace | 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, v) = 0 holds for all v ∈ V; and Non-degenerat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3467_chunk_0 | Lagrangian subspace | 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 transformation of the space. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3468_chunk_0 | Lagrangian subspace | 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 form behaves quite differently from a symmetric f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3469_chunk_0 | 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 phase space, and is called a canonical transformatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3470_chunk_0 | 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 x and y, then each equation of the system can be written where, i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3471_chunk_0 | 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 will have solutions in some cases, for example if some eq... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3472_chunk_0 | 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 are equal. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3473_chunk_0 | 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 the system has been underconstrained — that is, when the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3474_chunk_0 | Homogeneous equation | 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-z=1\\2x-2y+4z=-2\\-x+{\frac {1}{2}}y-z=0\end{cases}}} is a system of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3475_chunk_0 | Homogeneous equation | 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 play a prominent role in engineering, physics, chemistry, computer scie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3476_chunk_0 | Homogeneous equation | Very often, and in this article, the coefficients of the equations are real or complex numbers and the solutions are searched in the same set of numbers, but the theory and the algorithms apply for coefficients and solutions in any field. For solutions in an integral domain like the ring of the integers, or in other al... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3477_chunk_0 | Underdetermined system | In mathematics, a system of linear equations or a system of polynomial equations is considered underdetermined if there are fewer equations than unknowns (in contrast to an overdetermined system, where there are more equations than unknowns). The terminology can be explained using the concept of constraint counting. Ea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3478_chunk_0 | Underdetermined system | Therefore, the critical case (between overdetermined and underdetermined) occurs when the number of equations and the number of free variables are equal. For every variable giving a degree of freedom, there exists a corresponding constraint removing a degree of freedom. The underdetermined case, by contrast, occurs whe... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3479_chunk_0 | T-norm | In mathematics, a t-norm (also T-norm or, unabbreviated, triangular norm) is a kind of binary operation used in the framework of probabilistic metric spaces and in multi-valued logic, specifically in fuzzy logic. A t-norm generalizes intersection in a lattice and conjunction in logic. The name triangular norm refers to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3480_chunk_0 | Tangent Lie group | In mathematics, a tangent Lie group is a Lie group whose underlying space is the tangent bundle TG of a Lie group G. As a Lie group, the tangent bundle is a semidirect product of a normal abelian subgroup with underlying space the Lie algebra of G, and G itself. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3481_chunk_0 | Tangent vector | In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3482_chunk_0 | Tempered representation | In mathematics, a tempered representation of a linear semisimple Lie group is a representation that has a basis whose matrix coefficients lie in the Lp space L2+ε(G)for any ε > 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3483_chunk_0 | Tensor order | In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3484_chunk_0 | Tensor order | Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics (stress, elasticity, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3485_chunk_0 | Tensor order | This leads to the concept of a tensor field. In some areas, tensor fields are so ubiquitous that they are often simply called "tensors". Tullio Levi-Civita and Gregorio Ricci-Curbastro popularised tensors in 1900 – continuing the earlier work of Bernhard Riemann, Elwin Bruno Christoffel, and others – as part of the abs... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3486_chunk_0 | Ternary equivalence relation | In mathematics, a ternary equivalence relation is a kind of ternary relation analogous to a binary equivalence relation. A ternary equivalence relation is symmetric, reflexive, and transitive. The classic example is the relation of collinearity among three points in Euclidean space. In an abstract set, a ternary equiva... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3487_chunk_0 | Triadic relation | In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place. Just as a binary relation is formally defined as a set of pairs, i.e. a subset of the Cartesian pro... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3488_chunk_0 | Formal theorem | In mathematics, a theorem is a statement that has been proved, or can be proved. The proof of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3489_chunk_0 | Formal theorem | Moreover, many authors qualify as theorems only the most important results, and use the terms lemma, proposition and corollary for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formalized in order to allow mathematical reasoning about them. In this context, statements bec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3490_chunk_0 | Formal theorem | A theory consists of some basis statements called axioms, and some deducing rules (sometimes included in the axioms). The theorems of the theory are the statements that can be derived from the axioms by using the deducing rules. This formalization led to proof theory, which allows proving general theorems about theorem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3491_chunk_0 | Theta characteristic | In mathematics, a theta characteristic of a non-singular algebraic curve C is a divisor class Θ such that 2Θ is the canonical class. In terms of holomorphic line bundles L on a connected compact Riemann surface, it is therefore L such that L2 is the canonical bundle, here also equivalently the holomorphic cotangent bun... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3492_chunk_0 | Theta constant | In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction θm(τ) = θm(τ,0) of a theta function θm(τ,z) with rational characteristic m to z = 0. The variable τ may be a complex number in the upper half-plane in which case the theta constants are modular for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3493_chunk_0 | Thin set (Serre) | In mathematics, a thin set in the sense of Serre, named after Jean-Pierre Serre, is a certain kind of subset constructed in algebraic geometry over a given field K, by allowed operations that are in a definite sense 'unlikely'. The two fundamental ones are: solving a polynomial equation that may or may not be the case;... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3494_chunk_0 | Univariate time series | In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3495_chunk_0 | Univariate time series | Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, ma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3496_chunk_0 | Univariate time series | Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3497_chunk_0 | Univariate time series | Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3498_chunk_0 | Univariate time series | A stochastic model for a time series will generally reflect the fact that observations close together in time will be more closely related than observations further apart. In addition, time series models will often make use of the natural one-way ordering of time so that values for a given period will be expressed as d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_3499_chunk_0 | Stereotype algebra | In mathematics, a topological algebra A {\displaystyle A} is an algebra and at the same time a topological space, where the algebraic and the topological structures are coherent in a specified sense. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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