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c_0mb83va1st2l | 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... | Subbundle |
c_tfm1tgnw53uj | 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 hereditarily less than κ+) with critical point μ... | Subcompact cardinal |
c_g03gokmc1sgo | 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 superstrong cardinals. | Subcompact cardinal |
c_y3wxid3404ah | 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 inner models at the level of subcompact cardinals s... | Subcompact cardinal |
c_uj7k495apkbz | (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... | Subcompact cardinal |
c_hv0wjlwawcfj | 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 often have different definitions. | Slice chart |
c_lxntz2fjftb3 | 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... | Submodular valuation |
c_m2zms828ss10 | 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 rectangles", when it has no holes... | Subpaving |
c_whkurv3j9w8b | 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... | Subpaving |
c_jjburwoezgxr | 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... | Algebra of dual numbers |
c_za5a5800iw0z | 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 of ⟨ A , B , C , D , E , F ⟩ {\... | Subsequence |
c_4vv2qq74imkm | 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 ⟩ , {\displaystyle \langle A,B,C,D,E,F\... | Subsequence |
c_ts9o9jlnvvim | 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. | Subsequential limit |
c_6s39fu10rufd | 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... | Subsequential limit |
c_olc1qstd8cs4 | 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... | Subsequential limit |
c_zs0ziz6psnue | 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} . In particular, a universally measurable set of reals is nece... | Universally measurable set |
c_xdehenow5335 | 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 ∈ , {\displaystyle t\in ,} a 0 + t x ∈ A . ... | Radial set |
c_h56j1itxieip | 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 {\displaystyle B} that is "larger than a {\... | Cofinal sequence |
c_ic0um0uat3wo | 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... | Absolutely convex set |
c_bovkl6o1kqgh | 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... | Reduced residue system |
c_t69fhkday2ry | 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, but not under subtraction: 1 − 2 is not a natural number, although both 1 and ... | Reflexive transitive symmetric closure |
c_bjbr7b72jfvu | 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 span) or the generated set. | Reflexive transitive symmetric closure |
c_1vfqoodep9vx | 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... | Nowhere-dense set |
c_6kq6aongy1y2 | 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. This differs from a sum-free set, where on... | Sum-free sequence |
c_yj67wqtwcryu | 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 approximation of the identity; definitions of an ... | Summability kernel |
c_wfews9ndzl6u | 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... | Summation equation |
c_5yuqi40vyil5 | 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... | Supervector space |
c_qvfp2wtzxm18 | 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... | Superabundant number |
c_wy10g4iottlw | 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. In section 59 of that paper, Ramanujan defines g... | Superabundant number |
c_eoa4qyxo41z1 | 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... | Superadditive set function |
c_asirjoyz0uzj | 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... | Supercommutative algebra |
c_732d2u24gbat | 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... | Supercommutative algebra |
c_j094yrvghgdm | That is, even elements always commute. Odd elements, on the other hand, always anticommute. | Supercommutative algebra |
c_az3rzqblkt0z | 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... | Supercommutative algebra |
c_ybuw70qu1g4q | 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... | Superellipsoid |
c_o9keql2kwhvk | 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 robot grasping, collision detection, and m... | Superellipsoid |
c_cf0ssjqqzxqf | 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... | Superelliptic curve |
c_53mwqch5y6nq | 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 F i {\displaystyle F_{i}} of motion. Their level surfaces (invariant submanif... | Superintegrable Hamiltonian system |
c_9qkbl77f0kob | (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... | Superintegrable Hamiltonian system |
c_yzkth2go28vc | 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 neighbourhood U {\displaystyle U} of F {\displaystyle F} which is a trivial fiber b... | Superintegrable Hamiltonian system |
c_2rgxbvrqjevw | 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 ... | Superintegrable Hamiltonian system |
c_rtdmhymfoo47 | 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 positive power. For any possible exponent, whichever integer has the highes... | Superior highly composite number |
c_p2n5ghplc2tm | 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... | Superior highly composite number |
c_l9sh3x7sanq2 | 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 ... | Supermodule |
c_suvjeaj5mnjm | 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. | Supermodule |
c_ycbpby8vnxfm | 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 is a positive integer.Thus: A superparticular number ... | Superparticular number |
c_pe7xetio5ynr | 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 written about by Nicomachus in his treatise Introductio... | Superparticular number |
c_01g6htiazn6l | 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 the historical study of mathematics. Superpartie... | Superpartient ratio |
c_si10mefgwr7a | 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 ... | Superperfect number |
c_lqe8xpzq9avn | 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 as elliptic curves it is common to use various ad hoc defini... | Supersingular Abelian variety |
c_ee8f95qdb7c6 | 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... | Supersingular Abelian variety |
c_pj0ue4b34e2f | 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 abelian variety of some rank g whose endomorphism ring has ... | Supersingular Abelian variety |
c_edhxx2vjykkb | 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 called Artin supersingular if its formal Brauer group has infini... | Supersingular Abelian variety |
c_yzclqb7k0ed0 | 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... | Supersolvable arrangement |
c_mj55e2n85ijh | 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. | Surface bundle |
c_6tw30rbmkvar | 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. | Surface bundle over the circle |
c_kwixyl13rarp | 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. | Surface bundle over the circle |
c_8jz0fz7u1xnx | 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... | Surface bundle over the circle |
c_yc3ft5saumkd | 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... | Surface bundle over the circle |
c_x0bnz3ehmncr | 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... | Topological surface |
c_4s15gboiwdxd | 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 defined. For example, the surface of the Earth resembles (i... | Topological surface |
c_tgpf1t23yadp | 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 ... | Surface (geometry) |
c_3dittdiczqzk | 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 ... | Surface (geometry) |
c_rjpof6qvus18 | 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... | Surjective map |
c_jlc2nggphhzh | 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. | Surjective map |
c_zae9n2apgxfh | 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. | Surjunctive group |
c_euexm6x9cenn | 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. | Symbolic language (mathematics) |
c_n0y6yzi3nv1i | 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 ... | Symmetric Boolean function |
c_hr03nzqoe6oz | . . , n } → { 0 , 1 } {\displaystyle f:\{0,1,...,n\}\rightarrow \{0,1\}} . Symmetric Boolean functions are used to classify Boolean satisfiability problems. | Symmetric Boolean function |
c_4x6bzxrufdwd | 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... | Symmetric bilinear form |
c_0l50v6en4hzj | 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 ... | Symmetric bilinear form |
c_rlxff7emlm4h | 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... | Positive semidefinite matrices |
c_d1xk8ci0pus8 | 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. | Positive semidefinite matrices |
c_rkc36anlumuq | 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. | Positive semidefinite matrices |
c_p3la7q93hyb2 | 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. | Positive semidefinite matrices |
c_23jl1m78y1j8 | 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 . | Positive semidefinite matrices |
c_b9lw2jb7veyj | {\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... | Positive semidefinite matrices |
c_rlnv1nlcz648 | 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).... | Monomial symmetric polynomial |
c_jlbcjipi9p09 | 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... | Monomial symmetric polynomial |
c_hkf6pnj0xkeh | 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 ... | Monomial symmetric polynomial |
c_ggi9v618y4o1 | 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 ... | Riemann symmetric space |
c_yuo82da046vx | 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, since any geodesic can be extended indefinitely via sy... | Riemann symmetric space |
c_on56pmd4gsez | 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. | Riemann symmetric space |
c_2v3zaa98jeqx | 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}. ... | Symmetric tensor |
c_0azzqpc0q8c1 | 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... | Symmetry in mathematics |
c_pdk64sj9p058 | 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... | Symmetrizable compact operator |
c_lm08f7nw9aru | 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... | Symplectic integrator |
c_gtyhp5kb60wy | 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... | Symplectic matrix |
c_3xa5spfh7jjv | 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... | Lagrangian subspace |
c_yqrk80gv34bf | 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. | Lagrangian subspace |
c_69bc3tjde900 | 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... | Lagrangian subspace |
c_3oql4hnj7eh6 | 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... | Hamiltonian isotopy |
c_jyc56hdfpfqq | 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. | Syndetic set |
c_j4oessdlaeb2 | 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... | System of bilinear equations |
c_2dw3aie15bm9 | 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 equations. | System of differential equations |
c_nrw5mh0r837u | 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... | Over-determined system |
c_7dm04ko6a8d4 | 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. | Over-determined system |
c_92j36jcf44zj | 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... | Over-determined system |
c_ljerlu2t5bfk | 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... | Homogeneous equation |
c_svolj6p01nbe | 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... | Homogeneous equation |
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