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In mathematics, specifically in order theory and functional analysis, if C {\displaystyle C} is a cone at the origin in a topological vector space X {\displaystyle X} such that 0 ∈ C {\displaystyle 0\in C} and if U {\displaystyle {\mathcal {U}}} is the neighborhood filter at the origin, then C {\displaystyle C} is call... | https://en.wikipedia.org/wiki/Normal_cone_(functional_analysis) |
In mathematics, specifically in order theory and functional analysis, the order bound dual of an ordered vector space X {\displaystyle X} is the set of all linear functionals on X {\displaystyle X} that map order intervals, which are sets of the form := { x ∈ X: a ≤ x and x ≤ b } , {\displaystyle :=\{x\in X:a\leq x{\t... | https://en.wikipedia.org/wiki/Order_bound_dual |
In mathematics, specifically in order theory and functional analysis, the order dual of an ordered vector space X {\displaystyle X} is the set Pos ( X ∗ ) − Pos ( X ∗ ) {\displaystyle \operatorname {Pos} \left(X^{*}\right)-\operatorname {Pos} \left(X^{*}\right)} where Pos ( X ∗ ) {\displaystyle \operatorname {Pos... | https://en.wikipedia.org/wiki/Order_dual_(functional_analysis) |
In mathematics, specifically in order theory and functional analysis, the order topology of an ordered vector space ( X , ≤ ) {\displaystyle (X,\leq )} is the finest locally convex topological vector space (TVS) topology on X {\displaystyle X} for which every order interval is bounded, where an order interval in X {\di... | https://en.wikipedia.org/wiki/Order_topology_(functional_analysis) |
This allows for establishing intimate connections between this topology and the algebraic and order theoretic properties of ( X , ≤ ) . {\displaystyle (X,\leq ).} For many ordered topological vector spaces that occur in analysis, their topologies are identical to the order topology. | https://en.wikipedia.org/wiki/Order_topology_(functional_analysis) |
In mathematics, specifically in order theory and functional analysis, two elements x and y of a vector lattice X are lattice disjoint or simply disjoint if inf { | x | , | y | } = 0 {\displaystyle \inf \left\{|x|,|y|\right\}=0} , in which case we write x ⊥ y {\displaystyle x\perp y} , where the absolute value of x is d... | https://en.wikipedia.org/wiki/Lattice_disjoint |
In mathematics, specifically in order theory, a binary relation ≤ {\displaystyle \,\leq \,} on a vector space X {\displaystyle X} over the real or complex numbers is called Archimedean if for all x ∈ X , {\displaystyle x\in X,} whenever there exists some y ∈ X {\displaystyle y\in X} such that n x ≤ y {\displaystyle nx\... | https://en.wikipedia.org/wiki/Archimedean_ordered_vector_space |
In mathematics, specifically in real analysis, the Bolzano–Weierstrass theorem, named after Bernard Bolzano and Karl Weierstrass, is a fundamental result about convergence in a finite-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . The theorem states that each infinite bounded sequence in R n {\displ... | https://en.wikipedia.org/wiki/Bolzano-Weierstrass_theorem |
In mathematics, specifically in representation theory, a Borel subalgebra of a Lie algebra g {\displaystyle {\mathfrak {g}}} is a maximal solvable subalgebra. The notion is named after Armand Borel. If the Lie algebra g {\displaystyle {\mathfrak {g}}} is the Lie algebra of a complex Lie group, then a Borel subalgebra i... | https://en.wikipedia.org/wiki/Borel_subalgebra |
In mathematics, specifically in representation theory, a semisimple representation (also called a completely reducible representation) is a linear representation of a group or an algebra that is a direct sum of simple representations (also called irreducible representations). It is an example of the general mathematica... | https://en.wikipedia.org/wiki/Isotypic_decomposition |
In mathematics, specifically in representation theory, the Frobenius formula, introduced by G. Frobenius, computes the characters of irreducible representations of the symmetric group Sn. Among the other applications, the formula can be used to derive the hook length formula. | https://en.wikipedia.org/wiki/Frobenius_formula |
In mathematics, specifically in ring theory, a nilpotent algebra over a commutative ring is an algebra over a commutative ring, in which for some positive integer n every product containing at least n elements of the algebra is zero. The concept of a nilpotent Lie algebra has a different definition, which depends upon ... | https://en.wikipedia.org/wiki/Nilpotent_algebra |
In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. A torsion module is a module that equals its torsion submodule. A module is t... | https://en.wikipedia.org/wiki/Torsion_point |
This terminology is more commonly used for modules over a domain, that is, when the regular elements of the ring are all its nonzero elements. This terminology applies to abelian groups (with "module" and "submodule" replaced by "group" and "subgroup"). This is allowed by the fact that the abelian groups are the module... | https://en.wikipedia.org/wiki/Torsion_point |
In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M. Simple modules form build... | https://en.wikipedia.org/wiki/Simple_module |
In mathematics, specifically in spectral theory, an eigenvalue of a closed linear operator is called normal if the space admits a decomposition into a direct sum of a finite-dimensional generalized eigenspace and an invariant subspace where A − λ I {\displaystyle A-\lambda I} has a bounded inverse. The set of normal ei... | https://en.wikipedia.org/wiki/Normal_eigenvalue |
In mathematics, specifically in surgery theory, the surgery obstructions define a map θ: N ( X ) → L n ( π 1 ( X ) ) {\displaystyle \theta \colon {\mathcal {N}}(X)\to L_{n}(\pi _{1}(X))} from the normal invariants to the L-groups which is in the first instance a set-theoretic map (that means not necessarily a homomorph... | https://en.wikipedia.org/wiki/Surgery_obstruction |
In mathematics, specifically in symplectic geometry, the momentum map (or, by false etymology, moment map) is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angul... | https://en.wikipedia.org/wiki/Hamiltonian_action |
In mathematics, specifically in symplectic geometry, the symplectic cut is a geometric modification on symplectic manifolds. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the symplectic sum, that glues two manifolds together into one. The symplectic cut can also be viewed a... | https://en.wikipedia.org/wiki/Symplectic_cut |
In mathematics, specifically in symplectic geometry, the symplectic sum is a geometric modification on symplectic manifolds, which glues two given manifolds into a single new one. It is a symplectic version of connected summation along a submanifold, often called a fiber sum. The symplectic sum is the inverse of the sy... | https://en.wikipedia.org/wiki/Symplectic_sum |
In mathematics, specifically in symplectic topology and algebraic geometry, Gromov–Witten (GW) invariants are rational numbers that, in certain situations, count pseudoholomorphic curves meeting prescribed conditions in a given symplectic manifold. The GW invariants may be packaged as a homology or cohomology class in ... | https://en.wikipedia.org/wiki/Gromov-Witten_theory |
They also play a crucial role in closed type IIA string theory. They are named after Mikhail Gromov and Edward Witten. The rigorous mathematical definition of Gromov–Witten invariants is lengthy and difficult, so it is treated separately in the stable map article. This article attempts a more intuitive explanation of w... | https://en.wikipedia.org/wiki/Gromov-Witten_theory |
In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former.... | https://en.wikipedia.org/wiki/Quantum_cohomology_ring |
More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov–Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product. | https://en.wikipedia.org/wiki/Quantum_cohomology_ring |
Because it expresses a structure or pattern for Gromov–Witten invariants, quantum cohomology has important implications for enumerative geometry. It also connects to many ideas in mathematical physics and mirror symmetry. In particular, it is ring-isomorphic to symplectic Floer homology. Throughout this article, X is a... | https://en.wikipedia.org/wiki/Quantum_cohomology_ring |
In mathematics, specifically in symplectic topology and algebraic geometry, one can construct the moduli space of stable maps, satisfying specified conditions, from Riemann surfaces into a given symplectic manifold. This moduli space is the essence of the Gromov–Witten invariants, which find application in enumerative ... | https://en.wikipedia.org/wiki/Stable_map |
In mathematics, specifically in the area of hyperbolic geometry, Hilbert's arithmetic of ends is a method for endowing a geometric set, the set of ideal points or "ends" of a hyperbolic plane, with an algebraic structure as a field. It was introduced by German mathematician David Hilbert. | https://en.wikipedia.org/wiki/Hilbert's_arithmetic_of_ends |
In mathematics, specifically in the calculus of variations, a variation δf of a function f can be concentrated on an arbitrarily small interval, but not a single point. Accordingly, the necessary condition of extremum (functional derivative equal zero) appears in a weak formulation (variational form) integrated with an... | https://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations |
Several versions of the lemma are in use. Basic versions are easy to formulate and prove. More powerful versions are used when needed. | https://en.wikipedia.org/wiki/Fundamental_lemma_of_calculus_of_variations |
In mathematics, specifically in the field known as category theory, a monoidal category where the monoidal ("tensor") product is the categorical product is called a cartesian monoidal category. Any category with finite products (a "finite product category") can be thought of as a cartesian monoidal category. In any car... | https://en.wikipedia.org/wiki/Cartesian_monoidal_category |
In mathematics, specifically in the field of differential topology, Morse homology is a homology theory defined for any smooth manifold. It is constructed using the smooth structure and an auxiliary metric on the manifold, but turns out to be topologically invariant, and is in fact isomorphic to singular homology. Mors... | https://en.wikipedia.org/wiki/Morse_homology |
In mathematics, specifically in the field of finite group theory, the Sylow theorems are a collection of theorems named after the Norwegian mathematician Peter Ludwig Sylow that give detailed information about the number of subgroups of fixed order that a given finite group contains. The Sylow theorems form a fundament... | https://en.wikipedia.org/wiki/Sylow_p-subgroup |
The Sylow theorems assert a partial converse to Lagrange's theorem. Lagrange's theorem states that for any finite group G {\displaystyle G} the order (number of elements) of every subgroup of G {\displaystyle G} divides the order of G {\displaystyle G} . The Sylow theorems state that for every prime factor p {\displays... | https://en.wikipedia.org/wiki/Sylow_p-subgroup |
In mathematics, specifically in the field of group theory, the McKay conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number p {\displaystyle p} to that of the normalizer of a Sylow p {\displaystyle p} -subgroup. It is named after Canadian ma... | https://en.wikipedia.org/wiki/McKay_conjecture |
In mathematics, specifically in the field of numerical analysis, Kummer's transformation of series is a method used to accelerate the convergence of an infinite series. The method was first suggested by Ernst Kummer in 1837. | https://en.wikipedia.org/wiki/Kummer's_transformation_of_series |
In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically n... | https://en.wikipedia.org/wiki/Monotonically_normal_space |
In mathematics, specifically in the field of topology, a topological space is said to be a door space if every subset is open or closed (or both). The term comes from the introductory topology mnemonic that "a subset is not like a door: it can be open, closed, both, or neither". | https://en.wikipedia.org/wiki/Door_space |
In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation ( ρ , V ) {\displaystyle (\rho ,V)} or irrep of an algebraic structure A {\displaystyle A} is a nonzero representation that has no proper nontrivial subrepresentation ( ρ | W , W ) {\displaystyle (\rho |_{W},... | https://en.wikipedia.org/wiki/Irreducible_representation |
In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system ev... | https://en.wikipedia.org/wiki/Positive_semi-orbit |
In mathematics, specifically in the study of ordinary differential equations, the Peano existence theorem, Peano theorem or Cauchy–Peano theorem, named after Giuseppe Peano and Augustin-Louis Cauchy, is a fundamental theorem which guarantees the existence of solutions to certain initial value problems. | https://en.wikipedia.org/wiki/Cauchy–Peano_theorem |
In mathematics, specifically in the study of topology and open covers of a topological space X, a star refinement is a particular kind of refinement of an open cover of X. A related concept is the notion of barycentric refinement. Star refinements are used in the definition of fully normal space and in one definition o... | https://en.wikipedia.org/wiki/Star_refinement |
In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theo... | https://en.wikipedia.org/wiki/Nakano_vanishing_theorem |
In mathematics, specifically in the theory of Markovian stochastic processes in probability theory, the Chapman–Kolmogorov equation(CKE) is an identity relating the joint probability distributions of different sets of coordinates on a stochastic process. The equation was derived independently by both the British mathem... | https://en.wikipedia.org/wiki/Chapman–Kolmogorov_equation |
In mathematics, specifically in the theory of generalized functions, the limit of a sequence of distributions is the distribution that sequence approaches. The distance, suitably quantified, to the limiting distribution can be made arbitrarily small by selecting a distribution sufficiently far along the sequence. This ... | https://en.wikipedia.org/wiki/Limit_of_a_distribution |
In mathematics, specifically in topology and functional analysis, a subspace S of a uniform space X is said to be sequentially complete or semi-complete if every Cauchy sequence in S converges to an element in S. X is called sequentially complete if it is a sequentially complete subset of itself. | https://en.wikipedia.org/wiki/Sequentially_complete_space |
In mathematics, specifically in topology and geometry, a pseudoholomorphic curve (or J-holomorphic curve) is a smooth map from a Riemann surface into an almost complex manifold that satisfies the Cauchy–Riemann equation. Introduced in 1985 by Mikhail Gromov, pseudoholomorphic curves have since revolutionized the study ... | https://en.wikipedia.org/wiki/Pseudoholomorphic_curve |
In mathematics, specifically in topology of manifolds, a compact codimension-one submanifold F {\displaystyle F} of a manifold M {\displaystyle M} is said to be 2-sided in M {\displaystyle M} when there is an embedding h: F × → M {\displaystyle h\colon F\times \to M} with h ( x , 0 ) = x {\displaystyle h(x,0)=x} for e... | https://en.wikipedia.org/wiki/2-sided |
In mathematics, specifically in topology, a pseudo-Anosov map is a type of a diffeomorphism or homeomorphism of a surface. It is a generalization of a linear Anosov diffeomorphism of the torus. Its definition relies on the notion of a measured foliation introduced by William Thurston, who also coined the term "pseudo-A... | https://en.wikipedia.org/wiki/Measured_foliation |
In mathematics, specifically in topology, the interior of a subset S of a topological space X is the union of all subsets of S that are open in X. A point that is in the interior of S is an interior point of S. The interior of S is the complement of the closure of the complement of S. In this sense interior and closure... | https://en.wikipedia.org/wiki/Interior_points |
In mathematics, specifically in topology, the operation of connected sum is a geometric modification on manifolds. Its effect is to join two given manifolds together near a chosen point on each. This construction plays a key role in the classification of closed surfaces. More generally, one can also join manifolds toge... | https://en.wikipedia.org/wiki/Knot_sum |
In mathematics, specifically in transcendental number theory and Diophantine approximation, Siegel's lemma refers to bounds on the solutions of linear equations obtained by the construction of auxiliary functions. The existence of these polynomials was proven by Axel Thue; Thue's proof used Dirichlet's box principle. C... | https://en.wikipedia.org/wiki/Siegel's_lemma |
In mathematics, specifically linear algebra and geometry, relative dimension is the dual notion to codimension. In linear algebra, given a quotient map V → Q {\displaystyle V\to Q} , the difference dim V − dim Q is the relative dimension; this equals the dimension of the kernel. In fiber bundles, the relative dimension... | https://en.wikipedia.org/wiki/Relative_dimension |
These are dual in that the inclusion of a subspace V → W {\displaystyle V\to W} of codimension k dualizes to yield a quotient map W ∗ → V ∗ {\displaystyle W^{*}\to V^{*}} of relative dimension k, and conversely. The additivity of codimension under intersection corresponds to the additivity of relative dimension in a fi... | https://en.wikipedia.org/wiki/Relative_dimension |
In mathematics, specifically linear algebra, a degenerate bilinear form f (x, y ) on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V ) given by v ↦ (x ↦ f (x, v )) is not an isomorphism. An equivalent definition when V is finite-dimensional is that it has a non-trivial kernel: th... | https://en.wikipedia.org/wiki/Nondegenerate_bilinear_form |
In mathematics, specifically linear algebra, a real matrix A is copositive if x T A x ≥ 0 {\displaystyle x^{T}Ax\geq 0} for every nonnegative vector x ≥ 0 {\displaystyle x\geq 0} . The collection of all copositive matrices is a proper cone; it includes as a subset the collection of real positive-definite matrices. Copo... | https://en.wikipedia.org/wiki/Copositive_matrix |
In mathematics, specifically linear algebra, the Cauchy–Binet formula, named after Augustin-Louis Cauchy and Jacques Philippe Marie Binet, is an identity for the determinant of the product of two rectangular matrices of transpose shapes (so that the product is well-defined and square). It generalizes the statement that... | https://en.wikipedia.org/wiki/Cauchy-Binet_formula |
In mathematics, specifically measure theory, a complex measure generalizes the concept of measure by letting it have complex values. In other words, one allows for sets whose size (length, area, volume) is a complex number. | https://en.wikipedia.org/wiki/Complex_measure |
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity ∞ {\displaystyle \infty } if the subset is infinite.The counting measure c... | https://en.wikipedia.org/wiki/Counting_measure |
In mathematics, specifically module theory, given a ring R and an R-module M with a submodule N, the module M is said to be an essential extension of N (or N is said to be an essential submodule or large submodule of M) if for every submodule H of M, H ∩ N = { 0 } {\displaystyle H\cap N=\{0\}\,} implies that H = { 0 } ... | https://en.wikipedia.org/wiki/Essential_extension |
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two Euclidean vectors. The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovect... | https://en.wikipedia.org/wiki/Dyadics |
The dyadic product takes in two vectors and returns a second order tensor called a dyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it. The dyadic product is distributive over vector addition, and associat... | https://en.wikipedia.org/wiki/Dyadics |
Therefore, the dyadic product is linear in both of its operands. In general, two dyadics can be added to get another dyadic, and multiplied by numbers to scale the dyadic. However, the product is not commutative; changing the order of the vectors results in a different dyadic. | https://en.wikipedia.org/wiki/Dyadics |
The formalism of dyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics. It also has... | https://en.wikipedia.org/wiki/Dyadics |
Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents. The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. | https://en.wikipedia.org/wiki/Dyadics |
The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations. Dyadic notation was first established by Josiah Willard Gibbs in 1884. | https://en.wikipedia.org/wiki/Dyadics |
The notation and terminology are relatively obsolete today. Its uses in physics include continuum mechanics and electromagnetism. In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over... | https://en.wikipedia.org/wiki/Dyadics |
In mathematics, specifically number theory, Granville numbers, also known as S {\displaystyle {\mathcal {S}}} -perfect numbers, are an extension of the perfect numbers. | https://en.wikipedia.org/wiki/Granville_number |
In mathematics, specifically order theory, a partially ordered set is chain-complete if every chain in it has a least upper bound. It is ω-complete when every increasing sequence of elements (a type of countable chain) has a least upper bound; the same notion can be extended to other cardinalities of chains. | https://en.wikipedia.org/wiki/Chain-complete_partial_order |
In mathematics, specifically order theory, a well-quasi-ordering or wqo on a set X {\displaystyle X} is a quasi-ordering of X {\displaystyle X} for which every infinite sequence of elements x 0 , x 1 , x 2 , … {\displaystyle x_{0},x_{1},x_{2},\ldots } from X {\displaystyle X} contains an increasing pair x i ≤ x j {\dis... | https://en.wikipedia.org/wiki/Well-quasi-ordering |
In mathematics, specifically order theory, the Dedekind–MacNeille completion of a partially ordered set is the smallest complete lattice that contains it. It is named after Holbrook Mann MacNeille whose 1937 paper first defined and constructed it, and after Richard Dedekind because its construction generalizes the Dede... | https://en.wikipedia.org/wiki/Dedekind–MacNeille_completion |
In mathematics, specifically order theory, the join of a subset S {\displaystyle S} of a partially ordered set P {\displaystyle P} is the supremum (least upper bound) of S , {\displaystyle S,} denoted ⋁ S , {\textstyle \bigvee S,} and similarly, the meet of S {\displaystyle S} is the infimum (greatest lower bound), den... | https://en.wikipedia.org/wiki/Join_and_meet |
A partially ordered set in which all pairs have a join is a join-semilattice. Dually, a partially ordered set in which all pairs have a meet is a meet-semilattice. A partially ordered set that is both a join-semilattice and a meet-semilattice is a lattice. | https://en.wikipedia.org/wiki/Join_and_meet |
A lattice in which every subset, not just every pair, possesses a meet and a join is a complete lattice. It is also possible to define a partial lattice, in which not all pairs have a meet or join but the operations (when defined) satisfy certain axioms.The join/meet of a subset of a totally ordered set is simply the m... | https://en.wikipedia.org/wiki/Join_and_meet |
In mathematics, specifically projective geometry, a configuration in the plane consists of a finite set of points, and a finite arrangement of lines, such that each point is incident to the same number of lines and each line is incident to the same number of points.Although certain specific configurations had been stud... | https://en.wikipedia.org/wiki/Geometric_configuration |
In mathematics, specifically real analysis and functional analysis, the Kirszbraun theorem states that if U is a subset of some Hilbert space H1, and H2 is another Hilbert space, and f: U → H 2 {\displaystyle f:U\rightarrow H_{2}} is a Lipschitz-continuous map, then there is a Lipschitz-continuous map F: H 1 → H 2 {\di... | https://en.wikipedia.org/wiki/Kirszbraun_theorem |
It is for instance possible to construct counterexamples where the domain is a subset of R n {\displaystyle \mathbb {R} ^{n}} with the maximum norm and R m {\displaystyle \mathbb {R} ^{m}} carries the Euclidean norm. More generally, the theorem fails for R m {\displaystyle \mathbb {R} ^{m}} equipped with any ℓ p {\disp... | https://en.wikipedia.org/wiki/Kirszbraun_theorem |
In mathematics, specifically representation theory, tilting theory describes a way to relate the module categories of two algebras using so-called tilting modules and associated tilting functors. Here, the second algebra is the endomorphism algebra of a tilting module over the first algebra. Tilting theory was motivate... | https://en.wikipedia.org/wiki/Coxeter_functor |
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. | https://en.wikipedia.org/wiki/Primitive_spectrum |
Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring. For commutative rings the primitive ideals are maximal, and so commutative primitive rings are all fields. | https://en.wikipedia.org/wiki/Primitive_spectrum |
In mathematics, specifically ring theory, a principal ideal is an ideal I {\displaystyle I} in a ring R {\displaystyle R} that is generated by a single element a {\displaystyle a} of R {\displaystyle R} through multiplication by every element of R . {\displaystyle R.} The term also has another, similar meaning in order... | https://en.wikipedia.org/wiki/Principal_ideal |
In mathematics, specifically set theory and model theory, a stationary set is a set that is not too small in the sense that it intersects all club sets and is analogous to a set of non-zero measure in measure theory. There are at least three closely related notions of stationary set, depending on whether one is looking... | https://en.wikipedia.org/wiki/Stationary_set |
In mathematics, specifically set theory, a cumulative hierarchy is a family of sets W α {\displaystyle W_{\alpha }} indexed by ordinals α {\displaystyle \alpha } such that W α ⊆ W α + 1 {\displaystyle W_{\alpha }\subseteq W_{\alpha +1}} If λ {\displaystyle \lambda } is a limit ordinal, then W λ = ⋃ α < λ W α {\textstyl... | https://en.wikipedia.org/wiki/Cumulative_hierarchy |
In mathematics, specifically set theory, a dimensional operator on a set E is a function from the subsets of E to the subsets of E. | https://en.wikipedia.org/wiki/Dimensional_operator |
In mathematics, specifically set theory, the Cartesian product of two sets A and B, denoted A × B, is the set of all ordered pairs (a, b) where a is in A and b is in B. In terms of set-builder notation, that is A × B = { ( a , b ) ∣ a ∈ A and b ∈ B } . {\displaystyle A\times B=\{(a,b)\mid a\in A\ {\mbox{ and}}\ b\in B\... | https://en.wikipedia.org/wiki/Cylinder_(algebra) |
If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value).One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be represented by an n-dimensional array, where each element is an n-tuple. ... | https://en.wikipedia.org/wiki/Cylinder_(algebra) |
In mathematics, specifically set theory, the continuum hypothesis (abbreviated CH) is a hypothesis about the possible sizes of infinite sets. It states that there is no set whose cardinality is strictly between that of the integers and the real numbers, or equivalently, that any subset of the real numbers is finite, is... | https://en.wikipedia.org/wiki/Aleph_hypothesis |
In mathematics, specifically statistics and information geometry, a Bregman divergence or Bregman distance is a measure of difference between two points, defined in terms of a strictly convex function; they form an important class of divergences. When the points are interpreted as probability distributions – notably as... | https://en.wikipedia.org/wiki/Bregman_distance |
However, they satisfy a generalization of the Pythagorean theorem, and in information geometry the corresponding statistical manifold is interpreted as a (dually) flat manifold. This allows many techniques of optimization theory to be generalized to Bregman divergences, geometrically as generalizations of least squares... | https://en.wikipedia.org/wiki/Bregman_distance |
In mathematics, specifically the algebraic theory of fields, a normal basis is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The normal basis theorem states that any finite Galois extension of fields has a normal basis. In algebraic number ... | https://en.wikipedia.org/wiki/Normal_basis |
In mathematics, specifically the area of Diophantine approximation, the Davenport–Schmidt theorem tells us how well a certain kind of real number can be approximated by another kind. Specifically it tells us that we can get a good approximation to irrational numbers that are not quadratic by using either quadratic irra... | https://en.wikipedia.org/wiki/Davenport–Schmidt_theorem |
In mathematics, specifically the area of algebraic number theory, a cubic field is an algebraic number field of degree three. | https://en.wikipedia.org/wiki/Cyclic_cubic_field |
In mathematics, specifically the field of abstract algebra, Bergman's Diamond Lemma (after George Bergman) is a method for confirming whether a given set of monomials of an algebra forms a k {\displaystyle k} -basis. It is an extension of Gröbner bases to non-commutative rings. The proof of the lemma gives rise to an a... | https://en.wikipedia.org/wiki/Bergman's_diamond_lemma |
In mathematics, specifically the field of algebra, Sklyanin algebras are a class of noncommutative algebra named after Evgeny Sklyanin. This class of algebras was first studied in the classification of Artin-Schelter regular algebras of global dimension 3 in the 1980s. Sklyanin algebras can be grouped into two differen... | https://en.wikipedia.org/wiki/Sklyanin_algebra |
In mathematics, specifically the field of algebraic number theory, a Minkowski space is a Euclidean space associated with an algebraic number field.If K is a number field of degree d then there are d distinct embeddings of K into C. We let KC be the image of K in the product Cd, considered as equipped with the usual He... | https://en.wikipedia.org/wiki/Minkowski_space_(number_field) |
In mathematics, specifically the field of transcendental number theory, the four exponentials conjecture is a conjecture which, given the right conditions on the exponents, would guarantee the transcendence of at least one of four exponentials. The conjecture, along with two related, stronger conjectures, is at the top... | https://en.wikipedia.org/wiki/Four_exponentials_conjecture |
In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem. The theorem is named ... | https://en.wikipedia.org/wiki/Cauchy-Lipschitz_theorem |
In mathematics, specifically the theory of Lie algebras, Lie's theorem states that, over an algebraically closed field of characteristic zero, if π: g → g l ( V ) {\displaystyle \pi :{\mathfrak {g}}\to {\mathfrak {gl}}(V)} is a finite-dimensional representation of a solvable Lie algebra, then there's a flag V = V 0 ⊃ V... | https://en.wikipedia.org/wiki/Lie's_theorem |
In mathematics, specifically the theory of quadratic forms, an ε-quadratic form is a generalization of quadratic forms to skew-symmetric settings and to *-rings; ε = ±1, accordingly for symmetric or skew-symmetric. They are also called ( − ) n {\displaystyle (-)^{n}} -quadratic forms, particularly in the context of sur... | https://en.wikipedia.org/wiki/Quadratic_refinement |
In mathematics, specifically topology, a sequence covering map is any of a class of maps between topological spaces whose definitions all somehow relate sequences in the codomain with sequences in the domain. Examples include sequentially quotient maps, sequence coverings, 1-sequence coverings, and 2-sequence coverings... | https://en.wikipedia.org/wiki/Sequence_covering_map |
In mathematics, specifically transcendental number theory, Schanuel's conjecture is a conjecture made by Stephen Schanuel in the 1960s concerning the transcendence degree of certain field extensions of the rational numbers. | https://en.wikipedia.org/wiki/Schanuel's_conjecture |
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