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Cameron Gordon conjectured that 10 is the largest possible number of exceptional surgeries of any hyperbolic knot complement. This was proved by Marc Lackenby and Rob Meyerhoff, who show that the number of exceptional slopes is 10 for any compact orientable 3-manifold with boundary a torus and interior finite-volume hyperbolic. Their proof relies on the proof of the geometrization conjecture originated by Grigori Perelman and on computer assistance.
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https://en.wikipedia.org/wiki/Dehn_filling
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However, it is not currently known whether the figure-eight knot is the only one that achieves the bound of 10. A well-known conjecture is that the bound (except for the two knots mentioned) is 6. Agol has shown that there are only finitely many cases in which the number of exceptional slopes is 9 or 10.
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https://en.wikipedia.org/wiki/Dehn_filling
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In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal to -1. Its underlying Riemannian manifold has non-constant negative curvature, pinched between -1 and -1/4 (or -4 and -1, according to the choice of a normalization of the metric): in particular, it is a CAT(-1/4) space. Complex hyperbolic spaces are also the symmetric spaces associated with the Lie groups P U ( n , 1 ) {\displaystyle PU(n,1)} . They constitute one of the three families of rank one symmetric spaces of noncompact type, together with real and quaternionic hyperbolic spaces, classification to which must be added one exceptional space, the Cayley plane.
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https://en.wikipedia.org/wiki/Complex_hyperbolic_space
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In mathematics, hyperbolic coordinates are a method of locating points in quadrant I of the Cartesian plane { ( x , y ): x > 0 , y > 0 } = Q {\displaystyle \{(x,y)\ :\ x>0,\ y>0\ \}=Q} .Hyperbolic coordinates take values in the hyperbolic plane defined as: H P = { ( u , v ): u ∈ R , v > 0 } {\displaystyle HP=\{(u,v):u\in \mathbb {R} ,v>0\}} .These coordinates in HP are useful for studying logarithmic comparisons of direct proportion in Q and measuring deviations from direct proportion. For ( x , y ) {\displaystyle (x,y)} in Q {\displaystyle Q} take u = ln x y {\displaystyle u=\ln {\sqrt {\frac {x}{y}}}} and v = x y {\displaystyle v={\sqrt {xy}}} .The parameter u is the hyperbolic angle to (x, y) and v is the geometric mean of x and y. The inverse mapping is x = v e u , y = v e − u {\displaystyle x=ve^{u},\quad y=ve^{-u}} .The function Q → H P {\displaystyle Q\rightarrow HP} is a continuous mapping, but not an analytic function.
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https://en.wikipedia.org/wiki/Hyperbolic_coordinates
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In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points (cos t, sin t) form a circle with a unit radius, the points (cosh t, sinh t) form the right half of the unit hyperbola. Also, similarly to how the derivatives of sin(t) and cos(t) are cos(t) and –sin(t) respectively, the derivatives of sinh(t) and cosh(t) are cosh(t) and +sinh(t) respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: hyperbolic sine "sinh" (), hyperbolic cosine "cosh" (),from which are derived: hyperbolic tangent "tanh" (), hyperbolic cosecant "csch" or "cosech" () hyperbolic secant "sech" (), hyperbolic cotangent "coth" (),corresponding to the derived trigonometric functions.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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The inverse hyperbolic functions are: area hyperbolic sine "arsinh" (also denoted "sinh−1", "asinh" or sometimes "arcsinh") area hyperbolic cosine "arcosh" (also denoted "cosh−1", "acosh" or sometimes "arccosh") and so on.The hyperbolic functions take a real argument called a hyperbolic angle. The size of a hyperbolic angle is twice the area of its hyperbolic sector. The hyperbolic functions may be defined in terms of the legs of a right triangle covering this sector.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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In complex analysis, the hyperbolic functions arise when applying the ordinary sine and cosine functions to an imaginary angle. The hyperbolic sine and the hyperbolic cosine are entire functions. As a result, the other hyperbolic functions are meromorphic in the whole complex plane.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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By Lindemann–Weierstrass theorem, the hyperbolic functions have a transcendental value for every non-zero algebraic value of the argument.Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert. Riccati used Sc.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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(sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today. The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.
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https://en.wikipedia.org/wiki/Hyperbolic_cotangent
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In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai–Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (Compare the above with Playfair's axiom, the modern version of Euclid's parallel postulate.) The hyperbolic plane is a plane where every point is a saddle point.
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https://en.wikipedia.org/wiki/Hyperbolic_plane
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Hyperbolic plane geometry is also the geometry of pseudospherical surfaces, surfaces with a constant negative Gaussian curvature. Saddle surfaces have negative Gaussian curvature in at least some regions, where they locally resemble the hyperbolic plane. A modern use of hyperbolic geometry is in the theory of special relativity, particularly the Minkowski model.
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https://en.wikipedia.org/wiki/Hyperbolic_plane
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When geometers first realised they were working with something other than the standard Euclidean geometry, they described their geometry under many different names; Felix Klein finally gave the subject the name hyperbolic geometry to include it in the now rarely used sequence elliptic geometry (spherical geometry), parabolic geometry (Euclidean geometry), and hyperbolic geometry. In the former Soviet Union, it is commonly called Lobachevskian geometry, named after one of its discoverers, the Russian geometer Nikolai Lobachevsky. This page is mainly about the 2-dimensional (planar) hyperbolic geometry and the differences and similarities between Euclidean and hyperbolic geometry. See hyperbolic space for more information on hyperbolic geometry extended to three and more dimensions.
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https://en.wikipedia.org/wiki/Hyperbolic_plane
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In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. There are many ways to construct it as an open subset of R n {\displaystyle \mathbb {R} ^{n}} with an explicitly written Riemannian metric; such constructions are referred to as models. Hyperbolic 2-space, H2, which was the first instance studied, is also called the hyperbolic plane.
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https://en.wikipedia.org/wiki/Hyperbolic_space
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It is also sometimes referred to as Lobachevsky space or Bolyai–Lobachevsky space after the names of the author who first published on the topic of hyperbolic geometry. Sometimes the qualificative "real" is added to differentiate it from complex hyperbolic spaces, quaternionic hyperbolic spaces and the octononic hyperbolic plane which are the other symmetric spaces of negative curvature. Hyperbolic space serves as the prototype of a Gromov hyperbolic space which is a far-reaching notion including differential-geometric as well as more combinatorial spaces via a synthetic approach to negative curvature. Another generalisation is the notion of a CAT(-1) space.
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https://en.wikipedia.org/wiki/Hyperbolic_space
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In mathematics, hypercomplex analysis is the extension of complex analysis to the hypercomplex numbers. The first instance is functions of a quaternion variable, where the argument is a quaternion (in this case, the sub-field of hypercomplex analysis is called quaternionic analysis). A second instance involves functions of a motor variable where arguments are split-complex numbers. In mathematical physics, there are hypercomplex systems called Clifford algebras.
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https://en.wikipedia.org/wiki/Hypercomplex_analysis
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The study of functions with arguments from a Clifford algebra is called Clifford analysis. A matrix may be considered a hypercomplex number. For example, the study of functions of 2 × 2 real matrices shows that the topology of the space of hypercomplex numbers determines the function theory.
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https://en.wikipedia.org/wiki/Hypercomplex_analysis
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Functions such as square root of a matrix, matrix exponential, and logarithm of a matrix are basic examples of hypercomplex analysis. The function theory of diagonalizable matrices is particularly transparent since they have eigendecompositions. Suppose T = ∑ i = 1 N λ i E i {\displaystyle \textstyle T=\sum _{i=1}^{N}\lambda _{i}E_{i}} where the Ei are projections.
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https://en.wikipedia.org/wiki/Hypercomplex_analysis
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Then for any polynomial f {\displaystyle f} , f ( T ) = ∑ i = 1 N f ( λ i ) E i . {\displaystyle f(T)=\sum _{i=1}^{N}f(\lambda _{i})E_{i}.}
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https://en.wikipedia.org/wiki/Hypercomplex_analysis
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The modern terminology for a "system of hypercomplex numbers" is an algebra over the real numbers, and the algebras used in applications are often Banach algebras since Cauchy sequences can be taken to be convergent. Then the function theory is enriched by sequences and series. In this context the extension of holomorphic functions of a complex variable is developed as the holomorphic functional calculus. Hypercomplex analysis on Banach algebras is called functional analysis.
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https://en.wikipedia.org/wiki/Hypercomplex_analysis
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In mathematics, hypercomplex number is a traditional term for an element of a finite-dimensional unital algebra over the field of real numbers. The study of hypercomplex numbers in the late 19th century forms the basis of modern group representation theory.
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https://en.wikipedia.org/wiki/Hypercomplex_numbers
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In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work by Laurent Schwartz, Grothendieck and others.
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https://en.wikipedia.org/wiki/Hyperfunction
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In mathematics, hypergeometric identities are equalities involving sums over hypergeometric terms, i.e. the coefficients occurring in hypergeometric series. These identities occur frequently in solutions to combinatorial problems, and also in the analysis of algorithms. These identities were traditionally found 'by hand'. There exist now several algorithms which can find and prove all hypergeometric identities.
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https://en.wikipedia.org/wiki/Hypergeometric_identities
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In mathematics, ideal theory is the theory of ideals in commutative rings. While the notion of an ideal exists also for non-commutative rings, a much more substantial theory exists only for commutative rings (and this article therefore only considers ideals in commutative rings.) Throughout the articles, rings refer to commutative rings. See also the article ideal (ring theory) for basic operations such as sum or products of ideals.
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https://en.wikipedia.org/wiki/Ideal_theory
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In mathematics, if A {\displaystyle A} is a subset of B , {\displaystyle B,} then the inclusion map (also inclusion function, insertion, or canonical injection) is the function ι {\displaystyle \iota } that sends each element x {\displaystyle x} of A {\displaystyle A} to x , {\displaystyle x,} treated as an element of B: {\displaystyle B:} A "hooked arrow" (U+21AA ↪ RIGHTWARDS ARROW WITH HOOK) is sometimes used in place of the function arrow above to denote an inclusion map; thus: (However, some authors use this hooked arrow for any embedding.) This and other analogous injective functions from substructures are sometimes called natural injections. Given any morphism f {\displaystyle f} between objects X {\displaystyle X} and Y {\displaystyle Y} , if there is an inclusion map into the domain ι: A → X , {\displaystyle \iota :A\to X,} then one can form the restriction f ι {\displaystyle f\,\iota } of f . {\displaystyle f.} In many instances, one can also construct a canonical inclusion into the codomain R → Y {\displaystyle R\to Y} known as the range of f . {\displaystyle f.}
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https://en.wikipedia.org/wiki/Inclusion_mapping
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In mathematics, if G is a group and Π is a representation of it over the complex vector space V, then the complex conjugate representation Π is defined over the complex conjugate vector space V as follows: Π(g) is the conjugate of Π(g) for all g in G.Π is also a representation, as one may check explicitly. If g is a real Lie algebra and π is a representation of it over the vector space V, then the conjugate representation π is defined over the conjugate vector space V as follows: π(X) is the conjugate of π(X) for all X in g.π is also a representation, as one may check explicitly. If two real Lie algebras have the same complexification, and we have a complex representation of the complexified Lie algebra, their conjugate representations are still going to be different.
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https://en.wikipedia.org/wiki/Complex_conjugate_representation
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See spinor for some examples associated with spinor representations of the spin groups Spin(p + q) and Spin(p, q). If g {\displaystyle {\mathfrak {g}}} is a *-Lie algebra (a complex Lie algebra with a * operation which is compatible with the Lie bracket), π(X) is the conjugate of −π(X*) for all X in gFor a finite-dimensional unitary representation, the dual representation and the conjugate representation coincide. This also holds for pseudounitary representations.
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https://en.wikipedia.org/wiki/Complex_conjugate_representation
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In mathematics, if G is a group and ρ is a linear representation of it on the vector space V, then the dual representation ρ* is defined over the dual vector space V* as follows: ρ*(g) is the transpose of ρ(g−1), that is, ρ*(g) = ρ(g−1)T for all g ∈ G.The dual representation is also known as the contragredient representation. If g is a Lie algebra and π is a representation of it on the vector space V, then the dual representation π* is defined over the dual vector space V* as follows: π*(X) = −π(X)T for all X ∈ g.The motivation for this definition is that Lie algebra representation associated to the dual of a Lie group representation is computed by the above formula. But the definition of the dual of a Lie algebra representation makes sense even if it does not come from a Lie group representation. In both cases, the dual representation is a representation in the usual sense.
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https://en.wikipedia.org/wiki/Contragredient_representation
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In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g(x) with coefficients in K such that g(a) = 0. Elements of L which are not algebraic over K are called transcendental over K. These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, C being the field of complex numbers and Q being the field of rational numbers).
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https://en.wikipedia.org/wiki/Algebraic_element
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In the physics and engineering literature, it is common to use ∇2 to denote the Laplacian, rather than ∆. In mathematics as well as in physics and engineering, it is common to use Newton's notation for time derivatives, so that u ˙ {\displaystyle {\dot {u}}} is used to denote ∂u/∂t, so the equation can be written Note also that the ability to use either ∆ or ∇2 to denote the Laplacian, without explicit reference to the spatial variables, is a reflection of the fact that the Laplacian is independent of the choice of coordinate system. In mathematical terms, one would say that the Laplacian is "translationally and rotationally invariant."
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https://en.wikipedia.org/wiki/Particle_diffusion
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In fact, it is (loosely speaking) the simplest differential operator which has these symmetries. This can be taken as a significant (and purely mathematical) justification of the use of the Laplacian and of the heat equation in modeling any physical phenomena which are homogeneous and isotropic, of which heat diffusion is a principal example. The "diffusivity constant" α is often not present in mathematical studies of the heat equation, while its value can be very important in engineering.
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https://en.wikipedia.org/wiki/Particle_diffusion
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This is not a major difference, for the following reason. Let u be a function with ∂ u ∂ t = α Δ u . {\displaystyle {\frac {\partial u}{\partial t}}=\alpha \Delta u.}
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https://en.wikipedia.org/wiki/Particle_diffusion
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Define a new function v ( t , x ) = u ( t / α , x ) {\displaystyle v(t,x)=u(t/\alpha ,x)} . Then, according to the chain rule, one has Thus, there is a straightforward way of translating between solutions of the heat equation with a general value of α and solutions of the heat equation with α = 1. As such, for the sake of mathematical analysis, it is often sufficient to only consider the case α = 1. Since α > 0 {\displaystyle \alpha >0} there is another option to define a v {\displaystyle v} satisfying ∂ ∂ t v = Δ v {\textstyle {\frac {\partial }{\partial t}}v=\Delta v} as in (⁎) above by setting v ( t , x ) = u ( t , α 1 / 2 x ) {\displaystyle v(t,x)=u(t,\alpha ^{1/2}x)} . Note that the two possible means of defining the new function v {\displaystyle v} discussed here amount, in physical terms, to changing the unit of measure of time or the unit of measure of length.
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https://en.wikipedia.org/wiki/Particle_diffusion
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In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field F is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points.
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https://en.wikipedia.org/wiki/Conductor_of_an_abelian_variety
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In mathematics, in abstract algebra, a multivariate polynomial p over a field such that the Laplacian of p is zero is termed a harmonic polynomial.The harmonic polynomials form a vector subspace of the vector space of polynomials over the field. In fact, they form a graded subspace. For the real field, the harmonic polynomials are important in mathematical physics.The Laplacian is the sum of second partials with respect to all the variables, and is an invariant differential operator under the action of the orthogonal group via the group of rotations. The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radial polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials.
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https://en.wikipedia.org/wiki/Harmonic_polynomial
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In mathematics, in algebra, in the realm of group theory, a subgroup H {\displaystyle H} of a finite group G {\displaystyle G} is said to be semipermutable if H {\displaystyle H} commutes with every subgroup K {\displaystyle K} whose order is relatively prime to that of H {\displaystyle H} . Clearly, every permutable subgroup of a finite group is semipermutable. The converse, however, is not necessarily true.
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https://en.wikipedia.org/wiki/Semipermutable_subgroup
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In mathematics, in combinatorics, the Li Shanlan identity (also called Li Shanlan's summation formula) is a certain combinatorial identity attributed to the nineteenth century Chinese mathematician Li Shanlan. Since Li Shanlan is also known as Li Renshu (his courtesy name), this identity is also referred to as the Li Renshu identity. This identity appears in the third chapter of Duoji bilei (垛积比类 / 垛積比類, meaning summing finite series), a mathematical text authored by Li Shanlan and published in 1867 as part of his collected works.
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https://en.wikipedia.org/wiki/Li_Shanlan_identity
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A Czech mathematician Josef Kaucky published an elementary proof of the identity along with a history of the identity in 1964. Kaucky attributed the identity to a certain Li Jen-Shu. From the account of the history of the identity, it has been ascertained that Li Jen-Shu is in fact Li Shanlan. Western scholars had been studying Chinese mathematics for its historical value; but the attribution of this identity to a nineteenth century Chinese mathematician sparked a rethink on the mathematical value of the writings of Chinese mathematicians. "In the West Li is best remembered for a combinatoric formula, known as the 'Li Renshu identity', that he derived using only traditional Chinese mathematical methods."
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https://en.wikipedia.org/wiki/Li_Shanlan_identity
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In mathematics, in functional analysis, several different wavelets are known by the name Poisson wavelet. In one context, the term "Poisson wavelet" is used to denote a family of wavelets labeled by the set of positive integers, the members of which are associated with the Poisson probability distribution. These wavelets were first defined and studied by Karlene A. Kosanovich, Allan R. Moser and Michael J. Piovoso in 1995–96. In another context, the term refers to a certain wavelet which involves a form of the Poisson integral kernel. In still another context, the terminology is used to describe a family of complex wavelets indexed by positive integers which are connected with the derivatives of the Poisson integral kernel.
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https://en.wikipedia.org/wiki/Poisson_wavelet
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In mathematics, in general topology, compactification is the process or result of making a topological space into a compact space. A compact space is a space in which every open cover of the space contains a finite subcover. The methods of compactification are various, but each is a way of controlling points from "going off to infinity" by in some way adding "points at infinity" or preventing such an "escape".
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https://en.wikipedia.org/wiki/Compactification_(mathematics)
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In mathematics, in graph theory, the Seidel adjacency matrix of a simple undirected graph G is a symmetric matrix with a row and column for each vertex, having 0 on the diagonal, −1 for positions whose rows and columns correspond to adjacent vertices, and +1 for positions corresponding to non-adjacent vertices. It is also called the Seidel matrix or—its original name—the (−1,1,0)-adjacency matrix. It can be interpreted as the result of subtracting the adjacency matrix of G from the adjacency matrix of the complement of G. The multiset of eigenvalues of this matrix is called the Seidel spectrum. The Seidel matrix was introduced by J. H. van Lint and Johan Jacob Seidel in 1966 and extensively exploited by Seidel and coauthors.
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https://en.wikipedia.org/wiki/Seidel_adjacency_matrix
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The Seidel matrix of G is also the adjacency matrix of a signed complete graph KG in which the edges of G are negative and the edges not in G are positive. It is also the adjacency matrix of the two-graph associated with G and KG. The eigenvalue properties of the Seidel matrix are valuable in the study of strongly regular graphs.
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https://en.wikipedia.org/wiki/Seidel_adjacency_matrix
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In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
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https://en.wikipedia.org/wiki/Cyclic_decomposition_theorem
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In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices it commutes with. It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool for studying classes of commuting matrices. A square matrix is said to be in the Weyr canonical form if the matrix has the structure defining the Weyr canonical form. The Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885.
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https://en.wikipedia.org/wiki/Weyr_canonical_form
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The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name Jordan canonical form. The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885.
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https://en.wikipedia.org/wiki/Weyr_canonical_form
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This form has been variously called as modified Jordan form, reordered Jordan form, second Jordan form, and H-form. The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999.Recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of phylogenetic invariants in biomathematics.
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https://en.wikipedia.org/wiki/Weyr_canonical_form
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In mathematics, in number theory, Gauss composition law is a rule, invented by Carl Friedrich Gauss, for performing a binary operation on integral binary quadratic forms (IBQFs). Gauss presented this rule in his Disquisitiones Arithmeticae, a textbook on number theory published in 1801, in Articles 234 - 244. Gauss composition law is one of the deepest results in the theory of IBQFs and Gauss's formulation of the law and the proofs its properties as given by Gauss are generally considered highly complicated and very difficult. Several later mathematicians have simplified the formulation of the composition law and have presented it in a format suitable for numerical computations. The concept has also found generalisations in several directions.
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https://en.wikipedia.org/wiki/Gauss_composition_law
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In mathematics, in number theory, a Bhargava cube (also called Bhargava's cube) is a configuration consisting of eight integers placed at the eight corners of a cube. This configuration was extensively used by Manjul Bhargava, a Canadian-American Fields Medal winning mathematician, to study the composition laws of binary quadratic forms and other such forms. To each pair of opposite faces of a Bhargava cube one can associate an integer binary quadratic form thus getting three binary quadratic forms corresponding to the three pairs of opposite faces of the Bhargava cube. These three quadratic forms all have the same discriminant and Manjul Bhargava proved that their composition in the sense of Gauss is the identity element in the associated group of equivalence classes of primitive binary quadratic forms. (This formulation of Gauss composition was likely first due to Dedekind.) Using this property as the starting point for a theory of composition of binary quadratic forms Manjul Bhargava went on to define fourteen different composition laws using a cube.
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https://en.wikipedia.org/wiki/Bhargava_cube
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In mathematics, in particular abstract algebra and topology, a differential graded Lie algebra (or dg Lie algebra, or dgla) is a graded vector space with added Lie algebra and chain complex structures that are compatible. Such objects have applications in deformation theory and rational homotopy theory.
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https://en.wikipedia.org/wiki/Differential_graded_Lie_algebra
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In mathematics, in particular abstract algebra and topology, a homotopy Lie algebra (or L ∞ {\displaystyle L_{\infty }} -algebra) is a generalisation of the concept of a differential graded Lie algebra. To be a little more specific, the Jacobi identity only holds up to homotopy. Therefore, a differential graded Lie algebra can be seen as a homotopy Lie algebra where the Jacobi identity holds on the nose. These homotopy algebras are useful in classifying deformation problems over characteristic 0 in deformation theory because deformation functors are classified by quasi-isomorphism classes of L ∞ {\displaystyle L_{\infty }} -algebras. This was later extended to all characteristics by Jonathan Pridham.Homotopy Lie algebras have applications within mathematics and mathematical physics; they are linked, for instance, to the Batalin–Vilkovisky formalism much like differential graded Lie algebras are.
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https://en.wikipedia.org/wiki/Homotopy_Lie_algebra
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In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R i {\displaystyle R_{i}} such that R i R j ⊆ R i + j {\displaystyle R_{i}R_{j}\subseteq R_{i+j}} . The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid. The direct sum decomposition is usually referred to as gradation or grading. A graded module is defined similarly (see below for the precise definition).
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https://en.wikipedia.org/wiki/Graded_module
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It generalizes graded vector spaces. A graded module that is also a graded ring is called a graded algebra. A graded ring could also be viewed as a graded Z {\displaystyle \mathbb {Z} } -algebra. The associativity is not important (in fact not used at all) in the definition of a graded ring; hence, the notion applies to non-associative algebras as well; e.g., one can consider a graded Lie algebra.
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https://en.wikipedia.org/wiki/Graded_module
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In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects (e.g., the smooth algebraic curves of a fixed genus) can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces is formal moduli. Bernhard Riemann first used the term "moduli" in 1857.
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https://en.wikipedia.org/wiki/Moduli_functor
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In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was introduced in Dwyer & Wilkerson (1994), making precise earlier notions of a mod p finite loop space. A p-compact group has many Lie-like properties like maximal tori and Weyl groups, which are defined purely homotopically in terms of the classifying space, but with the important difference that the Weyl group, rather than being a finite reflection group over the integers, is now a finite p-adic reflection group. They admit a classification in terms of root data, which mirrors the classification of compact Lie groups, but with the integers replaced by the p-adic integers.
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https://en.wikipedia.org/wiki/P-compact_group
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In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed. In contexts where fractional ideals and ordinary ring ideals are both under discussion, the latter are sometimes termed integral ideals for clarity.
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https://en.wikipedia.org/wiki/Group_of_fractional_ideals
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In mathematics, in particular field theory, the conjugate elements or algebraic conjugates of an algebraic element α, over a field extension L/K, are the roots of the minimal polynomial pK,α(x) of α over K. Conjugate elements are commonly called conjugates in contexts where this is not ambiguous. Normally α itself is included in the set of conjugates of α. Equivalently, the conjugates of α are the images of α under the field automorphisms of L that leave fixed the elements of K. The equivalence of the two definitions is one of the starting points of Galois theory. The concept generalizes the complex conjugation, since the algebraic conjugates over R {\displaystyle \mathbb {R} } of a complex number are the number itself and its complex conjugate.
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https://en.wikipedia.org/wiki/Conjugate_element_(field_theory)
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In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator T: X → Y {\displaystyle T:X\rightarrow Y} acting between Hilbert spaces X {\displaystyle X} and Y {\displaystyle Y} , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T ∗ T {\displaystyle T^{*}T} (where T ∗ {\displaystyle T^{*}} denotes the adjoint of T {\displaystyle T} ). The singular values are non-negative real numbers, usually listed in decreasing order (σ1(T), σ2(T), …). The largest singular value σ1(T) is equal to the operator norm of T (see Min-max theorem). If T acts on Euclidean space R n {\displaystyle \mathbb {R} ^{n}} , there is a simple geometric interpretation for the singular values: Consider the image by T {\displaystyle T} of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T {\displaystyle T} (the figure provides an example in R 2 {\displaystyle \mathbb {R} ^{2}} ).
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https://en.wikipedia.org/wiki/Singular_values
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The singular values are the absolute values of the eigenvalues of a normal matrix A, because the spectral theorem can be applied to obtain unitary diagonalization of A {\displaystyle A} as A = U Λ U ∗ {\displaystyle A=U\Lambda U^{*}} . Therefore, A ∗ A = U Λ ∗ Λ U ∗ = U | Λ | U ∗ {\textstyle {\sqrt {A^{*}A}}={\sqrt {U\Lambda ^{*}\Lambda U^{*}}}=U\left|\Lambda \right|U^{*}} . Most norms on Hilbert space operators studied are defined using s-numbers.
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https://en.wikipedia.org/wiki/Singular_values
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For example, the Ky Fan-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence s-numbers are useful in classifying different operators. In the finite-dimensional case, a matrix can always be decomposed in the form U Σ V ∗ {\displaystyle \mathbf {U\Sigma V^{*}} } , where U {\displaystyle \mathbf {U} } and V ∗ {\displaystyle \mathbf {V^{*}} } are unitary matrices and Σ {\displaystyle \mathbf {\Sigma } } is a rectangular diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.
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https://en.wikipedia.org/wiki/Singular_values
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In mathematics, in particular homotopy theory, a continuous mapping between topological spaces i: A → X {\displaystyle i:A\to X} ,is a cofibration if it has the homotopy extension property with respect to all topological spaces S {\displaystyle S} . That is, i {\displaystyle i} is a cofibration if for each topological space S {\displaystyle S} , and for any continuous maps f , f ′: A → S {\displaystyle f,f':A\to S} and g: X → S {\displaystyle g:X\to S} with g ∘ i = f {\displaystyle g\circ i=f} , for any homotopy h: A × I → S {\displaystyle h:A\times I\to S} from f {\displaystyle f} to f ′ {\displaystyle f'} , there is a continuous map g ′: X → S {\displaystyle g':X\to S} and a homotopy h ′: X × I → S {\displaystyle h':X\times I\to S} from g {\displaystyle g} to g ′ {\displaystyle g'} such that h ′ ( i ( a ) , t ) = h ( a , t ) {\displaystyle h'(i(a),t)=h(a,t)} for all a ∈ A {\displaystyle a\in A} and t ∈ I {\displaystyle t\in I} . (Here, I {\displaystyle I} denotes the unit interval {\displaystyle } .)
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https://en.wikipedia.org/wiki/Cofibration
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This definition is formally dual to that of a fibration, which is required to satisfy the homotopy lifting property with respect to all spaces; this is one instance of the broader Eckmann–Hilton duality in topology. Cofibrations are a fundamental concept of homotopy theory. Quillen has proposed the notion of model category as a formal framework for doing homotopy theory in more general categories; a model category is endowed with three distinguished classes of morphisms called fibrations, cofibrations and weak equivalences satisfying certain lifting and factorization axioms.
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https://en.wikipedia.org/wiki/Cofibration
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In mathematics, in particular in algebra, polarization is a technique for expressing a homogeneous polynomial in a simpler fashion by adjoining more variables. Specifically, given a homogeneous polynomial, polarization produces a unique symmetric multilinear form from which the original polynomial can be recovered by evaluating along a certain diagonal. Although the technique is deceptively simple, it has applications in many areas of abstract mathematics: in particular to algebraic geometry, invariant theory, and representation theory. Polarization and related techniques form the foundations for Weyl's invariant theory.
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https://en.wikipedia.org/wiki/Polarization_formula
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In mathematics, in particular in algebraic geometry and differential geometry, Dolbeault cohomology (named after Pierre Dolbeault) is an analog of de Rham cohomology for complex manifolds. Let M be a complex manifold. Then the Dolbeault cohomology groups H p , q ( M , C ) {\displaystyle H^{p,q}(M,\mathbb {C} )} depend on a pair of integers p and q and are realized as a subquotient of the space of complex differential forms of degree (p,q).
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https://en.wikipedia.org/wiki/Dolbeault_cohomology
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In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism X × Y → Y {\displaystyle X\times Y\to Y} is a closed map (i.e. maps closed sets onto closed sets). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products. The image of a complete variety is closed and is a complete variety. A closed subvariety of a complete variety is complete.
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https://en.wikipedia.org/wiki/Completeness_of_projective_varieties
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A complex variety is complete if and only if it is compact as a complex-analytic variety. The most common example of a complete variety is a projective variety, but there do exist complete non-projective varieties in dimensions 2 and higher. While any complete nonsingular surface is projective, there exist nonsingular complete varieties in dimension 3 and higher which are not projective.
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https://en.wikipedia.org/wiki/Completeness_of_projective_varieties
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The first examples of non-projective complete varieties were given by Masayoshi Nagata and Heisuke Hironaka. An affine space of positive dimension is not complete. The morphism taking a complete variety to a point is a proper morphism, in the sense of scheme theory. An intuitive justification of "complete", in the sense of "no missing points", can be given on the basis of the valuative criterion of properness, which goes back to Claude Chevalley.
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https://en.wikipedia.org/wiki/Completeness_of_projective_varieties
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In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney classes are a set of topological invariants of a real vector bundle that describe the obstructions to constructing everywhere independent sets of sections of the vector bundle. Stiefel–Whitney classes are indexed from 0 to n, where n is the rank of the vector bundle. If the Stiefel–Whitney class of index i is nonzero, then there cannot exist ( n − i + 1 ) {\displaystyle (n-i+1)} everywhere linearly independent sections of the vector bundle. A nonzero nth Stiefel–Whitney class indicates that every section of the bundle must vanish at some point.
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https://en.wikipedia.org/wiki/Stiefel–Whitney_class
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A nonzero first Stiefel–Whitney class indicates that the vector bundle is not orientable. For example, the first Stiefel–Whitney class of the Möbius strip, as a line bundle over the circle, is not zero, whereas the first Stiefel–Whitney class of the trivial line bundle over the circle, S 1 × R {\displaystyle S^{1}\times \mathbb {R} } , is zero. The Stiefel–Whitney class was named for Eduard Stiefel and Hassler Whitney and is an example of a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } -characteristic class associated to real vector bundles. In algebraic geometry one can also define analogous Stiefel–Whitney classes for vector bundles with a non-degenerate quadratic form, taking values in etale cohomology groups or in Milnor K-theory. As a special case one can define Stiefel–Whitney classes for quadratic forms over fields, the first two cases being the discriminant and the Hasse–Witt invariant (Milnor 1970).
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https://en.wikipedia.org/wiki/Stiefel–Whitney_class
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In mathematics, in particular in algebraic topology, differential geometry and algebraic geometry, the Chern classes are characteristic classes associated with complex vector bundles. They have since become fundamental concepts in many branches of mathematics and physics, such as string theory, Chern–Simons theory, knot theory, Gromov–Witten invariants. Chern classes were introduced by Shiing-Shen Chern (1946).
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https://en.wikipedia.org/wiki/Total_Chern_class
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In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres.
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https://en.wikipedia.org/wiki/Hopf_invariant_one_problem
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In mathematics, in particular in category theory, the lifting property is a property of a pair of morphisms in a category. It is used in homotopy theory within algebraic topology to define properties of morphisms starting from an explicitly given class of morphisms. It appears in a prominent way in the theory of model categories, an axiomatic framework for homotopy theory introduced by Daniel Quillen. It is also used in the definition of a factorization system, and of a weak factorization system, notions related to but less restrictive than the notion of a model category. Several elementary notions may also be expressed using the lifting property starting from a list of (counter)examples.
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https://en.wikipedia.org/wiki/Right_lifting_property
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In mathematics, in particular in computational algebra, the Berlekamp–Zassenhaus algorithm is an algorithm for factoring polynomials over the integers, named after Elwyn Berlekamp and Hans Zassenhaus. As a consequence of Gauss's lemma, this amounts to solving the problem also over the rationals. The algorithm starts by finding factorizations over suitable finite fields using Hensel's lemma to lift the solution from modulo a prime p to a convenient power of p. After this the right factors are found as a subset of these. The worst case of this algorithm is exponential in the number of factors. Van Hoeij (2002) improved this algorithm by using the LLL algorithm, substantially reducing the time needed to choose the right subsets of mod p factors.
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https://en.wikipedia.org/wiki/Berlekamp–Zassenhaus_algorithm
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In mathematics, in particular in differential geometry, mathematical physics, and representation theory a Weitzenböck identity, named after Roland Weitzenböck, expresses a relationship between two second-order elliptic operators on a manifold with the same principal symbol. Usually Weitzenböck formulae are implemented for G-invariant self-adjoint operators between vector bundles associated to some principal G-bundle, although the precise conditions under which such a formula exists are difficult to formulate. This article focuses on three examples of Weitzenböck identities: from Riemannian geometry, spin geometry, and complex analysis.
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https://en.wikipedia.org/wiki/Weitzenböck_identity
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In mathematics, in particular in differential geometry, the minimal volume is a number that describes one aspect of a smooth manifold's topology. This diffeomorphism invariant was introduced by Mikhael Gromov. Given a smooth Riemannian manifold (M, g), one may consider its volume vol(M, g) and sectional curvature Kg. The minimal volume of a smooth manifold M is defined to be MinVol ( M ) := inf { vol ( M , g ): g a complete Riemannian metric with | K g | ≤ 1 } .
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https://en.wikipedia.org/wiki/Minimal_volume
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{\displaystyle \operatorname {MinVol} (M):=\inf\{\operatorname {vol} (M,g):g{\text{ a complete Riemannian metric with }}|K_{g}|\leq 1\}.} Any closed manifold can be given an arbitrarily small volume by scaling any choice of a Riemannian metric. The minimal volume removes the possibility of such scaling by the constraint on sectional curvatures.
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https://en.wikipedia.org/wiki/Minimal_volume
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So, if the minimal volume of M is zero, then a certain kind of nontrivial collapsing phenomena can be exhibited by Riemannian metrics on M. A trivial example, the only in which the possibility of scaling is present, is a closed flat manifold. The Berger spheres show that the minimal volume of the three-dimensional sphere is also zero. Gromov has conjectured that every closed simply connected odd-dimensional manifold has zero minimal volume.
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https://en.wikipedia.org/wiki/Minimal_volume
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By contrast, a positive lower bound for the minimal volume of M amounts to some (usually nontrivial) geometric inequality for the volume of an arbitrary complete Riemannian metric on M in terms of the size of its curvature. According to the Gauss-Bonnet theorem, if M is a closed and connected two-dimensional manifold, then MinVol(M) = 2π|χ(M)|. The infimum in the definition of minimal volume is realized by the metrics appearing from the uniformization theorem.
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https://en.wikipedia.org/wiki/Minimal_volume
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More generally, according to the Chern-Gauss-Bonnet formula, if M is a closed and connected manifold then MinVol ( M ) ≥ c ( n ) | χ ( M ) | . {\displaystyle \operatorname {MinVol} (M)\geq c(n){\big |}\chi (M){\big |}.}
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https://en.wikipedia.org/wiki/Minimal_volume
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Gromov, in 1982, showed that the volume of a complete Riemannian metric on a smooth manifold can always be estimated by the size of its curvature and by the simplicial volume of the manifold, via the inequality MinVol ( M ) ≥ ‖ M ‖ ( n − 1 ) n n ! . {\displaystyle \operatorname {MinVol} (M)\geq {\frac {\|M\|}{(n-1)^{n}n!}}.}
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https://en.wikipedia.org/wiki/Minimal_volume
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In mathematics, in particular in field theory and real algebra, a formally real field is a field that can be equipped with a (not necessarily unique) ordering that makes it an ordered field.
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https://en.wikipedia.org/wiki/Formally_real_field
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In mathematics, in particular in functional analysis and nonlinear analysis, it is possible to define the derivative of a function between two Fréchet spaces. This notion of differentiation, as it is Gateaux derivative between Fréchet spaces, is significantly weaker than the derivative in a Banach space, even between general topological vector spaces. Nevertheless, it is the weakest notion of differentiation for which many of the familiar theorems from calculus hold. In particular, the chain rule is true. With some additional constraints on the Fréchet spaces and functions involved, there is an analog of the inverse function theorem called the Nash–Moser inverse function theorem, having wide applications in nonlinear analysis and differential geometry.
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https://en.wikipedia.org/wiki/Differentiation_in_Fréchet_spaces
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In mathematics, in particular in functional analysis, the Rademacher system, named after Hans Rademacher, is an incomplete orthogonal system of functions on the unit interval of the following form: { t ↦ r n ( t ) = sgn ( sin 2 n + 1 π t ) ; t ∈ , n ∈ N } . {\displaystyle \{t\mapsto r_{n}(t)=\operatorname {sgn}(\sin 2^{n+1}\pi t);t\in ,n\in \mathbb {N} \}.} The Rademacher system is stochastically independent, and is closely related to the Walsh system. Specifically, the Walsh system can be constructed as a product of Rademacher functions.
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https://en.wikipedia.org/wiki/Rademacher_system
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In mathematics, in particular in homological algebra, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure.
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https://en.wikipedia.org/wiki/DG_Algebra
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In mathematics, in particular in homotopy theory within algebraic topology, the homotopy lifting property (also known as an instance of the right lifting property or the covering homotopy axiom) is a technical condition on a continuous function from a topological space E to another one, B. It is designed to support the picture of E "above" B by allowing a homotopy taking place in B to be moved "upstairs" to E. For example, a covering map has a property of unique local lifting of paths to a given sheet; the uniqueness is because the fibers of a covering map are discrete spaces. The homotopy lifting property will hold in many situations, such as the projection in a vector bundle, fiber bundle or fibration, where there need be no unique way of lifting.
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https://en.wikipedia.org/wiki/Homotopy_lifting_property
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In mathematics, in particular in mathematical analysis, the Whitney extension theorem is a partial converse to Taylor's theorem. Roughly speaking, the theorem asserts that if A is a closed subset of a Euclidean space, then it is possible to extend a given function of A in such a way as to have prescribed derivatives at the points of A. It is a result of Hassler Whitney.
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https://en.wikipedia.org/wiki/Whitney_extension_theorem
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In mathematics, in particular in measure theory, a content μ {\displaystyle \mu } is a real-valued function defined on a collection of subsets A {\displaystyle {\mathcal {A}}} such that μ ( A ) ∈ whenever A ∈ A . {\displaystyle \mu (A)\in \ {\text{ whenever }}A\in {\mathcal {A}}.} μ ( ∅ ) = 0. {\displaystyle \mu (\varnothing )=0.}
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https://en.wikipedia.org/wiki/Content_(measure_theory)
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μ ( A 1 ∪ A 2 ) = μ ( A 1 ) + μ ( A 2 ) whenever A 1 , A 2 , A 1 ∪ A 2 ∈ A and A 1 ∩ A 2 = ∅ . {\displaystyle \mu (A_{1}\cup A_{2})=\mu (A_{1})+\mu (A_{2}){\text{ whenever }}A_{1},A_{2},A_{1}\cup A_{2}\ \in {\mathcal {A}}{\text{ and }}A_{1}\cap A_{2}=\varnothing .} That is, a content is a generalization of a measure: while the latter must be countably additive, the former must only be finitely additive.
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https://en.wikipedia.org/wiki/Content_(measure_theory)
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In many important applications the A {\displaystyle {\mathcal {A}}} is chosen to be a ring of sets or to be at least a semiring of sets in which case some additional properties can be deduced which are described below. For this reason some authors prefer to define contents only for the case of semirings or even rings. If a content is additionally σ-additive it is called a pre-measure and if furthermore A {\displaystyle {\mathcal {A}}} is a σ-algebra, the content is called a measure. Therefore every (real-valued) measure is a content, but not vice versa. Contents give a good notion of integrating bounded functions on a space but can behave badly when integrating unbounded functions, while measures give a good notion of integrating unbounded functions.
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https://en.wikipedia.org/wiki/Content_(measure_theory)
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In mathematics, in particular in measure theory, an inner measure is a function on the power set of a given set, with values in the extended real numbers, satisfying some technical conditions. Intuitively, the inner measure of a set is a lower bound of the size of that set.
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https://en.wikipedia.org/wiki/Inner_measure
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In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space. More precisely, a Fréchet manifold consists of a Hausdorff space X {\displaystyle X} with an atlas of coordinate charts over Fréchet spaces whose transitions are smooth mappings. Thus X {\displaystyle X} has an open cover { U α } α ∈ I , {\displaystyle \left\{U_{\alpha }\right\}_{\alpha \in I},} and a collection of homeomorphisms ϕ α: U α → F α {\displaystyle \phi _{\alpha }:U_{\alpha }\to F_{\alpha }} onto their images, where F α {\displaystyle F_{\alpha }} are Fréchet spaces, such that is smooth for all pairs of indices α , β . {\displaystyle \alpha ,\beta .}
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https://en.wikipedia.org/wiki/Fréchet_manifold
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In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah-Singer index theorem and Atiyah-Bott fixed point theorem.
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https://en.wikipedia.org/wiki/Elliptic_complex
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In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by Erich Hecke (1937a,1937b), is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations.
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https://en.wikipedia.org/wiki/Hecke_operator
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In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P: O Y , f ( P ) → O X , P {\displaystyle f_{P}\colon {\mathcal {O}}_{Y,f(P)}\to {\mathcal {O}}_{X,P}} is a flat map for all P in X. A map of rings A → B {\displaystyle A\to B} is called flat if it is a homomorphism that makes B a flat A-module. A morphism of schemes is called faithfully flat if it is both surjective and flat.Two basic intuitions regarding flat morphisms are: flatness is a generic property; and the failure of flatness occurs on the jumping set of the morphism.The first of these comes from commutative algebra: subject to some finiteness conditions on f, it can be shown that there is a non-empty open subscheme Y ′ {\displaystyle Y'} of Y, such that f restricted to Y′ is a flat morphism (generic flatness). Here 'restriction' is interpreted by means of the fiber product of schemes, applied to f and the inclusion map of Y ′ {\displaystyle Y'} into Y. For the second, the idea is that morphisms in algebraic geometry can exhibit discontinuities of a kind that are detected by flatness. For instance, the operation of blowing down in the birational geometry of an algebraic surface, can give a single fiber that is of dimension 1 when all the others have dimension 0.
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https://en.wikipedia.org/wiki/Flat_map_(ring_theory)
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It turns out (retrospectively) that flatness in morphisms is directly related to controlling this sort of semicontinuity, or one-sided jumping. Flat morphisms are used to define (more than one version of) the flat topos, and flat cohomology of sheaves from it. This is a deep-lying theory, and has not been found easy to handle. The concept of étale morphism (and so étale cohomology) depends on the flat morphism concept: an étale morphism being flat, of finite type, and unramified.
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https://en.wikipedia.org/wiki/Flat_map_(ring_theory)
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In mathematics, in particular in topology, the de Groot dual (after Johannes de Groot) of a topology τ on a set X is the topology τ* whose closed sets are generated by compact saturated subsets of (X, τ).
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https://en.wikipedia.org/wiki/De_Groot_dual
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In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix A {\displaystyle A} and the outer product, v v T {\displaystyle vv^{T}} , of vector v {\displaystyle v} with itself.
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https://en.wikipedia.org/wiki/Bunch–Nielsen–Sorensen_formula
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In mathematics, in particular linear algebra, the Sherman–Morrison formula, named after Jack Sherman and Winifred J. Morrison, computes the inverse of the sum of an invertible matrix A {\displaystyle A} and the outer product, u v T {\displaystyle uv^{\textsf {T}}} , of vectors u {\displaystyle u} and v {\displaystyle v} . The Sherman–Morrison formula is a special case of the Woodbury formula. Though named after Sherman and Morrison, it appeared already in earlier publications.
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https://en.wikipedia.org/wiki/Sherman–Morrison_formula
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In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT.
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https://en.wikipedia.org/wiki/Matrix_determinant_lemma
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In mathematics, in particular number theory, an odd composite number N is a Somer–Lucas d-pseudoprime (with given d ≥ 1) if there exists a nondegenerate Lucas sequence U ( P , Q ) {\displaystyle U(P,Q)} with the discriminant D = P 2 − 4 Q , {\displaystyle D=P^{2}-4Q,} such that gcd ( N , D ) = 1 {\displaystyle \gcd(N,D)=1} and the rank appearance of N in the sequence U(P, Q) is 1 d ( N − ( D N ) ) , {\displaystyle {\frac {1}{d}}\left(N-\left({\frac {D}{N}}\right)\right),} where ( D N ) {\displaystyle \left({\frac {D}{N}}\right)} is the Jacobi symbol.
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https://en.wikipedia.org/wiki/Somer–Lucas_pseudoprime
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