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c_wn2hc6iccn9a | 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 hy... | Dehn filling |
c_xtgkbfhxo533 | 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. | Dehn filling |
c_wum9ti0t1cfo | 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 i... | Complex hyperbolic space |
c_lsosioqzgzjn | 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\... | Hyperbolic coordinates |
c_sdfui00mrn6u | 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 deriva... | Hyperbolic cotangent |
c_nko0m35d99j1 | 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 re... | Hyperbolic cotangent |
c_kc31zo6hdqdn | 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 ... | Hyperbolic cotangent |
c_9nvokexd2ag8 | 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. | Hyperbolic cotangent |
c_l654u9yjk3tl | 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. | Hyperbolic cotangent |
c_vo79o5eo3ofc | and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. | Hyperbolic cotangent |
c_pycipko16bcx | (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. | Hyperbolic cotangent |
c_ywzldbnqp760 | 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... | Hyperbolic plane |
c_9d31t10943k4 | 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 r... | Hyperbolic plane |
c_cl5ceplhipyb | 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), par... | Hyperbolic plane |
c_hgmce3rtmvrn | 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 \m... | Hyperbolic space |
c_k4hfcodfyz38 | 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 hyperb... | Hyperbolic space |
c_4zc0xpogw2r1 | 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 function... | Hypercomplex analysis |
c_wvfuhzecksze | 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. | Hypercomplex analysis |
c_whmiys7voy4x | 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}\... | Hypercomplex analysis |
c_3uuqzvv84373 | 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}.} | Hypercomplex analysis |
c_vanlgsj73avu | 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 holomorp... | Hypercomplex analysis |
c_ww546l0jcpxd | 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. | Hypercomplex numbers |
c_erh0ulj4cmap | 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 ... | Hyperfunction |
c_r8lg4rttkas9 | 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 han... | Hypergeometric identities |
c_la7mzh2uxozg | 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... | Ideal theory |
c_40thsy9i9n3o | 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 ... | Inclusion mapping |
c_xiwg3hzfk1lv | 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 re... | Complex conjugate representation |
c_0y6nj08hz4pe | 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-dimen... | Complex conjugate representation |
c_hx34f2ji5uw0 | 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 represen... | Contragredient representation |
c_pu25z44zeuv7 | 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 ge... | Algebraic element |
c_ir07j6xvo89a | 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 ... | Particle diffusion |
c_jn1v3fu5ppy5 | 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... | Particle diffusion |
c_ll8i5mp5f7dj | 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.} | Particle diffusion |
c_0bfb8zdvdn7s | 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... | Particle diffusion |
c_m8vqow4pygz2 | 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. | Conductor of an abelian variety |
c_aqwayw3jtxp4 | 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 pol... | Harmonic polynomial |
c_mb7owjuz3lad | 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 s... | Semipermutable subgroup |
c_oe0rmxnrcu8n | 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 R... | Li Shanlan identity |
c_biixwc30wdzp | 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... | Li Shanlan identity |
c_hvspj6zm86te | 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 wavele... | Poisson wavelet |
c_e9fgvauk9u9c | 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 "goin... | Compactification (mathematics) |
c_75jixogfnazv | 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 a... | Seidel adjacency matrix |
c_z6i713fn6scz | 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 regu... | Seidel adjacency matrix |
c_1fd65u31v0n3 | 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... | Cyclic decomposition theorem |
c_efs7fyny8i1f | 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 ... | Weyr canonical form |
c_3ca8qhc0ckve | 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. | Weyr canonical form |
c_u9bdvvqsjdbr | 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 par... | Weyr canonical form |
c_1a1d9oyod7n0 | 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 com... | Gauss composition law |
c_x7zpxv50batx | 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 bina... | Bhargava cube |
c_dsrkw4ycsu1k | 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. | Differential graded Lie algebra |
c_5gcz1794fodt | 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 alg... | Homotopy Lie algebra |
c_xzk13ayj9z6t | 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 int... | Graded module |
c_sp4xw4oi7c24 | 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 t... | Graded module |
c_j9zu0a4q3g7u | 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 sho... | Moduli functor |
c_w9z1kx4skveb | 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 ... | P-compact group |
c_ujwawrscv4yz | 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 frac... | Group of fractional ideals |
c_86emlxlxl1fk | 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 i... | Conjugate element (field theory) |
c_kswex3rxsm51 | 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... | Singular values |
c_w9uik835ctpf | 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\... | Singular values |
c_grz7fpwco7ro | 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 classifyin... | Singular values |
c_5ea6c3tue5r0 | 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 ... | Cofibration |
c_tsxpbsri7j78 | 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 cate... | Cofibration |
c_5v6b5dbqg4u4 | 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 e... | Polarization formula |
c_q88mwi52o6yv | 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 ... | Dolbeault cohomology |
c_ky6kxk1mm9du | 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 ... | Completeness of projective varieties |
c_bhuqi379mg3r | 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 ... | Completeness of projective varieties |
c_qbm028jki9al | 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 sens... | Completeness of projective varieties |
c_299er4cqxr4u | 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 t... | Stiefel–Whitney class |
c_fpw53xnhn27t | 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}\time... | Stiefel–Whitney class |
c_6kvln9iarllp | 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, kno... | Total Chern class |
c_dwiz9kr8eee2 | In mathematics, in particular in algebraic topology, the Hopf invariant is a homotopy invariant of certain maps between n-spheres. | Hopf invariant one problem |
c_g9gv6mj4sbo8 | 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 ... | Right lifting property |
c_72eo6x533zrx | 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... | Berlekamp–Zassenhaus algorithm |
c_pd6kjneu7ehw | 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 ... | Weitzenböck identity |
c_wfz2v2qgyggk | 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. T... | Minimal volume |
c_4ckbw91f8e1y | {\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 constrain... | Minimal volume |
c_peig63smiq4w | 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 sphe... | Minimal volume |
c_1wynahoxy7qb | 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, ... | Minimal volume |
c_wgherhva1u6j | 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 |}.} | Minimal volume |
c_h5gn1k2s04ju | 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)^{... | Minimal volume |
c_auuptvxxe7kk | 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. | Formally real field |
c_iekhun1aetvr | 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 g... | Differentiation in Fréchet spaces |
c_mnqb5oo3scvp | 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}(\s... | Rademacher system |
c_lzpzkbh74bpt | 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. | DG Algebra |
c_79vu972vlp8s | 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... | Homotopy lifting property |
c_dommhphrdhny | 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... | Whitney extension theorem |
c_7t8qpq809cqr | 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 (\var... | Content (measure theory) |
c_nfiztohyzhto | 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: w... | Content (measure theory) |
c_pajvfeuldcs0 | 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. ... | Content (measure theory) |
c_socqgoah9ptb | 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. | Inner measure |
c_4vxo3joxmlqc | 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 Fréch... | Fréchet manifold |
c_l2q29r26l80n | 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. T... | Elliptic complex |
c_20tt3xzmjan6 | 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. | Hecke operator |
c_5y2142frarpl | 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 (ring theory) |
c_u1oji4mphp0x | 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... | Flat map (ring theory) |
c_83k8z6x35qk6 | 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). | De Groot dual |
c_wdxlyh7e6kzm | 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} ... | Bunch–Nielsen–Sorensen formula |
c_seqltl4dri9p | 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... | Sherman–Morrison formula |
c_pwnmvjtkmpd1 | 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. | Matrix determinant lemma |
c_32mkzgaxdmd6 | 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... | Somer–Lucas pseudoprime |
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