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It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert–Weber space.
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https://en.wikipedia.org/wiki/3-manifold
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Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space. With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, and the dihedral angle between these pentagons is 72°. This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a hyperbolic manifold. It is a quotient space of the order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle.
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https://en.wikipedia.org/wiki/3-manifold
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In mathematics, Selberg's conjecture, also known as Selberg's eigenvalue conjecture, conjectured by Selberg (1965, p. 13), states that the eigenvalues of the Laplace operator on Maass wave forms of congruence subgroups are at least 1/4. Selberg showed that the eigenvalues are at least 3/16. Subsequent works improved the bound, and the best bound currently known is 975/4096≈0.238..., due to Kim & Sarnak (2003).
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https://en.wikipedia.org/wiki/Selberg_conjecture
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The generalized Ramanujan conjecture for the general linear group implies Selberg's conjecture. More precisely, Selberg's conjecture is essentially the generalized Ramanujan conjecture for the group GL2 over the rationals at the infinite place, and says that the component at infinity of the corresponding representation is a principal series representation of GL2(R) (rather than a complementary series representation). The generalized Ramanujan conjecture in turn follows from the Langlands functoriality conjecture, and this has led to some progress on Selberg's conjecture.
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https://en.wikipedia.org/wiki/Selberg_conjecture
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In mathematics, Sendov's conjecture, sometimes also called Ilieff's conjecture, concerns the relationship between the locations of roots and critical points of a polynomial function of a complex variable. It is named after Blagovest Sendov. The conjecture states that for a polynomial f ( z ) = ( z − r 1 ) ⋯ ( z − r n ) , ( n ≥ 2 ) {\displaystyle f(z)=(z-r_{1})\cdots (z-r_{n}),\qquad (n\geq 2)} with all roots r1, ..., rn inside the closed unit disk |z| ≤ 1, each of the n roots is at a distance no more than 1 from at least one critical point.
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https://en.wikipedia.org/wiki/Sendov's_conjecture
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The Gauss–Lucas theorem says that all of the critical points lie within the convex hull of the roots. It follows that the critical points must be within the unit disk, since the roots are. The conjecture has been proven for n < 9 by Brown-Xiang and for n sufficiently large by Tao.
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https://en.wikipedia.org/wiki/Sendov's_conjecture
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In mathematics, Serre and localizing subcategories form important classes of subcategories of an abelian category. Localizing subcategories are certain Serre subcategories. They are strongly linked to the notion of a quotient category.
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https://en.wikipedia.org/wiki/Localizing_subcategory
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In mathematics, Serre's modularity conjecture, introduced by Jean-Pierre Serre (1975, 1987), states that an odd, irreducible, two-dimensional Galois representation over a finite field arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by Chandrashekhar Khare in 2005, and a proof of the full conjecture was completed jointly by Khare and Jean-Pierre Wintenberger in 2008.
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https://en.wikipedia.org/wiki/Serre_modularity_conjecture
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In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain purely algebraic problems, in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil's initial definition of intersection numbers, around 1949, there had been a question of how to provide a more flexible and computable theory. Let R be a (Noetherian, commutative) regular local ring and P and Q be prime ideals of R. In 1958, Serre realized that classical algebraic-geometric ideas of multiplicity could be generalized using the concepts of homological algebra. Serre defined the intersection multiplicity of R/P and R/Q by means of the Tor functors of homological algebra, as χ ( R / P , R / Q ) := ∑ i = 0 ∞ ( − 1 ) i ℓ R ( Tor i R ( R / P , R / Q ) ) .
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https://en.wikipedia.org/wiki/Serre's_intersection_formula
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{\displaystyle \chi (R/P,R/Q):=\sum _{i=0}^{\infty }(-1)^{i}\ell _{R}(\operatorname {Tor} _{i}^{R}(R/P,R/Q)).} This requires the concept of the length of a module, denoted here by ℓ R {\displaystyle \ell _{R}} , and the assumption that ℓ R ( ( R / P ) ⊗ ( R / Q ) ) < ∞ . {\displaystyle \ell _{R}((R/P)\otimes (R/Q))<\infty .}
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https://en.wikipedia.org/wiki/Serre's_intersection_formula
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If this idea were to work, however, certain classical relationships would presumably have to continue to hold. Serre singled out four important properties. These then became conjectures, challenging in the general case. (There are more general statements of these conjectures where R/P and R/Q are replaced by finitely generated modules: see Serre's Local Algebra for more details.)
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https://en.wikipedia.org/wiki/Serre's_intersection_formula
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In mathematics, Shafarevich's theorem states that any finite solvable group is the Galois group of some finite extension of the rational numbers. It was first proved by Igor Shafarevich (1954), though Alexander Schmidt later pointed out a gap in the proof, which was fixed by Shafarevich (1989).
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https://en.wikipedia.org/wiki/Shafarevich's_theorem_on_solvable_Galois_groups
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In mathematics, Sharafutdinov's retraction is a construction that gives a retraction of an open non-negatively curved Riemannian manifold onto its soul. It was first used by Sharafutdinov to show that any two souls of a complete Riemannian manifold with non-negative sectional curvature are isometric. Perelman later showed that in this setting, Sharafutdinov's retraction is in fact a submersion, thereby essentially settling the soul conjecture.For open non-negatively curved Alexandrov space, Perelman also showed that there exists a Sharafutdinov retraction from the entire space to the soul. However it is not yet known whether this retraction is submetry or not. == References ==
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https://en.wikipedia.org/wiki/Sharafutdinov's_retraction
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In mathematics, Sharkovskii's theorem (also spelled Sharkovsky's theorem, Sharkovskiy's theorem, Šarkovskii's theorem or Sarkovskii's theorem), named after Oleksandr Mykolayovych Sharkovsky, who published it in 1964, is a result about discrete dynamical systems. One of the implications of the theorem is that if a discrete dynamical system on the real line has a periodic point of period 3, then it must have periodic points of every other period.
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https://en.wikipedia.org/wiki/Sharkovsky's_theorem
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In mathematics, Shimura's reciprocity law, introduced by Shimura (1971), describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field theory for such fields. There are also higher-dimensional generalizations.
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https://en.wikipedia.org/wiki/Shimura's_reciprocity_law
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In mathematics, Shintani's unit theorem introduced by Shintani (1976, proposition 4) is a refinement of Dirichlet's unit theorem and states that a subgroup of finite index of the totally positive units of a number field has a fundamental domain given by a rational polyhedric cone in the Minkowski space of the field (Neukirch 1999, p. 507).
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https://en.wikipedia.org/wiki/Shintani's_unit_theorem
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In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional elliptic modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric n × n matrices with positive definite imaginary part; the forms must satisfy an automorphy condition.
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https://en.wikipedia.org/wiki/Siegel_modular_form
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Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables. Siegel modular forms were first investigated by Carl Ludwig Siegel (1939) for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, such as arithmetic geometry and elliptic cohomology. Siegel modular forms have also been used in some areas of physics, such as conformal field theory and black hole thermodynamics in string theory.
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https://en.wikipedia.org/wiki/Siegel_modular_form
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In mathematics, Siegel's identity refers to one of two formulae that are used in the resolution of Diophantine equations.
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https://en.wikipedia.org/wiki/Siegel_identity
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In mathematics, Siegel's theorem on integral points states that for a smooth algebraic curve C of genus g defined over a number field K, presented in affine space in a given coordinate system, there are only finitely many points on C with coordinates in the ring of integers O of K, provided g > 0. The theorem was first proved in 1929 by Carl Ludwig Siegel and was the first major result on Diophantine equations that depended only on the genus and not any special algebraic form of the equations. For g > 1 it was superseded by Faltings's theorem in 1983.
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https://en.wikipedia.org/wiki/Siegel's_theorem_on_integral_points
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In mathematics, Sister Celine's polynomials are a family of hypergeometric polynomials introduced by Mary Celine Fasenmyer (1947). They include Legendre polynomials, Jacobi polynomials, and Bateman polynomials as special cases.
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https://en.wikipedia.org/wiki/Sister_Celine's_polynomials
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In mathematics, Smale's axiom A defines a class of dynamical systems which have been extensively studied and whose dynamics is relatively well understood. A prominent example is the Smale horseshoe map. The term "axiom A" originates with Stephen Smale. The importance of such systems is demonstrated by the chaotic hypothesis, which states that, 'for all practical purposes', a many-body thermostatted system is approximated by an Anosov system.
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https://en.wikipedia.org/wiki/Axiom_A
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In mathematics, Sobolev orthogonal polynomials are orthogonal polynomials with respect to a Sobolev inner product, i.e. an inner product with derivatives. By having conditions on the derivatives, the Sobolev orthogonal polynomials in general no longer share some of the nice features that classical orthogonal polynomials have. Sobolev orthogonal polynomials are named after Sergei Lvovich Sobolev.
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https://en.wikipedia.org/wiki/Sobolev_orthogonal_polynomials
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In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems.
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https://en.wikipedia.org/wiki/Sobolev_spaces_for_planar_domains
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In mathematics, Solèr's theorem is a result concerning certain infinite-dimensional vector spaces. It states that any orthomodular form that has an infinite orthonormal sequence is a Hilbert space over the real numbers, complex numbers or quaternions. Originally proved by Maria Pia Solèr, the result is significant for quantum logic and the foundations of quantum mechanics.
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https://en.wikipedia.org/wiki/Solèr_theorem
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In particular, Solèr's theorem helps to fill a gap in the effort to use Gleason's theorem to rederive quantum mechanics from information-theoretic postulates. It is also an important step in the Heunen-Kornell axiomatisation of the category of Hilbert spaces. Physicist John C. Baez notes,Nothing in the assumptions mentions the continuum: the hypotheses are purely algebraic. It therefore seems quite magical that is forced to be the real numbers, complex numbers or quaternions.Writing a decade after Solèr's original publication, Pitowsky calls her theorem "celebrated".
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https://en.wikipedia.org/wiki/Solèr_theorem
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In mathematics, Somos' quadratic recurrence constant, named after Michael Somos, is the number σ = 1 2 3 ⋯ = 1 1 / 2 2 1 / 4 3 1 / 8 ⋯ . {\displaystyle \sigma ={\sqrt {1{\sqrt {2{\sqrt {3\cdots }}}}}}=1^{1/2}\;2^{1/4}\;3^{1/8}\cdots .\,} This can be easily re-written into the far more quickly converging product representation σ = σ 2 / σ = ( 2 1 ) 1 / 2 ( 3 2 ) 1 / 4 ( 4 3 ) 1 / 8 ( 5 4 ) 1 / 16 ⋯ , {\displaystyle \sigma =\sigma ^{2}/\sigma =\left({\frac {2}{1}}\right)^{1/2}\left({\frac {3}{2}}\right)^{1/4}\left({\frac {4}{3}}\right)^{1/8}\left({\frac {5}{4}}\right)^{1/16}\cdots ,} which can then be compactly represented in infinite product form by: σ = ∏ k = 1 ∞ ( 1 + 1 k ) 1 2 k . {\displaystyle \sigma =\prod _{k=1}^{\infty }\left(1+{\frac {1}{k}}\right)^{\frac {1}{2^{k}}}.} The constant σ arises when studying the asymptotic behaviour of the sequence g 0 = 1 ; g n = n g n − 1 2 , n > 1 , {\displaystyle g_{0}=1\,;\,g_{n}=ng_{n-1}^{2},\qquad n>1,\,} with first few terms 1, 1, 2, 12, 576, 1658880, ... (sequence A052129 in the OEIS).
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https://en.wikipedia.org/wiki/Somos'_quadratic_recurrence_constant
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This sequence can be shown to have asymptotic behaviour as follows: g n ∼ σ 2 n n + 2 + O ( 1 n ) . {\displaystyle g_{n}\sim {\frac {\sigma ^{2^{n}}}{n+2+O({\frac {1}{n}})}}.} Guillera and Sondow give a representation in terms of the derivative of the Lerch transcendent: ln σ = − 1 2 ∂ Φ ∂ s ( 1 2 , 0 , 1 ) {\displaystyle \ln \sigma ={\frac {-1}{2}}{\frac {\partial \Phi }{\partial s}}\!\left({\frac {1}{2}},0,1\right)} where ln is the natural logarithm and Φ {\displaystyle \Phi } (z, s, q) is the Lerch transcendent. Finally, σ = 1.661687949633594121296 … {\displaystyle \sigma =1.661687949633594121296\dots \;} (sequence A112302 in the OEIS).
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https://en.wikipedia.org/wiki/Somos'_quadratic_recurrence_constant
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In mathematics, Sonine's formula is any of several formulas involving Bessel functions found by Nikolay Yakovlevich Sonin. One such formula is the following integral formula involving a product of three Bessel functions: ∫ 0 ∞ J z ( a t ) J z ( b t ) J z ( c t ) t 1 − z d t = 2 z − 1 Δ ( a , b , c ) 2 z − 1 π 1 / 2 Γ ( z + 1 2 ) ( a b c ) z {\displaystyle \int _{0}^{\infty }J_{z}(at)J_{z}(bt)J_{z}(ct)t^{1-z}\,dt={\frac {2^{z-1}\Delta (a,b,c)^{2z-1}}{\pi ^{1/2}\Gamma (z+{\tfrac {1}{2}})(abc)^{z}}}} where Δ is the area of a triangle with given sides.
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https://en.wikipedia.org/wiki/Sonine_formula
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In mathematics, Sophie Germain's identity is a polynomial factorization named after Sophie Germain stating that Beyond its use in elementary algebra, it can also be used in number theory to factorize integers of the special form x 4 + 4 y 4 {\displaystyle x^{4}+4y^{4}} , and it frequently forms the basis of problems in mathematics competitions.
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https://en.wikipedia.org/wiki/Sophie_Germain's_identity
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In mathematics, Spanier–Whitehead duality is a duality theory in homotopy theory, based on a geometrical idea that a topological space X may be considered as dual to its complement in the n-sphere, where n is large enough. Its origins lie in Alexander duality theory, in homology theory, concerning complements in manifolds. The theory is also referred to as S-duality, but this can now cause possible confusion with the S-duality of string theory.
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https://en.wikipedia.org/wiki/S-duality_(homotopy_theory)
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It is named for Edwin Spanier and J. H. C. Whitehead, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into stable homotopy theory.
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https://en.wikipedia.org/wiki/S-duality_(homotopy_theory)
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In mathematics, Specht's theorem gives a necessary and sufficient condition for two complex matrices to be unitarily equivalent. It is named after Wilhelm Specht, who proved the theorem in 1940.Two matrices A and B with complex number entries are said to be unitarily equivalent if there exists a unitary matrix U such that B = U *AU. Two matrices which are unitarily equivalent are also similar. Two similar matrices represent the same linear map, but with respect to a different basis; unitary equivalence corresponds to a change from an orthonormal basis to another orthonormal basis.
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https://en.wikipedia.org/wiki/Specht's_theorem
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If A and B are unitarily equivalent, then tr AA* = tr BB*, where tr denotes the trace (in other words, the Frobenius norm is a unitary invariant). This follows from the cyclic invariance of the trace: if B = U *AU, then tr BB* = tr U *AUU *A*U = tr AUU *A*UU * = tr AA*, where the second equality is cyclic invariance.Thus, tr AA* = tr BB* is a necessary condition for unitary equivalence, but it is not sufficient. Specht's theorem gives infinitely many necessary conditions which together are also sufficient.
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https://en.wikipedia.org/wiki/Specht's_theorem
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The formulation of the theorem uses the following definition. A word in two variables, say x and y, is an expression of the form W ( x , y ) = x m 1 y n 1 x m 2 y n 2 ⋯ x m p , {\displaystyle W(x,y)=x^{m_{1}}y^{n_{1}}x^{m_{2}}y^{n_{2}}\cdots x^{m_{p}},} where m1, n1, m2, n2, …, mp are non-negative integers. The degree of this word is m 1 + n 1 + m 2 + n 2 + ⋯ + m p .
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https://en.wikipedia.org/wiki/Specht's_theorem
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{\displaystyle m_{1}+n_{1}+m_{2}+n_{2}+\cdots +m_{p}.} Specht's theorem: Two matrices A and B are unitarily equivalent if and only if tr W(A, A*) = tr W(B, B*) for all words W.The theorem gives an infinite number of trace identities, but it can be reduced to a finite subset. Let n denote the size of the matrices A and B. For the case n = 2, the following three conditions are sufficient: tr A = tr B , tr A 2 = tr B 2 , and tr A A ∗ = tr B B ∗ .
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https://en.wikipedia.org/wiki/Specht's_theorem
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In mathematics, Spence's function, or dilogarithm, denoted as Li2(z), is a particular case of the polylogarithm. Two related special functions are referred to as Spence's function, the dilogarithm itself: Li 2 ( z ) = − ∫ 0 z ln ( 1 − u ) u d u , z ∈ C {\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\ln(1-u) \over u}\,du{\text{, }}z\in \mathbb {C} } and its reflection. For |z| < 1, an infinite series also applies (the integral definition constitutes its analytical extension to the complex plane): Li 2 ( z ) = ∑ k = 1 ∞ z k k 2 . {\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}
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https://en.wikipedia.org/wiki/Spence's_function
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Alternatively, the dilogarithm function is sometimes defined as ∫ 1 v ln t 1 − t d t = Li 2 ( 1 − v ) . {\displaystyle \int _{1}^{v}{\frac {\ln t}{1-t}}dt=\operatorname {Li} _{2}(1-v).} In hyperbolic geometry the dilogarithm can be used to compute the volume of an ideal simplex.
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https://en.wikipedia.org/wiki/Spence's_function
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Specifically, a simplex whose vertices have cross ratio z has hyperbolic volume D ( z ) = Im Li 2 ( z ) + arg ( 1 − z ) log | z | . {\displaystyle D(z)=\operatorname {Im} \operatorname {Li} _{2}(z)+\arg(1-z)\log |z|.} The function D(z) is sometimes called the Bloch-Wigner function.
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https://en.wikipedia.org/wiki/Spence's_function
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Lobachevsky's function and Clausen's function are closely related functions. William Spence, after whom the function was named by early writers in the field, was a Scottish mathematician working in the early nineteenth century. He was at school with John Galt, who later wrote a biographical essay on Spence.
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https://en.wikipedia.org/wiki/Spence's_function
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In mathematics, Spencer cohomology is cohomology of a manifold with coefficients in the sheaf of solutions of a linear partial differential operator. It was introduced by Donald C. Spencer in 1969.
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https://en.wikipedia.org/wiki/Spencer_cohomology
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In mathematics, Sperner's lemma is a combinatorial result on colorings of triangulations, analogous to the Brouwer fixed point theorem, which is equivalent to it. It states that every Sperner coloring (described below) of a triangulation of an n {\displaystyle n} -dimensional simplex contains a cell whose vertices all have different colors. The initial result of this kind was proved by Emanuel Sperner, in relation with proofs of invariance of domain. Sperner colorings have been used for effective computation of fixed points and in root-finding algorithms, and are applied in fair division (cake cutting) algorithms.
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https://en.wikipedia.org/wiki/Sperner_coloring
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According to the Soviet Mathematical Encyclopaedia (ed. I.M. Vinogradov), a related 1929 theorem (of Knaster, Borsuk and Mazurkiewicz) had also become known as the Sperner lemma – this point is discussed in the English translation (ed. M. Hazewinkel). It is now commonly known as the Knaster–Kuratowski–Mazurkiewicz lemma.
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https://en.wikipedia.org/wiki/Sperner_coloring
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In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator having degree at most n has length at most 2nπ.
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https://en.wikipedia.org/wiki/Spijker's_lemma
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In mathematics, Spouge's approximation is a formula for computing an approximation of the gamma function. It was named after John L. Spouge, who defined the formula in a 1994 paper. The formula is a modification of Stirling's approximation, and has the form Γ ( z + 1 ) = ( z + a ) z + 1 2 e − z − a ( c 0 + ∑ k = 1 a − 1 c k z + k + ε a ( z ) ) {\displaystyle \Gamma (z+1)=(z+a)^{z+{\frac {1}{2}}}e^{-z-a}\left(c_{0}+\sum _{k=1}^{a-1}{\frac {c_{k}}{z+k}}+\varepsilon _{a}(z)\right)} where a is an arbitrary positive integer and the coefficients are given by c 0 = 2 π c k = ( − 1 ) k − 1 ( k − 1 ) ! ( − k + a ) k − 1 2 e − k + a k ∈ { 1 , 2 , … , a − 1 } .
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https://en.wikipedia.org/wiki/Spouge's_approximation
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{\displaystyle {\begin{aligned}c_{0}&={\sqrt {2\pi }}\\c_{k}&={\frac {(-1)^{k-1}}{(k-1)! }}(-k+a)^{k-{\frac {1}{2}}}e^{-k+a}\qquad k\in \{1,2,\dots ,a-1\}.\end{aligned}}} Spouge has proved that, if Re(z) > 0 and a > 2, the relative error in discarding εa(z) is bounded by a − 1 2 ( 2 π ) − a − 1 2 . {\displaystyle a^{-{\frac {1}{2}}}(2\pi )^{-a-{\frac {1}{2}}}.}
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https://en.wikipedia.org/wiki/Spouge's_approximation
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The formula is similar to the Lanczos approximation, but has some distinct features. Whereas the Lanczos formula exhibits faster convergence, Spouge's coefficients are much easier to calculate and the error can be set arbitrarily low. The formula is therefore feasible for arbitrary-precision evaluation of the gamma function. However, special care must be taken to use sufficient precision when computing the sum due to the large size of the coefficients ck, as well as their alternating sign. For example, for a = 49, one must compute the sum using about 65 decimal digits of precision in order to obtain the promised 40 decimal digits of accuracy.
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https://en.wikipedia.org/wiki/Spouge's_approximation
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In mathematics, Steinhaus–Moser notation is a notation for expressing certain large numbers. It is an extension (devised by Leo Moser) of Hugo Steinhaus's polygon notation.
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https://en.wikipedia.org/wiki/Steinhaus–Moser_notation
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In mathematics, Stickelberger's theorem is a result of algebraic number theory, which gives some information about the Galois module structure of class groups of cyclotomic fields. A special case was first proven by Ernst Kummer (1847) while the general result is due to Ludwig Stickelberger (1890).
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https://en.wikipedia.org/wiki/Stickelberger's_theorem
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In mathematics, Stinespring's dilation theorem, also called Stinespring's factorization theorem, named after W. Forrest Stinespring, is a result from operator theory that represents any completely positive map on a C*-algebra A as a composition of two completely positive maps each of which has a special form: A *-representation of A on some auxiliary Hilbert space K followed by An operator map of the form T ↦ V*TV.Moreover, Stinespring's theorem is a structure theorem from a C*-algebra into the algebra of bounded operators on a Hilbert space. Completely positive maps are shown to be simple modifications of *-representations, or sometimes called *-homomorphisms.
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https://en.wikipedia.org/wiki/Stinespring_factorization_theorem
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In mathematics, Stirling numbers arise in a variety of analytic and combinatorial problems. They are named after James Stirling, who introduced them in a purely algebraic setting in his book Methodus differentialis (1730). They were rediscovered and given a combinatorial meaning by Masanobu Saka in 1782.Two different sets of numbers bear this name: the Stirling numbers of the first kind and the Stirling numbers of the second kind.
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https://en.wikipedia.org/wiki/Karamata_notation
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Additionally, Lah numbers are sometimes referred to as Stirling numbers of the third kind. Each kind is detailed in its respective article, this one serving as a description of relations between them. A common property of all three kinds is that they describe coefficients relating three different sequences of polynomials that frequently arise in combinatorics. Moreover, all three can be defined as the number of partitions of n elements into k non-empty subsets, where each subset is endowed with a certain kind of order (no order, cyclical, or linear).
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https://en.wikipedia.org/wiki/Karamata_notation
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In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a good approximation, leading to accurate results even for small values of n {\displaystyle n} . It is named after James Stirling, though a related but less precise result was first stated by Abraham de Moivre.One way of stating the approximation involves the logarithm of the factorial: where the big O notation means that, for all sufficiently large values of n {\displaystyle n} , the difference between ln ( n ! ) {\displaystyle \ln(n!)}
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https://en.wikipedia.org/wiki/Stirling's_approximation
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and n ln n − n {\displaystyle n\ln n-n} will be at most proportional to the logarithm. In computer science applications such as the worst-case lower bound for comparison sorting, it is convenient to use instead the binary logarithm, giving the equivalent form The error term in either base can be expressed more precisely as 1 2 log ( 2 π n ) + O ( 1 n ) {\displaystyle {\tfrac {1}{2}}\log(2\pi n)+O({\tfrac {1}{n}})} , corresponding to an approximate formula for the factorial itself, Here the sign ∼ {\displaystyle \sim } means that the two quantities are asymptotic, that is, that their ratio tends to 1 as n {\displaystyle n} tends to infinity. The following version of the bound holds for all n ≥ 1 {\displaystyle n\geq 1} , rather than only asymptotically:
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https://en.wikipedia.org/wiki/Stirling's_approximation
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In mathematics, Stone's representation theorem for Boolean algebras states that every Boolean algebra is isomorphic to a certain field of sets. The theorem is fundamental to the deeper understanding of Boolean algebra that emerged in the first half of the 20th century. The theorem was first proved by Marshall H. Stone. Stone was led to it by his study of the spectral theory of operators on a Hilbert space.
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https://en.wikipedia.org/wiki/Representation_theorem_for_Boolean_algebras
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In mathematics, Stone's theorem on one-parameter unitary groups is a basic theorem of functional analysis that establishes a one-to-one correspondence between self-adjoint operators on a Hilbert space H {\displaystyle {\mathcal {H}}} and one-parameter families ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} of unitary operators that are strongly continuous, i.e., ∀ t 0 ∈ R , ψ ∈ H: lim t → t 0 U t ( ψ ) = U t 0 ( ψ ) , {\displaystyle \forall t_{0}\in \mathbb {R} ,\psi \in {\mathcal {H}}:\qquad \lim _{t\to t_{0}}U_{t}(\psi )=U_{t_{0}}(\psi ),} and are homomorphisms, i.e., ∀ s , t ∈ R: U t + s = U t U s . {\displaystyle \forall s,t\in \mathbb {R} :\qquad U_{t+s}=U_{t}U_{s}.} Such one-parameter families are ordinarily referred to as strongly continuous one-parameter unitary groups.
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https://en.wikipedia.org/wiki/Stone's_theorem_on_one-parameter_unitary_groups
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The theorem was proved by Marshall Stone (1930, 1932), and John von Neumann (1932) showed that the requirement that ( U t ) t ∈ R {\displaystyle (U_{t})_{t\in \mathbb {R} }} be strongly continuous can be relaxed to say that it is merely weakly measurable, at least when the Hilbert space is separable. This is an impressive result, as it allows one to define the derivative of the mapping t ↦ U t , {\displaystyle t\mapsto U_{t},} which is only supposed to be continuous. It is also related to the theory of Lie groups and Lie algebras.
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https://en.wikipedia.org/wiki/Stone's_theorem_on_one-parameter_unitary_groups
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In mathematics, Strassmann's theorem is a result in field theory. It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes.
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https://en.wikipedia.org/wiki/Strassmann's_theorem
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In mathematics, Sullivan conjecture or Sullivan's conjecture on maps from classifying spaces can refer to any of several results and conjectures prompted by homotopy theory work of Dennis Sullivan. A basic theme and motivation concerns the fixed point set in group actions of a finite group G {\displaystyle G} . The most elementary formulation, however, is in terms of the classifying space B G {\displaystyle BG} of such a group.
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https://en.wikipedia.org/wiki/Sullivan_conjecture
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Roughly speaking, it is difficult to map such a space B G {\displaystyle BG} continuously into a finite CW complex X {\displaystyle X} in a non-trivial manner. Such a version of the Sullivan conjecture was first proved by Haynes Miller. Specifically, in 1984, Miller proved that the function space, carrying the compact-open topology, of base point-preserving mappings from B G {\displaystyle BG} to X {\displaystyle X} is weakly contractible.
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https://en.wikipedia.org/wiki/Sullivan_conjecture
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This is equivalent to the statement that the map X {\displaystyle X} → F ( B G , X ) {\displaystyle F(BG,X)} from X to the function space of maps B G {\displaystyle BG} → X {\displaystyle X} , not necessarily preserving the base point, given by sending a point x {\displaystyle x} of X {\displaystyle X} to the constant map whose image is x {\displaystyle x} is a weak equivalence. The mapping space F ( B G , X ) {\displaystyle F(BG,X)} is an example of a homotopy fixed point set. Specifically, F ( B G , X ) {\displaystyle F(BG,X)} is the homotopy fixed point set of the group G {\displaystyle G} acting by the trivial action on X {\displaystyle X} .
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https://en.wikipedia.org/wiki/Sullivan_conjecture
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In general, for a group G {\displaystyle G} acting on a space X {\displaystyle X} , the homotopy fixed points are the fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} of the mapping space F ( E G , X ) {\displaystyle F(EG,X)} of maps from the universal cover E G {\displaystyle EG} of B G {\displaystyle BG} to X {\displaystyle X} under the G {\displaystyle G} -action on F ( E G , X ) {\displaystyle F(EG,X)} given by g {\displaystyle g} in G {\displaystyle G} acts on a map f {\displaystyle f} in F ( E G , X ) {\displaystyle F(EG,X)} by sending it to g f g − 1 {\displaystyle gfg^{-1}} . The G {\displaystyle G} -equivariant map from E G {\displaystyle EG} to a single point ∗ {\displaystyle *} induces a natural map η: X G = F ( ∗ , X ) G {\displaystyle X^{G}=F(*,X)^{G}} → F ( E G , X ) G {\displaystyle F(EG,X)^{G}} from the fixed points to the homotopy fixed points of G {\displaystyle G} acting on X {\displaystyle X} . Miller's theorem is that η is a weak equivalence for trivial G {\displaystyle G} -actions on finite-dimensional CW complexes.
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https://en.wikipedia.org/wiki/Sullivan_conjecture
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An important ingredient and motivation for his proof is a result of Gunnar Carlsson on the homology of B Z / 2 {\displaystyle BZ/2} as an unstable module over the Steenrod algebra.Miller's theorem generalizes to a version of Sullivan's conjecture in which the action on X {\displaystyle X} is allowed to be non-trivial. In, Sullivan conjectured that η is a weak equivalence after a certain p-completion procedure due to A. Bousfield and D. Kan for the group G = Z / 2 {\displaystyle G=Z/2} . This conjecture was incorrect as stated, but a correct version was given by Miller, and proven independently by Dwyer-Miller-Neisendorfer, Carlsson, and Jean Lannes, showing that the natural map ( X G ) p {\displaystyle (X^{G})_{p}} → F ( E G , ( X ) p ) G {\displaystyle F(EG,(X)_{p})^{G}} is a weak equivalence when the order of G {\displaystyle G} is a power of a prime p, and where ( X ) p {\displaystyle (X)_{p}} denotes the Bousfield-Kan p-completion of X {\displaystyle X} . Miller's proof involves an unstable Adams spectral sequence, Carlsson's proof uses his affirmative solution of the Segal conjecture and also provides information about the homotopy fixed points F ( E G , X ) G {\displaystyle F(EG,X)^{G}} before completion, and Lannes's proof involves his T-functor.
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https://en.wikipedia.org/wiki/Sullivan_conjecture
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In mathematics, Surface fairing is an aspect of mesh smoothing. The goal of surface fairing is to compute shapes that are as smooth as possible. On an abstract level, mesh smoothing is concerned with the design and computation of smooth functions f: S → R d {\displaystyle f:S\rightarrow \mathbb {R} ^{d}} on a triangle mesh. Mesh fairing does not just slightly smooth the function f {\displaystyle f} in order to remove the high frequency noise.
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https://en.wikipedia.org/wiki/Surface_fairing
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It also smooths the function as much as possible in order to obtain, e.g., an as-smooth-as-possible surface patch or an as-smooth-as-possible shape deformation.How to actually measure smoothness or fairness obviously depends on the application, but in general fair surfaces should follow the principle of simplest shape: the surface should be free of any unnecessary details or oscillations. This can be modeled by a suitable energy that penalizes unaesthetic behavior of the surface.
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https://en.wikipedia.org/wiki/Surface_fairing
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A minimization of this fairness energy—subject to user-defined constraints—eventually yields the desired shape. Example applications include the construction of smooth blend surfaces and hole filling by smooth patches. == References ==
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https://en.wikipedia.org/wiki/Surface_fairing
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In mathematics, Suslin's problem is a question about totally ordered sets posed by Mikhail Yakovlevich Suslin (1920) and published posthumously. It has been shown to be independent of the standard axiomatic system of set theory known as ZFC; Solovay & Tennenbaum (1971) showed that the statement can neither be proven nor disproven from those axioms, assuming ZF is consistent. (Suslin is also sometimes written with the French transliteration as Souslin, from the Cyrillic Суслин.)
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https://en.wikipedia.org/wiki/Suslin's_hypothesis
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In mathematics, Sylvester’s criterion is a necessary and sufficient criterion to determine whether a Hermitian matrix is positive-definite. It is named after James Joseph Sylvester. Sylvester's criterion states that a n × n Hermitian matrix M is positive-definite if and only if all the following matrices have a positive determinant: the upper left 1-by-1 corner of M, the upper left 2-by-2 corner of M, the upper left 3-by-3 corner of M, ⋮ {\displaystyle {}\quad \vdots } M itself.In other words, all of the leading principal minors must be positive. By using appropriate permutations of rows and columns of M, it can also be shown that the positivity of any nested sequence of n principal minors of M is equivalent to M being positive-definite.An analogous theorem holds for characterizing positive-semidefinite Hermitian matrices, except that it is no longer sufficient to consider only the leading principal minors: a Hermitian matrix M is positive-semidefinite if and only if all principal minors of M are nonnegative.
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https://en.wikipedia.org/wiki/Sylvester's_criterion
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In mathematics, Symmetry-preserving observers, also known as invariant filters, are estimation techniques whose structure and design take advantage of the natural symmetries (or invariances) of the considered nonlinear model. As such, the main benefit is an expected much larger domain of convergence than standard filtering methods, e.g. Extended Kalman Filter (EKF) or Unscented Kalman Filter (UKF).
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https://en.wikipedia.org/wiki/Symmetry-preserving_filter
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In mathematics, Szymanski's conjecture, named after Ted H. Szymanski (1989), states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is, if the permutation σ matches each vertex v to another vertex σ(v), then for each v there exists a path in the hypercube graph from v to σ(v) such that no two paths for two different vertices u and v use the same edge in the same direction. Through computer experiments it has been verified that the conjecture is true for n ≤ 4 (Baudon, Fertin & Havel 2001). Although the conjecture remains open for n ≥ 5, in this case there exist permutations that require the use of paths that are not shortest paths in order to be routed (Lubiw 1990).
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https://en.wikipedia.org/wiki/Szymanski's_conjecture
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In mathematics, Tait's conjecture states that "Every 3-connected planar cubic graph has a Hamiltonian cycle (along the edges) through all its vertices". It was proposed by P. G. Tait (1884) and disproved by W. T. Tutte (1946), who constructed a counterexample with 25 faces, 69 edges and 46 vertices. Several smaller counterexamples, with 21 faces, 57 edges and 38 vertices, were later proved minimal by Holton & McKay (1988). The condition that the graph be 3-regular is necessary due to polyhedra such as the rhombic dodecahedron, which forms a bipartite graph with six degree-four vertices on one side and eight degree-three vertices on the other side; because any Hamiltonian cycle would have to alternate between the two sides of the bipartition, but they have unequal numbers of vertices, the rhombic dodecahedron is not Hamiltonian.
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https://en.wikipedia.org/wiki/Tait's_conjecture
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The conjecture was significant, because if true, it would have implied the four color theorem: as Tait described, the four-color problem is equivalent to the problem of finding 3-edge-colorings of bridgeless cubic planar graphs. In a Hamiltonian cubic planar graph, such an edge coloring is easy to find: use two colors alternately on the cycle, and a third color for all remaining edges. Alternatively, a 4-coloring of the faces of a Hamiltonian cubic planar graph may be constructed directly, using two colors for the faces inside the cycle and two more colors for the faces outside.
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https://en.wikipedia.org/wiki/Tait's_conjecture
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In mathematics, Takeuti's conjecture is the conjecture of Gaisi Takeuti that a sequent formalisation of second-order logic has cut-elimination (Takeuti 1953). It was settled positively: By Tait, using a semantic technique for proving cut-elimination, based on work by Schütte (Tait 1966); Independently by Prawitz (Prawitz 1968) and Takahashi (Takahashi 1967) by a similar technique (Takahashi 1967) - although Prawitz's and Takahashi's proofs are not limited to second-order logic, but concern higher-order logics in general; It is a corollary of Jean-Yves Girard's syntactic proof of strong normalization for System F.Takeuti's conjecture is equivalent to the 1-consistency of second-order arithmetic in the sense that each of the statements can be derived from each other in the weak system PRA. It is also equivalent to the strong normalization of the Girard/Reynold's System F.
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https://en.wikipedia.org/wiki/Takeuti_conjecture
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In mathematics, Tanaka's equation is an example of a stochastic differential equation which admits a weak solution but has no strong solution. It is named after the Japanese mathematician Hiroshi Tanaka (Tanaka Hiroshi). Tanaka's equation is the one-dimensional stochastic differential equation d X t = sgn ( X t ) d B t , {\displaystyle \mathrm {d} X_{t}=\operatorname {sgn}(X_{t})\,\mathrm {d} B_{t},} driven by canonical Brownian motion B, with initial condition X0 = 0, where sgn denotes the sign function sgn ( x ) = { + 1 , x ≥ 0 ; − 1 , x < 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}+1,&x\geq 0;\\-1,&x<0.\end{cases}}} (Note the unconventional value for sgn(0).)
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https://en.wikipedia.org/wiki/Tanaka_equation
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The signum function does not satisfy the Lipschitz continuity condition required for the usual theorems guaranteeing existence and uniqueness of strong solutions. The Tanaka equation has no strong solution, i.e. one for which the version B of Brownian motion is given in advance and the solution X is adapted to the filtration generated by B and the initial conditions. However, the Tanaka equation does have a weak solution, one for which the process X and version of Brownian motion are both specified as part of the solution, rather than the Brownian motion being given a priori.
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https://en.wikipedia.org/wiki/Tanaka_equation
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In this case, simply choose X to be any Brownian motion B ^ {\displaystyle {\hat {B}}} and define B ~ {\displaystyle {\tilde {B}}} by B ~ t = ∫ 0 t sgn ( B ^ s ) d B ^ s = ∫ 0 t sgn ( X s ) d X s , {\displaystyle {\tilde {B}}_{t}=\int _{0}^{t}\operatorname {sgn} {\big (}{\hat {B}}_{s}{\big )}\,\mathrm {d} {\hat {B}}_{s}=\int _{0}^{t}\operatorname {sgn} {\big (}X_{s}{\big )}\,\mathrm {d} X_{s},} i.e. d B ~ t = sgn ( X t ) d X t . {\displaystyle \mathrm {d} {\tilde {B}}_{t}=\operatorname {sgn}(X_{t})\,\mathrm {d} X_{t}.} Hence, d X t = sgn ( X t ) d B ~ t , {\displaystyle \mathrm {d} X_{t}=\operatorname {sgn}(X_{t})\,\mathrm {d} {\tilde {B}}_{t},} and so X is a weak solution of the Tanaka equation. Furthermore, this solution is weakly unique, i.e. any other weak solution must have the same law. Another counterexample of this type is Tsirelson's stochastic differential equation.
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https://en.wikipedia.org/wiki/Tanaka_equation
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In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named after Tadao Tannaka and Mark Grigorievich Krein.
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https://en.wikipedia.org/wiki/Tannaka–Krein_duality
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In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G. Duality theorems of Tannaka and Krein describe the converse passage from the category Π(G) back to the group G, allowing one to recover the group from its category of representations. Moreover, they in effect completely characterize all categories that can arise from a group in this fashion. Alexander Grothendieck later showed that by a similar process, Tannaka duality can be extended to the case of algebraic groups via Tannakian formalism. Meanwhile, the original theory of Tannaka and Krein continued to be developed and refined by mathematical physicists. A generalization of Tannaka–Krein theory provides the natural framework for studying representations of quantum groups, and is currently being extended to quantum supergroups, quantum groupoids and their dual Hopf algebroids.
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https://en.wikipedia.org/wiki/Tannaka–Krein_duality
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In mathematics, Tarski's plank problem is a question about coverings of convex regions in n-dimensional Euclidean space by "planks": regions between two hyperplanes. Tarski asked if the sum of the widths of the planks must be at least the minimum width of the convex region. The question was answered affirmatively by Thøger Bang (1950, 1951).
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https://en.wikipedia.org/wiki/Tarski's_plank_problem
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In mathematics, Tarski's theorem, proved by Alfred Tarski (1924), states that in ZF the theorem "For every infinite set A {\displaystyle A} , there is a bijective map between the sets A {\displaystyle A} and A × A {\displaystyle A\times A} " implies the axiom of choice. The opposite direction was already known, thus the theorem and axiom of choice are equivalent. Tarski told Jan Mycielski (2006) that when he tried to publish the theorem in Comptes Rendus de l'Académie des Sciences de Paris, Fréchet and Lebesgue refused to present it. Fréchet wrote that an implication between two well known propositions is not a new result. Lebesgue wrote that an implication between two false propositions is of no interest.
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https://en.wikipedia.org/wiki/Tarski's_theorem_about_choice
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In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by John Tate (1952, p. 297), and are used in class field theory.
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https://en.wikipedia.org/wiki/Tate_cohomology_group
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In mathematics, Tate duality or Poitou–Tate duality is a duality theorem for Galois cohomology groups of modules over the Galois group of an algebraic number field or local field, introduced by John Tate (1962) and Georges Poitou (1967).
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https://en.wikipedia.org/wiki/Poitou-Tate_duality
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In mathematics, Tate pairing is any of several closely related bilinear pairings involving elliptic curves or abelian varieties, usually over local or finite fields, based on the Tate duality pairings introduced by Tate (1958, 1963) and extended by Lichtenbaum (1969). Rück & Frey (1994) applied the Tate pairing over finite fields to cryptography.
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https://en.wikipedia.org/wiki/Tate_pairing
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In mathematics, Tate's isogeny theorem, proved by Tate (1966), states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).
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https://en.wikipedia.org/wiki/Tate's_isogeny_theorem
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In mathematics, Thaine's theorem is an analogue of Stickelberger's theorem for real abelian fields, introduced by Thaine (1988). Thaine's method has been used to shorten the proof of the Mazur–Wiles theorem (Washington 1997), to prove that some Tate–Shafarevich groups are finite, and in the proof of Mihăilescu's theorem (Schoof 2008).
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https://en.wikipedia.org/wiki/Thaine's_theorem
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In mathematics, The fundamental theorem of topos theory states that the slice E / X {\displaystyle \mathbf {E} /X} of a topos E {\displaystyle \mathbf {E} } over any one of its objects X {\displaystyle X} is itself a topos. Moreover, if there is a morphism f: A → B {\displaystyle f:A\rightarrow B} in E {\displaystyle \mathbf {E} } then there is a functor f ∗: E / B → E / A {\displaystyle f^{*}:\mathbf {E} /B\rightarrow \mathbf {E} /A} which preserves exponentials and the subobject classifier.
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https://en.wikipedia.org/wiki/Fundamental_theorem_of_topos_theory
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In mathematics, Theory of Lie groups is a series of books on Lie groups by Claude Chevalley (1946, 1951, 1955). The first in the series was one of the earliest books on Lie groups to treat them from the global point of view, and for many years was the standard text on Lie groups. The second and third volumes, on algebraic groups and Lie algebras, were written in French, and later reprinted bound together as one volume. Apparently further volumes were planned but not published, though his lectures (Chevalley 2005) on the classification of semisimple algebraic groups could be considered as a continuation of the series.
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https://en.wikipedia.org/wiki/Theory_of_Lie_groups
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In mathematics, Thomae's formula is a formula introduced by Carl Johannes Thomae (1870) relating theta constants to the branch points of a hyperelliptic curve (Mumford 1984, section 8).
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https://en.wikipedia.org/wiki/Thomae's_formula
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In mathematics, Thurston's classification theorem characterizes homeomorphisms of a compact orientable surface. William Thurston's theorem completes the work initiated by Jakob Nielsen (1944). Given a homeomorphism f: S → S, there is a map g isotopic to f such that at least one of the following holds: g is periodic, i.e. some power of g is the identity; g preserves some finite union of disjoint simple closed curves on S (in this case, g is called reducible); or g is pseudo-Anosov.The case where S is a torus (i.e., a surface whose genus is one) is handled separately (see torus bundle) and was known before Thurston's work. If the genus of S is two or greater, then S is naturally hyperbolic, and the tools of Teichmüller theory become useful.
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https://en.wikipedia.org/wiki/Nielsen–Thurston_classification
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In what follows, we assume S has genus at least two, as this is the case Thurston considered. (Note, however, that the cases where S has boundary or is not orientable are definitely still of interest.) The three types in this classification are not mutually exclusive, though a pseudo-Anosov homeomorphism is never periodic or reducible. A reducible homeomorphism g can be further analyzed by cutting the surface along the preserved union of simple closed curves Γ. Each of the resulting compact surfaces with boundary is acted upon by some power (i.e. iterated composition) of g, and the classification can again be applied to this homeomorphism.
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https://en.wikipedia.org/wiki/Nielsen–Thurston_classification
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In mathematics, Thurston's geometrization conjecture states that each of certain three-dimensional topological spaces has a unique geometric structure that can be associated with it. It is an analogue of the uniformization theorem for two-dimensional surfaces, which states that every simply connected Riemann surface can be given one of three geometries (Euclidean, spherical, or hyperbolic). In three dimensions, it is not always possible to assign a single geometry to a whole topological space. Instead, the geometrization conjecture states that every closed 3-manifold can be decomposed in a canonical way into pieces that each have one of eight types of geometric structure.
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https://en.wikipedia.org/wiki/Geometrisation_conjecture
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The conjecture was proposed by William Thurston (1982), and implies several other conjectures, such as the Poincaré conjecture and Thurston's elliptization conjecture. Thurston's hyperbolization theorem implies that Haken manifolds satisfy the geometrization conjecture. Thurston announced a proof in the 1980s and since then several complete proofs have appeared in print.
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https://en.wikipedia.org/wiki/Geometrisation_conjecture
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Grigori Perelman announced a proof of the full geometrization conjecture in 2003 using Ricci flow with surgery in two papers posted at the arxiv.org preprint server. Perelman's papers were studied by several independent groups that produced books and online manuscripts filling in the complete details of his arguments. Verification was essentially complete in time for Perelman to be awarded the 2006 Fields Medal for his work, and in 2010 the Clay Mathematics Institute awarded him its 1 million USD prize for solving the Poincare conjecture, though Perelman declined to accept either award. The Poincaré conjecture and the spherical space form conjecture are corollaries of the geometrization conjecture, although there are shorter proofs of the former that do not lead to the geometrization conjecture.
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https://en.wikipedia.org/wiki/Geometrisation_conjecture
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In mathematics, Thābit derived an equation for determining amicable numbers. His proof of this rule is presented in the Treatise on the Derivation of the Amicable Numbers in an Easy Way. This was done while writing on the theory of numbers, extending their use to describe the ratios between geometrical quantities, a step which the Greeks did not take. Thābit's work on amicable numbers and number theory helped him to invest more heavily into the Geometrical relations of numbers establishing his Transversal (geometry) theorem.Thābit described a generalized proof of the Pythagorean theorem.
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https://en.wikipedia.org/wiki/Thābit_ibn_Qurra
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He provided a strengthened extension of Pythagoras' proof which included the knowledge of Euclid's fifth postulate. This postulate states that the intersection between two straight line segments combine to create two interior angles which are less than 180 degrees. The method of reduction and composition used by Thābit resulted in a combination and extension of contemporary and ancient knowledge on this famous proof.
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https://en.wikipedia.org/wiki/Thābit_ibn_Qurra
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Thābit believed that geometry was tied with the equality and differences of magnitudes of lines and angles, as well as that ideas of motion (and ideas taken from physics more widely) should be integrated in geometry.The continued work done on geometric relations and the resulting exponential series allowed Thābit to calculate multiple solutions to chessboard problems. This problem was less to do with the game itself, and more to do with the number of solutions or the nature of solutions possible. In Thābit's case, he worked with combinatorics to work on the permutations needed to win a game of chess.In addition to Thābit's work on Euclidean geometry there is evidence that he was familiar with the geometry of Archimedes as well.
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https://en.wikipedia.org/wiki/Thābit_ibn_Qurra
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His work with conic sections and the calculation of a paraboloid shape (cupola) show his proficiency as an Archimedean geometer. This is further embossed by Thābit's use of the Archimedean property in order to produce a rudimentary approximation of the volume of a paraboloid. The use of uneven sections, while relatively simple, does show a critical understanding of both Euclidean and Archimedean geometry. Thābit was also responsible for a commentary on Archimedes' Liber Assumpta.
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https://en.wikipedia.org/wiki/Thābit_ibn_Qurra
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In mathematics, Toda–Smith complexes are spectra characterized by having a particularly simple BP-homology, and are useful objects in stable homotopy theory. Toda–Smith complexes provide examples of periodic self maps. These self maps were originally exploited in order to construct infinite families of elements in the homotopy groups of spheres. Their existence pointed the way towards the nilpotence and periodicity theorems.
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https://en.wikipedia.org/wiki/Toda–Smith_complex
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In mathematics, Tonelli's theorem in functional analysis is a fundamental result on the weak lower semicontinuity of nonlinear functionals on Lp spaces. As such, it has major implications for functional analysis and the calculus of variations. Roughly, it shows that weak lower semicontinuity for integral functionals is equivalent to convexity of the integral kernel. The result is attributed to the Italian mathematician Leonida Tonelli.
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https://en.wikipedia.org/wiki/Tonelli's_theorem_(functional_analysis)
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In mathematics, Tsen's theorem states that a function field K of an algebraic curve over an algebraically closed field is quasi-algebraically closed (i.e., C1). This implies that the Brauer group of any such field vanishes, and more generally that all the Galois cohomology groups H i(K, K*) vanish for i ≥ 1. This result is used to calculate the étale cohomology groups of an algebraic curve. The theorem was published by Chiungtze C. Tsen in 1933.
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https://en.wikipedia.org/wiki/Tsen's_theorem
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