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In mathematics, the Sierpiński space (or the connected two-point set) is a finite topological space with two points, only one of which is closed. It is the smallest example of a topological space which is neither trivial nor discrete. It is named after Wacław Sierpiński. The Sierpiński space has important relations to the theory of computation and semantics, because it is the classifying space for open sets in the Scott topology.
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https://en.wikipedia.org/wiki/Sierpinski_space
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In mathematics, the Silverman–Toeplitz theorem, first proved by Otto Toeplitz, is a result in summability theory characterizing matrix summability methods that are regular. A regular matrix summability method is a matrix transformation of a convergent sequence which preserves the limit.An infinite matrix ( a i , j ) i , j ∈ N {\displaystyle (a_{i,j})_{i,j\in \mathbb {N} }} with complex-valued entries defines a regular summability method if and only if it satisfies all of the following properties: lim i → ∞ a i , j = 0 j ∈ N (Every column sequence converges to 0.) lim i → ∞ ∑ j = 0 ∞ a i , j = 1 (The row sums converge to 1.) sup i ∑ j = 0 ∞ | a i , j | < ∞ (The absolute row sums are bounded.)
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https://en.wikipedia.org/wiki/Silverman–Toeplitz_theorem
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In mathematics, the Simon problems (or Simon's problems) are a series of fifteen questions posed in the year 2000 by Barry Simon, an American mathematical physicist. Inspired by other collections of mathematical problems and open conjectures, such as the famous list by David Hilbert, the Simon problems concern quantum operators. Eight of the problems pertain to anomalous spectral behavior of Schrödinger operators, and five concern operators that incorporate the Coulomb potential.In 2014, Artur Avila won a Fields Medal for work including the solution of three Simon problems. Among these was the problem of proving that the set of energy levels of one particular abstract quantum system was in fact the Cantor set, a challenge known as the "Ten Martini Problem" after the reward that Mark Kac offered for solving it.The 2000 list was a refinement of a similar set of problems that Simon had posed in 1984.
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https://en.wikipedia.org/wiki/Simon_problems
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In mathematics, the Sims conjecture is a result in group theory, originally proposed by Charles Sims. He conjectured that if G {\displaystyle G} is a primitive permutation group on a finite set S {\displaystyle S} and G α {\displaystyle G_{\alpha }} denotes the stabilizer of the point α {\displaystyle \alpha } in S {\displaystyle S} , then there exists an integer-valued function f {\displaystyle f} such that f ( d ) ≥ | G α | {\displaystyle f(d)\geq |G_{\alpha }|} for d {\displaystyle d} the length of any orbit of G α {\displaystyle G_{\alpha }} in the set S ∖ { α } {\displaystyle S\setminus \{\alpha \}} . The conjecture was proven by Peter Cameron, Cheryl Praeger, Jan Saxl, and Gary Seitz using the classification of finite simple groups, in particular the fact that only finitely many isomorphism types of sporadic groups exist. The theorem reads precisely as follows.
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https://en.wikipedia.org/wiki/Sims_conjecture
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Thus, in a primitive permutation group with "large" stabilizers, these stabilizers cannot have any small orbit. A consequence of their proof is that there exist only finitely many connected distance-transitive graphs having degree greater than 2. == References ==
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https://en.wikipedia.org/wiki/Sims_conjecture
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In mathematics, the Sister Beiter conjecture is a conjecture about the size of coefficients of ternary cyclotomic polynomials (i.e. where the index is the product of three prime numbers). It is named after Marion Beiter, a Catholic nun who first proposed it in 1968.
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https://en.wikipedia.org/wiki/Sister_Beiter_conjecture
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In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is not known whether there exists an algorithm that can solve this problem.A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation F(n) = F(n − 1) + F(n − 2)together with the initial values F(0) = 0 and F(1) = 1. The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients.
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https://en.wikipedia.org/wiki/Skolem_problem
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This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational numbers, and Mahler and Lech extended it to other systems of numbers. However, the proofs of the theorem do not show how to test whether there exist any zeros.
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https://en.wikipedia.org/wiki/Skolem_problem
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There does exist an algorithm to test whether a constant-recursive sequence has infinitely many zeros, and if so to construct a decomposition of the positions of those zeros into periodic subsequences, based on the algebraic properties of the roots of the characteristic polynomial of the given recurrence. The remaining difficult part of the Skolem problem is determining whether the finite set of non-repeating zeros is empty or not.Partial solutions to the Skolem problem are known, covering the special case of the problem for recurrences of degree at most four. However, these solutions do not apply to recurrences of degree five or more.For integer recurrences, the Skolem problem is known to be NP-hard.
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https://en.wikipedia.org/wiki/Skolem_problem
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In mathematics, the Smith conjecture states that if f is a diffeomorphism of the 3-sphere of finite order, then the fixed point set of f cannot be a nontrivial knot. Paul A. Smith (1939, remark after theorem 4) showed that a non-trivial orientation-preserving diffeomorphism of finite order with fixed points must have a fixed point set equal to a circle, and asked in (Eilenberg 1949, Problem 36) if the fixed point set could be knotted. Friedhelm Waldhausen (1969) proved the Smith conjecture for the special case of diffeomorphisms of order 2 (and hence any even order).
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https://en.wikipedia.org/wiki/Smith_conjecture
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The proof of the general case was described by John Morgan and Hyman Bass (1984) and depended on several major advances in 3-manifold theory, In particular the work of William Thurston on hyperbolic structures on 3-manifolds, and results by William Meeks and Shing-Tung Yau on minimal surfaces in 3-manifolds, with some additional help from Bass, Cameron Gordon, Peter Shalen, and Rick Litherland. Deane Montgomery and Leo Zippin (1954) gave an example of a continuous involution of the 3-sphere whose fixed point set is a wildly embedded circle, so the Smith conjecture is false in the topological (rather than the smooth or PL) category. Charles Giffen (1966) showed that the analogue of the Smith conjecture in higher dimensions is false: the fixed point set of a periodic diffeomorphism of a sphere of dimension at least 4 can be a knotted sphere of codimension 2.
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https://en.wikipedia.org/wiki/Smith_conjecture
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In mathematics, the Smith normal form (sometimes abbreviated SNF) is a normal form that can be defined for any matrix (not necessarily square) with entries in a principal ideal domain (PID). The Smith normal form of a matrix is diagonal, and can be obtained from the original matrix by multiplying on the left and right by invertible square matrices. In particular, the integers are a PID, so one can always calculate the Smith normal form of an integer matrix. The Smith normal form is very useful for working with finitely generated modules over a PID, and in particular for deducing the structure of a quotient of a free module. It is named after the Irish mathematician Henry John Stephen Smith.
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https://en.wikipedia.org/wiki/Smith_normal_form
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In mathematics, the Smith–Minkowski–Siegel mass formula (or Minkowski–Siegel mass formula) is a formula for the sum of the weights of the lattices (quadratic forms) in a genus, weighted by the reciprocals of the orders of their automorphism groups. The mass formula is often given for integral quadratic forms, though it can be generalized to quadratic forms over any algebraic number field. In 0 and 1 dimensions the mass formula is trivial, in 2 dimensions it is essentially equivalent to Dirichlet's class number formulas for imaginary quadratic fields, and in 3 dimensions some partial results were given by Gotthold Eisenstein. The mass formula in higher dimensions was first given by H. J. S. Smith (1867), though his results were forgotten for many years.
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https://en.wikipedia.org/wiki/Siegel_mass_formula
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It was rediscovered by H. Minkowski (1885), and an error in Minkowski's paper was found and corrected by C. L. Siegel (1935). Many published versions of the mass formula have errors; in particular the 2-adic densities are difficult to get right, and it is sometimes forgotten that the trivial cases of dimensions 0 and 1 are different from the cases of dimension at least 2. Conway & Sloane (1988) give an expository account and precise statement of the mass formula for integral quadratic forms, which is reliable because they check it on a large number of explicit cases. For recent proofs of the mass formula see (Kitaoka 1999) and (Eskin, Rudnick & Sarnak 1991). The Smith–Minkowski–Siegel mass formula is essentially the constant term of the Weil–Siegel formula.
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https://en.wikipedia.org/wiki/Siegel_mass_formula
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In mathematics, the Smith–Volterra–Cantor set (SVC), fat Cantor set, or ε-Cantor set is an example of a set of points on the real line that is nowhere dense (in particular it contains no intervals), yet has positive measure. The Smith–Volterra–Cantor set is named after the mathematicians Henry Smith, Vito Volterra and Georg Cantor. In an 1875 paper, Smith discussed a nowhere-dense set of positive measure on the real line, and Volterra introduced a similar example in 1881. The Cantor set as we know it today followed in 1883. The Smith–Volterra–Cantor set is topologically equivalent to the middle-thirds Cantor set.
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https://en.wikipedia.org/wiki/Smith–Volterra–Cantor_set
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In mathematics, the Springer resolution is a resolution of the variety of nilpotent elements in a semisimple Lie algebra, or the unipotent elements of a reductive algebraic group, introduced by Tonny Albert Springer in 1969. The fibers of this resolution are called Springer fibers.If U is the variety of unipotent elements in a reductive group G, and X the variety of Borel subgroups B, then the Springer resolution of U is the variety of pairs (u,B) of U×X such that u is in the Borel subgroup B. The map to U is the projection to the first factor. The Springer resolution for Lie algebras is similar, except that U is replaced by the nilpotent elements of the Lie algebra of G and X replaced by the variety of Borel subalgebras.The Grothendieck–Springer resolution is defined similarly, except that U is replaced by the whole group G (or the whole Lie algebra of G). When restricted to the unipotent elements of G it becomes the Springer resolution.
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https://en.wikipedia.org/wiki/Grothendieck–Springer_simultaneous_resolution
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In mathematics, the Stallings–Zeeman theorem is a result in algebraic topology, used in the proof of the Poincaré conjecture for dimension greater than or equal to five. It is named after the mathematicians John R. Stallings and Christopher Zeeman.
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https://en.wikipedia.org/wiki/Stallings–Zeeman_theorem
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In mathematics, the Steinberg representation, or Steinberg module or Steinberg character, denoted by St, is a particular linear representation of a reductive algebraic group over a finite field or local field, or a group with a BN-pair. It is analogous to the 1-dimensional sign representation ε of a Coxeter or Weyl group that takes all reflections to –1. For groups over finite fields, these representations were introduced by Robert Steinberg (1951, 1956, 1957), first for the general linear groups, then for classical groups, and then for all Chevalley groups, with a construction that immediately generalized to the other groups of Lie type that were discovered soon after by Steinberg, Suzuki and Ree.
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https://en.wikipedia.org/wiki/Steinberg_representation
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Over a finite field of characteristic p, the Steinberg representation has degree equal to the largest power of p dividing the order of the group. The Steinberg representation is the Alvis–Curtis dual of the trivial 1-dimensional representation. Matsumoto (1969), Shalika (1970), and Harish-Chandra (1973) defined analogous Steinberg representations (sometimes called special representations) for algebraic groups over local fields. For the general linear group GL(2), the dimension of the Jacquet module of a special representation is always one.
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https://en.wikipedia.org/wiki/Steinberg_representation
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In mathematics, the Steinberg triality groups of type 3D4 form a family of Steinberg or twisted Chevalley groups. They are quasi-split forms of D4, depending on a cubic Galois extension of fields K ⊂ L, and using the triality automorphism of the Dynkin diagram D4. Unfortunately the notation for the group is not standardized, as some authors write it as 3D4(K) (thinking of 3D4 as an algebraic group taking values in K) and some as 3D4(L) (thinking of the group as a subgroup of D4(L) fixed by an outer automorphism of order 3).
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https://en.wikipedia.org/wiki/3D4
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The group 3D4 is very similar to an orthogonal or spin group in dimension 8. Over finite fields these groups form one of the 18 infinite families of finite simple groups, and were introduced by Steinberg (1959). They were independently discovered by Jacques Tits in Tits (1958) and Tits (1959).
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https://en.wikipedia.org/wiki/3D4
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In mathematics, the Stein–Strömberg theorem or Stein–Strömberg inequality is a result in measure theory concerning the Hardy–Littlewood maximal operator. The result is foundational in the study of the problem of differentiation of integrals. The result is named after the mathematicians Elias M. Stein and Jan-Olov Strömberg.
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https://en.wikipedia.org/wiki/Stein–Strömberg_theorem
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In mathematics, the Stiefel manifold V k ( R n ) {\displaystyle V_{k}(\mathbb {R} ^{n})} is the set of all orthonormal k-frames in R n . {\displaystyle \mathbb {R} ^{n}.} That is, it is the set of ordered orthonormal k-tuples of vectors in R n . {\displaystyle \mathbb {R} ^{n}.}
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https://en.wikipedia.org/wiki/Stiefel_manifold
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It is named after Swiss mathematician Eduard Stiefel. Likewise one can define the complex Stiefel manifold V k ( C n ) {\displaystyle V_{k}(\mathbb {C} ^{n})} of orthonormal k-frames in C n {\displaystyle \mathbb {C} ^{n}} and the quaternionic Stiefel manifold V k ( H n ) {\displaystyle V_{k}(\mathbb {H} ^{n})} of orthonormal k-frames in H n {\displaystyle \mathbb {H} ^{n}} . More generally, the construction applies to any real, complex, or quaternionic inner product space. In some contexts, a non-compact Stiefel manifold is defined as the set of all linearly independent k-frames in R n , C n , {\displaystyle \mathbb {R} ^{n},\mathbb {C} ^{n},} or H n ; {\displaystyle \mathbb {H} ^{n};} this is homotopy equivalent, as the compact Stiefel manifold is a deformation retract of the non-compact one, by Gram–Schmidt. Statements about the non-compact form correspond to those for the compact form, replacing the orthogonal group (or unitary or symplectic group) with the general linear group.
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https://en.wikipedia.org/wiki/Stiefel_manifold
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In mathematics, the Stieltjes constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function: ζ ( 1 + s ) = 1 s + ∑ n = 0 ∞ ( − 1 ) n n ! γ n s n . {\displaystyle \zeta (1+s)={\frac {1}{s}}+\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n! }}\gamma _{n}s^{n}.} The constant γ 0 = γ = 0.577 … {\displaystyle \gamma _{0}=\gamma =0.577\dots } is known as the Euler–Mascheroni constant.
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https://en.wikipedia.org/wiki/Stieltjes_constants
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In mathematics, the Stieltjes moment problem, named after Thomas Joannes Stieltjes, seeks necessary and sufficient conditions for a sequence (m0, m1, m2, ...) to be of the form m n = ∫ 0 ∞ x n d μ ( x ) {\displaystyle m_{n}=\int _{0}^{\infty }x^{n}\,d\mu (x)} for some measure μ. If such a function μ exists, one asks whether it is unique. The essential difference between this and other well-known moment problems is that this is on a half-line , and in the Hamburger moment problem one considers the whole line (−∞, ∞).
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https://en.wikipedia.org/wiki/Stieltjes_moment_problem
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In mathematics, the Stieltjes polynomials En are polynomials associated to a family of orthogonal polynomials Pn. They are unrelated to the Stieltjes polynomial solutions of differential equations. Stieltjes originally considered the case where the orthogonal polynomials Pn are the Legendre polynomials. The Gauss–Kronrod quadrature formula uses the zeros of Stieltjes polynomials.
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https://en.wikipedia.org/wiki/Stieltjes_polynomials
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In mathematics, the Stirling polynomials are a family of polynomials that generalize important sequences of numbers appearing in combinatorics and analysis, which are closely related to the Stirling numbers, the Bernoulli numbers, and the generalized Bernoulli polynomials. There are multiple variants of the Stirling polynomial sequence considered below most notably including the Sheffer sequence form of the sequence, S k ( x ) {\displaystyle S_{k}(x)} , defined characteristically through the special form of its exponential generating function, and the Stirling (convolution) polynomials, σ n ( x ) {\displaystyle \sigma _{n}(x)} , which also satisfy a characteristic ordinary generating function and that are of use in generalizing the Stirling numbers (of both kinds) to arbitrary complex-valued inputs. We consider the "convolution polynomial" variant of this sequence and its properties second in the last subsection of the article. Still other variants of the Stirling polynomials are studied in the supplementary links to the articles given in the references.
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https://en.wikipedia.org/wiki/Stirling_polynomial
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In mathematics, the Stolarsky mean is a generalization of the logarithmic mean. It was introduced by Kenneth B. Stolarsky in 1975.
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https://en.wikipedia.org/wiki/Stolarsky_mean
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In mathematics, the Stolz–Cesàro theorem is a criterion for proving the convergence of a sequence. The theorem is named after mathematicians Otto Stolz and Ernesto Cesàro, who stated and proved it for the first time. The Stolz–Cesàro theorem can be viewed as a generalization of the Cesàro mean, but also as a l'Hôpital's rule for sequences.
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https://en.wikipedia.org/wiki/Stolz–Cesàro_theorem
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In mathematics, the Stone functor is a functor S: Topop → Bool, where Top is the category of topological spaces and Bool is the category of Boolean algebras and Boolean homomorphisms. It assigns to each topological space X the Boolean algebra S(X) of its clopen subsets, and to each morphism fop: X → Y in Topop (i.e., a continuous map f: Y → X) the homomorphism S(f): S(X) → S(Y) given by S(f)(Z) = f−1.
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https://en.wikipedia.org/wiki/Stone_functor
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In mathematics, the Stoneham numbers are a certain class of real numbers, named after mathematician Richard G. Stoneham (1920–1996). For coprime numbers b, c > 1, the Stoneham number αb,c is defined as α b , c = ∑ n = c k > 1 1 b n n = ∑ k = 1 ∞ 1 b c k c k {\displaystyle \alpha _{b,c}=\sum _{n=c^{k}>1}{\frac {1}{b^{n}n}}=\sum _{k=1}^{\infty }{\frac {1}{b^{c^{k}}c^{k}}}} It was shown by Stoneham in 1973 that αb,c is b-normal whenever c is an odd prime and b is a primitive root of c2. In 2002, Bailey & Crandall showed that coprimality of b, c > 1 is sufficient for b-normality of αb,c.
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https://en.wikipedia.org/wiki/Stoneham_number
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In mathematics, the Strahler number or Horton–Strahler number of a mathematical tree is a numerical measure of its branching complexity. These numbers were first developed in hydrology, as a way of measuring the complexity of rivers and streams, by Robert E. Horton (1945) and Arthur Newell Strahler (1952, 1957). In this application, they are referred to as the Strahler stream order and are used to define stream size based on a hierarchy of tributaries. The same numbers also arise in the analysis of L-systems and of hierarchical biological structures such as (biological) trees and animal respiratory and circulatory systems, in register allocation for compilation of high-level programming languages and in the analysis of social networks.
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https://en.wikipedia.org/wiki/Strahler_stream_order
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In mathematics, the Strömberg wavelet is a certain orthonormal wavelet discovered by Jan-Olov Strömberg and presented in a paper published in 1983. Even though the Haar wavelet was earlier known to be an orthonormal wavelet, Strömberg wavelet was the first smooth orthonormal wavelet to be discovered. The term wavelet had not been coined at the time of publishing the discovery of Strömberg wavelet and Strömberg's motivation was to find an orthonormal basis for the Hardy spaces.
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https://en.wikipedia.org/wiki/Strömberg_wavelet
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In mathematics, the Sturm sequence of a univariate polynomial p is a sequence of polynomials associated with p and its derivative by a variant of Euclid's algorithm for polynomials. Sturm's theorem expresses the number of distinct real roots of p located in an interval in terms of the number of changes of signs of the values of the Sturm sequence at the bounds of the interval. Applied to the interval of all the real numbers, it gives the total number of real roots of p.Whereas the fundamental theorem of algebra readily yields the overall number of complex roots, counted with multiplicity, it does not provide a procedure for calculating them. Sturm's theorem counts the number of distinct real roots and locates them in intervals.
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https://en.wikipedia.org/wiki/Sturm_theorem
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By subdividing the intervals containing some roots, it can isolate the roots into arbitrarily small intervals, each containing exactly one root. This yields the oldest real-root isolation algorithm, and arbitrary-precision root-finding algorithm for univariate polynomials.
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https://en.wikipedia.org/wiki/Sturm_theorem
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For computing over the reals, Sturm's theorem is less efficient than other methods based on Descartes' rule of signs. However, it works on every real closed field, and, therefore, remains fundamental for the theoretical study of the computational complexity of decidability and quantifier elimination in the first order theory of real numbers. The Sturm sequence and Sturm's theorem are named after Jacques Charles François Sturm, who discovered the theorem in 1829.
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https://en.wikipedia.org/wiki/Sturm_theorem
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In mathematics, the Sturm series associated with a pair of polynomials is named after Jacques Charles François Sturm.
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https://en.wikipedia.org/wiki/Sturm_series
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In mathematics, the Sugeno integral, named after M. Sugeno, is a type of integral with respect to a fuzzy measure. Let ( X , Ω ) {\displaystyle (X,\Omega )} be a measurable space and let h: X → {\displaystyle h:X\to } be an Ω {\displaystyle \Omega } -measurable function. The Sugeno integral over the crisp set A ⊆ X {\displaystyle A\subseteq X} of the function h {\displaystyle h} with respect to the fuzzy measure g {\displaystyle g} is defined by: ∫ A h ( x ) ∘ g = sup E ⊆ X = sup α ∈ {\displaystyle \int _{A}h(x)\circ g={\sup _{E\subseteq X}}\left={\sup _{\alpha \in }}\left} where F α = { x | h ( x ) ≥ α } {\displaystyle F_{\alpha }=\left\{x|h(x)\geq \alpha \right\}} . The Sugeno integral over the fuzzy set A ~ {\displaystyle {\tilde {A}}} of the function h {\displaystyle h} with respect to the fuzzy measure g {\displaystyle g} is defined by: ∫ A h ( x ) ∘ g = ∫ X ∘ g {\displaystyle \int _{A}h(x)\circ g=\int _{X}\left\circ g} where h A ( x ) {\displaystyle h_{A}(x)} is the membership function of the fuzzy set A ~ {\displaystyle {\tilde {A}}} .
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https://en.wikipedia.org/wiki/Sugeno_integral
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In mathematics, the Suita conjecture is a conjecture related to the theory of the Riemann surface, the boundary behavior of conformal maps, the theory of Bergman kernel, and the theory of the L2 extension. The conjecture states the following: Suita (1972): Let R be an Riemann surface, which admits a nontrivial Green function G R {\displaystyle G_{R}} . Let ω {\displaystyle \omega } be a local coordinate on a neighborhood V z 0 {\displaystyle V_{z_{0}}} of z 0 ∈ R {\displaystyle z_{0}\in R} satisfying w ( z 0 ) = 0 {\displaystyle w(z_{0})=0} . Let κ R {\displaystyle \kappa R} be the Bergman kernel for holomorphic (1, 0) forms on R. We define B R ( z ) | d w | 2 := κ R ( z ) | V z 0 {\displaystyle B_{R}(z)|dw|^{2}:=\kappa _{R}(z)|_{V_{z_{0}}}} , and B R ( z , t ¯ ) d ω ⊗ d t ¯ := κ R ( z , t ¯ ) {\displaystyle B_{R}(z,{\overline {t}})d\omega \otimes d{\overline {t}}:=\kappa _{R}(z,{\overline {t}})} .
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https://en.wikipedia.org/wiki/Suita_conjecture
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Let c β ( z ) {\displaystyle c_{\beta }(z)} be the logarithmic capacity which is locally defined by c β ( z 0 ) := exp lim ξ → z ( G R ( z , z 0 ) − log | ω ( z ) | ) {\displaystyle c_{\beta }(z_{0}):=\exp \lim _{\xi \to z}(G_{R}(z,z_{0})-\log |\omega (z)|)} on R. Then, the inequality ( c β ( z 0 ) ) 2 ≤ π B R ( z 0 ) {\displaystyle (c_{\beta }(z_{0}))^{2}\leq \pi B_{R}(z_{0})} holds on the every open Riemann surface R, and also, with equality, then B R ≡ 0 {\displaystyle B_{R}\equiv 0} or, R is conformally equivalent to the unit disc less a (possible) closed set of inner capacity zero. It was first proved by Błocki (2013) for the bounded plane domain and then completely in a more generalized version by Guan & Zhou (2015). Also, another proof of the Suita conjecture and some examples of its generalization to several complex variables (the multi (high) - dimensional Suita conjecture) were given in Błocki (2014a) and Błocki & Zwonek (2020). The multi (high) - dimensional Suita conjecture fails in non-pseudoconvex domains. This conjecture was proved through the optimal estimation of the Ohsawa–Takegoshi L2 extension theorem.
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https://en.wikipedia.org/wiki/Suita_conjecture
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In mathematics, the Suslin homology is a homology theory attached to algebraic varieties. It was proposed by Suslin in 1987, and developed by Suslin and Voevodsky (1996). It is sometimes called singular homology as it is analogous to the singular homology of topological spaces. By definition, given an abelian group A and a scheme X of finite type over a field k, the theory is given by H i ( X , A ) = Tor i Z ( C , A ) {\displaystyle H_{i}(X,A)=\operatorname {Tor} _{i}^{\mathbb {Z} }(C,A)} where C is a free graded abelian group whose degree n part is generated by integral subschemes of △ n × X {\displaystyle \triangle ^{n}\times X} , where △ n {\displaystyle \triangle ^{n}} is an n-simplex, that are finite and surjective over △ n {\displaystyle \triangle ^{n}} .
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https://en.wikipedia.org/wiki/Singular_homology_of_abstract_algebraic_varieties
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In mathematics, the Suslin operation 𝓐 is an operation that constructs a set from a collection of sets indexed by finite sequences of positive integers. The Suslin operation was introduced by Alexandrov (1916) and Suslin (1917). In Russia it is sometimes called the A-operation after Alexandrov. It is usually denoted by the symbol 𝓐 (a calligraphic capital letter A).
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https://en.wikipedia.org/wiki/Suslin_scheme
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In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie (SGA) was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris. (The name came from the small wood on the estate in Bures-sur-Yvette where the IHÉS was located from 1962.) The seminar notes were eventually published in twelve volumes, all except one in the Springer Lecture Notes in Mathematics series.
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https://en.wikipedia.org/wiki/Séminaire_de_Géométrie_Algébrique_du_Bois_Marie
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In mathematics, the T(1) theorem, first proved by David & Journé (1984), describes when an operator T given by a kernel can be extended to a bounded linear operator on the Hilbert space L2(Rn). The name T(1) theorem refers to a condition on the distribution T(1), given by the operator T applied to the function 1.
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https://en.wikipedia.org/wiki/T(1)_theorem
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In mathematics, the T-square is a two-dimensional fractal. It has a boundary of infinite length bounding a finite area. Its name comes from the drawing instrument known as a T-square.
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https://en.wikipedia.org/wiki/T-square_(fractal)
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In mathematics, the Tamagawa number τ ( G ) {\displaystyle \tau (G)} of a semisimple algebraic group defined over a global field k is the measure of G ( A ) / G ( k ) {\displaystyle G(\mathbb {A} )/G(k)} , where A {\displaystyle \mathbb {A} } is the adele ring of k. Tamagawa numbers were introduced by Tamagawa (1966), and named after him by Weil (1959). Tsuneo Tamagawa's observation was that, starting from an invariant differential form ω on G, defined over k, the measure involved was well-defined: while ω could be replaced by cω with c a non-zero element of k {\displaystyle k} , the product formula for valuations in k is reflected by the independence from c of the measure of the quotient, for the product measure constructed from ω on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory.
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https://en.wikipedia.org/wiki/Tamagawa_number
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In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by Langlands (1977) using an observation by Deligne, and named after Yutaka Taniyama. It was intended to be the group scheme whose representations correspond to the (hypothetical) CM motives over the field Q of rational numbers.
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https://en.wikipedia.org/wiki/Taniyama_group
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In mathematics, the Tarski–Seidenberg theorem states that a set in (n + 1)-dimensional space defined by polynomial equations and inequalities can be projected down onto n-dimensional space, and the resulting set is still definable in terms of polynomial identities and inequalities. The theorem—also known as the Tarski–Seidenberg projection property—is named after Alfred Tarski and Abraham Seidenberg. It implies that quantifier elimination is possible over the reals, that is that every formula constructed from polynomial equations and inequalities by logical connectives ∨ (or), ∧ (and), ¬ (not) and quantifiers ∀ (for all), ∃ (exists) is equivalent to a similar formula without quantifiers.
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https://en.wikipedia.org/wiki/Tarski–Seidenberg_theorem
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An important consequence is the decidability of the theory of real-closed fields. Although the original proof of the theorem was constructive, the resulting algorithm has a computational complexity that is too high for using the method on a computer. George E. Collins introduced the algorithm of cylindrical algebraic decomposition, which allows quantifier elimination over the reals in double exponential time. This complexity is optimal, as there are examples where the output has a double exponential number of connected components. This algorithm is therefore fundamental, and it is widely used in computational algebraic geometry.
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https://en.wikipedia.org/wiki/Tarski–Seidenberg_theorem
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In mathematics, the Tate curve is a curve defined over the ring of formal power series Z ] {\displaystyle \mathbb {Z} ]} with integer coefficients. Over the open subscheme where q is invertible, the Tate curve is an elliptic curve. The Tate curve can also be defined for q as an element of a complete field of norm less than 1, in which case the formal power series converge. The Tate curve was introduced by John Tate (1995) in a 1959 manuscript originally titled "Rational Points on Elliptic Curves Over Complete Fields"; he did not publish his results until many years later, and his work first appeared in Roquette (1970).
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https://en.wikipedia.org/wiki/Tate_curve
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In mathematics, the Tate topology is a Grothendieck topology of the space of maximal ideals of a k-affinoid algebra, whose open sets are the admissible open subsets and whose coverings are the admissible open coverings.
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https://en.wikipedia.org/wiki/Tate_topology
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In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century.
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https://en.wikipedia.org/wiki/Taylor_polynomial
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The partial sum formed by the first n + 1 terms of a Taylor series is a polynomial of degree n that is called the nth Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally more accurate as n increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations.
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https://en.wikipedia.org/wiki/Taylor_polynomial
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If the Taylor series of a function is convergent, its sum is the limit of the infinite sequence of the Taylor polynomials. A function may differ from the sum of its Taylor series, even if its Taylor series is convergent. A function is analytic at a point x if it is equal to the sum of its Taylor series in some open interval (or open disk in the complex plane) containing x. This implies that the function is analytic at every point of the interval (or disk).
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https://en.wikipedia.org/wiki/Taylor_polynomial
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In mathematics, the Teichmüller cocycle is a certain 3-cocycle associated to a simple algebra A over a field L which is a finite Galois extension of a field K and which has the property that any automorphism of L over K extends to an automorphism of A. The Teichmüller cocycle, or rather its cohomology class, is the obstruction to the algebra A coming from a simple algebra over K. It was introduced by Teichmüller (1940) and named by Eilenberg and MacLane (1948).
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https://en.wikipedia.org/wiki/Teichmüller_cocycle
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In mathematics, the Teichmüller space T ( S ) {\displaystyle T(S)} of a (real) topological (or differential) surface S {\displaystyle S} , is a space that parametrizes complex structures on S {\displaystyle S} up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüller spaces are named after Oswald Teichmüller. Each point in a Teichmüller space T ( S ) {\displaystyle T(S)} may be regarded as an isomorphism class of "marked" Riemann surfaces, where a "marking" is an isotopy class of homeomorphisms from S {\displaystyle S} to itself.
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https://en.wikipedia.org/wiki/Teichmüller_theory
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It can be viewed as a moduli space for marked hyperbolic structure on the surface, and this endows it with a natural topology for which it is homeomorphic to a ball of dimension 6 g − 6 {\displaystyle 6g-6} for a surface of genus g ≥ 2 {\displaystyle g\geq 2} . In this way Teichmüller space can be viewed as the universal covering orbifold of the Riemann moduli space. The Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The study of geometric features of these various structures is an active body of research. The sub-field of mathematics that studies the Teichmüller space is called Teichmüller theory.
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https://en.wikipedia.org/wiki/Teichmüller_theory
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In mathematics, the Teichmüller space TX of a (real) topological surface X, is a space that parameterizes complex structures on X up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Each point in TX may be regarded as an isomorphism class of 'marked' Riemann surfaces where a 'marking' is an isotopy class of homeomorphisms from X to X. The Teichmüller space is the universal covering orbifold of the (Riemann) moduli space. Teichmüller space has a canonical complex manifold structure and a wealth of natural metrics. The underlying topological space of Teichmüller space was studied by Fricke, and the Teichmüller metric on it was introduced by Oswald Teichmüller (1940).
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https://en.wikipedia.org/wiki/Low_dimensional_topology
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In mathematics, the Teichmüller–Tukey lemma (sometimes named just Tukey's lemma), named after John Tukey and Oswald Teichmüller, is a lemma that states that every nonempty collection of finite character has a maximal element with respect to inclusion. Over Zermelo–Fraenkel set theory, the Teichmüller–Tukey lemma is equivalent to the axiom of choice, and therefore to the well-ordering theorem, Zorn's lemma, and the Hausdorff maximal principle.
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https://en.wikipedia.org/wiki/Tukey's_lemma
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In mathematics, the Thom space, Thom complex, or Pontryagin–Thom construction (named after René Thom and Lev Pontryagin) of algebraic topology and differential topology is a topological space associated to a vector bundle, over any paracompact space.
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https://en.wikipedia.org/wiki/Thom_spectrum
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In mathematics, the Thomas–Fermi equation for the neutral atom is a second order non-linear ordinary differential equation, named after Llewellyn Thomas and Enrico Fermi, which can be derived by applying the Thomas–Fermi model to atoms. The equation reads d 2 y d x 2 = 1 x y 3 / 2 {\displaystyle {\frac {d^{2}y}{dx^{2}}}={\frac {1}{\sqrt {x}}}y^{3/2}} subject to the boundary conditions y ( 0 ) = 1 ; y ( + ∞ ) = 0 {\displaystyle y(0)=1\quad ;\quad y(+\infty )=0} If y {\displaystyle y} approaches zero as x {\displaystyle x} becomes large, this equation models the charge distribution of a neutral atom as a function of radius x {\displaystyle x} . Solutions where y {\displaystyle y} becomes zero at finite x {\displaystyle x} model positive ions. For solutions where y {\displaystyle y} becomes large and positive as x {\displaystyle x} becomes large, it can be interpreted as a model of a compressed atom, where the charge is squeezed into a smaller space. In this case the atom ends at the value of x {\displaystyle x} for which d y / d x = y / x {\displaystyle dy/dx=y/x} .
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https://en.wikipedia.org/wiki/Thomas–Fermi_equation
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In mathematics, the Thompson groups (also called Thompson's groups, vagabond groups or chameleon groups) are three groups, commonly denoted F ⊆ T ⊆ V {\displaystyle F\subseteq T\subseteq V} , that were introduced by Richard Thompson in some unpublished handwritten notes in 1965 as a possible counterexample to the von Neumann conjecture. Of the three, F is the most widely studied, and is sometimes referred to as the Thompson group or Thompson's group. The Thompson groups, and F in particular, have a collection of unusual properties that have made them counterexamples to many general conjectures in group theory. All three Thompson groups are infinite but finitely presented.
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https://en.wikipedia.org/wiki/Thompson_groups
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The groups T and V are (rare) examples of infinite but finitely-presented simple groups. The group F is not simple but its derived subgroup is and the quotient of F by its derived subgroup is the free abelian group of rank 2. F is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. It is conjectured that F is not amenable and hence a further counterexample to the long-standing but recently disproved von Neumann conjecture for finitely-presented groups: it is known that F is not elementary amenable. Higman (1974) introduced an infinite family of finitely presented simple groups, including Thompson's group V as a special case.
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https://en.wikipedia.org/wiki/Thompson_groups
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In mathematics, the Thue–Morse sequence or Prouhet–Thue–Morse sequence or parity sequence is the binary sequence (an infinite sequence of 0s and 1s) obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. The first few steps of this procedure yield the strings 0 then 01, 0110, 01101001, 0110100110010110, and so on, which are prefixes of the Thue–Morse sequence. The full sequence begins: 01101001100101101001011001101001.... The sequence is named after Axel Thue and Marston Morse.
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https://en.wikipedia.org/wiki/Thue-Morse_Sequence
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In mathematics, the Thurston boundary of Teichmüller space of a surface is obtained as the boundary of its closure in the projective space of functionals on simple closed curves on the surface. The Thurston boundary can be interpreted as the space of projective measured foliations on the surface. The Thurston boundary of the Teichmüller space of a closed surface of genus g {\displaystyle g} is homeomorphic to a sphere of dimension 6 g − 7 {\displaystyle 6g-7} . The action of the mapping class group on the Teichmüller space extends continuously over the union with the boundary.
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https://en.wikipedia.org/wiki/Thurston_boundary
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In mathematics, the Thurston norm is a function on the second homology group of an oriented 3-manifold introduced by William Thurston, which measures in a natural way the topological complexity of homology classes represented by surfaces.
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https://en.wikipedia.org/wiki/Thurston_norm
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In mathematics, the Tits alternative, named after Jacques Tits, is an important theorem about the structure of finitely generated linear groups.
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https://en.wikipedia.org/wiki/Tits_alternative
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In mathematics, the Toda bracket is an operation on homotopy classes of maps, in particular on homotopy groups of spheres, named after Hiroshi Toda, who defined them and used them to compute homotopy groups of spheres in (Toda 1962).
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https://en.wikipedia.org/wiki/Toda_bracket
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In mathematics, the Todd class is a certain construction now considered a part of the theory in algebraic topology of characteristic classes. The Todd class of a vector bundle can be defined by means of the theory of Chern classes, and is encountered where Chern classes exist — most notably in differential topology, the theory of complex manifolds and algebraic geometry. In rough terms, a Todd class acts like a reciprocal of a Chern class, or stands in relation to it as a conormal bundle does to a normal bundle. The Todd class plays a fundamental role in generalising the classical Riemann–Roch theorem to higher dimensions, in the Hirzebruch–Riemann–Roch theorem and the Grothendieck–Hirzebruch–Riemann–Roch theorem.
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https://en.wikipedia.org/wiki/Todd_class
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In mathematics, the Tonelli–Hobson test gives sufficient criteria for a function ƒ on R2 to be an integrable function. It is often used to establish that Fubini's theorem may be applied to ƒ. It is named for Leonida Tonelli and E. W. Hobson. More precisely, the Tonelli–Hobson test states that if ƒ is a real-valued measurable function on R2, and either of the two iterated integrals ∫ R ( ∫ R | f ( x , y ) | d x ) d y {\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dx\right)\,dy} or ∫ R ( ∫ R | f ( x , y ) | d y ) d x {\displaystyle \int _{\mathbb {R} }\left(\int _{\mathbb {R} }|f(x,y)|\,dy\right)\,dx} is finite, then ƒ is Lebesgue-integrable on R2.
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https://en.wikipedia.org/wiki/Tonelli–Hobson_test
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In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the torsion subgroup of an abelian group.
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https://en.wikipedia.org/wiki/Tor_functor
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In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Algebra.
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https://en.wikipedia.org/wiki/Tor_functor
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In mathematics, the Torelli theorem, named after Ruggiero Torelli, is a classical result of algebraic geometry over the complex number field, stating that a non-singular projective algebraic curve (compact Riemann surface) C is determined by its Jacobian variety J(C), when the latter is given in the form of a principally polarized abelian variety. In other words, the complex torus J(C), with certain 'markings', is enough to recover C. The same statement holds over any algebraically closed field. From more precise information on the constructed isomorphism of the curves it follows that if the canonically principally polarized Jacobian varieties of curves of genus ≥ 2 {\displaystyle \geq 2} are k-isomorphic for k any perfect field, so are the curves.This result has had many important extensions. It can be recast to read that a certain natural morphism, the period mapping, from the moduli space of curves of a fixed genus, to a moduli space of abelian varieties, is injective (on geometric points).
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https://en.wikipedia.org/wiki/Torelli's_theorem
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Generalizations are in two directions. Firstly, to geometric questions about that morphism, for example the local Torelli theorem. Secondly, to other period mappings. A case that has been investigated deeply is for K3 surfaces (by Viktor S. Kulikov, Ilya Pyatetskii-Shapiro, Igor Shafarevich and Fedor Bogomolov) and hyperkähler manifolds (by Misha Verbitsky, Eyal Markman and Daniel Huybrechts).
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https://en.wikipedia.org/wiki/Torelli's_theorem
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In mathematics, the Trefftz method is a method for the numerical solution of partial differential equations named after the German mathematician Erich Trefftz(de) (1888–1937). It falls within the class of finite element methods.
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https://en.wikipedia.org/wiki/Trefftz_method
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In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by Tricomi (1951) and Carlitz (1958) and Karlin and McGregor (1959), related to random walks on the positive integers. They are given in terms of Laguerre polynomials by ℓ n ( x ) = ( − 1 ) n L n ( x − n ) ( x ) . {\displaystyle \ell _{n}(x)=(-1)^{n}L_{n}^{(x-n)}(x).} They are special cases of the Chihara–Ismail polynomials.
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https://en.wikipedia.org/wiki/Tricomi–Carlitz_polynomial
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In mathematics, the Trombi–Varadarajan theorem, introduced by Trombi and Varadarjan (1971), gives an isomorphism between a certain space of spherical functions on a semisimple Lie group, and a certain space of holomorphic functions defined on a tubular neighborhood of the dual of a Cartan subalgebra.
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https://en.wikipedia.org/wiki/Trombi–Varadarajan_theorem
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In mathematics, the Turán number T(n,k,r) for r-uniform hypergraphs of order n is the smallest number of r-edges such that every induced subgraph on k vertices contains an edge. This number was determined for r = 2 by Turán (1941), and the problem for general r was introduced in Turán (1961). The paper (Sidorenko 1995) gives a survey of Turán numbers.
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https://en.wikipedia.org/wiki/Turán_number
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In mathematics, the Tutte homotopy theorem, introduced by Tutte (1958), generalises the concept of "path" from graphs to matroids, and states roughly that closed paths can be written as compositions of elementary closed paths, so that in some sense they are homotopic to the trivial closed path.
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https://en.wikipedia.org/wiki/Tutte_homotopy_theorem
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In mathematics, the Twisted Hessian curve represents a generalization of Hessian curves; it was introduced in elliptic curve cryptography to speed up the addition and doubling formulas and to have strongly unified arithmetic. In some operations (see the last sections), it is close in speed to Edwards curves.
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https://en.wikipedia.org/wiki/Twisted_Hessian_curves
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In mathematics, the Ulam numbers comprise an integer sequence devised by and named after Stanislaw Ulam, who introduced it in 1964. The standard Ulam sequence (the (1, 2)-Ulam sequence) starts with U1 = 1 and U2 = 2. Then for n > 2, Un is defined to be the smallest integer that is the sum of two distinct earlier terms in exactly one way and larger than all earlier terms.
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https://en.wikipedia.org/wiki/Ulam_numbers
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In mathematics, the Valentiner group is the perfect triple cover of the alternating group on 6 points, and is a group of order 1080. It was found by Herman Valentiner (1889) in the form of an action of A6 on the complex projective plane, and was studied further by Wiman (1896). All perfect alternating groups have perfect double covers. In most cases this is the universal central extension. The two exceptions are A6 (whose perfect triple cover is the Valentiner group) and A7, whose universal central extensions have centers of order 6.
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https://en.wikipedia.org/wiki/Valentiner_group
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In mathematics, the Veblen functions are a hierarchy of normal functions (continuous strictly increasing functions from ordinals to ordinals), introduced by Oswald Veblen in Veblen (1908). If φ0 is any normal function, then for any non-zero ordinal α, φα is the function enumerating the common fixed points of φβ for β<α. These functions are all normal.
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https://en.wikipedia.org/wiki/Veblen_functions
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In mathematics, the Veblen–Young theorem, proved by Oswald Veblen and John Wesley Young (1908, 1910, 1917), states that a projective space of dimension at least 3 can be constructed as the projective space associated to a vector space over a division ring. Non-Desarguesian planes give examples of 2-dimensional projective spaces that do not arise from vector spaces over division rings, showing that the restriction to dimension at least 3 is necessary. Jacques Tits generalized the Veblen–Young theorem to Tits buildings, showing that those of rank at least 3 arise from algebraic groups. John von Neumann (1998) generalized the Veblen–Young theorem to continuous geometry, showing that a complemented modular lattice of order at least 4 is isomorphic to the principal right ideals of a von Neumann regular ring.
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https://en.wikipedia.org/wiki/Veblen–Young_theorem
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In mathematics, the Veronese surface is an algebraic surface in five-dimensional projective space, and is realized by the Veronese embedding, the embedding of the projective plane given by the complete linear system of conics. It is named after Giuseppe Veronese (1854–1917). Its generalization to higher dimension is known as the Veronese variety. The surface admits an embedding in the four-dimensional projective space defined by the projection from a general point in the five-dimensional space. Its general projection to three-dimensional projective space is called a Steiner surface.
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https://en.wikipedia.org/wiki/Veronese_surface
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In mathematics, the Verschiebung or Verschiebung operator V is a homomorphism between affine commutative group schemes over a field of nonzero characteristic p. For finite group schemes it is the Cartier dual of the Frobenius homomorphism. It was introduced by Witt (1937) as the shift operator on Witt vectors taking (a0, a1, a2, ...) to (0, a0, a1, ...). ("Verschiebung" is German for "shift", but the term "Verschiebung" is often used for this operator even in other languages.) The Verschiebung operator V and the Frobenius operator F are related by FV = VF = , where is the pth power homomorphism of an abelian group scheme.
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https://en.wikipedia.org/wiki/Verschiebung_operator
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In mathematics, the Virasoro algebra (named after the physicist Miguel Ángel Virasoro) is a complex Lie algebra and the unique central extension of the Witt algebra. It is widely used in two-dimensional conformal field theory and in string theory.
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https://en.wikipedia.org/wiki/Virasoro_algebra
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In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
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https://en.wikipedia.org/wiki/Vitali_covering_lemma
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In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory.
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https://en.wikipedia.org/wiki/Vitali–Carathéodory_theorem
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In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure.
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https://en.wikipedia.org/wiki/Vitali–Hahn–Saks_theorem
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In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates. It was introduced by Vogel (1999), and is related by some observations made by Deligne (1996). Landsberg & Manivel (2006) generalized Vogel's work to higher symmetric powers. The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces A, B, C, where the symmetric square of the Lie algebra (usually) decomposes as a sum of the complex numbers and 3 irreducible spaces A, B, C.
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https://en.wikipedia.org/wiki/Vogel_plane
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In mathematics, the Volterra integral equations are a special type of integral equations. They are divided into two groups referred to as the first and the second kind. A linear Volterra equation of the first kind is f ( t ) = ∫ a t K ( t , s ) x ( s ) d s {\displaystyle f(t)=\int _{a}^{t}K(t,s)\,x(s)\,ds} where f is a given function and x is an unknown function to be solved for. A linear Volterra equation of the second kind is x ( t ) = f ( t ) + ∫ a t K ( t , s ) x ( s ) d s .
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https://en.wikipedia.org/wiki/Volterra_integral_equation
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{\displaystyle x(t)=f(t)+\int _{a}^{t}K(t,s)x(s)\,ds.} In operator theory, and in Fredholm theory, the corresponding operators are called Volterra operators. A useful method to solve such equations, the Adomian decomposition method, is due to George Adomian.
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https://en.wikipedia.org/wiki/Volterra_integral_equation
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A linear Volterra integral equation is a convolution equation if x ( t ) = f ( t ) + ∫ t 0 t K ( t − s ) x ( s ) d s . {\displaystyle x(t)=f(t)+\int _{t_{0}}^{t}K(t-s)x(s)\,ds.} The function K {\displaystyle K} in the integral is called the kernel.
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https://en.wikipedia.org/wiki/Volterra_integral_equation
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Such equations can be analyzed and solved by means of Laplace transform techniques. For a weakly singular kernel of the form K ( t , s ) = ( t 2 − s 2 ) − α {\displaystyle K(t,s)=(t^{2}-s^{2})^{-\alpha }} with 0 < α < 1 {\displaystyle 0<\alpha <1} , Volterra integral equation of the first kind can conveniently be transformed into a classical Abel integral equation. The Volterra integral equations were introduced by Vito Volterra and then studied by Traian Lalescu in his 1908 thesis, Sur les équations de Volterra, written under the direction of Émile Picard. In 1911, Lalescu wrote the first book ever on integral equations. Volterra integral equations find application in demography as Lotka's integral equation, the study of viscoelastic materials, in actuarial science through the renewal equation, and in fluid mechanics to describe the flow behavior near finite-sized boundaries.
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https://en.wikipedia.org/wiki/Volterra_integral_equation
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In mathematics, the Volterra lattice, also known as the discrete KdV equation, the Kac–van Moerbeke lattice, and the Langmuir lattice, is a system of ordinary differential equations with variables indexed by some of the points of a 1-dimensional lattice. It was introduced by Marc Kac and Pierre van Moerbeke (1975) and Jürgen Moser (1975) and is named after Vito Volterra. The Volterra lattice is a special case of the generalized Lotka–Volterra equation describing predator–prey interactions, for a sequence of species with each species preying on the next in the sequence.
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https://en.wikipedia.org/wiki/Volterra_lattice
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The Volterra lattice also behaves like a discrete version of the KdV equation. The Volterra lattice is an integrable system, and is related to the Toda lattice. It is also used as a model for Langmuir waves in plasmas.
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https://en.wikipedia.org/wiki/Volterra_lattice
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In mathematics, the Voorhoeve index is a non-negative real number associated with certain functions on the complex numbers, named after Marc Voorhoeve. It may be used to extend Rolle's theorem from real functions to complex functions, taking the role that for real functions is played by the number of zeros of the function in an interval.
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https://en.wikipedia.org/wiki/Voorhoeve_index
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In mathematics, the Vámos matroid or Vámos cube is a matroid over a set of eight elements that cannot be represented as a matrix over any field. It is named after English mathematician Peter Vámos, who first described it in an unpublished manuscript in 1968.
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https://en.wikipedia.org/wiki/Vámos_matroid
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