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In mathematics, a Niemeier lattice is one of the 24 positive definite even unimodular lattices of rank 24, which were classified by Hans-Volker Niemeier (1973). Venkov (1978) gave a simplified proof of the classification. In the 1970s, Witt (1941) has a sentence mentioning that he found more than 10 such lattices in the 1940s, but gives no further details. One example of a Niemeier lattice is the Leech lattice found in 1967.
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https://en.wikipedia.org/wiki/Niemeier_lattice
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In mathematics, a Noetherian ring is a ring that satisfies the ascending chain condition on left and right ideals; if the chain condition is satisfied only for left ideals or for right ideals, then the ring is said left-Noetherian or right-Noetherian respectively. That is, every increasing sequence I 1 ⊆ I 2 ⊆ I 3 ⊆ ⋯ {\displaystyle I_{1}\subseteq I_{2}\subseteq I_{3}\subseteq \cdots } of left (or right) ideals has a largest element; that is, there exists an n such that: I n = I n + 1 = ⋯ . {\displaystyle I_{n}=I_{n+1}=\cdots .}
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https://en.wikipedia.org/wiki/Noetherian_rings
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Equivalently, a ring is left-Noetherian (resp. right-Noetherian) if every left ideal (resp. right-ideal) is finitely generated.
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https://en.wikipedia.org/wiki/Noetherian_rings
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A ring is Noetherian if it is both left- and right-Noetherian. Noetherian rings are fundamental in both commutative and noncommutative ring theory since many rings that are encountered in mathematics are Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily on Noetherian property (for example, the Lasker–Noether theorem and the Krull intersection theorem). Noetherian rings are named after Emmy Noether, but the importance of the concept was recognized earlier by David Hilbert, with the proof of Hilbert's basis theorem (which asserts that polynomial rings are Noetherian) and Hilbert's syzygy theorem.
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https://en.wikipedia.org/wiki/Noetherian_rings
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In mathematics, a Noetherian topological space, named for Emmy Noether, is a topological space in which closed subsets satisfy the descending chain condition. Equivalently, we could say that the open subsets satisfy the ascending chain condition, since they are the complements of the closed subsets. The Noetherian property of a topological space can also be seen as a strong compactness condition, namely that every open subset of such a space is compact, and in fact it is equivalent to the seemingly stronger statement that every subset is compact.
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https://en.wikipedia.org/wiki/Noetherian_space
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In mathematics, a Nori semistable vector bundle is a particular type of vector bundle whose first definition has been first implicitly suggested by Madhav V. Nori, as one of the main ingredients for the construction of the fundamental group scheme. The original definition given by Nori was obviously not called Nori semistable. Also, Nori's definition was different from the one suggested nowadays. The category of Nori semistable vector bundles contains the Tannakian category of essentially finite vector bundles, whose naturally associated group scheme is the fundamental group scheme π 1 ( X , x ) {\displaystyle \pi _{1}(X,x)} .
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https://en.wikipedia.org/wiki/Nori-semistable_vector_bundle
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In mathematics, a Novikov–Shubin invariant, introduced by Sergei Novikov and Mikhail Shubin (1986), is an invariant of a compact Riemannian manifold related to the spectrum of the Laplace operator acting on square-integrable differential forms on its universal cover. The Novikov–Shubin invariant gives a measure of the density of eigenvalues around zero. It can be computed from a triangulation of the manifold, and it is a homotopy invariant. In particular, it does not depend on the chosen Riemannian metric on the manifold.
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https://en.wikipedia.org/wiki/Novikov–Shubin_invariant
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In mathematics, a Néron differential, named after André Néron, is an almost canonical choice of 1-form on an elliptic curve or abelian variety defined over a local field or global field. The Néron differential behaves well on the Néron minimal models. For an elliptic curve of the form y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 {\displaystyle y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}} the Néron differential is d x 2 y + a 1 x + a 3 {\displaystyle {\frac {dx}{2y+a_{1}x+a_{3}}}}
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https://en.wikipedia.org/wiki/Néron_differential
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In mathematics, a P-matrix is a complex square matrix with every principal minor is positive. A closely related class is that of P 0 {\displaystyle P_{0}} -matrices, which are the closure of the class of P-matrices, with every principal minor ≥ {\displaystyle \geq } 0.
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https://en.wikipedia.org/wiki/P-matrix
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In mathematics, a P-multimagic cube is a magic cube that remains magic even if all its numbers are replaced by their k th powers for 1 ≤ k ≤ P. 2-multimagic cubes are called bimagic, 3-multimagic cubes are called trimagic, and 4-multimagic cubes tetramagic. A P-multimagic cube is said to be semi-perfect if the k th power cubes are perfect for 1 ≤ k < P, and the P th power cube is semiperfect. If all P of the power cubes are perfect, the P-multimagic cube is said to be perfect.
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https://en.wikipedia.org/wiki/Tetramagic_cube
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The first known example of a bimagic cube was given by John Hendricks in 2000; it is a semiperfect cube of order 25 and magic constant 195325. In 2003, C. Bower discovered two semi-perfect bimagic cubes of order 16, and a perfect bimagic cube of order 32.MathWorld reports that only two trimagic cubes are known, discovered by C. Bower in 2003; a semiperfect cube of order 64 and a perfect cube of order 256. It also reports that he discovered the only two known tetramagic cubes, a semiperfect cube of order 1024, and perfect cube of order 8192.
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https://en.wikipedia.org/wiki/Tetramagic_cube
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In mathematics, a P-multimagic square (also known as a satanic square) is a magic square that remains magic even if all its numbers are replaced by their kth powers for 1 ≤ k ≤ P. 2-multimagic squares are called bimagic, 3-multimagic squares are called trimagic, 4-multimagic squares tetramagic, and 5-multimagic squares pentamagic.
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https://en.wikipedia.org/wiki/Tetramagic_square
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In mathematics, a P2-irreducible manifold is a 3-manifold that is irreducible and contains no 2-sided R P 2 {\displaystyle \mathbb {R} P^{2}} (real projective plane). An orientable manifold is P2-irreducible if and only if it is irreducible. Every non-orientable P2-irreducible manifold is a Haken manifold. == References ==
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https://en.wikipedia.org/wiki/P2-irreducible_manifold
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In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge.
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https://en.wikipedia.org/wiki/Padé_approximant
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For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since Padé approximant is a rational function, an artificial singular point may occur as an approximation, but this can be avoided by Borel–Padé analysis. The reason the Padé approximant tends to be a better approximation than a truncating Taylor series is clear from the viewpoint of the multi-point summation method. Since there are many cases in which the asymptotic expansion at infinity becomes 0 or a constant, it can be interpreted as the "incomplete two-point Padé approximation", in which the ordinary Padé approximation improves the method truncating a Taylor series.
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https://en.wikipedia.org/wiki/Padé_approximant
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In mathematics, a Paley–Wiener theorem is any theorem that relates decay properties of a function or distribution at infinity with analyticity of its Fourier transform. The theorem is named for Raymond Paley (1907–1933) and Norbert Wiener (1894–1964). The original theorems did not use the language of distributions, and instead applied to square-integrable functions. The first such theorem using distributions was due to Laurent Schwartz. These theorems heavily rely on the triangle inequality (to interchange the absolute value and integration).
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https://en.wikipedia.org/wiki/Paley–Wiener_theorem
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In mathematics, a Perron number is an algebraic integer α which is real and exceeds 1, but such that its conjugate elements are all less than α in absolute value. For example, the larger of the two roots of the irreducible polynomial x 2 − 3 x + 1 {\displaystyle x^{2}-3x+1} is a Perron number. Perron numbers are named after Oskar Perron; the Perron–Frobenius theorem asserts that, for a real square matrix with positive algebraic coefficients whose largest eigenvalue is greater than one, this eigenvalue is a Perron number. As a closely related case, the Perron number of a graph is defined to be the spectral radius of its adjacency matrix. Any Pisot number or Salem number is a Perron number, as is the Mahler measure of a monic integer polynomial.
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https://en.wikipedia.org/wiki/Perron_number
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In mathematics, a Petersson algebra is a composition algebra over a field constructed from an order-3 automorphism of a Hurwitz algebra. They were first constructed by Petersson (1969).
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https://en.wikipedia.org/wiki/Petersson_algebra
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In mathematics, a Petrovsky lacuna, named for the Russian mathematician I. G. Petrovsky, is a region where the fundamental solution of a linear hyperbolic partial differential equation vanishes. They were studied by Petrovsky (1945) who found topological conditions for their existence. Petrovsky's work was generalized and updated by Atiyah, Bott, and Gårding (1970, 1973).
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https://en.wikipedia.org/wiki/Petrovsky_lacuna
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In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field F of characteristic not 2. For a natural number n, an n-fold Pfister form over F is a quadratic form of dimension 2n that can be written as a tensor product of quadratic forms ⟨ ⟨ a 1 , a 2 , … , a n ⟩ ⟩ ≅ ⟨ 1 , − a 1 ⟩ ⊗ ⟨ 1 , − a 2 ⟩ ⊗ ⋯ ⊗ ⟨ 1 , − a n ⟩ , {\displaystyle \langle \!\langle a_{1},a_{2},\ldots ,a_{n}\rangle \!\rangle \cong \langle 1,-a_{1}\rangle \otimes \langle 1,-a_{2}\rangle \otimes \cdots \otimes \langle 1,-a_{n}\rangle ,} for some nonzero elements a1, ..., an of F. (Some authors omit the signs in this definition; the notation here simplifies the relation to Milnor K-theory, discussed below.)
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https://en.wikipedia.org/wiki/Pfister_form
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An n-fold Pfister form can also be constructed inductively from an (n−1)-fold Pfister form q and a nonzero element a of F, as q ⊕ ( − a ) q {\displaystyle q\oplus (-a)q} . So the 1-fold and 2-fold Pfister forms look like: ⟨ ⟨ a ⟩ ⟩ ≅ ⟨ 1 , − a ⟩ = x 2 − a y 2 {\displaystyle \langle \!\langle a\rangle \!\rangle \cong \langle 1,-a\rangle =x^{2}-ay^{2}} . ⟨ ⟨ a , b ⟩ ⟩ ≅ ⟨ 1 , − a , − b , a b ⟩ = x 2 − a y 2 − b z 2 + a b w 2 .
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https://en.wikipedia.org/wiki/Pfister_form
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{\displaystyle \langle \!\langle a,b\rangle \!\rangle \cong \langle 1,-a,-b,ab\rangle =x^{2}-ay^{2}-bz^{2}+abw^{2}.} For n ≤ 3, the n-fold Pfister forms are norm forms of composition algebras. In that case, two n-fold Pfister forms are isomorphic if and only if the corresponding composition algebras are isomorphic. In particular, this gives the classification of octonion algebras. The n-fold Pfister forms additively generate the n-th power I n of the fundamental ideal of the Witt ring of F.
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https://en.wikipedia.org/wiki/Pfister_form
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In mathematics, a Picard modular group, studied by Picard (1881), is a group of the form SU(J,L), where L is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and J is a hermitian form on L of signature (2, 1). Picard modular groups act on the unit sphere in C2 and the quotient is called a Picard modular surface.
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https://en.wikipedia.org/wiki/Picard_modular_group
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In mathematics, a Picard modular surface, studied by Picard (1881), is a complex surface constructed as a quotient of the unit ball in C2 by a Picard modular group. Picard modular surfaces are some of the simplest examples of Shimura varieties and are sometimes used as a test case for the general theory of Shimura varieties.
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https://en.wikipedia.org/wiki/Picard_modular_surface
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In mathematics, a Pisot–Vijayaraghavan number, also called simply a Pisot number or a PV number, is a real algebraic integer greater than 1, all of whose Galois conjugates are less than 1 in absolute value. These numbers were discovered by Axel Thue in 1912 and rediscovered by G. H. Hardy in 1919 within the context of diophantine approximation. They became widely known after the publication of Charles Pisot's dissertation in 1938. They also occur in the uniqueness problem for Fourier series.
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https://en.wikipedia.org/wiki/PV_number
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Tirukkannapuram Vijayaraghavan and Raphael Salem continued their study in the 1940s. Salem numbers are a closely related set of numbers. A characteristic property of PV numbers is that their powers approach integers at an exponential rate.
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https://en.wikipedia.org/wiki/PV_number
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Pisot proved a remarkable converse: if α > 1 is a real number such that the sequence ‖ α n ‖ {\displaystyle \|\alpha ^{n}\|} measuring the distance from its consecutive powers to the nearest integer is square-summable, or ℓ 2, then α is a Pisot number (and, in particular, algebraic). Building on this characterization of PV numbers, Salem showed that the set S of all PV numbers is closed. Its minimal element is a cubic irrationality known as the plastic number. Much is known about the accumulation points of S. The smallest of them is the golden ratio.
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https://en.wikipedia.org/wiki/PV_number
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In mathematics, a Plücker formula, named after Julius Plücker, is one of a family of formulae, of a type first developed by Plücker in the 1830s, that relate certain numeric invariants of algebraic curves to corresponding invariants of their dual curves. The invariant called the genus, common to both the curve and its dual, is connected to the other invariants by similar formulae. These formulae, and the fact that each of the invariants must be a positive integer, place quite strict limitations on their possible values.
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https://en.wikipedia.org/wiki/Plücker_formula
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In mathematics, a Poincaré–Steklov operator (after Henri Poincaré and Vladimir Steklov) maps the values of one boundary condition of the solution of an elliptic partial differential equation in a domain to the values of another boundary condition. Usually, either of the boundary conditions determines the solution. Thus, a Poincaré–Steklov operator encapsulates the boundary response of the system modelled by the partial differential equation. When the partial differential equation is discretized, for example by finite elements or finite differences, the discretization of the Poincaré–Steklov operator is the Schur complement obtained by eliminating all degrees of freedom inside the domain. Note that there may be many suitable different boundary conditions for a given partial differential equation and the direction in which a Poincaré–Steklov operator maps the values of one into another is given only by a convention.
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https://en.wikipedia.org/wiki/Dirichlet-to-Neumann_operator
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In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also a derivation. Poisson algebras appear naturally in Hamiltonian mechanics, and are also central in the study of quantum groups. Manifolds with a Poisson algebra structure are known as Poisson manifolds, of which the symplectic manifolds and the Poisson–Lie groups are a special case. The algebra is named in honour of Siméon Denis Poisson.
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https://en.wikipedia.org/wiki/Poisson_algebra
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In mathematics, a Poisson ring is a commutative ring on which an anticommutative and distributive binary operation {\displaystyle } satisfying the Jacobi identity and the product rule is defined. Such an operation is then known as the Poisson bracket of the Poisson ring. Many important operations and results of symplectic geometry and Hamiltonian mechanics may be formulated in terms of the Poisson bracket and, hence, apply to Poisson algebras as well. This observation is important in studying the classical limit of quantum mechanics—the non-commutative algebra of operators on a Hilbert space has the Poisson algebra of functions on a symplectic manifold as a singular limit, and properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra.
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https://en.wikipedia.org/wiki/Poisson_ring
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In mathematics, a Poisson superalgebra is a Z2-graded generalization of a Poisson algebra. Specifically, a Poisson superalgebra is an (associative) superalgebra A with a Lie superbracket : A ⊗ A → A {\displaystyle :A\otimes A\to A} such that (A, ) is a Lie superalgebra and the operator : A → A {\displaystyle :A\to A} is a superderivation of A: = z + ( − 1 ) | x | | y | y . {\displaystyle =z+(-1)^{|x||y|}y.\,} A supercommutative Poisson algebra is one for which the (associative) product is supercommutative. This is one possible way of "super"izing the Poisson algebra.
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https://en.wikipedia.org/wiki/Poisson_superbracket
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This gives the classical dynamics of fermion fields and classical spin-1/2 particles. The other is to define an antibracket algebra instead. This is used in the BRST and Batalin-Vilkovisky formalism.
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https://en.wikipedia.org/wiki/Poisson_superbracket
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In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal counterpart of a Poisson–Lie group is a Lie bialgebra, in analogy to Lie algebras as the infinitesimal counterparts of Lie groups. Many quantum groups are quantizations of the Poisson algebra of functions on a Poisson–Lie group.
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https://en.wikipedia.org/wiki/Poisson–Lie_group
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In mathematics, a Pontryagin cohomology operation is a cohomology operation taking cohomology classes in H2n(X,Z/prZ) to H2pn(X,Z/pr+1Z) for some prime number p. When p=2 these operations were introduced by Pontryagin (1942) and were named Pontrjagin squares by Whitehead (1949) (with the term "Pontryagin square" also being used). They were generalized to arbitrary primes by Thomas (1956).
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https://en.wikipedia.org/wiki/Pontryagin_cohomology_operation
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In mathematics, a Priestley space is an ordered topological space with special properties. Priestley spaces are named after Hilary Priestley who introduced and investigated them. Priestley spaces play a fundamental role in the study of distributive lattices. In particular, there is a duality ("Priestley duality") between the category of Priestley spaces and the category of bounded distributive lattices.
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https://en.wikipedia.org/wiki/Priestley_duality
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In mathematics, a Prym differential of a Riemann surface is a differential form on the universal covering space that transforms according to some complex character of the fundamental group. Equivalently it is a section of a certain line bundle on the Riemann surface in the same component as the canonical bundle. Prym differentials were introduced by Friedrich Prym (1869). The space of Prym differentials on a compact Riemann surface of genus g has dimension g – 1, unless the character of the fundamental group is trivial, in which case Prym differentials are the same as ordinary differentials and form a space of dimension g.
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https://en.wikipedia.org/wiki/Prym_differential
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In mathematics, a Prüfer domain is a type of commutative ring that generalizes Dedekind domains in a non-Noetherian context. These rings possess the nice ideal and module theoretic properties of Dedekind domains, but usually only for finitely generated modules. Prüfer domains are named after the German mathematician Heinz Prüfer.
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https://en.wikipedia.org/wiki/Prüfer_domain
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In mathematics, a Q-category or almost quotient category is a category that is a "milder version of a Grothendieck site." A Q-category is a coreflective subcategory. The Q stands for a quotient. The concept of Q-categories was introduced by Alexander Rosenberg in 1988. The motivation for the notion was its use in noncommutative algebraic geometry; in this formalism, noncommutative spaces are defined as sheaves on Q-categories.
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https://en.wikipedia.org/wiki/Q-category
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In mathematics, a Q-matrix is a square matrix whose associated linear complementarity problem LCP(M,q) has a solution for every vector q.
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https://en.wikipedia.org/wiki/Q-matrix
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In mathematics, a Rajchman measure, studied by Rajchman (1928), is a regular Borel measure on a locally compact group such as the circle, whose Fourier transform vanishes at infinity.
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https://en.wikipedia.org/wiki/Rajchman_measure
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In mathematics, a Ramanujan prime is a prime number that satisfies a result proven by Srinivasa Ramanujan relating to the prime-counting function.
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https://en.wikipedia.org/wiki/Ramanujan_prime
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In mathematics, a Ramanujan–Sato series generalizes Ramanujan’s pi formulas such as, 1 π = 2 2 99 2 ∑ k = 0 ∞ ( 4 k ) ! k ! 4 26390 k + 1103 396 4 k {\displaystyle {\frac {1}{\pi }}={\frac {2{\sqrt {2}}}{99^{2}}}\sum _{k=0}^{\infty }{\frac {(4k)!
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https://en.wikipedia.org/wiki/Ramanujan–Sato_series
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}{k!^{4}}}{\frac {26390k+1103}{396^{4k}}}} to the form 1 π = ∑ k = 0 ∞ s ( k ) A k + B C k {\displaystyle {\frac {1}{\pi }}=\sum _{k=0}^{\infty }s(k){\frac {Ak+B}{C^{k}}}} by using other well-defined sequences of integers s ( k ) {\displaystyle s(k)} obeying a certain recurrence relation, sequences which may be expressed in terms of binomial coefficients ( n k ) {\displaystyle {\tbinom {n}{k}}} , and A , B , C {\displaystyle A,B,C} employing modular forms of higher levels. Ramanujan made the enigmatic remark that there were "corresponding theories", but it was only recently that H. H. Chan and S. Cooper found a general approach that used the underlying modular congruence subgroup Γ 0 ( n ) {\displaystyle \Gamma _{0}(n)} , while G. Almkvist has experimentally found numerous other examples also with a general method using differential operators.Levels 1–4A were given by Ramanujan (1914), level 5 by H. H. Chan and S. Cooper (2012), 6A by Chan, Tanigawa, Yang, and Zudilin, 6B by Sato (2002), 6C by H. Chan, S. Chan, and Z. Liu (2004), 6D by H. Chan and H. Verrill (2009), level 7 by S. Cooper (2012), part of level 8 by Almkvist and Guillera (2012), part of level 10 by Y. Yang, and the rest by H. H. Chan and S. Cooper. The notation jn(τ) is derived from Zagier and Tn refers to the relevant McKay–Thompson series.
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https://en.wikipedia.org/wiki/Ramanujan–Sato_series
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In mathematics, a Ramsey cardinal is a certain kind of large cardinal number introduced by Erdős & Hajnal (1962) and named after Frank P. Ramsey, whose theorem establishes that ω enjoys a certain property that Ramsey cardinals generalize to the uncountable case. Let <ω denote the set of all finite subsets of κ. A cardinal number κ is called Ramsey if, for every function f: <ω → {0, 1}there is a set A of cardinality κ that is homogeneous for f. That is, for every n, the function f is constant on the subsets of cardinality n from A. A cardinal κ is called ineffably Ramsey if A can be chosen to be a stationary subset of κ. A cardinal κ is called virtually Ramsey if for every function f: <ω → {0, 1}there is C, a closed and unbounded subset of κ, so that for every λ in C of uncountable cofinality, there is an unbounded subset of λ that is homogenous for f; slightly weaker is the notion of almost Ramsey where homogenous sets for f are required of order type λ, for every λ < κ. The existence of any of these kinds of Ramsey cardinal is sufficient to prove the existence of 0#, or indeed that every set with rank less than κ has a sharp. Every measurable cardinal is a Ramsey cardinal, and every Ramsey cardinal is a Rowbottom cardinal. A property intermediate in strength between Ramseyness and measurability is existence of a κ-complete normal non-principal ideal I on κ such that for every A ∉ I and for every function f: <ω → {0, 1}there is a set B ⊂ A not in I that is homogeneous for f. This is strictly stronger than κ being ineffably Ramsey. The existence of a Ramsey cardinal implies the existence of 0# and this in turn implies the falsity of the Axiom of Constructibility of Kurt Gödel.
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https://en.wikipedia.org/wiki/Ramsey_cardinal
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In mathematics, a Raynaud surface is a particular kind of algebraic surface that was introduced in William E. Lang (1979) and named for Michel Raynaud (1978). To be precise, a Raynaud surface is a quasi-elliptic surface over an algebraic curve of genus g greater than 1, such that all fibers are irreducible and the fibration has a section. The Kodaira vanishing theorem fails for such surfaces; in other words the Kodaira theorem, valid in algebraic geometry over the complex numbers, has such surfaces as counterexamples, and these can only exist in characteristic p. Generalized Raynaud surfaces were introduced in (Lang 1983), and give examples of surfaces of general type with global vector fields.
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https://en.wikipedia.org/wiki/Raynaud_surface
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In mathematics, a Redheffer matrix, often denoted A n {\displaystyle A_{n}} as studied by Redheffer (1977), is a square (0,1) matrix whose entries aij are 1 if i divides j or if j = 1; otherwise, aij = 0. It is useful in some contexts to express Dirichlet convolution, or convolved divisors sums, in terms of matrix products involving the transpose of the n t h {\displaystyle n^{th}} Redheffer matrix.
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https://en.wikipedia.org/wiki/Redheffer_matrix
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In mathematics, a Ree group is a group of Lie type over a finite field constructed by Ree (1960, 1961) from an exceptional automorphism of a Dynkin diagram that reverses the direction of the multiple bonds, generalizing the Suzuki groups found by Suzuki using a different method. They were the last of the infinite families of finite simple groups to be discovered. Unlike the Steinberg groups, the Ree groups are not given by the points of a connected reductive algebraic group defined over a finite field; in other words, there is no "Ree algebraic group" related to the Ree groups in the same way that (say) unitary groups are related to Steinberg groups.
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https://en.wikipedia.org/wiki/Ree_group
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However, there are some exotic pseudo-reductive algebraic groups over non-perfect fields whose construction is related to the construction of Ree groups, as they use the same exotic automorphisms of Dynkin diagrams that change root lengths. Tits (1960) defined Ree groups over infinite fields of characteristics 2 and 3. Tits (1989) and Hée (1990) introduced Ree groups of infinite-dimensional Kac–Moody algebras.
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https://en.wikipedia.org/wiki/Ree_group
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In mathematics, a Relevance Vector Machine (RVM) is a machine learning technique that uses Bayesian inference to obtain parsimonious solutions for regression and probabilistic classification. The RVM has an identical functional form to the support vector machine, but provides probabilistic classification. It is actually equivalent to a Gaussian process model with covariance function: k ( x , x ′ ) = ∑ j = 1 N 1 α j φ ( x , x j ) φ ( x ′ , x j ) {\displaystyle k(\mathbf {x} ,\mathbf {x'} )=\sum _{j=1}^{N}{\frac {1}{\alpha _{j}}}\varphi (\mathbf {x} ,\mathbf {x} _{j})\varphi (\mathbf {x} ',\mathbf {x} _{j})} where φ {\displaystyle \varphi } is the kernel function (usually Gaussian), α j {\displaystyle \alpha _{j}} are the variances of the prior on the weight vector w ∼ N ( 0 , α − 1 I ) {\displaystyle w\sim N(0,\alpha ^{-1}I)} , and x 1 , … , x N {\displaystyle \mathbf {x} _{1},\ldots ,\mathbf {x} _{N}} are the input vectors of the training set.Compared to that of support vector machines (SVM), the Bayesian formulation of the RVM avoids the set of free parameters of the SVM (that usually require cross-validation-based post-optimizations). However RVMs use an expectation maximization (EM)-like learning method and are therefore at risk of local minima. This is unlike the standard sequential minimal optimization (SMO)-based algorithms employed by SVMs, which are guaranteed to find a global optimum (of the convex problem). The relevance vector machine was patented in the United States by Microsoft (patent expired September 4, 2019).
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https://en.wikipedia.org/wiki/Relevance_vector_machine
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In mathematics, a Riccati equation in the narrowest sense is any first-order ordinary differential equation that is quadratic in the unknown function. In other words, it is an equation of the form y ′ ( x ) = q 0 ( x ) + q 1 ( x ) y ( x ) + q 2 ( x ) y 2 ( x ) {\displaystyle y'(x)=q_{0}(x)+q_{1}(x)\,y(x)+q_{2}(x)\,y^{2}(x)} where q 0 ( x ) ≠ 0 {\displaystyle q_{0}(x)\neq 0} and q 2 ( x ) ≠ 0 {\displaystyle q_{2}(x)\neq 0} . If q 0 ( x ) = 0 {\displaystyle q_{0}(x)=0} the equation reduces to a Bernoulli equation, while if q 2 ( x ) = 0 {\displaystyle q_{2}(x)=0} the equation becomes a first order linear ordinary differential equation. The equation is named after Jacopo Riccati (1676–1754).More generally, the term Riccati equation is used to refer to matrix equations with an analogous quadratic term, which occur in both continuous-time and discrete-time linear-quadratic-Gaussian control. The steady-state (non-dynamic) version of these is referred to as the algebraic Riccati equation.
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https://en.wikipedia.org/wiki/Riccati_differential_equation
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In mathematics, a Rickart space (after Charles Earl Rickart), also called a basically disconnected space, is a topological space in which open σ-compact subsets have compact open closures. Grove & Pedersen (1984) named them after C. E. Rickart (1946), who showed that Rickart spaces are related to monotone σ-complete C*-algebras in the same way that Stonean spaces are related to AW*-algebras. Rickart spaces are totally disconnected and sub-Stonean spaces.
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https://en.wikipedia.org/wiki/Rickart_space
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In mathematics, a Riemann form in the theory of abelian varieties and modular forms, is the following data: A lattice Λ in a complex vector space Cg. An alternating bilinear form α from Λ to the integers satisfying the following Riemann bilinear relations:the real linear extension αR:Cg × Cg→R of α satisfies αR(iv, iw)=αR(v, w) for all (v, w) in Cg × Cg; the associated hermitian form H(v, w)=αR(iv, w) + iαR(v, w) is positive-definite. (The hermitian form written here is linear in the first variable.) Riemann forms are important because of the following: The alternatization of the Chern class of any factor of automorphy is a Riemann form. Conversely, given any Riemann form, we can construct a factor of automorphy such that the alternatization of its Chern class is the given Riemann form.
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https://en.wikipedia.org/wiki/Riemann_relations
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In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lines on a graph, but also the length of curves and other approximations. The sum is calculated by partitioning the region into shapes (rectangles, trapezoids, parabolas, or cubics) that together form a region that is similar to the region being measured, then calculating the area for each of these shapes, and finally adding all of these small areas together.
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https://en.wikipedia.org/wiki/Rectangle_method
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This approach can be used to find a numerical approximation for a definite integral even if the fundamental theorem of calculus does not make it easy to find a closed-form solution. Because the region by the small shapes is usually not exactly the same shape as the region being measured, the Riemann sum will differ from the area being measured. This error can be reduced by dividing up the region more finely, using smaller and smaller shapes. As the shapes get smaller and smaller, the sum approaches the Riemann integral.
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https://en.wikipedia.org/wiki/Rectangle_method
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In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°. The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of Bieberbach (1911, 1912) that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by Schoenflies (1891).
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https://en.wikipedia.org/wiki/Flat_metric
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In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.
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https://en.wikipedia.org/wiki/Riemann–Roch_theorem_for_smooth_manifolds
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In mathematics, a Riesel number is an odd natural number k for which k × 2 n − 1 {\displaystyle k\times 2^{n}-1} is composite for all natural numbers n (sequence A101036 in the OEIS). In other words, when k is a Riesel number, all members of the following set are composite: { k × 2 n − 1: n ∈ N } . {\displaystyle \left\{\,k\times 2^{n}-1:n\in \mathbb {N} \,\right\}.} If the form is instead k × 2 n + 1 {\displaystyle k\times 2^{n}+1} , then k is a Sierpinski number.
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https://en.wikipedia.org/wiki/Riesel_problem
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In mathematics, a Riesz space, lattice-ordered vector space or vector lattice is a partially ordered vector space where the order structure is a lattice. Riesz spaces are named after Frigyes Riesz who first defined them in his 1928 paper Sur la décomposition des opérations fonctionelles linéaires. Riesz spaces have wide-ranging applications. They are important in measure theory, in that important results are special cases of results for Riesz spaces. For example, the Radon–Nikodym theorem follows as a special case of the Freudenthal spectral theorem. Riesz spaces have also seen application in mathematical economics through the work of Greek-American economist and mathematician Charalambos D. Aliprantis.
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https://en.wikipedia.org/wiki/Vector_lattice
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In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by Claus Michael Ringel (1990). It has a basis of equivalence classes of objects of an abelian category, and the structure constants for this basis are related to the numbers of extensions of objects in the category.
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https://en.wikipedia.org/wiki/Ringel–Hall_algebra
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In mathematics, a Rosati involution, named after Carlo Rosati, is an involution of the rational endomorphism ring of an abelian variety induced by a polarization. Let A {\displaystyle A} be an abelian variety, let A ^ = P i c 0 ( A ) {\displaystyle {\hat {A}}=\mathrm {Pic} ^{0}(A)} be the dual abelian variety, and for a ∈ A {\displaystyle a\in A} , let T a: A → A {\displaystyle T_{a}:A\to A} be the translation-by- a {\displaystyle a} map, T a ( x ) = x + a {\displaystyle T_{a}(x)=x+a} . Then each divisor D {\displaystyle D} on A {\displaystyle A} defines a map ϕ D: A → A ^ {\displaystyle \phi _{D}:A\to {\hat {A}}} via ϕ D ( a ) = {\displaystyle \phi _{D}(a)=} . The map ϕ D {\displaystyle \phi _{D}} is a polarization if D {\displaystyle D} is ample.
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https://en.wikipedia.org/wiki/Rosati_involution
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In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.In the 1980s, the Rota-Baxter operator of weight 0 in the context of Lie algebras was rediscovered as the operator form of the classical Yang–Baxter equation, named after the well-known physicists Chen-Ning Yang and Rodney Baxter. The study of Rota–Baxter algebras experienced a renaissance this century, beginning with several developments, in the algebraic approach to renormalization of perturbative quantum field theory, dendriform algebras, associative analogue of the classical Yang–Baxter equation and mixable shuffle product constructions.
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https://en.wikipedia.org/wiki/Rota–Baxter_algebra
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In mathematics, a Rothberger space is a topological space that satisfies a certain a basic selection principle. A Rothberger space is a space in which for every sequence of open covers U 1 , U 2 , … {\displaystyle {\mathcal {U}}_{1},{\mathcal {U}}_{2},\ldots } of the space there are sets U 1 ∈ U 1 , U 2 ∈ U 2 , … {\displaystyle U_{1}\in {\mathcal {U}}_{1},U_{2}\in {\mathcal {U}}_{2},\ldots } such that the family { U n: n ∈ N } {\displaystyle \{U_{n}:n\in \mathbb {N} \}} covers the space.
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https://en.wikipedia.org/wiki/Rothberger_space
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In mathematics, a Ruth–Aaron pair consists of two consecutive integers (e.g., 714 and 715) for which the sums of the prime factors of each integer are equal: 714 = 2 × 3 × 7 × 17, 715 = 5 × 11 × 13,and 2 + 3 + 7 + 17 = 5 + 11 + 13 = 29.There are different variations in the definition, depending on how many times to count primes that appear multiple times in a factorization. The name was given by Carl Pomerance for Babe Ruth and Hank Aaron, as Ruth's career regular-season home run total was 714, a record which Aaron eclipsed on April 8, 1974, when he hit his 715th career home run. Pomerance was a mathematician at the University of Georgia at the time Aaron (a member of the nearby Atlanta Braves) broke Ruth's record, and the student of one of Pomerance's colleagues noticed that the sums of the prime factors of 714 and 715 were equal.
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https://en.wikipedia.org/wiki/Ruth–Aaron_pair
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In mathematics, a Salem number is a real algebraic integer α > 1 whose conjugate roots all have absolute value no greater than 1, and at least one of which has absolute value exactly 1. Salem numbers are of interest in Diophantine approximation and harmonic analysis. They are named after Raphaël Salem.
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https://en.wikipedia.org/wiki/Salem_number
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In mathematics, a Sastry automorphism, is an automorphism of a field of characteristic 2 satisfying some rather complicated conditions related to the problem of embedding Ree groups of type 2F4 into Chevalley groups of type F4. They were introduced by Sastry (1995), and named and classified by Bombieri (2002) who showed that there are 22 families of Sastry automorphisms, together with 22 exceptional ones over some finite fields of orders up to 210.
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https://en.wikipedia.org/wiki/Sastry_automorphism
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In mathematics, a Schauder basis or countable basis is similar to the usual (Hamel) basis of a vector space; the difference is that Hamel bases use linear combinations that are finite sums, while for Schauder bases they may be infinite sums. This makes Schauder bases more suitable for the analysis of infinite-dimensional topological vector spaces including Banach spaces. Schauder bases were described by Juliusz Schauder in 1927, although such bases were discussed earlier. For example, the Haar basis was given in 1909, and Georg Faber discussed in 1910 a basis for continuous functions on an interval, sometimes called a Faber–Schauder system.
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https://en.wikipedia.org/wiki/Schauder_dimension
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In mathematics, a Scherk surface (named after Heinrich Scherk) is an example of a minimal surface. Scherk described two complete embedded minimal surfaces in 1834; his first surface is a doubly periodic surface, his second surface is singly periodic. They were the third non-trivial examples of minimal surfaces (the first two were the catenoid and helicoid). The two surfaces are conjugates of each other. Scherk surfaces arise in the study of certain limiting minimal surface problems and in the study of harmonic diffeomorphisms of hyperbolic space.
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https://en.wikipedia.org/wiki/Scherk_surface
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In mathematics, a Schottky group is a special sort of Kleinian group, first studied by Friedrich Schottky (1877).
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https://en.wikipedia.org/wiki/Schottky_group
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In mathematics, a Schur-convex function, also known as S-convex, isotonic function and order-preserving function is a function f: R d → R {\displaystyle f:\mathbb {R} ^{d}\rightarrow \mathbb {R} } that for all x , y ∈ R d {\displaystyle x,y\in \mathbb {R} ^{d}} such that x {\displaystyle x} is majorized by y {\displaystyle y} , one has that f ( x ) ≤ f ( y ) {\displaystyle f(x)\leq f(y)} . Named after Issai Schur, Schur-convex functions are used in the study of majorization. Every function that is convex and symmetric is also Schur-convex. The opposite implication is not true, but all Schur-convex functions are symmetric (under permutations of the arguments).
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https://en.wikipedia.org/wiki/Schur-concave_function
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In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space. A tempered distribution is defined as a continuous linear functional on the space of Schwartz–Bruhat functions.
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https://en.wikipedia.org/wiki/Schwartz–Bruhat_function
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In mathematics, a Segal category is a model of an infinity category introduced by Hirschowitz & Simpson (1998), based on work of Graeme Segal in 1974.
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https://en.wikipedia.org/wiki/Segal_category
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In mathematics, a Segal space is a simplicial space satisfying some pullback conditions, making it look like a homotopical version of a category. More precisely, a simplicial set, considered as a simplicial discrete space, satisfies the Segal conditions iff it is the nerve of a category. The condition for Segal spaces is a homotopical version of this. Complete Segal spaces were introduced by Rezk (2001) as models for (∞, 1)-categories.
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https://en.wikipedia.org/wiki/Segal_space
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In mathematics, a Seifert surface (named after German mathematician Herbert Seifert) is an orientable surface whose boundary is a given knot or link. Such surfaces can be used to study the properties of the associated knot or link. For example, many knot invariants are most easily calculated using a Seifert surface. Seifert surfaces are also interesting in their own right, and the subject of considerable research.
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https://en.wikipedia.org/wiki/Seifert_matrix
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Specifically, let L be a tame oriented knot or link in Euclidean 3-space (or in the 3-sphere). A Seifert surface is a compact, connected, oriented surface S embedded in 3-space whose boundary is L such that the orientation on L is just the induced orientation from S. Note that any compact, connected, oriented surface with nonempty boundary in Euclidean 3-space is the Seifert surface associated to its boundary link. A single knot or link can have many different inequivalent Seifert surfaces. A Seifert surface must be oriented. It is possible to associate surfaces to knots which are not oriented nor orientable, as well.
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https://en.wikipedia.org/wiki/Seifert_matrix
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In mathematics, a Severi variety is an algebraic variety in a Hilbert scheme that parametrizes curves in projective space with given degree and geometric genus and at most node singularities. Its dimension is 3d + g − 1. It is a theorem that Severi varieties are algebraic varieties, i.e. it is irreducible.
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https://en.wikipedia.org/wiki/Severi_variety_(Hilbert_scheme)
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In mathematics, a Severi–Brauer variety over a field K is an algebraic variety V which becomes isomorphic to a projective space over an algebraic closure of K. The varieties are associated to central simple algebras in such a way that the algebra splits over K if and only if the variety has a rational point over K. Francesco Severi (1932) studied these varieties, and they are also named after Richard Brauer because of their close relation to the Brauer group. In dimension one, the Severi–Brauer varieties are conics. The corresponding central simple algebras are the quaternion algebras. The algebra (a,b)K corresponds to the conic C(a,b) with equation z 2 = a x 2 + b y 2 {\displaystyle z^{2}=ax^{2}+by^{2}\ } and the algebra (a,b)K splits, that is, (a,b)K is isomorphic to a matrix algebra over K, if and only if C(a,b) has a point defined over K: this is in turn equivalent to C(a,b) being isomorphic to the projective line over K.Such varieties are of interest not only in diophantine geometry, but also in Galois cohomology.
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https://en.wikipedia.org/wiki/Severi–Brauer_variety
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They represent (at least if K is a perfect field) Galois cohomology classes in H1(PGLn), where PGLn is the projective linear group, and n is the dimension of the variety V. There is a short exact sequence 1 → GL1 → GLn → PGLn → 1of algebraic groups. This implies a connecting homomorphism H1(PGLn) → H2(GL1)at the level of cohomology. Here H2(GL1) is identified with the Brauer group of K, while the kernel is trivial because H1(GLn) = {1} by an extension of Hilbert's Theorem 90.
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https://en.wikipedia.org/wiki/Severi–Brauer_variety
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Therefore, Severi–Brauer varieties can be faithfully represented by Brauer group elements, i.e. classes of central simple algebras. Lichtenbaum showed that if X is a Severi–Brauer variety over K then there is an exact sequence 0 → P i c ( X ) → Z → δ B r ( K ) → B r ( K ) / ( X ) → 0 . {\displaystyle 0\rightarrow \mathrm {Pic} (X)\rightarrow \mathbb {Z} {\stackrel {\delta }{\rightarrow }}\mathrm {Br} (K)\rightarrow \mathrm {Br} (K)/(X)\rightarrow 0\ .} Here the map δ sends 1 to the Brauer class corresponding to X.As a consequence, we see that if the class of X has order d in the Brauer group then there is a divisor class of degree d on X. The associated linear system defines the d-dimensional embedding of X over a splitting field L.
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https://en.wikipedia.org/wiki/Severi–Brauer_variety
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In mathematics, a Sheffer sequence or poweroid is a polynomial sequence, i.e., a sequence (pn(x): n = 0, 1, 2, 3, ...) of polynomials in which the index of each polynomial equals its degree, satisfying conditions related to the umbral calculus in combinatorics. They are named for Isador M. Sheffer.
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https://en.wikipedia.org/wiki/Sheffer_sequence
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In mathematics, a Shintani zeta function or Shintani L-function is a generalization of the Riemann zeta function. They were first studied by Takuro Shintani (1976). They include Hurwitz zeta functions and Barnes zeta functions.
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https://en.wikipedia.org/wiki/Shintani_zeta_function
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In mathematics, a Shioda modular surface is one of the elliptic surfaces studied by Shioda (1972).
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https://en.wikipedia.org/wiki/Shioda_modular_surface
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In mathematics, a Siegel domain or Piatetski-Shapiro domain is a special open subset of complex affine space generalizing the Siegel upper half plane studied by Siegel (1939). They were introduced by Piatetski-Shapiro (1959, 1969) in his study of bounded homogeneous domains.
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https://en.wikipedia.org/wiki/Siegel_domain
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In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943.Siegel modular varieties are the most basic examples of Shimura varieties. Siegel modular varieties generalize moduli spaces of elliptic curves to higher dimensions and play a central role in the theory of Siegel modular forms, which generalize classical modular forms to higher dimensions. They also have applications to black hole entropy and conformal field theory.
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https://en.wikipedia.org/wiki/Moduli_space_of_abelian_varieties
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In mathematics, a Smarandache–Wellin number is an integer that in a given base is the concatenation of the first n prime numbers written in that base. Smarandache–Wellin numbers are named after Florentin Smarandache and Paul R. Wellin. The first decimal Smarandache–Wellin numbers are: 2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, ... (sequence A019518 in the OEIS).
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https://en.wikipedia.org/wiki/Smarandache–Wellin_number
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In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations and partial differential equations, including the theory of harmonic maps.
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https://en.wikipedia.org/wiki/Sobolev_mapping
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In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of Lp-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense.
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https://en.wikipedia.org/wiki/Sobolev_space
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In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form f ( 2 m ) {\displaystyle f(2^{m})} , where f ( x ) {\displaystyle f(x)} is a low-degree polynomial with small integer coefficients. These primes allow fast modular reduction algorithms and are widely used in cryptography. They are named after Jerome Solinas. This class of numbers encompasses a few other categories of prime numbers: Mersenne primes, which have the form 2 k − 1 {\displaystyle 2^{k}-1} , Crandall or pseudo-Mersenne primes, which have the form 2 k − c {\displaystyle 2^{k}-c} for small odd c {\displaystyle c} .
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https://en.wikipedia.org/wiki/Generalized_Mersenne_prime
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In mathematics, a Somos sequence is a sequence of numbers defined by a certain recurrence relation, described below. They were discovered by mathematician Michael Somos. From the form of their defining recurrence (which involves division), one would expect the terms of the sequence to be fractions, but nevertheless many Somos sequences have the property that all of their members are integers.
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https://en.wikipedia.org/wiki/Somos_sequence
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In mathematics, a Specht module is one of the representations of symmetric groups studied by Wilhelm Specht (1935). They are indexed by partitions, and in characteristic 0 the Specht modules of partitions of n form a complete set of irreducible representations of the symmetric group on n points.
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https://en.wikipedia.org/wiki/Specht_modules
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In mathematics, a Spin(7)-manifold is an eight-dimensional Riemannian manifold whose holonomy group is contained in Spin(7). Spin(7)-manifolds are Ricci-flat and admit a parallel spinor. They also admit a parallel 4-form, known as the Cayley form, which is a calibrating form for a special class of submanifolds called Cayley cycles.
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https://en.wikipedia.org/wiki/Spin(7)_manifold
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In mathematics, a Stanley sequence is an integer sequence generated by a greedy algorithm that chooses the sequence members to avoid arithmetic progressions. If S {\displaystyle S} is a finite set of non-negative integers on which no three elements form an arithmetic progression (that is, a Salem–Spencer set), then the Stanley sequence generated from S {\displaystyle S} starts from the elements of S {\displaystyle S} , in sorted order, and then repeatedly chooses each successive element of the sequence to be a number that is larger than the already-chosen numbers and does not form any three-term arithmetic progression with them. These sequences are named after Richard P. Stanley.
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https://en.wikipedia.org/wiki/Stanley_sequence
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In mathematics, a Stanley–Reisner ring, or face ring, is a quotient of a polynomial algebra over a field by a square-free monomial ideal. Such ideals are described more geometrically in terms of finite simplicial complexes. The Stanley–Reisner ring construction is a basic tool within algebraic combinatorics and combinatorial commutative algebra. Its properties were investigated by Richard Stanley, Melvin Hochster, and Gerald Reisner in the early 1970s.
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https://en.wikipedia.org/wiki/Stanley–Reisner_ring
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In mathematics, a Stone algebra, or Stone lattice, is a pseudo-complemented distributive lattice such that a* ∨ a** = 1. They were introduced by Grätzer & Schmidt (1957) and named after Marshall Harvey Stone. Boolean algebras are Stone algebras, and Stone algebras are Ockham algebras. Examples: The open-set lattice of an extremally disconnected space is a Stone algebra. The lattice of positive divisors of a given positive integer is a Stone lattice.
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https://en.wikipedia.org/wiki/Stone_algebra
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In mathematics, a Sturmian word (Sturmian sequence or billiard sequence), named after Jacques Charles François Sturm, is a certain kind of infinitely long sequence of characters. Such a sequence can be generated by considering a game of English billiards on a square table. The struck ball will successively hit the vertical and horizontal edges labelled 0 and 1 generating a sequence of letters. This sequence is a Sturmian word.
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https://en.wikipedia.org/wiki/Sturmian_word
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In mathematics, a Størmer number or arc-cotangent irreducible number is a positive integer n {\displaystyle n} for which the greatest prime factor of n 2 + 1 {\displaystyle n^{2}+1} is greater than or equal to 2 n {\displaystyle 2n} . They are named after Carl Størmer.
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https://en.wikipedia.org/wiki/Størmer_number
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In mathematics, a Suslin algebra is a Boolean algebra that is complete, atomless, countably distributive, and satisfies the countable chain condition. They are named after Mikhail Yakovlevich Suslin.The existence of Suslin algebras is independent of the axioms of ZFC, and is equivalent to the existence of Suslin trees or Suslin lines.
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https://en.wikipedia.org/wiki/Suslin_algebra
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In mathematics, a Suslin representation of a set of reals (more precisely, elements of Baire space) is a tree whose projection is that set of reals. More generally, a subset A of κω is λ-Suslin if there is a tree T on κ × λ such that A = p. By a tree on κ × λ we mean here a subset T of the union of κi × λi for all i ∈ N (or i < ω in set-theoretical notation). Here, p = { f | ∃g: (f,g) ∈ } is the projection of T, where = { (f, g ) | ∀n ∈ ω: (f(n), g(n)) ∈ T } is the set of branches through T. Since is a closed set for the product topology on κω × λω where κ and λ are equipped with the discrete topology (and all closed sets in κω × λω come in this way from some tree on κ × λ), λ-Suslin subsets of κω are projections of closed subsets in κω × λω. When one talks of Suslin sets without specifying the space, then one usually means Suslin subsets of R, which descriptive set theorists usually take to be the set ωω.
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https://en.wikipedia.org/wiki/Suslin_set
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In mathematics, a Suslin tree is a tree of height ω1 such that every branch and every antichain is at most countable. They are named after Mikhail Yakovlevich Suslin. Every Suslin tree is an Aronszajn tree. The existence of a Suslin tree is independent of ZFC, and is equivalent to the existence of a Suslin line (shown by Kurepa (1935)) or a Suslin algebra.
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https://en.wikipedia.org/wiki/Suslin_tree
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The diamond principle, a consequence of V=L, implies that there is a Suslin tree, and Martin's axiom MA(ℵ1) implies that there are no Suslin trees. More generally, for any infinite cardinal κ, a κ-Suslin tree is a tree of height κ such that every branch and antichain has cardinality less than κ. In particular a Suslin tree is the same as a ω1-Suslin tree. Jensen (1972) showed that if V=L then there is a κ-Suslin tree for every infinite successor cardinal κ. Whether the Generalized Continuum Hypothesis implies the existence of an ℵ2-Suslin tree, is a longstanding open problem.
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https://en.wikipedia.org/wiki/Suslin_tree
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