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Paley graphs are named after Raymond Paley. They are closely related to the Paley construction for constructing Hadamard matrices from quadratic residues (Paley 1933). They were introduced as graphs independently by Sachs (1962) and Erdős & Rényi (1963). | https://en.wikipedia.org/wiki/Paley_digraph |
Sachs was interested in them for their self-complementarity properties, while Erdős and Rényi studied their symmetries. Paley digraphs are directed analogs of Paley graphs that yield antisymmetric conference matrices. They were introduced by Graham & Spencer (1971) (independently of Sachs, Erdős, and Rényi) as a way of... | https://en.wikipedia.org/wiki/Paley_digraph |
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's sta... | https://en.wikipedia.org/wiki/Pappus's_centroid_theorem |
In mathematics, Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of collinear points A , B , C , {\displaystyle A,B,C,} and another set of collinear points a , b , c , {\displaystyle a,b,c,} then the intersection points X , Y , Z {\displaystyle X,Y,Z} of line pairs A b {\displayst... | https://en.wikipedia.org/wiki/Pappian_plane |
If the Pappus line u {\displaystyle u} and the lines g , h {\displaystyle g,h} have a point in common, one gets the so-called little version of Pappus's theorem.The dual of this incidence theorem states that given one set of concurrent lines A , B , C {\displaystyle A,B,C} , and another set of concurrent lines a , b , ... | https://en.wikipedia.org/wiki/Pappian_plane |
Pascal's theorem is in turn a special case of the Cayley–Bacharach theorem. The Pappus configuration is the configuration of 9 lines and 9 points that occurs in Pappus's theorem, with each line meeting 3 of the points and each point meeting 3 lines. In general, the Pappus line does not pass through the point of interse... | https://en.wikipedia.org/wiki/Pappian_plane |
This configuration is self dual. Since, in particular, the lines B c , b C , X Y {\displaystyle Bc,bC,XY} have the properties of the lines x , y , z {\displaystyle x,y,z} of the dual theorem, and collinearity of X , Y , Z {\displaystyle X,Y,Z} is equivalent to concurrence of B c , b C , X Y {\displaystyle Bc,bC,XY} , t... | https://en.wikipedia.org/wiki/Pappian_plane |
In mathematics, Parseval's theorem usually refers to the result that the Fourier transform is unitary; loosely, that the sum (or integral) of the square of a function is equal to the sum (or integral) of the square of its transform. It originates from a 1799 theorem about series by Marc-Antoine Parseval, which was late... | https://en.wikipedia.org/wiki/Rayleigh's_energy_theorem |
In mathematics, Pascal's pyramid is a three-dimensional arrangement of the trinomial numbers, which are the coefficients of the trinomial expansion and the trinomial distribution. Pascal's pyramid is the three-dimensional analog of the two-dimensional Pascal's triangle, which contains the binomial numbers and relates t... | https://en.wikipedia.org/wiki/Pascal's_pyramid |
In mathematics, Pascal's rule (or Pascal's formula) is a combinatorial identity about binomial coefficients. It states that for positive natural numbers n and k, where ( n k ) {\displaystyle {\tbinom {n}{k}}} is a binomial coefficient; one interpretation of the coefficient of the xk term in the expansion of (1 + x)n. T... | https://en.wikipedia.org/wiki/Pascal's_rule |
In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem. | https://en.wikipedia.org/wiki/Pascal's_simplex |
In mathematics, Pascal's triangle is a triangular array of the binomial coefficients arising in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, G... | https://en.wikipedia.org/wiki/Pascal's_Triangle |
In mathematics, Peetre's inequality, named after Jaak Peetre, says that for any real number t {\displaystyle t} and any vectors x {\displaystyle x} and y {\displaystyle y} in R n , {\displaystyle \mathbb {R} ^{n},} the following inequality holds: The inequality was proved by J. Peetre in 1959 and has founds application... | https://en.wikipedia.org/wiki/Peetre's_inequality |
In mathematics, Petzval stressed practical applicability. He said, "Mankind does not exist for science's sake, but science should be used to improve the conditions of mankind." He worked on applications of the Laplace transformation. His work was very thorough, but not completely satisfying, since he could not use an e... | https://en.wikipedia.org/wiki/Joseph_Petzval |
Petzval wrote a paper in two volumes as well as a long work on this subject. A controversy with the student Simon Spritzer, who accused Petzval of plagiarism of Pierre-Simon Laplace, led the Spritzer-influenced mathematicians George Boole and Jules Henri Poincaré to later name the transformation after Laplace. Petzval ... | https://en.wikipedia.org/wiki/Joseph_Petzval |
In mathematics, Pfaffian functions are a certain class of functions whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff. | https://en.wikipedia.org/wiki/Pfaffian_function |
In mathematics, Philo tackled the problem of doubling the cube. The doubling of the cube was necessitated by the following problem: given a catapult, construct a second catapult that is capable of firing a projectile twice as heavy as the projectile of the first catapult. His solution was to find the point of intersect... | https://en.wikipedia.org/wiki/Philo_of_Byzantium |
In mathematics, Picard–Lefschetz theory studies the topology of a complex manifold by looking at the critical points of a holomorphic function on the manifold. It was introduced by Émile Picard for complex surfaces in his book Picard & Simart (1897), and extended to higher dimensions by Solomon Lefschetz (1924). It is ... | https://en.wikipedia.org/wiki/Picard–Lefschetz_theory |
In mathematics, Pieri's formula, named after Mario Pieri, describes the product of a Schubert cycle by a special Schubert cycle in the Schubert calculus, or the product of a Schur polynomial by a complete symmetric function. In terms of Schur functions sλ indexed by partitions λ, it states that s μ h r = ∑ λ s λ {\disp... | https://en.wikipedia.org/wiki/Pieri's_formula |
Pieri's formula implies Giambelli's formula. The Littlewood–Richardson rule is a generalization of Pieri's formula giving the product of any two Schur functions. Monk's formula is an analogue of Pieri's formula for flag manifolds. | https://en.wikipedia.org/wiki/Pieri's_formula |
In mathematics, Pisier–Ringrose inequality is an inequality in the theory of C*-algebras which was proved by Gilles Pisier in 1978 affirming a conjecture of John Ringrose. It is an extension of the Grothendieck inequality. | https://en.wikipedia.org/wiki/Pisier–Ringrose_inequality |
In mathematics, Plateau's problem is to show the existence of a minimal surface with a given boundary, a problem raised by Joseph-Louis Lagrange in 1760. However, it is named after Joseph Plateau who experimented with soap films. The problem is considered part of the calculus of variations. The existence and regularity... | https://en.wikipedia.org/wiki/Plateau's_problem |
In mathematics, Poinsot's spirals are two spirals represented by the polar equations r = a csch ( n θ ) {\displaystyle r=a\ \operatorname {csch} (n\theta )} r = a sech ( n θ ) {\displaystyle r=a\ \operatorname {sech} (n\theta )} where csch is the hyperbolic cosecant, and sech is the hyperbolic secant. They are name... | https://en.wikipedia.org/wiki/Poinsot's_spirals |
In mathematics, Pontryagin duality is a duality between locally compact abelian groups that allows generalizing Fourier transform to all such groups, which include the circle group (the multiplicative group of complex numbers of modulus one), the finite abelian groups (with the discrete topology), and the additive grou... | https://en.wikipedia.org/wiki/Pontryagin_dual |
The subject is named after Lev Pontryagin who laid down the foundations for the theory of locally compact abelian groups and their duality during his early mathematical works in 1934. Pontryagin's treatment relied on the groups being second-countable and either compact or discrete. This was improved to cover the genera... | https://en.wikipedia.org/wiki/Pontryagin_dual |
In mathematics, Porter's constant C arises in the study of the efficiency of the Euclidean algorithm. It is named after J. W. Porter of University College, Cardiff. Euclid's algorithm finds the greatest common divisor of two positive integers m and n. Hans Heilbronn proved that the average number of iterations of Eucli... | https://en.wikipedia.org/wiki/Porter's_constant |
In mathematics, Probabilistic number theory is a subfield of number theory, which explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers are, in some serious sense, like independent random variables. This however is n... | https://en.wikipedia.org/wiki/Probabilistic_number_theory |
In mathematics, Proizvolov's identity is an identity concerning sums of differences of positive integers. The identity was posed by Vyacheslav Proizvolov as a problem in the 1985 All-Union Soviet Student Olympiads (Savchev & Andreescu 2002, p. 66). | https://en.wikipedia.org/wiki/Proizvolov's_identity |
To state the identity, take the first 2N positive integers, 1, 2, 3, ..., 2N − 1, 2N,and partition them into two subsets of N numbers each. Arrange one subset in increasing order: A 1 < A 2 < ⋯ < A N . {\displaystyle A_{1} B 2 > ⋯ > B N . {\displaystyle B_{1}>B_{2}>\cdots >B_{N}.} Then the sum | A 1 − B 1 | + | A 2 − B... | https://en.wikipedia.org/wiki/Proizvolov's_identity |
In mathematics, Property B is a certain set theoretic property. Formally, given a finite set X, a collection C of subsets of X has Property B if we can partition X into two disjoint subsets Y and Z such that every set in C meets both Y and Z. The property gets its name from mathematician Felix Bernstein, who first intr... | https://en.wikipedia.org/wiki/Property_B |
In mathematics, Property FA is a property of groups first defined by Jean-Pierre Serre. A group G is said to have property FA if every action of G on a tree has a global fixed point. Serre shows that if a group has property FA, then it cannot split as an amalgamated product or HNN extension; indeed, if G is contained i... | https://en.wikipedia.org/wiki/Serre's_property_FA |
Property FA is equivalent for countable G to the three properties: G is not an amalgamated product; G does not have Z as a quotient group; G is finitely generated. For general groups G the third condition may be replaced by requiring that G not be the union of a strictly increasing sequence of subgroup. Examples of gro... | https://en.wikipedia.org/wiki/Serre's_property_FA |
The group SL2(Z) is an exception, since it is isomorphic to the amalgamated product of the cyclic groups C4 and C6 along C2. Any quotient group of a group with property FA has property FA. If some subgroup of finite index in G has property FA then so does G, but the converse does not hold in general. If N is a normal s... | https://en.wikipedia.org/wiki/Serre's_property_FA |
In mathematics, Pugh's closing lemma is a result that links periodic orbit solutions of differential equations to chaotic behaviour. It can be formally stated as follows: Let f: M → M {\displaystyle f:M\to M} be a C 1 {\displaystyle C^{1}} diffeomorphism of a compact smooth manifold M {\displaystyle M} . Given a nonwan... | https://en.wikipedia.org/wiki/Pugh's_closing_lemma |
In mathematics, Puiseux series are a generalization of power series that allow for negative and fractional exponents of the indeterminate. For example, the series x − 2 + 2 x − 1 / 2 + x 1 / 3 + 2 x 11 / 6 + x 8 / 3 + x 5 + ⋯ = x − 12 / 6 + 2 x − 3 / 6 + x 2 / 6 + 2 x 11 / 6 + x 16 / 6 + x 30 / 6 + ⋯ {\displaystyle {\b... | https://en.wikipedia.org/wiki/Puiseux_series |
{\displaystyle x^{1/6}.} Because a complex number has n nth roots, a convergent Puiseux series typically defines n functions in a neighborhood of 0. Puiseux's theorem, sometimes also called the Newton–Puiseux theorem, asserts that, given a polynomial equation P ( x , y ) = 0 {\displaystyle P(x,y)=0} with complex coeffi... | https://en.wikipedia.org/wiki/Puiseux_series |
In other words, every branch of an algebraic curve may be locally described by a Puiseux series in x (or in x − x0 when considering branches above a neighborhood of x0 ≠ 0). Using modern terminology, Puiseux's theorem asserts that the set of Puiseux series over an algebraically closed field of characteristic 0 is itsel... | https://en.wikipedia.org/wiki/Puiseux_series |
In mathematics, Pythagorean addition is a binary operation on the real numbers that computes the length of the hypotenuse of a right triangle, given its two sides. According to the Pythagorean theorem, for a triangle with sides a {\displaystyle a} and b {\displaystyle b} , this length can be calculated as where ⊕ {\dis... | https://en.wikipedia.org/wiki/Addition_in_quadrature |
In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups. (Thus, a Lie algebroid can be thought of as 'a Lie algebra with many objects '). | https://en.wikipedia.org/wiki/R-algebroid |
In mathematics, Racah polynomials are orthogonal polynomials named after Giulio Racah, as their orthogonality relations are equivalent to his orthogonality relations for Racah coefficients. The Racah polynomials were first defined by Wilson (1978) and are given by p n ( x ( x + γ + δ + 1 ) ) = 4 F 3 . {\displaystyle p... | https://en.wikipedia.org/wiki/Racah_polynomials |
In mathematics, Ramanujan's Master Theorem, named after Srinivasa Ramanujan, is a technique that provides an analytic expression for the Mellin transform of an analytic function. The result is stated as follows: If a complex-valued function f ( x ) {\textstyle f(x)} has an expansion of the form f ( x ) = ∑ k = 0 ∞ φ ( ... | https://en.wikipedia.org/wiki/Ramanujan's_master_theorem |
}}(-x)^{k}} then the Mellin transform of f ( x ) {\textstyle f(x)} is given by ∫ 0 ∞ x s − 1 f ( x ) d x = Γ ( s ) φ ( − s ) {\displaystyle \int _{0}^{\infty }x^{s-1}f(x)\,dx=\Gamma (s)\,\varphi (-s)} where Γ ( s ) {\textstyle \Gamma (s)} is the gamma function. It was widely used by Ramanujan to calculate definite inte... | https://en.wikipedia.org/wiki/Ramanujan's_master_theorem |
In mathematics, Ramanujan's congruences are some remarkable congruences for the partition function p(n). The mathematician Srinivasa Ramanujan discovered the congruences p ( 5 k + 4 ) ≡ 0 ( mod 5 ) , p ( 7 k + 5 ) ≡ 0 ( mod 7 ) , p ( 11 k + 6 ) ≡ 0 ( mod 11 ) . {\displaystyle {\begin{aligned}p(5k+4)&\equiv 0{\pmod {5}}... | https://en.wikipedia.org/wiki/Ramanujan's_congruences |
. then the number of its partitions is a multiple of 5.If a number is 5 more than a multiple of 7, i.e. it is in the sequence5, 12, 19, 26, 33, 40, . . | https://en.wikipedia.org/wiki/Ramanujan's_congruences |
. then the number of its partitions is a multiple of 7.If a number is 6 more than a multiple of 11, i.e. it is in the sequence6, 17, 28, 39, 50, 61, . . . then the number of its partitions is a multiple of 11. | https://en.wikipedia.org/wiki/Ramanujan's_congruences |
In mathematics, Rathjen's ψ {\displaystyle \psi } psi function is an ordinal collapsing function developed by Michael Rathjen. It collapses weakly Mahlo cardinals M {\displaystyle M} to generate large countable ordinals. A weakly Mahlo cardinal is a cardinal such that the set of regular cardinals below M {\displaystyle... | https://en.wikipedia.org/wiki/Rathjen's_psi_function |
In mathematics, Ratner's theorems are a group of major theorems in ergodic theory concerning unipotent flows on homogeneous spaces proved by Marina Ratner around 1990. The theorems grew out of Ratner's earlier work on horocycle flows. The study of the dynamics of unipotent flows played a decisive role in the proof of t... | https://en.wikipedia.org/wiki/Ratner's_theorems |
In mathematics, Raynaud's isogeny theorem, proved by Raynaud (1985), relates the Faltings heights of two isogeneous elliptic curves. | https://en.wikipedia.org/wiki/Raynaud's_isogeny_theorem |
In mathematics, Reeb sphere theorem, named after Georges Reeb, states that A closed oriented connected manifold M n that admits a singular foliation having only centers is homeomorphic to the sphere Sn and the foliation has exactly two singularities. | https://en.wikipedia.org/wiki/Reeb_sphere_theorem |
In mathematics, Reeb stability theorem, named after Georges Reeb, asserts that if one leaf of a codimension-one foliation is closed and has finite fundamental group, then all the leaves are closed and have finite fundamental group. | https://en.wikipedia.org/wiki/Reeb_stability_theorem |
In mathematics, Reidemeister torsion (or R-torsion, or Reidemeister–Franz torsion) is a topological invariant of manifolds introduced by Kurt Reidemeister (Reidemeister 1935) for 3-manifolds and generalized to higher dimensions by Wolfgang Franz (1935) and Georges de Rham (1936). Analytic torsion (or Ray–Singer torsion... | https://en.wikipedia.org/wiki/Ray–Singer_torsion |
It can be used to classify lens spaces. Reidemeister torsion is closely related to Whitehead torsion; see (Milnor 1966). It has also given some important motivation to arithmetic topology; see (Mazur). For more recent work on torsion see the books (Turaev 2002) and (Nicolaescu 2002, 2003). | https://en.wikipedia.org/wiki/Ray–Singer_torsion |
In mathematics, Ribet's lemma gives conditions for a subgroup of a product of groups to be the whole product group. It was introduced by Ribet (1976, lemma 5.2.2). | https://en.wikipedia.org/wiki/Ribet's_lemma |
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be called the absolute differential calculus (the foundation of tensor calculus),... | https://en.wikipedia.org/wiki/Absolute_differential_calculus |
Tensors and tensor fields can be expressed in terms of their components, and operations on tensors and tensor fields can be expressed in terms of operations on their components. The description of tensor fields and operations on them in terms of their components is the focus of the Ricci calculus. This notation allows ... | https://en.wikipedia.org/wiki/Absolute_differential_calculus |
While much of the notation may be applied with any tensors, operations relating to a differential structure are only applicable to tensor fields. Where needed, the notation extends to components of non-tensors, particularly multidimensional arrays. | https://en.wikipedia.org/wiki/Absolute_differential_calculus |
A tensor may be expressed as a linear sum of the tensor product of vector and covector basis elements. The resulting tensor components are labelled by indices of the basis. Each index has one possible value per dimension of the underlying vector space. | https://en.wikipedia.org/wiki/Absolute_differential_calculus |
The number of indices equals the degree (or order) of the tensor. For compactness and convenience, the Ricci calculus incorporates Einstein notation, which implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generall... | https://en.wikipedia.org/wiki/Absolute_differential_calculus |
In mathematics, Richardson's theorem establishes the undecidability of the equality of real numbers defined by expressions involving integers, π, ln 2 , {\displaystyle \ln 2,} and exponential and sine functions. It was proved in 1968 by mathematician and computer scientist Daniel Richardson of the University of Bath.... | https://en.wikipedia.org/wiki/Richardson's_theorem |
In mathematics, Riemann's differential equation, named after Bernhard Riemann, is a generalization of the hypergeometric differential equation, allowing the regular singular points to occur anywhere on the Riemann sphere, rather than merely at 0, 1, and ∞ {\displaystyle \infty } . The equation is also known as the Papp... | https://en.wikipedia.org/wiki/Riemann's_differential_equation |
Let α, β and γ be the exponents of one solution at 0, 1 and ∞ {\displaystyle \infty } respectively; and let α', β' and γ' be those of the other. Then α + α ′ + β + β ′ + γ + γ ′ = 1. {\displaystyle \alpha +\alpha '+\beta +\beta '+\gamma +\gamma '=1.} By applying suitable changes of variable, it is possible to transform... | https://en.wikipedia.org/wiki/Riemann's_differential_equation |
In mathematics, Riemann–Hilbert problems, named after Bernhard Riemann and David Hilbert, are a class of problems that arise in the study of differential equations in the complex plane. Several existence theorems for Riemann–Hilbert problems have been produced by Mark Krein, Israel Gohberg and others (see the book by C... | https://en.wikipedia.org/wiki/Riemann–Hilbert_problem |
In mathematics, Robinson arithmetic is a finitely axiomatized fragment of first-order Peano arithmetic (PA), first set out by Raphael M. Robinson in 1950. It is usually denoted Q. Q is almost PA without the axiom schema of mathematical induction. Q is weaker than PA but it has the same language, and both theories are i... | https://en.wikipedia.org/wiki/Robinson_arithmetic |
In mathematics, Rodrigues' formula (formerly called the Ivory–Jacobi formula) is a formula for the Legendre polynomials independently introduced by Olinde Rodrigues (1816), Sir James Ivory (1824) and Carl Gustav Jacobi (1827). The name "Rodrigues formula" was introduced by Heine in 1878, after Hermite pointed out in 18... | https://en.wikipedia.org/wiki/Rodrigues_formula |
In mathematics, Roth's theorem or Thue–Siegel–Roth theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of very good here was r... | https://en.wikipedia.org/wiki/Roth's_theorem |
In mathematics, Ruffini's rule is a method for computation of the Euclidean division of a polynomial by a binomial of the form x – r. It was described by Paolo Ruffini in 1804. The rule is a special case of synthetic division in which the divisor is a linear factor. | https://en.wikipedia.org/wiki/Ruffini's_rule |
In mathematics, S-space is a regular topological space that is hereditarily separable but is not a Lindelöf space. L-space is a regular topological space that is hereditarily Lindelöf but not separable. A space is separable if it has a countable dense set and hereditarily separable if every subspace is separable. It ha... | https://en.wikipedia.org/wiki/S_and_L_spaces |
It was shown in the early 1980s that the existence of S-space is independent of the usual axioms of ZFC. This means that to prove the existence of an S-space or to prove the non-existence of S-space, we need to assume axioms beyond those of ZFC. The L-space problem (whether an L-space can exist without assuming additio... | https://en.wikipedia.org/wiki/S_and_L_spaces |
Todorcevic proved that under PFA there are no S-spaces. This means that every regular T 1 {\displaystyle T_{1}} hereditarily separable space is Lindelöf. For some time, it was believed the L-space problem would have a similar solution (that its existence would be independent of ZFC). | https://en.wikipedia.org/wiki/S_and_L_spaces |
Todorcevic showed that there is a model of set theory with Martin's axiom where there is an L-space but there are no S-spaces. Further, Todorcevic found a compact S-space from a Cohen real. In 2005, Moore solved the L-space problem by constructing an L-space without assuming additional axioms and by combining Todorcevi... | https://en.wikipedia.org/wiki/S_and_L_spaces |
In mathematics, S2S is the monadic second order theory with two successors. It is one of the most expressive natural decidable theories known, with many decidable theories interpretable in S2S. Its decidability was proved by Rabin in 1969. | https://en.wikipedia.org/wiki/S2S_(mathematics) |
In mathematics, SO(5), also denoted SO5(R) or SO(5,R), is the special orthogonal group of degree 5 over the field R of real numbers, i.e. (isomorphic to) the group of orthogonal 5×5 matrices of determinant 1. | https://en.wikipedia.org/wiki/SO(5) |
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space. It could be either a real or complex simple Lie group of rank 4 and dimension 28. | https://en.wikipedia.org/wiki/SO(8) |
In mathematics, Salomon Bochner proved in 1946 that any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Consequently the isometry group of the manifold must be finite. | https://en.wikipedia.org/wiki/Bochner's_theorem_(Riemannian_geometry) |
In mathematics, Sard's theorem, also known as Sard's lemma or the Morse–Sard theorem, is a result in mathematical analysis that asserts that the set of critical values (that is, the image of the set of critical points) of a smooth function f from one Euclidean space or manifold to another is a null set, i.e., it has Le... | https://en.wikipedia.org/wiki/Sard's_lemma |
In mathematics, Sazonov's theorem, named after Vyacheslav Vasilievich Sazonov (Вячесла́в Васи́льевич Сазо́нов), is a theorem in functional analysis. It states that a bounded linear operator between two Hilbert spaces is γ-radonifying if it is a Hilbert–Schmidt operator. The result is also important in the study of stoc... | https://en.wikipedia.org/wiki/Sazonov's_theorem |
In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. It states that, if f n {\displaystyle f_{n}} is a sequence of integrable functions on a measure space ( X , Σ , μ ) {\displaystyle (X,\Sigma ,\mu )} that converges almost everywhere to ano... | https://en.wikipedia.org/wiki/Scheffé's_lemma |
In mathematics, Scheinerman's conjecture, now a theorem, states that every planar graph is the intersection graph of a set of line segments in the plane. This conjecture was formulated by E. R. Scheinerman in his Ph.D. thesis (1984), following earlier results that every planar graph could be represented as the intersec... | https://en.wikipedia.org/wiki/Scheinerman's_conjecture |
It was proven by Jeremie Chalopin and Daniel Gonçalves (2009). For instance, the graph G shown below to the left may be represented as the intersection graph of the set of segments shown below to the right. Here, vertices of G are represented by straight line segments and edges of G are represented by intersection poin... | https://en.wikipedia.org/wiki/Scheinerman's_conjecture |
Scheinerman also conjectured that segments with only three directions would be sufficient to represent 3-colorable graphs, and West (1991) conjectured that analogously every planar graph could be represented using four directions. If a graph is represented with segments having only k directions and no two segments belo... | https://en.wikipedia.org/wiki/Scheinerman's_conjecture |
Hartman, Newman & Ziv (1991) and de Fraysseix, Ossona de Mendez & Pach (1991) proved that every bipartite planar graph can be represented as an intersection graph of horizontal and vertical line segments; for this result see also Czyzowicz, Kranakis & Urrutia (1998). De Castro et al. (2002) proved that every triangle-f... | https://en.wikipedia.org/wiki/Scheinerman's_conjecture |
Chalopin, Gonçalves & Ochem (2007) proved that planar graphs are in 1-STRING, the class of intersection graphs of simple curves in the plane that intersect each other in at most one crossing point per pair. This class is intermediate between the intersection graphs of segments appearing in Scheinerman's conjecture and ... | https://en.wikipedia.org/wiki/Scheinerman's_conjecture |
In mathematics, Schilder's theorem is a generalization of the Laplace method from integrals on R n {\displaystyle \mathbb {R} ^{n}} to functional Wiener integration. The theorem is used in the large deviations theory of stochastic processes. Roughly speaking, out of Schilder's theorem one gets an estimate for the proba... | https://en.wikipedia.org/wiki/Schilder's_theorem |
In mathematics, Schinzel's hypothesis H is one of the most famous open problems in the topic of number theory. It is a very broad generalization of widely open conjectures such as the twin prime conjecture. The hypothesis is named after Andrzej Schinzel. | https://en.wikipedia.org/wiki/Schinzel's_hypothesis_H |
In mathematics, Scholz's reciprocity law is a reciprocity law for quadratic residue symbols of real quadratic number fields discovered by Theodor Schönemann (1839) and rediscovered by Arnold Scholz (1929). | https://en.wikipedia.org/wiki/Scholz's_reciprocity_law |
In mathematics, Schreier's lemma is a theorem in group theory used in the Schreier–Sims algorithm and also for finding a presentation of a subgroup. | https://en.wikipedia.org/wiki/Schreier's_lemma |
In mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry (part of enumerative geometry). It was a precursor of several more modern theories, for example characteristic classes, and in ... | https://en.wikipedia.org/wiki/Schubert's_enumerative_calculus |
Even more generally, "Schubert calculus" is often understood to encompass the study of analogous questions in generalized cohomology theories. The objects introduced by Schubert are the Schubert cells, which are locally closed sets in a Grassmannian defined by conditions of incidence of a linear subspace in projective ... | https://en.wikipedia.org/wiki/Schubert's_enumerative_calculus |
The intersection theory of these cells, which can be seen as the product structure in the cohomology ring of the Grassmannian of associated cohomology classes, in principle allows the prediction of the cases where intersections of cells results in a finite set of points, which are potentially concrete answers to enumer... | https://en.wikipedia.org/wiki/Schubert's_enumerative_calculus |
In mathematics, Schubert polynomials are generalizations of Schur polynomials that represent cohomology classes of Schubert cycles in flag varieties. They were introduced by Lascoux & Schützenberger (1982) and are named after Hermann Schubert. | https://en.wikipedia.org/wiki/Schubert_polynomials |
In mathematics, Schur algebras, named after Issai Schur, are certain finite-dimensional algebras closely associated with Schur–Weyl duality between general linear and symmetric groups. They are used to relate the representation theories of those two groups. Their use was promoted by the influential monograph of J. A. G... | https://en.wikipedia.org/wiki/Schur_algebra |
In mathematics, Schur polynomials, named after Issai Schur, are certain symmetric polynomials in n variables, indexed by partitions, that generalize the elementary symmetric polynomials and the complete homogeneous symmetric polynomials. In representation theory they are the characters of polynomial irreducible represe... | https://en.wikipedia.org/wiki/Skew_Schur_function |
In mathematics, Schur's inequality, named after Issai Schur, establishes that for all non-negative real numbers x, y, z, and t>0, ∑ c y c x t ( x − y ) ( x − z ) = x t ( x − y ) ( x − z ) + y t ( y − z ) ( y − x ) + z t ( z − x ) ( z − y ) ≥ 0 {\displaystyle \sum _{cyc}x^{t}(x-y)(x-z)=x^{t}(x-y)(x-z)+y^{t}(y-z)(y-x)+z^... | https://en.wikipedia.org/wiki/Schur's_Inequality |
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations of a group G and φ is a linear map from M to N that commutes with the action of the group, then ... | https://en.wikipedia.org/wiki/Schur's_Lemma |
In mathematics, Schwartz space S {\displaystyle {\mathcal {S}}} is the function space of all functions whose derivatives are rapidly decreasing. This space has the important property that the Fourier transform is an automorphism on this space. This property enables one, by duality, to define the Fourier transform for e... | https://en.wikipedia.org/wiki/Schwartz_function |
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds. | https://en.wikipedia.org/wiki/Seifert–Weber_space |
It is constructed by gluing each face of a dodecahedron to its opposite in a way that produces a closed 3-manifold. There are three ways to do this gluing consistently. Opposite faces are misaligned by 1/10 of a turn, so to match them they must be rotated by 1/10, 3/10 or 5/10 turn; a rotation of 3/10 gives the Seifert... | https://en.wikipedia.org/wiki/Seifert–Weber_space |
Rotation of 1/10 gives the Poincaré homology sphere, and rotation by 5/10 gives 3-dimensional real projective space. With the 3/10-turn gluing pattern, the edges of the original dodecahedron are glued to each other in groups of five. Thus, in the Seifert–Weber space, each edge is surrounded by five pentagonal faces, an... | https://en.wikipedia.org/wiki/Seifert–Weber_space |
This does not match the 117° dihedral angle of a regular dodecahedron in Euclidean space, but in hyperbolic space there exist regular dodecahedra with any dihedral angle between 60° and 117°, and the hyperbolic dodecahedron with dihedral angle 72° may be used to give the Seifert–Weber space a geometric structure as a h... | https://en.wikipedia.org/wiki/Seifert–Weber_space |
In mathematics, Seifert–Weber space (introduced by Herbert Seifert and Constantin Weber) is a closed hyperbolic 3-manifold. It is also known as Seifert–Weber dodecahedral space and hyperbolic dodecahedral space. It is one of the first discovered examples of closed hyperbolic 3-manifolds. | https://en.wikipedia.org/wiki/3-manifold |
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