thewordsmiths/wiki_stem · Datasets at Hugging Face

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c_ntthbh1ikq82	Paley digraph	Summary	Paley_digraph	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).
c_3g7xjfirb3u2	Paley digraph	Summary	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 constructing tournaments with a property previously known to be held only by random tournaments: in a Paley digraph, every small subset of vertices is dominated by some other vertex.
c_smc3jsz39x5r	Pappus's centroid theorem	Summary	Guldinus_theorem	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 statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.
c_oxc88m2v3lx3	Pappian plane	Summary	Pappus's_hexagon_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 {\displaystyle Ab} and a B , A c {\displaystyle aB,Ac} and a C , B c {\displaystyle aC,Bc} and b C {\displaystyle bC} are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon A b C a B c {\displaystyle AbCaBc} .It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one restricts the projective plane such that the Pappus line u {\displaystyle u} is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.
c_95wyhfyzlqnu	Pappian plane	Summary	Pappus's_hexagon_theorem	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 , c {\displaystyle a,b,c} , then the lines x , y , z {\displaystyle x,y,z} defined by pairs of points resulting from pairs of intersections A ∩ b {\displaystyle A\cap b} and a ∩ B , A ∩ c {\displaystyle a\cap B,\;A\cap c} and a ∩ C , B ∩ c {\displaystyle a\cap C,\;B\cap c} and b ∩ C {\displaystyle b\cap C} are concurrent. (Concurrent means that the lines pass through one point.) Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines.
c_2i8w72rwnyc4	Pappian plane	Summary	Pappus's_hexagon_theorem	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 intersection of A B C {\displaystyle ABC} and a b c {\displaystyle abc} .
c_ul3wztt2des2	Pappian plane	Summary	Pappus's_hexagon_theorem	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} , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.
c_gpnivbjx2yyi	Rayleigh's energy theorem	Summary	Rayleigh's_energy_theorem	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 later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.
c_a7y9yhdprxon	Pascal's pyramid	Summary	Pascal's_pyramid	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 to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.
c_78rykobzstge	Pascal's rule	Summary	Pascal's_rule	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. There is no restriction on the relative sizes of n and k, since, if n < k the value of the binomial coefficient is zero and the identity remains valid. Pascal's rule can also be viewed as a statement that the formula solves the linear two-dimensional difference equation over the natural numbers. Thus, Pascal's rule is also a statement about a formula for the numbers appearing in Pascal's triangle. Pascal's rule can also be generalized to apply to multinomial coefficients.
c_t8xmhxrw1sie	Pascal's simplex	Summary	Pascal's_simplex	In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.
c_j9mmsmlcxl26	Pascal's Triangle	Summary	Pascal_triangle	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, Germany, and Italy.The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 {\displaystyle n=0} at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 {\displaystyle k=0} and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.
c_axnzh9h3jkud	Peetre's inequality	Summary	Peetre's_inequality	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 applications in functional analysis and Sobolev spaces.
c_yxqpq1wyvfv6	Joseph Petzval	Mathematics	Joseph_Petzval > Discoveries and inventions > Mathematics	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 edge integration in order to invert the transformation.
c_pu8zon5g200y	Joseph Petzval	Mathematics	Joseph_Petzval > Discoveries and inventions > Mathematics	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 tried to represent practically everything in his environment mathematically. Thus he tried to mathematically model fencing or the course of the horse. His obsession with mathematics finally led to the discovery of the portrait objective.
c_2v4jbaanz8kw	Pfaffian function	Summary	Pfaffian_function	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.
c_i2wztj8g0xza	Philo of Byzantium	Mathematics	Philo_of_Byzantium > Works > Mathematics	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 intersection of a rectangular hyperbola and a circle, a solution that is similar to the solution given by Hero of Alexandria several centuries later.
c_evpslfvi7czi	Picard–Lefschetz theory	Summary	Picard–Lefschetz_formula	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 a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. Pierre Deligne and Nicholas Katz (1973) extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures.
c_z5w2wq6lgen5	Pieri's formula	Summary	Pieri's_formula	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 {\displaystyle \displaystyle s_{\mu }h_{r}=\sum _{\lambda }s_{\lambda }} where hr is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding r elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for multiplying an elementary symmetric polynomial with a Schur polynomial: s μ e r = ∑ λ s λ {\displaystyle \displaystyle s_{\mu }e_{r}=\sum _{\lambda }s_{\lambda }} The sum is now taken over all partitions λ obtained from μ by adding r elements, no two in the same row.
c_qycas88ya5d7	Pieri's formula	Summary	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.
c_o10lqxqk312n	Pisier–Ringrose inequality	Summary	Pisier–Ringrose_inequality	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.
c_gid1ybkb1xtx	Plateau's problem	Summary	Plateau_problem	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 problems are part of geometric measure theory.
c_dabb2yyctybn	Poinsot's spirals	Summary	Poinsot's_spirals	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 named after the French mathematician Louis Poinsot.
c_b4mtxoscmjgk	Pontryagin dual	Summary	Dual_group_(Quantum_Computing)	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 group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this theorem.
c_4i53tlel5bwi	Pontryagin dual	Summary	Dual_group_(Quantum_Computing)	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 general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.
c_y0mx57n1xiwo	Porter's constant	Summary	Porter's_constant	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 Euclid's algorithm, for fixed n and averaged over all choices of relatively prime integers m < n, is 12 ln ⁡ 2 π 2 ln ⁡ n + o ( ln ⁡ n ) . {\displaystyle {\frac {12\ln 2}{\pi ^{2}}}\ln n+o(\ln n).} Porter showed that the error term in this estimate is a constant, plus a polynomially-small correction, and Donald Knuth evaluated this constant to high accuracy. It is: C = 6 ln ⁡ 2 π 2 − 1 2 = 6 ln ⁡ 2 ( ( 48 ln ⁡ A ) − ( ln ⁡ 2 ) − ( 4 ln ⁡ π ) − 2 ) π 2 − 1 2 = 1.4670780794 … {\displaystyle {\begin{aligned}C&={{6\ln 2} \over {\pi ^{2}}}\left-{{1} \over {2}}\\&={{{6\ln 2}((48\ln A)-(\ln 2)-(4\ln \pi )-2)} \over {\pi ^{2}}}-{{1} \over {2}}\\&=1.4670780794\ldots \end{aligned}}} where γ {\displaystyle \gamma } is the Euler–Mascheroni constant ζ {\displaystyle \zeta } is the Riemann zeta function A {\displaystyle A} is the Glaisher–Kinkelin constant(sequence A086237 in the OEIS) − ζ ′ ( 2 ) = π 2 6 = ∑ k = 2 ∞ ln ⁡ k k 2 {\displaystyle -\zeta ^{\prime }(2)={{\pi ^{2}} \over 6}\left=\sum _{k=2}^{\infty }{{\ln k} \over {k^{2}}}} {\displaystyle }
c_menywn0eaq5s	Probabilistic number theory	Summary	Probabilistic_number_theory	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 not an idea that has a unique useful formal expression. The founders of the theory were Paul Erdős, Aurel Wintner and Mark Kac during the 1930s, one of the periods of investigation in analytic number theory. Foundational results include the Erdős–Wintner theorem and the Erdős–Kac theorem on additive functions.
c_exoa6hldzmi0	Proizvolov's identity	Summary	Proizvolov's_identity	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).
c_669uvfikshtm	Proizvolov's identity	Summary	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 2 \| + ⋯ + \| A N − B N \| {\displaystyle \|A_{1}-B_{1}\|+\|A_{2}-B_{2}\|+\cdots +\|A_{N}-B_{N}\|} is always equal to N2.
c_fx0c759xfhye	Property B	Summary	Property_B	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 introduced the property in 1908.Property B is equivalent to 2-coloring the hypergraph described by the collection C. A hypergraph with property B is also called 2-colorable. : 468 Sometimes it is also called bipartite, by analogy to the bipartite graphs. Property B is often studied for uniform hypergraphs (set systems in which all subsets of the system have the same cardinality) but it has also been considered in the non-uniform case.The problem of checking whether a collection C has Property B is called the set splitting problem.
c_17ewdcupr4d0	Serre's property FA	Summary	Serre's_property_FA	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 in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.
c_thlhy8pq5eux	Serre's property FA	Summary	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 groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2.
c_6j57tnls2kf7	Serre's property FA	Summary	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 subgroup of G and both N and G/N have property FA, then so does G. It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.
c_azmvocvteodx	Pugh's closing lemma	Summary	Pugh's_closing_lemma	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 nonwandering point x {\displaystyle x} of f {\displaystyle f} , there exists a diffeomorphism g {\displaystyle g} arbitrarily close to f {\displaystyle f} in the C 1 {\displaystyle C^{1}} topology of Diff 1 ⁡ ( M ) {\displaystyle \operatorname {Diff} ^{1}(M)} such that x {\displaystyle x} is a periodic point of g {\displaystyle g} .
c_qv3z4m2wpdlw	Puiseux series	Summary	Puiseux_series	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 {\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}} is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an nth root of the indeterminate. For example, the example above is a Laurent series in x 1 / 6 .
c_jfcuum70yxmp	Puiseux series	Summary	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 coefficients, its solutions in y, viewed as functions of x, may be expanded as Puiseux series in x that are convergent in some neighbourhood of 0.
c_ntr7r3rf7gry	Puiseux series	Summary	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 itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.
c_qnberac9casu	Addition in quadrature	Summary	Addition_in_quadrature	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 ⊕ {\displaystyle \oplus } denotes the Pythagorean addition operation.This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature.
c_0xob7q8jx6v8	R-algebroid	Summary	R-algebroid	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 ').
c_rvgtsrok5pqc	Racah polynomials	Summary	Racah_polynomials	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_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left.}
c_1p7yir4fwvp6	Ramanujan's master theorem	Summary	Ramanujan's_master_theorem	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 ( k ) k ! ( − x ) k {\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!
c_18baq8hxpguu	Ramanujan's master theorem	Summary	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 integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).A similar result was also obtained by Glaisher.
c_zk0xc8n9flxg	Ramanujan's congruences	Summary	Ramanujan_congruences	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}},\\p(7k+5)&\equiv 0{\pmod {7}},\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}} This means that: If a number is 4 more than a multiple of 5, i.e. it is in the sequence4, 9, 14, 19, 24, 29, . .
c_br23gg5t43re	Ramanujan's congruences	Summary	Ramanujan_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, . .
c_k6x26nyejmpo	Ramanujan's congruences	Summary	Ramanujan_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.
c_ko3gj8w4n21z	Rathjen's psi function	Summary	Rathjen's_psi_function	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 M} is closed under M {\displaystyle M} (i.e. all normal functions closed in M {\displaystyle M} are closed under some regular ordinal < M {\displaystyle
c_1qv62o9ls3t5	Ratner's theorems	Summary	Ratner's_theorems	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 the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field.
c_aergkdjz6sdn	Raynaud's isogeny theorem	Summary	Raynaud's_isogeny_theorem	In mathematics, Raynaud's isogeny theorem, proved by Raynaud (1985), relates the Faltings heights of two isogeneous elliptic curves.
c_9drm5ooifrui	Reeb sphere theorem	Summary	Reeb_sphere_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.
c_2kmappf6c1vp	Reeb stability theorem	Summary	Reeb_stability_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.
c_a1f8tludcfns	Ray–Singer torsion	Summary	Analytic_torsion	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) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field.
c_in8ukz7oc0t9	Ray–Singer torsion	Summary	Analytic_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).
c_u9l0gn6h1wk4	Ribet's lemma	Summary	Ribet's_lemma	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).
c_6h1i4u6vgj27	Absolute differential calculus	Summary	Absolute_differential_calculus	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), developed by Gregorio Ricci-Curbastro in 1887–1896, and subsequently popularized in a paper written with his pupil Tullio Levi-Civita in 1900. Jan Arnoldus Schouten developed the modern notation and formalism for this mathematical framework, and made contributions to the theory, during its applications to general relativity and differential geometry in the early twentieth century.A component of a tensor is a real number that is used as a coefficient of a basis element for the tensor space. The tensor is the sum of its components multiplied by their corresponding basis elements.
c_0e72zl65l19g	Absolute differential calculus	Summary	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 an efficient expression of such tensor fields and operations.
c_74kghctmamq8	Absolute differential calculus	Summary	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.
c_508xibxk56i1	Absolute differential calculus	Summary	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.
c_gxibj50bwy7z	Absolute differential calculus	Summary	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 generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold. This allows intuitive manipulation of expressions with familiarity of only a limited set of rules.
c_6kmvfeyaiau9	Richardson's theorem	Summary	Richardson's_theorem	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. Specifically, the class of expressions for which the theorem holds is that generated by rational numbers, the number π, the number ln 2, the variable x, the operations of addition, subtraction, multiplication, composition, and the sin, exp, and abs functions. For some classes of expressions (generated by other primitives than in Richardson's theorem) there exist algorithms that can determine whether an expression is zero.
c_1znji00in5d8	Riemann's differential equation	Summary	Riemann's_differential_equation	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 Papperitz equation.The hypergeometric differential equation is a second-order linear differential equation which has three regular singular points, 0, 1 and ∞ {\displaystyle \infty } . That equation admits two linearly independent solutions; near a singularity z s {\displaystyle z_{s}} , the solutions take the form x s f ( x ) {\displaystyle x^{s}f(x)} , where x = z − z s {\displaystyle x=z-z_{s}} is a local variable, and f {\displaystyle f} is locally holomorphic with f ( 0 ) ≠ 0 {\displaystyle f(0)\neq 0} . The real number s {\displaystyle s} is called the exponent of the solution at z s {\displaystyle z_{s}} .
c_u66tqeohpby5	Riemann's differential equation	Summary	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 the hypergeometric equation: Applying Möbius transformations will adjust the positions of the regular singular points, while other transformations (see below) can change the exponents at the regular singular points, subject to the exponents adding up to 1.
c_bjnw2nbtxilx	Riemann–Hilbert problem	Summary	Riemann-Hilbert_problem	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 Clancey and Gohberg (1981)).
c_8zrqovgljayw	Robinson arithmetic	Summary	Robinson_arithmetic	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 incomplete. Q is important and interesting because it is a finitely axiomatized fragment of PA that is recursively incompletable and essentially undecidable.
c_kcevv7397583	Rodrigues formula	Summary	Rodrigues_formula	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 1865 that Rodrigues was the first to discover it. The term is also used to describe similar formulas for other orthogonal polynomials. Askey (2005) describes the history of the Rodrigues formula in detail.
c_jybcmf9qwp0a	Roth's theorem	Summary	Roth's_theorem	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 refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).
c_qx35a6ry5rut	Ruffini's rule	Summary	Ruffini's_rule	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.
c_xx1zh4bkhyc7	S and L spaces	Summary	S_and_L_spaces	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 had been believed for a long time that S-space problem and L-space problem are dual, i.e. if there is an S-space in some model of set theory then there is an L-space in the same model and vice versa – which is not true.
c_meiep44u2rt8	S and L spaces	Summary	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 additional set-theoretic assumptions beyond those of ZFC) was not resolved until recently.
c_nd3sl38buokc	S and L spaces	Summary	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).
c_tn1wfnirtbh8	S and L spaces	Summary	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 Todorcevic's rho functions with number theory.
c_u9318kozh782	S2S (mathematics)	Summary	S2S_(mathematics)	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.
c_b1271dvwyc8a	SO(5)	Summary	SO(5)	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.
c_b3eqmzgeqhem	SO(8)	Summary	SO(8)	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.
c_c7w3udgfjnd1	Bochner's theorem (Riemannian geometry)	Summary	Bochner's_theorem_(Riemannian_geometry)	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.
c_wrqzyuva042q	Sard's lemma	Summary	Sard's_theorem	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 Lebesgue measure 0. This makes the set of critical values "small" in the sense of a generic property. The theorem is named for Anthony Morse and Arthur Sard.
c_kvki73zytvz5	Sazonov's theorem	Summary	Sazonov's_theorem	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 stochastic processes and the Malliavin calculus, since results concerning probability measures on infinite-dimensional spaces are of central importance in these fields. Sazonov's theorem also has a converse: if the map is not Hilbert–Schmidt, then it is not γ-radonifying.
c_st85lb51kt1n	Scheffé's lemma	Summary	Scheffé's_lemma	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 another integrable function f {\displaystyle f} , then ∫ \| f n − f \| d μ → 0 {\displaystyle \int \|f_{n}-f\|\,d\mu \to 0} if and only if ∫ \| f n \| d μ → ∫ \| f \| d μ {\displaystyle \int \|f_{n}\|\,d\mu \to \int \|f\|\,d\mu } .In probability theory, almost sure convergence can be weakened to requiring only convergence in probability.
c_7myr59xbu1nx	Scheinerman's conjecture	Summary	Scheinerman's_conjecture	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 intersection graph of a set of simple curves in the plane (Ehrlich, Even & Tarjan 1976).
c_09njfn5gltal	Scheinerman's conjecture	Summary	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 points.
c_gqt39j94r5lr	Scheinerman's conjecture	Summary	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 belong to the same line, then the graph can be colored using k colors, one color for each direction. Therefore, if every planar graph can be represented in this way with only four directions, then the four color theorem follows.
c_3ijyak6eilf4	Scheinerman's conjecture	Summary	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-free planar graph can be represented as an intersection graph of line segments having only three directions; this result implies Grötzsch's theorem (Grötzsch 1959) that triangle-free planar graphs can be colored with three colors. de Fraysseix & Ossona de Mendez (2005) proved that if a planar graph G can be 4-colored in such a way that no separating cycle uses all four colors, then G has a representation as an intersection graph of segments.
c_foh9oc29twf7	Scheinerman's conjecture	Summary	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 the intersection graphs of unrestricted simple curves from the result of Ehrlich et al. It can also be viewed as a generalization of the circle packing theorem, which shows the same result when curves are allowed to intersect in a tangent. The proof of the conjecture by Chalopin & Gonçalves (2009) was based on an improvement of this result.
c_vq3uvwsmp27q	Schilder's theorem	Summary	Schilder's_theorem	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 probability that a (scaled-down) sample path of Brownian motion will stray far from the mean path (which is constant with value 0). This statement is made precise using rate functions. Schilder's theorem is generalized by the Freidlin–Wentzell theorem for Itō diffusions.
c_t9p8ikcvvhak	Schinzel's hypothesis H	Summary	Schinzel's_hypothesis_H	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.
c_0hyu80y9btba	Scholz's reciprocity law	Summary	Scholz's_reciprocity_law	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).
c_t1fxgh7zmpwa	Schreier's lemma	Summary	Schreier's_lemma	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.
c_a2tzfgnkcqr1	Schubert's enumerative calculus	Summary	Schubert's_enumerative_calculus	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 particular its algorithmic aspects are still of current interest. The phrase "Schubert calculus" is sometimes used to mean the enumerative geometry of linear subspaces, roughly equivalent to describing the cohomology ring of Grassmannians, and sometimes used to mean the more general enumerative geometry of nonlinear varieties.
c_csdj62cbdlh3	Schubert's enumerative calculus	Summary	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 space with a given flag. For details see Schubert variety.
c_dbkh7ouki3pa	Schubert's enumerative calculus	Summary	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 enumerative questions. A supporting theoretical result is that the Schubert cells (or rather, their classes) span the whole cohomology ring. In detailed calculations the combinatorial aspects enter as soon as the cells have to be indexed. Lifted from the Grassmannian, which is a homogeneous space, to the general linear group that acts on it, similar questions are involved in the Bruhat decomposition and classification of parabolic subgroups (by block matrix). Putting Schubert's system on a rigorous footing is Hilbert's fifteenth problem.
c_8layndjoszoa	Schubert polynomials	Summary	Schubert_polynomial	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.
c_5puq8mznrpr3	Schur algebra	Summary	Schur_algebra	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. Green first published in 1980. The name "Schur algebra" is due to Green. In the modular case (over infinite fields of positive characteristic) Schur algebras were used by Gordon James and Karin Erdmann to show that the (still open) problems of computing decomposition numbers for general linear groups and symmetric groups are actually equivalent. Schur algebras were used by Friedlander and Suslin to prove finite generation of cohomology of finite group schemes.
c_i2708g716gq5	Skew Schur function	Summary	Skew_Schur_function	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 representations of the general linear groups. The Schur polynomials form a linear basis for the space of all symmetric polynomials. Any product of Schur polynomials can be written as a linear combination of Schur polynomials with non-negative integral coefficients; the values of these coefficients is given combinatorially by the Littlewood–Richardson rule. More generally, skew Schur polynomials are associated with pairs of partitions and have similar properties to Schur polynomials.
c_uy9vtgcumeo2	Schur's Inequality	Summary	Schur's_Inequality	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^{t}(z-x)(z-y)\geq 0} with equality if and only if x = y = z or two of them are equal and the other is zero. When t is an even positive integer, the inequality holds for all real numbers x, y and z. When t = 1 {\displaystyle t=1} , the following well-known special case can be derived: x 3 + y 3 + z 3 + 3 x y z ≥ x y ( x + y ) + x z ( x + z ) + y z ( y + z ) {\displaystyle x^{3}+y^{3}+z^{3}+3xyz\geq xy(x+y)+xz(x+z)+yz(y+z)}
c_wqxycpin9vrm	Schur's Lemma	Summary	Schur_lemma	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 either φ is invertible, or φ = 0. An important special case occurs when M = N, i.e. φ is a self-map; in particular, any element of the center of a group must act as a scalar operator (a scalar multiple of the identity) on M. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.
c_2igji4jsphn4	Schwartz function	Summary	Schwartz_functions	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 elements in the dual space S ∗ {\displaystyle {\mathcal {S}}^{*}} of S {\displaystyle {\mathcal {S}}} , that is, for tempered distributions. A function in the Schwartz space is sometimes called a Schwartz function. Schwartz space is named after French mathematician Laurent Schwartz.
c_9tyq9jhiwvxr	Seifert–Weber space	Summary	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.
c_33gpsv6y76k4	Seifert–Weber space	Summary	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–Weber space.
c_6eut8v0vh51c	Seifert–Weber space	Summary	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, and the dihedral angle between these pentagons is 72°.
c_v8y6k20uofe8	Seifert–Weber space	Summary	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 hyperbolic manifold. It is a (finite volume) quotient space of the (non-finite volume) order-5 dodecahedral honeycomb, a regular tessellation of hyperbolic 3-space by dodecahedra with this dihedral angle. The Seifert–Weber space is a rational homology sphere, and its first homology group is isomorphic to Z 5 3 {\displaystyle \mathbb {Z} _{5}^{3}} . William Thurston conjectured that the Seifert–Weber space is not a Haken manifold, that is, it does not contain any incompressible surfaces; Burton, Rubinstein & Tillmann (2012) proved the conjecture with the aid of their computer software Regina.
c_w95xi1kyljs2	3-manifold	Seifert–Weber space	3-manifold > Important examples of 3-manifolds > 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.

c_ntthbh1ikq82

Paley digraph

Summary

Paley_digraph

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).

c_3g7xjfirb3u2

Paley digraph

Summary

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 constructing tournaments with a property previously known to be held only by random tournaments: in a Paley digraph, every small subset of vertices is dominated by some other vertex.

c_smc3jsz39x5r

Pappus's centroid theorem

Summary

Guldinus_theorem

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 statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.

c_oxc88m2v3lx3

Pappian plane

Summary

Pappus's_hexagon_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 {\displaystyle Ab} and a B , A c {\displaystyle aB,Ac} and a C , B c {\displaystyle aC,Bc} and b C {\displaystyle bC} are collinear, lying on the Pappus line. These three points are the points of intersection of the "opposite" sides of the hexagon A b C a B c {\displaystyle AbCaBc} .It holds in a projective plane over any field, but fails for projective planes over any noncommutative division ring. Projective planes in which the "theorem" is valid are called pappian planes. If one restricts the projective plane such that the Pappus line u {\displaystyle u} is the line at infinity, one gets the affine version of Pappus's theorem shown in the second diagram.

c_95wyhfyzlqnu

Pappian plane

Summary

Pappus's_hexagon_theorem

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 , c {\displaystyle a,b,c} , then the lines x , y , z {\displaystyle x,y,z} defined by pairs of points resulting from pairs of intersections A ∩ b {\displaystyle A\cap b} and a ∩ B , A ∩ c {\displaystyle a\cap B,\;A\cap c} and a ∩ C , B ∩ c {\displaystyle a\cap C,\;B\cap c} and b ∩ C {\displaystyle b\cap C} are concurrent. (Concurrent means that the lines pass through one point.) Pappus's theorem is a special case of Pascal's theorem for a conic—the limiting case when the conic degenerates into 2 straight lines.

c_2i8w72rwnyc4

Pappian plane

Summary

Pappus's_hexagon_theorem

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 intersection of A B C {\displaystyle ABC} and a b c {\displaystyle abc} .

c_ul3wztt2des2

Pappian plane

Summary

Pappus's_hexagon_theorem

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} , the dual theorem is therefore just the same as the theorem itself. The Levi graph of the Pappus configuration is the Pappus graph, a bipartite distance-regular graph with 18 vertices and 27 edges.

c_gpnivbjx2yyi

Rayleigh's energy theorem

Summary

Rayleigh's_energy_theorem

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 later applied to the Fourier series. It is also known as Rayleigh's energy theorem, or Rayleigh's identity, after John William Strutt, Lord Rayleigh.Although the term "Parseval's theorem" is often used to describe the unitarity of any Fourier transform, especially in physics, the most general form of this property is more properly called the Plancherel theorem.

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Pascal's pyramid

Summary

Pascal's_pyramid

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 to the binomial expansion and the binomial distribution. The binomial and trinomial numbers, coefficients, expansions, and distributions are subsets of the multinomial constructs with the same names.

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Pascal's rule

Summary

Pascal's_rule

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. There is no restriction on the relative sizes of n and k, since, if n < k the value of the binomial coefficient is zero and the identity remains valid. Pascal's rule can also be viewed as a statement that the formula solves the linear two-dimensional difference equation over the natural numbers. Thus, Pascal's rule is also a statement about a formula for the numbers appearing in Pascal's triangle. Pascal's rule can also be generalized to apply to multinomial coefficients.

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Pascal's simplex

Summary

Pascal's_simplex

In mathematics, Pascal's simplex is a generalisation of Pascal's triangle into arbitrary number of dimensions, based on the multinomial theorem.

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Pascal's Triangle

Summary

Pascal_triangle

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, Germany, and Italy.The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 {\displaystyle n=0} at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 {\displaystyle k=0} and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas the numbers 1 and 3 in row 3 are added to produce the number 4 in row 4.

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Peetre's inequality

Summary

Peetre's_inequality

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 applications in functional analysis and Sobolev spaces.

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Joseph Petzval

Mathematics

Joseph_Petzval > Discoveries and inventions > Mathematics

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 edge integration in order to invert the transformation.

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Joseph Petzval

Mathematics

Joseph_Petzval > Discoveries and inventions > Mathematics

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 tried to represent practically everything in his environment mathematically. Thus he tried to mathematically model fencing or the course of the horse. His obsession with mathematics finally led to the discovery of the portrait objective.

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Pfaffian function

Summary

Pfaffian_function

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.

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Philo of Byzantium

Mathematics

Philo_of_Byzantium > Works > Mathematics

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 intersection of a rectangular hyperbola and a circle, a solution that is similar to the solution given by Hero of Alexandria several centuries later.

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Picard–Lefschetz theory

Summary

Picard–Lefschetz_formula

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 a complex analog of Morse theory that studies the topology of a real manifold by looking at the critical points of a real function. Pierre Deligne and Nicholas Katz (1973) extended Picard–Lefschetz theory to varieties over more general fields, and Deligne used this generalization in his proof of the Weil conjectures.

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Pieri's formula

Summary

Pieri's_formula

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 {\displaystyle \displaystyle s_{\mu }h_{r}=\sum _{\lambda }s_{\lambda }} where hr is a complete homogeneous symmetric polynomial and the sum is over all partitions λ obtained from μ by adding r elements, no two in the same column. By applying the ω involution on the ring of symmetric functions, one obtains the dual Pieri rule for multiplying an elementary symmetric polynomial with a Schur polynomial: s μ e r = ∑ λ s λ {\displaystyle \displaystyle s_{\mu }e_{r}=\sum _{\lambda }s_{\lambda }} The sum is now taken over all partitions λ obtained from μ by adding r elements, no two in the same row.

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Pieri's formula

Summary

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.

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Pisier–Ringrose inequality

Summary

Pisier–Ringrose_inequality

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.

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Plateau's problem

Summary

Plateau_problem

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 problems are part of geometric measure theory.

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Poinsot's spirals

Summary

Poinsot's_spirals

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 named after the French mathematician Louis Poinsot.

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Pontryagin dual

Summary

Dual_group_(Quantum_Computing)

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 group of the integers (also with the discrete topology), the real numbers, and every finite dimensional vector space over the reals or a p-adic field. The Pontryagin dual of a locally compact abelian group is the locally compact abelian topological group formed by the continuous group homomorphisms from the group to the circle group with the operation of pointwise multiplication and the topology of uniform convergence on compact sets. The Pontryagin duality theorem establishes Pontryagin duality by stating that any locally compact abelian group is naturally isomorphic with its bidual (the dual of its dual). The Fourier inversion theorem is a special case of this theorem.

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Pontryagin dual

Summary

Dual_group_(Quantum_Computing)

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 general locally compact abelian groups by Egbert van Kampen in 1935 and André Weil in 1940.

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Porter's constant

Summary

Porter's_constant

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 Euclid's algorithm, for fixed n and averaged over all choices of relatively prime integers m < n, is 12 ln ⁡ 2 π 2 ln ⁡ n + o ( ln ⁡ n ) . {\displaystyle {\frac {12\ln 2}{\pi ^{2}}}\ln n+o(\ln n).} Porter showed that the error term in this estimate is a constant, plus a polynomially-small correction, and Donald Knuth evaluated this constant to high accuracy. It is: C = 6 ln ⁡ 2 π 2 − 1 2 = 6 ln ⁡ 2 ( ( 48 ln ⁡ A ) − ( ln ⁡ 2 ) − ( 4 ln ⁡ π ) − 2 ) π 2 − 1 2 = 1.4670780794 … {\displaystyle {\begin{aligned}C&={{6\ln 2} \over {\pi ^{2}}}\left-{{1} \over {2}}\\&={{{6\ln 2}((48\ln A)-(\ln 2)-(4\ln \pi )-2)} \over {\pi ^{2}}}-{{1} \over {2}}\\&=1.4670780794\ldots \end{aligned}}} where γ {\displaystyle \gamma } is the Euler–Mascheroni constant ζ {\displaystyle \zeta } is the Riemann zeta function A {\displaystyle A} is the Glaisher–Kinkelin constant(sequence A086237 in the OEIS) − ζ ′ ( 2 ) = π 2 6 = ∑ k = 2 ∞ ln ⁡ k k 2 {\displaystyle -\zeta ^{\prime }(2)={{\pi ^{2}} \over 6}\left=\sum _{k=2}^{\infty }{{\ln k} \over {k^{2}}}} {\displaystyle }

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Probabilistic number theory

Summary

Probabilistic_number_theory

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 not an idea that has a unique useful formal expression. The founders of the theory were Paul Erdős, Aurel Wintner and Mark Kac during the 1930s, one of the periods of investigation in analytic number theory. Foundational results include the Erdős–Wintner theorem and the Erdős–Kac theorem on additive functions.

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Proizvolov's identity

Summary

Proizvolov's_identity

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).

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Proizvolov's identity

Summary

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 2 | + ⋯ + | A N − B N | {\displaystyle |A_{1}-B_{1}|+|A_{2}-B_{2}|+\cdots +|A_{N}-B_{N}|} is always equal to N2.

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Property B

Summary

Property_B

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 introduced the property in 1908.Property B is equivalent to 2-coloring the hypergraph described by the collection C. A hypergraph with property B is also called 2-colorable. : 468 Sometimes it is also called bipartite, by analogy to the bipartite graphs. Property B is often studied for uniform hypergraphs (set systems in which all subsets of the system have the same cardinality) but it has also been considered in the non-uniform case.The problem of checking whether a collection C has Property B is called the set splitting problem.

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Serre's property FA

Summary

Serre's_property_FA

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 in an amalgamated product then it is contained in one of the factors. In particular, a finitely generated group with property FA has finite abelianization.

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Serre's property FA

Summary

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 groups with property FA include SL3(Z) and more generally G(Z) where G is a simply-connected simple Chevalley group of rank at least 2.

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Serre's property FA

Summary

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 subgroup of G and both N and G/N have property FA, then so does G. It is a theorem of Watatani that Kazhdan's property (T) implies property FA, but not conversely. Indeed, any subgroup of finite index in a T-group has property FA.

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Pugh's closing lemma

Summary

Pugh's_closing_lemma

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 nonwandering point x {\displaystyle x} of f {\displaystyle f} , there exists a diffeomorphism g {\displaystyle g} arbitrarily close to f {\displaystyle f} in the C 1 {\displaystyle C^{1}} topology of Diff 1 ⁡ ( M ) {\displaystyle \operatorname {Diff} ^{1}(M)} such that x {\displaystyle x} is a periodic point of g {\displaystyle g} .

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Puiseux series

Summary

Puiseux_series

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 {\begin{aligned}x^{-2}&+2x^{-1/2}+x^{1/3}+2x^{11/6}+x^{8/3}+x^{5}+\cdots \\&=x^{-12/6}+2x^{-3/6}+x^{2/6}+2x^{11/6}+x^{16/6}+x^{30/6}+\cdots \end{aligned}}} is a Puiseux series in the indeterminate x. Puiseux series were first introduced by Isaac Newton in 1676 and rediscovered by Victor Puiseux in 1850.The definition of a Puiseux series includes that the denominators of the exponents must be bounded. So, by reducing exponents to a common denominator n, a Puiseux series becomes a Laurent series in an nth root of the indeterminate. For example, the example above is a Laurent series in x 1 / 6 .

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Puiseux series

Summary

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 coefficients, its solutions in y, viewed as functions of x, may be expanded as Puiseux series in x that are convergent in some neighbourhood of 0.

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Puiseux series

Summary

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 itself an algebraically closed field, called the field of Puiseux series. It is the algebraic closure of the field of formal Laurent series, which itself is the field of fractions of the ring of formal power series.

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Addition in quadrature

Summary

Addition_in_quadrature

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 ⊕ {\displaystyle \oplus } denotes the Pythagorean addition operation.This operation can be used in the conversion of Cartesian coordinates to polar coordinates. It also provides a simple notation and terminology for some formulas when its summands are complicated; for example, the energy-momentum relation in physics becomes It is implemented in many programming libraries as the hypot function, in a way designed to avoid errors arising due to limited-precision calculations performed on computers. In its applications to signal processing and propagation of measurement uncertainty, the same operation is also called addition in quadrature.

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R-algebroid

Summary

R-algebroid

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 ').

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Racah polynomials

Summary

Racah_polynomials

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_{n}(x(x+\gamma +\delta +1))={}_{4}F_{3}\left.}

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Ramanujan's master theorem

Summary

Ramanujan's_master_theorem

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 ( k ) k ! ( − x ) k {\displaystyle f(x)=\sum _{k=0}^{\infty }{\frac {\,\varphi (k)\,}{k!

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Ramanujan's master theorem

Summary

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 integrals and infinite series. Higher-dimensional versions of this theorem also appear in quantum physics (through Feynman diagrams).A similar result was also obtained by Glaisher.

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Ramanujan's congruences

Summary

Ramanujan_congruences

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}},\\p(7k+5)&\equiv 0{\pmod {7}},\\p(11k+6)&\equiv 0{\pmod {11}}.\end{aligned}}} This means that: If a number is 4 more than a multiple of 5, i.e. it is in the sequence4, 9, 14, 19, 24, 29, . .

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Ramanujan's congruences

Summary

Ramanujan_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, . .

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Ramanujan's congruences

Summary

Ramanujan_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.

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Rathjen's psi function

Summary

Rathjen's_psi_function

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 M} is closed under M {\displaystyle M} (i.e. all normal functions closed in M {\displaystyle M} are closed under some regular ordinal < M {\displaystyle

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Ratner's theorems

Summary

Ratner's_theorems

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 the Oppenheim conjecture by Grigory Margulis. Ratner's theorems have guided key advances in the understanding of the dynamics of unipotent flows. Their later generalizations provide ways to both sharpen the results and extend the theory to the setting of arbitrary semisimple algebraic groups over a local field.

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Raynaud's isogeny theorem

Summary

Raynaud's_isogeny_theorem

In mathematics, Raynaud's isogeny theorem, proved by Raynaud (1985), relates the Faltings heights of two isogeneous elliptic curves.

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Reeb sphere theorem

Summary

Reeb_sphere_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.

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Reeb stability theorem

Summary

Reeb_stability_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.

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Ray–Singer torsion

Summary

Analytic_torsion

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) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger (1977, 1979) and Werner Müller (1978) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the same for compact Riemannian manifolds. Reidemeister torsion was the first invariant in algebraic topology that could distinguish between closed manifolds which are homotopy equivalent but not homeomorphic, and can thus be seen as the birth of geometric topology as a distinct field.

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Ray–Singer torsion

Summary

Analytic_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).

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Ribet's lemma

Summary

Ribet's_lemma

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).

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Absolute differential calculus

Summary