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wiki_1800_chunk_0 | Manin matrices | In mathematics, Manin matrices, named after Yuri Manin who introduced them around 1987–88, are a class of matrices with elements in a not-necessarily commutative ring, which in a certain sense behave like matrices whose elements commute. In particular there is natural definition of the determinant for them and most lin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1801_chunk_0 | Manin matrices | Manin matrices are particular examples of Manin's general construction of "non-commutative symmetries" which can be applied to any algebra. From this point of view they are "non-commutative endomorphisms" of polynomial algebra C. Taking (q)-(super)-commuting variables one will get (q)-(super)-analogs of Manin matrices,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1802_chunk_0 | Marden's theorem | In mathematics, Marden's theorem, named after Morris Marden but proved about 100 years earlier by Jörg Siebeck, gives a geometric relationship between the zeroes of a third-degree polynomial with complex coefficients and the zeroes of its derivative. See also geometrical properties of polynomial roots. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1803_chunk_0 | Maschke's theorem | In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces. Maschke's theorem allows one to make general conclusions about representations of a finite group G without actually c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1804_chunk_0 | Mathieu differential equation | In mathematics, Mathieu functions, sometimes called angular Mathieu functions, are solutions of Mathieu's differential equation d 2 y d x 2 + ( a − 2 q cos ( 2 x ) ) y = 0 , {\displaystyle {\frac {d^{2}y}{dx^{2}}}+(a-2q\cos(2x))y=0,} where a, q are real-valued parameters. Since we may add π/2 to x to change the sign ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1805_chunk_0 | Matsumoto zeta function | In mathematics, Matsumoto zeta functions are a type of zeta function introduced by Kohji Matsumoto in 1990. They are functions of the form ϕ ( s ) = ∏ p 1 A p ( p − s ) {\displaystyle \phi (s)=\prod _{p}{\frac {1}{A_{p}(p^{-s})}}} where p is a prime and Ap is a polynomial. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1806_chunk_0 | Matsushima's formula | In mathematics, Matsushima's formula, introduced by Matsushima (1967), is a formula for the Betti numbers of a quotient of a symmetric space G/H by a discrete group, in terms of unitary representations of the group G. The Matsushima–Murakami formula is a generalization giving dimensions of spaces of automorphic forms, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1807_chunk_0 | Mazur's lemma | In mathematics, Mazur's lemma is a result in the theory of normed vector spaces. It shows that any weakly convergent sequence in a normed space has a sequence of convex combinations of its members that converges strongly to the same limit, and is used in the proof of Tonelli's theorem. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1808_chunk_0 | Midy's Theorem | In mathematics, Midy's theorem, named after French mathematician E. Midy, is a statement about the decimal expansion of fractions a/p where p is a prime and a/p has a repeating decimal expansion with an even period (sequence A028416 in the OEIS). If the period of the decimal representation of a/p is 2n, so that a p = 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1809_chunk_0 | Midy's Theorem | {\displaystyle a_{1}\dots a_{n}+a_{n+1}\dots a_{2n}=10^{n}-1.} For example, 1 13 = 0. 076923 ¯ and 076 + 923 = 999. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1810_chunk_0 | Midy's Theorem | {\displaystyle {\frac {1}{13}}=0. {\overline {076923}}{\text{ and }}076+923=999.} 1 17 = 0. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1811_chunk_0 | Midy's Theorem | 0588235294117647 ¯ and 05882352 + 94117647 = 99999999. {\displaystyle {\frac {1}{17}}=0. {\overline {0588235294117647}}{\text{ and }}05882352+94117647=99999999.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1812_chunk_0 | Milliken's tree theorem | In mathematics, Milliken's tree theorem in combinatorics is a partition theorem generalizing Ramsey's theorem to infinite trees, objects with more structure than sets. Let T be a finitely splitting rooted tree of height ω, n a positive integer, and S T n {\displaystyle \mathbb {S} _{T}^{n}} the collection of all strong... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1813_chunk_0 | Milnor fiber | In mathematics, Milnor maps are named in honor of John Milnor, who introduced them to topology and algebraic geometry in his book Singular Points of Complex Hypersurfaces (Princeton University Press, 1968) and earlier lectures. The most studied Milnor maps are actually fibrations, and the phrase Milnor fibration is mor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1814_chunk_0 | Minkowski's first inequality for convex bodies | In mathematics, Minkowski's first inequality for convex bodies is a geometrical result due to the German mathematician Hermann Minkowski. The inequality is closely related to the Brunn–Minkowski inequality and the isoperimetric inequality. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1815_chunk_0 | Minkowski's question-mark function | In mathematics, Minkowski's question-mark function, denoted ? (x), is a function with unusual fractal properties, defined by Hermann Minkowski in 1904. It maps quadratic irrational numbers to rational numbers on the unit interval, via an expression relating the continued fraction expansions of the quadratics to the bin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1816_chunk_0 | Minkowski's second theorem | In mathematics, Minkowski's second theorem is a result in the geometry of numbers about the values taken by a norm on a lattice and the volume of its fundamental cell. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1817_chunk_0 | Minkowski's theorem | In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the origin and which has volume greater than 2 n {\displaystyle 2^{n}} contains a non-zero integer point (meaning a point in Z n {\displaystyle \mathbb {Z} ^{n}} that is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1818_chunk_0 | Mitchell's group | In mathematics, Mitchell's group is a complex reflection group in 6 complex dimensions of order 108 × 9!, introduced by Mitchell (1914). It has the structure 6.PSU4(F3).2. As a complex reflection group it has 126 reflections of order 2, and its ring of invariants is a polynomial algebra with generators of degrees 6, 12... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1819_chunk_0 | Mittag-Leffler summation | In mathematics, Mittag-Leffler summation is any of several variations of the Borel summation method for summing possibly divergent formal power series, introduced by Mittag-Leffler (1908) | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1820_chunk_0 | Molien's formula | In mathematics, Molien's formula computes the generating function attached to a linear representation of a group G on a finite-dimensional vector space, that counts the homogeneous polynomials of a given total degree that are invariants for G. It is named for Theodor Molien. Precisely, it says: given a finite-dimension... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1821_chunk_0 | Monk's formula | In mathematics, Monk's formula, found by Monk (1959), is an analogue of Pieri's formula that describes the product of a linear Schubert polynomial by a Schubert polynomial. Equivalently, it describes the product of a special Schubert cycle by a Schubert cycle in the cohomology of a flag manifold. Write tij for the tran... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1822_chunk_0 | MISER algorithm | In mathematics, Monte Carlo integration is a technique for numerical integration using random numbers. It is a particular Monte Carlo method that numerically computes a definite integral. While other algorithms usually evaluate the integrand at a regular grid, Monte Carlo randomly chooses points at which the integrand ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1823_chunk_0 | Pair correlation conjecture | In mathematics, Montgomery's pair correlation conjecture is a conjecture made by Hugh Montgomery (1973) that the pair correlation between pairs of zeros of the Riemann zeta function (normalized to have unit average spacing) is 1 − ( sin ( π u ) π u ) 2 + δ ( u ) , {\displaystyle 1-\left({\frac {\sin(\pi u)}{\pi u}}\r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1824_chunk_0 | Moreau's theorem | In mathematics, Moreau's theorem is a result in convex analysis named after French mathematician Jean-Jacques Moreau. It shows that sufficiently well-behaved convex functionals on Hilbert spaces are differentiable and the derivative is well-approximated by the so-called Yosida approximation, which is defined in terms o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1825_chunk_0 | Sweedler's Hopf algebra | In mathematics, Moss E. Sweedler (1969, p. 89–90) introduced an example of an infinite-dimensional Hopf algebra, and Sweedler's Hopf algebra H4 is a certain 4-dimensional quotient of it that is neither commutative nor cocommutative. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1826_chunk_0 | Mostow's rigidity theorem | In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique. The theorem was proven for closed manifol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1827_chunk_0 | Mostow's rigidity theorem | Besson, Courtois & Gallot (1996) gave the simplest available proof. While the theorem shows that the deformation space of (complete) hyperbolic structures on a finite volume hyperbolic n {\displaystyle n} -manifold (for n > 2 {\displaystyle n>2} ) is a point, for a hyperbolic surface of genus g > 1 {\displaystyle g>1} ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1828_chunk_0 | Moufang polygon | In mathematics, Moufang polygons are a generalization by Jacques Tits of the Moufang planes studied by Ruth Moufang, and are irreducible buildings of rank two that admit the action of root groups. In a book on the topic, Tits and Richard Weiss classify them all. An earlier theorem, proved independently by Tits and Weis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1829_chunk_0 | Muirhead's Inequality | In mathematics, Muirhead's inequality, named after Robert Franklin Muirhead, also known as the "bunching" method, generalizes the inequality of arithmetic and geometric means. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1830_chunk_0 | Mumford's compactness theorem | In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by David Mumford (1971) as a consequence of a theorem about the compactness of sets of discr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1831_chunk_0 | Nakayama's conjecture | In mathematics, Nakayama's conjecture is a conjecture about Artinian rings, introduced by Nakayama (1958). The generalized Nakayama conjecture is an extension to more general rings, introduced by Auslander and Reiten (1975). Leuschke & Huneke (2004) proved some cases of the generalized Nakayama conjecture. Nakayama's c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1832_chunk_0 | Nambu dynamics | In mathematics, Nambu mechanics is a generalization of Hamiltonian mechanics involving multiple Hamiltonians. Recall that Hamiltonian mechanics is based upon the flows generated by a smooth Hamiltonian over a symplectic manifold. The flows are symplectomorphisms and hence obey Liouville's theorem. This was soon general... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1833_chunk_0 | Nesbitt's inequality | In mathematics, Nesbitt's inequality states that for positive real numbers a, b and c, a b + c + b a + c + c a + b ≥ 3 2 . {\displaystyle {\frac {a}{b+c}}+{\frac {b}{a+c}}+{\frac {c}{a+b}}\geq {\frac {3}{2}}.} It is an elementary special case (N = 3) of the difficult and much studied Shapiro inequality, and was publish... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1834_chunk_0 | Neville's schema | In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation that was derived by the mathematician Eric Harold Neville in 1934. Given n + 1 points, there is a unique polynomial of degree ≤ n which goes through the given points. Neville's algorithm evaluates this polynomial. Neville's algorithm ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1835_chunk_0 | Newton's identities | In mathematics, Newton's identities, also known as the Girard–Newton formulae, give relations between two types of symmetric polynomials, namely between power sums and elementary symmetric polynomials. Evaluated at the roots of a monic polynomial P in one variable, they allow expressing the sums of the k-th powers of a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1836_chunk_0 | Newton's theorem about ovals | In mathematics, Newton's theorem about ovals states that the area cut off by a secant of a smooth convex oval is not an algebraic function of the secant. Isaac Newton stated it as lemma 28 of section VI of book 1 of Newton's Principia, and used it to show that the position of a planet moving in an orbit is not an algeb... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1837_chunk_0 | Nirenberg's conjecture | In mathematics, Nirenberg's conjecture, now Osserman's theorem, states that if a neighborhood of the sphere is omitted by the Gauss map of a complete minimal surface, then the surface in question is a plane. It was proved by Robert Osserman in 1959. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1838_chunk_0 | Niven's theorem | In mathematics, Niven's theorem, named after Ivan Niven, states that the only rational values of θ in the interval 0° ≤ θ ≤ 90° for which the sine of θ degrees is also a rational number are: sin 0 ∘ = 0 , sin 30 ∘ = 1 2 , sin 90 ∘ = 1. {\displaystyle {\begin{aligned}\sin 0^{\circ }&=0,\\\sin 30^{\circ }&={\frac {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1839_chunk_0 | Noether identities | Yang–Mills gauge theory and gauge gravitation theory exemplify irreducible Lagrangian field theories. Different variants of second Noether’s theorem state the one-to-one correspondence between the non-trivial reducible Noether identities and the non-trivial reducible gauge symmetries. Formulated in a very general setti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1840_chunk_0 | Noether's theorem on rationality for surfaces | In mathematics, Noether's theorem on rationality for surfaces is a classical result of Max Noether on complex algebraic surfaces, giving a criterion for a rational surface. Let S be an algebraic surface that is non-singular and projective. Suppose there is a morphism φ from S to the projective line, with general fibre ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1841_chunk_0 | Novikov's compact leaf theorem | In mathematics, Novikov's compact leaf theorem, named after Sergei Novikov, states that A codimension-one foliation of a compact 3-manifold whose universal covering space is not contractible must have a compact leaf. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1842_chunk_0 | Oka's lemma | In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in C n {\displaystyle \mathbb {C} ^{n}} , the function − log d ( z ) {\displaystyle -\log d(z)} is plurisubharmonic, where d {\displaystyle d} is the distance to the boundary. This property shows that the domain is pseudoconvex.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1843_chunk_0 | Ono's inequality | In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, as shown by F. Balitrand in 1916. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1844_chunk_0 | Osgood's lemma | In mathematics, Osgood's lemma, introduced by William Fogg Osgood (1899), is a proposition in complex analysis. It states that a continuous function of several complex variables that is holomorphic in each variable separately is holomorphic. The assumption that the function is continuous can be dropped, but that form o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1845_chunk_0 | Osgood's lemma | If we assume that a function f: R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is globally continuous and separately differentiable on each variable (all partial derivatives exist everywhere), it is not true that f {\displaystyle f} will necessarily be differentiable. A counterexample in two dimensions is g... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1846_chunk_0 | Owen's T function | In mathematics, Owen's T function T(h, a), named after statistician Donald Bruce Owen, is defined by T ( h , a ) = 1 2 π ∫ 0 a e − 1 2 h 2 ( 1 + x 2 ) 1 + x 2 d x ( − ∞ < h , a < + ∞ ) . {\displaystyle T(h,a)={\frac {1}{2\pi }}\int _{0}^{a}{\frac {e^{-{\frac {1}{2}}h^{2}(1+x^{2})}}{1+x^{2}}}dx\quad \left(-\infty | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1847_chunk_0 | Padovan polynomials | {\displaystyle P_{11}(x)=x^{5}+6x^{2}.\,} The Padovan numbers are recovered by evaluating the polynomials Pn−3(x) at x = 1. Evaluating Pn−3(x) at x = 2 gives the nth Fibonacci number plus (−1)n. (sequence A008346 in the OEIS) The ordinary generating function for the sequence is ∑ n = 1 ∞ P n ( x ) t n = t 1 − x t 2 − t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1848_chunk_0 | Paley digraph | In mathematics, Paley graphs are dense undirected graphs constructed from the members of a suitable finite field by connecting pairs of elements that differ by a quadratic residue. The Paley graphs form an infinite family of conference graphs, which yield an infinite family of symmetric conference matrices. Paley graph... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1849_chunk_0 | Pappus's centroid 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 sta... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1850_chunk_0 | Pappian plane | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1851_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1852_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1853_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1854_chunk_0 | 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 late... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1855_chunk_0 | 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1856_chunk_0 | 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. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1857_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1858_chunk_0 | Pascal's 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, G... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1859_chunk_0 | 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 application... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1860_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1861_chunk_0 | Picard–Lefschetz theory | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1862_chunk_0 | 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 {\disp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1863_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1864_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1865_chunk_0 | Pontryagin dual | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1866_chunk_0 | 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 Eucli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1867_chunk_0 | 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1868_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1869_chunk_0 | 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 {\b... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1870_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1871_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1872_chunk_0 | 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 ⊕ {\dis... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1873_chunk_0 | 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 '). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1874_chunk_0 | 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 ( ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1875_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1876_chunk_0 | Ramanujan's 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}}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1877_chunk_0 | 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1878_chunk_0 | 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 t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1879_chunk_0 | Raynaud's isogeny theorem | In mathematics, Raynaud's isogeny theorem, proved by Raynaud (1985), relates the Faltings heights of two isogeneous elliptic curves. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1880_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1881_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1882_chunk_0 | 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),... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1883_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1884_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1885_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1886_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1887_chunk_0 | 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.... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1888_chunk_0 | 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 Papp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1889_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1890_chunk_0 | 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 i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1891_chunk_0 | 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 18... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1892_chunk_0 | 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1893_chunk_0 | 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 ha... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1894_chunk_0 | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1895_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1896_chunk_0 | Sard's lemma | 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://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1897_chunk_0 | 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 stoc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1898_chunk_0 | 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 ano... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1899_chunk_0 | 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 intersec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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