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wiki_7100_chunk_0 | Oscillation (mathematics) | In mathematics, the oscillation of a function or a sequence is a number that quantifies how much that sequence or function varies between its extreme values as it approaches infinity or a point. As is the case with limits, there are several definitions that put the intuitive concept into a form suitable for a mathemati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7101_chunk_0 | Weyl calculus | On an infinitesimal level the semigroup is described by a cone in the Lie algebra of SU(1,1) that can be identified with a light cone. The same framework generalizes to the symplectic group in higher dimensions, including its analogue in infinite dimensions. This article explains the theory for SU(1,1) in detail and su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7102_chunk_0 | P-adic gamma function | In mathematics, the p-adic gamma function Γp is a function of a p-adic variable analogous to the gamma function. It was first explicitly defined by Morita (1975), though Boyarsky (1980) pointed out that Dwork (1964) implicitly used the same function. Diamond (1977) defined a p-adic analog Gp of log Γ. Overholtzer (1952... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7103_chunk_0 | Parabolic cylinder function | In mathematics, the parabolic cylinder functions are special functions defined as solutions to the differential equation This equation is found when the technique of separation of variables is used on Laplace's equation when expressed in parabolic cylindrical coordinates. The above equation may be brought into two dist... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7104_chunk_0 | Partition topology | {\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.} The deleted integer topology is defined by letting X = ⋃ n ∈ N ( n − 1 , n ) ⊆ R {\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}} and P = { ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) , … } . {\displaystyle P={\left\{(0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7105_chunk_0 | Peak algebra | In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by Nyman (2003). It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2,...... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7106_chunk_0 | Pentagonal number theorem | In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 − x n ) = ∑ k = − ∞ ∞ ( − 1 ) k x k ( 3 k − 1 ) / 2 = 1 + ∑ k = 1 ∞ ( − 1 ) k ( x k ( 3 k + 1 ) / 2 + x k ( 3 k − 1 ) / 2 ) . {\displaystyle \prod _{... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7107_chunk_0 | Pentagonal number theorem | {\displaystyle (1-x)(1-x^{2})(1-x^{3})\cdots =1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}-\cdots .} The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula gk = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers (sequence A001318 in the OEIS). (The c... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7108_chunk_0 | Pentagram map | In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7109_chunk_0 | Permutoassociahedron | In mathematics, the permutoassociahedron is an n {\displaystyle n} -dimensional polytope whose vertices correspond to the bracketings of the permutations of n + 1 {\displaystyle n+1} terms and whose edges connect two bracketings that can be obtained from one another either by moving a pair of brackets using associativi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7110_chunk_0 | Persistence of a number | In the remainder of this article, base ten is assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7111_chunk_0 | Pigeonhole Principle | Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7112_chunk_0 | Pin group | In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is not surjective or a universal covering... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7113_chunk_0 | Plactic monoid | In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by Donald Knuth (1970) (who called it the tableau algebra), using an operation given by Craige Schensted (1961) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7114_chunk_0 | Plastic number | In mathematics, the plastic number ρ (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, le nombre radiant) is a mathematical constant which is the unique real solution of the cubic equation x 3 = x + 1. {\displaystyle x^{3}=x+1.} It has th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7115_chunk_0 | Plethystic exponential | In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as a concise relation between the generating series for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7116_chunk_0 | Canonical model (algebraic geometry) | In mathematics, the pluricanonical ring of an algebraic variety V (which is nonsingular), or of a complex manifold, is the graded ring R ( V , K ) = R ( V , K V ) {\displaystyle R(V,K)=R(V,K_{V})\,} of sections of powers of the canonical bundle K. Its nth graded component (for n ≥ 0 {\displaystyle n\geq 0} ) is: R n :=... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7117_chunk_0 | Plus construction | The direct limit of these groups via these maps is denoted GL ( R ) {\displaystyle \operatorname {GL} (R)} and its classifying space is denoted B GL ( R ) {\displaystyle B\operatorname {GL} (R)} . The plus construction may then be applied to the perfect normal subgroup E ( R ) {\displaystyle E(R)} of GL ( R ) = π... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7118_chunk_0 | Logical equality | In mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of algebraic structure that is known as a field. For boolean algebra, this means that the logical operation signified by "+" is not the same as the inclusive disjunction signified by "∨"... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7119_chunk_0 | Pointwise product | In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied (for instance, Y could be some set of numbers), then the pointw... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7120_chunk_0 | Polar coordinates system | Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and orbital... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7121_chunk_0 | Polar decomposition | In mathematics, the polar decomposition of a square real or complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix and P {\displaystyle P} is a positive semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7122_chunk_0 | Polar decomposition | This decomposition is useful in computing the fundamental group of (matrix) Lie groups.The polar decomposition can also be defined as A = P ′ U {\displaystyle A=P'U} where P ′ = U P U − 1 {\displaystyle P'=UPU^{-1}} is a symmetric positive-definite matrix with the same eigenvalues as P {\displaystyle P} but different e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7123_chunk_0 | Polar decomposition | The definition A = U P {\displaystyle A=UP} may be extended to rectangular matrices A ∈ C m × n {\displaystyle A\in \mathbb {C} ^{m\times n}} by requiring U ∈ C m × n {\displaystyle U\in \mathbb {C} ^{m\times n}} to be a semi-unitary matrix and P ∈ C n × n {\displaystyle P\in \mathbb {C} ^{n\times n}} to be a positive-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7124_chunk_0 | Polygamma function | In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gamma function: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . {\displaystyle \psi ^{(m)}(z):={\frac {\math... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7125_chunk_0 | Polygamma function | They are holomorphic on C ∖ Z ≤ 0 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}} . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7126_chunk_0 | Polylogarithm ladder | In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7127_chunk_0 | Polylogarithm ladder | Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the same notation without the subscript. Different polylogarithm functions in the complex plane The polylogarithm function is defined by a power series in z, which is also a Dirichlet series i... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7128_chunk_0 | Polylogarithm ladder | The special case s = 1 involves the ordinary natural logarithm, Li1(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7129_chunk_0 | Polynomial method in combinatorics | In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several l... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7130_chunk_0 | Negative part | In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\mbox{ if }}f(x)>0\\0&{\mbox{ otherwise. }}\end{cases}}} Intuitively, the graph of f + {... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7131_chunk_0 | Negative part | A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f+ and f− as f = f + − f − . {\displaystyle f=f^{+}-f^{-}.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7132_chunk_0 | Negative part | {\displaystyle f^{-}=-f.} One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7133_chunk_0 | Series solution of differential equations | In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7134_chunk_0 | Power set | In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7135_chunk_0 | Prime decomposition (3-manifold) | In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7136_chunk_0 | Prime decomposition (3-manifold) | So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of S 2 {\displaystyle S^{2}} over S 1 . {\displaystyle S^{1}.} The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7137_chunk_0 | Prime decomposition (3-manifold) | {\displaystyle S^{1}.} The proof is based on normal surface techniques originated by Hellmuth Kneser. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by John Milnor. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7138_chunk_0 | Double prime | In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with x′ generally meaning something related to (or derived from) x. For example, if a point is represented by the Cartesian coordinates (x, y), then that point rotated, translated or reflected... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7139_chunk_0 | Double prime | The negation of an event in probability theory: Pr(A′) = 1 − Pr(A) (other notation also exists). The result of a transformation: Tx = x′ The transpose of a matrix (other notation also exists) The dual of a vector spaceThe prime is said to "decorate" the letter to which it applies. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7140_chunk_0 | Double prime | The same convention is adopted in functional programming, particularly in Haskell. In geometry, geography and astronomy, prime and double prime are used as abbreviations for minute and second of arc (and thus latitude, longitude, elevation and right ascension). In physics, the prime is used to denote variables after an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7141_chunk_0 | Double prime | For example, vA′ would indicate the velocity of object A after an event. It is also commonly used in relativity: the event at (x, y, z, t) in frame S, has coordinates (x′, y′, z′, t′) in frame S′. In chemistry, it is used to distinguish between different functional groups connected to an atom in a molecule, such as R a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7142_chunk_0 | Double prime | The carbonyl carbon in proteins is denoted as C′, which distinguishes it from the other backbone carbon, the alpha carbon, which is denoted as Cα. In physical chemistry, it is used to distinguish between the lower state and the upper state of a quantum number during a transition. For example, J ′ denotes the upper stat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7143_chunk_0 | Double prime | The prime distinguishes places on these two chemicals, rather than places on other parts of DNA or RNA, like phosphate groups or nucleic acids. Thus, when indicating the direction of movement of an enzyme along a string of DNA, biologists will say that it moves from the 5′ end to the 3′ end, because these carbons are o... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7144_chunk_0 | Prime number race | In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jac... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7145_chunk_0 | Prime zeta function | In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for ℜ ( s ) > 1 {\displaystyle \Re (s)>1}: P ( s ) = ∑ p ∈ p r i m e s 1 p s = 1 2 s + 1 3 s + 1 5 s + 1 7 s + 1 11 s + ⋯ . {\displaystyle P(s)... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7146_chunk_0 | Prime-counting function | In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x) (unrelated to the number π). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7147_chunk_0 | Principalisation property | In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomeno... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7148_chunk_0 | Principal orbit type | In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7149_chunk_0 | Principal series representation | In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group. There, by analogy with spectral theory, one expects that the regular representation of G will decompose according to some kind of continuous spectrum, of representations involvi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7150_chunk_0 | Principal series representation | The discrete series consists of 'atoms' of the unitary dual (points carrying a Plancherel measure > 0). In the earliest examples studied, the rest (or most) of the unitary dual could be parametrised by starting with a subgroup H of G, simpler but not compact, and building up induced representations using representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7151_chunk_0 | Principal series representation | For the case of a semisimple Lie group G, the subgroup H is constructed starting from the Iwasawa decomposition G = KANwith K a maximal compact subgroup. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as H := MANwith M the centralizer in K of A. Representations ρ of H are consid... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7152_chunk_0 | Principal series representation | The induced representations of such ρ make up the principal series. The spherical principal series consists of representations induced from 1-dimensional representations of MAN obtained by extending characters of A using the homomorphism of MAN onto A. There may be other continuous series of representations relevant to... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7153_chunk_0 | Probabilistic method | In mathematics, the probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7154_chunk_0 | Differentiation of integrals | In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7155_chunk_0 | Fourier slice theorem | In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7156_chunk_0 | Fourier slice theorem | This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal orga... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7157_chunk_0 | Inversive ring geometry | In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7158_chunk_0 | Inversive ring geometry | The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(U), the center of U, then the group action of matrix ( c 0 0 c ) {\displaystyle {\begin{pmatrix}c&0\\0&c\end{pmatrix}}} on P(A) is the same as the action of the identity matrix. Such matrices represent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7159_chunk_0 | Inversive ring geometry | {\displaystyle U{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U\thicksim U.} Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography: ( u 0 0 1 ) ( 0 1 1 0 ) ( v 0 0 1 ) ( 0 1 1 0 ) = ( u 0 0 v ) . {\displaystyle {\begin{pmatrix}u&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7160_chunk_0 | Inversive ring geometry | U ( v 0 0 u ) = U ∼ U . {\displaystyle U{\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U\thicksim U.} Since u is arbitrary, it may be substituted for u−1. Homographies on P(A) are called linear-fractional transformations since U ( a c b d ) = U ∼ U . {\displaystyle U{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U\thicksim U.} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7161_chunk_0 | PSL(2,7) | In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7162_chunk_0 | Projective unitary group | In mathematics, the projective unitary group PU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projectiv... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7163_chunk_0 | Pseudoisotopy theorem | In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7164_chunk_0 | Q-theta function | In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 − q n + 1 / z ) {\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)} where on... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7165_chunk_0 | Quadratic bottleneck assignment problem | In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of the facilities location problems.It is related to the quadratic assignment problem in the same way as the linear bot... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7166_chunk_0 | Quadratic eigenvalue problem | In mathematics, the quadratic eigenvalue problem (QEP), is to find scalar eigenvalues λ {\displaystyle \lambda } , left eigenvectors y {\displaystyle y} and right eigenvectors x {\displaystyle x} such that Q ( λ ) x = 0 and y ∗ Q ( λ ) = 0 , {\displaystyle Q(\lambda )x=0~{\text{ and }}~y^{\ast }Q(\lambda )=0,} where Q ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7167_chunk_0 | Quadruple product | In mathematics, the quadruple product is a product of four vectors in three-dimensional Euclidean space. The name "quadruple product" is used for two different products, the scalar-valued scalar quadruple product and the vector-valued vector quadruple product or vector product of four vectors. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7168_chunk_0 | Pointwise | In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f ( x ) {\displaystyle f(x)} of some function f . {\displaystyle f.} An important class of pointwise concepts are the pointwise operations, that is, operations defined on functions by applying the op... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7169_chunk_0 | Quantum dilogarithm | In mathematics, the quantum dilogarithm is a special function defined by the formula ϕ ( x ) ≡ ( x ; q ) ∞ = ∏ n = 0 ∞ ( 1 − x q n ) , | q | < 1 {\displaystyle \phi (x)\equiv (x;q)_{\infty }=\prod _{n=0}^{\infty }(1-xq^{n}),\quad |q|<1} It is the same as the q-exponential function E q ( x ) {\displaystyle E_{q}(x)} . L... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7170_chunk_0 | Quantum dilogarithm | The latter is known to be a quantum generalization of Rogers' five term dilogarithm identity. Faddeev's quantum dilogarithm Φ b ( w ) {\displaystyle \Phi _{b}(w)} is defined by the following formula: Φ b ( z ) = exp ( 1 4 ∫ C e − 2 i z w sinh ( w b ) sinh ( w / b ) d w w ) , {\displaystyle \Phi _{b}(z)=\exp \left... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7171_chunk_0 | Quasi-derivative | In mathematics, the quasi-derivative is one of several generalizations of the derivative of a function between two Banach spaces. The quasi-derivative is a slightly stronger version of the Gateaux derivative, though weaker than the Fréchet derivative. Let f: A → F be a continuous function from an open set A in a Banach... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7172_chunk_0 | Quasi-derivative | If such a linear map u exists, then f is said to be quasi-differentiable at x0. Continuity of u need not be assumed, but it follows instead from the definition of the quasi-derivative. If f is Fréchet differentiable at x0, then by the chain rule, f is also quasi-differentiable and its quasi-derivative is equal to its F... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7173_chunk_0 | Quasi-dihedral group | The groups of order pn and nilpotency class n − 1 were the beginning of the classification of all p-groups via coclass. The modular maximal-cyclic group of order 2n always has nilpotency class 2. This makes the modular maximal-cyclic group less interesting, since most groups of order pn for large n have nilpotency clas... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7174_chunk_0 | Norm of a quaternion | In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space, or, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7175_chunk_0 | Norm of a quaternion | In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, and therefore a ring, being both a division ring and a domain. The algebra of quaternions is often denoted by H (for Hamilton), or in blackboard bold by H . {\displaystyle \mathbb {H} .} | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7176_chunk_0 | Norm of a quaternion | It can also be given by the Clifford algebra classifications Cl 0 , 2 ( R ) ≅ Cl 3 , 0 + ( R ) . {\displaystyle \operatorname {Cl} _{0,2}(\mathbb {R} )\cong \operatorname {Cl} _{3,0}^{+}(\mathbb {R} ).} In fact, it was the first noncommutative division algebra to be discovered. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7177_chunk_0 | Norm of a quaternion | According to the Frobenius theorem, the algebra H {\displaystyle \mathbb {H} } is one of only two finite-dimensional division rings containing a proper subring isomorphic to the real numbers; the other being the complex numbers. These rings are also Euclidean Hurwitz algebras, of which the quaternions are the largest a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7178_chunk_0 | Convergence of Fourier series | In mathematics, the question of whether the Fourier series of a periodic function converges to a given function is researched by a field known as classical harmonic analysis, a branch of pure mathematics. Convergence is not necessarily given in the general case, and certain criteria must be met for convergence to occur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7179_chunk_0 | Quotient of subspace theorem | In mathematics, the quotient of subspace theorem is an important property of finite-dimensional normed spaces, discovered by Vitali Milman.Let (X, ||·||) be an N-dimensional normed space. There exist subspaces Z ⊂ Y ⊂ X such that the following holds: The quotient space E = Y / Z is of dimension dim E ≥ c N, where c > 0... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7180_chunk_0 | Quotient of subspace theorem | The statement is relative easy to prove by induction on the dimension of Z (even for Y=Z, X=0, c=1) with a K that depends only on N; the point of the theorem is that K is independent of N. In fact, the constant c can be made arbitrarily close to 1, at the expense of the constant K becoming large. The original proof all... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7181_chunk_0 | Disc of convergence | In mathematics, the radius of convergence of a power series is the radius of the largest disk at the center of the series in which the series converges. It is either a non-negative real number or ∞ {\displaystyle \infty } . When it is positive, the power series converges absolutely and uniformly on compact sets inside ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7182_chunk_0 | Embree–Trefethen constant | In mathematics, the random Fibonacci sequence is a stochastic analogue of the Fibonacci sequence defined by the recurrence relation f n = f n − 1 ± f n − 2 {\displaystyle f_{n}=f_{n-1}\pm f_{n-2}} , where the signs + or − are chosen at random with equal probability 1 2 {\displaystyle {\tfrac {1}{2}}} , independently fo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7183_chunk_0 | Range of a function | In mathematics, the range of a function may refer to either of two closely related concepts: The codomain of the function The image of the functionGiven two sets X and Y, a binary relation f between X and Y is a (total) function (from X to Y) if for every x in X there is exactly one y in Y such that f relates x to y. T... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7184_chunk_0 | Rank (differential topology) | In mathematics, the rank of a differentiable map f: M → N {\displaystyle f:M\to N} between differentiable manifolds at a point p ∈ M {\displaystyle p\in M} is the rank of the derivative of f {\displaystyle f} at p {\displaystyle p} . Recall that the derivative of f {\displaystyle f} at p {\displaystyle p} is a linear m... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7185_chunk_0 | Average rank of elliptic curves | In mathematics, the rank of an elliptic curve is the rational Mordell–Weil rank of an elliptic curve E {\displaystyle E} defined over the field of rational numbers. Mordell's theorem says the group of rational points on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subse... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7186_chunk_0 | Average rank of elliptic curves | The number of independent basis points with infinite order is the rank of the curve. The rank is related to several outstanding problems in number theory, most notably the Birch–Swinnerton-Dyer conjecture. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7187_chunk_0 | Torsion-free rank | In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the rational numbers of dimensio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7188_chunk_0 | Ratio test | In mathematics, the ratio test is a test (or "criterion") for the convergence of a series ∑ n = 1 ∞ a n , {\displaystyle \sum _{n=1}^{\infty }a_{n},} where each term is a real or complex number and an is nonzero when n is large. The test was first published by Jean le Rond d'Alembert and is sometimes known as d'Alember... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7189_chunk_0 | Rational sieve | In mathematics, the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is less efficient than the general algorithm, it is conceptually simpler. It serves as a helpful first step in understanding how the general number field ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7190_chunk_0 | Real coordinate plane | In mathematics, the real coordinate space of dimension n, denoted Rn or R n {\displaystyle \mathbb {R} ^{n}} , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. Special cases are called the real line R1 and the real coordinate plane R2. With component-wise addition and scal... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7191_chunk_0 | Real coordinate plane | Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension n form a real coordinate space of dimension n. These one to one correspondences between vectors, points and coordinate vectors explain the names of coordinate space and coordinate vector. It allows using geometric terms and methods for... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7192_chunk_0 | Real projective plane | In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has basic applications to geometry, since the common construction of the real proje... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7193_chunk_0 | Real rank (C*-algebras) | In mathematics, the real rank of a C*-algebra is a noncommutative analogue of Lebesgue covering dimension. The notion was first introduced by Lawrence G. Brown and Gert K. Pedersen. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7194_chunk_0 | Rearrangement inequality | In mathematics, the rearrangement inequality states that for every choice of real numbers and every permutation y σ ( 1 ) , … , y σ ( n ) {\displaystyle y_{\sigma (1)},\ldots ,y_{\sigma (n)}} of y 1 , … , y n . {\displaystyle y_{1},\ldots ,y_{n}.} If the numbers x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} are dif... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7195_chunk_0 | Reciprocal Gamma function | In mathematics, the reciprocal gamma function is the function f ( z ) = 1 Γ ( z ) , {\displaystyle f(z)={\frac {1}{\Gamma (z)}},} where Γ(z) denotes the gamma function. Since the gamma function is meromorphic and nonzero everywhere in the complex plane, its reciprocal is an entire function. As an entire function, it is... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7196_chunk_0 | Reduced derivative | In mathematics, the reduced derivative is a generalization of the notion of derivative that is well-suited to the study of functions of bounded variation. Although functions of bounded variation have derivatives in the sense of Radon measures, it is desirable to have a derivative that takes values in the same space as ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7197_chunk_0 | False position | In mathematics, the regula falsi, method of false position, or false position method is a very old method for solving an equation with one unknown; this method, in modified form, is still in use. In simple terms, the method is the trial and error technique of using test ("false") values for the variable and then adjust... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7198_chunk_0 | False position | As an example, consider problem 26 in the Rhind papyrus, which asks for a solution of (written in modern notation) the equation x + x/4 = 15. This is solved by false position. First, guess that x = 4 to obtain, on the left, 4 + 4/4 = 5. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_7199_chunk_0 | False position | This guess is a good choice since it produces an integer value. However, 4 is not the solution of the original equation, as it gives a value which is three times too small. To compensate, multiply x (currently set to 4) by 3 and substitute again to get 12 + 12/4 = 15, verifying that the solution is x = 12. Modern versi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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