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Paratingent cone Summary Paratingent_cone In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connected two-point set Summary Connected_two-point_set In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection T = { S ⊆ X ∣ p ∈ S } ∪ { ∅ } {\... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Connected two-point set Summary Connected_two-point_set If X is uncountable, the topology on X is called the uncountable particular point topology.A generalization of the particular point topology is the closed extension topology. In the case when X \ {p} has the discrete topology, the closed extension topology is the ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partition topology Summary Odd–even_topology In mathematics, the partition topology is a topology that can be induced on any set X {\displaystyle X} by partitioning X {\displaystyle X} into disjoint subsets P ; {\displaystyle P;} these subsets form the basis for the topology. There are two important examples which have... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partition topology Summary Odd–even_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 ) , ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partition topology Summary Odd–even_topology The trivial partitions yield the discrete topology (each point of X {\displaystyle X} is a set in P , {\displaystyle P,} so P = { { x }: x ∈ X } {\displaystyle P=\{~\{x\}~:~x\in X~\}} ) or indiscrete topology (the entire set X {\displaystyle X} is in P , {\displaystyle P,} s... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Partition topology Summary Odd–even_topology Unless P {\displaystyle P} is trivial, at least one set in P {\displaystyle P} contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X {\displaystyle X} is not a Kolmogorov space, nor a T1... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Peak algebra Summary 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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pentagonal number theorem Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pentagonal number theorem Summary 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 (generalize... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pentagram map Summary 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. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Permutation category Summary Permutation_category In mathematics, the permutation category is a category where the objects are the natural numbers, the morphisms from a natural number n to itself are the elements of the symmetric group S n {\displaystyle S_{n}} and there are no morphisms from m to n if m ≠ n {\displays... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Permutoassociahedron Summary 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 eit... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Permutohedron Summary Permutohedron In mathematics, the permutohedron of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are the permutations of the first n natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Permutohedron Summary Permutohedron The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Persistence of a number Summary Additive_persistence In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number. Usually, this involves additive or multiplicative persistence of a non-... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Persistence of a number Summary Additive_persistence 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 dig... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Of the form Summary Of_the_form In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Complete partial order Summary Directed_complete_partial_order In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Arbitrarily large Summary Arbitrarily_large In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the conte... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pigeonhole Principle Summary Pigeon_hole_principle In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there must be at least t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pigeonhole Principle Summary Pigeon_hole_principle Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pigeonhole Principle Summary Pigeon_hole_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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pin group Summary 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 surjec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Ping-pong lemma Summary Ping-pong_lemma In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plactic monoid Summary 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 opera... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plastic number Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plethystic exponential Summary 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, a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Canonical model (algebraic geometry) Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plus construction Summary Plus_construction In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. Explicitly, if X {\displaystyle X} is a based connected CW complex and P {\displaystyle P} is a perfect normal subgroup of π... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plus construction Summary Plus_construction The most common application of the plus construction is in algebraic K-theory. If R {\displaystyle R} is a unital ring, we denote by GL n ( R ) {\displaystyle \operatorname {GL} _{n}(R)} the group of invertible n {\displaystyle n} -by- n {\displaystyle n} matrices with elem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Plus construction Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Logical equality Inequality Logical_equality > Inequality 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pointwise product Summary 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 co... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polar coordinates system Summary 2D_polar_angle In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polar coordinates system Summary 2D_polar_angle 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 po... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polar decomposition Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polar decomposition Summary 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 ei... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polar decomposition Summary 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polygamma function Summary 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 ) ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polygamma function Summary 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 |
Polylogarithm ladder Summary 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 r... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polylogarithm ladder Summary 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 pow... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polylogarithm ladder Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Polynomial method in combinatorics Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order complex Summary Order_complex In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order complex Summary Order_complex :\ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta .} Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset Γ ⊆ Δ {\displaystyle \Gamma \subseteq \Delta } be closed if and only if Γ is a simplicial complex, i.e. ∀ ρ ∀ σ: ρ ⊆ σ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Order complex Summary Order_complex :\ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma .} This is the Alexandrov topology on the poset of faces of Δ. The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative part Summary 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{case... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative part Summary Negative_part {\displaystyle f^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\mbox{ if }}f(x)<0\\0&{\mbox{ otherwise. }}\end{cases}}} Note that both f+ and f− are non-negative functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative part Summary 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 |
Negative part Summary Negative_part Also note that | f | = f + + f − {\displaystyle |f|=f^{+}+f^{-}} .Using these two equations one may express the positive and negative parts as f + = | f | + f 2 {\displaystyle f^{+}={\frac {|f|+f}{2}}} f − = | f | − f 2 . {\displaystyle f^{-}={\frac {|f|-f}{2}}.} Another representati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Negative part Summary 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 |
Series solution of differential equations Summary Power_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 solutio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Power set Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Predual Summary Predual In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable func... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime decomposition (3-manifold) Summary 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 pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime decomposition (3-manifold) Summary 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-orie... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime decomposition (3-manifold) Summary Prime_decomposition_(3-manifold) {\displaystyle S^{1}.} This sum is unique as long as we specify that each summand is either irreducible or a non-orientable S 2 {\displaystyle S^{2}} bundle over S 1 . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime decomposition (3-manifold) Summary 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 |
Prime end Summary Prime_end In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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 exam... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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 chem... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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 ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Double prime Use in mathematics, statistics, and science Double_prime > Use in mathematics, statistics, and science 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime number race Summary Distribution_of_primes 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 oc... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime number race Summary Distribution_of_primes Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), w... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime signature Summary Prime_signature In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having prime factorization p 1 m 1 p 2 m 2 … p n m n {\displaystyle p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{n}^{m_{n}}} is the multiset { ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime zeta function Summary 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 + ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Prime-counting function Summary 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 |
Principalisation property Summary Principal_ideal_theorem 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 idea... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal orbit type Summary Principal_orbit_type_theorem 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 |
Principal series representation Summary 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal series representation Summary 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 n... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal series representation Summary 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 :... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Principal series representation Summary 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 ont... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Probabilistic method Summary 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 spec... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Differentiation of integrals Summary 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 form... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier slice theorem Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Fourier slice theorem Summary 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 transfo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inversive ring geometry Summary Projective_line_over_a_ring 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; pa... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inversive ring geometry Summary Projective_line_over_a_ring 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 a... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inversive ring geometry Summary Projective_line_over_a_ring {\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{pma... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Inversive ring geometry Summary Projective_line_over_a_ring 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 . {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
PSL(2,7) Summary 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 e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Projective unitary group Summary Projective_special_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 p... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Pseudoisotopy theorem Summary 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 |
Q-Bessel polynomials Summary Q-Bessel_polynomials In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Charlier polynomials Summary Q-Charlier_polynomials In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Hahn polynomials Summary Q-Hahn_polynomials In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Konhauser polynomials Summary Q-Konhauser_polynomials In mathematics, the q-Konhauser polynomials are a q-analog of the Konhauser polynomials, introduced by Al-Salam & Verma (1983). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Krawtchouk polynomials Summary Q-Krawtchouk_polynomials In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14). give a detailed list of their properties. Stanton (1981) showed t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Laguerre polynomials Summary Q-Laguerre_polynomials In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(α)n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Daniel S. Moak (1981). Roelof Koekoek, Peter A. Lesky, and René F. ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Meixner polynomials Summary Q-Meixner_polynomials In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Meixner–Pollaczek polynomials Summary Q-Meixner–Pollaczek_polynomials In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-Racah polynomials Summary Q-Racah_polynomials In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-expansion principle Summary Q-expansion_principle In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients in M. It was introduced by Katz (1973, corollaries 1.6.2, 1.12.2). | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
Q-theta function Summary 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... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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