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In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones.
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https://en.wikipedia.org/wiki/Paratingent_cone
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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 } ∪ { ∅ } {\displaystyle T=\{S\subseteq X\mid p\in S\}\cup \{\emptyset \}} of subsets of X is the particular point topology on X. There are a variety of cases that are individually named: If X has two points, the particular point topology on X is the Sierpiński space. If X is finite (with at least 3 points), the topology on X is called the finite particular point topology. If X is countably infinite, the topology on X is called the countable particular point topology.
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https://en.wikipedia.org/wiki/Connected_two-point_set
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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 same as the particular point topology. This topology is used to provide interesting examples and counterexamples.
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https://en.wikipedia.org/wiki/Connected_two-point_set
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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 their own names: The odd–even topology is the topology where X = N {\displaystyle X=\mathbb {N} } and P = { { 2 k − 1 , 2 k }: k ∈ N } . {\displaystyle P={\left\{~\{2k-1,2k\}:k\in \mathbb {N} \right\}}.} Equivalently, P = { { 1 , 2 } , { 3 , 4 } , { 5 , 6 } , … } .
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https://en.wikipedia.org/wiki/Partition_topology
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{\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\{(0,1),(1,2),(2,3),\ldots \right\}}.}
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https://en.wikipedia.org/wiki/Partition_topology
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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,} so P = { X } {\displaystyle P=\{X\}} ). Any set X {\displaystyle X} with a partition topology generated by a partition P {\displaystyle P} can be viewed as a pseudometric space with a pseudometric given by: This is not a metric unless P {\displaystyle P} yields the discrete topology. The partition topology provides an important example of the independence of various separation axioms.
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https://en.wikipedia.org/wiki/Partition_topology
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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 space, a Hausdorff space or an Urysohn space. In a partition topology the complement of every open set is also open, and therefore a set is open if and only if it is closed. Therefore, X {\displaystyle X} is regular, completely regular, normal and completely normal. X / P {\displaystyle X/P} is the discrete topology.
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https://en.wikipedia.org/wiki/Partition_topology
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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,...,n} is an index i such that σ(i–1)<σ(i)>σ(i+1).) It is a left ideal of the descent algebra. The direct sum of the peak algebras for all n has a natural structure of a Hopf algebra.
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https://en.wikipedia.org/wiki/Peak_algebra
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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 _{n=1}^{\infty }\left(1-x^{n}\right)=\sum _{k=-\infty }^{\infty }\left(-1\right)^{k}x^{k\left(3k-1\right)/2}=1+\sum _{k=1}^{\infty }(-1)^{k}\left(x^{k(3k+1)/2}+x^{k(3k-1)/2}\right).} In other words, ( 1 − x ) ( 1 − x 2 ) ( 1 − x 3 ) ⋯ = 1 − x − x 2 + x 5 + x 7 − x 12 − x 15 + x 22 + x 26 − ⋯ .
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https://en.wikipedia.org/wiki/Pentagonal_number_theorem
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{\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 constant term 1 corresponds to k = 0 {\displaystyle k=0} .) This holds as an identity of convergent power series for | x | < 1 {\displaystyle |x|<1} , and also as an identity of formal power series. A striking feature of this formula is the amount of cancellation in the expansion of the product.
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https://en.wikipedia.org/wiki/Pentagonal_number_theorem
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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 pentagram map for a general polygon in a 1992 paper though it seems that the special case, in which the map is defined for pentagons only, goes back to an 1871 paper of Alfred Clebsch and a 1945 paper of Theodore Motzkin. The pentagram map is similar in spirit to the constructions underlying Desargues' theorem and Poncelet's porism. It echoes the rationale and construction underlying a conjecture of Branko Grünbaum concerning the diagonals of a polygon.
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https://en.wikipedia.org/wiki/Pentagram_map
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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 {\displaystyle m\neq n} .It is equivalent as a category to the category of finite sets and bijections between them.
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https://en.wikipedia.org/wiki/Permutation_category
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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 associativity or by transposing two consecutive terms that are not separated by a bracket. The permutoassociahedron was first defined as a CW complex by Mikhail Kapranov who noted that this structure appears implicitly in Mac Lane's coherence theorem for symmetric and braided categories as well as in Vladimir Drinfeld's work on the Knizhnik–Zamolodchikov equations. It was constructed as a convex polytope by Victor Reiner and Günter M. Ziegler.
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https://en.wikipedia.org/wiki/Permutoassociahedron
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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 connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one transposition), and the numbers on these places are neighbors (differ in value by 1).
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https://en.wikipedia.org/wiki/Permutohedron
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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 in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.)
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https://en.wikipedia.org/wiki/Permutohedron
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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-negative integer, which is how often one has to replace the number by the sum or product of its digits until one reaches a single digit. Because the numbers are broken down into their digits, the additive or multiplicative persistence depends on the radix.
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https://en.wikipedia.org/wiki/Persistence_of_a_number
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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.
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https://en.wikipedia.org/wiki/Persistence_of_a_number
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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.
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https://en.wikipedia.org/wiki/Of_the_form
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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 theoretical computer science: in denotational semantics and domain theory.
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https://en.wikipedia.org/wiki/Complete_partial_order
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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 context of real numbers (and its subsets thereof), though its meaning can differ from that of "sufficiently" and "infinitely".
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https://en.wikipedia.org/wiki/Arbitrarily_large
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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 two right-handed gloves, or at least two left-handed gloves, because there are three objects, but only two categories of handedness to put them into. This seemingly obvious statement, a type of counting argument, can be used to demonstrate possibly unexpected results. For example, given that the population of London is greater than the maximum number of hairs that can be present on a human's head, then the pigeonhole principle requires that there must be at least two people in London who have the same number of hairs on their heads.
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https://en.wikipedia.org/wiki/Pigeonhole_Principle
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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 Schubfachprinzip ("drawer principle" or "shelf principle").The principle has several generalizations and can be stated in various ways. In a more quantified version: for natural numbers k and m, if n = km + 1 objects are distributed among m sets, then the pigeonhole principle asserts that at least one of the sets will contain at least k + 1 objects. For arbitrary n and m, this generalizes to k + 1 = ⌊ ( n − 1 ) / m ⌋ + 1 = ⌈ n / m ⌉ , {\displaystyle k+1=\lfloor (n-1)/m\rfloor +1=\lceil n/m\rceil ,} where ⌊ ⋯ ⌋ {\displaystyle \lfloor \cdots \rfloor } and ⌈ ⋯ ⌉ {\displaystyle \lceil \cdots \rceil } denote the floor and ceiling functions, respectively.
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https://en.wikipedia.org/wiki/Pigeonhole_Principle
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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 smaller than its domain". Advanced mathematical proofs like Siegel's lemma build upon this more general concept.
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https://en.wikipedia.org/wiki/Pigeonhole_Principle
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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 space, but if the quadratic form is definite (and dimension is greater than 2), it is both. The non-trivial element of the kernel is denoted − 1 , {\displaystyle -1,} which should not be confused with the orthogonal transform of reflection through the origin, generally denoted − I . {\displaystyle -I.}
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https://en.wikipedia.org/wiki/Pin_group
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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.
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https://en.wikipedia.org/wiki/Ping-pong_lemma
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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) in his study of the longest increasing subsequence of a permutation. It was named the "monoïde plaxique" by Lascoux & Schützenberger (1981), who allowed any totally ordered alphabet in the definition. The etymology of the word "plaxique" is unclear; it may refer to plate tectonics ("tectonique des plaques" in French), as elementary relations that generate the equivalence allow conditional commutation of generator symbols: they can sometimes slide across each other (in apparent analogy to tectonic plates), but not freely.
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https://en.wikipedia.org/wiki/Plactic_monoid
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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 the exact value ρ = 9 + 69 18 3 + 9 − 69 18 3 . {\displaystyle \rho ={\sqrt{\frac {9+{\sqrt {69}}}{18}}}+{\sqrt{\frac {9-{\sqrt {69}}}{18}}}.} Its decimal expansion begins with 1.324717957244746025960908854....
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https://en.wikipedia.org/wiki/Plastic_number
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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 elementary, complete and power sums homogeneous symmetric polynomials in many variables. Its name comes from the operation called plethysm, defined in the context of so-called lambda rings. In combinatorics, the plethystic exponential is a generating function for many well studied sequences of integers, polynomials or power series, such as the number of integer partitions. It is also an important technique in the enumerative combinatorics of unlabelled graphs, and many other combinatorial objects.In geometry and topology, the plethystic exponential of a certain geometric/topologic invariant of a space, determines the corresponding invariant of its symmetric products.
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https://en.wikipedia.org/wiki/Plethystic_exponential
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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 := H 0 ( V , K n ) , {\displaystyle R_{n}:=H^{0}(V,K^{n}),\ } that is, the space of sections of the n-th tensor product Kn of the canonical bundle K. The 0th graded component R 0 {\displaystyle R_{0}} is sections of the trivial bundle, and is one-dimensional as V is projective. The projective variety defined by this graded ring is called the canonical model of V, and the dimension of the canonical model is called the Kodaira dimension of V. One can define an analogous ring for any line bundle L over V; the analogous dimension is called the Iitaka dimension. A line bundle is called big if the Iitaka dimension equals the dimension of the variety.
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https://en.wikipedia.org/wiki/Canonical_model_(algebraic_geometry)
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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 1 ( X ) {\displaystyle \pi _{1}(X)} then a map f: X → Y {\displaystyle f\colon X\to Y} is called a +-construction relative to P {\displaystyle P} if f {\displaystyle f} induces an isomorphism on homology, and P {\displaystyle P} is the kernel of π 1 ( X ) → π 1 ( Y ) {\displaystyle \pi _{1}(X)\to \pi _{1}(Y)} .The plus construction was introduced by Michel Kervaire (1969), and was used by Daniel Quillen to define algebraic K-theory. Given a perfect normal subgroup of the fundamental group of a connected CW complex X {\displaystyle X} , attach two-cells along loops in X {\displaystyle X} whose images in the fundamental group generate the subgroup. This operation generally changes the homology of the space, but these changes can be reversed by the addition of three-cells.
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https://en.wikipedia.org/wiki/Plus_construction
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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 elements in R {\displaystyle R} . GL n ( R ) {\displaystyle \operatorname {GL} _{n}(R)} embeds in GL n + 1 ( R ) {\displaystyle \operatorname {GL} _{n+1}(R)} by attaching a 1 {\displaystyle 1} along the diagonal and 0 {\displaystyle 0} s elsewhere.
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https://en.wikipedia.org/wiki/Plus_construction
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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 1 ( B GL ( R ) ) {\displaystyle \operatorname {GL} (R)=\pi _{1}(B\operatorname {GL} (R))} , generated by matrices which only differ from the identity matrix in one off-diagonal entry. For n > 0 {\displaystyle n>0} , the n {\displaystyle n} -th homotopy group of the resulting space, B GL ( R ) + {\displaystyle B\operatorname {GL} (R)^{+}} , is isomorphic to the n {\displaystyle n} -th K {\displaystyle K} -group of R {\displaystyle R} , that is, π n ( B GL ( R ) + ) ≅ K n ( R ) . {\displaystyle \pi _{n}\left(B\operatorname {GL} (R)^{+}\right)\cong K_{n}(R).}
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https://en.wikipedia.org/wiki/Plus_construction
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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 "∨" but is actually equivalent to the logical inequality operator signified by "≠", or what amounts to the same thing, the exclusive disjunction signified by "XOR" or "⊕". Naturally, these variations in usage have caused some failures to communicate between mathematicians and switching engineers over the years. At any rate, one has the following array of corresponding forms for the symbols associated with logical inequality: x + y x ≢ y J x y x X O R y x ≠ y {\displaystyle {\begin{aligned}x&+y&x&\not \equiv y&Jxy\\x&\mathrm {~XOR~} y&x&\neq y\end{aligned}}} This explains why "EQ" is often called "XNOR" in the combinational logic of circuit engineers, since it is the negation of the XOR operation; "NXOR" is a less commonly used alternative. Another rationalization of the admittedly circuitous name "XNOR" is that one begins with the "both false" operator NOR and then adds the eXception "or both true".
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https://en.wikipedia.org/wiki/Logical_equality
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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 pointwise product of f and g is another function from X to Y which maps x in X to f (x)g(x) in Y.
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https://en.wikipedia.org/wiki/Pointwise_product
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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 coordinate system) is called the pole, and the ray from the pole in the reference direction is the polar axis. The distance from the pole is called the radial coordinate, radial distance or simply radius, and the angle is called the angular coordinate, polar angle, or azimuth. Angles in polar notation are generally expressed in either degrees or radians (2π rad being equal to 360°).
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https://en.wikipedia.org/wiki/Polar_coordinates_system
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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 motion. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more intuitive to model using polar coordinates. The polar coordinate system is extended to three dimensions in two ways: the cylindrical and spherical coordinate systems.
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https://en.wikipedia.org/wiki/Polar_coordinates_system
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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 and P {\displaystyle P} is a positive semi-definite symmetric matrix in the real case), both square and of the same size.Intuitively, if a real n × n {\displaystyle n\times n} matrix A {\displaystyle A} is interpreted as a linear transformation of n {\displaystyle n} -dimensional space R n {\displaystyle \mathbb {R} ^{n}} , the polar decomposition separates it into a rotation or reflection U {\displaystyle U} of R n {\displaystyle \mathbb {R} ^{n}} , and a scaling of the space along a set of n {\displaystyle n} orthogonal axes. The polar decomposition of a square matrix A {\displaystyle A} always exists. If A {\displaystyle A} is invertible, the decomposition is unique, and the factor P {\displaystyle P} will be positive-definite. In that case, A {\displaystyle A} can be written uniquely in the form A = U e X {\displaystyle A=Ue^{X}} , where U {\displaystyle U} is unitary and X {\displaystyle X} is the unique self-adjoint logarithm of the matrix P {\displaystyle P} .
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https://en.wikipedia.org/wiki/Polar_decomposition
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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 eigenvectors. The polar decomposition of a matrix can be seen as the matrix analog of the polar form of a complex number z {\displaystyle z} as z = u r {\displaystyle z=ur} , where r {\displaystyle r} is its absolute value (a non-negative real number), and u {\displaystyle u} is a complex number with unit norm (an element of the circle group).
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https://en.wikipedia.org/wiki/Polar_decomposition
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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-semidefinite Hermitian matrix. The decomposition always exists and P {\displaystyle P} is always unique. The matrix U {\displaystyle U} is unique if and only if A {\displaystyle A} has full rank.
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https://en.wikipedia.org/wiki/Polar_decomposition
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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 {\mathrm {d} ^{m}}{\mathrm {d} z^{m}}}\psi (z)={\frac {\mathrm {d} ^{m+1}}{\mathrm {d} z^{m+1}}}\ln \Gamma (z).} Thus ψ ( 0 ) ( z ) = ψ ( z ) = Γ ′ ( z ) Γ ( z ) {\displaystyle \psi ^{(0)}(z)=\psi (z)={\frac {\Gamma '(z)}{\Gamma (z)}}} holds where ψ(z) is the digamma function and Γ(z) is the gamma function.
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https://en.wikipedia.org/wiki/Polygamma_function
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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.
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https://en.wikipedia.org/wiki/Polygamma_function
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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 polylogarithm function appears as the closed form of integrals of the Fermi–Dirac distribution and the Bose–Einstein distribution, and is also known as the Fermi–Dirac integral or the Bose–Einstein integral. In quantum electrodynamics, polylogarithms of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent to the Hurwitz zeta function — either function can be expressed in terms of the other — and both functions are special cases of the Lerch transcendent.
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https://en.wikipedia.org/wiki/Polylogarithm_ladder
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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 in s: Li s ( z ) = ∑ k = 1 ∞ z k k s = z + z 2 2 s + z 3 3 s + ⋯ {\displaystyle \operatorname {Li} _{s}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{s}}=z+{z^{2} \over 2^{s}}+{z^{3} \over 3^{s}}+\cdots } This definition is valid for arbitrary complex order s and for all complex arguments z with |z| < 1; it can be extended to |z| ≥ 1 by the process of analytic continuation. (Here the denominator ks is understood as exp(s ln k)).
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https://en.wikipedia.org/wiki/Polylogarithm_ladder
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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 of itself: Li s + 1 ( z ) = ∫ 0 z Li s ( t ) t d t {\displaystyle \operatorname {Li} _{s+1}(z)=\int _{0}^{z}{\frac {\operatorname {Li} _{s}(t)}{t}}dt} thus the dilogarithm is an integral of a function involving the logarithm, and so on. For nonpositive integer orders s, the polylogarithm is a rational function.
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https://en.wikipedia.org/wiki/Polylogarithm_ladder
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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 long-standing open problems. The polynomial method encompasses a wide range of specific techniques for using polynomials and ideas from areas such as algebraic geometry to solve combinatorics problems. While a few techniques that follow the framework of the polynomial method, such as Alon's Combinatorial Nullstellensatz, have been known since the 1990s, it was not until around 2010 that a broader framework for the polynomial method has been developed.
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https://en.wikipedia.org/wiki/Polynomial_method_in_combinatorics
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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 as faces σ ⊆ V {\displaystyle \sigma \subseteq V} , such that ∀ ρ ∀ σ: ρ ⊆ σ ∈ Δ ⇒ ρ ∈ Δ . {\displaystyle \forall \rho \,\forall \sigma \!
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https://en.wikipedia.org/wiki/Order_complex
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:\ \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 ∈ Γ ⇒ ρ ∈ Γ . {\displaystyle \forall \rho \,\forall \sigma \!
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https://en.wikipedia.org/wiki/Order_complex
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:\ \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 (S, ≤) is then the Alexandrov topology on the order complex associated to (S, ≤).
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https://en.wikipedia.org/wiki/Order_complex
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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 + {\displaystyle f^{+}} is obtained by taking the graph of f {\displaystyle f} , chopping off the part under the x-axis, and letting f + {\displaystyle f^{+}} take the value zero there. Similarly, the negative part of f is defined as f − ( x ) = max ( − f ( x ) , 0 ) = − min ( f ( x ) , 0 ) = { − f ( x ) if f ( x ) < 0 0 otherwise.
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https://en.wikipedia.org/wiki/Negative_part
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{\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.
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https://en.wikipedia.org/wiki/Negative_part
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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^{-}.}
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https://en.wikipedia.org/wiki/Negative_part
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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 representation, using the Iverson bracket is f + = f {\displaystyle f^{+}=f} f − = − f .
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https://en.wikipedia.org/wiki/Negative_part
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{\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.
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https://en.wikipedia.org/wiki/Negative_part
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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.
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https://en.wikipedia.org/wiki/Series_solution_of_differential_equations
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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 as P(S), 𝒫(S), P(S), P ( S ) {\displaystyle \mathbb {P} (S)} , ℘ ( S ) {\displaystyle \wp (S)} , or 2S. The notation 2S, meaning the set of all functions from S to a given set of two elements (e.g., {0, 1}), is used because the powerset of S can be identified with, equivalent to, or bijective to the set of all the functions from S to the given two elements set.Any subset of P(S) is called a family of sets over S.
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https://en.wikipedia.org/wiki/Power_set
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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 functions.
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https://en.wikipedia.org/wiki/Predual
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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 sphere of the same dimension. This condition is necessary since for any manifold M of dimension n {\displaystyle n} it is true that (where M # S n {\displaystyle M\#S^{n}} means the connected sum of M {\displaystyle M} and S n {\displaystyle S^{n}} ). If P {\displaystyle P} is a prime 3-manifold then either it is S 2 × S 1 {\displaystyle S^{2}\times S^{1}} or the non-orientable S 2 {\displaystyle S^{2}} bundle over S 1 , {\displaystyle S^{1},} or it is irreducible, which means that any embedded 2-sphere bounds a ball.
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https://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)
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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: every compact, non-orientable 3-manifold is a connected sum of irreducible 3-manifolds and non-orientable S 2 {\displaystyle S^{2}} bundles over S 1 .
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https://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)
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{\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 .
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https://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)
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{\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.
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https://en.wikipedia.org/wiki/Prime_decomposition_(3-manifold)
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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.
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https://en.wikipedia.org/wiki/Prime_end
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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 might be represented as (x′, y′). Usually, the meaning of x′ is defined when it is first used, but sometimes, its meaning is assumed to be understood: A derivative or differentiated function: in Lagrange's notation, f ′(x) and f ″(x) are the first and second derivatives of f (x) with respect to x. Likewise are f ‴(x) and f ⁗(x) . Similarly, if y = f (x), then y′ and y″ are the first and second derivatives of y with respect to x. Other notation for derivatives also exists (see Notation for differentiation). Set complement: A′ is the complement of the set A (other notation also exists).
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https://en.wikipedia.org/wiki/Double_prime
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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.
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https://en.wikipedia.org/wiki/Double_prime
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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 event.
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https://en.wikipedia.org/wiki/Double_prime
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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 and R′, representing different alkyl groups in an organic compound.
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https://en.wikipedia.org/wiki/Double_prime
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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 state of the quantum number J while J ″ denotes the lower state of the quantum number J.In molecular biology, the prime is used to denote the positions of carbon on a ring of deoxyribose or ribose.
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https://en.wikipedia.org/wiki/Double_prime
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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 on the ends of the DNA molecule. The chemistry of this reaction demands that the 3′ OH be extended by DNA synthesis. Prime can also be used to indicate which position a molecule has attached to, such as 5′-monophosphate.
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https://en.wikipedia.org/wiki/Double_prime
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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 Jacques Hadamard and Charles Jean de la Vallée Poussin in 1896 using ideas introduced by Bernhard Riemann (in particular, the Riemann zeta function). The first such distribution found is π(N) ~ N/log(N), where π(N) is the prime-counting function (the number of primes less than or equal to N) and log(N) is the natural logarithm of N. This means that for large enough N, the probability that a random integer not greater than N is prime is very close to 1 / log(N).
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https://en.wikipedia.org/wiki/Prime_number_race
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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), whereas among positive integers of at most 2000 digits, about one in 4600 is prime (log(102000) ≈ 4605.2). In other words, the average gap between consecutive prime numbers among the first N integers is roughly log(N).
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https://en.wikipedia.org/wiki/Prime_number_race
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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 { m 1 , m 2 , … , m n } {\displaystyle \left\{m_{1},m_{2},\dots ,m_{n}\right\}} . For example, all prime numbers have a prime signature of {1}, the squares of primes have a prime signature of {2}, the products of 2 distinct primes have a prime signature of {1, 1} and the products of a square of a prime and a different prime (e.g. 12, 18, 20, ...) have a prime signature of {2, 1}.
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https://en.wikipedia.org/wiki/Prime_signature
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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)=\sum _{p\,\in \mathrm {\,primes} }{\frac {1}{p^{s}}}={\frac {1}{2^{s}}}+{\frac {1}{3^{s}}}+{\frac {1}{5^{s}}}+{\frac {1}{7^{s}}}+{\frac {1}{11^{s}}}+\cdots .}
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https://en.wikipedia.org/wiki/Prime_zeta_function
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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).
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https://en.wikipedia.org/wiki/Prime-counting_function
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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 phenomenon has also been called principalization, or sometimes capitulation.
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https://en.wikipedia.org/wiki/Principalisation_property
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In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type.
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https://en.wikipedia.org/wiki/Principal_orbit_type
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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 involving a continuous parameter, as well as a discrete spectrum. The principal series representations are some induced representations constructed in a uniform way, in order to fill out the continuous part of the spectrum. In more detail, the unitary dual is the space of all representations relevant to decomposing the regular representation.
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https://en.wikipedia.org/wiki/Principal_series_representation
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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 representations of H which were accessible, in the sense of being easy to write down, and involving a parameter. (Such an induction process may produce representations that are not unitary.)
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https://en.wikipedia.org/wiki/Principal_series_representation
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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 considered that are irreducible, and unitary, and are the trivial representation on the subgroup N. (Assuming the case M a trivial group, such ρ are analogues of the representations of the group of diagonal matrices inside the special linear group.)
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https://en.wikipedia.org/wiki/Principal_series_representation
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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 the unitary dual: as their name implies, the principal series are the 'main' contribution. This type of construction has been found to have application to groups G that are not Lie groups (for example, finite groups of Lie type, groups over p-adic fields).
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https://en.wikipedia.org/wiki/Principal_series_representation
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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 the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for certain, without any possible error. This method has now been applied to other areas of mathematics such as number theory, linear algebra, and real analysis, as well as in computer science (e.g. randomized rounding), and information theory.
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https://en.wikipedia.org/wiki/Probabilistic_method
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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 for what functions f: X → R does lim r → 0 1 μ ( B r ( x ) ) ∫ B r ( x ) f ( y ) d μ ( y ) = f ( x ) {\displaystyle \lim _{r\to 0}{\frac {1}{\mu {\big (}B_{r}(x){\big )}}}\int _{B_{r}(x)}f(y)\,\mathrm {d} \mu (y)=f(x)} for all (or at least μ-almost all) x ∈ X? (Here, as in the rest of the article, Br(x) denotes the open ball in X with d-radius r and centre x.) This is a natural question to ask, especially in view of the heuristic construction of the Riemann integral, in which it is almost implicit that f(x) is a "good representative" for the values of f near x.
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https://en.wikipedia.org/wiki/Differentiation_of_integrals
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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 of that projection. Take that same function, but do a two-dimensional Fourier transform first, and then slice it through its origin, which is parallel to the projection line.In operator terms, if F1 and F2 are the 1- and 2-dimensional Fourier transform operators mentioned above, P1 is the projection operator (which projects a 2-D function onto a 1-D line), S1 is a slice operator (which extracts a 1-D central slice from a function),then F 1 P 1 = S 1 F 2 . {\displaystyle F_{1}P_{1}=S_{1}F_{2}.}
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https://en.wikipedia.org/wiki/Fourier_slice_theorem
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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 organ, and these slices can be interpolated to build up a complete Fourier transform of that density. The inverse Fourier transform is then used to recover the 3-dimensional density of the object. This technique was first derived by Ronald N. Bracewell in 1956 for a radio-astronomy problem.
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https://en.wikipedia.org/wiki/Fourier_slice_theorem
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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 u in U such that ua = c and ub = d. This relation is an equivalence relation. A typical equivalence class is written U. P(A) = { U: aA + bA = A }, that is, U is in the projective line if the ideal generated by a and b is all of A. The projective line P(A) is equipped with a group of homographies.
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https://en.wikipedia.org/wiki/Inversive_ring_geometry
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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 a normal subgroup N of V. The homographies of P(A) correspond to elements of the quotient group V / N . P(A) is considered an extension of the ring A since it contains a copy of A due to the embedding E: a → U. The multiplicative inverse mapping u → 1/u, ordinarily restricted to the group of units U of A, is expressed by a homography on P(A): U ( 0 1 1 0 ) = U ∼ U .
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https://en.wikipedia.org/wiki/Inversive_ring_geometry
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{\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}v&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}={\begin{pmatrix}u&0\\0&v\end{pmatrix}}.}
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https://en.wikipedia.org/wiki/Inversive_ring_geometry
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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.}
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https://en.wikipedia.org/wiki/Inversive_ring_geometry
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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 smallest nonabelian simple group after the alternating group A5 with 60 elements, isomorphic to PSL(2, 5).
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https://en.wikipedia.org/wiki/PSL(2,7)
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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 projective space. In terms of matrices, elements of U(n) are complex n×n unitary matrices, and elements of the center are diagonal matrices equal to eiθ multiplied by the identity matrix. Thus, elements of PU(n) correspond to equivalence classes of unitary matrices under multiplication by a constant phase θ. Abstractly, given a Hermitian space V, the group PU(V) is the image of the unitary group U(V) in the automorphism group of the projective space P(V).
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https://en.wikipedia.org/wiki/Projective_unitary_group
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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.
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https://en.wikipedia.org/wiki/Pseudoisotopy_theorem
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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.
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https://en.wikipedia.org/wiki/Q-Bessel_polynomials
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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.
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https://en.wikipedia.org/wiki/Q-Charlier_polynomials
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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.
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https://en.wikipedia.org/wiki/Q-Hahn_polynomials
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In mathematics, the q-Konhauser polynomials are a q-analog of the Konhauser polynomials, introduced by Al-Salam & Verma (1983).
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https://en.wikipedia.org/wiki/Q-Konhauser_polynomials
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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 that the q-Krawtchouk polynomials are spherical functions for 3 different Chevalley groups over finite fields, and Koornwinder et al. (2010–2022) showed that they are related to representations of the quantum group SU(2).
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https://en.wikipedia.org/wiki/Q-Krawtchouk_polynomials
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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. Swarttouw (2010, 14) give a detailed list of their properties.
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https://en.wikipedia.org/wiki/Q-Laguerre_polynomials
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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.
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https://en.wikipedia.org/wiki/Q-Meixner_polynomials
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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.
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https://en.wikipedia.org/wiki/Q-Meixner–Pollaczek_polynomials
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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.
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https://en.wikipedia.org/wiki/Q-Racah_polynomials
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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).
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https://en.wikipedia.org/wiki/Q-expansion_principle
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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 one takes 0 ≤ |q| < 1. It obeys the identities θ ( z ; q ) = θ ( q z ; q ) = − z θ ( 1 z ; q ) . {\displaystyle \theta (z;q)=\theta \left({\frac {q}{z}};q\right)=-z\theta \left({\frac {1}{z}};q\right).} It may also be expressed as: θ ( z ; q ) = ( z ; q ) ∞ ( q / z ; q ) ∞ {\displaystyle \theta (z;q)=(z;q)_{\infty }(q/z;q)_{\infty }} where ( ⋅ ⋅ ) ∞ {\displaystyle (\cdot \cdot )_{\infty }} is the q-Pochhammer symbol.
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https://en.wikipedia.org/wiki/Q-theta_function
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