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In mathematics, a nonlocal operator is a mapping which maps functions on a topological space to functions, in such a way that the value of the output function at a given point cannot be determined solely from the values of the input function in any neighbourhood of any point. An example of a nonlocal operator is the Fourier transform.
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https://en.wikipedia.org/wiki/Nonlocal_operator
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In mathematics, a nonnegative matrix, written X ≥ 0 , {\displaystyle \mathbf {X} \geq 0,} is a matrix in which all the elements are equal to or greater than zero, that is, x i j ≥ 0 ∀ i , j . {\displaystyle x_{ij}\geq 0\qquad \forall {i,j}.} A positive matrix is a matrix in which all the elements are strictly greater than zero.
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https://en.wikipedia.org/wiki/Nonnegative_matrices
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The set of positive matrices is a subset of all non-negative matrices. While such matrices are commonly found, the term is only occasionally used due to the possible confusion with positive-definite matrices, which are different. A matrix which is both non-negative and is positive semidefinite is called a doubly non-negative matrix. A rectangular non-negative matrix can be approximated by a decomposition with two other non-negative matrices via non-negative matrix factorization. Eigenvalues and eigenvectors of square positive matrices are described by the Perron–Frobenius theorem.
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https://en.wikipedia.org/wiki/Nonnegative_matrices
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In mathematics, a nonrecursive filter only uses input values like x, unlike recursive filter where it uses previous output values like y. In signal processing, non-recursive digital filters are often known as Finite Impulse Response (FIR) filters, as a non-recursive digital filter has a finite number of coefficients in the impulse response h. Examples: Non-recursive filter: y = 0.5x + 0.5x Recursive filter: y = 0.5y + 0.5x An important property of non-recursive filters is, that they will always be stable. This is not always the case for recursive filters.
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https://en.wikipedia.org/wiki/Nonrecursive_filter
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In mathematics, a nonstandard integer may refer to Hyperinteger, the integer part of a hyperreal number an integer in a non-standard model of arithmetic
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https://en.wikipedia.org/wiki/Nonstandard_integer
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In mathematics, a norm form is a homogeneous form in n variables constructed from the field norm of a field extension L/K of degree n. That is, writing N for the norm mapping to K, and selecting a basis e1, ..., en for L as a vector space over K, the form is given by N(x1e1 + ... + xnen)in variables x1, ..., xn. In number theory norm forms are studied as Diophantine equations, where they generalize, for example, the Pell equation. For this application the field K is usually the rational number field, the field L is an algebraic number field, and the basis is taken of some order in the ring of integers OL of L.
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https://en.wikipedia.org/wiki/Norm_form
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In mathematics, a norm is a function from a real or complex vector space to the non-negative real numbers that behaves in certain ways like the distance from the origin: it commutes with scaling, obeys a form of the triangle inequality, and is zero only at the origin. In particular, the Euclidean distance in a Euclidean space is defined by a norm on the associated Euclidean vector space, called the Euclidean norm, the 2-norm, or, sometimes, the magnitude of the vector. This norm can be defined as the square root of the inner product of a vector with itself. A seminorm satisfies the first two properties of a norm, but may be zero for vectors other than the origin.
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https://en.wikipedia.org/wiki/Vector_length
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A vector space with a specified norm is called a normed vector space. In a similar manner, a vector space with a seminorm is called a seminormed vector space. The term pseudonorm has been used for several related meanings.
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https://en.wikipedia.org/wiki/Vector_length
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It may be a synonym of "seminorm". A pseudonorm may satisfy the same axioms as a norm, with the equality replaced by an inequality " ≤ {\displaystyle \,\leq \,} " in the homogeneity axiom. It can also refer to a norm that can take infinite values, or to certain functions parametrised by a directed set.
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https://en.wikipedia.org/wiki/Vector_length
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In mathematics, a norm variety is a particular type of algebraic variety V over a field F, introduced for the purposes of algebraic K-theory by Voevodsky. The idea is to relate Milnor K-theory of F to geometric objects V, having function fields F(V) that 'split' given 'symbols' (elements of Milnor K-groups).The formulation is that p is a given prime number, different from the characteristic of F, and a symbol is the class mod p of an element { a 1 , … , a n } {\displaystyle \{a_{1},\dots ,a_{n}\}\ } of the n-th Milnor K-group. A field extension is said to split the symbol, if its image in the K-group for that field is 0.
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https://en.wikipedia.org/wiki/Norm_variety
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The conditions on a norm variety V are that V is irreducible and a non-singular complete variety. Further it should have dimension d equal to p n − 1 − 1. {\displaystyle p^{n-1}-1.\ } The key condition is in terms of the d-th Newton polynomial sd, evaluated on the (algebraic) total Chern class of the tangent bundle of V. This number s d ( V ) {\displaystyle s_{d}(V)\ } should not be divisible by p2, it being known it is divisible by p.
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https://en.wikipedia.org/wiki/Norm_variety
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In mathematics, a normal map is a concept in geometric topology due to William Browder which is of fundamental importance in surgery theory. Given a Poincaré complex X (more geometrically a Poincaré space), a normal map on X endows the space, roughly speaking, with some of the homotopy-theoretic global structure of a closed manifold. In particular, X has a good candidate for a stable normal bundle and a Thom collapse map, which is equivalent to there being a map from a manifold M to X matching the fundamental classes and preserving normal bundle information.
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https://en.wikipedia.org/wiki/Normal_invariants
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If the dimension of X is ≥ {\displaystyle \geq } 5 there is then only the algebraic topology surgery obstruction due to C. T. C. Wall to X actually being homotopy equivalent to a closed manifold. Normal maps also apply to the study of the uniqueness of manifold structures within a homotopy type, which was pioneered by Sergei Novikov. The cobordism classes of normal maps on X are called normal invariants.
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https://en.wikipedia.org/wiki/Normal_invariants
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Depending on the category of manifolds (differentiable, piecewise-linear, or topological), there are similarly defined, but inequivalent, concepts of normal maps and normal invariants. It is possible to perform surgery on normal maps, meaning surgery on the domain manifold, and preserving the map. Surgery on normal maps allows one to systematically kill elements in the relative homotopy groups by representing them as embeddings with trivial normal bundle.
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https://en.wikipedia.org/wiki/Normal_invariants
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In mathematics, a normal surface is a surface inside a triangulated 3-manifold that intersects each tetrahedron so that each component of intersection is a triangle or a quad (see figure). A triangle cuts off a vertex of the tetrahedron while a quad separates pairs of vertices. A normal surface may have many components of intersection, called normal disks, with one tetrahedron, but no two normal disks can be quads that separate different pairs of vertices since that would lead to the surface self-intersecting.
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https://en.wikipedia.org/wiki/Normal_surface
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Dually, a normal surface can be considered to be a surface that intersects each handle of a given handle structure on the 3-manifold in a prescribed manner similar to the above. The concept of normal surface can be generalized to arbitrary polyhedra. There are also related notions of almost normal surface and spun normal surface.
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https://en.wikipedia.org/wiki/Normal_surface
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The concept of normal surface is due to Hellmuth Kneser, who utilized it in his proof of the prime decomposition theorem for 3-manifolds. Later Wolfgang Haken extended and refined the notion to create normal surface theory, which is at the basis of many of the algorithms in 3-manifold theory. The notion of almost normal surfaces is due to Hyam Rubinstein. The notion of spun normal surface is due to Bill Thurston. Regina is software which enumerates normal and almost-normal surfaces in triangulated 3-manifolds, implementing Rubinstein's 3-sphere recognition algorithm, among other things.
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https://en.wikipedia.org/wiki/Normal_surface
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In mathematics, a normed algebra A is an algebra over a field which has a sub-multiplicative norm: ∀ x , y ∈ A ‖ x y ‖ ≤ ‖ x ‖ ‖ y ‖ . {\displaystyle \forall x,y\in A\qquad \|xy\|\leq \|x\|\|y\|.} Some authors require it to have a multiplicative identity 1A such that ║1A║ = 1.
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https://en.wikipedia.org/wiki/Normed_algebra
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In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers on which a norm is defined. A norm is a generalization of the intuitive notion of "length" in the physical world. If V {\displaystyle V} is a vector space over K {\displaystyle K} , where K {\displaystyle K} is a field equal to R {\displaystyle \mathbb {R} } or to C {\displaystyle \mathbb {C} } , then a norm on V {\displaystyle V} is a map V → R {\displaystyle V\to \mathbb {R} } , typically denoted by ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } , satisfying the following four axioms: Non-negativity: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ ≥ 0 {\displaystyle \;\lVert x\rVert \geq 0} . Positive definiteness: for every x ∈ V {\displaystyle x\in V} , ‖ x ‖ = 0 {\displaystyle \;\lVert x\rVert =0} if and only if x {\displaystyle x} is the zero vector.
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https://en.wikipedia.org/wiki/Normed_spaces
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Absolute homogeneity: for every λ ∈ K {\displaystyle \lambda \in K} and x ∈ V {\displaystyle x\in V} , Triangle inequality: for every x ∈ V {\displaystyle x\in V} and y ∈ V {\displaystyle y\in V} ,If V {\displaystyle V} is a real or complex vector space as above, and ‖ ⋅ ‖ {\displaystyle \lVert \cdot \rVert } is a norm on V {\displaystyle V} , then the ordered pair ( V , ‖ ⋅ ‖ ) {\displaystyle (V,\lVert \cdot \rVert )} is called a normed vector space. If it is clear from context which norm is intended, then it is common to denote the normed vector space simply by V {\displaystyle V} . A norm induces a distance, called its (norm) induced metric, by the formula which makes any normed vector space into a metric space and a topological vector space.
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https://en.wikipedia.org/wiki/Normed_spaces
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If this metric space is complete then the normed space is a Banach space. Every normed vector space can be "uniquely extended" to a Banach space, which makes normed spaces intimately related to Banach spaces. Every Banach space is a normed space but converse is not true.
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https://en.wikipedia.org/wiki/Normed_spaces
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For example, the set of the finite sequences of real numbers can be normed with the Euclidean norm, but it is not complete for this norm. An inner product space is a normed vector space whose norm is the square root of the inner product of a vector and itself. The Euclidean norm of a Euclidean vector space is a special case that allows defining Euclidean distance by the formula The study of normed spaces and Banach spaces is a fundamental part of functional analysis, a major subfield of mathematics.
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https://en.wikipedia.org/wiki/Normed_spaces
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In mathematics, a nowhere commutative semigroup is a semigroup S such that, for all a and b in S, if ab = ba then a = b. A semigroup S is nowhere commutative if and only if any two elements of S are inverses of each other.
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https://en.wikipedia.org/wiki/Nowhere_commutative_semigroup
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In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain. If f {\displaystyle f} is a function from real numbers to real numbers, then f {\displaystyle f} is nowhere continuous if for each point x {\displaystyle x} there is some ε > 0 {\displaystyle \varepsilon >0} such that for every δ > 0 , {\displaystyle \delta >0,} we can find a point y {\displaystyle y} such that | x − y | < δ {\displaystyle |x-y|<\delta } and | f ( x ) − f ( y ) | ≥ ε {\displaystyle |f(x)-f(y)|\geq \varepsilon } . Therefore, no matter how close we get to any fixed point, there are even closer points at which the function takes not-nearby values. More general definitions of this kind of function can be obtained, by replacing the absolute value by the distance function in a metric space, or by using the definition of continuity in a topological space.
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https://en.wikipedia.org/wiki/Nowhere_continuous_function
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In mathematics, a null semigroup (also called a zero semigroup) is a semigroup with an absorbing element, called zero, in which the product of any two elements is zero. If every element of a semigroup is a left zero then the semigroup is called a left zero semigroup; a right zero semigroup is defined analogously. According to Clifford and Preston, "In spite of their triviality, these semigroups arise naturally in a number of investigations."
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https://en.wikipedia.org/wiki/Null_semigroup
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In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by Hilbert (1893). (Dieudonné & Carrell 1970, 1971, p.57).
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https://en.wikipedia.org/wiki/Nullform
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In mathematics, a number of concepts employ the word harmonic. The similarity of this terminology to that of music is not accidental: the equations of motion of vibrating strings, drums and columns of air are given by formulas involving Laplacians; the solutions to which are given by eigenvalues corresponding to their modes of vibration. Thus, the term "harmonic" is applied when one is considering functions with sinusoidal variations, or solutions of Laplace's equation and related concepts.Mathematical terms whose names include "harmonic" include: Projective harmonic conjugate Cross-ratio Harmonic analysis Harmonic conjugate Harmonic form Harmonic function Harmonic mean Harmonic mode Harmonic number Harmonic series Alternating harmonic series Harmonic tremor Spherical harmonics
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https://en.wikipedia.org/wiki/Harmonic_(mathematics)
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In mathematics, a number of fixed-point theorems in infinite-dimensional spaces generalise the Brouwer fixed-point theorem. They have applications, for example, to the proof of existence theorems for partial differential equations. The first result in the field was the Schauder fixed-point theorem, proved in 1930 by Juliusz Schauder (a previous result in a different vein, the Banach fixed-point theorem for contraction mappings in complete metric spaces was proved in 1922). Quite a number of further results followed.
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https://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces
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One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension. For example, the research of Jean Leray who founded sheaf theory came out of efforts to extend Schauder's work. Schauder fixed-point theorem: Let C be a nonempty closed convex subset of a Banach space V. If f: C → C is continuous with a compact image, then f has a fixed point.
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https://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces
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Tikhonov (Tychonoff) fixed-point theorem: Let V be a locally convex topological vector space. For any nonempty compact convex set X in V, any continuous function f: X → X has a fixed point. Browder fixed-point theorem: Let K be a nonempty closed bounded convex set in a uniformly convex Banach space.
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https://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces
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Then any non-expansive function f: K → K has a fixed point. (A function f {\displaystyle f} is called non-expansive if ‖ f ( x ) − f ( y ) ‖ ≤ ‖ x − y ‖ {\displaystyle \|f(x)-f(y)\|\leq \|x-y\|} for each x {\displaystyle x} and y {\displaystyle y} .) Other results include the Markov–Kakutani fixed-point theorem (1936-1938) and the Ryll-Nardzewski fixed-point theorem (1967) for continuous affine self-mappings of compact convex sets, as well as the Earle–Hamilton fixed-point theorem (1968) for holomorphic self-mappings of open domains. Kakutani fixed-point theorem: Every correspondence that maps a compact convex subset of a locally convex space into itself with a closed graph and convex nonempty images has a fixed point.
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https://en.wikipedia.org/wiki/Fixed-point_theorems_in_infinite-dimensional_spaces
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In mathematics, a numerical semigroup is a special kind of a semigroup. Its underlying set is the set of all nonnegative integers except a finite number and the binary operation is the operation of addition of integers. Also, the integer 0 must be an element of the semigroup. For example, while the set {0, 2, 3, 4, 5, 6, ...} is a numerical semigroup, the set {0, 1, 3, 5, 6, ...} is not because 1 is in the set and 1 + 1 = 2 is not in the set.
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https://en.wikipedia.org/wiki/Numerical_monoid
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Numerical semigroups are commutative monoids and are also known as numerical monoids.The definition of numerical semigroup is intimately related to the problem of determining nonnegative integers that can be expressed in the form x1n1 + x2 n2 + ... + xr nr for a given set {n1, n2, ..., nr} of positive integers and for arbitrary nonnegative integers x1, x2, ..., xr. This problem had been considered by several mathematicians like Frobenius (1849–1917) and Sylvester (1814–1897) at the end of the 19th century. During the second half of the twentieth century, interest in the study of numerical semigroups resurfaced because of their applications in algebraic geometry.
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https://en.wikipedia.org/wiki/Numerical_monoid
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In mathematics, a one-dimensional array corresponds to a vector, a two-dimensional array resembles a matrix; more generally, a tensor may be represented as an n-dimensional data cube.
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https://en.wikipedia.org/wiki/Data_cube
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In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism φ: R → G {\displaystyle \varphi :\mathbb {R} \rightarrow G} from the real line R {\displaystyle \mathbb {R} } (as an additive group) to some other topological group G {\displaystyle G} . If φ {\displaystyle \varphi } is injective then φ ( R ) {\displaystyle \varphi (\mathbb {R} )} , the image, will be a subgroup of G {\displaystyle G} that is isomorphic to R {\displaystyle \mathbb {R} } as an additive group. One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an infinitesimal transformation is an infinitely small transformation of the one-parameter group that it generates.
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https://en.wikipedia.org/wiki/1-parameter_group
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It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension. The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a local flow - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of a vector field is used to define the Lie derivative of tensor fields along the vector field.
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https://en.wikipedia.org/wiki/1-parameter_group
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In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of p-adic numbers.
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https://en.wikipedia.org/wiki/P-adic_distribution
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In mathematics, a p-adic modular form is a p-adic analog of a modular form, with coefficients that are p-adic numbers rather than complex numbers. Serre (1973) introduced p-adic modular forms as limits of ordinary modular forms, and Katz (1973) shortly afterwards gave a geometric and more general definition. Katz's p-adic modular forms include as special cases classical p-adic modular forms, which are more or less p-adic linear combinations of the usual "classical" modular forms, and overconvergent p-adic modular forms, which in turn include Hida's ordinary modular forms as special cases.
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https://en.wikipedia.org/wiki/P-adic_modular_form
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In mathematics, a p-adic zeta function, or more generally a p-adic L-function, is a function analogous to the Riemann zeta function, or more general L-functions, but whose domain and target are p-adic (where p is a prime number). For example, the domain could be the p-adic integers Zp, a profinite p-group, or a p-adic family of Galois representations, and the image could be the p-adic numbers Qp or its algebraic closure. The source of a p-adic L-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a p-adic L-function (Kubota & Leopoldt 1964)—is via the p-adic interpolation of special values of L-functions.
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https://en.wikipedia.org/wiki/P-adic_zeta_function
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For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a p-adic L-function, the p-adic Riemann zeta function ζp(s), whose values at negative odd integers are those of the Riemann zeta function at negative odd integers (up to an explicit correction factor). p-adic L-functions arising in this fashion are typically referred to as analytic p-adic L-functions. The other major source of p-adic L-functions—first discovered by Kenkichi Iwasawa—is from the arithmetic of cyclotomic fields, or more generally, certain Galois modules over towers of cyclotomic fields or even more general towers.
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https://en.wikipedia.org/wiki/P-adic_zeta_function
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A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module involved. The main conjecture of Iwasawa theory (now a theorem due to Barry Mazur and Andrew Wiles) is the statement that the Kubota–Leopoldt p-adic L-function and an arithmetic analogue constructed by Iwasawa theory are essentially the same. In more general situations where both analytic and arithmetic p-adic L-functions are constructed (or expected), the statement that they agree is called the main conjecture of Iwasawa theory for that situation. Such conjectures represent formal statements concerning the philosophy that special values of L-functions contain arithmetic information.
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https://en.wikipedia.org/wiki/P-adic_zeta_function
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In mathematics, a p-adically closed field is a field that enjoys a closure property that is a close analogue for p-adic fields to what real closure is to the real field. They were introduced by James Ax and Simon B. Kochen in 1965.
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https://en.wikipedia.org/wiki/P-adically_closed_field
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In mathematics, a p-constrained group is a finite group resembling the centralizer of an element of prime order p in a group of Lie type over a finite field of characteristic p. They were introduced by Gorenstein and Walter (1964, p.169) in order to extend some of Thompson's results about odd groups to groups with dihedral Sylow 2-subgroups.
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https://en.wikipedia.org/wiki/P-constrained_group
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In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy algorithm which was proposed by Joel Spencer. He used a branching process to formally prove the optimal achievable bound under some side conditions. The other algorithm is called the Rödl nibble and was proposed by Vojtěch Rödl et al. They showed that the achievable packing by the Rödl nibble is in some sense close to that of the random greedy algorithm.
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https://en.wikipedia.org/wiki/Packing_in_a_hypergraph
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In mathematics, a pair of pants is a surface which is homeomorphic to the three-holed sphere. The name comes from considering one of the removed disks as the waist and the two others as the cuffs of a pair of pants. Pairs of pants are used as building blocks for compact surfaces in various theories. Two important applications are to hyperbolic geometry, where decompositions of closed surfaces into pairs of pants are used to construct the Fenchel-Nielsen coordinates on Teichmüller space, and in topological quantum field theory where they are the simplest non-trivial cobordisms between 1-dimensional manifolds.
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https://en.wikipedia.org/wiki/Pair_of_pants_(mathematics)
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In mathematics, a pairing function is a process to uniquely encode two natural numbers into a single natural number.Any pairing function can be used in set theory to prove that integers and rational numbers have the same cardinality as natural numbers.
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https://en.wikipedia.org/wiki/Cantor's_pairing_function
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In mathematics, a pairing is an R-bilinear map from the Cartesian product of two R-modules, where the underlying ring R is commutative.
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https://en.wikipedia.org/wiki/Pairing
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In mathematics, a palindromic prime (sometimes called a palprime) is a prime number that is also a palindromic number. Palindromicity depends on the base of the number system and its notational conventions, while primality is independent of such concerns. The first few decimal palindromic primes are: 2, 3, 5, 7, 11, 101, 131, 151, 181, 191, 313, 353, 373, 383, 727, 757, 787, 797, 919, 929, … (sequence A002385 in the OEIS)Except for 11, all palindromic primes have an odd number of digits, because the divisibility test for 11 tells us that every palindromic number with an even number of digits is a multiple of 11. It is not known if there are infinitely many palindromic primes in base 10. The largest known as of October 2021 is 101888529 - 10944264 - 1.which has 1,888,529 digits, and was found on 18 October 2021 by Ryan Propper and Serge Batalov. On the other hand, it is known that, for any base, almost all palindromic numbers are composite, i.e. the ratio between palindromic composites and all palindromes less than n tends to 1.
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https://en.wikipedia.org/wiki/Palindromic_prime
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In mathematics, a pandigital number is an integer that in a given base has among its significant digits each digit used in the base at least once. For example, 1234567890 (one billion two hundred thirty four million five hundred sixty seven thousand eight hundred ninety) is a pandigital number in base 10. The first few pandigital base 10 numbers are given by (sequence A171102 in the OEIS): 1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689The smallest pandigital number in a given base b is an integer of the form b b − 1 + ∑ d = 2 b − 1 d b b − 1 − d = b b − b ( b − 1 ) 2 + ( b − 1 ) × b b − 2 − 1 {\displaystyle b^{b-1}+\sum _{d=2}^{b-1}db^{b-1-d}={\frac {b^{b}-b}{(b-1)^{2}}}+(b-1)\times b^{b-2}-1} The following table lists the smallest pandigital numbers of a few selected bases. OEIS: A049363 gives the base 10 values for the first 18 bases.
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https://en.wikipedia.org/wiki/Pandigital_number
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In a trivial sense, all positive integers are pandigital in unary (or tallying). In binary, all integers are pandigital except for 0 and numbers of the form 2 n − 1 {\displaystyle 2^{n}-1} (the Mersenne numbers). The larger the base, the rarer pandigital numbers become, though one can always find runs of b x {\displaystyle b^{x}} consecutive pandigital numbers with redundant digits by writing all the digits of the base together (but not putting the zero first as the most significant digit) and adding x + 1 zeroes at the end as least significant digits.
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https://en.wikipedia.org/wiki/Pandigital_number
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Conversely, the smaller the base, the fewer pandigital numbers without redundant digits there are. 2 is the only such pandigital number in base 2, while there are more of these in base 10. Sometimes, the term is used to refer only to pandigital numbers with no redundant digits.
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https://en.wikipedia.org/wiki/Pandigital_number
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In some cases, a number might be called pandigital even if it doesn't have a zero as a significant digit, for example, 923456781 (these are sometimes referred to as "zeroless pandigital numbers"). No base 10 pandigital number can be a prime number if it doesn't have redundant digits. The sum of the digits 0 to 9 is 45, passing the divisibility rule for both 3 and 9.
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https://en.wikipedia.org/wiki/Pandigital_number
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The first base 10 pandigital prime is 10123457689; OEIS: A050288 lists more. For different reasons, redundant digits are also required for a pandigital number (in any base except unary) to also be a palindromic number in that base. The smallest pandigital palindromic number in base 10 is 1023456789876543201.
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https://en.wikipedia.org/wiki/Pandigital_number
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The largest pandigital number without redundant digits to be also a square number is 9814072356 = 990662. Two of the zeroless pandigital Friedman numbers are: 123456789 = ((86 + 2 × 7)5 − 91) / 34, and 987654321 = (8 × (97 + 6/2)5 + 1) / 34. A pandigital Friedman number without redundant digits is the square: 2170348569 = 465872 + (0 × 139).
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https://en.wikipedia.org/wiki/Pandigital_number
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While much of what has been said does not apply to Roman numerals, there are pandigital numbers: MCDXLIV, MCDXLVI, MCDLXIV, MCDLXVI, MDCXLIV, MDCXLVI, MDCLXIV, MDCLXVI. These, listed in OEIS: A105416, use each of the digits just once, while OEIS: A105417 has pandigital Roman numerals with repeats. Pandigital numbers are useful in fiction and in advertising. The Social Security number 987-65-4321 is a zeroless pandigital number reserved for use in advertising. Some credit card companies use pandigital numbers with redundant digits as fictitious credit card numbers (while others use strings of zeroes).
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https://en.wikipedia.org/wiki/Pandigital_number
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In mathematics, a pantachy or pantachie (from the Greek word πανταχη meaning everywhere) is a maximal totally ordered subset of a partially ordered set, especially a set of equivalence classes of sequences of real numbers. The term was introduced by du Bois-Reymond (1879, 1882) to mean a dense subset of an ordered set, and he also introduced "infinitary pantachies" to mean the ordered set of equivalence classes of real functions ordered by domination, but as Felix Hausdorff pointed out this is not a totally ordered set. Hausdorff (1907) redefined a pantachy to be a maximal totally ordered subset of this set.
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https://en.wikipedia.org/wiki/Pantachy
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In mathematics, a parabola is a plane curve which is mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One description of a parabola involves a point (the focus) and a line (the directrix). The focus does not lie on the directrix.
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https://en.wikipedia.org/wiki/Parabola
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The parabola is the locus of points in that plane that are equidistant from the directrix and the focus. Another description of a parabola is as a conic section, created from the intersection of a right circular conical surface and a plane parallel to another plane that is tangential to the conical surface.The line perpendicular to the directrix and passing through the focus (that is, the line that splits the parabola through the middle) is called the "axis of symmetry". The point where the parabola intersects its axis of symmetry is called the "vertex" and is the point where the parabola is most sharply curved.
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https://en.wikipedia.org/wiki/Parabola
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The distance between the vertex and the focus, measured along the axis of symmetry, is the "focal length". The "latus rectum" is the chord of the parabola that is parallel to the directrix and passes through the focus. Parabolas can open up, down, left, right, or in some other arbitrary direction.
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https://en.wikipedia.org/wiki/Parabola
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Any parabola can be repositioned and rescaled to fit exactly on any other parabola—that is, all parabolas are geometrically similar. Parabolas have the property that, if they are made of material that reflects light, then light that travels parallel to the axis of symmetry of a parabola and strikes its concave side is reflected to its focus, regardless of where on the parabola the reflection occurs. Conversely, light that originates from a point source at the focus is reflected into a parallel ("collimated") beam, leaving the parabola parallel to the axis of symmetry.
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https://en.wikipedia.org/wiki/Parabola
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The same effects occur with sound and other waves. This reflective property is the basis of many practical uses of parabolas. The parabola has many important applications, from a parabolic antenna or parabolic microphone to automobile headlight reflectors and the design of ballistic missiles. It is frequently used in physics, engineering, and many other areas.
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https://en.wikipedia.org/wiki/Parabola
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In mathematics, a paracompact space is a topological space in which every open cover has an open refinement that is locally finite. These spaces were introduced by Dieudonné (1944). Every compact space is paracompact. Every paracompact Hausdorff space is normal, and a Hausdorff space is paracompact if and only if it admits partitions of unity subordinate to any open cover.
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https://en.wikipedia.org/wiki/Paracompact_manifold
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Sometimes paracompact spaces are defined so as to always be Hausdorff. Every closed subspace of a paracompact space is paracompact. While compact subsets of Hausdorff spaces are always closed, this is not true for paracompact subsets.
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https://en.wikipedia.org/wiki/Paracompact_manifold
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A space such that every subspace of it is a paracompact space is called hereditarily paracompact. This is equivalent to requiring that every open subspace be paracompact. The notion of paracompact space is also studied in pointless topology, where it is more well-behaved.
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https://en.wikipedia.org/wiki/Paracompact_manifold
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For example, the product of any number of paracompact locales is a paracompact locale, but the product of two paracompact spaces may not be paracompact. Compare this to Tychonoff's theorem, which states that the product of any collection of compact topological spaces is compact. However, the product of a paracompact space and a compact space is always paracompact. Every metric space is paracompact. A topological space is metrizable if and only if it is a paracompact and locally metrizable Hausdorff space.
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https://en.wikipedia.org/wiki/Paracompact_manifold
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In mathematics, a parallelization of a manifold M {\displaystyle M\,} of dimension n is a set of n global smooth linearly independent vector fields.
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https://en.wikipedia.org/wiki/Parallelization_(mathematics)
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In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, called parametric curve and parametric surface, respectively. In such cases, the equations are collectively called a parametric representation, or parametric system, or parameterization (alternatively spelled as parametrisation) of the object.For example, the equations x = cos t y = sin t {\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}} form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors: ( x , y ) = ( cos t , sin t ) .
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https://en.wikipedia.org/wiki/Parametric_formula
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{\displaystyle (x,y)=(\cos t,\sin t).} Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).
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https://en.wikipedia.org/wiki/Parametric_formula
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Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.
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https://en.wikipedia.org/wiki/Parametric_formula
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In mathematics, a paramodular group is a special sort of arithmetic subgroup of the symplectic group. It is a generalization of the Siegel modular group, and has the same relation to polarized abelian varieties that the Siegel modular group has to principally polarized abelian varieties. It is the group of automorphisms of Z2n preserving a non-degenerate skew symmetric form. The name "paramodular group" is often used to mean one of several standard matrix representations of this group.
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https://en.wikipedia.org/wiki/Paramodular_group
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The corresponding group over the reals is called the parasymplectic group and is conjugate to a (real) symplectic group. A paramodular form is a Siegel modular form for a paramodular group. Paramodular groups were introduced by Conforto (1952) and named by Shimura (1958, section 8).
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https://en.wikipedia.org/wiki/Paramodular_group
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In mathematics, a paraproduct is a non-commutative bilinear operator acting on functions that in some sense is like the product of the two functions it acts on. According to Svante Janson and Jaak Peetre, in an article from 1988, "the name 'paraproduct' denotes an idea rather than a unique definition; several versions exist and can be used for the same purposes." The concept emerged in J.-M. Bony’s theory of paradifferential operators.This said, for a given operator Λ {\displaystyle \Lambda } to be defined as a paraproduct, it is normally required to satisfy the following properties: It should "reconstruct the product" in the sense that for any pair of functions ( f , g ) {\displaystyle (f,g)} in its domain, f g = Λ ( f , g ) + Λ ( g , f ) .
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https://en.wikipedia.org/wiki/Paraproduct
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{\displaystyle fg=\Lambda (f,g)+\Lambda (g,f).} For any appropriate functions f {\displaystyle f} and h {\displaystyle h} with h ( 0 ) = 0 {\displaystyle h(0)=0} , it is the case that h ( f ) = Λ ( f , h ′ ( f ) ) {\displaystyle h(f)=\Lambda (f,h'(f))} . It should satisfy some form of the Leibniz rule.A paraproduct may also be required to satisfy some form of Hölder's inequality.
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https://en.wikipedia.org/wiki/Paraproduct
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In mathematics, a paratopological group is a topological semigroup that is algebraically a group. In other words, it is a group G with a topology such that the group's product operation is a continuous function from G × G to G. This differs from the definition of a topological group in that the group inverse is not required to be continuous. As with topological groups, some authors require the topology to be Hausdorff.Compact paratopological groups are automatically topological groups. == References ==
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https://en.wikipedia.org/wiki/Paratopological_group
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In mathematics, a parent function is the core representation of a function type without manipulations such as translation and dilation. For example, for the family of quadratic functions having the general form y = a x 2 + b x + c , {\displaystyle y=ax^{2}+bx+c\,,} the simplest function is y = x 2 {\displaystyle y=x^{2}} .This is therefore the parent function of the family of quadratic equations. For linear and quadratic functions, the graph of any function can be obtained from the graph of the parent function by simple translations and stretches parallel to the axes.
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https://en.wikipedia.org/wiki/Parent_function
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For example, the graph of y = x2 − 4x + 7 can be obtained from the graph of y = x2 by translating +2 units along the X axis and +3 units along Y axis. This is because the equation can also be written as y − 3 = (x − 2)2. For many trigonometric functions, the parent function is usually a basic sin(x), cos(x), or tan(x).
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https://en.wikipedia.org/wiki/Parent_function
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For example, the graph of y = A sin(x) + B cos(x) can be obtained from the graph of y = sin(x) by translating it through an angle α along the positive X axis (where tan(α) = A⁄B), then stretching it parallel to the Y axis using a stretch factor R, where R2 = A2 + B2. This is because A sin(x) + B cos(x) can be written as R sin(x−α) (see List of trigonometric identities). The concept of parent function is less clear for polynomials of higher power because of the extra turning points, but for the family of n-degree polynomial functions for any given n, the parent function is sometimes taken as xn, or, to simplify further, x2 when n is even and x3 for odd n. Turning points may be established by differentiation to provide more detail of the graph.
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https://en.wikipedia.org/wiki/Parent_function
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In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.
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https://en.wikipedia.org/wiki/Partial_cyclic_order
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In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry. The partial derivative of a function f ( x , y , … ) {\displaystyle f(x,y,\dots )} with respect to the variable x {\displaystyle x} is variously denoted by It can be thought of as the rate of change of the function in the x {\displaystyle x} -direction. Sometimes, for z = f ( x , y , … ) {\displaystyle z=f(x,y,\ldots )} , the partial derivative of z {\displaystyle z} with respect to x {\displaystyle x} is denoted as ∂ z ∂ x .
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https://en.wikipedia.org/wiki/Partial_derivatives
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{\displaystyle {\tfrac {\partial z}{\partial x}}.} Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in: The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.
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https://en.wikipedia.org/wiki/Partial_derivatives
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In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similar to how x is thought of as an unknown number to be solved for in an algebraic equation like x2 − 3x + 2 = 0. However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers.
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https://en.wikipedia.org/wiki/Linear_partial_differential_equation
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Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000. Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering.
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https://en.wikipedia.org/wiki/Linear_partial_differential_equation
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For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation, etc.). They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology. Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise.
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https://en.wikipedia.org/wiki/Linear_partial_differential_equation
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As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.Ordinary differential equations form a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.
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https://en.wikipedia.org/wiki/Linear_partial_differential_equation
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In mathematics, a partial equivalence relation (often abbreviated as PER, in older literature also called restricted equivalence relation) is a homogeneous binary relation that is symmetric and transitive. If the relation is also reflexive, then the relation is an equivalence relation.
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https://en.wikipedia.org/wiki/Partial_equivalence_relation
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In mathematics, a partial function f from a set X to a set Y is a function from a subset S of X (possibly the whole X itself) to Y. The subset S, that is, the domain of f viewed as a function, is called the domain of definition or natural domain of f. If S equals X, that is, if f is defined on every element in X, then f is said to be a total function. More technically, a partial function is a binary relation over two sets that associates every element of the first set to at most one element of the second set; it is thus a functional binary relation. It generalizes the concept of a (total) function by not requiring every element of the first set to be associated to exactly one element of the second set. A partial function is often used when its exact domain of definition is not known or difficult to specify.
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https://en.wikipedia.org/wiki/Partial_functions
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This is the case in calculus, where, for example, the quotient of two functions is a partial function whose domain of definition cannot contain the zeros of the denominator. For this reason, in calculus, and more generally in mathematical analysis, a partial function is generally called simply a function.
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https://en.wikipedia.org/wiki/Partial_functions
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In computability theory, a general recursive function is a partial function from the integers to the integers; no algorithm can exist for deciding whether an arbitrary such function is in fact total. When arrow notation is used for functions, a partial function f {\displaystyle f} from X {\displaystyle X} to Y {\displaystyle Y} is sometimes written as f: X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} f: X ↛ Y , {\displaystyle f:X\nrightarrow Y,} or f: X ↪ Y . {\displaystyle f:X\hookrightarrow Y.} However, there is no general convention, and the latter notation is more commonly used for inclusion maps or embeddings.Specifically, for a partial function f: X ⇀ Y , {\displaystyle f:X\rightharpoonup Y,} and any x ∈ X , {\displaystyle x\in X,} one has either: f ( x ) = y ∈ Y {\displaystyle f(x)=y\in Y} (it is a single element in Y), or f ( x ) {\displaystyle f(x)} is undefined.For example, if f {\displaystyle f} is the square root function restricted to the integers f: Z → N , {\displaystyle f:\mathbb {Z} \to \mathbb {N} ,} defined by: f ( n ) = m {\displaystyle f(n)=m} if, and only if, m 2 = n , {\displaystyle m^{2}=n,} m ∈ N , n ∈ Z , {\displaystyle m\in \mathbb {N} ,n\in \mathbb {Z} ,} then f ( n ) {\displaystyle f(n)} is only defined if n {\displaystyle n} is a perfect square (that is, 0 , 1 , 4 , 9 , 16 , … {\displaystyle 0,1,4,9,16,\ldots } ). So f ( 25 ) = 5 {\displaystyle f(25)=5} but f ( 26 ) {\displaystyle f(26)} is undefined.
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https://en.wikipedia.org/wiki/Partial_functions
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In mathematics, a partial group algebra is an associative algebra related to the partial representations of a group.
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https://en.wikipedia.org/wiki/Partial_group_algebra
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In mathematics, a partial order or total order < on a set X {\displaystyle X} is said to be dense if, for all x {\displaystyle x} and y {\displaystyle y} in X {\displaystyle X} for which x < y {\displaystyle x
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https://en.wikipedia.org/wiki/Dense_relation
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In mathematics, a partially ordered set P is said to have Knaster's condition upwards (sometimes property (K)) if any uncountable subset A of P has an upwards-linked uncountable subset. An analogous definition applies to Knaster's condition downwards. The property is named after Polish mathematician Bronisław Knaster.
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https://en.wikipedia.org/wiki/Knaster's_condition
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Knaster's condition implies the countable chain condition (ccc), and it is sometimes used in conjunction with a weaker form of Martin's axiom, where the ccc requirement is replaced with Knaster's condition. Not unlike ccc, Knaster's condition is also sometimes used as a property of a topological space, in which case it means that the topology (as in, the family of all open sets) with inclusion satisfies the condition. Furthermore, assuming MA( ω 1 {\displaystyle \omega _{1}} ), ccc implies Knaster's condition, making the two equivalent.
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https://en.wikipedia.org/wiki/Knaster's_condition
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In mathematics, a partially ordered space (or pospace) is a topological space X {\displaystyle X} equipped with a closed partial order ≤ {\displaystyle \leq } , i.e. a partial order whose graph { ( x , y ) ∈ X 2 ∣ x ≤ y } {\displaystyle \{(x,y)\in X^{2}\mid x\leq y\}} is a closed subset of X 2 {\displaystyle X^{2}} . From pospaces, one can define dimaps, i.e. continuous maps between pospaces which preserve the order relation.
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https://en.wikipedia.org/wiki/Partially_ordered_space
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In mathematics, a partition matroid or partitional matroid is a matroid that is a direct sum of uniform matroids. It is defined over a base set in which the elements are partitioned into different categories. For each category, there is a capacity constraint - a maximum number of allowed elements from this category. The independent sets of a partition matroid are exactly the sets in which, for each category, the number of elements from this category is at most the category capacity.
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https://en.wikipedia.org/wiki/Partition_matroid
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In mathematics, a partition of a set is a grouping of its elements into non-empty subsets, in such a way that every element is included in exactly one subset. Every equivalence relation on a set defines a partition of this set, and every partition defines an equivalence relation. A set equipped with an equivalence relation or a partition is sometimes called a setoid, typically in type theory and proof theory.
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https://en.wikipedia.org/wiki/Set_partition
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In mathematics, a partition of an interval on the real line is a finite sequence x0, x1, x2, …, xn of real numbers such that a = x0 < x1 < x2 < … < xn = b.In other terms, a partition of a compact interval I is a strictly increasing sequence of numbers (belonging to the interval I itself) starting from the initial point of I and arriving at the final point of I. Every interval of the form is referred to as a subinterval of the partition x.
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https://en.wikipedia.org/wiki/Partition_of_an_interval
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In mathematics, a partition of unity of a topological space X {\displaystyle X} is a set R {\displaystyle R} of continuous functions from X {\displaystyle X} to the unit interval such that for every point x ∈ X {\displaystyle x\in X}: there is a neighbourhood of x {\displaystyle x} where all but a finite number of the functions of R {\displaystyle R} are 0, and the sum of all the function values at x {\displaystyle x} is 1, i.e., ∑ ρ ∈ R ρ ( x ) = 1. {\textstyle \sum _{\rho \in R}\rho (x)=1.} Partitions of unity are useful because they often allow one to extend local constructions to the whole space. They are also important in the interpolation of data, in signal processing, and the theory of spline functions.
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https://en.wikipedia.org/wiki/Partition_of_unity
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In mathematics, a path in a topological space X {\displaystyle X} is a continuous function from the closed unit interval {\displaystyle } into X . {\displaystyle X.} Paths play an important role in the fields of topology and mathematical analysis. For example, a topological space for which there exists a path connecting any two points is said to be path-connected.
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https://en.wikipedia.org/wiki/Arc_(topology)
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Any space may be broken up into path-connected components. The set of path-connected components of a space X {\displaystyle X} is often denoted π 0 ( X ) . {\displaystyle \pi _{0}(X).}
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https://en.wikipedia.org/wiki/Arc_(topology)
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One can also define paths and loops in pointed spaces, which are important in homotopy theory. If X {\displaystyle X} is a topological space with basepoint x 0 , {\displaystyle x_{0},} then a path in X {\displaystyle X} is one whose initial point is x 0 {\displaystyle x_{0}} . Likewise, a loop in X {\displaystyle X} is one that is based at x 0 {\displaystyle x_{0}} .
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https://en.wikipedia.org/wiki/Arc_(topology)
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