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In mathematics, a pedal curve of a given curve results from the orthogonal projection of a fixed point on the tangent lines of this curve. More precisely, for a plane curve C and a given fixed pedal point P, the pedal curve of C is the locus of points X so that the line PX is perpendicular to a tangent T to the curve passing through the point X. Conversely, at any point R on the curve C, let T be the tangent line at that point R; then there is a unique point X on the tangent T which forms with the pedal point P a line perpendicular to the tangent T (for the special case when the fixed point P lies on the tangent T, the points X and P coincide) – the pedal curve is the set of such points X, called the foot of the perpendicular to the tangent T from the fixed point P, as the variable point R ranges over the curve C. Complementing the pedal curve, there is a unique point Y on the line normal to C at R so that PY is perpendicular to the normal, so PXRY is a (possibly degenerate) rectangle. The locus of points Y is called the contrapedal curve. The orthotomic of a curve is its pedal magnified by a factor of 2 so that the center of similarity is P. This is locus of the reflection of P through the tangent line T. The pedal curve is the first in a series of curves C1, C2, C3, etc., where C1 is the pedal of C, C2 is the pedal of C1, and so on. In this scheme, C1 is known as the first positive pedal of C, C2 is the second positive pedal of C, and so on. Going the other direction, C is the first negative pedal of C1, the second negative pedal of C2, etc.
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https://en.wikipedia.org/wiki/Pedal_curve
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In mathematics, a percentage (from Latin per centum 'by a hundred') is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign (%), although the abbreviations pct., pct, and sometimes pc are also used. A percentage is a dimensionless number (pure number), primarily used for expressing proportions, but percent is nonetheless a unit of measurement in its orthography and usage.
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https://en.wikipedia.org/wiki/Percentage
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In mathematics, a perfect lattice (or perfect form) is a lattice in a Euclidean vector space, that is completely determined by the set S of its minimal vectors in the sense that there is only one positive definite quadratic form taking value 1 at all points of S. Perfect lattices were introduced by Korkine & Zolotareff (1877). A strongly perfect lattice is one whose minimal vectors form a spherical 4-design. This notion was introduced by Venkov (2001).
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https://en.wikipedia.org/wiki/Perfect_lattice
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Voronoi (1908) proved that a lattice is extreme if and only if it is both perfect and eutactic. The number of perfect lattices in dimensions 1, 2, 3, 4, 5, 6, 7, 8 is given by 1, 1, 1, 2, 3, 7, 33, 10916 (sequence A004026 in the OEIS). Conway & Sloane (1988) summarize the properties of perfect lattices of dimension up to 7. Sikirić, Schürmann & Vallentin (2007) verified that the list of 10916 perfect lattices in dimension 8 found by Martinet and others is complete. It was proven by Riener (2006) that only 2408 of these 10916 perfect lattices in dimension 8 are actually extreme lattices.
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https://en.wikipedia.org/wiki/Perfect_lattice
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In mathematics, a perfect magic cube is a magic cube in which not only the columns, rows, pillars, and main space diagonals, but also the cross section diagonals sum up to the cube's magic constant.Perfect magic cubes of order one are trivial; cubes of orders two to four can be proven not to exist, and cubes of orders five and six were first discovered by Walter Trump and Christian Boyer on November 13 and September 1, 2003, respectively. A perfect magic cube of order seven was given by A. H. Frost in 1866, and on March 11, 1875, an article was published in the Cincinnati Commercial newspaper on the discovery of a perfect magic cube of order 8 by Gustavus Frankenstein. Perfect magic cubes of orders nine and eleven have also been constructed. The first perfect cube of order 10 was constructed in 1988 (Li Wen, China).
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https://en.wikipedia.org/wiki/Perfect_magic_cube
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In mathematics, a perfect matrix is an m-by-n binary matrix that has no possible k-by-k submatrix K that satisfies the following conditions: k > 3 the row and column sums of K are each equal to b, where b ≥ 2 there exists no row of the (m − k)-by-k submatrix formed by the rows not included in K with a row sum greater than b.The following is an example of a K submatrix where k = 5 and b = 2: . {\displaystyle {\begin{bmatrix}1&1&0&0&0\\0&1&1&0&0\\0&0&1&1&0\\0&0&0&1&1\\1&0&0&0&1\end{bmatrix}}.} == References ==
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https://en.wikipedia.org/wiki/Perfect_matrix
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In mathematics, a perfect power is a natural number that is a product of equal natural factors, or, in other words, an integer that can be expressed as a square or a higher integer power of another integer greater than one. More formally, n is a perfect power if there exist natural numbers m > 1, and k > 1 such that mk = n. In this case, n may be called a perfect kth power. If k = 2 or k = 3, then n is called a perfect square or perfect cube, respectively. Sometimes 0 and 1 are also considered perfect powers (0k = 0 for any k > 0, 1k = 1 for any k).
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https://en.wikipedia.org/wiki/Perfect_power
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In mathematics, a period domain is a parameter space for a polarized Hodge structure. They can often be represented as the quotient of a Lie group by a compact subgroup.
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https://en.wikipedia.org/wiki/Period_domain
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In mathematics, a periodic sequence (sometimes called a cycle) is a sequence for which the same terms are repeated over and over: a1, a2, ..., ap, a1, a2, ..., ap, a1, a2, ..., ap, ...The number p of repeated terms is called the period (period).
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https://en.wikipedia.org/wiki/Periodic_sequence
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In mathematics, a periodic travelling wave (or wavetrain) is a periodic function of one-dimensional space that moves with constant speed. Consequently, it is a special type of spatiotemporal oscillation that is a periodic function of both space and time. Periodic travelling waves play a fundamental role in many mathematical equations, including self-oscillatory systems,excitable systems and reaction–diffusion–advection systems. Equations of these types are widely used as mathematical models of biology, chemistry and physics, and many examples in phenomena resembling periodic travelling waves have been found empirically. The mathematical theory of periodic travelling waves is most fully developed for partial differential equations, but these solutions also occur in a number of other types of mathematical system, including integrodifferential equations, integrodifference equations, coupled map lattices and cellular automataAs well as being important in their own right, periodic travelling waves are significant as the one-dimensional equivalent of spiral waves and target patterns in two-dimensional space, and of scroll waves in three-dimensional space.
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https://en.wikipedia.org/wiki/Periodic_travelling_wave
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In mathematics, a permutation group G acting on a non-empty finite set X is called primitive if G acts transitively on X and the only partitions the G-action preserves are the trivial partitions into either a single set or into |X| singleton sets. Otherwise, if G is transitive and G does preserve a nontrivial partition, G is called imprimitive. While primitive permutation groups are transitive, not all transitive permutation groups are primitive.
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https://en.wikipedia.org/wiki/Primitive_permutation_group
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The simplest example is the Klein four-group acting on the vertices of a square, which preserves the partition into diagonals. On the other hand, if a permutation group preserves only trivial partitions, it is transitive, except in the case of the trivial group acting on a 2-element set. This is because for a non-transitive action, either the orbits of G form a nontrivial partition preserved by G, or the group action is trivial, in which case all nontrivial partitions of X (which exists for |X| ≥ 3) are preserved by G. This terminology was introduced by Évariste Galois in his last letter, in which he used the French term équation primitive for an equation whose Galois group is primitive.
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https://en.wikipedia.org/wiki/Primitive_permutation_group
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In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself). The group of all permutations of a set M is the symmetric group of M, often written as Sym(M). The term permutation group thus means a subgroup of the symmetric group. If M = {1, 2, ..., n} then Sym(M) is usually denoted by Sn, and may be called the symmetric group on n letters.
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https://en.wikipedia.org/wiki/Permutation_group
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By Cayley's theorem, every group is isomorphic to some permutation group. The way in which the elements of a permutation group permute the elements of the set is called its group action. Group actions have applications in the study of symmetries, combinatorics and many other branches of mathematics, physics and chemistry.
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https://en.wikipedia.org/wiki/Permutation_group
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In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set.Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set {1, 2, 3}, namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set.
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https://en.wikipedia.org/wiki/Circular_notation
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Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics and in many other fields of science.
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https://en.wikipedia.org/wiki/Circular_notation
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In computer science, they are used for analyzing sorting algorithms; in quantum physics, for describing states of particles; and in biology, for describing RNA sequences. The number of permutations of n distinct objects is n factorial, usually written as n!, which means the product of all positive integers less than or equal to n. Technically, a permutation of a set S is defined as a bijection from S to itself. That is, it is a function from S to S for which every element occurs exactly once as an image value.
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https://en.wikipedia.org/wiki/Circular_notation
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This is related to the rearrangement of the elements of S in which each element s is replaced by the corresponding f(s). For example, the permutation (3, 1, 2) mentioned above is described by the function α {\displaystyle \alpha } defined as α ( 1 ) = 3 , α ( 2 ) = 1 , α ( 3 ) = 2 {\displaystyle \alpha (1)=3,\quad \alpha (2)=1,\quad \alpha (3)=2} .The collection of all permutations of a set form a group called the symmetric group of the set. The group operation is the composition (performing two given rearrangements in succession), which results in another rearrangement.
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https://en.wikipedia.org/wiki/Circular_notation
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As properties of permutations do not depend on the nature of the set elements, it is often the permutations of the set { 1 , 2 , … , n } {\displaystyle \{1,2,\ldots ,n\}} that are considered for studying permutations. In elementary combinatorics, the k-permutations, or partial permutations, are the ordered arrangements of k distinct elements selected from a set. When k is equal to the size of the set, these are the permutations of the set.
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https://en.wikipedia.org/wiki/Circular_notation
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In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map x ↦ g ( x ) {\displaystyle x\mapsto g(x)} is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function. In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms.
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https://en.wikipedia.org/wiki/Permutation_polynomial
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In mathematics, a phase line is a diagram that shows the qualitative behaviour of an autonomous ordinary differential equation in a single variable, d y d x = f ( y ) {\displaystyle {\tfrac {dy}{dx}}=f(y)} . The phase line is the 1-dimensional form of the general n {\displaystyle n} -dimensional phase space, and can be readily analyzed.
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https://en.wikipedia.org/wiki/Phase_line_(mathematics)
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In mathematics, a phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Each set of initial conditions is represented by a different point or curve. Phase portraits are an invaluable tool in studying dynamical systems. They consist of a plot of typical trajectories in the phase space.
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https://en.wikipedia.org/wiki/Phase_portrait
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This reveals information such as whether an attractor, a repellor or limit cycle is present for the chosen parameter value. The concept of topological equivalence is important in classifying the behaviour of systems by specifying when two different phase portraits represent the same qualitative dynamic behavior. An attractor is a stable point which is also called a "sink".
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https://en.wikipedia.org/wiki/Phase_portrait
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The repeller is considered as an unstable point, which is also known as a "source". A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space. The axes are of state variables.
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https://en.wikipedia.org/wiki/Phase_portrait
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In mathematics, a piecewise algebraic space is a generalization of a semialgebraic set, introduced by Maxim Kontsevich and Yan Soibelman. The motivation was for the proof of Deligne's conjecture on Hochschild cohomology. Robert Hardt, Pascal Lambrechts, Victor Turchin, and Ismar Volić later developed the theory.
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https://en.wikipedia.org/wiki/Piecewise_algebraic_space
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In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pass from chart to chart in it by piecewise linear functions. This is slightly stronger than the topological notion of a triangulation.An isomorphism of PL manifolds is called a PL homeomorphism.
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https://en.wikipedia.org/wiki/Piecewise-linear_manifold
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In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Piecewise definition is actually a way of expressing the function, rather than a characteristic of the function itself. A distinct, but related notion is that of a property holding piecewise for a function, used when the domain can be divided into intervals on which the property holds. Unlike for the notion above, this is actually a property of the function itself. A piecewise linear function (which happens to be also continuous) is depicted as an example.
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https://en.wikipedia.org/wiki/Piecewise_smooth
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In mathematics, a planar Riemann surface (or schlichtartig Riemann surface) is a Riemann surface sharing the topological properties of a connected open subset of the Riemann sphere. They are characterized by the topological property that the complement of every closed Jordan curve in the Riemann surface has two connected components. An equivalent characterization is the differential geometric property that every closed differential 1-form of compact support is exact. Every simply connected Riemann surface is planar. The class of planar Riemann surfaces was studied by Koebe who proved in 1910, as a generalization of the uniformization theorem, that every such surface is conformally equivalent to either the Riemann sphere or the complex plane with slits parallel to the real axis removed.
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https://en.wikipedia.org/wiki/Planar_Riemann_surface
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In mathematics, a planar lamina (or plane lamina) is a figure representing a thin, usually uniform, flat layer of the solid. It serves also as an idealized model of a planar cross section of a solid body in integration. Planar laminas can be used to determine moments of inertia, or center of mass of flat figures, as well as an aid in corresponding calculations for 3D bodies.
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https://en.wikipedia.org/wiki/Planar_lamina
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In mathematics, a plane curve is a curve in a plane that may be either a Euclidean plane, an affine plane or a projective plane. The most frequently studied cases are smooth plane curves (including piecewise smooth plane curves), and algebraic plane curves. Plane curves also include the Jordan curves (curves that enclose a region of the plane but need not be smooth) and the graphs of continuous functions.
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https://en.wikipedia.org/wiki/Plane_curve
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In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the whole space. Many fundamental tasks in mathematics, geometry, trigonometry, graph theory, and graphing are performed in a two-dimensional or planar space.
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https://en.wikipedia.org/wiki/Two-dimensional_space
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In mathematics, a point process is a random element whose values are "point patterns" on a set S. While in the exact mathematical definition a point pattern is specified as a locally finite counting measure, it is sufficient for more applied purposes to think of a point pattern as a countable subset of S that has no limit points.
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https://en.wikipedia.org/wiki/Point_process
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In mathematics, a point source is a singularity from which flux or flow is emanating. Although singularities such as this do not exist in the observable universe, mathematical point sources are often used as approximations to reality in physics and other fields.
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https://en.wikipedia.org/wiki/Point_source
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In mathematics, a point x is called an isolated point of a subset S (in a topological space X) if x is an element of S and there exists a neighborhood of x that does not contain any other points of S. This is equivalent to saying that the singleton {x} is an open set in the topological space S (considered as a subspace of X). Another equivalent formulation is: an element x of S is an isolated point of S if and only if it is not a limit point of S. If the space X is a metric space, for example a Euclidean space, then an element x of S is an isolated point of S if there exists an open ball around x that contains only finitely many elements of S.
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https://en.wikipedia.org/wiki/Discrete_subset
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In mathematics, a pointed set (also based set or rooted set) is an ordered pair ( X , x 0 ) {\displaystyle (X,x_{0})} where X {\displaystyle X} is a set and x 0 {\displaystyle x_{0}} is an element of X {\displaystyle X} called the base point, also spelled basepoint. : 10–11 Maps between pointed sets ( X , x 0 ) {\displaystyle (X,x_{0})} and ( Y , y 0 ) {\displaystyle (Y,y_{0})} —called based maps, pointed maps, or point-preserving maps—are functions from X {\displaystyle X} to Y {\displaystyle Y} that map one basepoint to another, i.e. maps f: X → Y {\displaystyle f\colon X\to Y} such that f ( x 0 ) = y 0 {\displaystyle f(x_{0})=y_{0}} . Based maps are usually denoted f: ( X , x 0 ) → ( Y , y 0 ) {\displaystyle f\colon (X,x_{0})\to (Y,y_{0})} .Pointed sets are very simple algebraic structures. In the sense of universal algebra, a pointed set is a set X {\displaystyle X} together with a single nullary operation ∗: X 0 → X , {\displaystyle *:X^{0}\to X,} which picks out the basepoint.
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https://en.wikipedia.org/wiki/Pointed_set
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Pointed maps are the homomorphisms of these algebraic structures. The class of all pointed sets together with the class of all based maps form a category. In this category the pointed singleton sets ( { a } , a ) {\displaystyle (\{a\},a)} are initial objects and terminal objects, i.e. they are zero objects.
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https://en.wikipedia.org/wiki/Pointed_set
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: 226 There is a faithful functor from pointed sets to usual sets, but it is not full and these categories are not equivalent. : 44 In particular, the empty set is not a pointed set because it has no element that can be chosen as the basepoint.The category of pointed sets and based maps is equivalent to the category of sets and partial functions. The base point serves as a "default value" for those arguments for which the partial function is not defined.
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https://en.wikipedia.org/wiki/Pointed_set
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One textbook notes that "This formal completion of sets and partial maps by adding 'improper', 'infinite' elements was reinvented many times, in particular, in topology (one-point compactification) and in theoretical computer science. "The category of pointed sets and pointed maps is isomorphic to the coslice category ( 1 ↓ S e t {\displaystyle \mathbf {1} \downarrow \mathbf {Set} } ), where 1 {\displaystyle \mathbf {1} } is (a functor that selects) a singleton set, and S e t {\displaystyle \scriptstyle {\mathbf {Set} }} (the identity functor of) the category of sets. : 46 This coincides with the algebraic characterization, since the unique map 1 → 1 {\displaystyle \mathbf {1} \to \mathbf {1} } extends the commutative triangles defining arrows of the coslice category to form the commutative squares defining homomorphisms of the algebras.
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https://en.wikipedia.org/wiki/Pointed_set
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The category of pointed sets and pointed maps has both products and coproducts, but it is not a distributive category. It is also an example of a category where 0 × A {\displaystyle 0\times A} is not isomorphic to 0 {\displaystyle 0} .Many algebraic structures are pointed sets in a rather trivial way. For example, groups are pointed sets by choosing the identity element as the basepoint, so that group homomorphisms are point-preserving maps. : 24 This observation can be restated in category theoretic terms as the existence of a forgetful functor from groups to pointed sets. : 582 A pointed set may be seen as a pointed space under the discrete topology or as a vector space over the field with one element.As "rooted set" the notion naturally appears in the study of antimatroids and transportation polytopes.
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https://en.wikipedia.org/wiki/Pointed_set
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In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x 0 , {\displaystyle x_{0},} that remains unchanged during subsequent discussion, and is kept track of during all operations. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i.e., a map f {\displaystyle f} between a pointed space X {\displaystyle X} with basepoint x 0 {\displaystyle x_{0}} and a pointed space Y {\displaystyle Y} with basepoint y 0 {\displaystyle y_{0}} is a based map if it is continuous with respect to the topologies of X {\displaystyle X} and Y {\displaystyle Y} and if f ( x 0 ) = y 0 . {\displaystyle f\left(x_{0}\right)=y_{0}.}
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https://en.wikipedia.org/wiki/Based_space
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This is usually denoted f: ( X , x 0 ) → ( Y , y 0 ) . {\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).}
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https://en.wikipedia.org/wiki/Based_space
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Pointed spaces are important in algebraic topology, particularly in homotopy theory, where many constructions, such as the fundamental group, depend on a choice of basepoint. The pointed set concept is less important; it is anyway the case of a pointed discrete space. Pointed spaces are often taken as a special case of the relative topology, where the subset is a single point. Thus, much of homotopy theory is usually developed on pointed spaces, and then moved to relative topologies in algebraic topology.
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https://en.wikipedia.org/wiki/Based_space
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In mathematics, a polar action is a proper and isometric action of a Lie group G on a complete Riemannian manifold M for which there exists a complete submanifold Σ that meets all the orbits and meets them always orthogonally; such a submanifold is called a section. A section is necessarily totally geodesic. If the sections of a polar action are flat with respect to the induced metric, then the action is called hyperpolar. In the case of linear orthogonal actions on Euclidean spaces, polar actions are called polar representations. The isotropy representations of Riemannian symmetric spaces are basic examples of polar representations. Conversely, Dadok has classified polar representations of compact Lie groups on Euclidean spaces, and it follows from his classification that such a representation has the same orbits as the isotropy representation of a symmetric space.
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https://en.wikipedia.org/wiki/Polar_action
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In mathematics, a polyadic space is a topological space that is the image under a continuous function of a topological power of an Alexandroff one-point compactification of a discrete space.
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https://en.wikipedia.org/wiki/Polyadic_space
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In mathematics, a polycyclic group is a solvable group that satisfies the maximal condition on subgroups (that is, every subgroup is finitely generated). Polycyclic groups are finitely presented, which makes them interesting from a computational point of view.
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https://en.wikipedia.org/wiki/Polycyclic_group
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In mathematics, a polygonal number is a number represented as dots or pebbles arranged in the shape of a regular polygon. The dots are thought of as alphas (units). These are one type of 2-dimensional figurate numbers.
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https://en.wikipedia.org/wiki/Polygonal_number
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In mathematics, a polyhedral complex is a set of polyhedra in a real vector space that fit together in a specific way. Polyhedral complexes generalize simplicial complexes and arise in various areas of polyhedral geometry, such as tropical geometry, splines and hyperplane arrangements.
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https://en.wikipedia.org/wiki/Polyhedral_complex
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In mathematics, a polylogarithmic function in n is a polynomial in the logarithm of n, a k ( log n ) k + a k − 1 ( log n ) k − 1 + ⋯ + a 1 ( log n ) + a 0 . {\displaystyle a_{k}(\log n)^{k}+a_{k-1}(\log n)^{k-1}+\cdots +a_{1}(\log n)+a_{0}.} The notation logkn is often used as a shorthand for (log n)k, analogous to sin2θ for (sin θ)2.
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https://en.wikipedia.org/wiki/Polylogarithmic_function
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In computer science, polylogarithmic functions occur as the order of time or memory used by some algorithms (e.g., "it has polylogarithmic order"), such as in the definition of QPTAS (see PTAS). All polylogarithmic functions of n are o(nε) for every exponent ε > 0 (for the meaning of this symbol, see small o notation), that is, a polylogarithmic function grows more slowly than any positive exponent. This observation is the basis for the soft O notation Õ(n).
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https://en.wikipedia.org/wiki/Polylogarithmic_function
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In mathematics, a polymatroid is a polytope associated with a submodular function. The notion was introduced by Jack Edmonds in 1970. It is also described as the multiset analogue of the matroid.
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https://en.wikipedia.org/wiki/Polymatroid
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In mathematics, a polynomial Diophantine equation is an indeterminate polynomial equation for which one seeks solutions restricted to be polynomials in the indeterminate. A Diophantine equation, in general, is one where the solutions are restricted to some algebraic system, typically integers. (In another usage ) Diophantine refers to the Hellenistic mathematician of the 3rd century, Diophantus of Alexandria, who made initial studies of integer Diophantine equations. An important type of polynomial Diophantine equations takes the form: s a + t b = c {\displaystyle sa+tb=c} where a, b, and c are known polynomials, and we wish to solve for s and t. A simple example (and a solution) is: s ( x 2 + 1 ) + t ( x 3 + 1 ) = 2 x {\displaystyle s(x^{2}+1)+t(x^{3}+1)=2x} s = − x 3 − x 2 + x {\displaystyle s=-x^{3}-x^{2}+x} t = x 2 + x .
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https://en.wikipedia.org/wiki/Polynomial_Diophantine_equation
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{\displaystyle t=x^{2}+x.} A necessary and sufficient condition for a polynomial Diophantine equation to have a solution is for c to be a multiple of the GCD of a and b. In the example above, the GCD of a and b was 1, so solutions would exist for any value of c. Solutions to polynomial Diophantine equations are not unique. Any multiple of a b {\displaystyle ab} (say r a b {\displaystyle rab} ) can be used to transform s {\displaystyle s} and t {\displaystyle t} into another solution s ′ = s + r b {\displaystyle s'=s+rb} t ′ = t − r a {\displaystyle t'=t-ra}: ( s + r b ) a + ( t − r a ) b = c . {\displaystyle (s+rb)a+(t-ra)b=c.} Some polynomial Diophantine equations can be solved using the extended Euclidean algorithm, which works as well with polynomials as it does with integers.
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https://en.wikipedia.org/wiki/Polynomial_Diophantine_equation
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In mathematics, a polynomial P(X) over a given field K is separable if its roots are distinct in an algebraic closure of K, that is, the number of distinct roots is equal to the degree of the polynomial.This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts coincide. In general, P(X) is separable if and only if it is square-free over any field that contains K, which holds if and only if P(X) is coprime to its formal derivative D P(X).
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https://en.wikipedia.org/wiki/Separable_polynomial
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In mathematics, a polynomial decomposition expresses a polynomial f as the functional composition g ∘ h {\displaystyle g\circ h} of polynomials g and h, where g and h have degree greater than 1; it is an algebraic functional decomposition. Algorithms are known for decomposing univariate polynomials in polynomial time. Polynomials which are decomposable in this way are composite polynomials; those which are not are indecomposable polynomials or sometimes prime polynomials (not to be confused with irreducible polynomials, which cannot be factored into products of polynomials). The degree of a composite polynomial is always a composite number, the product of the degrees of the composed polynomials. The rest of this article discusses only univariate polynomials; algorithms also exist for multivariate polynomials of arbitrary degree.
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https://en.wikipedia.org/wiki/Indecomposable_polynomial
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In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An example of a polynomial of a single indeterminate x is x2 − 4x + 7. An example with three indeterminates is x3 + 2xyz2 − yz + 1.
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https://en.wikipedia.org/wiki/Polynomial_multiplication
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Polynomials appear in many areas of mathematics and science. For example, they are used to form polynomial equations, which encode a wide range of problems, from elementary word problems to complicated scientific problems; they are used to define polynomial functions, which appear in settings ranging from basic chemistry and physics to economics and social science; they are used in calculus and numerical analysis to approximate other functions. In advanced mathematics, polynomials are used to construct polynomial rings and algebraic varieties, which are central concepts in algebra and algebraic geometry.
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https://en.wikipedia.org/wiki/Polynomial_multiplication
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In mathematics, a polynomial lemniscate or polynomial level curve is a plane algebraic curve of degree 2n, constructed from a polynomial p with complex coefficients of degree n. For any such polynomial p and positive real number c, we may define a set of complex numbers by | p ( z ) | = c . {\displaystyle |p(z)|=c.} This set of numbers may be equated to points in the real Cartesian plane, leading to an algebraic curve ƒ(x, y) = c2 of degree 2n, which results from expanding out p ( z ) p ¯ ( z ¯ ) {\displaystyle p(z){\bar {p}}({\bar {z}})} in terms of z = x + iy. When p is a polynomial of degree 1 then the resulting curve is simply a circle whose center is the zero of p. When p is a polynomial of degree 2 then the curve is a Cassini oval.
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https://en.wikipedia.org/wiki/Polynomial_lemniscate
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In mathematics, a polynomial matrix or matrix of polynomials is a matrix whose elements are univariate or multivariate polynomials. Equivalently, a polynomial matrix is a polynomial whose coefficients are matrices. A univariate polynomial matrix P of degree p is defined as: P = ∑ n = 0 p A ( n ) x n = A ( 0 ) + A ( 1 ) x + A ( 2 ) x 2 + ⋯ + A ( p ) x p {\displaystyle P=\sum _{n=0}^{p}A(n)x^{n}=A(0)+A(1)x+A(2)x^{2}+\cdots +A(p)x^{p}} where A ( i ) {\displaystyle A(i)} denotes a matrix of constant coefficients, and A ( p ) {\displaystyle A(p)} is non-zero.
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https://en.wikipedia.org/wiki/Polynomial_matrix
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An example 3×3 polynomial matrix, degree 2: P = ( 1 x 2 x 0 2 x 2 3 x + 2 x 2 − 1 0 ) = ( 1 0 0 0 0 2 2 − 1 0 ) + ( 0 0 1 0 2 0 3 0 0 ) x + ( 0 1 0 0 0 0 0 1 0 ) x 2 . {\displaystyle P={\begin{pmatrix}1&x^{2}&x\\0&2x&2\\3x+2&x^{2}-1&0\end{pmatrix}}={\begin{pmatrix}1&0&0\\0&0&2\\2&-1&0\end{pmatrix}}+{\begin{pmatrix}0&0&1\\0&2&0\\3&0&0\end{pmatrix}}x+{\begin{pmatrix}0&1&0\\0&0&0\\0&1&0\end{pmatrix}}x^{2}.} We can express this by saying that for a ring R, the rings M n ( R ) {\displaystyle M_{n}(R)} and ( M n ( R ) ) {\displaystyle (M_{n}(R))} are isomorphic.
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https://en.wikipedia.org/wiki/Polynomial_matrix
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In mathematics, a polynomial sequence is a sequence of polynomials indexed by the nonnegative integers 0, 1, 2, 3, ..., in which each index is equal to the degree of the corresponding polynomial. Polynomial sequences are a topic of interest in enumerative combinatorics and algebraic combinatorics, as well as applied mathematics.
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https://en.wikipedia.org/wiki/Polynomial_sequence
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In mathematics, a polynomial sequence { p n ( z ) } {\displaystyle \{p_{n}(z)\}} has a generalized Appell representation if the generating function for the polynomials takes on a certain form: K ( z , w ) = A ( w ) Ψ ( z g ( w ) ) = ∑ n = 0 ∞ p n ( z ) w n {\displaystyle K(z,w)=A(w)\Psi (zg(w))=\sum _{n=0}^{\infty }p_{n}(z)w^{n}} where the generating function or kernel K ( z , w ) {\displaystyle K(z,w)} is composed of the series A ( w ) = ∑ n = 0 ∞ a n w n {\displaystyle A(w)=\sum _{n=0}^{\infty }a_{n}w^{n}\quad } with a 0 ≠ 0 {\displaystyle a_{0}\neq 0} and Ψ ( t ) = ∑ n = 0 ∞ Ψ n t n {\displaystyle \Psi (t)=\sum _{n=0}^{\infty }\Psi _{n}t^{n}\quad } and all Ψ n ≠ 0 {\displaystyle \Psi _{n}\neq 0} and g ( w ) = ∑ n = 1 ∞ g n w n {\displaystyle g(w)=\sum _{n=1}^{\infty }g_{n}w^{n}\quad } with g 1 ≠ 0. {\displaystyle g_{1}\neq 0.} Given the above, it is not hard to show that p n ( z ) {\displaystyle p_{n}(z)} is a polynomial of degree n {\displaystyle n} . Boas–Buck polynomials are a slightly more general class of polynomials.
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https://en.wikipedia.org/wiki/Generalized_Appell_polynomials
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In mathematics, a polynomial sequence, i.e., a sequence of polynomials indexed by non-negative integers { 0 , 1 , 2 , 3 , … } {\textstyle \left\{0,1,2,3,\ldots \right\}} in which the index of each polynomial equals its degree, is said to be of binomial type if it satisfies the sequence of identities p n ( x + y ) = ∑ k = 0 n ( n k ) p k ( x ) p n − k ( y ) . {\displaystyle p_{n}(x+y)=\sum _{k=0}^{n}{n \choose k}\,p_{k}(x)\,p_{n-k}(y).} Many such sequences exist.
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https://en.wikipedia.org/wiki/Binomial_type
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The set of all such sequences forms a Lie group under the operation of umbral composition, explained below. Every sequence of binomial type may be expressed in terms of the Bell polynomials. Every sequence of binomial type is a Sheffer sequence (but most Sheffer sequences are not of binomial type). Polynomial sequences put on firm footing the vague 19th century notions of umbral calculus.
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https://en.wikipedia.org/wiki/Binomial_type
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In mathematics, a polynomial transformation consists of computing the polynomial whose roots are a given function of the roots of a polynomial. Polynomial transformations such as Tschirnhaus transformations are often used to simplify the solution of algebraic equations.
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https://en.wikipedia.org/wiki/Polynomial_transformation
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In mathematics, a polynomially reflexive space is a Banach space X, on which the space of all polynomials in each degree is a reflexive space. Given a multilinear functional Mn of degree n (that is, Mn is n-linear), we can define a polynomial p as p ( x ) = M n ( x , … , x ) {\displaystyle p(x)=M_{n}(x,\dots ,x)} (that is, applying Mn on the diagonal) or any finite sum of these. If only n-linear functionals are in the sum, the polynomial is said to be n-homogeneous. We define the space Pn as consisting of all n-homogeneous polynomials. The P1 is identical to the dual space, and is thus reflexive for all reflexive X. This implies that reflexivity is a prerequisite for polynomial reflexivity.
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https://en.wikipedia.org/wiki/Polynomially_reflexive_space
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In mathematics, a polyphase sequence is a sequence whose terms are complex roots of unity: a n = e i 2 π q x n {\displaystyle a_{n}=e^{i{\frac {2\pi }{q}}x_{n}}\,} where xn is an integer. Polyphase sequences are an important class of sequences and play important roles in synchronizing sequence design.
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https://en.wikipedia.org/wiki/Polyphase_sequence
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In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous sets are not equivalent to either meagre sets or measure zero sets, as shown below.
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https://en.wikipedia.org/wiki/Porous_set
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In mathematics, a positive (or signed) measure μ defined on a σ-algebra Σ of subsets of a set X is called a finite measure if μ(X) is a finite real number (rather than ∞). A set A in Σ is of finite measure if μ(A) < ∞. The measure μ is called σ-finite if X is a countable union of measurable sets each with finite measure.
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https://en.wikipedia.org/wiki/Sigma_finite_measure
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A set in a measure space is said to have σ-finite measure if it is a countable union of measurable sets with finite measure. A measure being σ-finite is a weaker condition than being finite, i.e. all finite measures are σ-finite but there are (many) σ-finite measures that are not finite. A different but related notion that should not be confused with σ-finiteness is s-finiteness.
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https://en.wikipedia.org/wiki/Sigma_finite_measure
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In mathematics, a positive harmonic function on the unit disc in the complex numbers is characterized as the Poisson integral of a finite positive measure on the circle. This result, the Herglotz-Riesz representation theorem, was proved independently by Gustav Herglotz and Frigyes Riesz in 1911. It can be used to give a related formula and characterization for any holomorphic function on the unit disc with positive real part. Such functions had already been characterized in 1907 by Constantin Carathéodory in terms of the positive definiteness of their Taylor coefficients.
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https://en.wikipedia.org/wiki/Herglotz_representation_theorem
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In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively non-negative) on that set. Precisely, Let p be a polynomial in n variables with real coefficients and let S be a subset of the n-dimensional Euclidean space ℝn. We say that: p is positive on S if p(x) > 0 for every x in S. p is non-negative on S if p(x) ≥ 0 for every x in S.
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https://en.wikipedia.org/wiki/Positive_polynomial
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In mathematics, a positive-definite function is, depending on the context, either of two types of function.
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https://en.wikipedia.org/wiki/Positive-semidefinite_function
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In mathematics, a power of three is a number of the form 3n where n is an integer, that is, the result of exponentiation with number three as the base and integer n as the exponent. In a context where only integers are considered, n is restricted to non-negative values, so there are 1, 3, and 3 multiplied by itself a certain number of times. The first ten powers of 3 for non-negative values of n are: 1, 3, 9, 27, 81, 243, 729, 2187, 6561, 19683, ... (sequence A000244 in the OEIS)
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https://en.wikipedia.org/wiki/Power_of_three
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In mathematics, a power series (in one variable) is an infinite series of the form where an represents the coefficient of the nth term and c is a constant. Power series are useful in mathematical analysis, where they arise as Taylor series of infinitely differentiable functions. In fact, Borel's theorem implies that every power series is the Taylor series of some smooth function.
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https://en.wikipedia.org/wiki/Power_series
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In many situations, c (the center of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form Beyond their role in mathematical analysis, power series also occur in combinatorics as generating functions (a kind of formal power series) and in electronic engineering (under the name of the Z-transform). The familiar decimal notation for real numbers can also be viewed as an example of a power series, with integer coefficients, but with the argument x fixed at 1⁄10. In number theory, the concept of p-adic numbers is also closely related to that of a power series.
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https://en.wikipedia.org/wiki/Power_series
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In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space. The notion of pre-Lie algebra has been introduced by Murray Gerstenhaber in his work on deformations of algebras. Pre-Lie algebras have been considered under some other names, among which one can cite left-symmetric algebras, right-symmetric algebras or Vinberg algebras.
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https://en.wikipedia.org/wiki/Pre-Lie_algebra
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In mathematics, a prehomogeneous vector space (PVS) is a finite-dimensional vector space V together with a subgroup G of the general linear group GL(V) such that G has an open dense orbit in V. Prehomogeneous vector spaces were introduced by Mikio Sato in 1970 and have many applications in geometry, number theory and analysis, as well as representation theory. The irreducible PVS were classified by Sato and Tatsuo Kimura in 1977, up to a transformation known as "castling". They are subdivided into two types, according to whether the semisimple part of G acts prehomogeneously or not. If it doesn't then there is a homogeneous polynomial on V which is invariant under the semisimple part of G.
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https://en.wikipedia.org/wiki/Prehomogeneous_vector_space
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In mathematics, a preordered class is a class equipped with a preorder.
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https://en.wikipedia.org/wiki/Preordered_class
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In mathematics, a preradical is a subfunctor of the identity functor in the category of left modules over a ring with identity. The class of all preradicals over R-mod is denoted by R-pr. There is a natural order in R-pr given by, for any two preradicals σ {\displaystyle \sigma } and τ {\displaystyle \tau } , σ ≤ τ {\displaystyle \sigma \leq \tau } , if for any left R-module M, σ M ≤ τ M {\displaystyle \sigma M\leq \tau M} . With this order R-pr becomes a big lattice.
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https://en.wikipedia.org/wiki/Preradical
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In mathematics, a presentation is one method of specifying a group. A presentation of a group G comprises a set S of generators—so that every element of the group can be written as a product of powers of some of these generators—and a set R of relations among those generators. We then say G has presentation ⟨ S ∣ R ⟩ . {\displaystyle \langle S\mid R\rangle .}
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https://en.wikipedia.org/wiki/Finitely_presented_group
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Informally, G has the above presentation if it is the "freest group" generated by S subject only to the relations R. Formally, the group G is said to have the above presentation if it is isomorphic to the quotient of a free group on S by the normal subgroup generated by the relations R. As a simple example, the cyclic group of order n has the presentation ⟨ a ∣ a n = 1 ⟩ , {\displaystyle \langle a\mid a^{n}=1\rangle ,} where 1 is the group identity. This may be written equivalently as ⟨ a ∣ a n ⟩ , {\displaystyle \langle a\mid a^{n}\rangle ,} thanks to the convention that terms that do not include an equals sign are taken to be equal to the group identity. Such terms are called relators, distinguishing them from the relations that do include an equals sign. Every group has a presentation, and in fact many different presentations; a presentation is often the most compact way of describing the structure of the group. A closely related but different concept is that of an absolute presentation of a group.
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https://en.wikipedia.org/wiki/Finitely_presented_group
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In mathematics, a primary cyclic group is a group that is both a cyclic group and a p-primary group for some prime number p. That is, it is a cyclic group of order pm, Cpm, for some prime number p, and natural number m. Every finite abelian group G may be written as a finite direct sum of primary cyclic groups, as stated in the fundamental theorem of finite abelian groups: G = ⨁ 1 ≤ i ≤ n C p i m i . {\displaystyle G=\bigoplus _{1\leq i\leq n}\mathrm {C} _{{p_{i}}^{m_{i}}}.} This expression is essentially unique: there is a bijection between the sets of groups in two such expressions, which maps each group to one that is isomorphic.
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https://en.wikipedia.org/wiki/Primary_cyclic_group
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Primary cyclic groups are characterised among finitely generated abelian groups as the torsion groups that cannot be expressed as a direct sum of two non-trivial groups. As such they, along with the group of integers, form the building blocks of finitely generated abelian groups. The subgroups of a primary cyclic group are linearly ordered by inclusion. The only other groups that have this property are the quasicyclic groups.
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https://en.wikipedia.org/wiki/Primary_cyclic_group
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In mathematics, a prime geodesic on a hyperbolic surface is a primitive closed geodesic, i.e. a geodesic which is a closed curve that traces out its image exactly once. Such geodesics are called prime geodesics because, among other things, they obey an asymptotic distribution law similar to the prime number theorem.
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https://en.wikipedia.org/wiki/Prime_geodesic
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In mathematics, a prime number p is called a Chen prime if p + 2 is either a prime or a product of two primes (also called a semiprime). The even number 2p + 2 therefore satisfies Chen's theorem. The Chen primes are named after Chen Jingrun, who proved in 1966 that there are infinitely many such primes. This result would also follow from the truth of the twin prime conjecture as the lower member of a pair of twin primes is by definition a Chen prime.
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https://en.wikipedia.org/wiki/Chen_prime
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The first few Chen primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 47, 53, 59, 67, 71, 83, 89, 101, … (sequence A109611 in the OEIS).The first few Chen primes that are not the lower member of a pair of twin primes are 2, 7, 13, 19, 23, 31, 37, 47, 53, 67, 83, 89, 109, 113, 127, ... (sequence A063637 in the OEIS).The first few non-Chen primes are 43, 61, 73, 79, 97, 103, 151, 163, 173, 193, 223, 229, 241, … (sequence A102540 in the OEIS).All of the supersingular primes are Chen primes. Rudolf Ondrejka discovered the following 3 × 3 magic square of nine Chen primes: As of March 2018, the largest known Chen prime is 2996863034895 × 21290000 − 1, with 388342 decimal digits. The sum of the reciprocals of Chen primes converges.
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https://en.wikipedia.org/wiki/Chen_prime
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In mathematics, a prime power is a positive integer which is a positive integer power of a single prime number. For example: 7 = 71, 9 = 32 and 64 = 26 are prime powers, while 6 = 2 × 3, 12 = 22 × 3 and 36 = 62 = 22 × 32 are not. The sequence of prime powers begins: 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 27, 29, 31, 32, 37, 41, 43, 47, 49, 53, 59, 61, 64, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 121, 125, 127, 128, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 243, 251, … (sequence A246655 in the OEIS). The prime powers are those positive integers that are divisible by exactly one prime number; in particular, the number 1 is not a prime power. Prime powers are also called primary numbers, as in the primary decomposition.
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https://en.wikipedia.org/wiki/Prime_power
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In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it usually means a sequence defined by the same recurrence relation as the Fibonacci numbers, but with different initial conditions causing all members of the sequence to be composite numbers that do not all have a common divisor. To put it algebraically, a sequence of this type is defined by an appropriate choice of two composite numbers a1 and a2, such that the greatest common divisor g c d ( a 1 , a 2 ) {\displaystyle \mathrm {gcd} (a_{1},a_{2})} is equal to 1, and such that for n > 2 {\displaystyle n>2} there are no primes in the sequence of numbers calculated from the formula a n = a n − 1 + a n − 2 {\displaystyle a_{n}=a_{n-1}+a_{n-2}} .The first primefree sequence of this type was published by Ronald Graham in 1964.
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https://en.wikipedia.org/wiki/Primefree_sequence
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In mathematics, a primorial prime is a prime number of the form pn# ± 1, where pn# is the primorial of pn (i.e. the product of the first n primes).Primality tests show that pn# − 1 is prime for n = 2, 3, 5, 6, 13, 24, ... (sequence A057704 in the OEIS) pn# + 1 is prime for n = 0, 1, 2, 3, 4, 5, 11, ... (sequence A014545 in the OEIS)The first term of the second sequence is 0 because p0# = 1 is the empty product, and thus p0# + 1 = 2, which is prime. Similarly, the first term of the first sequence is not 1, because p1# = 2, and 2 − 1 = 1 is not prime. The first few primorial primes are 2, 3, 5, 7, 29, 31, 211, 2309, 2311, 30029, 200560490131, 304250263527209, 23768741896345550770650537601358309 (sequence A228486 in the OEIS)As of October 2021, the largest known primorial prime (of the form pn# − 1) is 3267113# − 1 (n = 234,725) with 1,418,398 digits, found by the PrimeGrid project.As of 2022, the largest known prime of the form pn# + 1 is 392113# + 1 (n = 33,237) with 169,966 digits, found in 2001 by Daniel Heuer. Euclid's proof of the infinitude of the prime numbers is commonly misinterpreted as defining the primorial primes, in the following manner: Assume that the first n consecutive primes including 2 are the only primes that exist. If either pn# + 1 or pn# − 1 is a primorial prime, it means that there are larger primes than the nth prime (if neither is a prime, that also proves the infinitude of primes, but less directly; each of these two numbers has a remainder of either p − 1 or 1 when divided by any of the first n primes, and hence all its prime factors are larger than pn).
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https://en.wikipedia.org/wiki/Primorial_Prime
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In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X × G {\displaystyle X\times G} of a space X {\displaystyle X} with a group G {\displaystyle G} . In the same way as with the Cartesian product, a principal bundle P {\displaystyle P} is equipped with An action of G {\displaystyle G} on P {\displaystyle P} , analogous to ( x , g ) h = ( x , g h ) {\displaystyle (x,g)h=(x,gh)} for a product space. A projection onto X {\displaystyle X} . For a product space, this is just the projection onto the first factor, ( x , g ) ↦ x {\displaystyle (x,g)\mapsto x} .Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of ( x , e ) {\displaystyle (x,e)} .
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https://en.wikipedia.org/wiki/Principal_bundles
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Likewise, there is not generally a projection onto G {\displaystyle G} generalizing the projection onto the second factor, X × G → G {\displaystyle X\times G\to G} that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a principal bundle is the frame bundle F ( E ) {\displaystyle F(E)} of a vector bundle E {\displaystyle E} , which consists of all ordered bases of the vector space attached to each point.
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https://en.wikipedia.org/wiki/Principal_bundles
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The group G , {\displaystyle G,} in this case, is the general linear group, which acts on the right in the usual way: by changes of basis. Since there is no natural way to choose an ordered basis of a vector space, a frame bundle lacks a canonical choice of identity cross-section. Principal bundles have important applications in topology and differential geometry and mathematical gauge theory. They have also found application in physics where they form part of the foundational framework of physical gauge theories.
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https://en.wikipedia.org/wiki/Principal_bundles
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In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively (meaning that, for any x, y in X, there exists a unique g in G such that x·g = y, where · denotes the (right) action of G on X). An analogous definition holds in other categories, where, for example, G is a topological group, X is a topological space and the action is continuous, G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety and the action is regular.
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https://en.wikipedia.org/wiki/Principal_homogeneous_space
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In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors (e.g., Bourbaki) refer to PIDs as principal rings. The distinction is that a principal ideal ring may have zero divisors whereas a principal ideal domain cannot.
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https://en.wikipedia.org/wiki/Principal_ideal_domain
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Principal ideal domains are thus mathematical objects that behave somewhat like the integers, with respect to divisibility: any element of a PID has a unique decomposition into prime elements (so an analogue of the fundamental theorem of arithmetic holds); any two elements of a PID have a greatest common divisor (although it may not be possible to find it using the Euclidean algorithm). If x and y are elements of a PID without common divisors, then every element of the PID can be written in the form ax + by. Principal ideal domains are noetherian, they are integrally closed, they are unique factorization domains and Dedekind domains. All Euclidean domains and all fields are principal ideal domains. Principal ideal domains appear in the following chain of class inclusions: rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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https://en.wikipedia.org/wiki/Principal_ideal_domain
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In mathematics, a principal n-th root of unity (where n is a positive integer) of a ring is an element α {\displaystyle \alpha } satisfying the equations α n = 1 ∑ j = 0 n − 1 α j k = 0 for 1 ≤ k < n {\displaystyle {\begin{aligned}&\alpha ^{n}=1\\&\sum _{j=0}^{n-1}\alpha ^{jk}=0{\text{ for }}1\leq k
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https://en.wikipedia.org/wiki/Principal_root_of_unity
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In mathematics, a principal right (left) ideal ring is a ring R in which every right (left) ideal is of the form xR (Rx) for some element x of R. (The right and left ideals of this form, generated by one element, are called principal ideals.) When this is satisfied for both left and right ideals, such as the case when R is a commutative ring, R can be called a principal ideal ring, or simply principal ring. If only the finitely generated right ideals of R are principal, then R is called a right Bézout ring.
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https://en.wikipedia.org/wiki/Principal_ideal_ring
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Left Bézout rings are defined similarly. These conditions are studied in domains as Bézout domains. A commutative principal ideal ring which is also an integral domain is said to be a principal ideal domain (PID). In this article the focus is on the more general concept of a principal ideal ring which is not necessarily a domain.
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https://en.wikipedia.org/wiki/Principal_ideal_ring
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In mathematics, a principal subalgebra of a complex simple Lie algebra is a 3-dimensional simple subalgebra whose non-zero elements are regular. A finite-dimensional complex simple Lie algebra has a unique conjugacy class of principal subalgebras, each of which is the span of an sl2-triple.
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https://en.wikipedia.org/wiki/Principal_subalgebra
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In mathematics, a pro-p group (for some prime number p) is a profinite group G {\displaystyle G} such that for any open normal subgroup N ◃ G {\displaystyle N\triangleleft G} the quotient group G / N {\displaystyle G/N} is a p-group. Note that, as profinite groups are compact, the open subgroups are exactly the closed subgroups of finite index, so that the discrete quotient group is always finite. Alternatively, one can define a pro-p group to be the inverse limit of an inverse system of discrete finite p-groups. The best-understood (and historically most important) class of pro-p groups is the p-adic analytic groups: groups with the structure of an analytic manifold over Q p {\displaystyle \mathbb {Q} _{p}} such that group multiplication and inversion are both analytic functions.
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https://en.wikipedia.org/wiki/Pro-p_group
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