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c_jsqj58ezppkk | 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 p... | Pedal curve |
c_ye0fm1r4wo01 | 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 expressi... | Percentage |
c_kimqrz6n2xnd | 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... | Perfect lattice |
c_gra9jur9wjwd | 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 ... | Perfect lattice |
c_zjvbc778ufsa | 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 ... | Perfect magic cube |
c_b4bidn9z740p | 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 t... | Perfect matrix |
c_urds4lcfl8vr | 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... | Perfect power |
c_dtu96n8wis2m | 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. | Period domain |
c_85s9kqdsk9e6 | 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). | Periodic sequence |
c_r9gh43bbn9ix | 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 mathema... | Periodic travelling wave |
c_chhxv1eur9wv | 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... | Primitive permutation group |
c_6d2w4lba5034 | 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-trans... | Primitive permutation group |
c_v3k7vf5xl9ay | 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 ... | Permutation group |
c_ay6e26jwqvff | 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 chemist... | Permutation group |
c_cw9u0t979ve3 | 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 ... | Circular notation |
c_r48uxb5s8qu4 | 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... | Circular notation |
c_qgmnpowumu4m | 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... | Circular notation |
c_xtb525pgnl83 | 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 \alp... | Circular notation |
c_odipljqc305z | 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... | Circular notation |
c_nzeoi7hi1ixe | 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, ... | Permutation polynomial |
c_1n6b3k019gq8 | 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... | Phase line (mathematics) |
c_fl6xpkt2lkce | 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 i... | Phase portrait |
c_fltk7yqsjjxl | 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 att... | Phase portrait |
c_bgjqn1mdhhwp | 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. | Phase portrait |
c_uyep10a99zbk | 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. | Piecewise algebraic space |
c_ni7y7naqlrwh | 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 tr... | Piecewise-linear manifold |
c_bxde5prtde55 | 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 tha... | Piecewise smooth |
c_emqvoq7nmeif | 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 connect... | Planar Riemann surface |
c_p8nntlaws43m | 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 we... | Planar lamina |
c_6y8s481tbru2 | 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 enclo... | Plane curve |
c_ljfx2b2rc51v | 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 Euclid... | Two-dimensional space |
c_h2x15cd7mev1 | 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 li... | Point process |
c_9ul6bbjchee3 | 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. | Point source |
c_jag6g0ulaerz | 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... | Discrete subset |
c_kjzpu85y5ur7 | 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 ) {\displ... | Pointed set |
c_xt7jgum3gc4n | 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. | Pointed set |
c_hq2l09wxq8ap | : 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... | Pointed set |
c_12hhyx42g4pu | 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 ... | Pointed set |
c_ufx4ts57qzin | 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, grou... | Pointed set |
c_2vfrmhnaiuku | 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... | Based space |
c_200hzoku0njd | This is usually denoted f: ( X , x 0 ) → ( Y , y 0 ) . {\displaystyle f:\left(X,x_{0}\right)\to \left(Y,y_{0}\right).} | Based space |
c_tilc3c987f5t | 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... | Based space |
c_ytomp92b3gfz | 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 sec... | Polar action |
c_zqjiumkfw5yr | 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. | Polyadic space |
c_ej8ct0c6uixs | 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. | Polycyclic group |
c_d3ex84kbquss | 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. | Polygonal number |
c_4uasunydkbm0 | 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. | Polyhedral complex |
c_qfya80hzuz4s | 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 ... | Polylogarithmic function |
c_t9ae76whg2c6 | 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),... | Polylogarithmic function |
c_1xrwun1swuw8 | 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. | Polymatroid |
c_lygnujul6gm0 | 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 ) Dioph... | Polynomial Diophantine equation |
c_k1t0vzehrot4 | {\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 uni... | Polynomial Diophantine equation |
c_e6159gbemg0g | 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... | Separable polynomial |
c_w5tpkqulngnw | 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. ... | Indecomposable polynomial |
c_ofg4adhkmjn0 | 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 wit... | Polynomial multiplication |
c_vhlymp9ffndw | 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 chemist... | Polynomial multiplication |
c_lqwmt4zwpghy | 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.} Th... | Polynomial lemniscate |
c_o6i4360jo4cd | 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 )... | Polynomial matrix |
c_uy4f5milck6l | 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&... | Polynomial matrix |
c_m4ebhmi6ij86 | 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 ma... | Polynomial sequence |
c_ekrwq19ck3mu | 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_{... | Generalized Appell polynomials |
c_uadlqd0bymmh | 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... | Binomial type |
c_hwsfwqcn60di | 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... | Binomial type |
c_vp55gkoxsyyt | 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. | Polynomial transformation |
c_05fq2rydc3e0 | 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... | Polynomially reflexive space |
c_zmotkyv402os | 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. | Polyphase sequence |
c_j0cvzz50jouz | 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. | Porous set |
c_q2t9hmuondp0 | 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 measur... | Sigma finite measure |
c_7ms71azkucrj | 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 ... | Sigma finite measure |
c_4eppbdoscfhw | 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 ... | Herglotz representation theorem |
c_aktbdmsho3y8 | 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... | Positive polynomial |
c_49tut3lnt8y8 | In mathematics, a positive-definite function is, depending on the context, either of two types of function. | Positive-semidefinite function |
c_3a5jziwwcla0 | 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 c... | Power of three |
c_3vu8jb4l82hh | 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 ev... | Power series |
c_e1522iqo9tyx | 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 ... | Power series |
c_w2ue41erkqgk | 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 con... | Pre-Lie algebra |
c_cnajmglu4n1k | 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 a... | Prehomogeneous vector space |
c_jla7ldjdk1xa | In mathematics, a preordered class is a class equipped with a preorder. | Preordered class |
c_njvhnqdybfnd | 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 {\d... | Preradical |
c_yco6kenrryay | 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 ⟩ .... | Finitely presented group |
c_8c9ye3cs265p | 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 ... | Finitely presented group |
c_0saujonu718b | 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 stat... | Primary cyclic group |
c_svvwq3sl08cy | 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... | Primary cyclic group |
c_v2e7h7ueao1e | 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. | Prime geodesic |
c_zcr60xr3a68h | 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 wou... | Chen prime |
c_u4pbgmhqzaah | 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 O... | Chen prime |
c_91cok1w1bypv | 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, ... | Prime power |
c_53bb6rn8s2r5 | 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 ... | Primefree sequence |
c_q70y4etve62i | 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 A01454... | Primorial Prime |
c_dy89xe4lvhhn | 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 e... | Principal bundles |
c_1chofxfqueai | 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 ... | Principal bundles |
c_6pjegknzv113 | 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 appli... | Principal bundles |
c_0toc4s3wy4d3 | 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, ther... | Principal homogeneous space |
c_0chd5ubwxb6l | 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.... | Principal ideal domain |
c_dw0109sx9j0r | 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 (altho... | Principal ideal domain |
c_cxr9xaqquq06 | 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 | Principal root of unity |
c_hy24kd7709dp | 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 ... | Principal ideal ring |
c_ufqgu3lwkqy4 | 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 necessaril... | Principal ideal ring |
c_s6zffdehhng6 | 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. | Principal subalgebra |
c_lual3cz7ld84 | 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 ... | Pro-p group |
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