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c_ld07l52mi7lk | In mathematics, the paratingent cone and contingent cone were introduced by Bouligand (1932), and are closely related to tangent cones. | Paratingent cone |
c_ghio0s5kmqr3 | In mathematics, the particular point topology (or included point topology) is a topology where a set is open if it contains a particular point of the topological space. Formally, let X be any non-empty set and p ∈ X. The collection T = { S ⊆ X ∣ p ∈ S } ∪ { ∅ } {\displaystyle T=\{S\subseteq X\mid p\in S\}\cup \{\emptys... | Connected two-point set |
c_0z77t2sjsztz | If X is uncountable, the topology on X is called the uncountable particular point topology.A generalization of the particular point topology is the closed extension topology. In the case when X \ {p} has the discrete topology, the closed extension topology is the same as the particular point topology. This topology is ... | Connected two-point set |
c_xgkr270b8vif | In mathematics, the partition topology is a topology that can be induced on any set X {\displaystyle X} by partitioning X {\displaystyle X} into disjoint subsets P ; {\displaystyle P;} these subsets form the basis for the topology. There are two important examples which have their own names: The odd–even topology is th... | Partition topology |
c_yh6mbhmwjogu | {\displaystyle P=\{~\{1,2\},\{3,4\},\{5,6\},\ldots \}.} The deleted integer topology is defined by letting X = ⋃ n ∈ N ( n − 1 , n ) ⊆ R {\displaystyle X={\begin{matrix}\bigcup _{n\in \mathbb {N} }(n-1,n)\subseteq \mathbb {R} \end{matrix}}} and P = { ( 0 , 1 ) , ( 1 , 2 ) , ( 2 , 3 ) , … } . {\displaystyle P={\left\{(0... | Partition topology |
c_1owl1yhlbbx3 | The trivial partitions yield the discrete topology (each point of X {\displaystyle X} is a set in P , {\displaystyle P,} so P = { { x }: x ∈ X } {\displaystyle P=\{~\{x\}~:~x\in X~\}} ) or indiscrete topology (the entire set X {\displaystyle X} is in P , {\displaystyle P,} so P = { X } {\displaystyle P=\{X\}} ). Any se... | Partition topology |
c_8ip6qofhrfn9 | Unless P {\displaystyle P} is trivial, at least one set in P {\displaystyle P} contains more than one point, and the elements of this set are topologically indistinguishable: the topology does not separate points. Hence X {\displaystyle X} is not a Kolmogorov space, nor a T1 space, a Hausdorff space or an Urysohn space... | Partition topology |
c_jexfyv8nnt6z | In mathematics, the peak algebra is a (non-unital) subalgebra of the group algebra of the symmetric group Sn, studied by Nyman (2003). It consists of the elements of the group algebra of the symmetric group whose coefficients are the same for permutations with the same peaks. (Here a peak of a permutation σ on {1,2,...... | Peak algebra |
c_aagzk20mqxe0 | In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that ∏ n = 1 ∞ ( 1 − x n ) = ∑ k = − ∞ ∞ ( − 1 ) k x k ( 3 k − 1 ) / 2 = 1 + ∑ k = 1 ∞ ( − 1 ) k ( x k ( 3 k + 1 ) / 2 + x k ( 3 k − 1 ) / 2 ) . {\displaystyle \prod _{... | Pentagonal number theorem |
c_b3ipiilxt9e0 | {\displaystyle (1-x)(1-x^{2})(1-x^{3})\cdots =1-x-x^{2}+x^{5}+x^{7}-x^{12}-x^{15}+x^{22}+x^{26}-\cdots .} The exponents 1, 2, 5, 7, 12, ... on the right hand side are given by the formula gk = k(3k − 1)/2 for k = 1, −1, 2, −2, 3, ... and are called (generalized) pentagonal numbers (sequence A001318 in the OEIS). (The c... | Pentagonal number theorem |
c_pmgavbgbi2yi | In mathematics, the pentagram map is a discrete dynamical system on the moduli space of polygons in the projective plane. The pentagram map takes a given polygon, finds the intersections of the shortest diagonals of the polygon, and constructs a new polygon from these intersections. Richard Schwartz introduced the pent... | Pentagram map |
c_ktplvp9hopx6 | In mathematics, the permutation category is a category where the objects are the natural numbers, the morphisms from a natural number n to itself are the elements of the symmetric group S n {\displaystyle S_{n}} and there are no morphisms from m to n if m ≠ n {\displaystyle m\neq n} .It is equivalent as a category to t... | Permutation category |
c_p2g13ml0rgun | In mathematics, the permutoassociahedron is an n {\displaystyle n} -dimensional polytope whose vertices correspond to the bracketings of the permutations of n + 1 {\displaystyle n+1} terms and whose edges connect two bracketings that can be obtained from one another either by moving a pair of brackets using associativi... | Permutoassociahedron |
c_4chorymlmt7p | In mathematics, the permutohedron of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are the permutations of the first n natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connect two vertices (permutations). Two p... | Permutohedron |
c_szn7rkvr59h2 | The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate in which two places the connected p... | Permutohedron |
c_vhk9ye5qooit | In mathematics, the persistence of a number is the number of times one must apply a given operation to an integer before reaching a fixed point at which the operation no longer alters the number. Usually, this involves additive or multiplicative persistence of a non-negative integer, which is how often one has to repla... | Persistence of a number |
c_myf6ngo7jwu0 | In the remainder of this article, base ten is assumed. The single-digit final state reached in the process of calculating an integer's additive persistence is its digital root. Put another way, a number's additive persistence counts how many times we must sum its digits to arrive at its digital root. | Persistence of a number |
c_5dhyyjyu2ojd | In mathematics, the phrase "of the form" indicates that a mathematical object, or (more frequently) a collection of objects, follows a certain pattern of expression. It is frequently used to reduce the formality of mathematical proofs. | Of the form |
c_27l1wpd8kgzn | In mathematics, the phrase complete partial order is variously used to refer to at least three similar, but distinct, classes of partially ordered sets, characterized by particular completeness properties. Complete partial orders play a central role in theoretical computer science: in denotational semantics and domain ... | Complete partial order |
c_1n2fztnd0efe | In mathematics, the phrases arbitrarily large, arbitrarily small and arbitrarily long are used in statements to make clear of the fact that an object is large, small and long with little limitation or restraint, respectively. The use of "arbitrarily" often occurs in the context of real numbers (and its subsets thereof)... | Arbitrarily large |
c_ry4iiwfx4mfm | In mathematics, the pigeonhole principle states that if n items are put into m containers, with n > m, then at least one container must contain more than one item. For example, if one has three gloves (and none is ambidextrous/reversible), then there must be at least two right-handed gloves, or at least two left-handed... | Pigeonhole Principle |
c_x3ar1q0gsyt5 | Although the pigeonhole principle appears as early as 1624 in a book attributed to Jean Leurechon, it is commonly called Dirichlet's box principle or Dirichlet's drawer principle after an 1834 treatment of the principle by Peter Gustav Lejeune Dirichlet under the name Schubfachprinzip ("drawer principle" or "shelf prin... | Pigeonhole Principle |
c_gr9t8zsk8i4l | Though the most straightforward application is to finite sets (such as pigeons and boxes), it is also used with infinite sets that cannot be put into one-to-one correspondence. To do so requires the formal statement of the pigeonhole principle, which is "there does not exist an injective function whose codomain is smal... | Pigeonhole Principle |
c_d5pyhwvqiuee | In mathematics, the pin group is a certain subgroup of the Clifford algebra associated to a quadratic space. It maps 2-to-1 to the orthogonal group, just as the spin group maps 2-to-1 to the special orthogonal group. In general the map from the Pin group to the orthogonal group is not surjective or a universal covering... | Pin group |
c_awshty21lvia | In mathematics, the ping-pong lemma, or table-tennis lemma, is any of several mathematical statements that ensure that several elements in a group acting on a set freely generates a free subgroup of that group. | Ping-pong lemma |
c_119ua557ildx | In mathematics, the plactic monoid is the monoid of all words in the alphabet of positive integers modulo Knuth equivalence. Its elements can be identified with semistandard Young tableaux. It was discovered by Donald Knuth (1970) (who called it the tableau algebra), using an operation given by Craige Schensted (1961) ... | Plactic monoid |
c_2ao28lw3vkbr | In mathematics, the plastic number ρ (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, le nombre radiant) is a mathematical constant which is the unique real solution of the cubic equation x 3 = x + 1. {\displaystyle x^{3}=x+1.} It has th... | Plastic number |
c_ldg4c7hp5ocs | In mathematics, the plethystic exponential is a certain operator defined on (formal) power series which, like the usual exponential function, translates addition into multiplication. This exponential operator appears naturally in the theory of symmetric functions, as a concise relation between the generating series for... | Plethystic exponential |
c_toforu7hj7tm | In mathematics, the pluricanonical ring of an algebraic variety V (which is nonsingular), or of a complex manifold, is the graded ring R ( V , K ) = R ( V , K V ) {\displaystyle R(V,K)=R(V,K_{V})\,} of sections of powers of the canonical bundle K. Its nth graded component (for n ≥ 0 {\displaystyle n\geq 0} ) is: R n :=... | Canonical model (algebraic geometry) |
c_a09z7gfvdhc8 | In mathematics, the plus construction is a method for simplifying the fundamental group of a space without changing its homology and cohomology groups. Explicitly, if X {\displaystyle X} is a based connected CW complex and P {\displaystyle P} is a perfect normal subgroup of π 1 ( X ) {\displaystyle \pi _{1}(X)} then a ... | Plus construction |
c_rhelfy1uzpd9 | The most common application of the plus construction is in algebraic K-theory. If R {\displaystyle R} is a unital ring, we denote by GL n ( R ) {\displaystyle \operatorname {GL} _{n}(R)} the group of invertible n {\displaystyle n} -by- n {\displaystyle n} matrices with elements in R {\displaystyle R} . GL n ( R ) {... | Plus construction |
c_w0pl44588v0h | The direct limit of these groups via these maps is denoted GL ( R ) {\displaystyle \operatorname {GL} (R)} and its classifying space is denoted B GL ( R ) {\displaystyle B\operatorname {GL} (R)} . The plus construction may then be applied to the perfect normal subgroup E ( R ) {\displaystyle E(R)} of GL ( R ) = π... | Plus construction |
c_9trs6xgd29bm | In mathematics, the plus sign "+" almost invariably indicates an operation that satisfies the axioms assigned to addition in the type of algebraic structure that is known as a field. For boolean algebra, this means that the logical operation signified by "+" is not the same as the inclusive disjunction signified by "∨"... | Logical equality |
c_a2her0uayqc4 | In mathematics, the pointwise product of two functions is another function, obtained by multiplying the images of the two functions at each value in the domain. If f and g are both functions with domain X and codomain Y, and elements of Y can be multiplied (for instance, Y could be some set of numbers), then the pointw... | Pointwise product |
c_brb53076g27y | In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the origin of a Cartesian coordinate system) is called the pole, and the ray fro... | Polar coordinates system |
c_m82defp6i3zi | Grégoire de Saint-Vincent and Bonaventura Cavalieri independently introduced the concepts in the mid-17th century, though the actual term "polar coordinates" has been attributed to Gregorio Fontana in the 18th century. The initial motivation for the introduction of the polar system was the study of circular and orbital... | Polar coordinates system |
c_gwqp7yve7sp4 | In mathematics, the polar decomposition of a square real or complex matrix A {\displaystyle A} is a factorization of the form A = U P {\displaystyle A=UP} , where U {\displaystyle U} is a unitary matrix and P {\displaystyle P} is a positive semi-definite Hermitian matrix ( U {\displaystyle U} is an orthogonal matrix an... | Polar decomposition |
c_1itaz18i5l43 | This decomposition is useful in computing the fundamental group of (matrix) Lie groups.The polar decomposition can also be defined as A = P ′ U {\displaystyle A=P'U} where P ′ = U P U − 1 {\displaystyle P'=UPU^{-1}} is a symmetric positive-definite matrix with the same eigenvalues as P {\displaystyle P} but different e... | Polar decomposition |
c_q5xbro6afxx7 | The definition A = U P {\displaystyle A=UP} may be extended to rectangular matrices A ∈ C m × n {\displaystyle A\in \mathbb {C} ^{m\times n}} by requiring U ∈ C m × n {\displaystyle U\in \mathbb {C} ^{m\times n}} to be a semi-unitary matrix and P ∈ C n × n {\displaystyle P\in \mathbb {C} ^{n\times n}} to be a positive-... | Polar decomposition |
c_x4m6qcsw27cj | In mathematics, the polygamma function of order m is a meromorphic function on the complex numbers C {\displaystyle \mathbb {C} } defined as the (m + 1)th derivative of the logarithm of the gamma function: ψ ( m ) ( z ) := d m d z m ψ ( z ) = d m + 1 d z m + 1 ln Γ ( z ) . {\displaystyle \psi ^{(m)}(z):={\frac {\math... | Polygamma function |
c_u5l8284u3k36 | They are holomorphic on C ∖ Z ≤ 0 {\displaystyle \mathbb {C} \backslash \mathbb {Z} _{\leq 0}} . At all the nonpositive integers these polygamma functions have a pole of order m + 1. The function ψ(1)(z) is sometimes called the trigamma function. | Polygamma function |
c_ri2xshwxxfyb | In mathematics, the polylogarithm (also known as Jonquière's function, for Alfred Jonquière) is a special function Lis(z) of order s and argument z. Only for special values of s does the polylogarithm reduce to an elementary function such as the natural logarithm or a rational function. In quantum statistics, the polyl... | Polylogarithm ladder |
c_45bppulp2m4d | Polylogarithms should not be confused with polylogarithmic functions, nor with the offset logarithmic integral Li(z), which has the same notation without the subscript. Different polylogarithm functions in the complex plane The polylogarithm function is defined by a power series in z, which is also a Dirichlet series i... | Polylogarithm ladder |
c_zkjtduuz8lop | The special case s = 1 involves the ordinary natural logarithm, Li1(z) = −ln(1−z), while the special cases s = 2 and s = 3 are called the dilogarithm (also referred to as Spence's function) and trilogarithm respectively. The name of the function comes from the fact that it may also be defined as the repeated integral o... | Polylogarithm ladder |
c_i3zldxzw0vcn | In mathematics, the polynomial method is an algebraic approach to combinatorics problems that involves capturing some combinatorial structure using polynomials and proceeding to argue about their algebraic properties. Recently, the polynomial method has led to the development of remarkably simple solutions to several l... | Polynomial method in combinatorics |
c_hid6b11tx2tk | In mathematics, the poset topology associated to a poset (S, ≤) is the Alexandrov topology (open sets are upper sets) on the poset of finite chains of (S, ≤), ordered by inclusion. Let V be a set of vertices. An abstract simplicial complex Δ is a set of finite sets of vertices, known as faces σ ⊆ V {\displaystyle \sigm... | Order complex |
c_0s8zp087xq4a | :\ \rho \subseteq \sigma \in \Delta \Rightarrow \rho \in \Delta .} Given a simplicial complex Δ as above, we define a (point set) topology on Δ by declaring a subset Γ ⊆ Δ {\displaystyle \Gamma \subseteq \Delta } be closed if and only if Γ is a simplicial complex, i.e. ∀ ρ ∀ σ: ρ ⊆ σ ∈ Γ ⇒ ρ ∈ Γ . {\displaystyle \foral... | Order complex |
c_61478lkq4ae1 | :\ \rho \subseteq \sigma \in \Gamma \Rightarrow \rho \in \Gamma .} This is the Alexandrov topology on the poset of faces of Δ. The order complex associated to a poset (S, ≤) has the set S as vertices, and the finite chains of (S, ≤) as faces. The poset topology associated to a poset (S, ≤) is then the Alexandrov topolo... | Order complex |
c_rtvpqnzxvz0z | In mathematics, the positive part of a real or extended real-valued function is defined by the formula f + ( x ) = max ( f ( x ) , 0 ) = { f ( x ) if f ( x ) > 0 0 otherwise. {\displaystyle f^{+}(x)=\max(f(x),0)={\begin{cases}f(x)&{\mbox{ if }}f(x)>0\\0&{\mbox{ otherwise. }}\end{cases}}} Intuitively, the graph of f + {... | Negative part |
c_n46858klwtu5 | {\displaystyle f^{-}(x)=\max(-f(x),0)=-\min(f(x),0)={\begin{cases}-f(x)&{\mbox{ if }}f(x)<0\\0&{\mbox{ otherwise. }}\end{cases}}} Note that both f+ and f− are non-negative functions. | Negative part |
c_9c1fflsk3i5l | A peculiarity of terminology is that the 'negative part' is neither negative nor a part (like the imaginary part of a complex number is neither imaginary nor a part). The function f can be expressed in terms of f+ and f− as f = f + − f − . {\displaystyle f=f^{+}-f^{-}.} | Negative part |
c_1qaqy7sthbmr | Also note that | f | = f + + f − {\displaystyle |f|=f^{+}+f^{-}} .Using these two equations one may express the positive and negative parts as f + = | f | + f 2 {\displaystyle f^{+}={\frac {|f|+f}{2}}} f − = | f | − f 2 . {\displaystyle f^{-}={\frac {|f|-f}{2}}.} Another representation, using the Iverson bracket is f +... | Negative part |
c_xckqakpq936e | {\displaystyle f^{-}=-f.} One may define the positive and negative part of any function with values in a linearly ordered group. The unit ramp function is the positive part of the identity function. | Negative part |
c_9lmu96mf4zfw | In mathematics, the power series method is used to seek a power series solution to certain differential equations. In general, such a solution assumes a power series with unknown coefficients, then substitutes that solution into the differential equation to find a recurrence relation for the coefficients. | Series solution of differential equations |
c_6qb1qo58yue2 | In mathematics, the power set (or powerset) of a set S is the set of all subsets of S, including the empty set and S itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is postulated by the axiom of power set. The powerset of S is variously denoted a... | Power set |
c_s17pujwc5u9q | In mathematics, the predual of an object D is an object P whose dual space is D. For example, the predual of the space of bounded operators is the space of trace class operators, and the predual of the space L∞(R) of essentially bounded functions on R is the Banach space L1(R) of integrable functions. | Predual |
c_tdi1mp9u6rm0 | In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-manifolds. A manifold is prime if it cannot be presented as a connected sum of more than one manifold, none of which is the... | Prime decomposition (3-manifold) |
c_v6uvq7ezj19p | So the theorem can be restated to say that there is a unique connected sum decomposition into irreducible 3-manifolds and fiber bundles of S 2 {\displaystyle S^{2}} over S 1 . {\displaystyle S^{1}.} The prime decomposition holds also for non-orientable 3-manifolds, but the uniqueness statement must be modified slightly... | Prime decomposition (3-manifold) |
c_kqtnq70m200l | {\displaystyle S^{1}.} This sum is unique as long as we specify that each summand is either irreducible or a non-orientable S 2 {\displaystyle S^{2}} bundle over S 1 . | Prime decomposition (3-manifold) |
c_8lnsr9dssfmr | {\displaystyle S^{1}.} The proof is based on normal surface techniques originated by Hellmuth Kneser. Existence was proven by Kneser, but the exact formulation and proof of the uniqueness was done more than 30 years later by John Milnor. | Prime decomposition (3-manifold) |
c_nebhak34w17s | In mathematics, the prime end compactification is a method to compactify a topological disc (i.e. a simply connected open set in the plane) by adding the boundary circle in an appropriate way. | Prime end |
c_h1777uzhvw53 | In mathematics, the prime is generally used to generate more variable names for similar things without resorting to subscripts, with x′ generally meaning something related to (or derived from) x. For example, if a point is represented by the Cartesian coordinates (x, y), then that point rotated, translated or reflected... | Double prime |
c_mylfr4cycaj1 | The negation of an event in probability theory: Pr(A′) = 1 − Pr(A) (other notation also exists). The result of a transformation: Tx = x′ The transpose of a matrix (other notation also exists) The dual of a vector spaceThe prime is said to "decorate" the letter to which it applies. | Double prime |
c_s4s0ewgm325i | The same convention is adopted in functional programming, particularly in Haskell. In geometry, geography and astronomy, prime and double prime are used as abbreviations for minute and second of arc (and thus latitude, longitude, elevation and right ascension). In physics, the prime is used to denote variables after an... | Double prime |
c_d6glzwssttt0 | For example, vA′ would indicate the velocity of object A after an event. It is also commonly used in relativity: the event at (x, y, z, t) in frame S, has coordinates (x′, y′, z′, t′) in frame S′. In chemistry, it is used to distinguish between different functional groups connected to an atom in a molecule, such as R a... | Double prime |
c_3dl0166z7nkf | The carbonyl carbon in proteins is denoted as C′, which distinguishes it from the other backbone carbon, the alpha carbon, which is denoted as Cα. In physical chemistry, it is used to distinguish between the lower state and the upper state of a quantum number during a transition. For example, J ′ denotes the upper stat... | Double prime |
c_8a96utliyiag | The prime distinguishes places on these two chemicals, rather than places on other parts of DNA or RNA, like phosphate groups or nucleic acids. Thus, when indicating the direction of movement of an enzyme along a string of DNA, biologists will say that it moves from the 5′ end to the 3′ end, because these carbons are o... | Double prime |
c_cjpdpzz0cm40 | In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying the rate at which this occurs. The theorem was proved independently by Jac... | Prime number race |
c_acbopvub6ozg | Consequently, a random integer with at most 2n digits (for large enough n) is about half as likely to be prime as a random integer with at most n digits. For example, among the positive integers of at most 1000 digits, about one in 2300 is prime (log(101000) ≈ 2302.6), whereas among positive integers of at most 2000 di... | Prime number race |
c_lwr8k32nl7ct | In mathematics, the prime signature of a number is the multiset of (nonzero) exponents of its prime factorization. The prime signature of a number having prime factorization p 1 m 1 p 2 m 2 … p n m n {\displaystyle p_{1}^{m_{1}}p_{2}^{m_{2}}\dots p_{n}^{m_{n}}} is the multiset { m 1 , m 2 , … , m n } {\displaystyle \le... | Prime signature |
c_13hdtvn5y72m | In mathematics, the prime zeta function is an analogue of the Riemann zeta function, studied by Glaisher (1891). It is defined as the following infinite series, which converges for ℜ ( s ) > 1 {\displaystyle \Re (s)>1}: P ( s ) = ∑ p ∈ p r i m e s 1 p s = 1 2 s + 1 3 s + 1 5 s + 1 7 s + 1 11 s + ⋯ . {\displaystyle P(s)... | Prime zeta function |
c_30lg1jtc3sn7 | In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number x. It is denoted by π(x) (unrelated to the number π). | Prime-counting function |
c_957xgwlh3xqi | In mathematics, the principal ideal theorem of class field theory, a branch of algebraic number theory, says that extending ideals gives a mapping on the class group of an algebraic number field to the class group of its Hilbert class field, which sends all ideal classes to the class of a principal ideal. The phenomeno... | Principalisation property |
c_2r7ish0mrwt6 | In mathematics, the principal orbit type theorem states that compact Lie group acting smoothly on a connected differentiable manifold has a principal orbit type. | Principal orbit type |
c_xb6bw0i00lc2 | In mathematics, the principal series representations of certain kinds of topological group G occur in the case where G is not a compact group. There, by analogy with spectral theory, one expects that the regular representation of G will decompose according to some kind of continuous spectrum, of representations involvi... | Principal series representation |
c_9lkbk1tkcmc5 | The discrete series consists of 'atoms' of the unitary dual (points carrying a Plancherel measure > 0). In the earliest examples studied, the rest (or most) of the unitary dual could be parametrised by starting with a subgroup H of G, simpler but not compact, and building up induced representations using representation... | Principal series representation |
c_y0hqd60w2571 | For the case of a semisimple Lie group G, the subgroup H is constructed starting from the Iwasawa decomposition G = KANwith K a maximal compact subgroup. Then H is chosen to contain AN (which is a non-compact solvable Lie group), being taken as H := MANwith M the centralizer in K of A. Representations ρ of H are consid... | Principal series representation |
c_md6u2vztz03m | The induced representations of such ρ make up the principal series. The spherical principal series consists of representations induced from 1-dimensional representations of MAN obtained by extending characters of A using the homomorphism of MAN onto A. There may be other continuous series of representations relevant to... | Principal series representation |
c_zs9omdbbymxi | In mathematics, the probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős, for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of... | Probabilistic method |
c_n8hxs1katbzq | In mathematics, the problem of differentiation of integrals is that of determining under what circumstances the mean value integral of a suitable function on a small neighbourhood of a point approximates the value of the function at that point. More formally, given a space X with a measure μ and a metric d, one asks fo... | Differentiation of integrals |
c_jrrzd7xunmlo | In mathematics, the projection-slice theorem, central slice theorem or Fourier slice theorem in two dimensions states that the results of the following two calculations are equal: Take a two-dimensional function f(r), project (e.g. using the Radon transform) it onto a (one-dimensional) line, and do a Fourier transform ... | Fourier slice theorem |
c_wzab30tndsxx | This idea can be extended to higher dimensions. This theorem is used, for example, in the analysis of medical CT scans where a "projection" is an x-ray image of an internal organ. The Fourier transforms of these images are seen to be slices through the Fourier transform of the 3-dimensional density of the internal orga... | Fourier slice theorem |
c_rsxzn88wjbqt | In mathematics, the projective line over a ring is an extension of the concept of projective line over a field. Given a ring A with 1, the projective line P(A) over A consists of points identified by projective coordinates. Let U be the group of units of A; pairs (a, b) and (c, d) from A × A are related when there is a... | Inversive ring geometry |
c_nupsd135i4xz | The homographies are expressed through use of the matrix ring over A and its group of units V as follows: If c is in Z(U), the center of U, then the group action of matrix ( c 0 0 c ) {\displaystyle {\begin{pmatrix}c&0\\0&c\end{pmatrix}}} on P(A) is the same as the action of the identity matrix. Such matrices represent... | Inversive ring geometry |
c_jst885cre613 | {\displaystyle U{\begin{pmatrix}0&1\\1&0\end{pmatrix}}=U\thicksim U.} Furthermore, for u,v ∈ U, the mapping a → uav can be extended to a homography: ( u 0 0 1 ) ( 0 1 1 0 ) ( v 0 0 1 ) ( 0 1 1 0 ) = ( u 0 0 v ) . {\displaystyle {\begin{pmatrix}u&0\\0&1\end{pmatrix}}{\begin{pmatrix}0&1\\1&0\end{pmatrix}}{\begin{pmatrix}... | Inversive ring geometry |
c_064q6uih2cj3 | U ( v 0 0 u ) = U ∼ U . {\displaystyle U{\begin{pmatrix}v&0\\0&u\end{pmatrix}}=U\thicksim U.} Since u is arbitrary, it may be substituted for u−1. Homographies on P(A) are called linear-fractional transformations since U ( a c b d ) = U ∼ U . {\displaystyle U{\begin{pmatrix}a&c\\b&d\end{pmatrix}}=U\thicksim U.} | Inversive ring geometry |
c_jfann3b3blf7 | In mathematics, the projective special linear group PSL(2, 7), isomorphic to GL(3, 2), is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group of the Fano plane. With 168 elements, PSL(2, 7) is the ... | PSL(2,7) |
c_cqdcclwqcsg1 | In mathematics, the projective unitary group PU(n) is the quotient of the unitary group U(n) by the right multiplication of its center, U(1), embedded as scalars. Abstractly, it is the holomorphic isometry group of complex projective space, just as the projective orthogonal group is the isometry group of real projectiv... | Projective unitary group |
c_6torb2bvpfbr | In mathematics, the pseudoisotopy theorem is a theorem of Jean Cerf's which refers to the connectivity of a group of diffeomorphisms of a manifold. | Pseudoisotopy theorem |
c_nyopyue4lpr1 | In mathematics, the q-Bessel polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Bessel polynomials |
c_xhuc5i28ufbh | In mathematics, the q-Charlier polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Charlier polynomials |
c_iy7yzpxu03os | In mathematics, the q-Hahn polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Hahn polynomials |
c_tjywvu568r64 | In mathematics, the q-Konhauser polynomials are a q-analog of the Konhauser polynomials, introduced by Al-Salam & Verma (1983). | Q-Konhauser polynomials |
c_2zcqc3k3tbja | In mathematics, the q-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14). give a detailed list of their properties. Stanton (1981) showed that the q-Krawtchouk polynomials are spherical functions f... | Q-Krawtchouk polynomials |
c_8snid0d6arb8 | In mathematics, the q-Laguerre polynomials, or generalized Stieltjes–Wigert polynomials P(α)n(x;q) are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme introduced by Daniel S. Moak (1981). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their pro... | Q-Laguerre polynomials |
c_xxvmzl050ty4 | In mathematics, the q-Meixner polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Meixner polynomials |
c_ok2h3fdzhuz2 | In mathematics, the q-Meixner–Pollaczek polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Meixner–Pollaczek polynomials |
c_x40ezksh5aqd | In mathematics, the q-Racah polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Askey & Wilson (1979). Roelof Koekoek, Peter A. Lesky, and René F. Swarttouw (2010, 14) give a detailed list of their properties. | Q-Racah polynomials |
c_kctgv1af3a6y | In mathematics, the q-expansion principle states that a modular form f has coefficients in a module M if its q-expansion at enough cusps resembles the q-expansion of a modular form g with coefficients in M. It was introduced by Katz (1973, corollaries 1.6.2, 1.12.2). | Q-expansion principle |
c_h4ix00q1ek7p | In mathematics, the q-theta function (or modified Jacobi theta function) is a type of q-series which is used to define elliptic hypergeometric series. It is given by θ ( z ; q ) := ∏ n = 0 ∞ ( 1 − q n z ) ( 1 − q n + 1 / z ) {\displaystyle \theta (z;q):=\prod _{n=0}^{\infty }(1-q^{n}z)\left(1-q^{n+1}/z\right)} where on... | Q-theta function |
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