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{\displaystyle \det {\begin{bmatrix}{\dot {\mathbf {x} }},&{\ddot {\mathbf {x} }},&\dots ,&{\mathbf {x} }^{(n)}\end{bmatrix}}=\pm 1.} Such a curve is said to be parametrized by its affine arclength. For such a parameterization, t ↦ {\displaystyle t\mapsto } determines a mapping into the special affine group, known as ... | Wikipedia - Affine geometry of curves | null | null | null |
Let x, y and z denote arbitrary elements of the algebra A over the field K. Let powers to positive (non-zero) integer be recursively defined by x1 ≝ x and either xn+1 ≝ xnx (right powers) or xn+1 ≝ xxn (left powers) depending on authors. Unital: there exist an element e so that ex = x = xe; in that case we can define x... | Wikipedia - Quadratic representation | null | null | null |
Jacobi identity: (xy)z + (yz)x + (zx)y = 0 or x(yz) + y(zx) + z(xy) = 0 depending on authors. Jordan identity: (x2y)x = x2(yx) or (xy)x2 = x(yx2) depending on authors. Alternative: (xx)y = x(xy) (left alternative) and (yx)x = y(xx) (right alternative). | Wikipedia - Quadratic representation | null | null | null |
Flexible: (xy)x = x(yx). nth power associative with n ≥ 2: xn−kxk = xn for all integers k so that 0 < k < n. Third power associative: x2x = xx2. Fourth power associative: x3x = x2x2 = xx3 (compare with fourth power commutative below). | Wikipedia - Quadratic representation | null | null | null |
Power associative: the subalgebra generated by any element is associative, i.e., nth power associative for all n ≥ 2. nth power commutative with n ≥ 2: xn−kxk = xkxn−k for all integers k so that 0 < k < n. Third power commutative: x2x = xx2. Fourth power commutative: x3x = xx3 (compare with fourth power associative abo... | Wikipedia - Quadratic representation | null | null | null |
Let x, y ∈ M. We say that x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive property and so can be written x < y. x causally precedes y if y − x is future-directed null or future-directed timelike. It gives a partial ordering of spacetime and so can be written x ≤ y.Sup... | Wikipedia - Minkowski space-time | null | null | null |
Then the simultaneous hyperplane for x is { y: η ( x , y ) = 0 } . {\displaystyle \{y:\eta (x,y)=0\}.} Since this hyperplane varies as x varies, there is a relativity of simultaneity in Minkowski space. | Wikipedia - Minkowski space-time | null | null | null |
Let x1 through xn be a set of data points, with no assumptions made about their internal structure, and let s be a function that quantifies the similarity between any two points, such that s(i, j) > s(i, k) iff xi is more similar to xj than to xk. For this example, the negative squared distance of two data points was u... | Wikipedia - Affinity propagation | null | null | null |
Let z = f(x, y) be a function of two real variables. This is a parametric surface, parametrized as x = t y = u z = f ( t , u ) . {\displaystyle {\begin{aligned}x&=t\\y&=u\\z&=f(t,u)\,.\end{aligned}}} Every point of this surface is regular, as the two first columns of the Jacobian matrix form the identity matrix of rank... | Wikipedia - Surface (mathematics) | null | null | null |
Let z be a Boolean; then we can form an equation with no solutions, z = ¬ z {\displaystyle z=\neg z} To solve this equation by recursion, we introduce a new function f defined by, f n = ¬ ( n n ) {\displaystyle f\ n=\neg (n\ n)} where n is an auxiliary variable to hold the recursion value. (We take it that ¬ {\displays... | Wikipedia - Deductive lambda calculus | null | null | null |
¬ ( x x ) ) ( λ x . ¬ ( x x ) ) = ¬ ( ( λ x . ¬ ( x x ) ) ( λ x . ¬ ( x x ) ) ) = ¬ ( f f ) {\displaystyle {\begin{aligned}f\ f&=(\lambda x.\neg (x\ x))(\lambda x.\neg (x\ x))\\&=\neg ((\lambda x.\neg (x\ x))(\lambda x.\neg (x\ x)))\\&=\neg (f\ f)\end{aligned}}} Then f f is neither true nor false, and as f f is a Boole... | Wikipedia - Deductive lambda calculus | null | null | null |
The commutation relations of the affine Lie algebra are = f c a b J m + n c + k n δ a b δ n + m , 0 . {\displaystyle =f_{c}^{ab}J_{m+n}^{c}+kn\delta ^{ab}\delta _{n+m,0}.} This affine Lie algebra is the chiral symmetry algebra associated to the left-moving currents K ( t a , ∂ z g g − 1 ) {\displaystyle {\mathcal {K}}... | Wikipedia - Wess–Zumino–Witten model | null | null | null |
A second copy of the same affine Lie algebra is associated to the right-moving currents K ( t a , g − 1 ∂ z ¯ g ) {\displaystyle {\mathcal {K}}(t^{a},g^{-1}\partial _{\bar {z}}g)} . The generators J ¯ a ( z ) {\displaystyle {\bar {J}}^{a}(z)} of that second copy are antiholomorphic. The full symmetry algebra of the WZW... | Wikipedia - Wess–Zumino–Witten model | null | null | null |
Let { ( M i , b i ): i ∈ I } {\displaystyle \left\{\left(M_{i},b_{i}\right):i\in I\right\}} be a family indexed by I {\displaystyle I} of modules equipped with bilinear forms. The orthogonal direct sum is the module direct sum with bilinear form B {\displaystyle B} defined by in which the summation makes sense even for... | Wikipedia - Complementary subspace | null | null | null |
Let { E k } k {\displaystyle \{E_{k}\}_{k}} be an arbitrary positive operator-valued measure (POVM); that is, a set of positive semidefinite operators E k {\displaystyle E_{k}} satisfying ∑ k E k = I {\displaystyle \sum _{k}E_{k}=I} . Then, for any pair of states ρ {\displaystyle \rho } and σ {\displaystyle \sigma } , ... | Wikipedia - Quantum fidelity | null | null | null |
Let { X α: α ∈ A } {\displaystyle \left\{X_{\alpha }:\alpha \in A\right\}} be a family of (not necessarily disjoint) topological spaces such that the induced topologies agree on each intersection X α ∩ X β . {\displaystyle X_{\alpha }\cap X_{\beta }.} Assume further that X α ∩ X β {\displaystyle X_{\alpha }\cap X_{\bet... | Wikipedia - Coherent topology | null | null | null |
Then the topological union X {\displaystyle X} is the set-theoretic union endowed with the final topology coinduced by the inclusion maps i α: X α → X s e t {\displaystyle i_{\alpha }:X_{\alpha }\to X^{set}} . The inclusion maps will then be topological embeddings and X {\displaystyle X} will be coherent with the subsp... | Wikipedia - Coherent topology | null | null | null |
Conversely, if X {\displaystyle X} is a topological space and is coherent with a family of subspaces { C α } {\displaystyle \left\{C_{\alpha }\right\}} that cover X , {\displaystyle X,} then X {\displaystyle X} is homeomorphic to the topological union of the family { C α } . {\displaystyle \left\{C_{\alpha }\right\}.} ... | Wikipedia - Coherent topology | null | null | null |
One can also describe the topological union by means of the disjoint union. Specifically, if X {\displaystyle X} is a topological union of the family { X α } , {\displaystyle \left\{X_{\alpha }\right\},} then X {\displaystyle X} is homeomorphic to the quotient of the disjoint union of the family { X α } {\displaystyle ... | Wikipedia - Coherent topology | null | null | null |
; that is, If the spaces { X α } {\displaystyle \left\{X_{\alpha }\right\}} are all disjoint then the topological union is just the disjoint union. Assume now that the set A is directed, in a way compatible with inclusion: α ≤ β {\displaystyle \alpha \leq \beta } whenever X α ⊂ X β {\displaystyle X_{\alpha }\subset X_{... | Wikipedia - Coherent topology | null | null | null |
Let { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} be a basis of V {\displaystyle V} , i.e. a set of n {\displaystyle n} linearly independent vectors that span the n {\displaystyle n} -dimensional vector space V {\displaystyle V} . The basis that is dual to { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_... | Wikipedia - Geometric algebra | null | null | null |
The dual basis vectors may be constructed as e i = ( − 1 ) i − 1 ( e 1 ∧ ⋯ ∧ e ˇ i ∧ ⋯ ∧ e n ) I − 1 , {\displaystyle e^{i}=(-1)^{i-1}(e_{1}\wedge \cdots \wedge {\check {e}}_{i}\wedge \cdots \wedge e_{n})I^{-1},} where the e ˇ i {\displaystyle {\check {e}}_{i}} denotes that the i {\displaystyle i} th basis vector is om... | Wikipedia - Geometric algebra | null | null | null |
Given a vector a {\displaystyle a} , scalar components a i {\displaystyle a^{i}} can be defined as a i = a ⋅ e i , {\displaystyle a^{i}=a\cdot e^{i}\ ,} in terms of which a {\displaystyle a} can be separated into vector components as a = ∑ i a i e i . {\displaystyle a=\sum _{i}a^{i}e_{i}\ .} We can also define scalar c... | Wikipedia - Geometric algebra | null | null | null |
{\displaystyle a=\sum _{i}a_{i}e^{i}\ .} A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra. For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing J = ( j 1 , … , j n ) , {\displaystyle J=(j_{... | Wikipedia - Geometric algebra | null | null | null |
Let { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} be a set of basis vectors that span an n {\displaystyle n} -dimensional vector space. From geometric algebra, we interpret the pseudoscalar e 1 ∧ e 2 ∧ ⋯ ∧ e n {\displaystyle e_{1}\wedge e_{2}\wedge \cdots \wedge e_{n}} to be the signed volume of the n {\dis... | Wikipedia - Geometric calculus | null | null | null |
We denote these selected basis vectors by { e i 1 , … , e i k } {\displaystyle \{e_{i_{1}},\ldots ,e_{i_{k}}\}} . A general k {\displaystyle k} -volume of the k {\displaystyle k} -parallelotope subtended by these basis vectors is the grade k {\displaystyle k} multivector e i 1 ∧ e i 2 ∧ ⋯ ∧ e i k {\displaystyle e_{i_{1... | Wikipedia - Geometric calculus | null | null | null |
We are free to choose components as infinitesimally small as we wish as long as they remain nonzero. Since the outer product of these terms can be interpreted as a k {\displaystyle k} -volume, a natural way to define a measure is d k X = ( d x i 1 e i 1 ) ∧ ( d x i 2 e i 2 ) ∧ ⋯ ∧ ( d x i k e i k ) = ( e i 1 ∧ e i 2 ∧ ... | Wikipedia - Geometric calculus | null | null | null |
Compare the Riemannian volume form in the theory of differential forms. The integral is taken with respect to this measure: ∫ V F ( x ) d k X = ∫ V F ( x ) ( e i 1 ∧ e i 2 ∧ ⋯ ∧ e i k ) d x i 1 d x i 2 ⋯ d x i k . | Wikipedia - Geometric calculus | null | null | null |
{\displaystyle \int _{V}F(x)\,d^{k}X=\int _{V}F(x)\left(e_{i_{1}}\wedge e_{i_{2}}\wedge \cdots \wedge e_{i_{k}}\right)dx^{i_{1}}dx^{i_{2}}\cdots dx^{i_{k}}.} More formally, consider some directed volume V {\displaystyle V} of the subspace. We may divide this volume into a sum of simplices. | Wikipedia - Geometric calculus | null | null | null |
Let { x i } {\displaystyle \{x_{i}\}} be the coordinates of the vertices. At each vertex we assign a measure Δ U i ( x ) {\displaystyle \Delta U_{i}(x)} as the average measure of the simplices sharing the vertex. Then the integral of F ( x ) {\displaystyle F(x)} with respect to U ( x ) {\displaystyle U(x)} over this vo... | Wikipedia - Geometric calculus | null | null | null |
Let { n B } {\displaystyle \{nB\}} , n = 0 , 1 , … {\displaystyle n=0,1,\ldots } , be a family of shapes, where B is a structuring element, n B = B ⊕ ⋯ ⊕ B ⏟ n times {\displaystyle nB=\underbrace {B\oplus \cdots \oplus B} _{n{\mbox{ times}}}} , and 0 B = { o } {\displaystyle 0B=\{o\}} , where o denotes the origin.The v... | Wikipedia - Morphological skeleton | null | null | null |
Let {Hx}x ∈ X be a measurable family of Hilbert spaces. A family of von Neumann algebras {Ax}x ∈ X with A x ⊆ L ( H x ) {\displaystyle A_{x}\subseteq \operatorname {L} (H_{x})} is measurable if and only if there is a countable set D of measurable operator families that pointwise generate {Ax} x ∈ X as a von Neumann a... | Wikipedia - Direct integral | null | null | null |
Theorem. If {Ax}x ∈ X is a measurable family of von Neumann algebras and μ is standard, then the family of operator commutants is also measurable and ′ = ∫ X ⊕ A x ′ d μ ( x ) . {\displaystyle {\bigg }'=\int _{X}^{\oplus }A'_{x}d\mu (x).} | Wikipedia - Direct integral | null | null | null |
Let {P1 ,P2, P3, P4} be the points of a tetrahedron. Let Δi be the area of the face opposite vertex Pi and let θij be the dihedral angle between the two faces of the tetrahedron adjacent to the edge PiPj. The law of cosines for this tetrahedron, which relates the areas of the faces of the tetrahedron to the dihedral an... | Wikipedia - Triangular pyramid | null | null | null |
Let {a, a, ..., a} be a Prüfer sequence: The tree will have n+2 nodes, numbered from 1 to n+2. For each node set its degree to the number of times it appears in the sequence plus 1. For instance, in pseudo-code: Convert-Prüfer-to-Tree(a) 1 n ← length 2 T ← a graph with n + 2 isolated nodes, numbered 1 to n + 2 3 degree... | Wikipedia - Prüfer sequence | null | null | null |
Let Γ < I s o m ( H n ) {\displaystyle \Gamma <\mathrm {Isom} (\mathbb {H} ^{n})} be a hyperbolic reflection group. Choose any point v 0 ∈ H n {\displaystyle v_{0}\in \mathbb {H} ^{n}} ; we shall call it the basic (or initial) point. The fundamental domain P 0 {\displaystyle P_{0}} of its stabilizer Γ v 0 {\displaystyl... | Wikipedia - Vinberg's algorithm | null | null | null |
, H m {\displaystyle H_{1},...,H_{m}} be the faces of this cone, and let a 1 , . . . | Wikipedia - Vinberg's algorithm | null | null | null |
, a m {\displaystyle a_{1},...,a_{m}} be outer normal vectors to it. Consider the half-spaces H k − = { x ∈ R n , 1 | ( x , a k ) ≤ 0 } . {\displaystyle H_{k}^{-}=\{x\in \mathbb {R} ^{n,1}|(x,a_{k})\leq 0\}.} | Wikipedia - Vinberg's algorithm | null | null | null |
There exists a unique fundamental polyhedron P {\displaystyle P} of Γ {\displaystyle \Gamma } contained in P 0 {\displaystyle P_{0}} and containing the point v 0 {\displaystyle v_{0}} . Its faces containing v 0 {\displaystyle v_{0}} are formed by faces H 1 , . . | Wikipedia - Vinberg's algorithm | null | null | null |
. , H m {\displaystyle H_{1},...,H_{m}} of the cone P 0 {\displaystyle P_{0}} . The other faces H m + 1 , . | Wikipedia - Vinberg's algorithm | null | null | null |
{\displaystyle H_{m+1},...} and the corresponding outward normals a m + 1 , . . . {\displaystyle a_{m+1},...} are constructed by induction. Namely, for H j {\displaystyle H_{j}} we take a mirror such that the root a j {\displaystyle a_{j}} orthogonal to it satisfies the conditions (1) ( v 0 , a j ) < 0 {\displaystyle (... | Wikipedia - Vinberg's algorithm | null | null | null |
Let Γ and Δ be discrete subgroups of the isometry group of hyperbolic n-space H, where n ≥ 3, whose quotients H/Γ and H/Δ have finite volume. If Γ and Δ are isomorphic as discrete groups then they are conjugate. | Wikipedia - Topological rigidity | null | null | null |
Let Γ {\displaystyle \Gamma } be a Fuchsian group and k {\displaystyle k} its trace field. Let A {\displaystyle A} be the k {\displaystyle k} -subalgebra of the matrix algebra M 2 ( R ) {\displaystyle M_{2}(\mathbb {R} )} generated by the preimages of elements of Γ {\displaystyle \Gamma } . The algebra A {\displaystyle... | Wikipedia - Trace field | null | null | null |
The quaternion algebra of Γ ( 2 ) {\displaystyle \Gamma ^{(2)}} is called the invariant quaternion algebra of Γ {\displaystyle \Gamma } , denoted by A Γ {\displaystyle A\Gamma } . As for trace fields, the former is not the same for all groups in the same commensurability class but the latter is. If Γ {\displaystyle \Ga... | Wikipedia - Trace field | null | null | null |
Let Γ {\displaystyle \Gamma } be a group and Z Γ {\displaystyle \mathbb {Z} \Gamma } its group ring. The group Γ {\displaystyle \Gamma } is said to be of type FPn if there exists a resolution of the trivial Z Γ {\displaystyle \mathbb {Z} \Gamma } -module Z {\displaystyle \mathbb {Z} } such that the n first terms are fi... | Wikipedia - Finiteness properties of groups | null | null | null |
It is also possible to define classes FPn(R) and FLn(R) for any commutative ring R, by replacing the group ring Z Γ {\displaystyle \mathbb {Z} \Gamma } by R Γ {\displaystyle R\Gamma } in the definitions above. Either of the conditions Fn or FHn imply FPn and FLn (over any commutative ring). A group is of type FP1 if an... | Wikipedia - Finiteness properties of groups | null | null | null |
Let Δ ( a , b , c ) = 1 / 2 {\displaystyle \Delta (a,b,c)=^{1/2}} be the usual triangular factor, then the Racah coefficient is a product of four of these by a sum over factorials, W ( a b c d ; e f ) = Δ ( a , b , e ) Δ ( c , d , e ) Δ ( a , c , f ) Δ ( b , d , f ) w ( a b c d ; e f ) {\displaystyle W(abcd;ef)=\Delta... | Wikipedia - Racah W-coefficient | null | null | null |
( z − α 4 ) ! ( β 1 − z ) ! ( β 2 − z ) ! | Wikipedia - Racah W-coefficient | null | null | null |
( β 3 − z ) ! {\displaystyle w(abcd;ef)\equiv \sum _{z}{\frac {(-1)^{z+\beta _{1}}(z+1)! }{(z-\alpha _{1})! | Wikipedia - Racah W-coefficient | null | null | null |
(z-\alpha _{2})! (z-\alpha _{3})! (z-\alpha _{4})! | Wikipedia - Racah W-coefficient | null | null | null |
(\beta _{1}-z)! (\beta _{2}-z)! (\beta _{3}-z)!}}} | Wikipedia - Racah W-coefficient | null | null | null |
and α 1 = a + b + e ; β 1 = a + b + c + d ; {\displaystyle \alpha _{1}=a+b+e;\quad \beta _{1}=a+b+c+d;} α 2 = c + d + e ; β 2 = a + d + e + f ; {\displaystyle \alpha _{2}=c+d+e;\quad \beta _{2}=a+d+e+f;} α 3 = a + c + f ; β 3 = b + c + e + f ; {\displaystyle \alpha _{3}=a+c+f;\quad \beta _{3}=b+c+e+f;} α 4 = b + d + f ... | Wikipedia - Racah W-coefficient | null | null | null |
Let Π {\displaystyle \Pi } be a minor-bidimensional problem such that for any graph G excluding some fixed graph as a minor and of treewidth at most t, deciding whether ( G , k ) ∈ Π {\displaystyle (G,k)\in \Pi } can be done in time 2 O ( t ) ⋅ | G | O ( 1 ) {\displaystyle 2^{O(t)}\cdot |G|^{O(1)}} . Then for every gra... | Wikipedia - Bidimensionality | null | null | null |
Let Σ be a finite set (an "alphabet") and let A be the set of all regular expressions over Σ. We consider two such regular expressions equal if they describe the same language. Then A forms a Kleene algebra. In fact, this is a free Kleene algebra in the sense that any equation among regular expressions follows from the... | Wikipedia - Regular algebra | null | null | null |
Again let Σ be an alphabet. Let A be the set of all regular languages over Σ (or the set of all context-free languages over Σ; or the set of all recursive languages over Σ; or the set of all languages over Σ). Then the union (written as +) and the concatenation (written as ·) of two elements of A again belong to A, and... | Wikipedia - Regular algebra | null | null | null |
Let M be a monoid with identity element e and let A be the set of all subsets of M. For two such subsets S and T, let S + T be the union of S and T and set ST = {st: s in S and t in T}. S* is defined as the submonoid of M generated by S, which can be described as {e} ∪ S ∪ SS ∪ SSS ∪ ... Then A forms a Kleene algebra w... | Wikipedia - Regular algebra | null | null | null |
The linear subspaces of a unital algebra over a field form a Kleene algebra. Given linear subspaces V and W, define V + W to be the sum of the two subspaces, and 0 to be the trivial subspace {0}. Define V · W = span {v · w | v ∈ V, w ∈ W}, the linear span of the product of vectors from V and W respectively. | Wikipedia - Regular algebra | null | null | null |
Define 1 = span {I}, the span of the unit of the algebra. The closure of V is the direct sum of all powers of V. Suppose M is a set and A is the set of all binary relations on M. Taking + to be the union, · to be the composition and * to be the reflexive transitive closure, we obtain a Kleene algebra. Every Boolean alg... | Wikipedia - Regular algebra | null | null | null |
Using the extended real number line, take a + b to be the minimum of a and b and ab to be the ordinary sum of a and b (with the sum of +∞ and −∞ being defined as +∞). a* is defined to be the real number zero for nonnegative a and −∞ for negative a. This is a Kleene algebra with zero element +∞ and one element the real ... | Wikipedia - Regular algebra | null | null | null |
Let Σ be a finite set of symbols (alternatively called characters), called the alphabet. No assumption is made about the nature of the symbols. A string (or word) over Σ is any finite sequence of symbols from Σ. For example, if Σ = {0, 1}, then 01011 is a string over Σ. The length of a string s is the number of symbols... | Wikipedia - Sequence of symbols | null | null | null |
For example, if Σ = {0, 1}, then Σ2 = {00, 01, 10, 11}. Note that Σ0 = {ε} for any alphabet Σ. The set of all strings over Σ of any length is the Kleene closure of Σ and is denoted Σ*. | Wikipedia - Sequence of symbols | null | null | null |
In terms of Σn, Σ ∗ = ⋃ n ∈ N ∪ { 0 } Σ n {\displaystyle \Sigma ^{*}=\bigcup _{n\in \mathbb {N} \cup \{0\}}\Sigma ^{n}} For example, if Σ = {0, 1}, then Σ* = {ε, 0, 1, 00, 01, 10, 11, 000, 001, 010, 011, ...}. Although the set Σ* itself is countably infinite, each element of Σ* is a string of finite length. A set of st... | Wikipedia - Sequence of symbols | null | null | null |
Let Σ {\displaystyle \Sigma } be the projective space P G ( 2 n + 1 , K ) {\displaystyle PG(2n+1,K)} for n ≥ 1 {\displaystyle n\geq 1} an integer, and K {\displaystyle K} a division ring. A regulus R {\displaystyle R} in Σ {\displaystyle \Sigma } is a collection of pairwise disjoint n {\displaystyle n} -dimensional sub... | Wikipedia - Spread (projective geometry) | null | null | null |
Let Φ ( r → , t , μ a = 0 ) {\displaystyle \Phi ({\vec {r}},t,\mu _{a}=0)} be the Green function solution to the diffusion equation for a non-absorbing homogeneous medium. Then, the Green function solution for the medium when its absorption coefficient is μ a {\displaystyle \mu _{a}} can be obtained as: Φ ( r → , t , μ... | Wikipedia - Diffusion theory | null | null | null |
Let Φ(A)− denote the weak-operator closure of Φ(A) in B(HΦ). Each bounded linear functional ρ on Φ(A) is weak-operator continuous and extends uniquely preserving norm, to a weak-operator continuous linear functional ρ on the von Neumann algebra Φ(A)−. If ρ is hermitian, or positive, the same is true of ρ. The mapping ρ... | Wikipedia - Universal representation (C*-algebra) | null | null | null |
Let Φ: V → W be a linear map between vector spaces V and W (i.e., Φ is an element of L(V, W), also denoted Hom(V, W)), and let be a multilinear form on W (also known as a tensor – not to be confused with a tensor field – of rank (0, s), where s is the number of factors of W in the product). Then the pullback Φ∗F of F b... | Wikipedia - Pullback (differential geometry) | null | null | null |
Let Χ be a set of objects called names. The abstract syntax for the π-calculus is built from the following BNF grammar (where x and y are any names from Χ): P , Q ::= x ( y ) . P Receive on channel x , bind the result to y , then run P | x ¯ ⟨ y ⟩ . P Send the value y over channel x , then run P | P | Q Run P and Q sim... | Wikipedia - Π-calculus | null | null | null |
P Repeatedly spawn copies of P | 0 Terminate the process {\displaystyle {\begin{aligned}P,Q::=&\;x(y).P\,\,\,\,\,&{\text{Receive on channel }}x{\text{, bind the result to }}y{\text{, then run }}P\\&\;|\;{\overline {x}}\langle y\rangle .P\,\,\,\,\,&{\text{Send the value }}y{\text{ over channel }}x{\text{, then run }}P\\... | Wikipedia - Π-calculus | null | null | null |
Let Ω be a subdomain (bounded or not) of R n {\displaystyle \mathbb {R} ^{n}} (with n an integer). Denote by Γ its boundary (assumed smooth). Consider the following heat equation on Ω × (0, T), for T > 0, u t − Δ u = 0 in Ω × ( 0 , T ) , u = 0 on Γ × ( 0 , T ) , {\displaystyle {\begin{array}{rcll}u_{t}-\Delta u&=&0&{\m... | Wikipedia - Flow (mathematics) | null | null | null |
The mathematical setting for this problem can be the semigroup approach. To use this tool, we introduce the unbounded operator ΔD defined on L 2 ( Ω ) {\displaystyle L^{2}(\Omega )} by its domain D ( Δ D ) = H 2 ( Ω ) ∩ H 0 1 ( Ω ) {\displaystyle D(\Delta _{D})=H^{2}(\Omega )\cap H_{0}^{1}(\Omega )} (see the classical ... | Wikipedia - Flow (mathematics) | null | null | null |
{\displaystyle \Delta _{D}v=\Delta v=\sum _{i=1}^{n}{\frac {\partial ^{2}}{\partial x_{i}^{2}}}v~.} With this operator, the heat equation becomes u ′ ( t ) = Δ D u ( t ) {\displaystyle u'(t)=\Delta _{D}u(t)} and u(0) = u0. Thus, the flow corresponding to this equation is (see notations above) φ ( u 0 , t ) = e t Δ D u ... | Wikipedia - Flow (mathematics) | null | null | null |
Let Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} denote the space of smooth m-forms with compact support on a smooth manifold M . {\displaystyle M.} A current is a linear functional on Ω c m ( M ) {\displaystyle \Omega _{c}^{m}(M)} which is continuous in the sense of distributions. Thus a linear functional is an m-di... | Wikipedia - Current (mathematics) | null | null | null |
The space D m ( M ) {\displaystyle {\mathcal {D}}_{m}(M)} of m-dimensional currents on M {\displaystyle M} is a real vector space with operations defined by Much of the theory of distributions carries over to currents with minimal adjustments. For example, one may define the support of a current T ∈ D m ( M ) {\display... | Wikipedia - Current (mathematics) | null | null | null |
Let Ω {\displaystyle \Omega } be a connected open set in the complex plane C {\displaystyle \mathbb {C} } and f: Ω → C {\displaystyle f:\Omega \to \mathbb {C} } a holomorphic function. If f {\displaystyle f} is not constant, then the set of the critical points of f {\displaystyle f} , that is, the zeros of the derivati... | Wikipedia - Branch point | null | null | null |
The winding number of f ( γ ) {\displaystyle f(\gamma )} with respect to the point f ( z 0 ) {\displaystyle f(z_{0})} is a positive integer called the ramification index of z 0 {\displaystyle z_{0}} . If the ramification index is greater than 1, then z 0 {\displaystyle z_{0}} is called a ramification point of f {\displ... | Wikipedia - Branch point | null | null | null |
Typically, one is not interested in f {\displaystyle f} itself, but in its inverse function. However, the inverse of a holomorphic function in the neighborhood of a ramification point does not properly exist, and so one is forced to define it in a multiple-valued sense as a global analytic function. It is common to abu... | Wikipedia - Branch point | null | null | null |
More general definitions of branch points are possible for other kinds of multiple-valued global analytic functions, such as those that are defined implicitly. A unifying framework for dealing with such examples is supplied in the language of Riemann surfaces below. | Wikipedia - Branch point | null | null | null |
In particular, in this more general picture, poles of order greater than 1 can also be considered ramification points. In terms of the inverse global analytic function f − 1 {\displaystyle f^{-1}} , branch points are those points around which there is nontrivial monodromy. For example, the function f ( z ) = z 2 {\disp... | Wikipedia - Branch point | null | null | null |
The inverse function is the square root f − 1 ( w ) = w 1 / 2 {\displaystyle f^{-1}(w)=w^{1/2}} , which has a branch point at w 0 = 0 {\displaystyle w_{0}=0} . Indeed, going around the closed loop w = e i θ {\displaystyle w=e^{i\theta }} , one starts at θ = 0 {\displaystyle \theta =0} and e i 0 / 2 = 1 {\displaystyle e... | Wikipedia - Branch point | null | null | null |
Let Ω {\displaystyle \Omega } be an open bounded domain of R d {\displaystyle \mathbb {R} ^{d}} , with d ≥ 2 {\displaystyle d\geq 2} , which is subject to a nonsmooth perturbation confined in a small region ω ε ( x ~ ) = x ~ + ε ω {\displaystyle \omega _{\varepsilon }({\tilde {x}})={\tilde {x}}+\varepsilon \omega } of ... | Wikipedia - Topological derivative | null | null | null |
Let Ω: Ψ(Rn+1)n+1 → R be a volume form defined on Rn+1. We can induce a volume form on M given by ω: Ψ(M)n → R given by ω(X1,...,Xn) := Ω(X1,...,Xn,ξ). This is a natural definition: in Euclidean differential geometry where ξ is the Euclidean unit normal then the standard Euclidean volume spanned by X1,...,Xn is always ... | Wikipedia - Affine differential geometry | null | null | null |
Let α 1 , … , α n {\displaystyle \alpha _{1},\dots ,\alpha _{n}} be the roots of P(X). The so-called Lagrange factors of P(X) are the cofactors of these roots, If all roots are different, then the Lagrange factors form a basis of the space of polynomials of degree at most n − 1. By analysis of the recursion procedure o... | Wikipedia - Jenkins-Traub Algorithm for Polynomial Zeros | null | null | null |
Let α i ( t ) = P ( Y 1 = y 1 , … , Y t = y t , X t = i ∣ θ ) {\displaystyle \alpha _{i}(t)=P(Y_{1}=y_{1},\ldots ,Y_{t}=y_{t},X_{t}=i\mid \theta )} , the probability of seeing the observations y 1 , y 2 , … , y t {\displaystyle y_{1},y_{2},\ldots ,y_{t}} and being in state i {\displaystyle i} at time t {\displaystyle t... | Wikipedia - Baum–Welch algorithm | null | null | null |
Let β i ( t ) = P ( Y t + 1 = y t + 1 , … , Y T = y T ∣ X t = i , θ ) {\displaystyle \beta _{i}(t)=P(Y_{t+1}=y_{t+1},\ldots ,Y_{T}=y_{T}\mid X_{t}=i,\theta )} that is the probability of the ending partial sequence y t + 1 , … , y T {\displaystyle y_{t+1},\ldots ,y_{T}} given starting state i {\displaystyle i} at time t... | Wikipedia - Baum–Welch algorithm | null | null | null |
Let δ be a derivation (background) of h and denote by g the one-dimensional Lie algebra spanned by δ. Define the Lie bracket on e = g ⊕ h by = = + λ δ ( H 1 ) − μ δ ( H 2 ) . {\displaystyle ==+\lambda \delta (H_{1})-\mu \delta (H_{2}).} It is obvious from the definition of the bracket that h is and ideal in e in and... | Wikipedia - Lie algebra extension | null | null | null |
Let ζ be a root of unity. Then the integral closure of Z in the cyclotomic field Q(ζ) is Z. This can be found by using the minimal polynomial and using Eisenstein's criterion. | Wikipedia - Integral element | null | null | null |
Let λ i {\displaystyle \lambda _{i}} be an eigenvalue of A {\displaystyle A} of algebraic multiplicity μ i {\displaystyle \mu _{i}} . First, find the ranks (matrix ranks) of the matrices ( A − λ i I ) , ( A − λ i I ) 2 , … , ( A − λ i I ) m i {\displaystyle (A-\lambda _{i}I),(A-\lambda _{i}I)^{2},\ldots ,(A-\lambda _{i... | Wikipedia - Canonical basis | null | null | null |
{\displaystyle \rho _{k}=\operatorname {rank} (A-\lambda _{i}I)^{k-1}-\operatorname {rank} (A-\lambda _{i}I)^{k}\qquad (k=1,2,\ldots ,m_{i}).} The variable ρ k {\displaystyle \rho _{k}} designates the number of linearly independent generalized eigenvectors of rank k (generalized eigenvector rank; see generalized eigenv... | Wikipedia - Canonical basis | null | null | null |
Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μA(λi) of the eigenvalue is its multiplicity as a root of the characteristic polynomial, that is, the largest integer k such that (λ − λi)k divides evenly that polynomial.Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas ... | Wikipedia - Characteristic root | null | null | null |
Let ξ i = ( E i , M i , p i ) {\displaystyle \xi _{i}=(E_{i},M_{i},p_{i})} be a rank n vector bundle over an n-dimensional smooth manifold M i {\displaystyle M_{i}} for i = 1,2. Denote by D ( E i ) {\displaystyle D(E_{i})} the total space of the associated (closed) disk bundle D ( ξ i ) {\displaystyle D(\xi _{i})} and ... | Wikipedia - Plumbing (mathematics) | null | null | null |
Let h: D 1 n → D 2 n {\displaystyle h:D_{1}^{n}\rightarrow D_{2}^{n}} and k: D 1 n → D 2 n {\displaystyle k:D_{1}^{n}\rightarrow D_{2}^{n}} be two diffeomorphisms (either both orientation preserving or reversing). The plumbing of D ( E 1 ) {\displaystyle D(E_{1})} and D ( E 2 ) {\displaystyle D(E_{2})} at x 1 {\display... | Wikipedia - Plumbing (mathematics) | null | null | null |
Let π {\displaystyle \pi } be an irreducible, finite-dimensional representation of a complex semisimple Lie algebra g {\displaystyle {\mathfrak {g}}} . Suppose h {\displaystyle {\mathfrak {h}}} is a Cartan subalgebra of g {\displaystyle {\mathfrak {g}}} . The character of π {\displaystyle \pi } is then the function ch ... | Wikipedia - Weyl character | null | null | null |
Let π {\displaystyle \pi } denote a probability density function on R d {\displaystyle \mathbb {R} ^{d}} , one from which it is desired to draw an ensemble of independent and identically distributed samples. We consider the overdamped Langevin Itô diffusion X ˙ = ∇ log π ( X ) + 2 W ˙ {\displaystyle {\dot {X}}=\nabla... | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
It turns out that, in fact, ρ ∞ = π {\displaystyle \rho _{\infty }=\pi } . Approximate sample paths of the Langevin diffusion can be generated by many discrete-time methods. One of the simplest is the Euler–Maruyama method with a fixed time step τ > 0 {\displaystyle \tau >0} . | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
We set X 0 := x 0 {\displaystyle X_{0}:=x_{0}} and then recursively define an approximation X k {\displaystyle X_{k}} to the true solution X ( k τ ) {\displaystyle X(k\tau )} by X k + 1 := X k + τ ∇ log π ( X k ) + 2 τ ξ k , {\displaystyle X_{k+1}:=X_{k}+\tau \nabla \log \pi (X_{k})+{\sqrt {2\tau }}\xi _{k},} where e... | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
We consider the above update rule as defining a proposal X ~ k + 1 {\displaystyle {\tilde {X}}_{k+1}} for a new state, X ~ k + 1 := X k + τ ∇ log π ( X k ) + 2 τ ξ k . {\displaystyle {\tilde {X}}_{k+1}:=X_{k}+\tau \nabla \log \pi (X_{k})+{\sqrt {2\tau }}\xi _{k}.} This proposal is accepted or rejected according to th... | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
Let u {\displaystyle u} be drawn from the continuous uniform distribution on the interval {\displaystyle } . If u ≤ α {\displaystyle u\leq \alpha } , then the proposal is accepted, and we set X k + 1 := X ~ k + 1 {\displaystyle X_{k+1}:={\tilde {X}}_{k+1}} ; otherwise, the proposal is rejected, and we set X k + 1 := X... | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
Compared to naive Metropolis–Hastings, MALA has the advantage that it usually proposes moves into regions of higher π {\displaystyle \pi } probability, which are then more likely to be accepted. On the other hand, when π {\displaystyle \pi } is strongly anisotropic (i.e. it varies much more quickly in some directions t... | Wikipedia - Metropolis-adjusted Langevin algorithm | null | null | null |
Let σ = { + , × , − , 0 , 1 } {\displaystyle \sigma =\{+,\times ,-,0,1\}} be again the standard signature for fields. When regarded as σ {\displaystyle \sigma } -structures in the natural way, the rational numbers form a substructure of the real numbers, and the real numbers form a substructure of the complex numbers. ... | Wikipedia - Interpretation function | null | null | null |
Indeed, the integers are the substructure of the real numbers generated by the empty set, using this signature. The notion in abstract algebra that corresponds to a substructure of a field, in this signature, is that of a subring, rather than that of a subfield. The most obvious way to define a graph is a structure wit... | Wikipedia - Interpretation function | null | null | null |
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