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So for short, from that time on we may represent Γ as one formula instead of a set. Another omission for convenience is when Γ is an empty set, in which case Γ may not appear. Negation introduction From ( p → q ) {\displaystyle (p\to q)} and ( p → ¬ q ) {\displaystyle (p\to \neg q)} , infer ¬ p {\displaystyle \neg p} . | Wikipedia - Sentential calculus | null | null | null |
That is, { ( p → q ) , ( p → ¬ q ) } ⊢ ¬ p {\displaystyle \{(p\to q),(p\to \neg q)\}\vdash \neg p} . Negation elimination From ¬ p {\displaystyle \neg p} , infer ( p → r ) {\displaystyle (p\to r)} . That is, { ¬ p } ⊢ ( p → r ) {\displaystyle \{\neg p\}\vdash (p\to r)} . | Wikipedia - Sentential calculus | null | null | null |
Double negation elimination From ¬ ¬ p {\displaystyle \neg \neg p} , infer p. That is, ¬ ¬ p ⊢ p {\displaystyle \neg \neg p\vdash p} . Conjunction introduction From p and q, infer ( p ∧ q ) {\displaystyle (p\land q)} . That is, { p , q } ⊢ ( p ∧ q ) {\displaystyle \{p,q\}\vdash (p\land q)} . | Wikipedia - Sentential calculus | null | null | null |
Conjunction elimination From ( p ∧ q ) {\displaystyle (p\land q)} , infer p. From ( p ∧ q ) {\displaystyle (p\land q)} , infer q. That is, ( p ∧ q ) ⊢ p {\displaystyle (p\land q)\vdash p} and ( p ∧ q ) ⊢ q {\displaystyle (p\land q)\vdash q} . Disjunction introduction From p, infer ( p ∨ q ) {\displaystyle (p\lor q)} . ... | Wikipedia - Sentential calculus | null | null | null |
That is, p ⊢ ( p ∨ q ) {\displaystyle p\vdash (p\lor q)} and q ⊢ ( p ∨ q ) {\displaystyle q\vdash (p\lor q)} . Disjunction elimination From ( p ∨ q ) {\displaystyle (p\lor q)} and ( p → r ) {\displaystyle (p\to r)} and ( q → r ) {\displaystyle (q\to r)} , infer r. That is, { p ∨ q , p → r , q → r } ⊢ r {\displaystyle \... | Wikipedia - Sentential calculus | null | null | null |
That is, { p → q , q → p } ⊢ ( p ↔ q ) {\displaystyle \{p\to q,q\to p\}\vdash (p\leftrightarrow q)} . Biconditional elimination From ( p ↔ q ) {\displaystyle (p\leftrightarrow q)} , infer ( p → q ) {\displaystyle (p\to q)} . From ( p ↔ q ) {\displaystyle (p\leftrightarrow q)} , infer ( q → p ) {\displaystyle (q\to p)} ... | Wikipedia - Sentential calculus | null | null | null |
That is, ( p ↔ q ) ⊢ ( p → q ) {\displaystyle (p\leftrightarrow q)\vdash (p\to q)} and ( p ↔ q ) ⊢ ( q → p ) {\displaystyle (p\leftrightarrow q)\vdash (q\to p)} . Modus ponens (conditional elimination) From p and ( p → q ) {\displaystyle (p\to q)} , infer q. That is, { p , p → q } ⊢ q {\displaystyle \{p,p\to q\}\vdash ... | Wikipedia - Sentential calculus | null | null | null |
Let L = (P, G, I) be an incidence structure, for which the elements of P are called points and the elements of G are called lines. L is a linear space if the following three axioms hold: (L1) two distinct points are incident with exactly one line. (L2) every line is incident to at least two distinct points. (L3) L cont... | Wikipedia - Linear space (geometry) | null | null | null |
Let L be a Lie algebra over a field K, then it is possible to define the structure of a chain complex on the exterior algebra of L. This is a K-linear mapping ∂: ⋀ p + 1 L → ⋀ p L {\displaystyle \partial :{\textstyle \bigwedge }^{p+1}L\to {\textstyle \bigwedge }^{p}L} defined on decomposable elements by ∂ ( x 1 ∧ ⋯ ∧ x... | Wikipedia - Exterior power | null | null | null |
Let L be a Lie algebra over a field k of characteristic p>0. A p operation on L is a map X ↦ X {\displaystyle X\mapsto X^{}} satisfying a d ( X ) = a d ( X ) p {\displaystyle \mathrm {ad} (X^{})=\mathrm {ad} (X)^{p}} for all X ∈ L {\displaystyle X\in L} , ( t X ) = t p X {\displaystyle (tX)^{}=t^{p}X^{}} for all t ... | Wikipedia - Restricted Lie algebra | null | null | null |
Let L be the maximum length any code word is permitted to have. Let p1, …, pn be the frequencies of the symbols of the alphabet to be encoded. We first sort the symbols so that pi ≤ pi+1. Create L coins for each symbol, of denominations 2−1, …, 2−L, each of numismatic value pi. | Wikipedia - Package-merge algorithm | null | null | null |
Use the package-merge algorithm to select the set of coins of minimum numismatic value whose denominations total n − 1. Let hi be the number of coins of numismatic value pi selected. The optimal length-limited Huffman code will encode symbol i with a bit string of length hi. The canonical Huffman code can easily be con... | Wikipedia - Package-merge algorithm | null | null | null |
Let L {\displaystyle {\mathcal {L}}} be a first-order language and T {\displaystyle T} be a theory over L . {\displaystyle {\mathcal {L}}.} For a model A {\displaystyle {\mathfrak {A}}} of T {\displaystyle T} one expands L {\displaystyle {\mathcal {L}}} to a new language L A := L ∪ { c a: a ∈ A } {\displaystyle {\mathc... | Wikipedia - Diagram (mathematical logic) | null | null | null |
Now one may expand A {\displaystyle {\mathfrak {A}}} to the model A A := ( A , a ) a ∈ A . {\displaystyle {\mathfrak {A}}_{A}:=({\mathfrak {A}},a)_{a\in A}.} The positive diagram of A {\displaystyle {\mathfrak {A}}} , sometimes denoted D + ( A ) {\displaystyle D^{+}({\mathfrak {A}})} , is the set of all those atomic se... | Wikipedia - Diagram (mathematical logic) | null | null | null |
The diagram D ( A ) {\displaystyle D({\mathfrak {A}})} of A {\displaystyle {\mathfrak {A}}} is the set of all atomic sentences and negations of atomic sentences of L A {\displaystyle {\mathcal {L}}_{A}} that hold in A A . {\displaystyle {\mathfrak {A}}_{A}.} Symbolically, D ( A ) = D + ( A ) ∪ ¬ D − ( A ) {\displaystyl... | Wikipedia - Diagram (mathematical logic) | null | null | null |
Let M = X n r {\displaystyle M=X^{nr}} be a differentiable manifold of dimension N=nr. A d-web W(d,n,r) of codimension r in an open set D ⊂ X n r {\displaystyle D\subset X^{nr}} is a set of d foliations of codimension r which are in general position. In the notation W(d,n,r) the number d is the number of foliations for... | Wikipedia - Web (differential geometry) | null | null | null |
Let M = x 1 a 1 ⋯ x n a n {\displaystyle M=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}} and N = x 1 b 1 ⋯ x n b n {\displaystyle N=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}} be two monomials, with exponent vectors A = {\displaystyle A=} and B = . {\displaystyle B=.} One says that M divides N, or that N is a multiple of M, if a i ≤ b i... | Wikipedia - Multivariate division algorithm | null | null | null |
{\textstyle {\frac {N}{M}}=x_{1}^{b_{1}-a_{1}}\cdots x_{n}^{b_{n}-a_{n}}.} In other words, the exponent vector of N M {\textstyle {\frac {N}{M}}} is the componentwise subtraction of the exponent vectors of N and M. The greatest common divisor gcd(M, N) of M and N is the monomial x 1 min ( a 1 , b 1 ) ⋯ x n min ( a n , ... | Wikipedia - Multivariate division algorithm | null | null | null |
Let M be a JBW factor. The inner automorphisms of M are those generated by the period two automorphisms Q(1 – 2p) where p is a projection. Two projections are equivalent if there is an inner automorphism carrying one onto the other. Given two projections in a factor, one of them is always equivalent to a sub-projection... | Wikipedia - Jordan operator algebra | null | null | null |
If each is equivalent to a sub-projection of the other, they are equivalent. A JBW factor can be classified into three mutually exclusive types as follows: It is type I if there is a minimal projection. It is type In if 1 can be written as a sum of n orthogonal minimal projections for 1 ≤ n ≤ ∞. | Wikipedia - Jordan operator algebra | null | null | null |
It is Type II if there are no minimal projections but the subprojections of some fixed projections e form a modular lattice, i.e. p ≤ q implies (p ∨ r) ∧ q = p ∨ (r ∧ q) for any projection r ≤ e. If e can be taken to be 1, it is Type II1. Otherwise it is type II≈. It is Type III if the projections do not form a modular... | Wikipedia - Jordan operator algebra | null | null | null |
Let M be a complex Kähler surface with c 1 = 0 {\displaystyle c_{1}=0} (K3 surface or a torus). The canonical bundle of M is trivial, as follows from the Kodaira classification of surfaces. Hence M admits a holomorphic symplectic form. It was observed by Akira Fujiki (for n = 2 {\displaystyle n=2} ) and Arnaud Beauvill... | Wikipedia - Hilbert scheme | null | null | null |
This is not very difficult to see, e.g., for n = 2 {\displaystyle n=2} . Indeed, M {\displaystyle M^{}} is a blow-up of a symmetric square of M. Singularities of Sym 2 M {\displaystyle \operatorname {Sym} ^{2}M} are locally isomorphic to C 2 × C 2 / { ± 1 } {\displaystyle \mathbb {C} ^{2}\times \mathbb {C} ^{2}/\{\p... | Wikipedia - Hilbert scheme | null | null | null |
This is used to show that the symplectic form is naturally extended to the smooth part of the exceptional divisors of M {\displaystyle M^{}} . It is extended to the rest of M {\displaystyle M^{}} by Hartogs' principle. A holomorphically symplectic, Kähler manifold is hyperkähler, as follows from the Calabi–Yau theore... | Wikipedia - Hilbert scheme | null | null | null |
Let M be a differentiable manifold and p a point of M. An affine connection on M allows one to define the notion of a straight line through the point p.Let v ∈ TpM be a tangent vector to the manifold at p. Then there is a unique geodesic γv: → M satisfying γv(0) = p with initial tangent vector γ′v(0) = v. The correspon... | Wikipedia - Exponential map (Riemannian geometry) | null | null | null |
Let M be a differentiable manifold and p a point of M. Suppose that f is a function defined in a neighborhood of p, and differentiable at p. If v is a tangent vector to M at p, then the directional derivative of f along v, denoted variously as df(v) (see Exterior derivative), ∇ v f ( p ) {\displaystyle \nabla _{\mathbf... | Wikipedia - Directional derivative | null | null | null |
Let M be a graded module over a graded algebra, which is generated over a field by its elements of positive degree. Then M has a free resolution in which the free modules Ei may be graded in such a way that the di and ε are graded linear maps. Among these graded free resolutions, the minimal free resolutions are those ... | Wikipedia - Flat resolution | null | null | null |
Let M be a real-analytic manifold of dimension n, and let X be its complexification. The sheaf of microlocal functions on M is given as H n ( μ M ( O X ) ⊗ o r M / X ) {\displaystyle {\mathcal {H}}^{n}(\mu _{M}({\mathcal {O}}_{X})\otimes {\mathcal {or}}_{M/X})} where μ M {\displaystyle \mu _{M}} denotes the microlocali... | Wikipedia - Microlocal calculus | null | null | null |
Let M be a smooth manifold. A (smooth) singular k-simplex in M is defined as a smooth map from the standard simplex in Rk to M. The group Ck(M, Z) of singular k-chains on M is defined to be the free abelian group on the set of singular k-simplices in M. These groups, together with the boundary map, ∂, define a chain co... | Wikipedia - Stokes' Theorem | null | null | null |
the singular cohomology group Hk(M, Z)), defined using continuous rather than smooth simplices in M. On the other hand, the differential forms, with exterior derivative, d, as the connecting map, form a cochain complex, which defines the de Rham cohomology groups H d R k ( M , R ) {\displaystyle H_{dR}^{k}(M,\mathbf {R... | Wikipedia - Stokes' Theorem | null | null | null |
This gives a linear map from the space of k-forms to the kth group of singular cochains, Ck(M, Z), the linear functionals on Ck(M, Z). In other words, a k-form ω defines a functional on the k-chains. Stokes' theorem says that this is a chain map from de Rham cohomology to singular cohomology with real coefficients; the... | Wikipedia - Stokes' Theorem | null | null | null |
This gives a homomorphism from de Rham cohomology to singular cohomology. On the level of forms, this means: closed forms, i.e., dω = 0, have zero integral over boundaries, i.e. over manifolds that can be written as ∂Σc Mc, and exact forms, i.e., ω = dσ, have zero integral over cycles, i.e. if the boundaries sum up to ... | Wikipedia - Stokes' Theorem | null | null | null |
In other words, if {ci} are cycles generating the kth homology group, then for any corresponding real numbers, {ai} , there exist a closed form, ω, such that and this form is unique up to exact forms. Stokes' theorem on smooth manifolds can be derived from Stokes' theorem for chains in smooth manifolds, and vice versa.... | Wikipedia - Stokes' Theorem | null | null | null |
Let M be a smooth manifold. A smooth differential form of degree k is a smooth section of the kth exterior power of the cotangent bundle of M. The set of all differential k-forms on a manifold M is a vector space, often denoted Ωk(M). The definition of a differential form may be restated as follows. At any point p ∈ M,... | Wikipedia - Differential forms | null | null | null |
This space is naturally isomorphic to the fiber at p of the dual bundle of the kth exterior power of the tangent bundle of M. That is, β is also a linear functional β p: ⋀ k T p M → R {\textstyle \beta _{p}\colon {\textstyle \bigwedge }^{k}T_{p}M\to \mathbf {R} } , i.e. the dual of the kth exterior power is isomorphic ... | Wikipedia - Differential forms | null | null | null |
In the presence of an inner product on TpM (induced by a Riemannian metric on M), αp may be represented as the inner product with a tangent vector Xp. Differential 1-forms are sometimes called covariant vector fields, covector fields, or "dual vector fields", particularly within physics. | Wikipedia - Differential forms | null | null | null |
The exterior algebra may be embedded in the tensor algebra by means of the alternation map. The alternation map is defined as a mapping Alt: ⨂ k T ∗ M → ⨂ k T ∗ M . {\displaystyle \operatorname {Alt} \colon {\bigotimes }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.} | Wikipedia - Differential forms | null | null | null |
For a tensor τ {\displaystyle \tau } at a point p, Alt ( τ p ) ( x 1 , … , x k ) = 1 k ! ∑ σ ∈ S k sgn ( σ ) τ p ( x σ ( 1 ) , … , x σ ( k ) ) , {\displaystyle \operatorname {Alt} (\tau _{p})(x_{1},\dots ,x_{k})={\frac {1}{k! }}\sum _{\sigma \in S_{k}}\operatorname {sgn}(\sigma )\tau _{p}(x_{\sigma (1)},\dots ,x_{\... | Wikipedia - Differential forms | null | null | null |
The alternation map is constant on the cosets of the ideal in the tensor algebra generated by the symmetric 2-forms, and therefore descends to an embedding Alt: ⋀ k T ∗ M → ⨂ k T ∗ M . {\displaystyle \operatorname {Alt} \colon {\textstyle \bigwedge }^{k}T^{*}M\to {\bigotimes }^{k}T^{*}M.} This map exhibits β as a total... | Wikipedia - Differential forms | null | null | null |
Let M {\displaystyle M} be a C*-algebra (more generally, an operator system in a C*-algebra A {\displaystyle A} ) with identity 1. {\displaystyle 1.} Let M + {\displaystyle M^{+}} denote the set of positive elements in M . {\displaystyle M.} | Wikipedia - Positive linear functional | null | null | null |
A linear functional ρ {\displaystyle \rho } on M {\displaystyle M} is said to be positive if ρ ( a ) ≥ 0 , {\displaystyle \rho (a)\geq 0,} for all a ∈ M + . {\displaystyle a\in M^{+}.} Theorem. A linear functional ρ {\displaystyle \rho } on M {\displaystyle M} is positive if and only if ρ {\displaystyle \rho } is bound... | Wikipedia - Positive linear functional | null | null | null |
Let M {\displaystyle M} be a Riemannian manifold and let G {\displaystyle G} be a Lie group of isometries of M {\displaystyle M} acting transitively on the sphere bundle S M . {\displaystyle SM.} Under these assumptions the space V ∞ ( M ) G {\displaystyle {\mathcal {V}}^{\infty }(M)^{G}} of G {\displaystyle G} -invari... | Wikipedia - Valuation (geometry) | null | null | null |
{\displaystyle M.} Then integrals of the form ∫ G ϕ i ( A ∩ g B ) d g {\displaystyle \int _{G}\phi _{i}(A\cap gB)dg} are expressible as linear combinations of ϕ k ( A ) ϕ l ( B ) {\displaystyle \phi _{k}(A)\phi _{l}(B)} with coefficients c i k l {\displaystyle c_{i}^{kl}} independent of A {\displaystyle A} and B {\disp... | Wikipedia - Valuation (geometry) | null | null | null |
Their existence in this generality was proved by Fu. For the three simply connected real space forms, that is, the sphere, Euclidean space, and hyperbolic space, they go back to Blaschke, Santaló, Chern, and Federer. Describing the kinematic formulas explicitly is typically a difficult problem. | Wikipedia - Valuation (geometry) | null | null | null |
In fact already in the step from real to complex space forms, considerable difficulties arise and these have only recently been resolved by Bernig, Fu, and Solanes. The key insight responsible for this progress is that the kinematic formulas contain the same information as the algebra of invariant valuations V ∞ ( M ) ... | Wikipedia - Valuation (geometry) | null | null | null |
For a precise statement, let be the kinematic operator, that is, the map determined by the kinematic formulas (2). Let denote the Alesker-Poincaré duality, which is a linear isomorphism. Finally let m G ∗ {\displaystyle m_{G}^{*}} be the adjoint of the product map The Fundamental theorem of algebraic integral geometry ... | Wikipedia - Valuation (geometry) | null | null | null |
Let M {\displaystyle M} be a closed vector subspace of a Hausdorff locally convex space X . {\displaystyle X.} If m ′ ∈ M ′ {\displaystyle m^{\prime }\in M^{\prime }} and if x ′ ∈ X ′ {\displaystyle x^{\prime }\in X^{\prime }} is a continuous linear extension of m ′ {\displaystyle m^{\prime }} to X {\displaystyle X} th... | Wikipedia - Algebraic adjoint | null | null | null |
Let M {\displaystyle M} be a closed vector subspace of a Hausdorff locally convex space X {\displaystyle X} and denote the canonical quotient map by Assume X / M {\displaystyle X/M} is endowed with the quotient topology induced by the quotient map π: X → X / M . {\displaystyle \pi :X\to X/M.} Then the transpose of the ... | Wikipedia - Transpose of a linear map | null | null | null |
{\displaystyle M^{\bot }.} If X {\displaystyle X} is a Banach space then t π: ( X / M ) ′ → M ⊥ {\displaystyle {}^{t}\pi :(X/M)^{\prime }\to M^{\bot }} is also an isometry. Using this transpose, every continuous linear functional on the quotient space X / M {\displaystyle X/M} is canonically identified with a continuou... | Wikipedia - Transpose of a linear map | null | null | null |
Let M {\displaystyle M} be a smooth manifold; a (smooth) distribution Δ {\displaystyle \Delta } assigns to any point x ∈ M {\displaystyle x\in M} a vector subspace Δ x ⊂ T x M {\displaystyle \Delta _{x}\subset T_{x}M} in a smooth way. More precisely, Δ {\displaystyle \Delta } consists in a collection { Δ x ⊂ T x M } x ... | Wikipedia - Singular distribution (differential geometry) | null | null | null |
{\displaystyle \{X_{1}(y),\ldots ,X_{k}(y)\}=\Delta _{y}.} The set of smooth vector fields { X 1 , … , X k } {\displaystyle \{X_{1},\ldots ,X_{k}\}} is also called a local basis of Δ {\displaystyle \Delta } . Note that the number k {\displaystyle k} may be different for different neighbourhoods. The notation Δ {\displa... | Wikipedia - Singular distribution (differential geometry) | null | null | null |
Let M {\displaystyle M} be a variety over an algebraically closed field k {\displaystyle k} , G {\displaystyle G} an algebraic group, for example G m {\displaystyle \mathbb {G} _{m}} . Recall that a G-torsor over M {\displaystyle M} is an algebraic space P {\displaystyle P} with an action of G {\displaystyle G} and a m... | Wikipedia - Gerbe | null | null | null |
Here B G {\displaystyle BG} denotes the classifying stack of G {\displaystyle G} , i.e. a quotient {\displaystyle } of a point by a trivial G {\displaystyle G} -action. There is no need to impose the compatibility with the group structure in that case since it is covered by the definition of a stack. The underlying to... | Wikipedia - Gerbe | null | null | null |
Let M {\displaystyle M} be an n × n {\displaystyle n\times n} square matrix over some finite field F, let x b a s e {\displaystyle x_{\mathrm {base} }} be a random vector of length n {\displaystyle n} , and let x = M x b a s e {\displaystyle x=Mx_{\mathrm {base} }} . Consider the sequence of vectors S = {\displaystyle... | Wikipedia - Block Wiedemann algorithm | null | null | null |
But the Berlekamp–Massey algorithm allows us to calculate relatively efficiently some sequence q 0 … q L {\displaystyle q_{0}\ldots q_{L}} with ∑ i = 0 L q i S y = 0 ∀ r {\displaystyle \sum _{i=0}^{L}q_{i}S_{y}=0\;\forall \;r} . Our hope is that this sequence, which by construction annihilates y ⋅ S {\displaystyle y\c... | Wikipedia - Block Wiedemann algorithm | null | null | null |
Let M {\displaystyle M} be the maximum possible transmission rate in bytes/second. Then T max = { b / ( M − r ) if r < M ∞ otherwise {\displaystyle T_{\text{max}}={\begin{cases}b/(M-r)&{\text{ if }}r | Wikipedia - Token bucket | null | null | null |
Let M {\displaystyle {\mathcal {M}}} be a structure and let V = { v 1 , … , v n } {\displaystyle V=\{v_{1},\ldots ,v_{n}\}} be a finite set of variables. Then a team over M {\displaystyle {\mathcal {M}}} with domain V {\displaystyle V} is a set of assignments over M {\displaystyle {\mathcal {M}}} with domain V {\displa... | Wikipedia - Independence-friendly logic | null | null | null |
Let M {\displaystyle {\mathcal {M}}} denote a finite von Neumann algebra with center Z {\displaystyle {\mathcal {Z}}} . One of the fundamental characterizing properties of finite von Neumann algebras is the existence of a center-valued trace. A von Neumann algebra M {\displaystyle {\mathcal {M}}} is finite if and only ... | Wikipedia - Finite von Neumann algebra | null | null | null |
Let Mink be the category of open subsets of Minkowski space M with inclusion maps as morphisms. We are given a covariant functor A {\displaystyle {\mathcal {A}}} from Mink to uC*alg, the category of unital C* algebras, such that every morphism in Mink maps to a monomorphism in uC*alg (isotony). The Poincaré group acts ... | Wikipedia - Local quantum field theory | null | null | null |
There exists a pullback of this action, which is continuous in the norm topology of A ( M ) {\displaystyle {\mathcal {A}}(M)} (Poincaré covariance). Minkowski space has a causal structure. If an open set V lies in the causal complement of an open set U, then the image of the maps A ( i U , U ∪ V ) {\displaystyle {\math... | Wikipedia - Local quantum field theory | null | null | null |
If U ¯ {\displaystyle {\bar {U}}} is the causal completion of an open set U, then A ( i U , U ¯ ) {\displaystyle {\mathcal {A}}(i_{U,{\bar {U}}})} is an isomorphism (primitive causality). A state with respect to a C*-algebra is a positive linear functional over it with unit norm. If we have a state over A ( M ) {\displ... | Wikipedia - Local quantum field theory | null | null | null |
The states over the open sets form a presheaf structure. According to the GNS construction, for each state, we can associate a Hilbert space representation of A ( M ) . | Wikipedia - Local quantum field theory | null | null | null |
{\displaystyle {\mathcal {A}}(M).} Pure states correspond to irreducible representations and mixed states correspond to reducible representations. Each irreducible representation (up to equivalence) is called a superselection sector. We assume there is a pure state called the vacuum such that the Hilbert space associat... | Wikipedia - Local quantum field theory | null | null | null |
Let N be the set of nonnegative integers. A subset S of N is called a numerical semigroup if the following conditions are satisfied. 0 is an element of S N − S, the complement of S in N, is finite. If x and y are in S then x + y is also in S.There is a simple method to construct numerical semigroups. | Wikipedia - Numerical monoid | null | null | null |
Let A = {n1, n2, ..., nr} be a nonempty set of positive integers. The set of all integers of the form x1 n1 + x2 n2 + ... + xr nr is the subset of N generated by A and is denoted by ⟨ A ⟩. The following theorem fully characterizes numerical semigroups. | Wikipedia - Numerical monoid | null | null | null |
Let N {\displaystyle N} be a network with e {\displaystyle e} connections, m {\displaystyle m} inputs and n {\displaystyle n} outputs. Below, x 1 , x 2 , … {\displaystyle x_{1},x_{2},\dots } denote vectors in R m {\displaystyle \mathbb {R} ^{m}} , y 1 , y 2 , … {\displaystyle y_{1},y_{2},\dots } vectors in R n {\displa... | Wikipedia - Mathematics of artificial neural networks | null | null | null |
In supervised learning, a sequence of training examples ( x 1 , y 1 ) , … , ( x p , y p ) {\displaystyle (x_{1},y_{1}),\dots ,(x_{p},y_{p})} produces a sequence of weights w 0 , w 1 , … , w p {\displaystyle w_{0},w_{1},\dots ,w_{p}} starting from some initial weight w 0 {\displaystyle w_{0}} , usually chosen at random.... | Wikipedia - Mathematics of artificial neural networks | null | null | null |
The computation is the same in each step, hence only the case i = 1 {\displaystyle i=1} is described. w 1 {\displaystyle w_{1}} is calculated from ( x 1 , y 1 , w 0 ) {\displaystyle (x_{1},y_{1},w_{0})} by considering a variable weight w {\displaystyle w} and applying gradient descent to the function w ↦ E ( f N ( w , ... | Wikipedia - Mathematics of artificial neural networks | null | null | null |
Let N ⊂ M {\displaystyle N\subset M} be a finite index subfactor, and P {\displaystyle {\mathcal {P}}} the corresponding subfactor planar algebra. Assume that P {\displaystyle {\mathcal {P}}} is irreducible (i.e. P 1 , + = N ′ ∩ M 1 = C {\displaystyle {\mathcal {P}}_{1,+}=N'\cap M_{1}=\mathbb {C} } ). Let N ⊂ K ⊂ M {\d... | Wikipedia - Planar algebra | null | null | null |
Note that e K M ∈ P 2 , + {\displaystyle e_{K}^{M}\in {\mathcal {P}}_{2,+}} . Let i d := e M M {\displaystyle id:=e_{M}^{M}} and e 1 := e N M {\displaystyle e_{1}:=e_{N}^{M}} . Note that t r ( e 1 ) = δ − 2 = − 1 {\displaystyle tr(e_{1})=\delta ^{-2}=^{-1}} and t r ( i d ) = 1 {\displaystyle tr(id)=1} . | Wikipedia - Planar algebra | null | null | null |
Let the bijective linear map F: P 2 , ± → P 2 , ∓ {\displaystyle {\mathcal {F}}:{\mathcal {P}}_{2,\pm }\to {\mathcal {P}}_{2,\mp }} be the Fourier transform, also called 1 {\displaystyle 1} -click (of the outer star) or 90 ∘ {\displaystyle 90^{\circ }} rotation; and let a ∗ b {\displaystyle a*b} be the coproduct of a {... | Wikipedia - Planar algebra | null | null | null |
The coproduct satisfies the equality a ∗ b = F ( F − 1 ( a ) F − 1 ( b ) ) . {\displaystyle a*b={\mathcal {F}}({\mathcal {F}}^{-1}(a){\mathcal {F}}^{-1}(b)).} For any positive operators a , b {\displaystyle a,b} , the coproduct a ∗ b {\displaystyle a*b} is also positive; this can be seen diagrammatically: Let a ¯ := F ... | Wikipedia - Planar algebra | null | null | null |
The map F 4 {\displaystyle {\mathcal {F}}^{4}} corresponds to four 1 {\displaystyle 1} -clicks of the outer star, so it's the identity map, and then a ¯ ¯ = a {\displaystyle {\overline {\overline {a}}}=a} . In the Kac algebra case, the contragredient is exactly the antipode, which, for a finite group, correspond to the... | Wikipedia - Planar algebra | null | null | null |
Note that e 1 = e N M {\displaystyle e_{1}=e_{N}^{M}} and i d = e M M {\displaystyle id=e_{M}^{M}} are biprojections; this can be seen as follows: A projection b {\displaystyle b} is a biprojection iff it is the Jones projection e K M {\displaystyle e_{K}^{M}} of an intermediate subfactor N ⊂ K ⊂ M {\displaystyle N\sub... | Wikipedia - Planar algebra | null | null | null |
Noncommutative uncertainty principle: Let x ∈ P 2 , ± {\displaystyle x\in {\mathcal {P}}_{2,\pm }} , nonzero. Then S ( x ) S ( F ( x ) ) ≥ δ 2 {\displaystyle {\mathcal {S}}(x){\mathcal {S}}({\mathcal {F}}(x))\geq \delta ^{2}} Assuming x {\displaystyle x} and F ( x ) {\displaystyle {\mathcal {F}}(x)} positive, the equal... | Wikipedia - Planar algebra | null | null | null |
Let O1 and O2 be the centers of the two circles, C1 and C2 and let r1 and r2 be their radii, with r1 > r2; in other words, circle C1 is defined as the larger of the two circles. Two different methods may be used to construct the external and internal tangent lines. External tangentsA new circle C3 of radius r1 − r2 is ... | Wikipedia - Tangent lines to circles | null | null | null |
Using the method above, two lines are drawn from O2 that are tangent to this new circle. These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking both circles C1 and C2 by a constant amount, r2, which shrinks C2 to a point. Two radial lines may be drawn from the center O1 th... | Wikipedia - Tangent lines to circles | null | null | null |
The desired external tangent lines are the lines perpendicular to these radial lines at those tangent points, which may be constructed as described above. Internal tangentsA new circle C3 of radius r1 + r2 is drawn centered on O1. Using the method above, two lines are drawn from O2 that are tangent to this new circle. | Wikipedia - Tangent lines to circles | null | null | null |
These lines are parallel to the desired tangent lines, because the situation corresponds to shrinking C2 to a point while expanding C1 by a constant amount, r2. Two radial lines may be drawn from the center O1 through the tangent points on C3; these intersect C1 at the desired tangent points. The desired internal tange... | Wikipedia - Tangent lines to circles | null | null | null |
Let P (propositions) and A (actions) be two finite sets of symbols, and let Var be a countably infinite set of variables. The set of formulas of (propositional, modal) μ-calculus is defined as follows: each proposition and each variable is a formula; if ϕ {\displaystyle \phi } and ψ {\displaystyle \psi } are formulas, ... | Wikipedia - Modal mu calculus | null | null | null |
Given the above definitions, we can enrich the syntax with: ϕ ∨ ψ {\displaystyle \phi \lor \psi } meaning ¬ ( ¬ ϕ ∧ ¬ ψ ) {\displaystyle \neg (\neg \phi \land \neg \psi )} ⟨ a ⟩ ϕ {\displaystyle \langle a\rangle \phi } (pronounced either: a {\displaystyle a} diamond ϕ {\displaystyle \phi } or after a {\displaystyle a} ... | Wikipedia - Modal mu calculus | null | null | null |
Let P 1 = ( 2 , 3 ) {\displaystyle P_{1}=(2,{\sqrt {3}})} be a point on the curve 2 y 2 = x 3 − x 2 + x {\displaystyle 2y^{2}=x^{3}-x^{2}+x} . In coordinates ( X 1: Z 1 ) {\displaystyle (X_{1}:Z_{1})} , with x 1 = X 1 / Z 1 {\displaystyle x_{1}=X_{1}/Z_{1}} , P 1 = ( 2: 1 ) {\displaystyle P_{1}=(2:1)} . Then: X X 1 = X... | Wikipedia - Montgomery curve | null | null | null |
Let P = ( a , b ) {\displaystyle P=(a,b)} be a point of the circle with equation x 2 + y 2 = r 2 . {\displaystyle x^{2}+y^{2}=r^{2}.} The tangent at P has equation a x + b y = r 2 , {\displaystyle ax+by=r^{2},} because P lies on both the curves and O P → = ( a , b ) T {\displaystyle {\vec {OP}}=(a,b)^{T}} is a normal v... | Wikipedia - Tangent lines to circles | null | null | null |
{\displaystyle ax_{0}=r^{2}.} Conversely, if one starts with point P 0 = ( x 0 , 0 ) , {\displaystyle P_{0}=(x_{0},0),} than the two tangents through P0 meet the circle at the two points P 1 / 2 = ( a , b ± ) {\displaystyle P_{1/2}=(a,b_{\pm })} with Written in vector form: If point P 0 = ( x 0 , y 0 ) {\displaystyle P... | Wikipedia - Tangent lines to circles | null | null | null |
{\textstyle {\vec {e}}_{1}={\frac {1}{d_{0}}}{\binom {x_{0}}{y_{0}}},\ {\vec {e}}_{2}={\frac {1}{d_{0}}}{\binom {-y_{0}}{x_{0}}}.} Then the tangents through point P0 touch the circle at the points For d0 < r no tangents exist. | Wikipedia - Tangent lines to circles | null | null | null |
For d0 = r point P0 lies on the circle and there is just one tangent with equation x 0 x + y 0 y = r 2 . {\displaystyle x_{0}x+y_{0}y=r^{2}.} In case of d0 > r there are 2 tangents with equations x 1 x + y 1 y = r 2 , x 2 x + y 2 y = r 2 . | Wikipedia - Tangent lines to circles | null | null | null |
{\displaystyle x_{1}x+y_{1}y=r^{2},\ x_{2}x+y_{2}y=r^{2}.} Relation to circle inversion: Equation a x 0 = r 2 {\displaystyle ax_{0}=r^{2}} describes the circle inversion of point ( x 0 , 0 ) . {\displaystyle (x_{0},0).} Relation to pole and polar: The polar of point ( x 0 , 0 ) {\displaystyle (x_{0},0)} has equation x ... | Wikipedia - Tangent lines to circles | null | null | null |
Let P = (X1: Y1: Z1) be a point, then the doubling formula is given by: A = X12 B = Y12 D = A + B G = (X1 + Y1)2 − D X3 = (2Y1 − G) × (X1 + A + 1) Y3 = (G − 2X1) × (Y1 + B + 1) Z3 = (X1 − Y1) × (G + 2D)The cost of this algorithm is three multiplications + three squarings + 11 additions + 3×2. ExampleIf P = ( − 1: − 1: ... | Wikipedia - Hessian form of an elliptic curve | null | null | null |
Let P = p 0 + p 1 X + ⋯ + p m X m , Q = q 0 + q 1 X + ⋯ + q n X n . {\displaystyle P=p_{0}+p_{1}X+\cdots +p_{m}X^{m},\quad Q=q_{0}+q_{1}X+\cdots +q_{n}X^{n}.} be two univariate polynomials with coefficients in a field K. Let us denote by P i {\displaystyle {\mathcal {P}}_{i}} the K vector space of dimension i of polyno... | Wikipedia - Euclidean algorithm for polynomials | null | null | null |
The resultant of P and Q is the determinant of the Sylvester matrix, which is the (square) matrix of φ 0 {\displaystyle \varphi _{0}} on the bases of the powers of X. Similarly, the i-subresultant polynomial is defined in term of determinants of submatrices of the matrix of φ i . {\displaystyle \varphi _{i}.} Let us de... | Wikipedia - Euclidean algorithm for polynomials | null | null | null |
{\displaystyle S={\begin{pmatrix}p_{m}&0&\cdots &0&q_{n}&0&\cdots &0\\p_{m-1}&p_{m}&\cdots &0&q_{n-1}&q_{n}&\cdots &0\\p_{m-2}&p_{m-1}&\ddots &0&q_{n-2}&q_{n-1}&\ddots &0\\\vdots &\vdots &\ddots &p_{m}&\vdots &\vdots &\ddots &q_{n}\\\vdots &\vdots &\cdots &p_{m-1}&\vdots &\vdots &\cdots &q_{n-1}\\p_{0}&p_{1}&\cdots &\v... | Wikipedia - Euclidean algorithm for polynomials | null | null | null |
Let Vi be the (m + n − 2i) × (m + n − i) matrix defined as follows. First we add (i + 1) columns of zeros to the right of the (m + n − 2i − 1) × (m + n − 2i − 1) identity matrix. | Wikipedia - Euclidean algorithm for polynomials | null | null | null |
Then we border the bottom of the resulting matrix by a row consisting in (m + n − i − 1) zeros followed by Xi, Xi−1, ..., X, 1: V i = ( 1 0 ⋯ 0 0 0 ⋯ 0 0 1 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋱ ⋮ ⋮ ⋱ ⋮ 0 0 0 ⋯ 1 0 0 ⋯ 0 0 0 ⋯ 0 X i X i − 1 ⋯ 1 ) . {\displaystyle V_{i}={\begin{pmatrix}1&0&\cdots &0&0&0&\cdots &0\\0&1&\cdots &0&0&0&\cdots &... | Wikipedia - Euclidean algorithm for polynomials | null | null | null |
Let P and Q be combinatorially equivalent 3-dimensional convex polytopes; that is, they are convex polytopes with isomorphic face lattices. Suppose further that each pair of corresponding faces from P and Q are congruent to each other, i.e. equal up to a rigid motion. Then P and Q are themselves congruent. To see that ... | Wikipedia - Cauchy's theorem (geometry) | null | null | null |
Let P and c be arrays of length n with 1-based indexing (i.e. the first entry of an array has index 1). The algorithm for generating all n! permutations of the set {1, 2, ..., n} is given by the following pseudocode: P ← ; yield P; c ← ; (the first entry of c is not used) i ← 2; while i ≤ n do left-rotate the first i e... | Wikipedia - Tompkins–Paige algorithm | null | null | null |
Let P be a configuration of n points in a projective plane, not all on a line. Let t be the number of lines determined by P. Then, t ≥ n, and if t = n, any two lines have exactly one point of P in common. In this case, P is either a projective plane or P is a near pencil, meaning that exactly n - 1 of the points are co... | Wikipedia - De Bruijn–Erdős theorem (incidence geometry) | null | null | null |
Let P be a connected, weighted graph. At every iteration of Prim's algorithm, an edge must be found that connects a vertex in a subgraph to a vertex outside the subgraph. Since P is connected, there will always be a path to every vertex. The output Y of Prim's algorithm is a tree, because the edge and vertex added to t... | Wikipedia - Prim's algorithm | null | null | null |
Let Y1 be a minimum spanning tree of graph P. If Y1=Y then Y is a minimum spanning tree. Otherwise, let e be the first edge added during the construction of tree Y that is not in tree Y1, and V be the set of vertices connected by the edges added before edge e. Then one endpoint of edge e is in set V and the other is no... | Wikipedia - Prim's algorithm | null | null | null |
As one travels along the path, one must encounter an edge f joining a vertex in set V to one that is not in set V. Now, at the iteration when edge e was added to tree Y, edge f could also have been added and it would be added instead of edge e if its weight was less than e, and since edge f was not added, we conclude t... | Wikipedia - Prim's algorithm | null | null | null |
Let tree Y2 be the graph obtained by removing edge f from and adding edge e to tree Y1. It is easy to show that tree Y2 is connected, has the same number of edges as tree Y1, and the total weights of its edges is not larger than that of tree Y1, therefore it is also a minimum spanning tree of graph P and it contains ed... | Wikipedia - Prim's algorithm | null | null | null |
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