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Let A {\displaystyle A} be an observable for some system S {\displaystyle {\mathcal {S}}} with self-Hamiltonian H S {\displaystyle H_{\mathcal {S}}} . The system S {\displaystyle {\mathcal {S}}} is measured by an apparatus R {\displaystyle {\mathcal {R}}} which is coupled to S {\displaystyle {\mathcal {S}}} through int... | Wikipedia - Quantum Nondemolition measurement | null | null | null |
Allow time-dependence to denote the Heisenberg picture observables: A ( t ) = e − i t H S A e + i t H S . {\displaystyle A(t)=e^{-itH_{\mathcal {S}}}Ae^{+itH_{\mathcal {S}}}.} A sequence of measurements of A {\displaystyle A} are said to be QND measurements if and only if = 0 {\displaystyle =0} for any t n {\displayst... | Wikipedia - Quantum Nondemolition measurement | null | null | null |
If this property holds for any choice of t n {\displaystyle t_{n}} and t m {\displaystyle t_{m}} , then A {\displaystyle A} is said to be a continuous QND variable. If this only holds for certain discrete times, then A {\displaystyle A} is said to be a stroboscopic QND variable. For example, in the case of a free parti... | Wikipedia - Quantum Nondemolition measurement | null | null | null |
On the other hand, for the harmonic oscillator the position and momentum satisfy periodic in time commutation relations which imply that x and p are not continuous QND observables. However, if one makes the measurements at times separated by an integral numbers of half-periods (τ = kπ/ω), then the commutators vanish. T... | Wikipedia - Quantum Nondemolition measurement | null | null | null |
Let A {\displaystyle \mathbf {A} } be an m × n {\displaystyle m\times n} matrix with r {\displaystyle r} linearly independent columns (i.e. Rank ( A ) = r {\displaystyle \operatorname {Rank} (\mathbf {A} )=r} ). We will show that: To do this, we will produce an n × ( n − r ) {\displaystyle n\times (n-r)} matrix X {\d... | Wikipedia - Rank theorem | null | null | null |
So, X {\displaystyle \mathbf {X} } is an n × ( n − r ) {\displaystyle n\times (n-r)} matrix such that Therefore, each of the n − r {\displaystyle n-r} columns of X {\displaystyle \mathbf {X} } are particular solutions of A x = 0 F m {\displaystyle \mathbf {Ax} ={0}_{{F}^{m}}} . Furthermore, the n − r {\displaystyle n-r... | Wikipedia - Rank theorem | null | null | null |
For this, let be any vector such that A u = 0 F m {\displaystyle \mathbf {Au} =\mathbf {0} _{{F}^{m}}} . Since the columns of A 1 {\displaystyle \mathbf {A} _{1}} are linearly independent, A 1 x = 0 F m {\displaystyle \mathbf {A} _{1}\mathbf {x} =\mathbf {0} _{{F}^{m}}} implies x = 0 F r {\displaystyle \mathbf {x} =\ma... | Wikipedia - Rank theorem | null | null | null |
And we have already seen that the columns of X {\displaystyle \mathbf {X} } are linearly independent. Hence, the columns of X {\displaystyle \mathbf {X} } constitute a basis for the null space of A {\displaystyle \mathbf {A} } . | Wikipedia - Rank theorem | null | null | null |
Therefore, the nullity of A {\displaystyle \mathbf {A} } is n − r {\displaystyle n-r} . Since r {\displaystyle r} equals rank of A {\displaystyle \mathbf {A} } , it follows that Rank ( A ) + Nullity ( A ) = n {\displaystyle \operatorname {Rank} (\mathbf {A} )+\operatorname {Nullity} (\mathbf {A} )=n} . This conclud... | Wikipedia - Rank theorem | null | null | null |
Let A {\displaystyle {\boldsymbol {A}}} and T {\displaystyle {\boldsymbol {T}}} be two second order tensors, then In index notation with respect to an orthonormal basis We also have In index notation If the tensor A {\displaystyle {\boldsymbol {A}}} is symmetric then | Wikipedia - Tensor derivative (continuum mechanics) | null | null | null |
Let A {\displaystyle {\boldsymbol {A}}} be a second order tensor. Then Therefore, Here I {\displaystyle {\boldsymbol {\mathsf {I}}}} is the fourth order identity tensor. In index notation with respect to an orthonormal basis This result implies that where Therefore, if the tensor A {\displaystyle {\boldsymbol {A}}} is ... | Wikipedia - Tensor derivative (continuum mechanics) | null | null | null |
Let A {\displaystyle {\mathcal {A}}} be a set of symbols, called attributes (or column names). For each α ∈ A {\displaystyle \alpha \in {\mathcal {A}}} let U α {\displaystyle U_{\alpha }} be a non-empty set, the set of all possible values of the attribute α {\displaystyle \alpha } . For example, if A = { name , age , i... | Wikipedia - Valuation algebra | null | null | null |
An x {\displaystyle x} -tuple is a function f {\displaystyle f} so that dom ( f ) = x {\displaystyle {\hbox{dom}}(f)=x} and f ( α ) ∈ U α {\displaystyle f(\alpha )\in U_{\alpha }} for each α ∈ x {\displaystyle \alpha \in x} The set of all x {\displaystyle x} -tuples is denoted by E x {\displaystyle E_{x}} . For an x {\... | Wikipedia - Valuation algebra | null | null | null |
The set of attributes x {\displaystyle x} is called the domain of R {\displaystyle R} and denoted by d ( R ) {\displaystyle d(R)} . For y ⊆ d ( R ) {\displaystyle y\subseteq d(R)} the projection of R {\displaystyle R} onto y {\displaystyle y} is defined as follows: π y ( R ) := { f ∣ f ∈ R } . {\displaystyle \pi _{y}(... | Wikipedia - Valuation algebra | null | null | null |
The operations are well defined since d ( R ⋈ S ) = d ( R ) ∪ d ( S ) {\displaystyle d(R\bowtie S)=d(R)\cup d(S)} If x ⊆ d ( R ) {\displaystyle x\subseteq d(R)} , then d ( π x ( R ) ) = x {\displaystyle d(\pi _{x}(R))=x} .It is easy to see that relational databases satisfy the axioms of a labeled information algebra: s... | Wikipedia - Valuation algebra | null | null | null |
Let A {\displaystyle {\mathcal {A}}} be a weak*-closed operator algebra contained in B(H), the set of all bounded operators on a Hilbert space H and for T any operator in B(H), let β ( T , A ) = sup { ‖ P ⊥ T P ‖: P is a projection and P ⊥ A P = ( 0 ) } . {\displaystyle \beta (T,{\mathcal {A}})=\sup \left\{\left\|P^{\p... | Wikipedia - Reflexive operator algebra | null | null | null |
{\displaystyle \beta (T,{\mathcal {A}})=0{\mbox{ implies that }}T{\mbox{ is in }}{\mathcal {A}}.} We note that for any T in B(H) the following inequality is satisfied: β ( T , A ) ≤ dist ( T , A ) . {\displaystyle \beta (T,{\mathcal {A}})\leq {\mbox{dist}}(T,{\mathcal {A}}).} | Wikipedia - Reflexive operator algebra | null | null | null |
Here dist ( T , A ) {\displaystyle {\mbox{dist}}(T,{\mathcal {A}})} is the distance of T from the algebra, namely the smallest norm of an operator T-A where A runs over the algebra. We call A {\displaystyle {\mathcal {A}}} hyperreflexive if there is a constant K such that for every operator T in B(H), dist ( T , A ) ≤ ... | Wikipedia - Reflexive operator algebra | null | null | null |
The smallest such K is called the distance constant for A {\displaystyle {\mathcal {A}}} . A hyper-reflexive operator algebra is automatically reflexive. In the case of a reflexive algebra of matrices with nonzero entries specified by a given pattern, the problem of finding the distance constant can be rephrased as a m... | Wikipedia - Reflexive operator algebra | null | null | null |
Let A ∈ Γ ( T M ) {\displaystyle A\in \Gamma (TM)} be a vector field, understood as a derivation by the C ∞ ( M ) {\displaystyle C^{\infty }(M)} -isomorphism Γ ( T M ) → Der R C ∞ ( M ) , A ↦ ( f ↦ A f ) {\displaystyle \Gamma (TM)\to \operatorname {Der} _{\mathbb {R} }C^{\infty }(M),\quad A\mapsto (f\mapsto Af)} for ... | Wikipedia - Stochastic differential geometry | null | null | null |
Let A ⊆ R ^ {\displaystyle A\subseteq {\widehat {\mathbb {R} }}} . Then p is a limit point of A if and only if every neighbourhood of p includes a point y ∈ A {\displaystyle y\in A} such that y ≠ p . {\displaystyle y\neq p.} Let f: R ^ → R ^ , A ⊆ R ^ , L ∈ R ^ , p ∈ R ^ {\displaystyle f:{\widehat {\mathbb {R} }}\to {\... | Wikipedia - Projectively extended real number system | null | null | null |
{\displaystyle f(x)\in B.} This corresponds to the regular topological definition of continuity, applied to the subspace topology on A ∪ { p } , {\displaystyle A\cup \lbrace p\rbrace ,} and the restriction of f to A ∪ { p } . {\displaystyle A\cup \lbrace p\rbrace .} | Wikipedia - Projectively extended real number system | null | null | null |
Let A(G) be the Fourier algebra of a compact group G. Building upon the work of Wiener, Lévy, Gelfand, and Beurling, in 1959 Helson, Kahane, Katznelson, and Rudin proved that, when G is compact and abelian, a function f defined on a closed convex subset of the plane operates in A(G) if and only if f is real analytic. I... | Wikipedia - Fourier algebra | null | null | null |
Let A, B and C be square matrices of order n × n. The following naive algorithm implements C = C + A * B: for i = 1 to n for j = 1 to n for k = 1 to n C(i,j) = C(i,j) + A(i,k) * B(k,j) Arithmetic cost (time-complexity): n2(2n − 1) for sufficiently large n or O(n3). Rewriting this algorithm with communication cost label... | Wikipedia - Communication-avoiding algorithm | null | null | null |
Let A, B, C, and D be sets. The Cartesian product A × B is not commutative, A × B ≠ B × A , {\displaystyle A\times B\neq B\times A,} because the ordered pairs are reversed unless at least one of the following conditions is satisfied: A is equal to B, or A or B is the empty set.For example: A = {1,2}; B = {3,4} A × B = ... | Wikipedia - Cartesian power | null | null | null |
Let ABC be a plane triangle and let ( x: y: z ) be the trilinear coordinates of an arbitrary point in the plane of triangle ABC. A straight line in the plane of triangle ABC whose equation in trilinear coordinates has the form f ( a, b, c ) x + g ( a, b, c ) y + h ( a, b, c ) z = 0where the point with trilinear coordin... | Wikipedia - Modern triangle geometry | null | null | null |
Let ABC be a plane triangle and let ( x: y: z ) {\displaystyle (x:y:z)} be the trilinear coordinates of an arbitrary point in the plane of triangle ABC. A straight line in the plane of triangle ABC whose equation in trilinear coordinates has the form f ( a , b , c ) x + g ( a , b , c ) y + h ( a , b , c ) z = 0 {\displ... | Wikipedia - Central line (geometry) | null | null | null |
Let ABC be a triangle, let G be its centroid, and let D, E, and F be the midpoints of BC, CA, and AB, respectively. For any point P in the plane of ABC then The centroid divides each median into parts in the ratio 2:1, with the centroid being twice as close to the midpoint of a side as it is to the opposite vertex. For... | Wikipedia - Median (geometry) | null | null | null |
The medians of a right triangle with hypotenuse c {\displaystyle c} satisfy m a 2 + m b 2 = 5 m c 2 . {\displaystyle m_{a}^{2}+m_{b}^{2}=5m_{c}^{2}.} Any triangle's area T can be expressed in terms of its medians m a , m b {\displaystyle m_{a},m_{b}} , and m c {\displaystyle m_{c}} as follows. If their semi-sum ( m a +... | Wikipedia - Median (geometry) | null | null | null |
Let AF be a vector space over a field F, and let L1 and L2 be two linear functionals on AF with the property L1(e) = L2(e) = 1F for some e in AF. We define multiplication of two elements x, y in AF by x ⋅ y = L 1 ( x ) y + L 2 ( y ) x − L 1 ( x ) L 2 ( y ) e . {\displaystyle x\cdot y=L_{1}(x)y+L_{2}(y)x-L_{1}(x)L_{2}(y... | Wikipedia - Functional-theoretic algebra | null | null | null |
So, AF forms an associative algebra with unit e and is called a functional theoretic algebra(FTA). Suppose the two linear functionals L1 and L2 are the same, say L. Then AF becomes a commutative algebra with multiplication defined by x ⋅ y = L ( x ) y + L ( y ) x − L ( x ) L ( y ) e . {\displaystyle x\cdot y=L(x)y+L(y)... | Wikipedia - Functional-theoretic algebra | null | null | null |
Let Ai be the ith element of the first sequence. Let Bj be the jth element of the second sequence. Let Pij be the length of the longest common subsequence for the first i elements of A and the first j elements B. P i j = { 0 if i = 0 or j = 0 1 + P i − 1 , j − 1 if A i = B j max ( P i − 1 , j , P i , j − 1 ) if A i ≠ B... | Wikipedia - Hunt–Szymanski algorithm | null | null | null |
The generalized Bochner theorem states that a measurable function on G {\displaystyle {\mathit {G}}} is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on G ^ {\displaystyle {\widehat {G}}} if and only if it is positive definite. Thus, B ( G ) {\displaystyle B({\mathit {G}}... | Wikipedia - Fourier algebra | null | null | null |
Let B ( X , Ω ) {\displaystyle {\mathcal {B}}(X,\Omega )} be the vector space of all bounded complex-valued Ω {\displaystyle \Omega } -measurable functions f: X → C , {\displaystyle f:X\to \mathbb {C} ,} which becomes a Banach algebra when normed by ‖ f ‖ ∞ := sup x ∈ X | f ( x ) | . {\displaystyle \|f\|_{\infty }:=\su... | Wikipedia - Spectral theory of normal C*-algebras | null | null | null |
{\displaystyle \left({\mathcal {B}}(X,\Omega ),\|\cdot \|_{\infty }\right).} Hence the quotient of B ( X , Ω ) {\displaystyle {\mathcal {B}}(X,\Omega )} by N ∞ {\displaystyle N^{\infty }} is also a Banach algebra, denoted by L ∞ ( π ) := B ( X , Ω ) / N ∞ {\displaystyle L^{\infty }(\pi ):={\mathcal {B}}(X,\Omega )/N^{\... | Wikipedia - Spectral theory of normal C*-algebras | null | null | null |
{\displaystyle f.} This article will follow the usual practice of writing f {\displaystyle f} rather than f + N ∞ {\displaystyle f+N^{\infty }} to represent elements of L ∞ ( π ) . {\displaystyle L^{\infty }(\pi ).} | Wikipedia - Spectral theory of normal C*-algebras | null | null | null |
Let B 0 {\displaystyle {\mathcal {B}}_{0}} be the class consisting of the C*-algebras C 0 ( R ) , C 0 ( R 2 ) , D n , S D n {\displaystyle C_{0}(\mathbb {R} ),C_{0}(\mathbb {R} ^{2}),D_{n},SD_{n}} for each n ≥ 2 {\displaystyle n\geq 2} , and let B {\displaystyle {\mathcal {B}}} be the class of all C*-algebras of the fo... | Wikipedia - Projectionless C*-algebra | null | null | null |
. . , k r {\displaystyle r,k_{1},...,k_{r}} are integers, and where B 1 , . | Wikipedia - Projectionless C*-algebra | null | null | null |
. . , B r {\displaystyle B_{1},...,B_{r}} belong to B 0 {\displaystyle {\mathcal {B}}_{0}} . Every C*-algebra A in B {\displaystyle {\mathcal {B}}} is projectionless, moreover, its only projection is 0. == References == | Wikipedia - Projectionless C*-algebra | null | null | null |
Let B = { b 1 , b 2 , … , b n } {\displaystyle \mathbf {B} =\{\mathbf {b} _{1},\mathbf {b} _{2},\dots ,\mathbf {b} _{n}\}} be a δ {\displaystyle \delta } -LLL-reduced basis of a lattice L {\displaystyle {\mathcal {L}}} . From the definition of LLL-reduced basis, we can derive several other useful properties about B {\d... | Wikipedia - Lenstra–Lenstra–Lovász lattice basis reduction algorithm | null | null | null |
The first vector in the basis is also bounded by the determinant of the lattice: ‖ b 1 ‖ ≤ ( 2 / ( 4 δ − 1 ) ) ( n − 1 ) / 2 ⋅ ( det ( L ) ) 1 / n {\displaystyle \Vert \mathbf {b} _{1}\Vert \leq (2/({\sqrt {4\delta -1}}))^{(n-1)/2}\cdot (\det({\mathcal {L}}))^{1/n}} . In particular, for δ = 3 / 4 {\displaystyle \delta ... | Wikipedia - Lenstra–Lenstra–Lovász lattice basis reduction algorithm | null | null | null |
Let B be a Hamel basis for g. Then each G ∈ g has a unique expression as G = ∑ α ∈ A c α G α , c α ∈ F , G α ∈ B {\displaystyle G=\sum _{\alpha \in A}c_{\alpha }G_{\alpha },\quad c_{\alpha }\in F,G_{\alpha }\in B} for some indexing set A of suitable size. In this expansion, only finitely many cα are nonzero. In the seq... | Wikipedia - Lie algebra extension | null | null | null |
The following theorem is useful: Theorem:There is a basis such that the structure constants are antisymmetric in all indices if and only if the Lie algebra is a direct sum of simple compact Lie algebras and u(1) Lie algebras. This is the case if and only if there is a real positive definite metric g on g satisfying the... | Wikipedia - Lie algebra extension | null | null | null |
in any basis. This last condition is necessary on physical grounds for non-Abelian gauge theories in quantum field theory. Thus one can produce an infinite list of possible gauge theories using the Cartan catalog of simple Lie algebras on their compact form (i.e., sl(n, C {\displaystyle \mathbb {C} } ) → su(n), etc. On... | Wikipedia - Lie algebra extension | null | null | null |
Let B be a complex Banach algebra containing a unit e. Then we define the spectrum σ(x) (or more explicitly σB(x)) of an element x of B to be the set of those complex numbers λ for which λe − x is not invertible in B. This extends the definition for bounded linear operators B(X) on a Banach space X, since B(X) is a uni... | Wikipedia - Point spectrum | null | null | null |
Let B be a relational structure with universe B and vocabulary τ. i) A set X ⊆ B is guarded in B if there exists a ground atom α(b_1, ..., b_k) in B such that X = {b_1, ..., b_k}. ii) A τ-structure A, in particular a substructure A ⊆ B, is guarded if its universe is a guarded set in A (in B). iii) A tuple (b_1, ..., b_... | Wikipedia - Guarded logic | null | null | null |
Let B be a subspace of A. Define B to be a quadratic ideal or an inner ideal if the image of Q(b) is contained in B for all b in B; define B to be an outer ideal if B is mapped into itself by every Q(a) for all a in A. An ideal of A is a subspace which is both an inner and an outer ideal. A quadratic Jordan algebra is ... | Wikipedia - Quadratic Jordan algebra | null | null | null |
Let B be an A-algebra and suppose B is given the I-adic topology, I an ideal of B. We say B is I-smooth over A if it satisfies the lifting property: given an A-algebra C, an ideal N of C whose square is zero and an A-algebra map u: B → C / N {\displaystyle u:B\to C/N} that is continuous when C / N {\displaystyle C/N} i... | Wikipedia - Smooth algebra | null | null | null |
A standard example is this: let A be a ring, B = A ] {\displaystyle B=A\!]} and I = ( t 1 , … , t n ) . {\displaystyle I=(t_{1},\ldots ,t_{n}).} Then B is I-smooth over A. Let A be a noetherian local k-algebra with maximal ideal m {\displaystyle {\mathfrak {m}}} . Then A is m {\displaystyle {\mathfrak {m}}} -smooth ov... | Wikipedia - Smooth algebra | null | null | null |
Let B {\displaystyle B} be a symmetric bilinear form on an n {\displaystyle n} -dimensional inner product space ( V , g ) . {\displaystyle (V,g).} Then Additionally, note that if B = κ g {\displaystyle B=\kappa g} for some number κ , {\displaystyle \kappa ,} then one automatically has κ = 1 n tr g B . {\displaystyle ... | Wikipedia - Schur's lemma (Riemannian geometry) | null | null | null |
{ With these observations in mind, one can restate the Schur lemma in the following form: Let ( M , g ) {\displaystyle (M,g)} be a connected smooth Riemannian manifold whose dimension is not equal to two. Then the following are equivalent: There is a function κ {\displaystyle \kappa } on M {\displaystyle M} such that R... | Wikipedia - Schur's lemma (Riemannian geometry) | null | null | null |
{\displaystyle |\operatorname {Ric} |_{g}^{2}\leq \textstyle {\frac {1}{n}}R^{2}.} If ( M , g ) {\displaystyle (M,g)} is a connected smooth pseudo-Riemannian manifold, then the first three conditions are equivalent, and they imply the fourth condition. Note that the dimensional restriction is important, since every two... | Wikipedia - Schur's lemma (Riemannian geometry) | null | null | null |
Let B {\displaystyle B} be derived from the d {\displaystyle d} regular graph G {\displaystyle G} . So, the number of variables of C ( B , S ) {\displaystyle C(B,S)} is ( d n 2 ) {\displaystyle \left({\dfrac {dn}{2}}\right)} and the number of constraints is n {\displaystyle n} . According to Alon - Chung, if X {\displa... | Wikipedia - Zemor's decoding algorithm | null | null | null |
In matrix G {\displaystyle G} , we can assume that λ / d {\displaystyle \lambda /d} is bounded away from 1 {\displaystyle 1} . For those values of d {\displaystyle d} in which d − 1 {\displaystyle d-1} is odd prime, there are explicit constructions of sequences of d {\displaystyle d} - regular bipartite graphs with arb... | Wikipedia - Zemor's decoding algorithm | null | null | null |
Let B {\displaystyle B} be the base of the number system you are using, and n {\displaystyle n} be the degree of the root to be extracted. Let x {\displaystyle x} be the radicand processed thus far, y {\displaystyle y} be the root extracted thus far, and r {\displaystyle r} be the remainder. Let α {\displaystyle \alpha... | Wikipedia - Shifting nth-root algorithm | null | null | null |
Let BordM be the category whose morphisms are n-dimensional submanifolds of M and whose objects are connected components of the boundaries of such submanifolds. Regard two morphisms as equivalent if they are homotopic via submanifolds of M, and so form the quotient category hBordM: The objects in hBordM are the objects... | Wikipedia - Topological field theory | null | null | null |
This is the composition law for morphisms in the cobordism category. Since functors are required to preserve composition, this says that the linear map corresponding to a sewn together morphism is just the composition of the linear map for each piece. There is an equivalence of categories between the category of 2-dime... | Wikipedia - Topological field theory | null | null | null |
Let C ( n ) {\displaystyle C(n)} denote the number of comparisons that merge-insertion sort makes, in the worst case, when sorting n {\displaystyle n} elements. This number of comparisons can be broken down as the sum of three terms: ⌊ n / 2 ⌋ {\displaystyle \lfloor n/2\rfloor } comparisons among the pairs of items, C ... | Wikipedia - Ford–Johnson algorithm | null | null | null |
Similarly, the elements y 6 {\displaystyle y_{6}} and y 5 {\displaystyle y_{5}} of the second group are each inserted into a subsequence of length at most seven, using three comparisons. More generally, the worst-case number of comparisons for the elements in the i {\displaystyle i} th group is i + 1 {\displaystyle i+1... | Wikipedia - Ford–Johnson algorithm | null | null | null |
By summing the number of comparisons used for all the elements and solving the resulting recurrence relation, this analysis can be used to compute the values of C ( n ) {\displaystyle C(n)} , giving the formula C ( n ) = ∑ i = 1 n ⌈ log 2 3 i 4 ⌉ ≈ n log 2 n − 1.415 n {\displaystyle C(n)=\sum _{i=1}^{n}\left\lceil ... | Wikipedia - Ford–Johnson algorithm | null | null | null |
Let C and D be concepts, a and b be individuals, and R be a role. If a is R-related to b, then b is called an R-successor of a. | Wikipedia - Description Logics | null | null | null |
Let C be a category with finite products. Let pt denote a terminal object of C (an empty product). A ring object in C is an object R equipped with morphisms R × R → a R {\displaystyle R\times R\;{\stackrel {a}{\to }}\,R} (addition), R × R → m R {\displaystyle R\times R\;{\stackrel {m}{\to }}\,R} (multiplication), pt → ... | Wikipedia - Ring of functions | null | null | null |
Let C be a positively oriented, piecewise smooth, simple closed curve in a plane, and let D be the region bounded by C. If L and M are functions of (x, y) defined on an open region containing D and have continuous partial derivatives there, then ∮ C ( L d x + M d y ) = ∬ D ( ∂ M ∂ x − ∂ L ∂ y ) d x d y {\displaystyle \... | Wikipedia - Euclidean plane | null | null | null |
Let C be a pseudo-disks-set with n objects and union complexity u. Using linear programming relaxation, it is possible to find a disjoint set of size at least n u ⋅ | M D S ( C ) | {\displaystyle {\frac {n}{u}}\cdot |MDS(C)|} . This is possible either with a randomized algorithm that has a high probability of success a... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n axis-parallel rectangles in the plane, all with the same height H but with varying lengths. The following algorithm finds a disjoint set with a size of at least |MDS(C)|/2 in time O(n log n): Draw m horizontal lines, such that: The separation between two lines is strictly more than H. Each line inte... | Wikipedia - Maximum disjoint set | null | null | null |
Therefore the lines partition the set of rectangles into m subsets ( R i , … , R m {\displaystyle R_{i},\ldots ,R_{m}} ) – each subset includes the rectangles intersected by a single line. For each subset R i {\displaystyle R_{i}} , compute an exact MDS M i {\displaystyle M_{i}} using the one-dimensional greedy algorit... | Wikipedia - Maximum disjoint set | null | null | null |
By construction, the rectangles in ( R i {\displaystyle R_{i}} ) can intersect only rectangles in R i + 1 {\displaystyle R_{i+1}} or in R i − 1 {\displaystyle R_{i-1}} . Therefore, each of the following two unions is a disjoint sets: Union of odd MDSs: M 1 ∪ M 3 ∪ ⋯ {\displaystyle M_{1}\cup M_{3}\cup \cdots } Union of ... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n axis-parallel rectangles in the plane, all with the same height but with varying lengths. There is an algorithm that finds a disjoint set with a size of at least |MDS(C)|/(1 + 1/k) in time O(n2k−1), for every constant k > 1.The algorithm is an improvement of the above-mentioned 2-approximation, by c... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n axis-parallel rectangles in the plane. The following algorithm finds a disjoint set with a size of at least | M D S ( C ) | log n {\displaystyle {\frac {|MDS(C)|}{\log {n}}}} in time O ( n log n ) {\displaystyle O(n\log {n})}: INITIALIZATION: sort the horizontal edges of the given rectangles by ... | Wikipedia - Maximum disjoint set | null | null | null |
Partition the input rectangles into three groups according to their relation to the line x = x m e d {\displaystyle x=x_{\mathrm {med} }}: those entirely to its left ( R l e f t {\displaystyle R_{\mathrm {left} }} ), those entirely to its right ( R r i g h t {\displaystyle R_{\mathrm {right} }} ), and those intersected... | Wikipedia - Maximum disjoint set | null | null | null |
By construction, the rectangles in M l e f t {\displaystyle M_{\mathrm {left} }} and M r i g h t {\displaystyle M_{\mathrm {right} }} are all disjoint, so M l e f t ∪ M r i g h t {\displaystyle M_{\mathrm {left} }\cup M_{\mathrm {right} }} is a disjoint set. Compute an exact MDS in R i n t {\displaystyle R_{\mathrm {in... | Wikipedia - Maximum disjoint set | null | null | null |
Return either M l e f t ∪ M r i g h t {\displaystyle M_{\mathrm {left} }\cup M_{\mathrm {right} }} or M i n t {\displaystyle M_{\mathrm {int} }} – whichever of them is larger.It is provable by induction that, at the last step, either M l e f t ∪ M r i g h t {\displaystyle M_{\mathrm {left} }\cup M_{\mathrm {right} }} o... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n disks, such that the ratio between the largest radius and the smallest radius is at most r. The following algorithm finds MDS(C) exactly in time 2 O ( r ⋅ n ) {\displaystyle 2^{O(r\cdot {\sqrt {n}})}} .The algorithm is based on a width-bounded geometric separator on the set Q of the centers of all d... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n fat objects, such as squares or circles, of arbitrary sizes. Chan described an algorithm finds a disjoint set with a size of at least (1 − O(√b))·|MDS(C)| in time nO(b), for every constant b > 1. The algorithm is based on the following geometric separator theorem, which can be proved similarly to th... | Wikipedia - Maximum disjoint set | null | null | null |
But since we can only approximate MDS(C) by a constant factor, the constant a must be larger. Fortunately, a remains a constant independent of |C|. | Wikipedia - Maximum disjoint set | null | null | null |
This separator theorem allows to build the following PTAS: Select a constant b. Check all possible combinations of up to b + 1 labels. If |MDS(C)| has a size of at most b (i.e. all sets of b + 1 labels are not disjoint) then just return that MDS and exit. This step takes nO(b) time. | Wikipedia - Maximum disjoint set | null | null | null |
Otherwise, use a geometric separator to separate C to two subsets. Find the approximate MDS in Cinside and Coutside separately, and return their combination as the approximate MDS in C.Let E(m) be the error of the above algorithm when the optimal MDS size is MDS(C) = m. When m ≤ b, the error is 0 because the maximum di... | Wikipedia - Maximum disjoint set | null | null | null |
Therefore the error function satisfies the following recurrence relation: E ( m ) = 0 if m ≤ b {\displaystyle E(m)=0\ \ \ \ {\text{ if }}m\leq b} E ( m ) = E ( a ⋅ m ) + E ( ( 1 − a ) ⋅ m ) + c ⋅ m if m > b {\displaystyle E(m)=E(a\cdot m)+E((1-a)\cdot m)+c\cdot {\sqrt {m}}{\text{ if }}m>b} The solution to this recurren... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n fat objects, such as squares or circles, of arbitrary sizes. There is a PTAS for finding an MDS based on multi-level grid alignment. It has been discovered by two groups in approximately the same time, and described in two different ways. | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a set of n squares or circles of identical size. Hochbaum and Maass presented a polynomial-time approximation scheme for finding an MDS using a simple shifted-grid strategy. It finds a solution within (1 − ε) of the maximum in time nO(1/ε2) time and linear space. The strategy generalizes to any collection of f... | Wikipedia - Maximum disjoint set | null | null | null |
Let C be a smooth complete curve and Pic ( C ) {\displaystyle \operatorname {Pic} (C)} the Picard group of it; i.e., the group of isomorphism classes of line bundles on C. Since C is smooth, Pic ( C ) {\displaystyle \operatorname {Pic} (C)} can be identified as the divisor class group of C and thus there is the deg... | Wikipedia - Algebraic set | null | null | null |
See equations defining abelian varieties); thus, Jac ( C ) {\displaystyle \operatorname {Jac} (C)} is a projective variety. The tangent space to Jac ( C ) {\displaystyle \operatorname {Jac} (C)} at the identity element is naturally isomorphic to H 1 ( C , O C ) ; {\displaystyle \operatorname {H} ^{1}(C,{\mathcal ... | Wikipedia - Algebraic set | null | null | null |
Let C be a smooth projective curve of genus g over the complex numbers C. The Betti numbers bi(ΣnC) of the symmetric products ΣnC for all n = 0, 1, 2, ... are given by the generating function ∑ n = 0 ∞ ∑ i = 0 2 n b i ( Σ n C ) y n u i − n = ( 1 + y ) 2 g ( 1 − u y ) ( 1 − u − 1 y ) {\displaystyle \sum _{n=0}^{\infty }... | Wikipedia - Symmetric products of algebraic curves | null | null | null |
Let C be a symmetric cone in the Euclidean space E. As above, Aut C denotes the closed subgroup of GL(E) taking C (or equivalently its closure) onto itself. Let G = Aut0 C be its identity component. K = G ∩ O(E). | Wikipedia - Structure group (Jordan algebra) | null | null | null |
It is a maximal compact subgroup of G and the stabilizer of a point e in C. It is connected. The group G is invariant under taking adjoints. Let σg =(g*)−1, period 2 automorphism. | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Thus K is the fixed point subgroup of σ. Let g {\displaystyle {\mathfrak {g}}} be the Lie algebra of G. Thus σ induces an involution of g {\displaystyle {\mathfrak {g}}} and hence a ±1 eigenspace decomposition g = k ⊕ p , {\displaystyle \displaystyle {{\mathfrak {g}}={\mathfrak {k}}\oplus {\mathfrak {p}},}} where k {\d... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
The product is commutative since ⊆ k {\displaystyle {\mathfrak {k}}} annihilates e, so that ab = L(a)L(b)e = L(b)L(a)e = ba. It remains to check the Jordan identity = 0. The associator is given by = b. Since lies in k {\displaystyle {\mathfrak {k}}} it follows that ,L(b)] = L(). | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Making both sides act on c yields = 2 b . {\displaystyle \displaystyle {=2b.}} On the other hand, ( , c ) = ( b 2 ( b a ) − b ( b 2 a ) , c ) = − ( b 2 , ) {\displaystyle \displaystyle {(,c)=(b^{2}(ba)-b(b^{2}a),c)=-(b^{2},)}} and likewise ( , c ) = ( b , ) . | Wikipedia - Structure group (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {(,c)=(b,).}} Combining these expressions gives ( , c ) = 0 , {\displaystyle \displaystyle {(,c)=0,}} which implies the Jordan identity. Finally the positive cone of E coincides with C. This depends on the fact that in any Euclidean Jordan algebra E Q ( e a ) = e 2 L ( a ) . | Wikipedia - Structure group (Jordan algebra) | null | null | null |
{\displaystyle \displaystyle {Q(e^{a})=e^{2L(a)}.}} In fact Q(ea) is a positive operator, Q(eta) is a one-parameter group of positive operators: this follows by continuity for rational t, where it is a consequence of the behaviour of powers So it has the form exp tX for some self-adjoint operator X. Taking the derivati... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let C be the group ring of a discrete group G. For a locally compact group G, the group C*-algebra C*(G) of G is defined to be the C*-enveloping algebra of L1(G), i.e. the completion of Cc(G) with respect to the largest C*-norm: ‖ f ‖ C ∗ := sup π ‖ π ( f ) ‖ , {\displaystyle \|f\|_{C^{*}}:=\sup _{\pi }\|\pi (f)\|,} wh... | Wikipedia - Group algebra of a topological group | null | null | null |
Let C be the outer end of the rod, and A, B be the pivots of the sliders. Let AB and BC be the distances from A to B and B to C, respectively. Let us assume that sliders A and B move along the y and x coordinate axes, respectively. When the rod makes an angle θ with the x-axis, the coordinates of point C are given by x... | Wikipedia - Trammel of Archimedes | null | null | null |
The further equation x 2 ( A B + B C ) 2 + y 2 ( B C ) 2 = 1 {\displaystyle {\frac {x^{2}}{(AB+BC)^{2}}}+{\frac {y^{2}}{(BC)^{2}}}=1} is immediate as well. The trammel of Archimedes is an example of a four-bar linkage with two sliders and two pivots, and is special case of the more general oblique trammel. The axes con... | Wikipedia - Trammel of Archimedes | null | null | null |
Let C be the positive cone in a simple Euclidean Jordan algebra E. Aut C is the closed subgroup of GL(E) taking C (or its closure) onto itself. Let G = Aut0 C be the identity component of Aut C and let K be the closed subgroup of G fixing 1. From the group theoretic properties of cones, K is a connected compact subgrou... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let p {\displaystyle {\mathfrak {p}}} be the −1 eigenspace of σ. k {\displaystyle {\mathfrak {k}}} consists of derivations of E that are skew-adjoint for the inner product defined by the trace form. ,L(b)] = L(). If a and b are in E, then D = is a derivation of E, so lies in k {\displaystyle {\mathfrak {k}}} . | Wikipedia - Structure group (Jordan algebra) | null | null | null |
These derivations span k {\displaystyle {\mathfrak {k}}} . If a is in C, then Q(a) lies in G. C is the connected component of the open set of invertible elements of E containing 1. It consists of exponentials of elements of E and the exponential map gives a diffeomorphism of E onto C. The map a ↦ L(a) gives an isomorph... | Wikipedia - Structure group (Jordan algebra) | null | null | null |
Let C denote the field of Complex numbers. A continuous function γ from the closed interval of real numbers to the field C is called a curve. The complex numbers γ(0) and γ(1) are, respectively, the initial and terminal points of the curve. If they coincide, the curve is called a loop. | Wikipedia - Functional-theoretic algebra | null | null | null |
The set V of all the curves is a vector space over C. We can make this vector space of curves into an algebra by defining multiplication as above. Choosing e ( t ) = 1 , ∀ ∈ {\displaystyle e(t)=1,\forall \in } we have for α,β in C, α ⋅ β = α ( 0 ) β + β ( 1 ) α − α ( 0 ) β ( 1 ) e {\displaystyle {\alpha }\cdot {\beta ... | Wikipedia - Functional-theoretic algebra | null | null | null |
Let C {\displaystyle {\mathcal {C}}} be the category of vector spaces K {\displaystyle K} -Vect over a field K {\displaystyle K} and let D {\displaystyle {\mathcal {D}}} be the category of algebras K {\displaystyle K} -Alg over K {\displaystyle K} (assumed to be unital and associative). Let U {\displaystyle U}: K {\dis... | Wikipedia - Universal construction | null | null | null |
The tensor algebra is characterized by the fact: “Any linear map from V {\displaystyle V} to an algebra A {\displaystyle A} can be uniquely extended to an algebra homomorphism from T ( V ) {\displaystyle T(V)} to A {\displaystyle A} .”This statement is an initial property of the tensor algebra since it expresses the fa... | Wikipedia - Universal construction | null | null | null |
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