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Let the blank page denote False, and let a Cross be read as Not. Then the primary arithmetic has the following sentential reading: = False = True = not False = Not True = FalseThe primary algebra interprets sentential logic as follows. A letter represents any given sentential expression. Thus: interprets Not A interpre... | Wikipedia - Boundary algebra | null | null | null |
Equivalently, the primary algebra interprets sentential logic. Given an assignment of every variable to the Marked or Unmarked states, this primary algebra translation reduces to a primary arithmetic expression, which can be simplified. Repeating this exercise for all possible assignments of the two primitive values to... | Wikipedia - Boundary algebra | null | null | null |
This is an example of a decision procedure, one more or less in the spirit of conventional truth tables. Given some primary algebra formula containing N variables, this decision procedure requires simplifying 2N primary arithmetic formulae. For a less tedious decision procedure more in the spirit of Quine's "truth valu... | Wikipedia - Boundary algebra | null | null | null |
Schwartz (1981) proved that the primary algebra is equivalent — syntactically, semantically, and proof theoretically — with the classical propositional calculus. Likewise, it can be shown that the primary algebra is syntactically equivalent with expressions built up in the usual way from the classical truth values true... | Wikipedia - Boundary algebra | null | null | null |
All that is required is that the interpretation of concatenation change from OR to AND. IF A THEN B now translates as instead of . More generally, the primary algebra is "self-dual", meaning that any primary algebra formula has two sentential or Boolean readings, each the dual of the other. | Wikipedia - Boundary algebra | null | null | null |
Another consequence of self-duality is the irrelevance of De Morgan's laws; those laws are built into the syntax of the primary algebra from the outset. The true nature of the distinction between the primary algebra on the one hand, and 2 and sentential logic on the other, now emerges. In the latter formalisms, complem... | Wikipedia - Boundary algebra | null | null | null |
But an empty Cross is a well-formed primary algebra expression, denoting the Marked state, a primitive value. Hence a nonempty Cross is an operator, while an empty Cross is an operand because it denotes a primitive value. Thus the primary algebra reveals that the heretofore distinct mathematical concepts of operator an... | Wikipedia - Boundary algebra | null | null | null |
Let the circles have centres c1 = (x1, y1) and c2 = (x2, y2) with radius r1 and r2 respectively. Expressing a line by the equation a x + b y + c = 0 , {\displaystyle ax+by+c=0,} with the normalization a 2 + b 2 = 1 , {\displaystyle a^{2}+b^{2}=1,} then a bitangent line satisfies: Solving for (a, b, c) by subtracting th... | Wikipedia - Tangent lines to two circles | null | null | null |
sin θ is then ± 1 − R 2 {\displaystyle \pm {\sqrt {1-R^{2}}}} (depending on the sign of θ, equivalently the direction of rotation), and the above equations are rotation of (X, Y) by ±θ using the rotation matrix: k = 1 is the tangent line to the right of the circles looking from c1 to c2. k = −1 is the tangent line to t... | Wikipedia - Tangent lines to two circles | null | null | null |
If r1 is positive and r2 negative then c1 will lie to the left of each line and c2 to the right, and the two tangent lines will cross. In this way all four solutions are obtained. Switching signs of both radii switches k = 1 and k = −1. | Wikipedia - Tangent lines to two circles | null | null | null |
Let the circumference of a circle be measured in degrees and let the radius R of the circle be also measured in degrees. Choosing a fixed diameter AB and an arbitrary point P on the circle and dropping the perpendicular PM to AB, we can compute the area of the triangle APB in two ways. Equating the two expressions for ... | Wikipedia - Bhāskara I's sine approximation formula | null | null | null |
{\displaystyle {\frac {1}{PM}}={\frac {AB}{AP\times BP}}.} Letting x be the length of the arc AP, the length of the arc BP is 180 − x. These arcs are much bigger than the respective chords. Hence one gets 1 P M > 2 R x ( 180 − x ) . | Wikipedia - Bhāskara I's sine approximation formula | null | null | null |
{\displaystyle {\frac {1}{PM}}>{\frac {2R}{x(180-x)}}.} One now seeks two constants α and β such that 1 P M = α 2 R x ( 180 − x ) + β . {\displaystyle {\frac {1}{PM}}=\alpha {\frac {2R}{x(180-x)}}+\beta .} It is indeed not possible to obtain such constants. However, one may choose values for α and β so that the above e... | Wikipedia - Bhāskara I's sine approximation formula | null | null | null |
Let the direction vector be (0,1) and the mirror be ( t , t 2 ) . {\displaystyle (t,t^{2}).} Then u ′ = 1 {\displaystyle u'=1} u ″ = 0 {\displaystyle u''=0} v ′ = 2 t {\displaystyle v'=2t} v ″ = 2 {\displaystyle v''=2} a = 0 {\displaystyle a=0} b = 1 {\displaystyle b=1} F ( x , y , t ) = ( x − t ) ( 1 − 4 t 2 ) + 4 t (... | Wikipedia - Caustic (mathematics) | null | null | null |
Let the homogeneous differential equation: m d 2 y d t 2 + γ d y d t + k y = 0 {\displaystyle m{\frac {d^{2}y}{dt^{2}}}+\gamma {\frac {dy}{dt}}+ky=0} model damped unforced oscillations of a weight on a spring. The displacement will then be of the form y ( t ) = A e − t / T cos ( μ t − δ ) {\displaystyle y(t)=Ae^{-t/T... | Wikipedia - Relaxation (physics) | null | null | null |
Let the motif M of interest be of length l. If M occurs in every input string then any substring of M also occurs in every input string. Here occurrence means occurrence within a Hamming distance of d. It follows that there are at least l-k+1 strings each of length k (for k ≤ l) such that each of these occurs in every ... | Wikipedia - Planted motif search | null | null | null |
Another k-mer of Q occurs starting from ij +1 and so on, with the last k-mer occurring at ij + l – k. An l-mer can be obtained by combining these k-mers that occur starting from each such ij. PMS2 works as follows. In the first phase find all the (k, d) motifs present in all the input strings (for some appropriate valu... | Wikipedia - Planted motif search | null | null | null |
Let the node at which we are starting be called the initial node. Let the distance of node Y be the distance from the initial node to Y. Dijkstra's algorithm will initially start with infinite distances and will try to improve them step by step. Mark all nodes unvisited. Create a set of all the unvisited nodes called t... | Wikipedia - Dijkstra’s Algorithm | null | null | null |
Assign to every node a tentative distance value: set it to zero for our initial node and to infinity for all other nodes. During the run of the algorithm, the tentative distance of a node v is the length of the shortest path discovered so far between the node v and the starting node. Since initially no path is known to... | Wikipedia - Dijkstra’s Algorithm | null | null | null |
Set the initial node as current. For the current node, consider all of its unvisited neighbors and calculate their tentative distances through the current node. Compare the newly calculated tentative distance to the one currently assigned to the neighbor and assign it the smaller one. | Wikipedia - Dijkstra’s Algorithm | null | null | null |
For example, if the current node A is marked with a distance of 6, and the edge connecting it with a neighbor B has length 2, then the distance to B through A will be 6 + 2 = 8. If B was previously marked with a distance greater than 8 then change it to 8. Otherwise, the current value will be kept. | Wikipedia - Dijkstra’s Algorithm | null | null | null |
When we are done considering all of the unvisited neighbors of the current node, mark the current node as visited and remove it from the unvisited set. A visited node will never be checked again (this is valid and optimal in connection with the behavior in step 6.: that the next nodes to visit will always be in the ord... | Wikipedia - Dijkstra’s Algorithm | null | null | null |
Let the parametric surface be given by the equation where r {\displaystyle \mathbf {r} } is a vector-valued function of the parameters (u, v) and the parameters vary within a certain domain D in the parametric uv-plane. The first partial derivatives with respect to the parameters are usually denoted r u := ∂ r ∂ u {\te... | Wikipedia - Parametrized surface | null | null | null |
Let the position of a point in space be characterized by three coordinate variables ( q 1 , q 2 , q 3 ) {\displaystyle (q^{1},q^{2},q^{3})} . The coordinate curve q1 represents a curve on which q2, q3 are constant. Let x be the position vector of the point relative to some origin. Then, assuming that such a mapping and... | Wikipedia - Tensors in curvilinear coordinates | null | null | null |
Let the vertices of our graph be partitioned in U and V, and consider a partial matching, as indicated by the Pair_U and Pair_V tables that contain the one vertex to which each vertex of U and of V is matched, or NIL for unmatched vertices. The key idea is to add two dummy vertices on each side of the graph: uDummy con... | Wikipedia - Hopcroft–Karp algorithm | null | null | null |
Observe in particular that BFS marks the unmatched nodes of U with distance 0, then increments the distance every time it comes back to U. This guarantees that the paths considered in the BFS are of minimal length to connect unmatched vertices of U to unmatched vertices of V while always going back from V to U on edges... | Wikipedia - Hopcroft–Karp algorithm | null | null | null |
If BFS returns true, then we can go ahead and update the pairing for vertices on the minimal-length paths found from U to V: we do so using a depth-first search (DFS). Note that each vertex in V on such a path, except for the last one, is currently matched. So we can explore with the DFS, making sure that the paths tha... | Wikipedia - Hopcroft–Karp algorithm | null | null | null |
We update along every such path by removing from the matching all edges of the path that are currently in the matching, and adding to the matching all edges of the path that are currently not in the matching: as this is an augmenting path (the first and last edges of the path were not part of the matching, and the path... | Wikipedia - Hopcroft–Karp algorithm | null | null | null |
Also observe that the DFS does not visit the same vertex multiple times. This is thanks to the following lines: Dist = ∞ return false When we were not able to find any shortest augmenting path from a vertex u, then the DFS marks vertex u by setting Dist to infinity, so that these vertices are not visited again. One las... | Wikipedia - Hopcroft–Karp algorithm | null | null | null |
Let there be a simple random sample X 1 , … , X n {\displaystyle X_{1},\ldots ,X_{n}} where each X i {\displaystyle X_{i}} is i.i.d from a Bernoulli(p) distribution and weight w i {\displaystyle w_{i}} is the weight for each observation. Standardize the (positive) weights w i {\displaystyle w_{i}} so they sum to 1. The... | Wikipedia - Clopper-Pearson interval | null | null | null |
Since the X i {\displaystyle X_{i}} are independent and each one has variance Var ( X i ) = p ( 1 − p ) {\displaystyle {\text{Var}}(X_{i})=p(1-p)} , the sampling variance of the proportion therefore is: Var ( p ^ ) = ∑ i = 1 n Var ( w i X i ) = p ( 1 − p ) ∑ i = 1 n w i 2 {\displaystyle {\text{Var}}({\hat {p}})=\sum _{... | Wikipedia - Clopper-Pearson interval | null | null | null |
Let three random variables form the Markov chain X → Y → Z {\displaystyle X\rightarrow Y\rightarrow Z} , implying that the conditional distribution of Z {\displaystyle Z} depends only on Y {\displaystyle Y} and is conditionally independent of X {\displaystyle X} . Specifically, we have such a Markov chain if the joint ... | Wikipedia - Data processing inequality | null | null | null |
Let u = , {\displaystyle u={\begin{bmatrix}0&\mathbf {a} &c\\0&0_{n}&\mathbf {b} \\0&0&0\end{bmatrix}},} which fulfills u 3 = 0 n + 2 {\displaystyle u^{3}=0_{n+2}} . The exponential map evaluates to exp ( u ) = ∑ k = 0 ∞ 1 k ! u k = I n + 2 + u + 1 2 u 2 = . {\displaystyle \exp(u)=\sum _{k=0}^{\infty }{\frac {1}{k! | Wikipedia - Heisenberg algebra | null | null | null |
}}u^{k}=I_{n+2}+u+{\tfrac {1}{2}}u^{2}={\begin{bmatrix}1&\mathbf {a} &c+{1 \over 2}\mathbf {a} \cdot \mathbf {b} \\0&I_{n}&\mathbf {b} \\0&0&1\end{bmatrix}}.} The exponential map of any nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connected, simply-connected Lie group. Thi... | Wikipedia - Heisenberg algebra | null | null | null |
Let us assume there is a set of basis functionals { P α } {\displaystyle \{P_{\alpha }\}} spanning the theory space under consideration so that any action functional, i.e. any point of this theory space, can be written as a linear combination of the P α {\displaystyle P_{\alpha }} 's. Then solutions Γ k {\displaystyle... | Wikipedia - Asymptotic safety in quantum gravity | null | null | null |
Together with the corresponding initial conditions these equations fix the evolution of the running couplings g α ( k ) {\displaystyle g_{\alpha }(k)} , and thus determine Γ k {\displaystyle \Gamma _{k}} completely. As one can see, the FRGE gives rise to a system of infinitely many coupled differential equations since ... | Wikipedia - Asymptotic safety in quantum gravity | null | null | null |
A possible way out is to restrict the analysis on a finite-dimensional subspace as an approximation of the full theory space. In other words, such a truncation of the theory space sets all but a finite number of couplings to zero, considering only the reduced basis { P α } {\displaystyle \{P_{\alpha }\}} with α = 1 , ... | Wikipedia - Asymptotic safety in quantum gravity | null | null | null |
Let us call a function y: ( S ∪ T ) → R {\displaystyle y:(S\cup T)\to \mathbb {R} } a potential if y ( i ) + y ( j ) ≤ c ( i , j ) {\displaystyle y(i)+y(j)\leq c(i,j)} for each i ∈ S , j ∈ T {\displaystyle i\in S,j\in T} . The value of potential y is the sum of the potential over all vertices: ∑ v ∈ S ∪ T y ( v ) {\dis... | Wikipedia - Hungarian method | null | null | null |
This proves that both of them are optimal. In fact, the Hungarian method finds a perfect matching of tight edges: an edge i j {\displaystyle ij} is called tight for a potential y if y ( i ) + y ( j ) = c ( i , j ) {\displaystyle y(i)+y(j)=c(i,j)} . Let us denote the subgraph of tight edges by G y {\displaystyle G_{y}} ... | Wikipedia - Hungarian method | null | null | null |
The cost of a perfect matching in G y {\displaystyle G_{y}} (if there is one) equals the value of y. During the algorithm we maintain a potential y and an orientation of G y {\displaystyle G_{y}} (denoted by G y → {\displaystyle {\overrightarrow {G_{y}}}} ) which has the property that the edges oriented from T to S for... | Wikipedia - Hungarian method | null | null | null |
We are done if M is a perfect matching. In a general step, let R S ⊆ S {\displaystyle R_{S}\subseteq S} and R T ⊆ T {\displaystyle R_{T}\subseteq T} be the vertices not covered by M (so R S {\displaystyle R_{S}} consists of the vertices in S with no incoming edge and R T {\displaystyle R_{T}} consists of the vertices i... | Wikipedia - Hungarian method | null | null | null |
This can be computed by breadth-first search. If R T ∩ Z {\displaystyle R_{T}\cap Z} is nonempty, then reverse the orientation of a directed path in G y → {\displaystyle {\overrightarrow {G_{y}}}} from R S {\displaystyle R_{S}} to R T {\displaystyle R_{T}} . Thus the size of the corresponding matching increases by 1. | Wikipedia - Hungarian method | null | null | null |
If R T ∩ Z {\displaystyle R_{T}\cap Z} is empty, then let Δ := min { c ( i , j ) − y ( i ) − y ( j ): i ∈ Z ∩ S , j ∈ T ∖ Z } . {\displaystyle \Delta :=\min\{c(i,j)-y(i)-y(j):i\in Z\cap S,j\in T\setminus Z\}.} Δ is well defined because at least one such edge i j {\displaystyle ij} must exist whenever the matching is no... | Wikipedia - Hungarian method | null | null | null |
Increase y by Δ on the vertices of Z ∩ S {\displaystyle Z\cap S} and decrease y by Δ on the vertices of Z ∩ T {\displaystyle Z\cap T} . The resulting y is still a potential, and although the graph G y {\displaystyle G_{y}} changes, it still contains M (see the next subsections). | Wikipedia - Hungarian method | null | null | null |
We orient the new edges from S to T. By the definition of Δ the set Z of vertices reachable from R S {\displaystyle R_{S}} increases (note that the number of tight edges does not necessarily increase). We repeat these steps until M is a perfect matching, in which case it gives a minimum cost assignment. The running tim... | Wikipedia - Hungarian method | null | null | null |
Let us consider an "infinite" DMRG algorithm for the S = 1 {\displaystyle S=1} antiferromagnetic quantum Heisenberg chain. The recipe can be applied for every translationally invariant one-dimensional lattice. DMRG is a renormalization-group technique because it offers an efficient truncation of the Hilbert space of on... | Wikipedia - Density Matrix Renormalization Group | null | null | null |
Let us consider degenerate energy eigenstates and a perturbation that completely lifts the degeneracy to first order of correction. The perturbed Hamiltonian is denoted as where H ^ 0 {\displaystyle {\hat {H}}_{0}} is the unperturbed Hamiltonian, V ^ {\displaystyle {\hat {V}}} is the perturbation operator, and 0 < λ < ... | Wikipedia - Perturbative expansion | null | null | null |
The eventual degeneracy among the other states with ∀ m ≠ n {\displaystyle \forall m\neq n} does not change our arguments. All states | ψ n k ( 0 ) ⟩ {\displaystyle \left|\psi _{nk}^{(0)}\right\rangle } with various values of k {\displaystyle k} share the same energy E n ( 0 ) {\displaystyle E_{n}^{(0)}} when there is ... | Wikipedia - Perturbative expansion | null | null | null |
By V n l , n k {\displaystyle V_{nl,nk}} and V m , n k {\displaystyle V_{m,nk}} , we denote the matrix elements of the perturbation operator V ^ {\displaystyle {\hat {V}}} in the basis of the unperturbed eigenstates. We assume that the basis vectors | ψ n k ( 0 ) ⟩ {\displaystyle \left|\psi _{nk}^{(0)}\right\rangle } i... | Wikipedia - Perturbative expansion | null | null | null |
Let us consider the decomposition of the density matrix to pure components as ϱ = ∑ k p k | Ψ k ⟩ ⟨ Ψ k | . {\displaystyle \varrho =\sum _{k}p_{k}\vert \Psi _{k}\rangle \langle \Psi _{k}\vert .} The Heisenberg uncertainty relation is valid for all | Ψ k ⟩ {\displaystyle \vert \Psi _{k}\rangle } ( Δ A ) Ψ k 2 ( Δ B ) Ψ ... | Wikipedia - Quantum Cramér–Rao bound | null | null | null |
From these, employing the Cauchy-Schwarz inequality we arrive at ( Δ θ ) A 2 ≥ 1 4 min { p k , Ψ k } . {\displaystyle (\Delta \theta )_{A}^{2}\geq {\frac {1}{4\min _{\{p_{k},\Psi _{k}\}}}}.} Here ( Δ θ ) A 2 = ( Δ A ) 2 | ∂ θ ⟨ A ⟩ | 2 = ( Δ A ) 2 | ⟨ i ⟩ | 2 {\displaystyle (\Delta \theta )_{A}^{2}={\frac {(\Delta A)... | Wikipedia - Quantum Cramér–Rao bound | null | null | null |
{\displaystyle A.} Moreover, the convex roof of the variance is given as min { p k , Ψ k } = 1 4 F Q , {\displaystyle \min _{\{p_{k},\Psi _{k}\}}\left={\frac {1}{4}}F_{Q},} where F Q {\displaystyle F_{Q}} is the quantum Fisher information. == References == | Wikipedia - Quantum Cramér–Rao bound | null | null | null |
Let us consider the example of a system of charged particles subject to a constant external magnetic field: in this case the canonical time reversal operation that reverses the velocities and the time t {\displaystyle t} and keeps the coordinates untouched is no more a symmetry for the system. Under this consideration,... | Wikipedia - Time-reversal invariance | null | null | null |
Let us continue with the example from above where A = k ⟨ x , y , z ⟩ / ( z 2 − x y − y x , z x − x z , z y − y z ) {\displaystyle A=k\langle x,y,z\rangle /(z^{2}-xy-yx,zx-xz,zy-yz)} . We found that the overlap ambiguity z 2 x {\displaystyle z^{2}x} does not resolve. This gives us h 1 = x y x + y x 2 {\displaystyle h_{... | Wikipedia - Bergman's diamond lemma | null | null | null |
Hence we do not need to scale it and can add it to our set of relations which is now { z 2 − x y − y x , z x − x z , z y − y z , y x 2 − x 2 y } {\displaystyle \{z^{2}-xy-yx,zx-xz,zy-yz,yx^{2}-x^{2}y\}} . The previous ambiguity now resolves to either h 1 {\displaystyle h_{1}} or h 2 {\displaystyle h_{2}} . Adding the n... | Wikipedia - Bergman's diamond lemma | null | null | null |
Let us try and resolve it with the relations we currently have. Again, resolving z 2 {\displaystyle z^{2}} first we obtain, z 2 y = ( x y + y x ) y = x y 2 + y x y . {\displaystyle z^{2}y=(xy+yx)y=xy^{2}+yxy.} | Wikipedia - Bergman's diamond lemma | null | null | null |
On the other hand resolving z y {\displaystyle zy} twice first and then z 2 {\displaystyle z^{2}} we find, z 2 y = z y z = y z 2 = y ( x y + y x ) = y x y + y 2 x . {\displaystyle z^{2}y=zyz=yz^{2}=y(xy+yx)=yxy+y^{2}x.} Thus we have h 3 = x y 2 + y x y {\displaystyle h_{3}=xy^{2}+yxy} and h 4 = y x y + y 2 x {\displays... | Wikipedia - Bergman's diamond lemma | null | null | null |
Let us define algebraic systems used in the forthcoming symmetry definition. | Wikipedia - Lie point symmetry | null | null | null |
Let us do some comparison among Bell nonlocality, quantum steering, and quantum entanglement. By definition, a Bell nonlocal which does not admit a local hidden variable model for some measurement setting, a quantum steering state is a state which does not admit a local hidden state model for some measurement assemblag... | Wikipedia - Quantum steering | null | null | null |
Let us first consider the clean, non-interacting system: J i = J , h i = h , J i n t = 0 {\displaystyle J_{i}=J,\;h_{i}=h,\;J_{\rm {int}}=0} . In equilibrium, the ground state is ferromagnetically ordered with spins aligned along the z {\displaystyle z} axis for J > h {\displaystyle J>h} , but is a paramagnet for J < h... | Wikipedia - Localization-protected quantum order | null | null | null |
Let us further restrict to the case that L/K is cyclic with Galois group G of order n generated by t. Let A be a crossed product (L,G,c) with factor set c. Let u = ut be the generator in A corresponding to t. We can define the other generators u t i = u i {\displaystyle u_{t^{i}}=u^{i}\,} and then we have un = a in K. ... | Wikipedia - Factor set | null | null | null |
Let us now assume that the condition ∇ × ( ∇ × ϵ ) T = 0 {\displaystyle {\boldsymbol {\nabla }}\times ({\boldsymbol {\nabla }}\times {\boldsymbol {\epsilon }})^{T}={\boldsymbol {0}}} is satisfied in a portion of a body. Is this condition sufficient to guarantee the existence of a continuous, single-valued displacement ... | Wikipedia - Compatibility (mechanics) | null | null | null |
Use these values to compute μ N {\displaystyle \mu _{N}} and a N . {\displaystyle a_{N}.} Initialize λ N {\displaystyle \lambda _{N}} to some arbitrary value. | Wikipedia - Variational Bayes | null | null | null |
Use the current value of λ N , {\displaystyle \lambda _{N},} along with the known values of the other parameters, to compute b N {\displaystyle b_{N}} . Use the current value of b N , {\displaystyle b_{N},} along with the known values of the other parameters, to compute λ N {\displaystyle \lambda _{N}} . Repeat the las... | Wikipedia - Variational Bayes | null | null | null |
Note also that the posterior distributions have the same form as the corresponding prior distributions. We did not assume this; the only assumption we made was that the distributions factorize, and the form of the distributions followed naturally. It turns out (see below) that the fact that the posterior distributions ... | Wikipedia - Variational Bayes | null | null | null |
Let us redesign the original library with extensibility in mind using the ideas from the paper Extensibility for the Masses. We use the same implementation as in the first code example but now add a new interface containing the functions over the type as well as a factory for the algebra. Notice that we now generate th... | Wikipedia - Expression problem | null | null | null |
Let us return to the classical Yang–Mills theory. It is possible to encode the gauge invariant information of the theory in terms of `loop-like' variables. We need the notion of a holonomy. | Wikipedia - Loop representation | null | null | null |
A holonomy is a measure of how much the initial and final values of a spinor or vector differ after parallel transport around a closed loop γ {\displaystyle \gamma } ; it is denoted h γ {\displaystyle h_{\gamma }} Knowledge of the holonomies is equivalent to knowledge of the connection, up to gauge equivalence. Holono... | Wikipedia - Loop representation | null | null | null |
For a closed loop x = y {\displaystyle x=y} if we take the trace of this, that is, putting α = β {\displaystyle \alpha =\beta } and summing we obtain ( h e ′ ) α α = U α γ − 1 ( x ) ( h e ) γ σ U σ α ( x ) = ( h e ) γ σ = δ σ γ ( h e ) γ σ = ( h e ) γ γ {\displaystyle (h'_{e})_{\alpha \alpha }=U_{\alpha \gamma }^{-1}(... | Wikipedia - Loop representation | null | null | null |
It is denoted W γ {\displaystyle W_{\gamma }} and is called a Wilson loop. The explicit form of the holonomy is h γ = P exp { − ∫ γ 0 γ 1 d s γ ˙ a A a i ( γ ( s ) ) T i } {\displaystyle h_{\gamma }={\mathcal {P}}\exp {\Big \{}-\int _{\gamma _{0}}^{\gamma _{1}}\,ds{\dot {\gamma }}^{a}A_{a}^{i}(\gamma (s))T_{i}{\Big... | Wikipedia - Loop representation | null | null | null |
It turns out that there are infinitely many more examples of sets of matrices that satisfy these relations, where each set comprises ( N + 1 ) × ( N + 1 ) {\displaystyle (N+1)\times (N+1)} matrices with N = 1 , 2 , 3 , … {\displaystyle N=1,2,3,\dots } , and where none of these can be thought to `decompose' into two or ... | Wikipedia - Loop representation | null | null | null |
Let us say we have an array of n elements. We choose the gap we intend to give. Then we would have a final array of size (1 + ε)n. The algorithm works in log n rounds. In each round we insert as many elements as there are in the final array already, before re-balancing the array. | Wikipedia - Library sort | null | null | null |
For finding the position of inserting, we apply Binary Search in the final array and then swap the following elements till we hit an empty space. Once the round is over, we re-balance the final array by inserting spaces between each element. Following are three important steps of the algorithm: Binary Search: Finding t... | Wikipedia - Library sort | null | null | null |
This can be done by linearly moving towards left or right side of the array if you hit an empty space in the middle element. Insertion: Inserting the element in the position found and swapping the following elements by 1 position till an empty space is hit. This is done in logarithmic time, with high probability. | Wikipedia - Library sort | null | null | null |
Re-Balancing: Inserting spaces between each pair of elements in the array. The cost of rebalancing is linear in the number of elements already inserted. As these lengths increase with the powers of 2 for each round, the total cost of rebalancing is also linear. | Wikipedia - Library sort | null | null | null |
Let us suppose there is a field ϕ {\displaystyle \phi } with a potential depending upon two parameters, a {\displaystyle a} and b {\displaystyle b} . V ( ϕ ) = − a ϕ α + b ϕ β {\displaystyle V\left(\phi \right)=-a\phi ^{\alpha }+b\phi ^{\beta }} Let us also suppose that a {\displaystyle a} is positive and nonzero and β... | Wikipedia - Dangerously irrelevant operator | null | null | null |
It is clear that if the scaling dimension of b {\displaystyle b} is negative, b {\displaystyle b} is an irrelevant parameter. However, the crucial point is, that the VEV ⟨ ϕ ⟩ = ( a α b β ) 1 β − α = ( a α β ) 1 β − α b − 1 β − α {\displaystyle \langle \phi \rangle =\left({\frac {a\alpha }{b\beta }}\right)^{\frac {1}{\... | Wikipedia - Dangerously irrelevant operator | null | null | null |
Let us take (1) the line segment joining the points (1, 0) and (0, 1) and (2) the unit circle with center at the origin. As curves in V, their equations can be obtained as f ( t ) = 1 − t + i t and g ( t ) = cos ( 2 π t ) + i sin ( 2 π t ) {\displaystyle f(t)=1-t+it{\mbox{ and }}g(t)=\cos(2\pi t)+i\sin(2\pi t)} Sin... | Wikipedia - Functional-theoretic algebra | null | null | null |
Let us take the case that G is the Galois group of a field extension L/K. A factor system c in H2(G,L*) gives rise to a crossed product algebra: 31 A, which is a K-algebra containing L as a subfield, generated by the elements λ in L and ug with multiplication λ u g = u g λ g , {\displaystyle \lambda u_{g}=u_{g}\lambda ... | Wikipedia - Crossed product algebra | null | null | null |
{\displaystyle A=(L,G,c).} The crossed product algebra A is a central simple algebra (CSA) of degree equal to . The converse holds: every central simple algebra over K that splits over L and such that deg A = arises in this way. The tensor product of algebras corresponds to multiplication of the corresponding elements... | Wikipedia - Crossed product algebra | null | null | null |
However, it can be shown that graph-changing, spatially diffeomorphism invariant operators such as the Master constraint cannot be defined on the kinematic Hilbert space H K i n {\displaystyle {\mathcal {H}}_{Kin}} . The way out is to define M ^ {\displaystyle {\hat {M}}} not on H K i n {\displaystyle {\mathcal {H}}_{K... | Wikipedia - Hamiltonian constraint of LQG | null | null | null |
What is done first is, we are able to compute the matrix elements of the would-be operator M ^ {\displaystyle {\hat {M}}} , that is, we compute the quadratic form Q M {\displaystyle Q_{M}} . We would like there to be a unique, positive, self-adjoint operator M ^ {\displaystyle {\hat {M}}} whose matrix elements reproduc... | Wikipedia - Hamiltonian constraint of LQG | null | null | null |
Let v 0 = f ( x , y ) {\displaystyle v_{0}=f(x,y)} , we attempt to improve f ( x , y ) {\displaystyle f(x,y)} by testing a sequence of candidate offsets at an exponentially decreasing distance from v 0 {\displaystyle v_{0}} u i = v 0 + w α i R i {\displaystyle u_{i}=v_{0}+w\alpha ^{i}R_{i}} where R i {\displaystyle R_{... | Wikipedia - PatchMatch | null | null | null |
Let w m a x = max { w i } {\displaystyle w_{max}=\max\{w_{i}\}} , be the maximum weight. In IWRR, each cycle is split into w m a x {\displaystyle w_{max}} rounds. A queue with weight w i {\displaystyle w_{i}} can emit one packet at round r {\displaystyle r} only if r ≤ w i {\displaystyle r\leq w_{i}} . | Wikipedia - Weighted round robin | null | null | null |
{\displaystyle {\begin{aligned}y(t)&=Ye^{j\omega t}=|Y|e^{j(\omega t+\arg(Y))},\\Y&=|Y|e^{j\arg(Y)}.\end{aligned}}} Note that, in a linear time-invariant system, the input frequency ω {\displaystyle \omega } has not changed, only the amplitude and the phase angle of the sinusoid has been changed by the system. The freq... | Wikipedia - Pulse-transfer function | null | null | null |
The phase delay (i.e., the frequency-dependent amount of delay introduced to the sinusoid by the transfer function) is: τ ϕ ( ω ) = − ϕ ( ω ) ω . {\displaystyle \tau _{\phi }(\omega )=-{\frac {\phi (\omega )}{\omega }}.} The group delay (i.e., the frequency-dependent amount of delay introduced to the envelope of the si... | Wikipedia - Pulse-transfer function | null | null | null |
Let x = ( x 1 , x 2 , … , x n ) {\displaystyle \mathbf {x} =(\mathbf {x} _{1},\mathbf {x} _{2},\ldots ,\mathbf {x} _{n})} be a sample of n {\displaystyle n} independent observations from a mixture of two multivariate normal distributions of dimension d {\displaystyle d} , and let z = ( z 1 , z 2 , … , z n ) {\displayst... | Wikipedia - Expectation-maximization algorithm | null | null | null |
Let x be a point in Xn given by a marking f: Rn → Γ with a volume-one metric graph structure L on Γ. Let T be the universal cover of Γ. Thus T is a simply connected graph, that is T is a topological tree. We can also lift the metric structure L to T by giving every edge of T the same length as the length of its image i... | Wikipedia - Outer space (mathematics) | null | null | null |
Moreover, every minimal free and discrete isometric action of Fn on a real tree with the quotient being a metric graph of volume one arises in this fashion from some point x of Xn. This defines a bijective correspondence between Xn and the set of equivalence classes of minimal free and discrete isometric actions of Fn ... | Wikipedia - Outer space (mathematics) | null | null | null |
Let x i {\displaystyle x^{i}} be the generalized coordinates and x ˙ i {\displaystyle {\dot {x}}^{i}} be the generalized velocities, then the kinetic energy for a unit mass is given by T = 1 2 g i k x ˙ i x ˙ k {\displaystyle T={\tfrac {1}{2}}g_{ik}{\dot {x}}^{i}{\dot {x}}^{k}} , where g i k {\displaystyle g_{ik}} is t... | Wikipedia - Christoffel symbols | null | null | null |
Let x {\displaystyle x\,\!} be a state value in a binary domain. A binomial opinion about the truth of state value x {\displaystyle x\,\!} is the ordered quadruple ω x = ( b x , d x , u x , a x ) {\displaystyle \omega _{x}=(b_{x},d_{x},u_{x},a_{x})\,\!} | Wikipedia - Subjective logic | null | null | null |
where: These components satisfy b x + d x + u x = 1 {\displaystyle b_{x}+d_{x}+u_{x}=1\,\!} and b x , d x , u x , a x ∈ {\displaystyle b_{x},d_{x},u_{x},a_{x}\in \,\!} . | Wikipedia - Subjective logic | null | null | null |
The characteristics of various opinion classes are listed below. The projected probability of a binomial opinion is defined as P x = b x + a x u x {\displaystyle \mathrm {P} _{x}=b_{x}+a_{x}u_{x}\,\!} . | Wikipedia - Subjective logic | null | null | null |
Binomial opinions can be represented on an equilateral triangle as shown below. A point inside the triangle represents a ( b x , d x , u x ) {\displaystyle (b_{x},d_{x},u_{x})\,\!} triple. | Wikipedia - Subjective logic | null | null | null |
The b,d,u-axes run from one edge to the opposite vertex indicated by the Belief, Disbelief or Uncertainty label. For example, a strong positive opinion is represented by a point towards the bottom right Belief vertex. The base rate, also called the prior probability, is shown as a red pointer along the base line, and t... | Wikipedia - Subjective logic | null | null | null |
, is formed by projecting the opinion onto the base, parallel to the base rate projector line. Opinions about three values/propositions X, Y and Z are visualized on the triangle to the left, and their equivalent Beta PDFs (Probability Density Functions) are visualized on the plots to the right. The numerical values and... | Wikipedia - Subjective logic | null | null | null |
The Beta PDF is normally denoted as B e t a ( p ( x ) ; α , β ) {\displaystyle \mathrm {Beta} (p(x);\alpha ,\beta )\,\!} where α {\displaystyle \alpha \,\!} and β {\displaystyle \beta \,\!} | Wikipedia - Subjective logic | null | null | null |
are its two strength parameters. The Beta PDF of a binomial opinion ω x = ( b x , d x , u x , a x ) {\displaystyle \omega _{x}=(b_{x},d_{x},u_{x},a_{x})\,\!} is the function B e t a ( p ( x ) ; α , β ) where { α = W b x u x + W a x β = W d x u x + W ( 1 − a x ) {\displaystyle \mathrm {Beta} (p(x);\alpha ,\beta ){\mbox{... | Wikipedia - Subjective logic | null | null | null |
Let x ∈ R ^ {\displaystyle x\in {\widehat {\mathbb {R} }}} and A ⊆ R ^ {\displaystyle A\subseteq {\widehat {\mathbb {R} }}} . A is a neighbourhood of x, if A contains an open interval B that contains x. A is a right-sided neighbourhood of x, if there is a real number y such that y ≠ x {\displaystyle y\neq x} and A cont... | Wikipedia - Projectively extended real number system | null | null | null |
Let x(t) be a curve in R n {\displaystyle \mathbb {R} ^{n}} . Assume, as one does in the Euclidean case, that the first n derivatives of x(t) are linearly independent so that, in particular, x(t) does not lie in any lower-dimensional affine subspace of R 2 {\displaystyle \mathbb {R} ^{2}} . Then the curve parameter t c... | Wikipedia - Affine geometry of curves | null | null | null |
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