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The emergence of computer-assisted proofs has allowed proof lengths to further expand, such as the 255-page Feit–Thompson theorem. The result of this trend is a philosophy of the quasi-empiricist proof that can not be considered infallible, but has a probability attached to it.The concept of rigor in mathematics dates ... | Wikipedia - Fields of mathematics | null | null | null |
The method of demonstrating rigorous proof was enhanced in the sixteenth century through the use of symbolic notation. In the 18th century, social transition led to mathematicians earning their keep through teaching, which led to more careful thinking about the underlying concepts of mathematics. This produced more rig... | Wikipedia - Fields of mathematics | null | null | null |
This was solved by the inclusion of axioms with the apodictic inference rules of mathematical theories; the re-introduction of axiomatic method pioneered by the ancient Greeks. It results that "rigor" is no more a relevant concept in mathematics, as a proof is either correct or erroneous, and a "rigorous proof" is simp... | Wikipedia - Fields of mathematics | null | null | null |
Mathematical relation Finitary relation Antisymmetric relation Asymmetric relation Bijection Bijection, injection and surjection Binary relation Composition of relations Congruence relation Connected relation Converse relation Coreflexive relation Covering relation Cyclic order Dense relation Dependence relation Depend... | Wikipedia - Outline of logic | null | null | null |
Mathematical results concerning cliques include the following. Turán's theorem gives a lower bound on the size of a clique in dense graphs. If a graph has sufficiently many edges, it must contain a large clique. For instance, every graph with n {\displaystyle n} vertices and more than ⌊ n 2 ⌋ ⋅ ⌈ n 2 ⌉ {\displaystyle \... | Wikipedia - Maximum clique | null | null | null |
Ramsey's theorem states that every graph or its complement graph contains a clique with at least a logarithmic number of vertices. According to a result of Moon & Moser (1965), a graph with 3n vertices can have at most 3n maximal cliques. The graphs meeting this bound are the Moon–Moser graphs K3,3,..., a special case ... | Wikipedia - Maximum clique | null | null | null |
Hadwiger's conjecture, still unproven, relates the size of the largest clique minor in a graph (its Hadwiger number) to its chromatic number. The Erdős–Faber–Lovász conjecture is another unproven statement relating graph coloring to cliques. The Erdős–Hajnal conjecture that families of graphs defined by forbidden graph... | Wikipedia - Maximum clique | null | null | null |
A block graph is a graph whose biconnected components are cliques. A chordal graph is a graph whose vertices can be ordered into a perfect elimination ordering, an ordering such that the neighbors of each vertex v that come later than v in the ordering form a clique. A cograph is a graph all of whose induced subgraphs ... | Wikipedia - Maximum clique | null | null | null |
An interval graph is a graph whose maximal cliques can be ordered in such a way that, for each vertex v, the cliques containing v are consecutive in the ordering. A line graph is a graph whose edges can be covered by edge-disjoint cliques in such a way that each vertex belongs to exactly two of the cliques in the cover... | Wikipedia - Maximum clique | null | null | null |
A split graph is a graph in which some clique contains at least one endpoint of every edge. A triangle-free graph is a graph that has no cliques other than its vertices and edges.Additionally, many other mathematical constructions involve cliques in graphs. Among them, The clique complex of a graph G is an abstract sim... | Wikipedia - Maximum clique | null | null | null |
It is an example of median graph, and is associated with a median algebra on the cliques of a graph: the median m(A,B,C) of three cliques A, B, and C is the clique whose vertices belong to at least two of the cliques A, B, and C. The clique-sum is a method for combining two graphs by merging them along a shared clique.... | Wikipedia - Maximum clique | null | null | null |
The intersection number of a graph is the minimum number of cliques needed to cover all the graph's edges. The clique graph of a graph is the intersection graph of its maximal cliques.Closely related concepts to complete subgraphs are subdivisions of complete graphs and complete graph minors. In particular, Kuratowski'... | Wikipedia - Maximum clique | null | null | null |
Mathematical rigour can apply to methods of mathematical proof and to methods of mathematical practice (thus relating to other interpretations of rigour). | Wikipedia - Scientific rigour | null | null | null |
Mathematical rigour is often cited as a kind of gold standard for mathematical proof. Its history traces back to Greek mathematics, especially to Euclid's Elements.Until the 19th century, Euclid's Elements was seen as extremely rigorous and profound, but in the late 19th century, Hilbert (among others) realized that th... | Wikipedia - Rigorous proof | null | null | null |
During the 19th century, the term "rigorous" began to be used to describe increasing levels of abstraction when dealing with calculus which eventually became known as mathematical analysis. The works of Cauchy added rigour to the older works of Euler and Gauss. The works of Riemann added rigour to the works of Cauchy. | Wikipedia - Rigorous proof | null | null | null |
The works of Weierstrass added rigour to the works of Riemann, eventually culminating in the arithmetization of analysis. Starting in the 1870s, the term gradually came to be associated with Cantorian set theory. Mathematical rigour can be modelled as amenability to algorithmic proof checking. | Wikipedia - Rigorous proof | null | null | null |
Indeed, with the aid of computers, it is possible to check some proofs mechanically. Formal rigour is the introduction of high degrees of completeness by means of a formal language where such proofs can be codified using set theories such as ZFC (see automated theorem proving). Published mathematical arguments have to ... | Wikipedia - Rigorous proof | null | null | null |
In this sense, written mathematical discourse is a prototype of formal proof. Often, a written proof is accepted as rigorous although it might not be formalised as yet. The reason often cited by mathematicians for writing informally is that completely formal proofs tend to be longer and more unwieldy, thereby obscuring... | Wikipedia - Rigorous proof | null | null | null |
An argument that appears obvious to human intuition may in fact require fairly long formal derivations from the axioms. A particularly well-known example is how in Principia Mathematica, Whitehead and Russell have to expend a number of lines of rather opaque effort in order to establish that, indeed, it is sensical to ... | Wikipedia - Rigorous proof | null | null | null |
Mathematical skills are important not only for the national economy but also for an individual's life chances: low numeracy increases the probability of arrest, depression, physical illnesses, unemployment. One of the main causes of low numeracy is a congenital condition called dyscalculia. As the Foresight report on M... | Wikipedia - Educational neuroscience | null | null | null |
Dyscalculia relates to numeracy and affects between 4–7% of children. It has a much lower profile than dyslexia but can also have substantial impacts: it can reduce lifetime earnings by £114,000 and reduce the probability of achieving five or more GCSEs (A*-C) by 7–20 percentage points. | Wikipedia - Educational neuroscience | null | null | null |
Home and school interventions have again been identified by the Project. Also, technological interventions are extremely promising, offering individualised instruction and help, although these need more development." (Executive Summary, Section 5.3) Understanding typical and atypical mathematical development is a cruci... | Wikipedia - Educational neuroscience | null | null | null |
Over the past ten years, a brain system for simple number processing has been identified and a handful of studies of children's brains that shed light on its development.An increasing convergence of evidence suggests that dyscalculia may be due to a deficit in an inherited core system for representing the number of obj... | Wikipedia - Educational neuroscience | null | null | null |
Despite these similarities in terms of the scientific progress, public awareness of dyscalculia is much lower than it is for dyslexia. The UK's Chief Scientific Advisor, John Beddington, notes that, "developmental dyscalculia is currently the poor relation of dyslexia, with a much lower public profile. But the conseque... | Wikipedia - Educational neuroscience | null | null | null |
"The application of neuroscience to understanding mathematical processing has already resulted in understanding beyond the early cognitive theories. Cognitive neuroscience research has revealed the existence of an innate ‘number sense’ system, present in animals and infants as well as adults, that is responsible for ba... | Wikipedia - Educational neuroscience | null | null | null |
This parietal system is active in children and adults during basic numerical tasks, but over the course of development it appears to become more specialised. Furthermore, children with mathematical learning disabilities (dyscalculia) show weaker activation in this region than typically developing children during basic ... | Wikipedia - Educational neuroscience | null | null | null |
In addition to this basic number sense, numerical information can be stored verbally in the language system, a system that neuroscience research is beginning to reveal as qualitatively different at the brain level to the number sense system. This system also stores information about other well learned verbal sequences,... | Wikipedia - Educational neuroscience | null | null | null |
Showing that these subsets of arithmetic skills are supported by different brain mechanisms offers the opportunity for a deeper understanding of the learning processes required to acquire arithmetic proficiency. Neuroimaging studies of mathematical learning disabilities are still rare but dyscalculia is an area of incr... | Wikipedia - Educational neuroscience | null | null | null |
For example, many children with dyscalculia also have dyslexia, and those that do may show different activation of the verbal networks that support maths, while those who have dyscalculia only, may show impairments of the parietal number sense system. Indeed, the few studies carried out on children with dyscalculia onl... | Wikipedia - Educational neuroscience | null | null | null |
Mathematical statistics is a key subset of the discipline of statistics. Statistical theorists study and improve statistical procedures with mathematics, and statistical research often raises mathematical questions. Mathematicians and statisticians like Gauss, Laplace, and C. S. Peirce used decision theory with probabi... | Wikipedia - Mathematical Statistics | null | null | null |
Mathematical statistics is the application of mathematics to statistics. Mathematical techniques used for this include mathematical analysis, linear algebra, stochastic analysis, differential equations, and measure-theoretic probability theory. | Wikipedia - Statistical Sciences | null | null | null |
Mathematical symbols are a type of ideogram. | Wikipedia - Ideogram | null | null | null |
Mathematical theory (aka formal theory) refers to the use of mathematics in constructing social theories. Mathematical sociology aims to sociological theory in formal terms, which such theories can be understood to lack. The benefits of this approach not only include increased clarity, but also, through mathematics, th... | Wikipedia - Sociological theory | null | null | null |
Mathematically Correct a website which supports traditional mathematics NYC HOLD a New York-based organization of teachers, professional mathematicians, parents and others which has been extremely active in recent years in working for adoption of mastery-based, traditional math programs Illinois Loop – extensive web co... | Wikipedia - Traditional mathematics | null | null | null |
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group SO(3) or SU(2) as ... | Wikipedia - Lorentz invariance in loop quantum gravity | null | null | null |
These are some of the many ways in which different quantizations of the same classical theory can result in inequivalent quantum theories, or even in the impossibility to carry quantization through. One can't distinguish between SO(3) and SU(2) or between SO(3,1) and SL(2,C) at this level: the respective Lie algebras a... | Wikipedia - Lorentz invariance in loop quantum gravity | null | null | null |
The physical interpretation of the Lie algebra is that of infinitesimally small group transformations, and gauge bosons (such as the graviton) are Lie algebra representations, not Lie group representations. What this means for the Lorentz group is that, for sufficiently small velocity parameters, all four complexified ... | Wikipedia - Lorentz invariance in loop quantum gravity | null | null | null |
At the level of the Lie algebra, this corresponds to what is called q-deforming the Lie algebra, and the parameter q is related to the value of the cosmological constant. The effect of replacing a Lie algebra by a q-deformed version is that the series of its representations is truncated (in the case of the rotation gro... | Wikipedia - Lorentz invariance in loop quantum gravity | null | null | null |
Mathematically the two lines are represented by the equations: { Y = C , X < P b Y = A X + B , X > P b {\displaystyle \left\{{\begin{array}{l}Y=C,&X P_{\rm {b}}\end{array}}\right.} where Y is the crop production or yield, C is the maximum yield, X is the soil salinity, A is the slope (regression coefficient) of the des... | Wikipedia - Maas–Hoffman model | null | null | null |
Mathematically, 1 is: in arithmetic (algebra) and calculus, the natural number that follows 0 and the multiplicative identity element of the integers, real numbers and complex numbers; more generally, in algebra, the multiplicative identity (also called unity), usually of a group or a ring.Formalizations of the natural... | Wikipedia - 1 | null | null | null |
However, 1 is especially common for the multiplicative identity of a ring, i.e., when an addition and 0 are also present. When such a ring has characteristic n not equal to 0, the element called 1 has the property that n1 = 1n = 0 (where this 0 is the additive identity of the ring). Important examples are finite fields... | Wikipedia - 1 | null | null | null |
By definition, 1 is the magnitude, absolute value, or norm of a unit complex number, unit vector, and a unit matrix (more usually called an identity matrix). The term unit matrix is sometimes used to mean a matrix composed entirely of 1s. By definition, 1 is the probability of an event that is absolutely or almost cert... | Wikipedia - 1 | null | null | null |
In category theory, 1 is sometimes used to denote the terminal object of a category. In number theory, 1 is the value of Legendre's constant, which was introduced in 1808 by Adrien-Marie Legendre in expressing the asymptotic behavior of the prime-counting function. Legendre's constant was originally conjectured to be a... | Wikipedia - 1 | null | null | null |
Mathematically, Kakuro puzzles can be represented as integer programming problems, and are NP-complete. See also Yato and Seta, 2004.There are two kinds of mathematical symmetry readily identifiable in Kakuro puzzles: minimum and maximum constraints are duals, as are missing and required values. All sum combinations ca... | Wikipedia - Kakuro | null | null | null |
Mathematically, QED is an abelian gauge theory with the symmetry group U(1), defined on Minkowski space (flat spacetime). The gauge field, which mediates the interaction between the charged spin-1/2 fields, is the electromagnetic field. The QED Lagrangian for a spin-1/2 field interacting with the electromagnetic field ... | Wikipedia - Quantum Electrodynamics | null | null | null |
ψ ¯ ≡ ψ † γ 0 {\displaystyle {\bar {\psi }}\equiv \psi ^{\dagger }\gamma ^{0}} , called "psi-bar", is sometimes referred to as the Dirac adjoint. D μ ≡ ∂ μ + i e A μ + i e B μ {\displaystyle D_{\mu }\equiv \partial _{\mu }+ieA_{\mu }+ieB_{\mu }} is the gauge covariant derivative. | Wikipedia - Quantum Electrodynamics | null | null | null |
e is the coupling constant, equal to the electric charge of the bispinor field. A μ {\displaystyle A_{\mu }} is the covariant four-potential of the electromagnetic field generated by the electron itself. It is also known as a gauge field or a U ( 1 ) {\displaystyle {\text{U}}(1)} connection. | Wikipedia - Quantum Electrodynamics | null | null | null |
B μ {\displaystyle B_{\mu }} is the external field imposed by external source. m is the mass of the electron or positron. F μ ν = ∂ μ A ν − ∂ ν A μ {\displaystyle F_{\mu \nu }=\partial _{\mu }A_{\nu }-\partial _{\nu }A_{\mu }} is the electromagnetic field tensor. | Wikipedia - Quantum Electrodynamics | null | null | null |
This is also known as the curvature of the gauge field.Expanding the covariant derivative reveals a second useful form of the Lagrangian (external field B μ {\displaystyle B_{\mu }} set to zero for simplicity) L = − 1 4 F μ ν F μ ν + ψ ¯ ( i γ μ ∂ μ − m ) ψ − e j μ A μ {\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\m... | Wikipedia - Quantum Electrodynamics | null | null | null |
Mathematically, a Yang–Mills instanton is a self-dual or anti-self-dual connection in a principal bundle over a four-dimensional Riemannian manifold that plays the role of physical space-time in non-abelian gauge theory. Instantons are topologically nontrivial solutions of Yang–Mills equations that absolutely minimize ... | Wikipedia - Instanton | null | null | null |
Yang–Mills instantons have been explicitly constructed in many cases by means of twistor theory, which relates them to algebraic vector bundles on algebraic surfaces, and via the ADHM construction, or hyperkähler reduction (see hyperkähler manifold), a sophisticated linear algebra procedure. The groundbreaking work of ... | Wikipedia - Instanton | null | null | null |
Mathematically, a box P ( a , b | x , y ) {\displaystyle P(a,b|x,y)} admits a quantum realization if and only if there exists a pair of Hilbert spaces H A , H B {\displaystyle H_{A},H_{B}} , a normalized vector | ψ ⟩ ∈ H A ⊗ H B {\displaystyle \left|\psi \right\rangle \in H_{A}\otimes H_{B}} and projection operators E ... | Wikipedia - Quantum non-locality | null | null | null |
Contrary to the classical set of correlations, when viewed in probability space, Q {\displaystyle Q} is not a polytope. On the contrary, it contains both straight and curved boundaries. In addition, Q {\displaystyle Q} is not closed: this means that there exist boxes P ( a , b | x , y ) {\displaystyle P(a,b|x,y)} which... | Wikipedia - Quantum non-locality | null | null | null |
In the above definition, the space-like separation of the two parties conducting the Bell experiment was modeled by imposing that their associated operator algebras act on different factors H A , H B {\displaystyle H_{A},H_{B}} of the overall Hilbert space H = H A ⊗ H B {\displaystyle H=H_{A}\otimes H_{B}} describing t... | Wikipedia - Quantum non-locality | null | null | null |
Namely, ∑ a E a x = I , ∑ b F b y = I {\displaystyle \sum _{a}E_{a}^{x}={\mathbb {I} },\sum _{b}F_{b}^{y}={\mathbb {I} }} . P ( a , b | x , y ) = ⟨ ψ | E a x F b y | ψ ⟩ {\displaystyle P(a,b|x,y)=\left\langle \psi \right|E_{a}^{x}F_{b}^{y}\left|\psi \right\rangle } , for all a , b , x , y {\displaystyle a,b,x,y} . | Wikipedia - Quantum non-locality | null | null | null |
= 0 {\displaystyle =0} , for all a , b , x , y {\displaystyle a,b,x,y} .Call Q c {\displaystyle Q_{c}} the set of all such correlations P ( a , b | x , y ) {\displaystyle P(a,b|x,y)} . How does this new set relate to the more conventional Q {\displaystyle Q} defined above? It can be proven that Q c {\displaystyle Q_{c}... | Wikipedia - Quantum non-locality | null | null | null |
Moreover, Q ¯ ⊆ Q c {\displaystyle {\bar {Q}}\subseteq Q_{c}} , where Q ¯ {\displaystyle {\bar {Q}}} denotes the closure of Q {\displaystyle Q} . Tsirelson's problem consists in deciding whether the inclusion relation Q ¯ ⊆ Q c {\displaystyle {\bar {Q}}\subseteq Q_{c}} is strict, i.e., whether or not Q ¯ = Q c {\displa... | Wikipedia - Quantum non-locality | null | null | null |
Mathematically, a coherent state | α ⟩ {\displaystyle |\alpha \rangle } is defined to be the (unique) eigenstate of the annihilation operator â with corresponding eigenvalue α. Formally, this reads, a ^ | α ⟩ = α | α ⟩ . {\displaystyle {\hat {a}}|\alpha \rangle =\alpha |\alpha \rangle ~.} Since â is not hermitian, α is... | Wikipedia - Coherent state | null | null | null |
The state | α ⟩ {\displaystyle |\alpha \rangle } is called a canonical coherent state in the literature, since there are many other types of coherent states, as can be seen in the companion article Coherent states in mathematical physics. Physically, this formula means that a coherent state remains unchanged by the ann... | Wikipedia - Coherent state | null | null | null |
A Poisson distribution is a necessary and sufficient condition that all detections are statistically independent. Contrast this to a single-particle state ( | 1 ⟩ {\displaystyle |1\rangle } Fock state): once one particle is detected, there is zero probability of detecting another. | Wikipedia - Coherent state | null | null | null |
The derivation of this will make use of (unconventionally normalized) dimensionless operators, X and P, normally called field quadratures in quantum optics. (See Nondimensionalization.) These operators are related to the position and momentum operators of a mass m on a spring with constant k, P = 1 2 ℏ m ω p ^ , X = m ... | Wikipedia - Coherent state | null | null | null |
{\displaystyle {P}={\sqrt {\frac {1}{2\hbar m\omega }}}\ {\hat {p}}{\text{,}}\quad {X}={\sqrt {\frac {m\omega }{2\hbar }}}\ {\hat {x}}{\text{,}}\quad \quad {\text{where }}\omega \equiv {\sqrt {k/m}}~.} For an optical field, E R = ( 2 ℏ ω ϵ 0 V ) 1 / 2 cos ( θ ) X and E I = ( 2 ℏ ω ϵ 0 V ) 1 / 2 sin ( θ ) X {\displa... | Wikipedia - Coherent state | null | null | null |
Erwin Schrödinger was searching for the most classical-like states when he first introduced minimum uncertainty Gaussian wave-packets. The quantum state of the harmonic oscillator that minimizes the uncertainty relation with uncertainty equally distributed between X and P satisfies the equation ( X − ⟨ X ⟩ ) | α ⟩ = − ... | Wikipedia - Coherent state | null | null | null |
Thus, given (∆X−∆P)2 ≥ 0, Schrödinger found that the minimum uncertainty states for the linear harmonic oscillator are the eigenstates of (X + iP). Since â is (X + iP), this is recognizable as a coherent state in the sense of the above definition. Using the notation for multi-photon states, Glauber characterized the st... | Wikipedia - Coherent state | null | null | null |
The name coherent state took hold after Glauber's work. If the uncertainty is minimized, but not necessarily equally balanced between X and P, the state is called a squeezed coherent state. The coherent state's location in the complex plane (phase space) is centered at the position and momentum of a classical oscillato... | Wikipedia - Coherent state | null | null | null |
As shown in Figure 5, the uncertainty, equally spread in all directions, is represented by a disk with diameter 1⁄2. As the phase varies, the coherent state circles around the origin and the disk neither distorts nor spreads. This is the most similar a quantum state can be to a single point in phase space. | Wikipedia - Coherent state | null | null | null |
Since the uncertainty (and hence measurement noise) stays constant at 1⁄2 as the amplitude of the oscillation increases, the state behaves increasingly like a sinusoidal wave, as shown in Figure 1. Moreover, since the vacuum state | 0 ⟩ {\displaystyle |0\rangle } is just the coherent state with α=0, all coherent states... | Wikipedia - Coherent state | null | null | null |
The notation | α ⟩ {\displaystyle |\alpha \rangle } does not refer to a Fock state. For example, when α = 1, one should not mistake | 1 ⟩ {\displaystyle |1\rangle } for the single-photon Fock state, which is also denoted | 1 ⟩ {\displaystyle |1\rangle } in its own notation. The expression | α ⟩ {\displaystyle |\alpha \... | Wikipedia - Coherent state | null | null | null |
The formal solution of the eigenvalue equation is the vacuum state displaced to a location α in phase space, i.e., it is obtained by letting the unitary displacement operator D(α) operate on the vacuum, | α ⟩ = e α a ^ † − α ∗ a ^ | 0 ⟩ = D ( α ) | 0 ⟩ {\displaystyle |\alpha \rangle =e^{\alpha {\hat {a}}^{\dagger }-\al... | Wikipedia - Coherent state | null | null | null |
}}}|n\rangle =e^{-{|\alpha |^{2} \over 2}}e^{\alpha {\hat {a}}^{\dagger }}e^{-{\alpha ^{*}{\hat {a}}}}|0\rangle ~,} where | n ⟩ {\displaystyle |n\rangle } are energy (number) eigenvectors of the Hamiltonian H = ℏ ω ( a ^ † a ^ + 1 2 ) . {\displaystyle H=\hbar \omega \left({\hat {a}}^{\dagger }{\hat {a}}+{\frac {1}{2}}\... | Wikipedia - Coherent state | null | null | null |
For the corresponding Poissonian distribution, the probability of detecting n photons is P ( n ) = | ⟨ n | α ⟩ | 2 = e − ⟨ n ⟩ ⟨ n ⟩ n n ! . | Wikipedia - Coherent state | null | null | null |
{\displaystyle P(n)=|\langle n|\alpha \rangle |^{2}=e^{-\langle n\rangle }{\frac {\langle n\rangle ^{n}}{n!}}~.} Similarly, the average photon number in a coherent state is ⟨ n ⟩ = ⟨ a ^ † a ^ ⟩ = | α | 2 {\displaystyle ~\langle n\rangle =\langle {\hat {a}}^{\dagger }{\hat {a}}\rangle =|\alpha |^{2}~} and the variance ... | Wikipedia - Coherent state | null | null | null |
These results apply to detection results at a single detector and thus relate to first order coherence (see degree of coherence). However, for measurements correlating detections at multiple detectors, higher-order coherence is involved (e.g., intensity correlations, second order coherence, at two detectors). Glauber's... | Wikipedia - Coherent state | null | null | null |
It is perfectly coherent to all orders. The second-order correlation coefficient g 2 ( 0 ) {\displaystyle g^{2}(0)} gives a direct measure of the degree of coherence of photon states in terms of the variance of the photon statistics in the beam under study. g 2 ( 0 ) = 1 + V a r ( a ^ † a ^ ) − ⟨ a ^ † a ^ ⟩ ( ⟨ a ^ † ... | Wikipedia - Coherent state | null | null | null |
In the case of a Poisson distribution, the variance is equal to the mean, i.e. V a r ( n ) = n ¯ {\displaystyle {\rm {Var}}(n)={\bar {n}}} g 2 ( 0 ) = 1 {\displaystyle g^{2}(0)=1} .A second-order correlation coefficient of 1 means that photons in coherent states are uncorrelated. Hanbury Brown and Twiss studied the cor... | Wikipedia - Coherent state | null | null | null |
Quanta that obey Fermi-Dirac statistics are anti-correlated. In this case the variance is V a r ( n ) = n ¯ − n ¯ 2 {\displaystyle {\rm {Var}}(n)={\bar {n}}-{\bar {n}}^{2}} g 2 ( 0 ) = 0 {\displaystyle g^{2}(0)=0} .Anti-correlation is characterized by a second-order correlation coefficient =0. Roy J. Glauber's work was... | Wikipedia - Coherent state | null | null | null |
(One can imagine, over very short durations, a near-instantaneous interference pattern from the two detectors, due to the narrow band filters, that dances around randomly due to the shifting relative phase difference. With a coincidence counter, the dancing interference pattern would be stronger at times of increased i... | Wikipedia - Coherent state | null | null | null |
The Hanbury-Brown and Twiss results prompted Glauber to look at higher order coherence, and he came up with a complete quantum-theoretic description of coherence to all orders in the electromagnetic field (and a quantum-theoretic description of signal-plus-noise). He coined the term coherent state and showed that they ... | Wikipedia - Coherent state | null | null | null |
Mathematically, a rotation is a rigid body movement which, unlike a translation, keeps at least one point fixed. This definition applies to rotations in two dimensions (in a plane), in which exactly one point is kept fixed; and also in three dimensions (in space), in which additional points may be kept fixed (as in rot... | Wikipedia - 180-degree rotation | null | null | null |
That common point lies within the axis of that motion. The axis is perpendicular to the plane of the motion. If a rotation around a point or axis is followed by a second rotation around the same point/axis, a third rotation results. | Wikipedia - 180-degree rotation | null | null | null |
The reverse (inverse) of a rotation is also a rotation. Thus, the rotations around a point/axis form a group. However, a rotation around a point or axis and a rotation around a different point/axis may result in something other than a rotation, e.g. a translation. | Wikipedia - 180-degree rotation | null | null | null |
Rotations around the x, y and z axes are called principal rotations. Rotation around any axis can be performed by taking a rotation around the x axis, followed by a rotation around the y axis, and followed by a rotation around the z axis. That is to say, any spatial rotation can be decomposed into a combination of prin... | Wikipedia - 180-degree rotation | null | null | null |
Mathematically, applying a Gaussian blur to an image is the same as convolving the image with a Gaussian function. This is also known as a two-dimensional Weierstrass transform. By contrast, convolving by a circle (i.e., a circular box blur) would more accurately reproduce the bokeh effect. Since the Fourier transform ... | Wikipedia - Gaussian blur | null | null | null |
The Gaussian blur is a type of image-blurring filter that uses a Gaussian function (which also expresses the normal distribution in statistics) for calculating the transformation to apply to each pixel in the image. The formula of a Gaussian function in one dimension is In two dimensions, it is the product of two such ... | Wikipedia - Gaussian blur | null | null | null |
When applied in two dimensions, this formula produces a surface whose contours are concentric circles with a Gaussian distribution from the center point. Values from this distribution are used to build a convolution matrix which is applied to the original image. This convolution process is illustrated visually in the f... | Wikipedia - Gaussian blur | null | null | null |
Each pixel's new value is set to a weighted average of that pixel's neighborhood. The original pixel's value receives the heaviest weight (having the highest Gaussian value) and neighboring pixels receive smaller weights as their distance to the original pixel increases. This results in a blur that preserves boundaries... | Wikipedia - Gaussian blur | null | null | null |
In theory, the Gaussian function at every point on the image will be non-zero, meaning that the entire image would need to be included in the calculations for each pixel. In practice, when computing a discrete approximation of the Gaussian function, pixels at a distance of more than 3σ have a small enough influence to ... | Wikipedia - Gaussian blur | null | null | null |
Typically, an image processing program need only calculate a matrix with dimensions ⌈ 6 σ ⌉ {\displaystyle \lceil 6\sigma \rceil } × ⌈ 6 σ ⌉ {\displaystyle \lceil 6\sigma \rceil } (where ⌈ ⋅ ⌉ {\displaystyle \lceil \cdot \rceil } is the ceiling function) to ensure a result sufficiently close to that obtained by the ent... | Wikipedia - Gaussian blur | null | null | null |
In computational terms, this is a useful property, since the calculation can be performed in O ( w kernel w image h image ) + O ( h kernel w image h image ) {\displaystyle O\left(w_{\text{kernel}}w_{\text{image}}h_{\text{image}}\right)+O\left(h_{\text{kernel}}w_{\text{image}}h_{\text{image}}\right)} time (where h is he... | Wikipedia - Gaussian blur | null | null | null |
Because of this relationship, processing time cannot be saved by simulating a Gaussian blur with successive, smaller blurs — the time required will be at least as great as performing the single large blur. Gaussian blurring is commonly used when reducing the size of an image. | Wikipedia - Gaussian blur | null | null | null |
When downsampling an image, it is common to apply a low-pass filter to the image prior to resampling. This is to ensure that spurious high-frequency information does not appear in the downsampled image (aliasing). Gaussian blurs have nice properties, such as having no sharp edges, and thus do not introduce ringing into... | Wikipedia - Gaussian blur | null | null | null |
Mathematically, consider a symmetric game with two players that each have payoff function Π ( x i , x j ) {\displaystyle \,\Pi (x_{i},x_{j})} , where x i {\displaystyle \,x_{i}} represents the player's own decision, and x j {\displaystyle \,x_{j}} represents the decision of the other player. Assume Π {\displaystyle \,\... | Wikipedia - Strategic complementarities | null | null | null |
Mathematically, for the spectral power distribution of a radiant exitance or irradiance one may write: M ( λ ) = ∂ 2 Φ ∂ A ∂ λ ≈ Φ A Δ λ {\displaystyle M(\lambda )={\frac {\partial ^{2}\Phi }{\partial A\,\partial \lambda }}\approx {\frac {\Phi }{A\,\Delta \lambda }}} where M(λ) is the spectral irradiance (or exitance) ... | Wikipedia - Spectral power distribution | null | null | null |
Mathematically, resilience can be approximated by the inverse of the return time to an equilibrium given by resilience ≡ − Re ( λ 1 ( A ) ) ) {\displaystyle {\text{resilience}}\equiv -{\text{Re}}(\lambda _{1}({\textbf {A}})))} where λ 1 {\textstyle \lambda _{1}} is the maximum eigenvalue of matrix A {\textstyle {\textb... | Wikipedia - Resilience (mathematics) | null | null | null |
Mathematically, the ability to break up a multiplication in this way is known as the distributive law, which can be expressed in algebra as the property that a(b+c) = ab + ac. The grid method uses the distributive property twice to expand the product, once for the horizontal factor, and once for the vertical factor. Hi... | Wikipedia - Partial products algorithm | null | null | null |
Mathematically, the basic setup is captured by a dagger symmetric monoidal category: composition of morphisms models sequential composition of processes, and the tensor product describes parallel composition of processes. The role of the dagger is to assign to each state a corresponding test. These can then be adorned ... | Wikipedia - Categorical quantization | null | null | null |
In the diagrammatic calculus, it allows wires to be bent, allowing for a less restricted transfer of information. In particular, it allows entangled states and measurements, and gives elegant descriptions of protocols such as quantum teleportation. In quantum theory, it being compact closed is related to the Choi-Jamio... | Wikipedia - Categorical quantization | null | null | null |
Considering only the morphisms that are completely positive maps, one can also handle mixed states, allowing the study of quantum channels categorically. Wires are always two-ended (and can never be split into a Y), reflecting the no-cloning and no-deleting theorems of quantum mechanics. Special commutative dagger Frob... | Wikipedia - Categorical quantization | null | null | null |
In early works, dagger biproducts were used to study both classical communication and the superposition principle. Later, these two features have been separated. Complementary Frobenius algebras embody the principle of complementarity, which is used to great effect in quantum computation, as in the ZX-calculus.A substa... | Wikipedia - Categorical quantization | null | null | null |
Mathematically, the estimated probabilities of each color ball can be represented as R, Y, and B. If the participant strictly prefers Gamble A to Gamble B, by utility theory, it is presumed this preference is reflected by the expected utilities of the two gambles. We reach a contradiction in our utility calculations. T... | Wikipedia - Ellsberg paradox | null | null | null |
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