text stringlengths 9 3.55k | source stringlengths 31 280 |
|---|---|
As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of directional statistics and of statistics of non-Euclidean spaces. This computatio... | https://en.wikipedia.org/wiki/Mean_of_circular_quantities |
In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by some quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of scale or con... | https://en.wikipedia.org/wiki/Fuzzy_concept |
In mathematics and statistics, a piecewise linear, PL or segmented function is a real-valued function of a real variable, whose graph is composed of straight-line segments. | https://en.wikipedia.org/wiki/Piecewise-linear_function |
In mathematics and statistics, a probability vector or stochastic vector is a vector with non-negative entries that add up to one. The positions (indices) of a probability vector represent the possible outcomes of a discrete random variable, and the vector gives us the probability mass function of that random variable,... | https://en.wikipedia.org/wiki/Probability_vector |
In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by measuring or counting, respectively. If it can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the varia... | https://en.wikipedia.org/wiki/Discrete_variables |
In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness. | https://en.wikipedia.org/wiki/Random_number |
In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over ... | https://en.wikipedia.org/wiki/Wide_sense_stationary |
The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend. In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. In the latter case of a deterministic trend, ... | https://en.wikipedia.org/wiki/Wide_sense_stationary |
A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process t... | https://en.wikipedia.org/wiki/Wide_sense_stationary |
For many applications strict-sense stationarity is too restrictive. Other forms of stationarity such as wide-sense stationarity or N-th-order stationarity are then employed. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology). | https://en.wikipedia.org/wiki/Wide_sense_stationary |
In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical est... | https://en.wikipedia.org/wiki/Asymptotic_distribution |
In mathematics and statistics, an error term is an additive type of error. Common examples include: errors and residuals in statistics, e.g. in linear regression the error term in numerical integration | https://en.wikipedia.org/wiki/Error_term |
In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude... | https://en.wikipedia.org/wiki/Absolute_deviation |
In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evolution of the process's distribution. This article, as opposed to the article t... | https://en.wikipedia.org/wiki/Kolmogorov_equations_(Markov_jump_process) |
In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are known for their power, simplicity, and low error rates when compared to other methods. According to experimental results, random projection pre... | https://en.wikipedia.org/wiki/Random_projections |
In mathematics and statistics, sums of powers occur in a number of contexts: Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the ana... | https://en.wikipedia.org/wiki/Power_sum |
Fermat's Last Theorem states that x k + y k = z k {\displaystyle x^{k}+y^{k}=z^{k}} is impossible in positive integers with k>2. The equation of a superellipse is | x / a | k + | y / b | k = 1 {\displaystyle |x/a|^{k}+|y/b|^{k}=1} . The squircle is the case k = 4 , a = b {\displaystyle k=4,a=b} . | https://en.wikipedia.org/wiki/Power_sum |
Euler's sum of powers conjecture (disproved) concerns situations in which the sum of n integers, each a kth power of an integer, equals another kth power. The Fermat-Catalan conjecture asks whether there are an infinitude of examples in which the sum of two coprime integers, each a power of an integer, with the powers ... | https://en.wikipedia.org/wiki/Power_sum |
The Jacobi–Madden equation is a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 {\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}} in integers. The Prouhet–Tarry–Escott problem considers sums of two sets of kth powers of integers that are equal for multiple values of k. A taxicab number is the smallest integer that can be... | https://en.wikipedia.org/wiki/Power_sum |
The Lander, Parkin, and Selfridge conjecture concerns the minimal value of m + n in ∑ i = 1 n a i k = ∑ j = 1 m b j k . {\displaystyle \sum _{i=1}^{n}a_{i}^{k}=\sum _{j=1}^{m}b_{j}^{k}.} | https://en.wikipedia.org/wiki/Power_sum |
Waring's problem asks whether for every natural number k there exists an associated positive integer s such that every natural number is the sum of at most s kth powers of natural numbers. The successive powers of the golden ratio φ obey the Fibonacci recurrence: φ n + 1 = φ n + φ n − 1 . {\displaystyle \varphi ^{n+1}=... | https://en.wikipedia.org/wiki/Power_sum |
Newton's identities express the sum of the kth powers of all the roots of a polynomial in terms of the coefficients in the polynomial. The sum of cubes of numbers in arithmetic progression is sometimes another cube. The Fermat cubic, in which the sum of three cubes equals another cube, has a general solution. | https://en.wikipedia.org/wiki/Power_sum |
The power sum symmetric polynomial is a building block for symmetric polynomials. The sum of the reciprocals of all perfect powers including duplicates (but not including 1) equals 1. | https://en.wikipedia.org/wiki/Power_sum |
The Erdős–Moser equation, 1 k + 2 k + ⋯ + m k = ( m + 1 ) k {\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}} where m {\displaystyle m} and k {\displaystyle k} are positive integers, is conjectured to have no solutions other than 11 + 21 = 31. The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown wh... | https://en.wikipedia.org/wiki/Power_sum |
In mathematics and statistics, the Fréchet mean is a generalization of centroids to metric spaces, giving a single representative point or central tendency for a cluster of points. It is named after Maurice Fréchet. Karcher mean is the renaming of the Riemannian Center of Mass construction developed by Karsten Grove an... | https://en.wikipedia.org/wiki/Fréchet_mean |
In mathematics and statistics, the arithmetic mean ( arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, ... | https://en.wikipedia.org/wiki/Statistical_mean |
For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distr... | https://en.wikipedia.org/wiki/Statistical_mean |
In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f {\displaystyle f} . It is also called Kolmogorov mean after Soviet mathematician And... | https://en.wikipedia.org/wiki/Quasi-arithmetic_mean |
In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion. There are several equivalent definitions of weak convergence of a sequence of measures, some of... | https://en.wikipedia.org/wiki/Portmanteau_theorem |
For example, the sequence where P n {\displaystyle P_{n}} is the Dirac measure located at 1 / n {\displaystyle 1/n} converges weakly to the Dirac measure located at 0 (if we view these as measures on R {\displaystyle \mathbf {R} } with the usual topology), but it does not converge setwise. This is intuitively clear: we... | https://en.wikipedia.org/wiki/Portmanteau_theorem |
It also defines a weak topology on P ( S ) {\displaystyle {\mathcal {P}}(S)} , the set of all probability measures defined on ( S , Σ ) {\displaystyle (S,\Sigma )} . The weak topology is generated by the following basis of open sets: { U ϕ , x , δ | ϕ: S → R is bounded and continuous, x ∈ R and δ > 0 } , {\displaystyle... | https://en.wikipedia.org/wiki/Portmanteau_theorem |
If S {\displaystyle S} is also separable, then P ( S ) {\displaystyle {\mathcal {P}}(S)} is metrizable and separable, for example by the Lévy–Prokhorov metric. If S {\displaystyle S} is also compact or Polish, so is P ( S ) {\displaystyle {\mathcal {P}}(S)} . If S {\displaystyle S} is separable, it naturally embeds int... | https://en.wikipedia.org/wiki/Portmanteau_theorem |
In mathematics and string theory, a conifold is a generalization of a manifold. Unlike manifolds, conifolds can contain conical singularities, i.e. points whose neighbourhoods look like cones over a certain base. In physics, in particular in flux compactifications of string theory, the base is usually a five-dimensiona... | https://en.wikipedia.org/wiki/Conifold |
In mathematics and supersymmetric gauge theory, spectral networks are "networks of trajectories on Riemann surfaces obeying certain local rules. Spectral networks arise naturally in four-dimensional N = 2 theories coupled to surface defects, particularly the theories of class S." == References == | https://en.wikipedia.org/wiki/Spectral_network |
In mathematics and telecommunications, stochastic geometry models of wireless networks refer to mathematical models based on stochastic geometry that are designed to represent aspects of wireless networks. The related research consists of analyzing these models with the aim of better understanding wireless communicatio... | https://en.wikipedia.org/wiki/Stochastic_geometry_models_of_wireless_networks |
This model is considered to be pioneering and the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireles... | https://en.wikipedia.org/wiki/Stochastic_geometry_models_of_wireless_networks |
In mathematics and the field of number theory, the Landau–Ramanujan constant is the positive real number b that occurs in a theorem proved by Edmund Landau in 1908, stating that for large x {\displaystyle x} , the number of positive integers below x {\displaystyle x} that are the sum of two square numbers behaves asymp... | https://en.wikipedia.org/wiki/Landau-Ramanujan_constant |
In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by J ( n , x ) = ∫ 0 x t n e t ( e t − 1 ) 2 d t . {\displaystyle J(n,x)=\int _{0}^{x}t^{n}{\frac {e^{t}}{(e^{t}-1)^{2}}}\,dt.} Note that e t ( e t − 1 ) 2 = ∑ k = 0 ∞ k e k t . {\displaystyle {\frac {e^{t}}{(e^{t}-1)^{2}}... | https://en.wikipedia.org/wiki/Transport_function |
In mathematics and the foundations of quantum mechanics, the projective Hilbert space P ( H ) {\displaystyle P(H)} of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes of non-zero vectors v {\displaystyle v} in H {\displaystyle H} , for the relation ∼ {\displaystyle \sim } on H {\displaystyl... | https://en.wikipedia.org/wiki/Ray_(quantum_theory) |
In mathematics and theoretical computer science the Lawson topology, named after Jimmie D. Lawson, is a topology on partially ordered sets used in the study of domain theory. The lower topology on a poset P is generated by the subbasis consisting of all complements of principal filters on P. The Lawson topology on P is... | https://en.wikipedia.org/wiki/Lawson_topology |
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant-recursive sequence is also known as a linear recurrence sequence, linear-rec... | https://en.wikipedia.org/wiki/Constant-recursive_sequence |
The square number sequence 0 , 1 , 4 , 9 , 16 , 25 , … {\displaystyle 0,1,4,9,16,25,\ldots } is also constant-recursive. However, not all sequences are constant-recursive; for example, the factorial sequence 1 , 1 , 2 , 6 , 24 , 120 , … {\displaystyle 1,1,2,6,24,120,\ldots } is not constant-recursive. All arithmetic pr... | https://en.wikipedia.org/wiki/Constant-recursive_sequence |
Formally, a sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is constant-recursive if it satisfies a recurrence relation where c i {\displaystyle c_{i}} are constants. For example, the Fibonacci sequence satisfies the recurrence relation F n = F n − 1 + F n − 2 , {\displayst... | https://en.wikipedia.org/wiki/Constant-recursive_sequence |
They also arise in algebraic number theory, due to the relation of the sequence to the roots of a polynomial; in the analysis of algorithms as the running time of simple recursive functions; and in formal language theory, where they count strings up to a given length in a regular language. Constant-recursive sequences ... | https://en.wikipedia.org/wiki/Constant-recursive_sequence |
In mathematics and theoretical computer science, a k-regular sequence is a sequence satisfying linear recurrence equations that reflect the base-k representations of the integers. The class of k-regular sequences generalizes the class of k-automatic sequences to alphabets of infinite size. | https://en.wikipedia.org/wiki/K-regular_sequence |
In mathematics and theoretical computer science, a k-synchronized sequence is an infinite sequence of terms s(n) characterized by a finite automaton taking as input two strings m and n, each expressed in some fixed base k, and accepting if m = s(n). The class of k-synchronized sequences lies between the classes of k-au... | https://en.wikipedia.org/wiki/K-synchronized_sequence |
In mathematics and theoretical computer science, a pattern is an unavoidable pattern if it is unavoidable on any finite alphabet. | https://en.wikipedia.org/wiki/Unavoidable_pattern |
In mathematics and theoretical computer science, a semiautomaton is a deterministic finite automaton having inputs but no output. It consists of a set Q of states, a set Σ called the input alphabet, and a function T: Q × Σ → Q called the transition function. Associated with any semiautomaton is a monoid called the char... | https://en.wikipedia.org/wiki/Semiautomaton |
In mathematics and theoretical computer science, a set constraint is an equation or an inequation between sets of terms. Similar to systems of (in)equations between numbers, methods are studied for solving systems of set constraints. Different approaches admit different operators (like "∪", "∩", "\", and function appli... | https://en.wikipedia.org/wiki/Set_constraint |
In mathematics and theoretical computer science, an automatic sequence (also called a k-automatic sequence or a k-recognizable sequence when one wants to indicate that the base of the numerals used is k) is an infinite sequence of terms characterized by a finite automaton. The n-th term of an automatic sequence a(n) is... | https://en.wikipedia.org/wiki/Automatic_sequence |
In mathematics and theoretical computer science, analysis of Boolean functions is the study of real-valued functions on { 0 , 1 } n {\displaystyle \{0,1\}^{n}} or { − 1 , 1 } n {\displaystyle \{-1,1\}^{n}} (such functions are sometimes known as pseudo-Boolean functions) from a spectral perspective. The functions studie... | https://en.wikipedia.org/wiki/Analysis_of_Boolean_functions |
In mathematics and theoretical computer science, entropy compression is an information theoretic method for proving that a random process terminates, originally used by Robin Moser to prove an algorithmic version of the Lovász local lemma. | https://en.wikipedia.org/wiki/Entropy_compression |
In mathematics and theoretical physics (especially twistor string theory), an amplituhedron is a geometric structure introduced in 2013 by Nima Arkani-Hamed and Jaroslav Trnka. It enables simplified calculation of particle interactions in some quantum field theories. In planar N = 4 supersymmetric Yang–Mills theory, al... | https://en.wikipedia.org/wiki/Amplituhedron |
Instead, they are treated as properties that emerge from an underlying phenomenon.The connection between the amplituhedron and scattering amplitudes is a conjecture that has passed many non-trivial checks, including an understanding of how locality and unitarity arise as consequences of positivity. Research has been le... | https://en.wikipedia.org/wiki/Amplituhedron |
In mathematics and theoretical physics (especially twistor theory), twistor space is the complex vector space of solutions of the twistor equation ∇ A ′ ( A Ω B ) = 0 {\displaystyle \nabla _{A'}^{(A}\Omega _{^{}}^{B)}=0} . It was described in the 1960s by Roger Penrose and Malcolm MacCallum. According to Andrew Hodges,... | https://en.wikipedia.org/wiki/Twistor_space |
In mathematics and theoretical physics, Noether's second theorem relates symmetries of an action functional with a system of differential equations. The action S of a physical system is an integral of a so-called Lagrangian function L, from which the system's behavior can be determined by the principle of least action.... | https://en.wikipedia.org/wiki/Noether's_second_theorem |
Noether's second theorem is sometimes used in gauge theory. Gauge theories are the basic elements of all modern field theories of physics, such as the prevailing Standard Model. The theorem is named after Emmy Noether. | https://en.wikipedia.org/wiki/Noether's_second_theorem |
In mathematics and theoretical physics, Wigner's classification is a classification of the nonnegative ( E ≥ 0 ) {\displaystyle ~(~E\geq 0~)~} energy irreducible unitary representations of the Poincaré group which have either finite or zero mass eigenvalues. (Since this group is noncompact, these unitary representation... | https://en.wikipedia.org/wiki/One_particle_Hilbert_space |
The Casimir invariants of the Poincaré group are C 1 = P μ P μ , {\displaystyle ~C_{1}=P^{\mu }\,P_{\mu }~,} (Einstein notation) where P is the 4-momentum operator, and C 2 = W α W α , {\displaystyle ~C_{2}=W^{\alpha }\,W_{\alpha }~,} where W is the Pauli–Lubanski pseudovector. The eigenvalues of these operators serve ... | https://en.wikipedia.org/wiki/One_particle_Hilbert_space |
The physically relevant representations may thus be classified according to whether m > 0 ; {\displaystyle ~m>0~;} m = 0 {\displaystyle ~m=0~} but P 0 > 0 ; {\displaystyle ~P_{0}>0~;\quad } or whether m = 0 {\displaystyle ~m=0~} with P μ = 0 , for μ = 0 , 1 , 2 , 3 . {\displaystyle ~P^{\mu }=0~,{\text{ for }}\mu =0,1,2... | https://en.wikipedia.org/wiki/One_particle_Hilbert_space |
For the first case Note that the eigenspace (see generalized eigenspaces of unbounded operators) associated with P = ( m , 0 , 0 , 0 ) {\displaystyle ~P=(m,0,0,0)~} is a representation of SO(3).In the ray interpretation, one can go over to Spin(3) instead. So, massive states are classified by an irreducible Spin(3) uni... | https://en.wikipedia.org/wiki/One_particle_Hilbert_space |
This is the double cover of SE(2) (see projective representation). We have two cases, one where irreps are described by an integral multiple of 1/2 called the helicity, and the other called the "continuous spin" representation. For the third case The only finite-dimensional unitary solution is the trivial representatio... | https://en.wikipedia.org/wiki/One_particle_Hilbert_space |
In mathematics and theoretical physics, a Gerstenhaber algebra (sometimes called an antibracket algebra or braid algebra) is an algebraic structure discovered by Murray Gerstenhaber (1963) that combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formali... | https://en.wikipedia.org/wiki/Gerstenhaber_algebra |
In mathematics and theoretical physics, a bifundamental representation is a representation obtained as a tensor product of two fundamental or antifundamental representations. For example, the MN-dimensional representation (M,N) of the group S U ( M ) × S U ( N ) {\displaystyle SU(M)\times SU(N)} is a bifundamental repr... | https://en.wikipedia.org/wiki/Bifundamental_representation |
In mathematics and theoretical physics, a large diffeomorphism is an equivalence class of diffeomorphisms under the equivalence relation where diffeomorphisms that can be continuously connected to each other are in the same equivalence class. For example, a two-dimensional real torus has a SL(2,Z) group of large diffeo... | https://en.wikipedia.org/wiki/Large_diffeomorphism |
In mathematics and theoretical physics, a locally compact quantum group is a relatively new C*-algebraic approach toward quantum groups that generalizes the Kac algebra, compact-quantum-group and Hopf-algebra approaches. Earlier attempts at a unifying definition of quantum groups using, for example, multiplicative unit... | https://en.wikipedia.org/wiki/Locally_compact_quantum_group |
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional real n-space together with a non-degenerate quadratic form q. Such a quadratic form can, given a suitable choice of basis (e1, …, en), be applied to a vector x = x1e1 + ⋯ + xnen, giving which is called the scalar square of the vect... | https://en.wikipedia.org/wiki/Pseudo-Euclidean_space |
In mathematics and theoretical physics, a representation of a Lie group is a linear action of a Lie group on a vector space. Equivalently, a representation is a smooth homomorphism of the group into the group of invertible operators on the vector space. Representations play an important role in the study of continuous ... | https://en.wikipedia.org/wiki/Representation_theory_of_Lie_groups |
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading. The prefix super- comes from the theory of supersymmetry in theoretical phy... | https://en.wikipedia.org/wiki/Associative_superalgebra |
Superalgebras and their representations, supermodules, provide an algebraic framework for formulating supersymmetry. The study of such objects is sometimes called super linear algebra. Superalgebras also play an important role in related field of supergeometry where they enter into the definitions of graded manifolds, ... | https://en.wikipedia.org/wiki/Associative_superalgebra |
In mathematics and theoretical physics, a supermatrix is a Z2-graded analog of an ordinary matrix. Specifically, a supermatrix is a 2×2 block matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann algebra) or an ordinar... | https://en.wikipedia.org/wiki/Supermatrix |
In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged. The index subset must generally either be all covariant or all contravariant. For example, holds when the tensor is antisymmetric wit... | https://en.wikipedia.org/wiki/Antisymmetric_tensor |
In mathematics and theoretical physics, an invariant differential operator is a kind of mathematical map from some objects to an object of similar type. These objects are typically functions on R n {\displaystyle \mathbb {R} ^{n}} , functions on a manifold, vector valued functions, vector fields, or, more generally, se... | https://en.wikipedia.org/wiki/Invariant_differential_operator |
The word invariant indicates that the operator contains some symmetry. This means that there is a group G {\displaystyle G} with a group action on the functions (or other objects in question) and this action is preserved by the operator: D ( g ⋅ f ) = g ⋅ ( D f ) . {\displaystyle D(g\cdot f)=g\cdot (Df).} Usually, the ... | https://en.wikipedia.org/wiki/Invariant_differential_operator |
In mathematics and theoretical physics, and especially gauge theory, the deformed Hermitian Yang–Mills (dHYM) equation is a differential equation describing the equations of motion for a D-brane in the B-model (commonly called a B-brane) of string theory. The equation was derived by Mariño-Minasian-Moore-Strominger in ... | https://en.wikipedia.org/wiki/Deformed_Hermitian_Yang–Mills_equation |
In mathematics and theoretical physics, braid statistics is a generalization of the spin statistics of bosons and fermions based on the concept of braid group. While for fermions (Bosons) the corresponding statistics is associated to a phase gain of π {\displaystyle \pi } ( 2 π {\displaystyle 2\pi } ) under the exchang... | https://en.wikipedia.org/wiki/Braid_statistics |
In mathematics and theoretical physics, fusion rules are rules that determine the exact decomposition of the tensor product of two representations of a group into a direct sum of irreducible representations. The term is often used in the context of two-dimensional conformal field theory where the relevant group is gene... | https://en.wikipedia.org/wiki/Fusion_rules |
In mathematics and theoretical physics, quasiperiodic motion is in rough terms the type of motion executed by a dynamical system containing a finite number (two or more) of incommensurable frequencies.That is, if we imagine that the phase space is modelled by a torus T (that is, the variables are periodic like angles),... | https://en.wikipedia.org/wiki/Quasi-periodic_motion |
In mathematics and theoretical physics, resummation is a procedure to obtain a finite result from a divergent sum (series) of functions. Resummation involves a definition of another (convergent) function in which the individual terms defining the original function are re-scaled, and an integral transformation of this n... | https://en.wikipedia.org/wiki/Resummation |
Feynman and H. Kleinert. In quantum mechanics it was extended to any order here, and in quantum field theory here. See also Chapters 16–20 in the textbook cited below. | https://en.wikipedia.org/wiki/Resummation |
In mathematics and theoretical physics, the Berezinian or superdeterminant is a generalization of the determinant to the case of supermatrices. The name is for Felix Berezin. The Berezinian plays a role analogous to the determinant when considering coordinate changes for integration on a supermanifold. | https://en.wikipedia.org/wiki/Berezinian |
In mathematics and theoretical physics, the Eguchi–Hanson space is a non-compact, self-dual, asymptotically locally Euclidean (ALE) metric on the cotangent bundle of the 2-sphere T*S2. The holonomy group of this 4-real-dimensional manifold is SU(2). The metric is generally attributed to the physicists Tohru Eguchi and ... | https://en.wikipedia.org/wiki/Eguchi–Hanson_space |
The even dimensional space of dimension d {\displaystyle d} can be described using complex coordinates w i ∈ C d / 2 {\displaystyle w_{i}\in \mathbb {C} ^{d/2}} with a metric g i j ¯ = ( 1 + ρ d r d ) 2 / d , {\displaystyle g_{i{\bar {j}}}={\bigg (}1+{\frac {\rho ^{d}}{r^{d}}}{\bigg )}^{2/d}{\bigg },} where ρ {\displa... | https://en.wikipedia.org/wiki/Eguchi–Hanson_space |
Similarly, the space can also be used to construct Calabi–Yau manifolds by replacing the orbifold singularities of T 6 / Z 3 {\displaystyle T^{6}/\mathbb {Z} _{3}} with Eguchi–Hanson spaces.The Eguchi–Hanson metric is the prototypical example of a gravitational instanton; detailed expressions for the metric are given i... | https://en.wikipedia.org/wiki/Eguchi–Hanson_space |
In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold that is induced from the metric tensor on a manifold into which the submanifold is embedded, through the pullback. It may be determined using the following formula (using the Einstein summation convention), which is... | https://en.wikipedia.org/wiki/Induced_metric |
In mathematics and theoretical physics, the term quantum group denotes one of a few different kinds of noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix quantum groups (which are structures on unital separable C*... | https://en.wikipedia.org/wiki/Quantum_group |
The same term is also used for other Hopf algebras that deform or are close to classical Lie groups or Lie algebras, such as a "bicrossproduct" class of quantum groups introduced by Shahn Majid a little after the work of Drinfeld and Jimbo. In Drinfeld's approach, quantum groups arise as Hopf algebras depending on an a... | https://en.wikipedia.org/wiki/Quantum_group |
In mathematics and theoretical physics, zeta function regularization is a type of regularization or summability method that assigns finite values to divergent sums or products, and in particular can be used to define determinants and traces of some self-adjoint operators. The technique is now commonly applied to proble... | https://en.wikipedia.org/wiki/Zeta_function_regularization |
In mathematics and transportation engineering, traffic flow is the study of interactions between travellers (including pedestrians, cyclists, drivers, and their vehicles) and infrastructure (including highways, signage, and traffic control devices), with the aim of understanding and developing an optimal transport netw... | https://en.wikipedia.org/wiki/Traffic_flow |
In mathematics and, in particular, differential geometry a circular surface is the image of a map ƒ: I × S1 → R3, where I ⊂ R is an open interval and S1 is the unit circle, defined by f ( t , θ ) := γ ( t ) + r ( t ) u ( t ) cos θ + r ( t ) v ( t ) sin θ , {\displaystyle f(t,\theta ):=\gamma (t)+r(t){\mathbf {u} }(... | https://en.wikipedia.org/wiki/Circular_surface |
In the case of circular surfaces the generators are circles; called the generating circles. In the case of ruled surface the generators are straight lines; called rulings. == References == | https://en.wikipedia.org/wiki/Circular_surface |
In mathematics and, specifically, real analysis, the Dini derivatives (or Dini derivates) are a class of generalizations of the derivative. They were introduced by Ulisse Dini, who studied continuous but nondifferentiable functions. The upper Dini derivative, which is also called an upper right-hand derivative, of a co... | https://en.wikipedia.org/wiki/Dini_derivative |
The lower Dini derivative, f′−, is defined by f − ′ ( t ) = lim inf h → 0 + f ( t ) − f ( t − h ) h , {\displaystyle f'_{-}(t)=\liminf _{h\to {0+}}{\frac {f(t)-f(t-h)}{h}},} where lim inf is the infimum limit. If f is defined on a vector space, then the upper Dini derivative at t in the direction d is defined by f + ′ ... | https://en.wikipedia.org/wiki/Dini_derivative |
In mathematics applied to analysis of social structures, homogeneity blockmodeling is an approach in blockmodeling, which is best suited for a preliminary or main approach to valued networks, when a prior knowledge about these networks is not available. This is due to the fact, that homogeneity blockmodeling emphasizes... | https://en.wikipedia.org/wiki/Homogeneity_blockmodeling |
In mathematics applied to computer science, Monge arrays, or Monge matrices, are mathematical objects named for their discoverer, the French mathematician Gaspard Monge. An m-by-n matrix is said to be a Monge array if, for all i , j , k , ℓ {\displaystyle \scriptstyle i,\,j,\,k,\,\ell } such that 1 ≤ i < k ≤ m and 1 ≤ ... | https://en.wikipedia.org/wiki/Monge_array |
In mathematics before the 1970s, the term umbral calculus referred to the surprising similarity between seemingly unrelated polynomial equations and certain shadowy techniques used to "prove" them. These techniques were introduced by John Blissard and are sometimes called Blissard's symbolic method. They are often attr... | https://en.wikipedia.org/wiki/Umbral_calculus |
In mathematics compact convergence (or uniform convergence on compact sets) is a type of convergence that generalizes the idea of uniform convergence. It is associated with the compact-open topology. | https://en.wikipedia.org/wiki/Topology_of_compact_convergence |
In mathematics complex analysis, the Sarason interpolation theorem, introduced by Sarason (1967), is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation. | https://en.wikipedia.org/wiki/Sarason_interpolation_theorem |
In mathematics differential geometry, an antifundamental representation of a Lie group is the complex conjugate of the fundamental representation, although the distinction between the fundamental and the antifundamental representation is a matter of convention. However, these two are often non-equivalent, because each ... | https://en.wikipedia.org/wiki/Antifundamental_representation |
In mathematics education at primary school level, a number bond (sometimes alternatively called an addition fact) is a simple addition sum which has become so familiar that a child can recognise it and complete it almost instantly, with recall as automatic as that of an entry from a multiplication table in multiplicati... | https://en.wikipedia.org/wiki/Number_bond |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.