thewordsmiths/wiki_stem · Datasets at Hugging Face

content_id stringlengths 14 14	page_title stringlengths 1 250	section_title stringlengths 1 1.26k ⌀	breadcrumb stringlengths 1 1.39k	text stringlengths 9 3.55k
c_8qzd6uudj5sw	Mean of circular quantities	Summary	Circular_mean	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 computation produces a different result than the arithmetic mean, with the difference being greater when the angles are widely distributed. For example, the arithmetic mean of the three angles 0°, 0°, and 90° is (0° + 0° + 90°) / 3 = 30°, but the vector mean is arctan(1/2) = 26.565°. Moreover, with the arithmetic mean the circular variance is only defined ±180°.
c_dcgv95tbcf4n	Fuzzy concept	Sciences	Fuzzy_concept > Applications > Sciences	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 conceptual hierarchy.
c_x92iqqfym8fj	Piecewise-linear function	Summary	Piecewise_linear_function	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.
c_t2aj6zdi1vs2	Probability vector	Summary	Stochastic_vector	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, which is the standard way of characterizing a discrete probability distribution.
c_0ve3sp9q03cy	Discrete variables	Summary	Discrete_value	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 variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts a variable can be discrete in some ranges of the number line and continuous in others.
c_7rhgxf0ewm59	Random number	Summary	Random_number	In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness.
c_tj7jl6escrby	Wide sense stationary	Summary	Stationary_stochastic_process	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 time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles around the trend line, but overall it does not trend up nor down. Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary.
c_3qr17wqd6m7c	Wide sense stationary	Summary	Stationary_stochastic_process	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, the process is called a trend-stationary process, and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.
c_abrr4hjgc20e	Wide sense stationary	Summary	Stationary_stochastic_process	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 that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.
c_ayz90t220y67	Wide sense stationary	Summary	Stationary_stochastic_process	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).
c_j032bv4esgv3	Asymptotic distribution	Summary	Asymptotic_variance	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 estimators.
c_0w8dgx49gazm	Error term	Summary	Error_term	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
c_ir7km8nnv2r3	Absolute deviation	Summary	Deviation_(statistics)	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 of the value indicates the size of the difference.
c_k0mugjs87odi	Kolmogorov equations (Markov jump process)	Summary	Kolmogorov_equations_(continuous-time_Markov_chains)	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 titled Kolmogorov equations, focuses on the scenario where we have a continuous-time Markov chain (so the state space Ω {\displaystyle \Omega } is countable). In this case, we can treat the Kolmogorov equations as a way to describe the probability P ( x , s ; y , t ) {\displaystyle P(x,s;y,t)} , where x , y ∈ Ω {\displaystyle x,y\in \Omega } (the state space) and t > s , t , s ∈ R ≥ 0 {\displaystyle t>s,t,s\in \mathbb {R} _{\geq 0}} are the final and initial times, respectively.
c_gzrfebexk962	Random projections	Summary	Random_projection	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 preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing.
c_l64i1gt6tqpq	Power sum	Summary	Sums_of_powers	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 analysis of variance involves summing the squares of quantities. Faulhaber's formula expresses 1 k + 2 k + 3 k + ⋯ + n k {\displaystyle 1^{k}+2^{k}+3^{k}+\cdots +n^{k}} as a polynomial in n, or alternatively in terms of a Bernoulli polynomial. Fermat's right triangle theorem states that there is no solution in positive integers for a 2 = b 4 + c 4 {\displaystyle a^{2}=b^{4}+c^{4}} and a 4 = b 4 + c 2 {\displaystyle a^{4}=b^{4}+c^{2}} .
c_ilpi2el5nhpt	Power sum	Summary	Sums_of_powers	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} .
c_k9xxowgxoq6c	Power sum	Summary	Sums_of_powers	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 not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1. Beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2.
c_2io59lq1ljbr	Power sum	Summary	Sums_of_powers	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 expressed as a sum of two positive third powers in n distinct ways. The Riemann zeta function is the sum of the reciprocals of the positive integers each raised to the power s, where s is a complex number whose real part is greater than 1.
c_2btgfl64i5z3	Power sum	Summary	Sums_of_powers	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}.}
c_leejx0yj6j6y	Power sum	Summary	Sums_of_powers	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}=\varphi ^{n}+\varphi ^{n-1}.}
c_idg7lc1jzlij	Power sum	Summary	Sums_of_powers	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.
c_i81sxr9dauov	Power sum	Summary	Sums_of_powers	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.
c_ouclspjevknz	Power sum	Summary	Sums_of_powers	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 whether all remaining integers can be expressed in this form. The sums of powers Sm(z, n) = zm + (z+1)m + ... + (z+n−1)m is related to the Bernoulli polynomials Bm(z) by (∂n−∂z) Sm(z, n) = Bm(z) and (∂2λ−∂Z) S2k+1(z, n) = Ŝ′k+1(Z) where Z = z(z−1), λ = S1(z, n), Ŝk+1(Z) ≡ S2k+1(0, z). The sum of the terms in the geometric series is ∑ k = i n z k = z i − z n + 1 1 − z . {\displaystyle \sum _{k=i}^{n}z^{k}={\frac {z^{i}-z^{n+1}}{1-z}}.}
c_o6554v34fuw6	Fréchet mean	Summary	Fréchet_mean	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 and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.
c_dd56fi4k1keh	Statistical mean	Summary	Arithmetic_average	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, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic. In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent.
c_alcxlg2n8qyp	Statistical mean	Summary	Arithmetic_average	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 distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.
c_h7x9kmw06fyp	Quasi-arithmetic mean	Summary	Generalized_f-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 Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.
c_6i6y8g4aw87a	Portmanteau theorem	Weak convergence of measures	Portmanteau_lemma > Weak convergence of measures	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 which are (apparently) more general than others.
c_uuo6mv57vqn5	Portmanteau theorem	Weak convergence of measures	Portmanteau_lemma > Weak convergence of measures	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 only know that 1 / n {\displaystyle 1/n} is "close" to 0 {\displaystyle 0} because of the topology of R {\displaystyle \mathbf {R} } . This definition of weak convergence can be extended for S {\displaystyle S} any metrizable topological space.
c_q5z055g03k6j	Portmanteau theorem	Weak convergence of measures	Portmanteau_lemma > Weak convergence of measures	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 \left\{\ U_{\phi ,x,\delta }\ \left\|\quad \phi \colon S\to \mathbf {R} {\text{ is bounded and continuous, }}x\in \mathbf {R} {\text{ and }}\delta >0\ \right.\right\},} where U ϕ , x , δ := { μ ∈ P ( S ) \| \| ∫ S ϕ d μ − x \| < δ } . {\displaystyle U_{\phi ,x,\delta }:=\left\{\ \mu \in {\mathcal {P}}(S)\ \left\|\quad \left\|\int _{S}\phi \,\mathrm {d} \mu -x\right\|<\delta \ \right.\right\}.}
c_sasswyy4j65a	Portmanteau theorem	Weak convergence of measures	Portmanteau_lemma > Weak convergence of measures	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 into P ( S ) {\displaystyle {\mathcal {P}}(S)} as the (closed) set of Dirac measures, and its convex hull is dense. There are many "arrow notations" for this kind of convergence: the most frequently used are P n ⇒ P {\displaystyle P_{n}\Rightarrow P} , P n ⇀ P {\displaystyle P_{n}\rightharpoonup P} , P n → w P {\displaystyle P_{n}\xrightarrow {w} P} and P n → D P {\displaystyle P_{n}\xrightarrow {\mathcal {D}} P} .
c_uza24plss7ex	Conifold	Summary	Conifold	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-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.
c_hu0j50sh4oia	Spectral network	Summary	Spectral_network	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 ==
c_z7zxfpjaxmxh	Stochastic geometry models of wireless networks	Summary	Stochastic_geometry_models_of_wireless_networks	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 communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.In the early 1960s a stochastic geometry model was developed to study wireless networks.
c_k2xiix9djqs1	Stochastic geometry models of wireless networks	Summary	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 wireless network technologies including mobile ad hoc networks, sensor networks, vehicular ad hoc networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks. Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.The principal idea underlying the research of these stochastic geometry models, also known as random spatial models, is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.
c_qijo0a4b9zg1	Landau-Ramanujan constant	Summary	Landau-Ramanujan_constant	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 asymptotically as b x log ⁡ ( x ) . {\displaystyle {\dfrac {bx}{\sqrt {\log(x)}}}.} This constant b was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.
c_lq2ky9fpg5v1	Transport function	Summary	Transport_function	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}}}=\sum _{k=0}^{\infty }k\,e^{kt}.}
c_phdmjxvuhwmh	Ray (quantum theory)	Summary	Projective_Hilbert_space	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 {\displaystyle H} given by w ∼ v {\displaystyle w\sim v} if and only if v = λ w {\displaystyle v=\lambda w} for some non-zero complex number λ {\displaystyle \lambda } .The equivalence classes of v {\displaystyle v} for the relation ∼ {\displaystyle \sim } are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space.
c_qjzu6z0xhbj4	Lawson topology	Summary	Lawson_topology	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 the smallest common refinement of the lower topology and the Scott topology on P.
c_95sxbcgrj3ae	Constant-recursive sequence	Summary	Linear_recursive_sequence	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-recursive sequence, linear-recurrent sequence, a C-finite sequence, or a solution to a linear recurrence with constant coefficients. The most famous example of a constant-recursive sequence is the Fibonacci sequence 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , … {\displaystyle 0,1,1,2,3,5,8,13,\ldots } , in which each number is the sum of the previous two. The power of two sequence 1 , 2 , 4 , 8 , 16 , … {\displaystyle 1,2,4,8,16,\ldots } is also constant-recursive because each number is the sum of twice the previous number.
c_vckvstx5that	Constant-recursive sequence	Summary	Linear_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 progressions, all geometric progressions, and all polynomials are constant-recursive.
c_0my7jnfr9xnf	Constant-recursive sequence	Summary	Linear_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 , {\displaystyle F_{n}=F_{n-1}+F_{n-2},} where F n {\displaystyle F_{n}} is the n {\displaystyle n} th Fibonacci number. Constant-recursive sequences are studied in combinatorics and the theory of finite differences.
c_gvlh412jwuhq	Constant-recursive sequence	Summary	Linear_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 are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product. The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. On the other hand, the Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, is a famous unsolved problem in mathematics.
c_snw7t548niwr	K-regular sequence	Summary	K-regular_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.
c_iss8dir2gonq	K-synchronized sequence	Summary	K-synchronized_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-automatic sequences and k-regular sequences.
c_gpcopkpmkwwm	Unavoidable pattern	Summary	Unavoidable_pattern	In mathematics and theoretical computer science, a pattern is an unavoidable pattern if it is unavoidable on any finite alphabet.
c_zkl3opnsqyon	Semiautomaton	Summary	Transition_monoid	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 characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors.
c_tsg43dfuwt8c	Set constraint	Summary	Set_constraint	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 application) on sets and different (in)equation relations (like "=", "⊆", and "⊈") between set expressions. Systems of set constraints are useful to describe (in particular infinite) sets of ground terms. They arise in program analysis, abstract interpretation, and type inference.
c_zy383qot4v7b	Automatic sequence	Summary	Automatic_sequence	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 a mapping of the final state reached in a finite automaton accepting the digits of the number n in some fixed base k.An automatic set is a set of non-negative integers S for which the sequence of values of its characteristic function χS is an automatic sequence; that is, S is k-automatic if χS(n) is k-automatic, where χS(n) = 1 if n ∈ {\displaystyle \in } S and 0 otherwise.
c_icfje5kh91hb	Analysis of Boolean functions	Summary	Analysis_of_Boolean_functions	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 studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning.
c_n2mnejj92abp	Entropy compression	Summary	Entropy_compression	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.
c_wyj35zvv69fy	Amplituhedron	Summary	Amplituhedron	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, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian.Amplituhedron theory challenges the notion that spacetime locality and unitarity are necessary components of a model of particle interactions.
c_j9c3pb9ne20z	Amplituhedron	Summary	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 led by Nima Arkani-Hamed. Edward Witten described the work as "very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be".
c_7grxoswpqsrl	Twistor space	Summary	Twistor_space	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, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.
c_m7x5e6x23pre	Noether's second theorem	Summary	Noether's_second_theorem	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. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations.
c_02n4vrw0tqgs	Noether's second theorem	Summary	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.
c_bv93prbo9oxh	One particle Hilbert space	Summary	Wigner's_classification	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 representations are infinite-dimensional.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.
c_6dxxqds49gjv	One particle Hilbert space	Summary	Wigner's_classification	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 to label the representations. The first is associated with mass-squared and the second with helicity or spin.
c_fwpy9ft22ggc	One particle Hilbert space	Summary	Wigner's_classification	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 =0,1,2,3~.} Wigner found that massless particles are fundamentally different from massive particles.
c_1o00yrewhu3g	One particle Hilbert space	Summary	Wigner's_classification	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) unitary representation that characterizes their spin, and a positive mass, m. For the second case Look at the stabilizer of P = ( k , 0 , 0 , − k ) . {\displaystyle ~P=(k,0,0,-k)~.}
c_9nepk9bjv0sh	One particle Hilbert space	Summary	Wigner's_classification	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 representation called the vacuum.
c_69jpwla3wuih	Gerstenhaber algebra	Summary	Antibracket_algebra	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 formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms.
c_vpzc5og0hnda	Bifundamental representation	Summary	Bifundamental_representation	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 representation. These representations occur in quiver diagrams.
c_8937a8dgywsv	Large diffeomorphism	Summary	Large_diffeomorphism	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 diffeomorphisms by which the one-cycles a , b {\displaystyle a,b} of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the modular group. More generally, for a surface S, the structure of self-homeomorphisms up to homotopy is known as the mapping class group. It is known (for compact, orientable S) that this is isomorphic with the automorphism group of the fundamental group of S. This is consistent with the genus 1 case, stated above, if one takes into account that then the fundamental group is Z2, on which the modular group acts as automorphisms (as a subgroup of index 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).
c_kcg9jrpabm2i	Locally compact quantum group	Summary	Locally_compact_quantum_group	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 unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group.
c_1geaftoweuya	Pseudo-Euclidean space	Summary	Pseudo-Euclidean_vector_space	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 vector x.: 3 For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic. Note that if 1 ≤ i ≤ k < j ≤ n, then q(ei + ej) = 0, so that ei + ej is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square. As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).
c_dhkfo6gi655a	Representation theory of Lie groups	Summary	Representation_theory_of_Lie_groups	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 symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.
c_3pi3ko8cf04c	Associative superalgebra	Summary	Even_subalgebra	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 physics.
c_mbj6lr404xlr	Associative superalgebra	Summary	Even_subalgebra	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, supermanifolds and superschemes.
c_nt2d8nrrrk07	Supermatrix	Summary	Supermatrix	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 ordinary field (thought of as a purely even commutative superalgebra). Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry.
c_m7bj7xs42od5	Antisymmetric tensor	Summary	Antisymmetric_tensor	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 with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order k {\displaystyle k} may be referred to as a differential k {\displaystyle k} -form, and a completely antisymmetric contravariant tensor field may be referred to as a k {\displaystyle k} -vector field.
c_9usftp4q1jpd	Invariant differential operator	Summary	Invariant_differential_operator	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, sections of a vector bundle. In an invariant differential operator D {\displaystyle D} , the term differential operator indicates that the value D f {\displaystyle Df} of the map depends only on f ( x ) {\displaystyle f(x)} and the derivatives of f {\displaystyle f} in x {\displaystyle x} .
c_jmzi3pq8j0as	Invariant differential operator	Summary	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 action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.
c_ekzfmv42r1gg	Deformed Hermitian Yang–Mills equation	Summary	Deformed_Hermitian_Yang–Mills_equations	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 the case of Abelian gauge group (the unitary group U ⁡ ( 1 ) {\displaystyle \operatorname {U} (1)} ), and by Leung–Yau–Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.
c_a0lpvwemj53f	Braid statistics	Summary	Braid_statistics	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 exchange of identical particles, a particle with braid statistics leads to a rational fraction of π {\displaystyle \pi } under such exchange or even a non-trivial unitary transformation in the Hilbert space (see non-Abelian anyons). A similar notion exists using a loop braid group. Braid statistics are applicable to theoretical particles such as the two-dimensional anyons and their higher-dimensional analogues known as plektons.
c_dj9e868xksp2	Fusion rules	Summary	Fusion_rules	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 generated by the Virasoro algebra, the relevant representations are the conformal families associated with a primary field and the tensor product is realized by operator product expansions. The fusion rules contain the information about the kind of families that appear on the right hand side of these OPEs, including the multiplicities. More generally, integrable models in 2 dimensions which aren't conformal field theories are also described by fusion rules for their charges. == References ==
c_8mcirxiuor59	Quasi-periodic motion	Summary	Quasiperiodic_motion	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), the trajectory of the system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. A quasiperiodic function on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve R → Twhich is linear (when lifted from T to its covering Euclidean space), by composition. It is therefore oscillating, with a finite number of underlying frequencies. (NB the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice is something distinct from this.) The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.
c_owmsvu5b6q17	Resummation	Summary	Resummation	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 new function to obtain the original function. Borel resummation is probably the most well-known example. The simplest method is an extension of a variational approach to higher order based on a paper by R.P.
c_7d6dujqdc2f6	Resummation	Summary	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.
c_2e9diy3tn8hp	Berezinian	Summary	Berezinian	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.
c_hz2ge5o3808g	Eguchi–Hanson space	Summary	Eguchi–Hanson_space	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 Andrew J. Hanson; it was discovered independently by the mathematician Eugenio Calabi around the same time in 1979.The Eguchi-Hanson metric has Ricci tensor equal to zero, making it a solution to the vacuum Einstein equations of general relativity, albeit with Riemannian rather than Lorentzian metric signature. It may be regarded as a resolution of the A1 singularity according to the ADE classification which is the singularity at the fixed point of the C2/Z2 orbifold where the Z2 group inverts the signs of both complex coordinates in C2.
c_c93cnonux8h9	Eguchi–Hanson space	Summary	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 {\displaystyle \rho } is a scale setting constant and r 2 = \| w \| C d / 2 2 {\displaystyle r^{2}=\|w\|_{\mathbb {C} ^{d/2}}^{2}} . Aside from its inherent importance in pure geometry, the space is important in string theory. Certain types of K3 surfaces can be approximated as a combination of several Eguchi–Hanson metrics since both have the same holonomy group.
c_wi2v6hr7eyip	Eguchi–Hanson space	Summary	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 in that article. It is then an example of a hyperkähler manifold. == References ==
c_tktvu3991die	Induced metric	Summary	Induced_metric	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 the component form of the pullback operation: g a b = ∂ a X μ ∂ b X ν g μ ν {\displaystyle g_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }\ } Here a {\displaystyle a} , b {\displaystyle b} describe the indices of coordinates ξ a {\displaystyle \xi ^{a}} of the submanifold while the functions X μ ( ξ a ) {\displaystyle X^{\mu }(\xi ^{a})} encode the embedding into the higher-dimensional manifold whose tangent indices are denoted μ {\displaystyle \mu } , ν {\displaystyle \nu } .
c_n4ckz5159xn0	Quantum group	Summary	Quantum_group	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*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra.
c_ldjrbj0dm63h	Quantum group	Summary	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 auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.
c_vlbkg3p10ro7	Zeta function regularization	Summary	Heat_kernel_regulator	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 problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.
c_49rlmqiigpyv	Traffic flow	Summary	Cumulative_vehicle_count_curve	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 network with efficient movement of traffic and minimal traffic congestion problems.
c_qsbvr1xz7s0x	Circular surface	Summary	Circular_surface	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} }(t)\cos \theta +r(t){\mathbf {v} }(t)\sin \theta ,\,} where γ, u, v: I → R3 and r: I → R>0, when R>0 := { x ∈ R: x > 0 }. Moreover, it is usually assumed that u · u = v · v = 1 and u · v = 0, where dot denotes the canonical scalar product on R3, i.e. u and v are unit length and mutually perpendicular. The map γ: I → R3 is called the base curve for the circular surface and the two maps u, v: I → R3 are called the direction frame for the circular surface. For a fixed t0 ∈ I the image of ƒ(t0, θ) is called a generating circle of the circular surface.Circular surfaces are an analogue of ruled surfaces.
c_ch3h70docvtv	Circular surface	Summary	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 ==
c_0mdkq7qiefmy	Dini derivative	Summary	Dini_derivative	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 continuous function f: R → R , {\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} },} is denoted by f′+ and defined by f + ′ ( t ) = lim sup h → 0 + f ( t + h ) − f ( t ) h , {\displaystyle f'_{+}(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},} where lim sup is the supremum limit and the limit is a one-sided limit.
c_3gdvg3edple8	Dini derivative	Summary	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 + ′ ( t , d ) = lim sup h → 0 + f ( t + h d ) − f ( t ) h . {\displaystyle f'_{+}(t,d)=\limsup _{h\to {0+}}{\frac {f(t+hd)-f(t)}{h}}.} If f is locally Lipschitz, then f′+ is finite. If f is differentiable at t, then the Dini derivative at t is the usual derivative at t.
c_ulf4n3taalhf	Homogeneity blockmodeling	Summary	Homogeneity_blockmodeling	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 the similarity of link (tie) strengths within the blocks over the pattern of links. In this approach, tie (link) values (or statistical data computed on them) are assumed to be equal (homogenous) within blocks.This approach to the generalized blockmodeling of valued networks was first proposed by Aleš Žiberna in 2007 with the basic idea, "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly–formed ideal blocks, which are appropriate for blockmodeling of valued networks, are then presented together with the definitions of their block inconsistencies. Similar approach to the homogeneity blockmodeling, dealing with direct approach for structural equivalence, was previously suggested by Stephen P. Borgatti and Martin G. Everett (1992).
c_o8t4cslti0ig	Monge array	Summary	Monge_array	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 ≤ j < ℓ ≤ n {\displaystyle 1\leq i
c_cp8l5p61qenr	Umbral calculus	Summary	Umbral_calculus	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 attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.
c_ljot45daaecx	Topology of compact convergence	Summary	Topology_of_compact_convergence	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.
c_gdu814se53w2	Sarason interpolation theorem	Summary	Sarason_interpolation_theorem	In mathematics complex analysis, the Sarason interpolation theorem, introduced by Sarason (1967), is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation.
c_o4lszvtblh7e	Antifundamental representation	Summary	Antifundamental_representation	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 of them is a complex representation. == References ==
c_uprj82y5f7lp	Number bond	Summary	Number_bond	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 multiplication. For example, a number bond looks like 5 + 2 = 7 {\displaystyle 5+2=7} A child who "knows" this number bond should be able to immediately fill in any one of these three numbers if it were missing, given the other two, without having to "work it out". Number bonds are often learned in sets for which the sum is a common round number such as 10 or 20. Having acquired some familiar number bonds, children should also soon learn how to use them to develop strategies to complete more complicated sums, for example by navigating from a new sum to an adjacent number bond they know, i.e. 5 + 2 and 4 + 3 are both number bonds that make 7; or by strategies like "making ten", for example recognising that 7 + 6 = (7 + 3) + 3 = 13.

c_8qzd6uudj5sw

Mean of circular quantities

Summary

Circular_mean

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 computation produces a different result than the arithmetic mean, with the difference being greater when the angles are widely distributed. For example, the arithmetic mean of the three angles 0°, 0°, and 90° is (0° + 0° + 90°) / 3 = 30°, but the vector mean is arctan(1/2) = 26.565°. Moreover, with the arithmetic mean the circular variance is only defined ±180°.

c_dcgv95tbcf4n

Fuzzy concept

Sciences

Fuzzy_concept > Applications > Sciences

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 conceptual hierarchy.

c_x92iqqfym8fj

Piecewise-linear function

Summary

Piecewise_linear_function

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.

c_t2aj6zdi1vs2

Probability vector

Summary

Stochastic_vector

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, which is the standard way of characterizing a discrete probability distribution.

c_0ve3sp9q03cy

Discrete variables

Summary

Discrete_value

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 variable is continuous in that interval. If it can take on a value such that there is a non-infinitesimal gap on each side of it containing no values that the variable can take on, then it is discrete around that value. In some contexts a variable can be discrete in some ranges of the number line and continuous in others.

c_7rhgxf0ewm59

Random number

Summary

Random_number

In mathematics and statistics, a random number is either Pseudo-random or a number generated for, or part of, a set exhibiting statistical randomness.

c_tj7jl6escrby

Wide sense stationary

Summary

Stationary_stochastic_process

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 time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles around the trend line, but overall it does not trend up nor down. Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data are often transformed to become stationary.

c_3qr17wqd6m7c

Wide sense stationary

Summary

Stationary_stochastic_process

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, the process is called a trend-stationary process, and stochastic shocks have only transitory effects after which the variable tends toward a deterministically evolving (non-constant) mean.

c_abrr4hjgc20e

Wide sense stationary

Summary

Stationary_stochastic_process

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 that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.

c_ayz90t220y67

Wide sense stationary

Summary

Stationary_stochastic_process

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).

c_j032bv4esgv3

Asymptotic distribution

Summary

Asymptotic_variance

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 estimators.

c_0w8dgx49gazm

Error term

Summary

Error_term

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

c_ir7km8nnv2r3

Absolute deviation

Summary

Deviation_(statistics)

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 of the value indicates the size of the difference.

c_k0mugjs87odi

Kolmogorov equations (Markov jump process)

Summary

Kolmogorov_equations_(continuous-time_Markov_chains)

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 titled Kolmogorov equations, focuses on the scenario where we have a continuous-time Markov chain (so the state space Ω {\displaystyle \Omega } is countable). In this case, we can treat the Kolmogorov equations as a way to describe the probability P ( x , s ; y , t ) {\displaystyle P(x,s;y,t)} , where x , y ∈ Ω {\displaystyle x,y\in \Omega } (the state space) and t > s , t , s ∈ R ≥ 0 {\displaystyle t>s,t,s\in \mathbb {R} _{\geq 0}} are the final and initial times, respectively.

c_gzrfebexk962

Random projections

Summary

Random_projection

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 preserves distances well, but empirical results are sparse. They have been applied to many natural language tasks under the name random indexing.

c_l64i1gt6tqpq

Power sum

Summary

Sums_of_powers

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 analysis of variance involves summing the squares of quantities. Faulhaber's formula expresses 1 k + 2 k + 3 k + ⋯ + n k {\displaystyle 1^{k}+2^{k}+3^{k}+\cdots +n^{k}} as a polynomial in n, or alternatively in terms of a Bernoulli polynomial. Fermat's right triangle theorem states that there is no solution in positive integers for a 2 = b 4 + c 4 {\displaystyle a^{2}=b^{4}+c^{4}} and a 4 = b 4 + c 2 {\displaystyle a^{4}=b^{4}+c^{2}} .

c_ilpi2el5nhpt

Power sum

Summary

Sums_of_powers

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} .

c_k9xxowgxoq6c

Power sum

Summary

Sums_of_powers

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 not necessarily equal, can equal another integer that is a power, with the reciprocals of the three powers summing to less than 1. Beal's conjecture concerns the question of whether the sum of two coprime integers, each a power greater than 2 of an integer, with the powers not necessarily equal, can equal another integer that is a power greater than 2.

c_2io59lq1ljbr

Power sum

Summary

Sums_of_powers

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 expressed as a sum of two positive third powers in n distinct ways. The Riemann zeta function is the sum of the reciprocals of the positive integers each raised to the power s, where s is a complex number whose real part is greater than 1.

c_2btgfl64i5z3

Power sum

Summary

Sums_of_powers

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}.}

c_leejx0yj6j6y

Power sum

Summary

Sums_of_powers

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}=\varphi ^{n}+\varphi ^{n-1}.}

c_idg7lc1jzlij

Power sum

Summary

Sums_of_powers

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.

c_i81sxr9dauov

Power sum

Summary

Sums_of_powers

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.

c_ouclspjevknz

Power sum

Summary

Sums_of_powers

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 whether all remaining integers can be expressed in this form. The sums of powers Sm(z, n) = zm + (z+1)m + ... + (z+n−1)m is related to the Bernoulli polynomials Bm(z) by (∂n−∂z) Sm(z, n) = Bm(z) and (∂2λ−∂Z) S2k+1(z, n) = Ŝ′k+1(Z) where Z = z(z−1), λ = S1(z, n), Ŝk+1(Z) ≡ S2k+1(0, z). The sum of the terms in the geometric series is ∑ k = i n z k = z i − z n + 1 1 − z . {\displaystyle \sum _{k=i}^{n}z^{k}={\frac {z^{i}-z^{n+1}}{1-z}}.}

c_o6554v34fuw6

Fréchet mean

Summary

Fréchet_mean

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 and Hermann Karcher. On the real numbers, the arithmetic mean, median, geometric mean, and harmonic mean can all be interpreted as Fréchet means for different distance functions.

c_dd56fi4k1keh

Statistical mean

Summary

Arithmetic_average

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, or a survey. The term "arithmetic mean" is preferred in some mathematics and statistics contexts because it helps distinguish it from other types of means, such as geometric and harmonic. In addition to mathematics and statistics, the arithmetic mean is frequently used in economics, anthropology, history, and almost every academic field to some extent.

c_alcxlg2n8qyp

Statistical mean

Summary

Arithmetic_average

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 distribution of income for which a few people's incomes are substantially higher than most people's, the arithmetic mean may not coincide with one's notion of "middle". In that case, robust statistics, such as the median, may provide a better description of central tendency.

c_h7x9kmw06fyp

Quasi-arithmetic mean

Summary

Generalized_f-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 Andrey Kolmogorov. It is a broader generalization than the regular generalized mean.

c_6i6y8g4aw87a

Portmanteau theorem

Weak convergence of measures

Portmanteau_lemma > Weak convergence of measures

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 which are (apparently) more general than others.

c_uuo6mv57vqn5

Portmanteau theorem

Weak convergence of measures

Portmanteau_lemma > Weak convergence of measures

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 only know that 1 / n {\displaystyle 1/n} is "close" to 0 {\displaystyle 0} because of the topology of R {\displaystyle \mathbf {R} } . This definition of weak convergence can be extended for S {\displaystyle S} any metrizable topological space.

c_q5z055g03k6j

Portmanteau theorem

Weak convergence of measures

Portmanteau_lemma > Weak convergence of measures

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 \left\{\ U_{\phi ,x,\delta }\ \left|\quad \phi \colon S\to \mathbf {R} {\text{ is bounded and continuous, }}x\in \mathbf {R} {\text{ and }}\delta >0\ \right.\right\},} where U ϕ , x , δ := { μ ∈ P ( S ) | | ∫ S ϕ d μ − x | < δ } . {\displaystyle U_{\phi ,x,\delta }:=\left\{\ \mu \in {\mathcal {P}}(S)\ \left|\quad \left|\int _{S}\phi \,\mathrm {d} \mu -x\right|<\delta \ \right.\right\}.}

c_sasswyy4j65a

Portmanteau theorem

Weak convergence of measures

Portmanteau_lemma > Weak convergence of measures

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 into P ( S ) {\displaystyle {\mathcal {P}}(S)} as the (closed) set of Dirac measures, and its convex hull is dense. There are many "arrow notations" for this kind of convergence: the most frequently used are P n ⇒ P {\displaystyle P_{n}\Rightarrow P} , P n ⇀ P {\displaystyle P_{n}\rightharpoonup P} , P n → w P {\displaystyle P_{n}\xrightarrow {w} P} and P n → D P {\displaystyle P_{n}\xrightarrow {\mathcal {D}} P} .

c_uza24plss7ex

Conifold

Summary

Conifold

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-dimensional real manifold, since the typically considered conifolds are complex 3-dimensional (real 6-dimensional) spaces.

c_hu0j50sh4oia

Spectral network

Summary

Spectral_network

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 ==

c_z7zxfpjaxmxh

Stochastic geometry models of wireless networks

Summary

Stochastic_geometry_models_of_wireless_networks

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 communication networks in order to predict and control various network performance metrics. The models require using techniques from stochastic geometry and related fields including point processes, spatial statistics, geometric probability, percolation theory, as well as methods from more general mathematical disciplines such as geometry, probability theory, stochastic processes, queueing theory, information theory, and Fourier analysis.In the early 1960s a stochastic geometry model was developed to study wireless networks.

c_k2xiix9djqs1

Stochastic geometry models of wireless networks

Summary

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 wireless network technologies including mobile ad hoc networks, sensor networks, vehicular ad hoc networks, cognitive radio networks and several types of cellular networks, such as heterogeneous cellular networks. Key performance and quality of service quantities are often based on concepts from information theory such as the signal-to-interference-plus-noise ratio, which forms the mathematical basis for defining network connectivity and coverage.The principal idea underlying the research of these stochastic geometry models, also known as random spatial models, is that it is best to assume that the locations of nodes or the network structure and the aforementioned quantities are random in nature due to the size and unpredictability of users in wireless networks. The use of stochastic geometry can then allow for the derivation of closed-form or semi-closed-form expressions for these quantities without resorting to simulation methods or (possibly intractable or inaccurate) deterministic models.

c_qijo0a4b9zg1

Landau-Ramanujan constant

Summary

Landau-Ramanujan_constant

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 asymptotically as b x log ⁡ ( x ) . {\displaystyle {\dfrac {bx}{\sqrt {\log(x)}}}.} This constant b was rediscovered in 1913 by Srinivasa Ramanujan, in the first letter he wrote to G.H. Hardy.

c_lq2ky9fpg5v1

Transport function

Summary

Transport_function

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}}}=\sum _{k=0}^{\infty }k\,e^{kt}.}

c_phdmjxvuhwmh

Ray (quantum theory)

Summary

Projective_Hilbert_space

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 {\displaystyle H} given by w ∼ v {\displaystyle w\sim v} if and only if v = λ w {\displaystyle v=\lambda w} for some non-zero complex number λ {\displaystyle \lambda } .The equivalence classes of v {\displaystyle v} for the relation ∼ {\displaystyle \sim } are also called rays or projective rays. This is the usual construction of projectivization, applied to a complex Hilbert space.

c_qjzu6z0xhbj4

Lawson topology

Summary

Lawson_topology

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 the smallest common refinement of the lower topology and the Scott topology on P.

c_95sxbcgrj3ae

Constant-recursive sequence

Summary

Linear_recursive_sequence

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-recursive sequence, linear-recurrent sequence, a C-finite sequence, or a solution to a linear recurrence with constant coefficients. The most famous example of a constant-recursive sequence is the Fibonacci sequence 0 , 1 , 1 , 2 , 3 , 5 , 8 , 13 , … {\displaystyle 0,1,1,2,3,5,8,13,\ldots } , in which each number is the sum of the previous two. The power of two sequence 1 , 2 , 4 , 8 , 16 , … {\displaystyle 1,2,4,8,16,\ldots } is also constant-recursive because each number is the sum of twice the previous number.

c_vckvstx5that

Constant-recursive sequence

Summary

Linear_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 progressions, all geometric progressions, and all polynomials are constant-recursive.

c_0my7jnfr9xnf

Constant-recursive sequence

Summary

Linear_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 , {\displaystyle F_{n}=F_{n-1}+F_{n-2},} where F n {\displaystyle F_{n}} is the n {\displaystyle n} th Fibonacci number. Constant-recursive sequences are studied in combinatorics and the theory of finite differences.

c_gvlh412jwuhq

Constant-recursive sequence

Summary

Linear_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 are closed under important mathematical operations such as term-wise addition, term-wise multiplication, and Cauchy product. The Skolem–Mahler–Lech theorem states that the zeros of a constant-recursive sequence have a regularly repeating (eventually periodic) form. On the other hand, the Skolem problem, which asks for an algorithm to determine whether a linear recurrence has at least one zero, is a famous unsolved problem in mathematics.

c_snw7t548niwr

K-regular sequence

Summary

K-regular_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.

c_iss8dir2gonq

K-synchronized sequence

Summary

K-synchronized_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-automatic sequences and k-regular sequences.

c_gpcopkpmkwwm

Unavoidable pattern

Summary

Unavoidable_pattern

In mathematics and theoretical computer science, a pattern is an unavoidable pattern if it is unavoidable on any finite alphabet.

c_zkl3opnsqyon

Semiautomaton

Summary

Transition_monoid

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 characteristic monoid, input monoid, transition monoid or transition system of the semiautomaton, which acts on the set of states Q. This may be viewed either as an action of the free monoid of strings in the input alphabet Σ, or as the induced transformation semigroup of Q. In older books like Clifford and Preston (1967) semigroup actions are called "operands". In category theory, semiautomata essentially are functors.

c_tsg43dfuwt8c

Set constraint

Summary

Set_constraint

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 application) on sets and different (in)equation relations (like "=", "⊆", and "⊈") between set expressions. Systems of set constraints are useful to describe (in particular infinite) sets of ground terms. They arise in program analysis, abstract interpretation, and type inference.

c_zy383qot4v7b

Automatic sequence

Summary

Automatic_sequence

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 a mapping of the final state reached in a finite automaton accepting the digits of the number n in some fixed base k.An automatic set is a set of non-negative integers S for which the sequence of values of its characteristic function χS is an automatic sequence; that is, S is k-automatic if χS(n) is k-automatic, where χS(n) = 1 if n ∈ {\displaystyle \in } S and 0 otherwise.

c_icfje5kh91hb

Analysis of Boolean functions

Summary

Analysis_of_Boolean_functions

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 studied are often, but not always, Boolean-valued, making them Boolean functions. The area has found many applications in combinatorics, social choice theory, random graphs, and theoretical computer science, especially in hardness of approximation, property testing, and PAC learning.

c_n2mnejj92abp

Entropy compression

Summary

Entropy_compression

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.

c_wyj35zvv69fy

Amplituhedron

Summary

Amplituhedron

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, also equivalent to the perturbative topological B model string theory in twistor space, an amplituhedron is defined as a mathematical space known as the positive Grassmannian.Amplituhedron theory challenges the notion that spacetime locality and unitarity are necessary components of a model of particle interactions.

c_j9c3pb9ne20z

Amplituhedron

Summary

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 led by Nima Arkani-Hamed. Edward Witten described the work as "very unexpected" and said that "it is difficult to guess what will happen or what the lessons will turn out to be".

c_7grxoswpqsrl

Twistor space

Summary

Twistor_space

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, twistor space is useful for conceptualizing the way photons travel through space, using four complex numbers. He also posits that twistor space may aid in understanding the asymmetry of the weak nuclear force.

c_m7x5e6x23pre

Noether's second theorem

Summary

Noether's_second_theorem

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. Specifically, the theorem says that if the action has an infinite-dimensional Lie algebra of infinitesimal symmetries parameterized linearly by k arbitrary functions and their derivatives up to order m, then the functional derivatives of L satisfy a system of k differential equations.

c_02n4vrw0tqgs

Noether's second theorem

Summary

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.

c_bv93prbo9oxh

One particle Hilbert space

Summary

Wigner's_classification

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 representations are infinite-dimensional.) It was introduced by Eugene Wigner, to classify particles and fields in physics—see the article particle physics and representation theory. It relies on the stabilizer subgroups of that group, dubbed the Wigner little groups of various mass states.

c_6dxxqds49gjv

One particle Hilbert space

Summary

Wigner's_classification

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 to label the representations. The first is associated with mass-squared and the second with helicity or spin.

c_fwpy9ft22ggc

One particle Hilbert space

Summary

Wigner's_classification

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 =0,1,2,3~.} Wigner found that massless particles are fundamentally different from massive particles.

c_1o00yrewhu3g

One particle Hilbert space

Summary

Wigner's_classification

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) unitary representation that characterizes their spin, and a positive mass, m. For the second case Look at the stabilizer of P = ( k , 0 , 0 , − k ) . {\displaystyle ~P=(k,0,0,-k)~.}

c_9nepk9bjv0sh

One particle Hilbert space

Summary

Wigner's_classification

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 representation called the vacuum.

c_69jpwla3wuih

Gerstenhaber algebra

Summary

Antibracket_algebra

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 formalism. It appears also in the generalization of Hamiltonian formalism known as the De Donder–Weyl theory as the algebra of generalized Poisson brackets defined on differential forms.

c_vpzc5og0hnda

Bifundamental representation

Summary

Bifundamental_representation

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 representation. These representations occur in quiver diagrams.

c_8937a8dgywsv

Large diffeomorphism

Summary

Large_diffeomorphism

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 diffeomorphisms by which the one-cycles a , b {\displaystyle a,b} of the torus are transformed into their integer linear combinations. This group of large diffeomorphisms is called the modular group. More generally, for a surface S, the structure of self-homeomorphisms up to homotopy is known as the mapping class group. It is known (for compact, orientable S) that this is isomorphic with the automorphism group of the fundamental group of S. This is consistent with the genus 1 case, stated above, if one takes into account that then the fundamental group is Z2, on which the modular group acts as automorphisms (as a subgroup of index 2 in all automorphisms, since the orientation may also be reverse, by a transformation with determinant −1).

c_kcg9jrpabm2i

Locally compact quantum group

Summary

Locally_compact_quantum_group

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 unitaries have enjoyed some success but have also encountered several technical problems. One of the main features distinguishing this new approach from its predecessors is the axiomatic existence of left and right invariant weights. This gives a noncommutative analogue of left and right Haar measures on a locally compact Hausdorff group.

c_1geaftoweuya

Pseudo-Euclidean space

Summary

Pseudo-Euclidean_vector_space

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 vector x.: 3 For Euclidean spaces, k = n, implying that the quadratic form is positive-definite. When 0 < k < n, q is an isotropic quadratic form, otherwise it is anisotropic. Note that if 1 ≤ i ≤ k < j ≤ n, then q(ei + ej) = 0, so that ei + ej is a null vector. In a pseudo-Euclidean space with k < n, unlike in a Euclidean space, there exist vectors with negative scalar square. As with the term Euclidean space, the term pseudo-Euclidean space may be used to refer to an affine space or a vector space depending on the author, with the latter alternatively being referred to as a pseudo-Euclidean vector space (see point–vector distinction).

c_dhkfo6gi655a

Representation theory of Lie groups

Summary

Representation_theory_of_Lie_groups

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 symmetry. A great deal is known about such representations, a basic tool in their study being the use of the corresponding 'infinitesimal' representations of Lie algebras.

c_3pi3ko8cf04c

Associative superalgebra

Summary

Even_subalgebra

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 physics.

c_mbj6lr404xlr

Associative superalgebra

Summary

Even_subalgebra

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, supermanifolds and superschemes.

c_nt2d8nrrrk07

Supermatrix

Summary

Supermatrix

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 ordinary field (thought of as a purely even commutative superalgebra). Supermatrices arise in the study of super linear algebra where they appear as the coordinate representations of a linear transformations between finite-dimensional super vector spaces or free supermodules. They have important applications in the field of supersymmetry.

c_m7bj7xs42od5

Antisymmetric tensor

Summary

Antisymmetric_tensor

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 with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor is completely (or totally) antisymmetric. A completely antisymmetric covariant tensor field of order k {\displaystyle k} may be referred to as a differential k {\displaystyle k} -form, and a completely antisymmetric contravariant tensor field may be referred to as a k {\displaystyle k} -vector field.

c_9usftp4q1jpd

Invariant differential operator

Summary

Invariant_differential_operator

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, sections of a vector bundle. In an invariant differential operator D {\displaystyle D} , the term differential operator indicates that the value D f {\displaystyle Df} of the map depends only on f ( x ) {\displaystyle f(x)} and the derivatives of f {\displaystyle f} in x {\displaystyle x} .

c_jmzi3pq8j0as

Invariant differential operator

Summary

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 action of the group has the meaning of a change of coordinates (change of observer) and the invariance means that the operator has the same expression in all admissible coordinates.

c_ekzfmv42r1gg

Deformed Hermitian Yang–Mills equation

Summary

Deformed_Hermitian_Yang–Mills_equations

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 the case of Abelian gauge group (the unitary group U ⁡ ( 1 ) {\displaystyle \operatorname {U} (1)} ), and by Leung–Yau–Zaslow using mirror symmetry from the corresponding equations of motion for D-branes in the A-model of string theory.

c_a0lpvwemj53f

Braid statistics

Summary

Braid_statistics

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 exchange of identical particles, a particle with braid statistics leads to a rational fraction of π {\displaystyle \pi } under such exchange or even a non-trivial unitary transformation in the Hilbert space (see non-Abelian anyons). A similar notion exists using a loop braid group. Braid statistics are applicable to theoretical particles such as the two-dimensional anyons and their higher-dimensional analogues known as plektons.

c_dj9e868xksp2

Fusion rules

Summary

Fusion_rules

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 generated by the Virasoro algebra, the relevant representations are the conformal families associated with a primary field and the tensor product is realized by operator product expansions. The fusion rules contain the information about the kind of families that appear on the right hand side of these OPEs, including the multiplicities. More generally, integrable models in 2 dimensions which aren't conformal field theories are also described by fusion rules for their charges. == References ==

c_8mcirxiuor59

Quasi-periodic motion

Summary

Quasiperiodic_motion

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), the trajectory of the system is modelled by a curve on T that wraps around the torus without ever exactly coming back on itself. A quasiperiodic function on the real line is the type of function (continuous, say) obtained from a function on T, by means of a curve R → Twhich is linear (when lifted from T to its covering Euclidean space), by composition. It is therefore oscillating, with a finite number of underlying frequencies. (NB the sense in which theta functions and the Weierstrass zeta function in complex analysis are said to have quasi-periods with respect to a period lattice is something distinct from this.) The theory of almost periodic functions is, roughly speaking, for the same situation but allowing T to be a torus with an infinite number of dimensions.

c_owmsvu5b6q17

Resummation

Summary

Resummation

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 new function to obtain the original function. Borel resummation is probably the most well-known example. The simplest method is an extension of a variational approach to higher order based on a paper by R.P.

c_7d6dujqdc2f6

Resummation

Summary

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.

c_2e9diy3tn8hp

Berezinian

Summary

Berezinian

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.

c_hz2ge5o3808g

Eguchi–Hanson space

Summary

Eguchi–Hanson_space

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 Andrew J. Hanson; it was discovered independently by the mathematician Eugenio Calabi around the same time in 1979.The Eguchi-Hanson metric has Ricci tensor equal to zero, making it a solution to the vacuum Einstein equations of general relativity, albeit with Riemannian rather than Lorentzian metric signature. It may be regarded as a resolution of the A1 singularity according to the ADE classification which is the singularity at the fixed point of the C2/Z2 orbifold where the Z2 group inverts the signs of both complex coordinates in C2.

c_c93cnonux8h9

Eguchi–Hanson space

Summary

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 {\displaystyle \rho } is a scale setting constant and r 2 = | w | C d / 2 2 {\displaystyle r^{2}=|w|_{\mathbb {C} ^{d/2}}^{2}} . Aside from its inherent importance in pure geometry, the space is important in string theory. Certain types of K3 surfaces can be approximated as a combination of several Eguchi–Hanson metrics since both have the same holonomy group.

c_wi2v6hr7eyip

Eguchi–Hanson space

Summary

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 in that article. It is then an example of a hyperkähler manifold. == References ==

c_tktvu3991die

Induced metric

Summary

Induced_metric

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 the component form of the pullback operation: g a b = ∂ a X μ ∂ b X ν g μ ν {\displaystyle g_{ab}=\partial _{a}X^{\mu }\partial _{b}X^{\nu }g_{\mu \nu }\ } Here a {\displaystyle a} , b {\displaystyle b} describe the indices of coordinates ξ a {\displaystyle \xi ^{a}} of the submanifold while the functions X μ ( ξ a ) {\displaystyle X^{\mu }(\xi ^{a})} encode the embedding into the higher-dimensional manifold whose tangent indices are denoted μ {\displaystyle \mu } , ν {\displaystyle \nu } .

c_n4ckz5159xn0

Quantum group

Summary

Quantum_group

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*-algebras), and bicrossproduct quantum groups. Despite their name, they do not themselves have a natural group structure, though they are in some sense 'close' to a group. The term "quantum group" first appeared in the theory of quantum integrable systems, which was then formalized by Vladimir Drinfeld and Michio Jimbo as a particular class of Hopf algebra.

c_ldjrbj0dm63h

Quantum group

Summary

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 auxiliary parameter q or h, which become universal enveloping algebras of a certain Lie algebra, frequently semisimple or affine, when q = 1 or h = 0. Closely related are certain dual objects, also Hopf algebras and also called quantum groups, deforming the algebra of functions on the corresponding semisimple algebraic group or a compact Lie group.

c_vlbkg3p10ro7

Zeta function regularization

Summary

Heat_kernel_regulator

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 problems in physics, but has its origins in attempts to give precise meanings to ill-conditioned sums appearing in number theory.

c_49rlmqiigpyv

Traffic flow

Summary

Cumulative_vehicle_count_curve

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 network with efficient movement of traffic and minimal traffic congestion problems.

c_qsbvr1xz7s0x

Circular surface

Summary

Circular_surface

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} }(t)\cos \theta +r(t){\mathbf {v} }(t)\sin \theta ,\,} where γ, u, v: I → R3 and r: I → R>0, when R>0 := { x ∈ R: x > 0 }. Moreover, it is usually assumed that u · u = v · v = 1 and u · v = 0, where dot denotes the canonical scalar product on R3, i.e. u and v are unit length and mutually perpendicular. The map γ: I → R3 is called the base curve for the circular surface and the two maps u, v: I → R3 are called the direction frame for the circular surface. For a fixed t0 ∈ I the image of ƒ(t0, θ) is called a generating circle of the circular surface.Circular surfaces are an analogue of ruled surfaces.

c_ch3h70docvtv

Circular surface

Summary

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 ==

c_0mdkq7qiefmy

Dini derivative

Summary

Dini_derivative

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 continuous function f: R → R , {\displaystyle f:{\mathbb {R} }\rightarrow {\mathbb {R} },} is denoted by f′+ and defined by f + ′ ( t ) = lim sup h → 0 + f ( t + h ) − f ( t ) h , {\displaystyle f'_{+}(t)=\limsup _{h\to {0+}}{\frac {f(t+h)-f(t)}{h}},} where lim sup is the supremum limit and the limit is a one-sided limit.

c_3gdvg3edple8

Dini derivative

Summary

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 + ′ ( t , d ) = lim sup h → 0 + f ( t + h d ) − f ( t ) h . {\displaystyle f'_{+}(t,d)=\limsup _{h\to {0+}}{\frac {f(t+hd)-f(t)}{h}}.} If f is locally Lipschitz, then f′+ is finite. If f is differentiable at t, then the Dini derivative at t is the usual derivative at t.

c_ulf4n3taalhf

Homogeneity blockmodeling

Summary

Homogeneity_blockmodeling

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 the similarity of link (tie) strengths within the blocks over the pattern of links. In this approach, tie (link) values (or statistical data computed on them) are assumed to be equal (homogenous) within blocks.This approach to the generalized blockmodeling of valued networks was first proposed by Aleš Žiberna in 2007 with the basic idea, "that the inconsistency of an empirical block with its ideal block can be measured by within block variability of appropriate values". The newly–formed ideal blocks, which are appropriate for blockmodeling of valued networks, are then presented together with the definitions of their block inconsistencies. Similar approach to the homogeneity blockmodeling, dealing with direct approach for structural equivalence, was previously suggested by Stephen P. Borgatti and Martin G. Everett (1992).

c_o8t4cslti0ig

Monge array

Summary

Monge_array

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 ≤ j < ℓ ≤ n {\displaystyle 1\leq i

c_cp8l5p61qenr

Umbral calculus

Summary

Umbral_calculus

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 attributed to Édouard Lucas (or James Joseph Sylvester), who used the technique extensively.

c_ljot45daaecx

Topology of compact convergence

Summary

Topology_of_compact_convergence

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.

c_gdu814se53w2

Sarason interpolation theorem

Summary

Sarason_interpolation_theorem

In mathematics complex analysis, the Sarason interpolation theorem, introduced by Sarason (1967), is a generalization of the Caratheodory interpolation theorem and Nevanlinna–Pick interpolation.

c_o4lszvtblh7e

Antifundamental representation

Summary

Antifundamental_representation

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 of them is a complex representation. == References ==

c_uprj82y5f7lp

Number bond

Summary

Number_bond

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 multiplication. For example, a number bond looks like 5 + 2 = 7 {\displaystyle 5+2=7} A child who "knows" this number bond should be able to immediately fill in any one of these three numbers if it were missing, given the other two, without having to "work it out". Number bonds are often learned in sets for which the sum is a common round number such as 10 or 20. Having acquired some familiar number bonds, children should also soon learn how to use them to develop strategies to complete more complicated sums, for example by navigating from a new sum to an adjacent number bond they know, i.e. 5 + 2 and 4 + 3 are both number bonds that make 7; or by strategies like "making ten", for example recognising that 7 + 6 = (7 + 3) + 3 = 13.