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c_4slfsb409ayg | The natural logarithm has the number e ≈ 2.718 as its base; its use is widespread in mathematics and physics, because of its very simple derivative. The binary logarithm uses base 2 and is frequently used in computer science. | Logarithm function |
c_0yp1no29xnr4 | Logarithms were introduced by John Napier in 1614 as a means of simplifying calculations. They were rapidly adopted by navigators, scientists, engineers, surveyors and others to perform high-accuracy computations more easily. Using logarithm tables, tedious multi-digit multiplication steps can be replaced by table look... | Logarithm function |
c_infrjzq9ovda | This is possible because the logarithm of a product is the sum of the logarithms of the factors: provided that b, x and y are all positive and b ≠ 1. The slide rule, also based on logarithms, allows quick calculations without tables, but at lower precision. The present-day notion of logarithms comes from Leonhard Euler... | Logarithm function |
c_8csm97qaq29d | For example, the decibel (dB) is a unit used to express ratio as logarithms, mostly for signal power and amplitude (of which sound pressure is a common example). In chemistry, pH is a logarithmic measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of th... | Logarithm function |
c_yuj9ydqfbt5x | They help to describe frequency ratios of musical intervals, appear in formulas counting prime numbers or approximating factorials, inform some models in psychophysics, and can aid in forensic accounting. The concept of logarithm as the inverse of exponentiation extends to other mathematical structures as well. However... | Logarithm function |
c_bylcldmr0a3l | In mathematics, the logarithmic integral function or integral logarithm li(x) is a special function. It is relevant in problems of physics and has number theoretic significance. In particular, according to the prime number theorem, it is a very good approximation to the prime-counting function, which is defined as the ... | Offset logarithmic integral |
c_xbjqi8yq81aw | In mathematics, the logarithmic mean is a function of two non-negative numbers which is equal to their difference divided by the logarithm of their quotient. This calculation is applicable in engineering problems involving heat and mass transfer. | Logarithmic average |
c_g1vho4zbtj8c | In mathematics, the logarithmic norm is a real-valued functional on operators, and is derived from either an inner product, a vector norm, or its induced operator norm. The logarithmic norm was independently introduced by Germund Dahlquist and Sergei Lozinskiĭ in 1958, for square matrices. It has since been extended to... | Logarithmic norm |
c_kg7g1j6bbz0f | In mathematics, the longest element of a Coxeter group is the unique element of maximal length in a finite Coxeter group with respect to the chosen generating set consisting of simple reflections. It is often denoted by w0. See (Humphreys 1992, Section 1.8: Simple transitivity and the longest element, pp. 15–16) and (D... | Longest element of a Coxeter group |
c_p6igrca3t4ad | In mathematics, the look-and-say sequence is the sequence of integers beginning as follows: 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, ... (sequence A005150 in the OEIS).To generate a member of the sequence from the previous member, read off the digits of the previous member, counting the nu... | Look-and-say sequence |
c_tiokbrxcnmk0 | 1211 is read off as "one 1, one 2, two 1s" or 111221. 111221 is read off as "three 1s, two 2s, one 1" or 312211.The look-and-say sequence was analyzed by John Conway after he was introduced to it by one of his students at a party.The idea of the look-and-say sequence is similar to that of run-length encoding. | Look-and-say sequence |
c_euvspqs8oc40 | If started with any digit d from 0 to 9 then d will remain indefinitely as the last digit of the sequence. For any d other than 1, the sequence starts as follows: d, 1d, 111d, 311d, 13211d, 111312211d, 31131122211d, …Ilan Vardi has called this sequence, starting with d = 3, the Conway sequence (sequence A006715 in the ... | Look-and-say sequence |
c_btlvqs4f4if0 | In mathematics, the lower convex envelope f ˘ {\displaystyle {\breve {f}}} of a function f {\displaystyle f} defined on an interval {\displaystyle } is defined at each point of the interval as the supremum of all convex functions that lie under that function, i.e. f ˘ ( x ) = sup { g ( x ) ∣ g is convex and g ≤ f over... | Lower convex envelope |
c_whw452yjcgkx | In mathematics, the lower envelope or pointwise minimum of a finite set of functions is the pointwise minimum of the functions, the function whose value at every point is the minimum of the values of the functions in the given set. The concept of a lower envelope can also be extended to partial functions by taking the ... | Pointwise maximum |
c_zcvp5znw35li | For functions of a single real variable whose graphs have a bounded number of intersection points, the complexity of the lower or upper envelope can be bounded using Davenport–Schinzel sequences, and these envelopes can be computed efficiently by a divide-and-conquer algorithm that computes and then merges the envelope... | Pointwise maximum |
c_3vxsakpaijly | The upper and lower envelopes of Lipschitz functions preserve the property of being Lipschitz. However, the lower and upper envelope operations do not necessarily preserve the property of being a continuous function. == References == | Pointwise maximum |
c_dh4sny7eg70i | In mathematics, the lower limit topology or right half-open interval topology is a topology defined on the set R {\displaystyle \mathbb {R} } of real numbers; it is different from the standard topology on R {\displaystyle \mathbb {R} } (generated by the open intervals) and has a number of interesting properties. It is ... | Sorgenfrey line |
c_gmx0063q6gfl | Like the Cantor set and the long line, the Sorgenfrey line often serves as a useful counterexample to many otherwise plausible-sounding conjectures in general topology. The product of R l {\displaystyle \mathbb {R} _{l}} with itself is also a useful counterexample, known as the Sorgenfrey plane. In complete analogy, on... | Sorgenfrey line |
c_898l2qdzmxc3 | In mathematics, the lowest common denominator or least common denominator (abbreviated LCD) is the lowest common multiple of the denominators of a set of fractions. It simplifies adding, subtracting, and comparing fractions. | Common denominator |
c_qu8gosag67ds | In mathematics, the magnitude or size of a mathematical object is a property which determines whether the object is larger or smaller than other objects of the same kind. More formally, an object's magnitude is the displayed result of an ordering (or ranking) of the class of objects to which it belongs. In physics, mag... | Magnitude (mathematics) |
c_c02cn1z5bwan | In mathematics, the main conjecture of Iwasawa theory is a deep relationship between p-adic L-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by Mazur and Wiles (1984). The Herbrand–Ribet theorem and the Gra... | Iwasawa main conjecture |
c_tftcu16f7rq7 | In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952). | Representation theory of SL2(R) |
c_o8jk6gvd3v0z | In mathematics, the mandelbox is a fractal with a boxlike shape found by Tom Lowe in 2010. It is defined in a similar way to the famous Mandelbrot set as the values of a parameter such that the origin does not escape to infinity under iteration of certain geometrical transformations. The mandelbox is defined as a map o... | Mandelbox |
c_94eus2vyq505 | In mathematics, the map segmentation problem is a kind of optimization problem. It involves a certain geographic region that has to be partitioned into smaller sub-regions in order to achieve a certain goal. Typical optimization objectives include: Minimizing the workload of a fleet of vehicles assigned to the sub-regi... | Map segmentation |
c_fg02y4gyiu92 | In mathematics, the mapping torus in topology of a homeomorphism f of some topological space X to itself is a particular geometric construction with f. Take the cartesian product of X with a closed interval I, and glue the boundary components together by the static homeomorphism: M f = ( I × X ) ( 1 , x ) ∼ ( 0 , f ( x... | Mapping torus |
c_6oc128hpa1j2 | In mathematics, the master stability function is a tool used to analyse the stability of the synchronous state in a dynamical system consisting of many identical oscillators which are coupled together, such as the Kuramoto model. The setting is as follows. Consider a system with N {\displaystyle N} identical oscillator... | Master stability function |
c_z6649jg0bdls | A synchronous state of the system of oscillators is where all the oscillators are in the same state. The coupling is defined by a coupling strength σ {\displaystyle \sigma } , a matrix A i j {\displaystyle A_{ij}} which describes how the oscillators are coupled together, and a function g {\displaystyle g} of the state ... | Master stability function |
c_dzxpq96zztrd | {\displaystyle {\dot {x}}_{i}=f(x_{i})+\sigma \sum _{j=1}^{N}A_{ij}g(x_{j}).} It is assumed that the row sums ∑ j A i j {\displaystyle \sum _{j}A_{ij}} vanish so that the manifold of synchronous states is neutrally stable. The master stability function is now defined as the function which maps the complex number γ {\di... | Master stability function |
c_c2ysko4xhmr3 | In mathematics, the matching distance is a metric on the space of size functions. The core of the definition of matching distance is the observation that the information contained in a size function can be combinatorially stored in a formal series of lines and points of the plane, called respectively cornerlines and co... | Matching distance |
c_ydx8qtw9hjhb | ℓ 2 {\displaystyle \ell _{2}} ) counted with their multiplicities, augmented by adding a countable infinity of points of the diagonal { ( x , y ) ∈ R 2: x = y } {\displaystyle \{(x,y)\in \mathbb {R} ^{2}:x=y\}} . The matching distance between ℓ 1 {\displaystyle \ell _{1}} and ℓ 2 {\displaystyle \ell _{2}} is given by d... | Matching distance |
c_ba3ooijk4my4 | {\displaystyle \delta \left((x,y),(x',y')\right)=\min \left\{\max\{|x-x'|,|y-y'|\},\max \left\{{\frac {y-x}{2}},{\frac {y'-x'}{2}}\right\}\right\}.} Roughly speaking, the matching distance d match {\displaystyle d_{\text{match}}} between two size functions is the minimum, over all the matchings between the cornerpoints... | Matching distance |
c_s8gzxzybnkik | In mathematics, the mathematician Sophus Lie ( LEE) initiated lines of study involving integration of differential equations, transformation groups, and contact of spheres that have come to be called Lie theory. For instance, the latter subject is Lie sphere geometry. This article addresses his approach to transformati... | Lie theory |
c_jhd4ibqehk44 | The subject is part of differential geometry since Lie groups are differentiable manifolds. Lie groups evolve out of the identity (1) and the tangent vectors to one-parameter subgroups generate the Lie algebra. The structure of a Lie group is implicit in its algebra, and the structure of the Lie algebra is expressed by... | Lie theory |
c_gycqtxzxaoze | In mathematics, the matrix ( 1 − 1 1 1 ) {\displaystyle {\begin{pmatrix}1&-1\\1&1\end{pmatrix}}} is sometimes called the quincunx matrix. It is a 2×2 Hadamard matrix, and its rows form the basis of a diagonal square lattice consisting of the integer points whose coordinates both have the same parity; this lattice is a ... | Quincunx matrix |
c_1ccenyjwz3l5 | In mathematics, the matrix exponential is a matrix function on square matrices analogous to the ordinary exponential function. It is used to solve systems of linear differential equations. In the theory of Lie groups, the matrix exponential gives the exponential map between a matrix Lie algebra and the corresponding Li... | Exponential of a matrix |
c_kcb2dn8sfajv | Let X be an n×n real or complex matrix. The exponential of X, denoted by eX or exp(X), is the n×n matrix given by the power series where X 0 {\displaystyle X^{0}} is defined to be the identity matrix I {\displaystyle I} with the same dimensions as X {\displaystyle X} . The series always converges, so the exponential of... | Exponential of a matrix |
c_fydx2ol4h0h6 | In mathematics, the matrix representation of conic sections permits the tools of linear algebra to be used in the study of conic sections. It provides easy ways to calculate a conic section's axis, vertices, tangents and the pole and polar relationship between points and lines of the plane determined by the conic. The ... | Matrix representation of conic sections |
c_m3j3131jcod6 | This equation can be written in matrix notation, in terms of a symmetric matrix to simplify some subsequent formulae, as The sum of the first three terms of this equation, namely is the quadratic form associated with the equation, and the matrix is called the matrix of the quadratic form. The trace and determinant of A... | Matrix representation of conic sections |
c_47xb1fxf4wsm | In mathematics, the matrix sign function is a matrix function on square matrices analogous to the complex sign function.It was introduced by J.D. Roberts in 1971 as a tool for model reduction and for solving Lyapunov and Algebraic Riccatia equation in a technical report of Cambridge University, which was later publishe... | Matrix sign function |
c_ou2njmr848as | In mathematics, the maximum modulus principle in complex analysis states that if f {\displaystyle f} is a holomorphic function, then the modulus | f | {\displaystyle |f|} cannot exhibit a strict local maximum that is properly within the domain of f {\displaystyle f} . In other words, either f {\displaystyle f} is local... | Maximum modulus principle |
c_ua6uroap0a7z | In mathematics, the maximum-minimums identity is a relation between the maximum element of a set S of n numbers and the minima of the 2n − 1 non-empty subsets of S. Let S = {x1, x2, ..., xn}. The identity states that max { x 1 , x 2 , … , x n } = ∑ i = 1 n x i − ∑ i < j min { x i , x j } + ∑ i < j < k min { x i , x j ,... | Maximum-minimums identity |
c_vft0cgqnhm5l | In mathematics, the max–min inequality is as follows: For any function f: Z × W → R , {\displaystyle \ f:Z\times W\to \mathbb {R} \ ,} sup z ∈ Z inf w ∈ W f ( z , w ) ≤ inf w ∈ W sup z ∈ Z f ( z , w ) . {\displaystyle \sup _{z\in Z}\inf _{w\in W}f(z,w)\leq \inf _{w\in W}\sup _{z\in Z}f(z,w)\ .} When equality holds one ... | Max–min inequality |
c_qnu89lwxhmw3 | In mathematics, the mean (topological) dimension of a topological dynamical system is a non-negative extended real number that is a measure of the complexity of the system. Mean dimension was first introduced in 1999 by Gromov. Shortly after it was developed and studied systematically by Lindenstrauss and Weiss. In par... | Mean dimension |
c_vw4902psufcj | For various topological dynamical systems with infinite topological entropy, the mean dimension can be calculated or at least bounded from below and above. This allows mean dimension to be used to distinguish between systems with infinite topological entropy. Mean dimension is also related to the problem of embedding t... | Mean dimension |
c_rr4fkw06483s | In mathematics, the mean curvature H {\displaystyle H} of a surface S {\displaystyle S} is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her ... | Mean curvature |
c_rdkl8nsgg0j7 | In mathematics, the mean value problem was posed by Stephen Smale in 1981. This problem is still open in full generality. The problem asks: For a given complex polynomial f {\displaystyle f} of degree d ≥ 2 {\displaystyle d\geq 2} A and a complex number z {\displaystyle z} , is there a critical point c {\displaystyle c... | Mean value problem |
c_y426eixjmb97 | In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It is one of the most important results in real analysis. This theorem is used to ... | Mean-value theorem |
c_yplqegtiylom | In mathematics, the measurable Riemann mapping theorem is a theorem proved in 1960 by Lars Ahlfors and Lipman Bers in complex analysis and geometric function theory. Contrary to its name, it is not a direct generalization of the Riemann mapping theorem, but instead a result concerning quasiconformal mappings and soluti... | Measurable Riemann mapping theorem |
c_3ntokn5mwbn6 | In mathematics, the mediant of two fractions, generally made up of four positive integers a c {\displaystyle {\frac {a}{c}}\quad } and b d {\displaystyle \quad {\frac {b}{d}}\quad } is defined as a + b c + d . {\displaystyle \quad {\frac {a+b}{c+d}}.} That is to say, the numerator and denominator of the mediant are the... | Mediant (mathematics) |
c_d0cuzf75s93o | It is sometimes called the freshman sum, as it is a common mistake in the early stages of learning about addition of fractions. Technically, this is a binary operation on valid fractions (nonzero denominator), considered as ordered pairs of appropriate integers, a priori disregarding the perspective on rational numbers... | Mediant (mathematics) |
c_q3rsk9q3eg1b | However, if the fraction 1/1 is replaced by the fraction 2/2, which is an equivalent fraction denoting the same rational number 1, the mediant of the fractions 2/2 and 1/2 is 3/4. For a stronger connection to rational numbers the fractions may be required to be reduced to lowest terms, thereby selecting unique represen... | Mediant (mathematics) |
c_iho0av09t1tw | In mathematics, the membership function of a fuzzy set is a generalization of the indicator function for classical sets. In fuzzy logic, it represents the degree of truth as an extension of valuation. Degrees of truth are often confused with probabilities, although they are conceptually distinct, because fuzzy truth re... | Fuzzy membership |
c_8j8oox5nzlb5 | In mathematics, the metaplectic group Mp2n is a double cover of the symplectic group Sp2n. It can be defined over either real or p-adic numbers. The construction covers more generally the case of an arbitrary local or finite field, and even the ring of adeles. The metaplectic group has a particularly significant infini... | Metaplectic group |
c_b0z5cmxquwpk | In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a second-order ordinary differential equation of the form with u ′ ≡ d u d z {\textstyle u'\equiv {\frac {du}{dz}}} and u ″ ≡ d 2 u d z 2 {\textstyle u''\equiv {\frac {d^{2}u}{dz^{2}}}} . in ... | Indicial equation |
c_6162rdsmn1lj | In mathematics, the method of characteristics is a technique for solving partial differential equations. Typically, it applies to first-order equations, although more generally the method of characteristics is valid for any hyperbolic partial differential equation. The method is to reduce a partial differential equatio... | Charpit method |
c_3v84gz3wui14 | In mathematics, the method of clearing denominators, also called clearing fractions, is a technique for simplifying an equation equating two expressions that each are a sum of rational expressions – which includes simple fractions. | Clearing denominators |
c_o7lra65upy34 | In mathematics, the method of descent is the term coined by the French mathematician Jacques Hadamard as a method for solving a partial differential equation in several real or complex variables, by regarding it as the specialisation of an equation in more variables, constant in the extra parameters. This method has be... | Hadamard's method of descent |
c_ve6n6059k5ld | In mathematics, the method of dominant balance is used to determine the asymptotic behavior of solutions to an ordinary differential equation without fully solving the equation. The process is iterative, in that the result obtained by performing the method once can be used as input when the method is repeated, to obtai... | Method of dominant balance |
c_4cu3tgx3kecc | Drop these terms and solve the resulting simpler ODE. Check that the solution is consistent with step 2. If this is the case, then one has the controlling factor of the asymptotic behavior; otherwise, one needs try dropping different terms in step 2, instead. Repeat the process to higher orders, relying on the above re... | Method of dominant balance |
c_60ctrbhul766 | In mathematics, the method of equating the coefficients is a way of solving a functional equation of two expressions such as polynomials for a number of unknown parameters. It relies on the fact that two expressions are identical precisely when corresponding coefficients are equal for each different type of term. The m... | Equating coefficients |
c_4vq33s0iuved | In mathematics, the method of matched asymptotic expansions is a common approach to finding an accurate approximation to the solution to an equation, or system of equations. It is particularly used when solving singularly perturbed differential equations. It involves finding several different approximate solutions, eac... | Method of matched asymptotic expansions |
c_nfcw75jg5ked | In mathematics, the method of steepest descent or saddle-point method is an extension of Laplace's method for approximating an integral, where one deforms a contour integral in the complex plane to pass near a stationary point (saddle point), in roughly the direction of steepest descent or stationary phase. The saddle-... | Saddle-point method |
c_1ybns5xewplq | One version of the method of steepest descent deforms the contour of integration C into a new path integration C′ so that the following conditions hold: C′ passes through one or more zeros of the derivative g′(z), the imaginary part of g(z) is constant on C′.The method of steepest descent was first published by Debye (... | Saddle-point method |
c_6kq0rkhxeg7b | In mathematics, the method of undetermined coefficients is an approach to finding a particular solution to certain nonhomogeneous ordinary differential equations and recurrence relations. It is closely related to the annihilator method, but instead of using a particular kind of differential operator (the annihilator) i... | Method of undetermined coefficients |
c_hkgk09i4vtfy | In mathematics, the metric derivative is a notion of derivative appropriate to parametrized paths in metric spaces. It generalizes the notion of "speed" or "absolute velocity" to spaces which have a notion of distance (i.e. metric spaces) but not direction (such as vector spaces). | Metric derivative |
c_35ndfw0q2abi | In mathematics, the mex ("minimum excluded value") of a subset of a well-ordered set is the smallest value from the whole set that does not belong to the subset. That is, it is the minimum value of the complement set. Beyond sets, subclasses of well-ordered classes have minimum excluded values. | Mex (mathematics) |
c_iqnfhue1cexx | Minimum excluded values of subclasses of the ordinal numbers are used in combinatorial game theory to assign nim-values to impartial games. According to the Sprague–Grundy theorem, the nim-value of a game position is the minimum excluded value of the class of values of the positions that can be reached in a single move... | Mex (mathematics) |
c_hgit63lwwve4 | In mathematics, the mice problem is a continuous pursuit–evasion problem in which a number of mice (or insects, dogs, missiles, etc.) are considered to be placed at the corners of a regular polygon. In the classic setup, each then begins to move towards its immediate neighbour (clockwise or anticlockwise). The goal is ... | Mice problem |
c_eh39hbt36xqd | The most common version has the mice starting at the corners of a unit square, moving at unit speed. In this case they meet after a time of one unit, because the distance between two neighboring mice always decreases at a speed of one unit. More generally, for a regular polygon of n {\displaystyle n} unit-length sides,... | Mice problem |
c_1bzpuj975zo5 | In mathematics, the minimum k-cut is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges are referred to as k-cut. The goal is to find the minimum-weight k-cut. This partitioning can have applications in VLSI d... | Minimum k-cut |
c_0xbkx0arphx5 | In mathematics, the minimum rank is a graph parameter mr ( G ) {\displaystyle \operatorname {mr} (G)} for a graph G. It was motivated by the Colin de Verdière graph invariant. | Minimum rank of a graph |
c_41anh9ixziqy | In mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties can be derived from their definitions, as linear maps or more generally; and the rules for manipulations of tensors arise as an extens... | Tensor (intrinsic definition) |
c_4d8zbxfdm420 | The same is true in general relativity, of tensor fields describing a physical property. The component-free approach is also used extensively in abstract algebra and homological algebra, where tensors arise naturally. Note: This article assumes an understanding of the tensor product of vector spaces without chosen base... | Tensor (intrinsic definition) |
c_tz5sz69agegx | In mathematics, the modular group is the projective special linear group PSL ( 2 , Z ) {\textstyle \operatorname {PSL} (2,\mathbb {Z} )} of 2 × 2 matrices with integer coefficients and determinant 1. The matrices A and −A are identified. The modular group acts on the upper-half of the complex plane by fractional line... | Hecke group |
c_iga2c20yex56 | In mathematics, the modular lambda function λ(τ) is a highly symmetric Holomorphic function on the complex upper half-plane. It is invariant under the fractional linear action of the congruence group Γ(2), and generates the function field of the corresponding quotient, i.e., it is a Hauptmodul for the modular curve X(2... | Elliptic modulus |
c_pbrrowym6z3j | In mathematics, the moduli stack of elliptic curves, denoted as M 1 , 1 {\displaystyle {\mathcal {M}}_{1,1}} or M ell {\displaystyle {\mathcal {M}}_{\textrm {ell}}} , is an algebraic stack over Spec ( Z ) {\displaystyle {\text{Spec}}(\mathbb {Z} )} classifying elliptic curves. Note that it is a special case of the modu... | Moduli stack of elliptic curves |
c_yrllbx27ko0b | In mathematics, the modulus of convexity and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same relationship to the ε-δ definition of uniform convexity as the modulus of continuity does to the ε-δ definition of continuity... | Modulus and characteristic of convexity |
c_q2huj4gx2qge | In mathematics, the moments of a function are certain quantitative measures related to the shape of the function's graph. If the function represents mass density, then the zeroth moment is the total mass, the first moment (normalized by total mass) is the center of mass, and the second moment is the moment of inertia. ... | Moment (statistics) |
c_fk4zbac8i4x8 | For a distribution of mass or probability on a bounded interval, the collection of all the moments (of all orders, from 0 to ∞) uniquely determines the distribution (Hausdorff moment problem). The same is not true on unbounded intervals (Hamburger moment problem). In the mid-nineteenth century, Pafnuty Chebyshev became... | Moment (statistics) |
c_knsngrldlpv7 | In mathematics, the monkey saddle is the surface defined by the equation z = x 3 − 3 x y 2 , {\displaystyle z=x^{3}-3xy^{2},\,} or in cylindrical coordinates z = ρ 3 cos ( 3 φ ) . {\displaystyle z=\rho ^{3}\cos(3\varphi ).} It belongs to the class of saddle surfaces, and its name derives from the observation that a s... | Monkey saddle |
c_zjxiwfb3fgce | The point ( 0 , 0 , 0 ) {\displaystyle (0,0,0)} on the monkey saddle corresponds to a degenerate critical point of the function z ( x , y ) {\displaystyle z(x,y)} at ( 0 , 0 ) {\displaystyle (0,0)} . The monkey saddle has an isolated umbilical point with zero Gaussian curvature at the origin, while the curvature is str... | Monkey saddle |
c_q3s94j44p9sa | {\displaystyle z=x^{3}-3xy^{2}=\operatorname {Re} =\operatorname {Re} =r^{3}\cos(3\varphi ).} By replacing 3 in the cylindrical equation with any integer k ≥ 1 , {\displaystyle k\geq 1,} one can create a saddle with k {\displaystyle k} depressions. Another orientation of the monkey saddle is the Smelt petal defined by ... | Monkey saddle |
c_gredun22kffb | In mathematics, the monopole moduli space is a space parametrizing monopoles (solutions of the Bogomolny equations). Atiyah and Hitchin (1988) studied the moduli space for 2 monopoles in detail and used it to describe the scattering of monopoles. | Monopole moduli space |
c_g1jtlqmx17g0 | In mathematics, the monster Lie algebra is an infinite-dimensional generalized Kac–Moody algebra acted on by the monster group, which was used to prove the monstrous moonshine conjectures. | Monster Lie algebra |
c_zunu236dl9fc | In mathematics, the mountain climbing problem is a mathematical problem that considers a two-dimensional mountain range (represented as a continuous function), and asks whether it is possible for two mountain climbers starting at sea level on the left and right sides of the mountain to meet at the summit, while maintai... | Mountain climbing problem |
c_exfaoozwidlh | In mathematics, the moving sofa problem or sofa problem is a two-dimensional idealisation of real-life furniture-moving problems and asks for the rigid two-dimensional shape of largest area that can be maneuvered through an L-shaped planar region with legs of unit width. The area thus obtained is referred to as the sof... | Moving sofa problem |
c_hqna2qn4751e | In mathematics, the multicomplex number systems C n {\displaystyle \mathbb {C} _{n}} are defined inductively as follows: Let C0 be the real number system. For every n > 0 let in be a square root of −1, that is, an imaginary unit. Then C n + 1 = { z = x + y i n + 1: x , y ∈ C n } {\displaystyle \mathbb {C} _{n+1}=\lbrac... | Multicomplex number |
c_h74bmr35n5tr | In the multicomplex number systems one also requires that i n i m = i m i n {\displaystyle i_{n}i_{m}=i_{m}i_{n}} (commutativity). Then C 1 {\displaystyle \mathbb {C} _{1}} is the complex number system, C 2 {\displaystyle \mathbb {C} _{2}} is the bicomplex number system, C 3 {\displaystyle \mathbb {C} _{3}} is the tric... | Multicomplex number |
c_34nbjfwi9lms | {\displaystyle \mathbb {C} _{n}.} The multicomplex number systems are not to be confused with Clifford numbers (elements of a Clifford algebra), since Clifford's square roots of −1 anti-commute ( i n i m + i m i n = 0 {\displaystyle i_{n}i_{m}+i_{m}i_{n}=0} when m ≠ n for Clifford). Because the multicomplex numbers hav... | Multicomplex number |
c_l9o15g4u9971 | Any product i n i m {\displaystyle i_{n}i_{m}} of two distinct multicomplex units behaves as the j {\displaystyle j} of the split-complex numbers, and therefore the multicomplex numbers contain a number of copies of the split-complex number plane. With respect to subalgebra C k {\displaystyle \mathbb {C} _{k}} , k = 0,... | Multicomplex number |
c_q5hfrfx68mwn | In mathematics, the multinomial theorem describes how to expand a power of a sum in terms of powers of the terms in that sum. It is the generalization of the binomial theorem from binomials to multinomials. | Multinomial formula |
c_szmwfna1lqfy | In mathematics, the multiple gamma function Γ N {\displaystyle \Gamma _{N}} is a generalization of the Euler gamma function and the Barnes G-function. The double gamma function was studied by Barnes (1901). At the end of this paper he mentioned the existence of multiple gamma functions generalizing it, and studied thes... | Multiple gamma function |
c_w4j4ls990lso | In mathematics, the multiple orthogonal polynomials (MOPs) are orthogonal polynomials in one variable that are orthogonal with respect to a finite family of measures. The polynomials are divided into two classes named type 1 and type 2.In the literature, MOPs are also called d {\displaystyle d} -orthogonal polynomials,... | Multiple orthogonal polynomials |
c_p8o1di0s9uf5 | In mathematics, the multiple zeta functions are generalizations of the Riemann zeta function, defined by ζ ( s 1 , … , s k ) = ∑ n 1 > n 2 > ⋯ > n k > 0 1 n 1 s 1 ⋯ n k s k = ∑ n 1 > n 2 > ⋯ > n k > 0 ∏ i = 1 k 1 n i s i , {\displaystyle \zeta (s_{1},\ldots ,s_{k})=\sum _{n_{1}>n_{2}>\cdots >n_{k}>0}\ {\frac {1}{n_{1}^... | Multiple zeta values |
c_6nxi16vyxaef | In mathematics, the multiplication theorem is a certain type of identity obeyed by many special functions related to the gamma function. For the explicit case of the gamma function, the identity is a product of values; thus the name. The various relations all stem from the same underlying principle; that is, the relati... | Multiplication theorem |
c_yxfid7pvi32u | In mathematics, the multiplicative ergodic theorem, or Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets (also spelled "Oseledec") in 1965 and reported at the International Mathematical Congress in Moscow in 196... | Oseledets theorem |
c_dcfgpa0ckear | In mathematics, the multiplicative semigroup, denoted by W0, generated by the set { 3 n + 2 2 n + 1: n ≥ 0 } {\displaystyle \left\{{\frac {3n+2}{2n+1}}:n\geq 0\right\}} is called the Wooley semigroup in honour of the American mathematician Trevor D. Wooley. The multiplicative semigroup, denoted by W, generated by the s... | Wild number |
c_6wyn729b3lz5 | It is called the Wooley integer semigroup and members of this semigroup are called Wooley integers. Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers. | Wild number |
c_f872kbjs14be | In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root. The notion of multiplicity is important to be able to count correctly without specifying excepti... | Multiple roots of a polynomial |
c_odpss27epyus | Hence the expression, "counted with multiplicity". If multiplicity is ignored, this may be emphasized by counting the number of distinct elements, as in "the number of distinct roots". However, whenever a set (as opposed to multiset) is formed, multiplicity is automatically ignored, without requiring use of the term "d... | Multiple roots of a polynomial |
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