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wiki_1200_chunk_0 | Traveling plane wave | This plane too travels along the direction of propagation n → {\displaystyle {\vec {n}}} with velocity c {\displaystyle c} ; and the value of the field is then the same, and constant in time, at every one of its points. The wave F {\displaystyle F} may be a scalar or vector field; its values are the values of G {\displ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1201_chunk_0 | Vector space | In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called vectors, may be added together and multiplied ("scaled") by numbers called scalars. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1202_chunk_0 | Vector space | The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a di... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1203_chunk_0 | Vector space | This provides a concise and synthetic way for manipulating and studying systems of linear equations. Vector spaces are characterized by their dimension, which, roughly speaking, specifies the number of independent directions in the space. This means that, for two vector spaces over a given field and with the same dimen... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1204_chunk_0 | Vector space | A vector space is finite-dimensional if its dimension is a natural number. Otherwise, it is infinite-dimensional, and its dimension is an infinite cardinal. Finite-dimensional vector spaces occur naturally in geometry and related areas. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1205_chunk_0 | Vector space | Infinite-dimensional vector spaces occur in many areas of mathematics. For example, polynomial rings are countably infinite-dimensional vector spaces, and many function spaces have the cardinality of the continuum as a dimension. Many vector spaces that are considered in mathematics are also endowed with other structur... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1206_chunk_0 | Acceleration vector | In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of the "bend". | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1207_chunk_0 | Equipotential | In mathematics and physics, an equipotential or isopotential refers to a region in space where every point is at the same potential. This usually refers to a scalar potential (in that case it is a level set of the potential), although it can also be applied to vector potentials. An equipotential of a scalar potential f... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1208_chunk_0 | Equipotential | An equipotential region might be referred as being 'of equipotential' or simply be called 'an equipotential'. An equipotential region of a scalar potential in three-dimensional space is often an equipotential surface (or potential isosurface), but it can also be a three-dimensional mathematical solid in space. The grad... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1209_chunk_0 | Equipotential | Extending this definition, an isopotential is the locus of all points that are of the same potential. Gravity is perpendicular to the equipotential surfaces of the gravity potential, and in electrostatics and steady electric currents, the electric field (and hence the current, if any) is perpendicular to the equipotent... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1210_chunk_0 | Equipotential | In electrostatics, a conductor is a three-dimensional equipotential region. In the case of a hollow conductor (Faraday cage), the equipotential region includes the space inside. A ball will not be accelerated left or right by the force of gravity if it is resting on a flat, horizontal surface, because it is an equipote... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1211_chunk_0 | D'Alembert–Euler condition | In mathematics and physics, especially the study of mechanics and fluid dynamics, the d'Alembert-Euler condition is a requirement that the streaklines of a flow are irrotational. Let x = x(X,t) be the coordinates of the point x into which X is carried at time t by a (fluid) flow. Let x ¨ = D 2 x D t {\displaystyle {\dd... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1212_chunk_0 | Asymptotic homogenization | In mathematics and physics, homogenization is a method of studying partial differential equations with rapidly oscillating coefficients, such as ∇ ⋅ ( A ( x → ϵ ) ∇ u ϵ ) = f {\displaystyle \nabla \cdot \left(A\left({\frac {\vec {x}}{\epsilon }}\right)\nabla u_{\epsilon }\right)=f} where ϵ {\displaystyle \epsilon } is ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1213_chunk_0 | Asymptotic homogenization | Under this assumption, materials such as fluids, solids, etc. can be treated as homogeneous materials and associated with these materials are material properties such as shear modulus, elastic moduli, etc. Frequently, inhomogeneous materials (such as composite materials) possess microstructure and therefore they are su... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1214_chunk_0 | Asymptotic homogenization | {\displaystyle \nabla _{y}\cdot \left(A({\vec {y}})\nabla w_{j}\right)=-\nabla _{y}\cdot \left(A({\vec {y}}){\vec {e}}_{j}\right).} This process of replacing an equation with a highly oscillatory coefficient with one with a homogeneous (uniform) coefficient is known as homogenization. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1215_chunk_0 | Asymptotic homogenization | This subject is inextricably linked with the subject of micromechanics for this very reason. In homogenization one equation is replaced by another if u ϵ ≈ u {\displaystyle u_{\epsilon }\approx u} for small enough ϵ {\displaystyle \epsilon } , provided u ϵ → u {\displaystyle u_{\epsilon }\to u} in some appropriate norm... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1216_chunk_0 | Asymptotic homogenization | Classical results of homogenization theory were obtained for media with periodic microstructure modeled by partial differential equations with periodic coefficients. These results were later generalized to spatially homogeneous random media modeled by differential equations with random coefficients which statistical pr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1217_chunk_0 | Warped model | In mathematics and physics, in particular differential geometry and general relativity, a warped geometry is a Riemannian or Lorentzian manifold whose metric tensor can be written in form d s 2 = g a b ( y ) d y a d y b + f ( y ) g i j ( x ) d x i d x j . {\displaystyle ds^{2}=g_{ab}(y)\,dy^{a}\,dy^{b}+f(y)g_{ij}(x)\,d... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1218_chunk_0 | Euler's Equation | In mathematics and physics, many topics are named in honor of Swiss mathematician Leonhard Euler (1707–1783), who made many important discoveries and innovations. Many of these items named after Euler include their own unique function, equation, formula, identity, number (single or sequence), or other mathematical enti... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1219_chunk_0 | Multiple-scale analysis | In mathematics and physics, multiple-scale analysis (also called the method of multiple scales) comprises techniques used to construct uniformly valid approximations to the solutions of perturbation problems, both for small as well as large values of the independent variables. This is done by introducing fast-scale and... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1220_chunk_0 | Anti-de Sitter space | Einstein's general theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space and time separately. The cases of spacetime of constant curvature are de Sitter space (positive), Minkowski space (zero), and anti-de Sitter space (negat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1221_chunk_0 | Toda field theory | In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1222_chunk_0 | Super Minkowski space | In mathematics and physics, super Minkowski space or Minkowski superspace is a supersymmetric extension of Minkowski space, sometimes used as the base manifold (or rather, supermanifold) for superfields. It is acted on by the super Poincaré algebra. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1223_chunk_0 | Kinetic Monte Carlo surface growth method | In mathematics and physics, surface growth refers to models used in the dynamical study of the growth of a surface, usually by means of a stochastic differential equation of a field. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1224_chunk_0 | Artin billiard | In mathematics and physics, the Artin billiard is a type of a dynamical billiard first studied by Emil Artin in 1924. It describes the geodesic motion of a free particle on the non-compact Riemann surface H / Γ , {\displaystyle \mathbb {H} /\Gamma ,} where H {\displaystyle \mathbb {H} } is the upper half-plane endowed ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1225_chunk_0 | Artin billiard | The quantum mechanical version of Artin's billiard is also exactly solvable. The eigenvalue spectrum consists of a bound state and a continuous spectrum above the energy E = 1 / 4 {\displaystyle E=1/4} . The wave functions are given by Bessel functions. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1226_chunk_0 | Kadomtsev–Petviashvili equation | In mathematics and physics, the Kadomtsev–Petviashvili equation (often abbreviated as KP equation) is a partial differential equation to describe nonlinear wave motion. Named after Boris Borisovich Kadomtsev and Vladimir Iosifovich Petviashvili, the KP equation is usually written as where λ = ± 1 {\displaystyle \lambda... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1227_chunk_0 | Kadomtsev–Petviashvili equation | Like the KdV equation, the KP equation is completely integrable. It can also be solved using the inverse scattering transform much like the nonlinear Schrödinger equation.In 2002, the regularized version of the KP equation, naturally referred to as the Benjamin–Bona–Mahony–Kadomtsev–Petviashvili equation (or simply the... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1228_chunk_0 | Magnus expansion | In mathematics and physics, the Magnus expansion, named after Wilhelm Magnus (1907–1990), provides an exponential representation of the solution of a first-order homogeneous linear differential equation for a linear operator. In particular, it furnishes the fundamental matrix of a system of linear ordinary differential... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1229_chunk_0 | Poincaré recurrence theorem | In mathematics and physics, the Poincaré recurrence theorem states that certain dynamical systems will, after a sufficiently long but finite time, return to a state arbitrarily close to (for continuous state systems), or exactly the same as (for discrete state systems), their initial state. The Poincaré recurrence time... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1230_chunk_0 | Poincaré recurrence theorem | The theorem is commonly discussed in the context of ergodic theory, dynamical systems and statistical mechanics. Systems to which the Poincaré recurrence theorem applies are called conservative systems. The theorem is named after Henri Poincaré, who discussed it in 1890 and proved by Constantin Carathéodory using measu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1231_chunk_0 | Geometric center | In mathematics and physics, the centroid, also known as geometric center or center of figure, of a plane figure or solid figure is the arithmetic mean position of all the points in the surface of the figure. The same definition extends to any object in n-dimensional Euclidean space.In geometry, one often assumes unifor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1232_chunk_0 | Diamagnetic inequality | In mathematics and physics, the diamagnetic inequality relates the Sobolev norm of the absolute value of a section of a line bundle to its covariant derivative. The diamagnetic inequality has an important physical interpretation, that a charged particle in a magnetic field has more energy in its ground state than it wo... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1233_chunk_0 | Heat diffusion | In mathematics and physics, the heat equation is a certain partial differential equation. Solutions of the heat equation are sometimes known as caloric functions. The theory of the heat equation was first developed by Joseph Fourier in 1822 for the purpose of modeling how a quantity such as heat diffuses through a give... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1234_chunk_0 | Heat diffusion | The heat equation can also be considered on Riemannian manifolds, leading to many geometric applications. Following work of Subbaramiah Minakshisundaram and Åke Pleijel, the heat equation is closely related with spectral geometry. A seminal nonlinear variant of the heat equation was introduced to differential geometry ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1235_chunk_0 | Heat diffusion | Certain solutions of the heat equation known as heat kernels provide subtle information about the region on which they are defined, as exemplified through their application to the Atiyah–Singer index theorem.The heat equation, along with variants thereof, is also important in many fields of science and applied mathemat... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1236_chunk_0 | Heat diffusion | In image analysis, the heat equation is sometimes used to resolve pixelation and to identify edges. Following Robert Richtmyer and John von Neumann's introduction of "artificial viscosity" methods, solutions of heat equations have been useful in the mathematical formulation of hydrodynamical shocks. Solutions of the he... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1237_chunk_0 | Inverse scattering | The nonlinear Schrödinger equation, the Korteweg–de Vries equation and the KP equation are examples of soliton equations. In one space dimension the inverse scattering problem is equivalent to a Riemann-Hilbert problem. Since its early statement for radiolocation, many applications have been found for inverse scatterin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1238_chunk_0 | Orientation entanglement | In mathematics and physics, the notion of orientation entanglement is sometimes used to develop intuition relating to the geometry of spinors or alternatively as a concrete realization of the failure of the special orthogonal groups to be simply connected. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1239_chunk_0 | Plate trick | In mathematics and physics, the plate trick, also known as Dirac's string trick, the belt trick, or the Balinese cup trick, is any of several demonstrations of the idea that rotating an object with strings attached to it by 360 degrees does not return the system to its original state, while a second rotation of 360 deg... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1240_chunk_0 | Right hand grip rule | The rule can be used to find the direction of the magnetic field, rotation, spirals, electromagnetic fields, mirror images, and enantiomers in mathematics and chemistry. The sequence is often: index finger along the first vector, then middle finger along the second, then thumb along the third. Two other sequences also ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1241_chunk_0 | Spectral asymmetry | In mathematics and physics, the spectral asymmetry is the asymmetry in the distribution of the spectrum of eigenvalues of an operator. In mathematics, the spectral asymmetry arises in the study of elliptic operators on compact manifolds, and is given a deep meaning by the Atiyah-Singer index theorem. In physics, it has... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1242_chunk_0 | Vector notation | In mathematics and physics, vector notation is a commonly used notation for representing vectors, which may be Euclidean vectors, or more generally, members of a vector space. For representing a vector, the common typographic convention is lower case, upright boldface type, as in v. The International Organization for S... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1243_chunk_0 | Skorokhod's embedding theorem | In mathematics and probability theory, Skorokhod's embedding theorem is either or both of two theorems that allow one to regard any suitable collection of random variables as a Wiener process (Brownian motion) evaluated at a collection of stopping times. Both results are named for the Ukrainian mathematician A. V. Skor... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1244_chunk_0 | Continuum percolation theory | As in discrete percolation, a common research focus of continuum percolation is studying the conditions of occurrence for infinite or giant components. Other shared concepts and analysis techniques exist in these two types of percolation theory as well as the study of random graphs and random geometric graphs. Continuu... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1245_chunk_0 | Borell–TIS inequality | In mathematics and probability, the Borell–TIS inequality is a result bounding the probability of a deviation of the uniform norm of a centered Gaussian stochastic process above its expected value. The result is named for Christer Borell and its independent discoverers Boris Tsirelson, Ildar Ibragimov, and Vladimir Sud... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1246_chunk_0 | Harmonic spinor | In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formally an operator for Minkowski space, to get a form of quantum theory compati... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1247_chunk_0 | Nonlinear differential equation | In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other scientists since most systems are inherently nonli... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1248_chunk_0 | Nonlinear differential equation | In other words, in a nonlinear system of equations, the equation(s) to be solved cannot be written as a linear combination of the unknown variables or functions that appear in them. Systems can be defined as nonlinear, regardless of whether known linear functions appear in the equations. In particular, a differential e... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1249_chunk_0 | Nonlinear differential equation | As nonlinear dynamical equations are difficult to solve, nonlinear systems are commonly approximated by linear equations (linearization). This works well up to some accuracy and some range for the input values, but some interesting phenomena such as solitons, chaos, and singularities are hidden by linearization. It fol... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1250_chunk_0 | Nonlinear differential equation | Although such chaotic behavior may resemble random behavior, it is in fact not random. For example, some aspects of the weather are seen to be chaotic, where simple changes in one part of the system produce complex effects throughout. This nonlinearity is one of the reasons why accurate long-term forecasts are impossib... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1251_chunk_0 | Hereditarily finite set | In mathematics and set theory, hereditarily finite sets are defined as finite sets whose elements are all hereditarily finite sets. In other words, the set itself is finite, and all of its elements are finite sets, recursively all the way down to the empty set. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1252_chunk_0 | Analytic representation | In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hilbert transform. The analytic representation of a real-valued function is an analyt... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1253_chunk_0 | Analytic representation | The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-va... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1254_chunk_0 | Hilbert Transform | In mathematics and signal processing, the Hilbert transform is a specific singular integral that takes a function, u(t) of a real variable and produces another function of a real variable H(u)(t). The Hilbert transform is given by the Cauchy principal value of the convolution with the function 1 / ( π t ) {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1255_chunk_0 | Z-transform | In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (the z-domain or z-plane) representation.It can be considered as a discrete-time equivalent of the Laplace transform (the s-domain or s-plane). This simi... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1256_chunk_0 | Advanced Z-transform | In mathematics and signal processing, the advanced z-transform is an extension of the z-transform, to incorporate ideal delays that are not multiples of the sampling time. It takes the form F ( z , m ) = ∑ k = 0 ∞ f ( k T + m ) z − k {\displaystyle F(z,m)=\sum _{k=0}^{\infty }f(kT+m)z^{-k}} where T is the sampling peri... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1257_chunk_0 | Constant-Q transform | In mathematics and signal processing, the constant-Q transform and variable-Q transform, simply known as CQT and VQT, transforms a data series to the frequency domain. It is related to the Fourier transform and very closely related to the complex Morlet wavelet transform. Its design is suited for musical representation... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1258_chunk_0 | Collaboration graph | In mathematics and social science, a collaboration graph is a graph modeling some social network where the vertices represent participants of that network (usually individual people) and where two distinct participants are joined by an edge whenever there is a collaborative relationship between them of a particular kin... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1259_chunk_0 | Constant-energy surface | In mathematics and solid state physics, the first Brillouin zone (named after Léon Brillouin) is a uniquely defined primitive cell in reciprocal space. In the same way the Bravais lattice is divided up into Wigner–Seitz cells in the real lattice, the reciprocal lattice is broken up into Brillouin zones. The boundaries ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1260_chunk_0 | Constant-energy surface | The first Brillouin zone is the locus of points in reciprocal space that are closer to the origin of the reciprocal lattice than they are to any other reciprocal lattice points (see the derivation of the Wigner–Seitz cell). Another definition is as the set of points in k-space that can be reached from the origin withou... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1261_chunk_0 | Constant-energy surface | There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. In general, the n-th Brillouin zon... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1262_chunk_0 | Constant-energy surface | A related concept is that of the irreducible Brillouin zone, which is the first Brillouin zone reduced by all of the symmetries in the point group of the lattice (point group of the crystal). The concept of a Brillouin zone was developed by Léon Brillouin (1889–1969), a French physicist.Within the Brillouin zone, a con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1263_chunk_0 | Dimension of an algebraic variety | In mathematics and specifically in algebraic geometry, the dimension of an algebraic variety may be defined in various equivalent ways. Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1264_chunk_0 | Formal space | A geometric application was the theorem of Sullivan and Micheline Vigué-Poirrier (1976): every simply connected closed Riemannian manifold X whose rational cohomology ring is not generated by one element has infinitely many geometrically distinct closed geodesics. The proof used rational homotopy theory to show that th... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1265_chunk_0 | Skorokhod's representation theorem | In mathematics and statistics, Skorokhod's representation theorem is a result that shows that a weakly convergent sequence of probability measures whose limit measure is sufficiently well-behaved can be represented as the distribution/law of a pointwise convergent sequence of random variables defined on a common probab... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1266_chunk_0 | Mean of circular quantities | In mathematics and statistics, a circular mean or angular mean is a mean designed for angles and similar cyclic quantities, such as times of day, and fractional parts of real numbers. This is necessary since most of the usual means may not be appropriate on angle-like quantities. For example, the arithmetic mean of 0° ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1267_chunk_0 | Mean of circular quantities | As another example, the "average time" between 11 PM and 1 AM is either midnight or noon, depending on whether the two times are part of a single night or part of a single calendar day. The circular mean is one of the simplest examples of directional statistics and of statistics of non-Euclidean spaces. This computatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1268_chunk_0 | Fuzzy concept | In mathematics and statistics, a fuzzy variable (such as "the temperature", "hot" or "cold") is a value which could lie in a probable range defined by some quantitative limits or parameters, and which can be usefully described with imprecise categories (such as "high", "medium" or "low") using some kind of scale or con... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1269_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1270_chunk_0 | Probability 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,... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1271_chunk_0 | Discrete variables | In mathematics and statistics, a quantitative variable may be continuous or discrete if they are typically obtained by measuring or counting, respectively. If it can take on two particular real values such that it can also take on all real values between them (even values that are arbitrarily close together), the varia... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1272_chunk_0 | Wide sense stationary | In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1273_chunk_0 | Wide sense stationary | The most common cause of violation of stationarity is a trend in the mean, which can be due either to the presence of a unit root or of a deterministic trend. In the former case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. In the latter case of a deterministic trend, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1274_chunk_0 | Wide sense stationary | A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1275_chunk_0 | Asymptotic distribution | In mathematics and statistics, an asymptotic distribution is a probability distribution that is in a sense the "limiting" distribution of a sequence of distributions. One of the main uses of the idea of an asymptotic distribution is in providing approximations to the cumulative distribution functions of statistical est... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1276_chunk_0 | 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 | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1277_chunk_0 | Absolute deviation | In mathematics and statistics, deviation is a measure of difference between the observed value of a variable and some other value, often that variable's mean. The sign of the deviation reports the direction of that difference (the deviation is positive when the observed value exceeds the reference value). The magnitude... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1278_chunk_0 | Kolmogorov equations (Markov jump process) | In mathematics and statistics, in the context of Markov processes, the Kolmogorov equations, including Kolmogorov forward equations and Kolmogorov backward equations, are a pair of systems of differential equations that describe the time evolution of the process's distribution. This article, as opposed to the article t... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1279_chunk_0 | Random projections | In mathematics and statistics, random projection is a technique used to reduce the dimensionality of a set of points which lie in Euclidean space. Random projection methods are known for their power, simplicity, and low error rates when compared to other methods. According to experimental results, random projection pre... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1280_chunk_0 | Power sum | In mathematics and statistics, sums of powers occur in a number of contexts: Sums of squares arise in many contexts. For example, in geometry, the Pythagorean theorem involves the sum of two squares; in number theory, there are Legendre's three-square theorem and Jacobi's four-square theorem; and in statistics, the ana... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1281_chunk_0 | Power sum | Fermat's Last Theorem states that x k + y k = z k {\displaystyle x^{k}+y^{k}=z^{k}} is impossible in positive integers with k>2. The equation of a superellipse is | x / a | k + | y / b | k = 1 {\displaystyle |x/a|^{k}+|y/b|^{k}=1} . The squircle is the case k = 4 , a = b {\displaystyle k=4,a=b} . | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1282_chunk_0 | Power sum | The Jacobi–Madden equation is a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 {\displaystyle a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}} in integers. The Prouhet–Tarry–Escott problem considers sums of two sets of kth powers of integers that are equal for multiple values of k. A taxicab number is the smallest integer that can be... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1283_chunk_0 | Power sum | The Erdős–Moser equation, 1 k + 2 k + ⋯ + m k = ( m + 1 ) k {\displaystyle 1^{k}+2^{k}+\cdots +m^{k}=(m+1)^{k}} where m {\displaystyle m} and k {\displaystyle k} are positive integers, is conjectured to have no solutions other than 11 + 21 = 31. The sums of three cubes cannot equal 4 or 5 modulo 9, but it is unknown wh... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1284_chunk_0 | 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 an... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1285_chunk_0 | Statistical mean | In mathematics and statistics, the arithmetic mean ( arr-ith-MET-ik), arithmetic average, or just the mean or average (when the context is clear) is the sum of a collection of numbers divided by the count of numbers in the collection. The collection is often a set of results from an experiment, an observational study, ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1286_chunk_0 | Statistical mean | For example, per capita income is the arithmetic average income of a nation's population. While the arithmetic mean is often used to report central tendencies, it is not a robust statistic: it is greatly influenced by outliers (values much larger or smaller than most others). For skewed distributions, such as the distr... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1287_chunk_0 | Quasi-arithmetic mean | In mathematics and statistics, the quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean and the geometric mean, using a function f {\displaystyle f} . It is also called Kolmogorov mean after Soviet mathematician And... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1288_chunk_0 | Portmanteau theorem | In mathematics and statistics, weak convergence is one of many types of convergence relating to the convergence of measures. It depends on a topology on the underlying space and thus is not a purely measure theoretic notion. There are several equivalent definitions of weak convergence of a sequence of measures, some of... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1289_chunk_0 | Portmanteau theorem | For example, the sequence where P n {\displaystyle P_{n}} is the Dirac measure located at 1 / n {\displaystyle 1/n} converges weakly to the Dirac measure located at 0 (if we view these as measures on R {\displaystyle \mathbf {R} } with the usual topology), but it does not converge setwise. This is intuitively clear: we... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1290_chunk_0 | Portmanteau theorem | It also defines a weak topology on P ( S ) {\displaystyle {\mathcal {P}}(S)} , the set of all probability measures defined on ( S , Σ ) {\displaystyle (S,\Sigma )} . The weak topology is generated by the following basis of open sets: { U ϕ , x , δ | ϕ: S → R is bounded and continuous, x ∈ R and δ > 0 } , {\displaystyle... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1291_chunk_0 | Portmanteau theorem | If S {\displaystyle S} is also separable, then P ( S ) {\displaystyle {\mathcal {P}}(S)} is metrizable and separable, for example by the Lévy–Prokhorov metric. If S {\displaystyle S} is also compact or Polish, so is P ( S ) {\displaystyle {\mathcal {P}}(S)} . If S {\displaystyle S} is separable, it naturally embeds int... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1292_chunk_0 | 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-dimensiona... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1293_chunk_0 | 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 communicatio... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1294_chunk_0 | Stochastic geometry models of wireless networks | This model is considered to be pioneering and the origin of continuum percolation. Network models based on geometric probability were later proposed and used in the late 1970s and continued throughout the 1980s for examining packet radio networks. Later their use increased significantly for studying a number of wireles... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1295_chunk_0 | 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 asymp... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1296_chunk_0 | Ray (quantum theory) | In mathematics and the foundations of quantum mechanics, the projective Hilbert space P ( H ) {\displaystyle P(H)} of a complex Hilbert space H {\displaystyle H} is the set of equivalence classes of non-zero vectors v {\displaystyle v} in H {\displaystyle H} , for the relation ∼ {\displaystyle \sim } on H {\displaystyl... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1297_chunk_0 | Constant-recursive sequence | Formally, a sequence of numbers s 0 , s 1 , s 2 , s 3 , … {\displaystyle s_{0},s_{1},s_{2},s_{3},\ldots } is constant-recursive if it satisfies a recurrence relation where c i {\displaystyle c_{i}} are constants. For example, the Fibonacci sequence satisfies the recurrence relation F n = F n − 1 + F n − 2 , {\displayst... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1298_chunk_0 | Constant-recursive sequence | They also arise in algebraic number theory, due to the relation of the sequence to the roots of a polynomial; in the analysis of algorithms as the running time of simple recursive functions; and in formal language theory, where they count strings up to a given length in a regular language. Constant-recursive sequences ... | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
wiki_1299_chunk_0 | 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. | https://www.kaggle.com/datasets/conjuring92/wiki-stem-corpus |
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