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c_f3ly1571y6p2 | In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) co... | Quantum graph |
c_tz0uc2ivxyor | In mathematics and physics, a recurrent tensor, with respect to a connection ∇ {\displaystyle \nabla } on a manifold M, is a tensor T for which there is a one-form ω on M such that ∇ T = ω ⊗ T . {\displaystyle \nabla T=\omega \otimes T.\,} | Recurrent tensor |
c_c5dtmsmblysf | In mathematics and physics, a scalar field is a function associating a single number to every point in a space – possibly physical space. The scalar may either be a pure mathematical number (dimensionless) or a scalar physical quantity (with units). In a physical context, scalar fields are required to be independent of... | Scalar fields |
c_kcmnd9szkkoj | In mathematics and physics, a soliton is a nonlinear, self-reinforcing, localized wave packet that is strongly stable, in that it preserves its shape while propagating freely, at constant velocity, and recovers it even after collisions with other such localized wave packets. Its remarkable stability can be traced to a ... | Soliton wave |
c_btup9ypi418r | The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the "Wave of Translation". The term soliton was coined by Zabusky and Kruskal to describe localized, strongly stable ... | Soliton wave |
c_hwxjxnd8llpg | In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in materials, and in numerous applications in the phy... | Tensor analysis |
c_k0n5a0qpq7ob | In mathematics and physics, a traveling plane wave is a special case of plane wave, namely a field whose evolution in time can be described as simple translation of its values at a constant wave speed c {\displaystyle c} , along a fixed direction of propagation n → {\displaystyle {\vec {n}}} . Such a field can be writt... | Traveling plane wave |
c_wdw9k51w5y5s | 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... | Traveling plane wave |
c_p4p97l1jvvom | 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... | Vector space |
c_boaix5xqd703 | 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... | Vector space |
c_z66rmk2e1dki | 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... | Vector space |
c_v5g8wi0e2nva | 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. | Vector space |
c_95iyq6agz3kk | 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... | Vector space |
c_o3a4l1zx0cwv | 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". | Acceleration vector |
c_30xfa66ld0gr | 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... | Equipotential |
c_z34bgi5lfbbn | 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... | Equipotential |
c_2xeo2d1hy0oj | Electrical conductors offer an intuitive example. If a and b are any two points within or at the surface of a given conductor, and given there is no flow of charge being exchanged between the two points, then the potential difference is zero between the two points. Thus, an equipotential would contain both points a and... | Equipotential |
c_i81svgzbxjpp | 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... | Equipotential |
c_9m08ebyhqf4t | 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... | Equipotential |
c_lvwbxzf05kae | 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... | D'Alembert–Euler condition |
c_hyoovzti80e7 | 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 ... | Asymptotic homogenization |
c_6b4l6afrepl5 | 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... | Asymptotic homogenization |
c_2ll7pnovch64 | {\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. | Asymptotic homogenization |
c_h8bkqavpng6y | 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... | Asymptotic homogenization |
c_prglxl4fe0jj | The analogue of the differential element in the continuum concept (which contains enough atom, or molecular structure to be representative of that material), is known as the "Representative Volume Element" in homogenization and micromechanics. This element contains enough statistical information about the inhomogeneous... | Asymptotic homogenization |
c_hlyw7p9pia00 | 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... | Asymptotic homogenization |
c_5v8p88co8hww | 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... | Warped model |
c_seh0x65jp5r3 | In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the (linear algebraic) properties of the Pauli matrices. Here, a few classes of such matrices are summarized. | Generalizations of Pauli matrices |
c_h1b35j16ax8v | 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... | Euler's Equation |
c_ia6rp1eg25un | 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... | Multiple-scale analysis |
c_tum3ojj8tki0 | In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. Anti-de Sitter space and de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observ... | Anti-de Sitter space |
c_b4p5n6549cbg | 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... | Anti-de Sitter space |
c_nvtetlrgunlp | 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. | Toda field theory |
c_n0mexcyue0o2 | 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. | Super Minkowski space |
c_k0x9n60m8ix6 | 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. | Kinetic Monte Carlo surface growth method |
c_3a3dzh9ttibq | 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 ... | Artin billiard |
c_7fi0o13sgevj | As such, it is an example of an Anosov flow. Artin's paper used symbolic dynamics for analysis of the system. | Artin billiard |
c_7vpamnw7nun6 | 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. | Artin billiard |
c_d9qcbhtgchls | In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface. In differential geometry, an affine conn... | Christoffel symbol |
c_josd06glal7p | Abstractly, one would say that the manifold has an associated (orthonormal) frame bundle, with each "frame" being a possible choice of a coordinate frame. An invariant metric implies that the structure group of the frame bundle is the orthogonal group O(p, q). As a result, such a manifold is necessarily a (pseudo-)Riem... | Christoffel symbol |
c_t41dc9qd01iq | The Christoffel symbols provide a concrete representation of the connection of (pseudo-)Riemannian geometry in terms of coordinates on the manifold. Additional concepts, such as parallel transport, geodesics, etc. can then be expressed in terms of Christoffel symbols. | Christoffel symbol |
c_hdz1y83qqguv | In general, there are an infinite number of metric connections for a given metric tensor; however, there is a unique connection that is free of torsion, the Levi-Civita connection. It is common in physics and general relativity to work almost exclusively with the Levi-Civita connection, by working in coordinate frames ... | Christoffel symbol |
c_0gejk67u1im8 | At each point of the underlying n-dimensional manifold, for any local coordinate system around that point, the Christoffel symbols are denoted Γijk for i, j, k = 1, 2, ..., n. Each entry of this n × n × n array is a real number. Under linear coordinate transformations on the manifold, the Christoffel symbols transform ... | Christoffel symbol |
c_fcbr0v56asjv | Christoffel symbols are used for performing practical calculations. For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gr... | Christoffel symbol |
c_6sy3onfsg7ij | 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... | Kadomtsev–Petviashvili equation |
c_xjlx77y364x5 | 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... | Kadomtsev–Petviashvili equation |
c_5l8yjl4lx49h | 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... | Magnus expansion |
c_46a91dfy7q0o | 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... | Poincaré recurrence theorem |
c_hc17h0thj9ow | 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... | Poincaré recurrence theorem |
c_mg5iywu2f3ow | 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... | Geometric center |
c_f2tyoo9j7zhu | 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... | Diamagnetic inequality |
c_rfex29mgveun | 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... | Heat diffusion |
c_pcwwmysnt982 | 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 ... | Heat diffusion |
c_6ttzwl5ju4jf | 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... | Heat diffusion |
c_df0v2vrwwlij | 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... | Heat diffusion |
c_zjcakn03qeug | In mathematics and physics, the inverse scattering problem is the problem of determining characteristics of an object, based on data of how it scatters incoming radiation or particles. It is the inverse problem to the direct scattering problem, which is to determine how radiation or particles are scattered based on the... | Inverse scattering |
c_3fphq55lqv9j | 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... | Inverse scattering |
c_2b44kqv2qooh | 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. | Orientation entanglement |
c_g8sz0159r1ls | 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... | Plate trick |
c_xhvl0jq3a2w5 | In mathematics and physics, the right-hand rule is a common mnemonic for understanding the orientation of axes in three-dimensional space. It is also a convenient method for quickly finding the direction of the cross product of two vectors. Rather than a mathematical fact, it is a convention, closely related to the con... | Right hand grip rule |
c_v6gks3y4qoz3 | One can see this by holding one's hands outward and together, palms up, with the thumbs out-stretched to the right and left, and the fingers making a curling motion from straight outward to pointing upward. If the curling motion of the fingers represents a movement from the first (x-axis) to the second (y-axis), then t... | Right hand grip rule |
c_kyyv6l9bn29p | 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 ... | Right hand grip rule |
c_lbj5keb7o06a | 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... | Spectral asymmetry |
c_vwnzs15qbnpp | For example, the vacuum expectation value of the baryon number is given by the spectral asymmetry of the Hamiltonian operator. The spectral asymmetry of the confined quark fields is an important property of the chiral bag model. For fermions, it is known as the Witten index, and can be understood as describing the Casi... | Spectral asymmetry |
c_amnuzi3iq22x | In mathematics and physics, the term generator or generating set may refer to any of a number of related concepts. The underlying concept in each case is that of a smaller set of objects, together with a set of operations that can be applied to it, that result in the creation of a larger collection of objects, called t... | Generator (mathematics) |
c_8e8dmsqlxp50 | 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... | Vector notation |
c_xfmer9gnawig | In mathematics and political science, the quota rule describes a desired property of a proportional apportionment or election method. It states that the number of seats that should be allocated to a given party should be between the upper or lower roundings (called upper and lower quotas) of its fractional proportional... | Quota rule |
c_0zeqa9o3d38o | 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... | Skorokhod's embedding theorem |
c_q04wq0ie3j1e | In mathematics and probability theory, continuum percolation theory is a branch of mathematics that extends discrete percolation theory to continuous space (often Euclidean space ℝn). More specifically, the underlying points of discrete percolation form types of lattices whereas the underlying points of continuum perco... | Continuum percolation theory |
c_ac1xo9wdumbt | 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... | Continuum percolation theory |
c_w906f2tqolrn | 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... | Borell–TIS inequality |
c_vred2azkmcdq | 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... | Harmonic spinor |
c_4yr8h6q6tvct | In mathematics and related subjects, understanding a mathematical expression depends on an understanding of symbols of grouping, such as parentheses (), brackets , and braces {}. These same symbols are also used in ways where they are not symbols of grouping. For example, in the expression 3(x+y) the parentheses are sy... | Symbols of grouping |
c_yytvpti8k34r | For example, to indicate the product of binomials, parentheses are usually used, thus: ( 2 x + 3 ) ( 3 x + 4 ) {\displaystyle (2x+3)(3x+4)} . But if one of the binomials itself contains parentheses, as in ( 2 ( a + b ) + 3 ) {\displaystyle (2(a+b)+3)} one or more pairs of parentheses may be replaced by brackets, thus: ... | Symbols of grouping |
c_u0a9ywtl2g3s | The usage of the word "parentheses" varies from country to country. In the United States, the word parentheses (singular "parenthesis") is used for the curved symbol of grouping, but in many other countries the curved symbol of grouping is called a "bracket" and the symbol of grouping with two right angles joined is ca... | Symbols of grouping |
c_epmxyyvbsilh | If two of these symbols are used, one on the left and the mirror image of it on the right, it almost always indicates a set, as in { a , b , c } {\displaystyle \{a,b,c\}} , the set containing three members, a {\displaystyle a} , b {\displaystyle b} , and c {\displaystyle c} . But if it is used only on the left, it grou... | Symbols of grouping |
c_wn596xqoqjil | One is the bar above an expression, as in the square root sign in which the bar is a symbol of grouping. For example √p+q is the square root of the sum. The bar is also a symbol of grouping in repeated decimal digits. | Symbols of grouping |
c_6ulc00xxix5n | A decimal point followed by one or more digits with a bar over them, for example 0.123, represents the repeating decimal 0.123123123... .A superscript is understood to be grouped as long as it continues in the form of a superscript. For example if an x has a superscript of the forma+b, the sum is the exponent. For exam... | Symbols of grouping |
c_880rtqmmi0b0 | 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... | Nonlinear differential equation |
c_9vzcuabimkb0 | 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... | Nonlinear differential equation |
c_drl49wysnx4i | 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... | Nonlinear differential equation |
c_lhtbbs89the1 | 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... | Nonlinear differential equation |
c_3m4euqt5ulqn | 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. | Hereditarily finite set |
c_yelrf7621wzi | 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... | Analytic representation |
c_7cwr8cjb08iu | 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... | Analytic representation |
c_ls5m83b47ws9 | 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... | Hilbert Transform |
c_hscs546823g3 | 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... | Z-transform |
c_ni60dxsxye5j | The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain... | Z-transform |
c_w2dt7ftvkzxv | 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... | Advanced Z-transform |
c_kpf8re6g68bs | 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... | Constant-Q transform |
c_t7nypcrs1j16 | 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... | Collaboration graph |
c_mdb3xj51kz37 | 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 ... | Constant-energy surface |
c_t8gy9vz8prai | 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... | Constant-energy surface |
c_f1erkmv81kmn | 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... | Constant-energy surface |
c_kxvcfstz9bi5 | 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... | Constant-energy surface |
c_sdsrctzfpqt2 | 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 ... | Dimension of an algebraic variety |
c_88xzfgk4r4v9 | In mathematics and specifically in topology, rational homotopy theory is a simplified version of homotopy theory for topological spaces, in which all torsion in the homotopy groups is ignored. It was founded by Dennis Sullivan (1977) and Daniel Quillen (1969). This simplification of homotopy theory makes certain calcul... | Formal space |
c_falm61b9hcdk | 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... | Formal space |
c_ngd2mjnbn8nb | 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... | Skorokhod's representation theorem |
c_oe1vyq8b28h0 | 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° ... | Mean of circular quantities |
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