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241 relations: Absolute continuity, Abstraction, Abuse of notation, Academic Press, Airy function, Almost everywhere, Analytic continuation, Analytic function, Applied mathematics, Arthur Mattuck, Associative algebra, Atom (measure theory), Augustin-Louis Cauchy, Beam (structure), Bending moment, Bending stiffness, Bessel function, Billiard ball, Birkhäuser, Bound state, Bra–ket notation, Brownian motion, Bump function, Cauchy distribution, Cauchy principal value, Cauchy problem, Cauchy's integral formula, Cauchy–Kowalevski theorem, Cesàro summation, Characteristic function (probability theory), Coarea formula, Complex analysis, Complex plane, Compressible flow, Continuous function, Continuous spectrum, Convergence of random variables, Convex combination, Convex hull, Convolution, Cumulative distribution function, Current (mathematics), Daniell integral, Deflection (engineering), Delta potential, Dense set, Densely defined operator, Derivative, Differentiable function, Differentiable manifold, ..., Differential operator, Differential structure, Diffusion process, Digital signal processing, Dipole, Dirac comb, Dirac measure, Directional derivative, Dirichlet kernel, Distribution (mathematics), Divergent series, Domain (mathematical analysis), Double layer potential, Dual space, Dyadics, Dynamics (mechanics), Eigenfunction, Eigenvalues and eigenvectors, Electric potential, Electromagnetism, Electron, Elliptic partial differential equation, Euclidean space, Euler–Bernoulli beam theory, Euler–Tricomi equation, Even and odd functions, Fejér kernel, Force, Fourier series, Fourier transform, Fritz John, Fubini's theorem, Function of a real variable, Fundamental solution, Generalized function, Geometric measure theory, Gradient, Graph of a function, Green's function, Gustav Kirchhoff, Hamiltonian (quantum mechanics), Hardy space, Harmonic oscillator, Harold R. Parks, Heat equation, Heat kernel, Heaviside step function, Hermann von Helmholtz, Heuristic, Hilbert space, Holomorphic function, Homogeneous function, Hooke's law, Hyperbolic partial differential equation, Hyperreal number, Identity element, Identity function, If and only if, Impulse (physics), Impulse response, Indicator function, Infinitesimal, Initial value problem, Integral, Integration by parts, Johann Radon, Joseph Fourier, Journal of Mathematical Physics, Kronecker delta, Laplace operator, Laplace transform, Laplace's equation, Laplacian of the indicator, Laurent Schwartz, Lazare Carnot, Lebesgue integration, Lebesgue measure, Limit of a function, Linear form, Linear time-invariant theory, Lipschitz continuity, Local time (mathematics), Locally compact space, Lp space, Markov property, Mathematical object, Mathematics, Measure (mathematics), Michel Plancherel, Minkowski content, Mixture distribution, Mollifier, Moment (mathematics), Moment-generating function, Momentum, Momentum operator, Motion (physics), Multi-index notation, Multiplier (Fourier analysis), Multipole expansion, Mutatis mutandis, Newtonian potential, Non-standard analysis, Norbert Wiener, Normal distribution, Numerical analysis, Observable, Oliver Heaviside, Open set, Operational calculus, Orthonormal basis, Oscillatory integral, Overshoot (signal), Parabolic partial differential equation, Partial derivative, Partial differential equation, Paul Dirac, Periodic function, Piecewise linear function, Plane wave, Point particle, Poisson kernel, Poisson summation formula, Polynomial, Position operator, Potential theory, Probability density function, Probability distribution, Probability measure, Probability theory, Product measure, Quantum mechanics, Radon measure, Radon transform, Radon–Nikodym theorem, Real number, Rectangular function, Reflection (mathematics), Riemann integral, Riemann–Stieltjes integral, Riesz representation theorem, Rigged Hilbert space, Rotation (mathematics), Salomon Bochner, Sampling (signal processing), Schwartz space, Self-adjoint, Semigroup, Sequence, Series (mathematics), Several complex variables, Sigma-algebra, Signal processing, Siméon Denis Poisson, Sinc function, Singular measure, Smoothness, Sobolev inequality, Sobolev space, Sokhotski–Plemelj theorem, Spectrum, Spherical measure, Springer Science+Business Media, Square-integrable function, Statistics, Steven G. Krantz, Structural mechanics, Submersion (mathematics), Support (mathematics), Szegő kernel, The Principles of Quantum Mechanics, Topology, Transonic, Triangular function, Unbounded operator, Undergraduate Texts in Mathematics, Uniform distribution (continuous), Unit doublet, Unit sphere, Vague topology, Vanish at infinity, Variance, Wave, Wave equation, Wave function, Wave propagation, Weak topology, Wigner semicircle distribution, William Thomson, 1st Baron Kelvin, Wrapped distribution, Zero of a function. Expand index (191 more) »
Absolute continuity
In calculus, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity.
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Abstraction
Abstraction in its main sense is a conceptual process where general rules and concepts are derived from the usage and classification of specific examples, literal ("real" or "concrete") signifiers, first principles, or other methods.
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Abuse of notation
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion).
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Academic Press
Academic Press is an academic book publisher.
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Airy function
In the physical sciences, the Airy function (or Airy function of the first kind) Ai(x) is a special function named after the British astronomer George Biddell Airy (1801–92).
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Almost everywhere
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities.
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Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Applied mathematics
Applied mathematics is the application of mathematical methods by different fields such as science, engineering, business, computer science, and industry.
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Arthur Mattuck
Arthur Paul Mattuck (born June 13, 1930) is a professor of mathematics at the Massachusetts Institute of Technology.
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Associative algebra
In mathematics, an associative algebra is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field.
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Atom (measure theory)
In mathematics, more precisely in measure theory, an atom is a measurable set which has positive measure and contains no set of smaller positive measure.
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Augustin-Louis Cauchy
Baron Augustin-Louis Cauchy FRS FRSE (21 August 178923 May 1857) was a French mathematician, engineer and physicist who made pioneering contributions to several branches of mathematics, including: mathematical analysis and continuum mechanics.
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Beam (structure)
A beam is a structural element that primarily resists loads applied laterally to the beam's axis.
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Bending moment
A bending moment is the reaction induced in a structural element when an external force or moment is applied to the element causing the element to bend.
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Bending stiffness
The bending stiffness (K) is the resistance of a member against bending deformation.
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Bessel function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are the canonical solutions of Bessel's differential equation for an arbitrary complex number, the order of the Bessel function.
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Billiard ball
A billiard ball is a small, hard ball used in cue sports, such as carom billiards, pool, and snooker.
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Birkhäuser
Birkhäuser is a former Swiss publisher founded in 1879 by Emil Birkhäuser.
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Bound state
In quantum physics, a bound state is a special quantum state of a particle subject to a potential such that the particle has a tendency to remain localised in one or more regions of space.
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Bra–ket notation
In quantum mechanics, bra–ket notation is a standard notation for describing quantum states.
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Brownian motion
Brownian motion or pedesis (from πήδησις "leaping") is the random motion of particles suspended in a fluid (a liquid or a gas) resulting from their collision with the fast-moving molecules in the fluid.
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Bump function
In mathematics, a bump function is a function f:\mathbf^n\rightarrow \mathbf on a Euclidean space \mathbf^n which is both smooth (in the sense of having continuous derivatives of all orders) and compactly supported.
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Cauchy distribution
The Cauchy distribution, named after Augustin Cauchy, is a continuous probability distribution.
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Cauchy principal value
In mathematics, the Cauchy principal value, named after Augustin Louis Cauchy, is a method for assigning values to certain improper integrals which would otherwise be undefined.
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Cauchy problem
A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain.
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Cauchy's integral formula
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis.
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Cauchy–Kowalevski theorem
In mathematics, the Cauchy–Kowalevski theorem (also written as the Cauchy–Kovalevskaya theorem) is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.
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Cesàro summation
In mathematical analysis, Cesàro summation (also known as the Cesàro mean) assigns values to some infinite sums that are not convergent in the usual sense.
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Characteristic function (probability theory)
In probability theory and statistics, the characteristic function of any real-valued random variable completely defines its probability distribution.
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Coarea formula
In the mathematical field of geometric measure theory, the coarea formula expresses the integral of a function over an open set in Euclidean space in terms of integrals over the level sets of another function.
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Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.
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Complex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the perpendicular imaginary axis.
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Compressible flow
Compressible flow (gas dynamics) is the branch of fluid mechanics that deals with flows having significant changes in fluid density.
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Continuous function
In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output.
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Continuous spectrum
In physics, a continuous spectrum usually means a set of attainable values for some physical quantity (such as energy or wavelength) that is best described as an interval of real numbers, as opposed to a discrete spectrum, a set of attainable values that is discrete in the mathematical sense, where there is a positive gap between each value and the next one.
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Convergence of random variables
In probability theory, there exist several different notions of convergence of random variables.
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Convex combination
In convex geometry, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1.
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Convex hull
In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.
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Convolution
In mathematics (and, in particular, functional analysis) convolution is a mathematical operation on two functions (f and g) to produce a third function, that is typically viewed as a modified version of one of the original functions, giving the integral of the pointwise multiplication of the two functions as a function of the amount that one of the original functions is translated.
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Cumulative distribution function
In probability theory and statistics, the cumulative distribution function (CDF, also cumulative density function) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. In the case of a continuous distribution, it gives the area under the probability density function from minus infinity to x. Cumulative distribution functions are also used to specify the distribution of multivariate random variables.
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Current (mathematics)
In mathematics, more particularly in functional analysis, differential topology, and geometric measure theory, a k-current in the sense of Georges de Rham is a functional on the space of compactly supported differential k-forms, on a smooth manifold M. Formally currents behave like Schwartz distributions on a space of differential forms.
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Daniell integral
In mathematics, the Daniell integral is a type of integration that generalizes the concept of more elementary versions such as the Riemann integral to which students are typically first introduced.
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Deflection (engineering)
In engineering, deflection is the degree to which a structural element is displaced under a load.
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Delta potential
In quantum mechanics the delta potential is a potential well mathematically described by the Dirac delta function - a generalized function.
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Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
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Densely defined operator
In mathematics – specifically, in operator theory – a densely defined operator or partially defined operator is a type of partially defined function.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Differentiable function
In calculus (a branch of mathematics), a differentiable function of one real variable is a function whose derivative exists at each point in its domain.
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Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
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Differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator.
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Differential structure
In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.
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Diffusion process
In probability theory and statistics, a diffusion process is a solution to a stochastic differential equation.
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Digital signal processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations.
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Dipole
In electromagnetism, there are two kinds of dipoles.
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Dirac comb
In mathematics, a Dirac comb (also known as an impulse train and sampling function in electrical engineering) is a periodic tempered distribution constructed from Dirac delta functions for some given period T. The symbol \operatorname(t), where the period is omitted, represents a Dirac comb of unit period.
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Dirac measure
In mathematics, a Dirac measure assigns a size to a set based solely on whether it contains a fixed element x or not.
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Directional derivative
In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.
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Dirichlet kernel
In mathematical analysis, the Dirichlet kernel is the collection of functions e^.
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Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
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Divergent series
In mathematics, a divergent series is an infinite series that is not convergent, meaning that the infinite sequence of the partial sums of the series does not have a finite limit.
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Domain (mathematical analysis)
In mathematical analysis, a domain is any connected open subset of a finite-dimensional vector space.
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Double layer potential
In potential theory, an area of mathematics, a double layer potential is a solution of Laplace's equation corresponding to the electrostatic or magnetic potential associated to a dipole distribution on a closed surface S in three-dimensions.
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Dual space
In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.
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Dyadics
In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
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Dynamics (mechanics)
Dynamics is the branch of applied mathematics (specifically classical mechanics) concerned with the study of forces and torques and their effect on motion, as opposed to kinematics, which studies the motion of objects without reference to these forces.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Electric potential
An electric potential (also called the electric field potential, potential drop or the electrostatic potential) is the amount of work needed to move a unit positive charge from a reference point to a specific point inside the field without producing any acceleration.
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Electromagnetism
Electromagnetism is a branch of physics involving the study of the electromagnetic force, a type of physical interaction that occurs between electrically charged particles.
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Electron
The electron is a subatomic particle, symbol or, whose electric charge is negative one elementary charge.
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Elliptic partial differential equation
Second order linear partial differential equations (PDEs) are classified as either elliptic, hyperbolic, or parabolic.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Euler–Bernoulli beam theory
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory)Timoshenko, S., (1953), History of strength of materials, McGraw-Hill New York is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams.
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Euler–Tricomi equation
In mathematics, the Euler–Tricomi equation is a linear partial differential equation useful in the study of transonic flow.
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Even and odd functions
In mathematics, even functions and odd functions are functions which satisfy particular symmetry relations, with respect to taking additive inverses.
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Fejér kernel
In mathematics, the Fejér kernel is a summability kernel used to express the effect of Cesàro summation on Fourier series.
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Force
In physics, a force is any interaction that, when unopposed, will change the motion of an object.
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Fourier series
In mathematics, a Fourier series is a way to represent a function as the sum of simple sine waves.
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Fourier transform
The Fourier transform (FT) decomposes a function of time (a signal) into the frequencies that make it up, in a way similar to how a musical chord can be expressed as the frequencies (or pitches) of its constituent notes.
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Fritz John
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems.
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Fubini's theorem
In mathematical analysis Fubini's theorem, introduced by, is a result that gives conditions under which it is possible to compute a double integral using iterated integrals.
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Function of a real variable
In mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers, or a subset of that contains an interval of positive length.
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Fundamental solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions).
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Generalized function
In mathematics, generalized functions, or distributions, are objects extending the notion of functions.
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Geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory.
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Gradient
In mathematics, the gradient is a multi-variable generalization of the derivative.
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Graph of a function
In mathematics, the graph of a function f is, formally, the set of all ordered pairs, and, in practice, the graphical representation of this set.
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Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential equation defined on a domain, with specified initial conditions or boundary conditions.
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Gustav Kirchhoff
Gustav Robert Kirchhoff (12 March 1824 – 17 October 1887) was a German physicist who contributed to the fundamental understanding of electrical circuits, spectroscopy, and the emission of black-body radiation by heated objects.
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Hamiltonian (quantum mechanics)
In quantum mechanics, a Hamiltonian is an operator corresponding to the total energy of the system in most of the cases.
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Hardy space
In complex analysis, the Hardy spaces (or Hardy classes) Hp are certain spaces of holomorphic functions on the unit disk or upper half plane.
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Harmonic oscillator
In classical mechanics, a harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force, F, proportional to the displacement, x: where k is a positive constant.
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Harold R. Parks
Harold Raymond Parks (born May 22, 1949) is an American mathematician and is a professor emeritus of mathematics at Oregon State University.
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Heat equation
The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.
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Heat kernel
In the mathematical study of heat conduction and diffusion, a heat kernel is the fundamental solution to the heat equation on a specified domain with appropriate boundary conditions.
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Heaviside step function
The Heaviside step function, or the unit step function, usually denoted by or (but sometimes, or), is a discontinuous function named after Oliver Heaviside (1850–1925), whose value is zero for negative argument and one for positive argument.
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Hermann von Helmholtz
Hermann Ludwig Ferdinand von Helmholtz (August 31, 1821 – September 8, 1894) was a German physician and physicist who made significant contributions in several scientific fields.
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Heuristic
A heuristic technique (εὑρίσκω, "find" or "discover"), often called simply a heuristic, is any approach to problem solving, learning, or discovery that employs a practical method, not guaranteed to be optimal, perfect, logical, or rational, but instead sufficient for reaching an immediate goal.
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Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.
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Holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain.
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Homogeneous function
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
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Hooke's law
Hooke's law is a principle of physics that states that the force needed to extend or compress a spring by some distance scales linearly with respect to that distance.
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Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives.
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Hyperreal number
The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities.
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Identity element
In mathematics, an identity element or neutral element is a special type of element of a set with respect to a binary operation on that set, which leaves other elements unchanged when combined with them.
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Identity function
Graph of the identity function on the real numbers In mathematics, an identity function, also called an identity relation or identity map or identity transformation, is a function that always returns the same value that was used as its argument.
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If and only if
In logic and related fields such as mathematics and philosophy, if and only if (shortened iff) is a biconditional logical connective between statements.
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Impulse (physics)
In classical mechanics, impulse (symbolized by J or Imp) is the integral of a force, F, over the time interval, t, for which it acts.
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Impulse response
In signal processing, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an impulse.
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Indicator function
In mathematics, an indicator function or a characteristic function is a function defined on a set X that indicates membership of an element in a subset A of X, having the value 1 for all elements of A and the value 0 for all elements of X not in A. It is usually denoted by a symbol 1 or I, sometimes in boldface or blackboard boldface, with a subscript specifying the subset.
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Infinitesimal
In mathematics, infinitesimals are things so small that there is no way to measure them.
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Initial value problem
In mathematics, the field of differential equations, an initial value problem (also called the Cauchy problem by some authors) is an ordinary differential equation together with a specified value, called the initial condition, of the unknown function at a given point in the domain of the solution.
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data.
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Integration by parts
In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of their derivative and antiderivative.
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Johann Radon
Johann Karl August Radon (16 December 1887 – 25 May 1956) was an Austrian mathematician.
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Joseph Fourier
Jean-Baptiste Joseph Fourier (21 March 1768 – 16 May 1830) was a French mathematician and physicist born in Auxerre and best known for initiating the investigation of Fourier series and their applications to problems of heat transfer and vibrations.
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Journal of Mathematical Physics
The Journal of Mathematical Physics is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics.
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Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers.
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Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space.
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Laplace transform
In mathematics, the Laplace transform is an integral transform named after its discoverer Pierre-Simon Laplace.
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Laplace's equation
In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties.
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Laplacian of the indicator
In mathematics, the Laplacian of the indicator of the domain D is a generalisation of the derivative of the Dirac delta function to higher dimensions, and is non-zero only on the surface of D. It can be viewed as the surface delta prime function.
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Laurent Schwartz
Laurent-Moïse Schwartz (5 March 1915 – 4 July 2002) was a French mathematician.
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Lazare Carnot
Lazare Nicolas Marguerite, Count Carnot (13 May 1753 – 2 August 1823) was a French mathematician, physicist and politician.
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Lebesgue integration
In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the -axis.
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Lebesgue measure
In measure theory, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of n-dimensional Euclidean space.
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Limit of a function
Although the function (sin x)/x is not defined at zero, as x becomes closer and closer to zero, (sin x)/x becomes arbitrarily close to 1.
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Linear form
In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.
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Linear time-invariant theory
Linear time-invariant theory, commonly known as LTI system theory, comes from applied mathematics and has direct applications in NMR spectroscopy, seismology, circuits, signal processing, control theory, and other technical areas.
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Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.
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Local time (mathematics)
In the mathematical theory of stochastic processes, local time is a stochastic process associated with diffusion processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level.
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Locally compact space
In topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space.
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Lp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the ''p''-norm for finite-dimensional vector spaces.
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Markov property
In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process.
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Mathematical object
A mathematical object is an abstract object arising in mathematics.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Measure (mathematics)
In mathematical analysis, a measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size.
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Michel Plancherel
Michel Plancherel (16 January 1885, Bussy, Fribourg4 March 1967, Zurich) was a Swiss mathematician.
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Minkowski content
The Minkowski content (named after Hermann Minkowski), or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary measurable sets.
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Mixture distribution
In probability and statistics, a mixture distribution is the probability distribution of a random variable that is derived from a collection of other random variables as follows: first, a random variable is selected by chance from the collection according to given probabilities of selection, and then the value of the selected random variable is realized.
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Mollifier
In mathematics, mollifiers (also known as approximations to the identity) are smooth functions with special properties, used for example in distribution theory to create sequences of smooth functions approximating nonsmooth (generalized) functions, via convolution.
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Moment (mathematics)
In mathematics, a moment is a specific quantitative measure, used in both mechanics and statistics, of the shape of a set of points.
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Moment-generating function
In probability theory and statistics, the moment-generating function of a real-valued random variable is an alternative specification of its probability distribution.
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Momentum
In Newtonian mechanics, linear momentum, translational momentum, or simply momentum (pl. momenta) is the product of the mass and velocity of an object.
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Momentum operator
In quantum mechanics, the momentum operator is an operator which maps the wave function in a Hilbert space representing a quantum state to another function.
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Motion (physics)
In physics, motion is a change in position of an object over time.
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Multi-index notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.
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Multiplier (Fourier analysis)
In Fourier analysis, a multiplier operator is a type of linear operator, or transformation of functions.
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Multipole expansion
A multipole expansion is a mathematical series representing a function that depends on angles—usually the two angles on a sphere.
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Mutatis mutandis
Mutatis mutandis is a Medieval Latin phrase meaning "the necessary changes having been made" or "once the necessary changes have been made".
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Newtonian potential
In mathematics, the Newtonian potential or Newton potential is an operator in vector calculus that acts as the inverse to the negative Laplacian, on functions that are smooth and decay rapidly enough at infinity.
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Non-standard analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers.
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Norbert Wiener
Norbert Wiener (November 26, 1894 – March 18, 1964) was an American mathematician and philosopher.
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Normal distribution
In probability theory, the normal (or Gaussian or Gauss or Laplace–Gauss) distribution is a very common continuous probability distribution.
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Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).
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Observable
In physics, an observable is a dynamic variable that can be measured.
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Oliver Heaviside
Oliver Heaviside FRS (18 May 1850 – 3 February 1925) was an English self-taught electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques for the solution of differential equations (equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis.
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Open set
In topology, an open set is an abstract concept generalizing the idea of an open interval in the real line.
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Operational calculus
Operational calculus, also known as operational analysis, is a technique by which problems in analysis, in particular differential equations, are transformed into algebraic problems, usually the problem of solving a polynomial equation.
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Orthonormal basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.
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Oscillatory integral
In mathematical analysis an oscillatory integral is a type of distribution.
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Overshoot (signal)
In signal processing, control theory, electronics, and mathematics, overshoot is the occurrence of a signal or function exceeding its target.
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Parabolic partial differential equation
A parabolic partial differential equation is a type of partial differential equation (PDE).
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Partial derivative
In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
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Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
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Paul Dirac
Paul Adrien Maurice Dirac (8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century.
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Periodic function
In mathematics, a periodic function is a function that repeats its values in regular intervals or periods.
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Piecewise linear function
In mathematics, a piecewise linear function is a real-valued function defined on the real numbers or a segment thereof, whose graph is composed of straight-line sections.
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Plane wave
In the physics of wave propagation, a plane wave (also spelled planewave) is a wave whose wavefronts (surfaces of constant phase) are infinite parallel planes.
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Point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics.
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Poisson kernel
In potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation, given Dirichlet boundary conditions on the unit disc.
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Poisson summation formula
In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform.
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Polynomial
In mathematics, a polynomial is an expression consisting of variables (also called indeterminates) and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables.
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Position operator
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.
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Potential theory
In mathematics and mathematical physics, potential theory is the study of harmonic functions.
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Probability density function
In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function, whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample.
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Probability distribution
In probability theory and statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment.
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Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.
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Probability theory
Probability theory is the branch of mathematics concerned with probability.
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Product measure
In mathematics, given two measurable spaces and measures on them, one can obtain a product measurable space and a product measure on that space.
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Quantum mechanics
Quantum mechanics (QM; also known as quantum physics, quantum theory, the wave mechanical model, or matrix mechanics), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.
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Radon measure
In mathematics (specifically in measure theory), a Radon measure, named after Johann Radon, is a measure on the σ-algebra of Borel sets of a Hausdorff topological space X that is locally finite and inner regular.
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Radon transform
In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line.
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Radon–Nikodym theorem
In mathematics, the Radon–Nikodym theorem is a result in measure theory.
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Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
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Rectangular function
The rectangular function (also known as the rectangle function, rect function, Pi function, gate function, unit pulse, or the normalized boxcar function) is defined as: 0 & \mbox |t| > \frac \\ \frac & \mbox |t|.
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Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
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Riemann integral
In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.
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Riemann–Stieltjes integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes.
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Riesz representation theorem
There are several well-known theorems in functional analysis known as the Riesz representation theorem.
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Rigged Hilbert space
In mathematics, a rigged Hilbert space (Gelfand triple, nested Hilbert space, equipped Hilbert space) is a construction designed to link the distribution and square-integrable aspects of functional analysis.
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Rotation (mathematics)
Rotation in mathematics is a concept originating in geometry.
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Salomon Bochner
Salomon Bochner (20 August 1899 – 2 May 1982) was an American mathematician, known for work in mathematical analysis, probability theory and differential geometry.
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Sampling (signal processing)
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal.
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Schwartz space
In mathematics, Schwartz space is the function space of all functions whose derivatives are rapidly decreasing (defined rigorously below).
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Self-adjoint
In mathematics, an element x of a *-algebra is self-adjoint if x^*.
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Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.
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Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed.
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Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity.
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Several complex variables
The theory of functions of several complex variables is the branch of mathematics dealing with complex valued functions on the n-tuples of complex numbers.
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Sigma-algebra
In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set X is a collection Σ of subsets of X that includes the empty subset, is closed under complement, and is closed under countable unions and countable intersections.
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Signal processing
Signal processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon", such as sound, images, and biological measurements.
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Siméon Denis Poisson
Baron Siméon Denis Poisson FRS FRSE (21 June 1781 – 25 April 1840) was a French mathematician, engineer, and physicist, who made several scientific advances.
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Sinc function
In mathematics, physics and engineering, the cardinal sine function or sinc function, denoted by, has two slightly different definitions.
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Singular measure
In mathematics, two positive (or signed or complex) measures μ and ν defined on a measurable space (Ω, &Sigma) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by \mu \perp \nu.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Sobolev inequality
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces.
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Sobolev space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function itself and its derivatives up to a given order.
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Sokhotski–Plemelj theorem
The Sokhotski–Plemelj theorem (Polish spelling is Sochocki) is a theorem in complex analysis, which helps in evaluating certain integrals.
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Spectrum
A spectrum (plural spectra or spectrums) is a condition that is not limited to a specific set of values but can vary, without steps, across a continuum.
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Spherical measure
In mathematics — specifically, in geometric measure theory — spherical measure σn is the “natural” Borel measure on the ''n''-sphere Sn.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Square-integrable function
In mathematics, a square-integrable function, also called a quadratically integrable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite.
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Statistics
Statistics is a branch of mathematics dealing with the collection, analysis, interpretation, presentation, and organization of data.
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Steven G. Krantz
Steven George Krantz (born February 3, 1951) is an American scholar, mathematician, and writer.
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Structural mechanics
Structural mechanics or Mechanics of structures is the computation of deformations, deflections, and internal forces or stresses (stress equivalents) within structures, either for design or for performance evaluation of existing structures.
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Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.
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Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the domain containing those elements which are not mapped to zero.
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Szegő kernel
In the mathematical study of several complex variables, the Szegő kernel is an integral kernel that gives rise to a reproducing kernel on a natural Hilbert space of holomorphic functions.
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The Principles of Quantum Mechanics
The Principles of Quantum Mechanics is an influential monograph on quantum mechanics written by Paul Dirac and first published by Oxford University Press in 1930.
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Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
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Transonic
In aeronautics, transonic (or transsonic) flight is flying at or near the speed of sound (at sea level under average conditions), relative to the air through which the vehicle is traveling.
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Triangular function
A triangular function (also known as a triangle function, hat function, or tent function) is a function whose graph takes the shape of a triangle.
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Unbounded operator
In mathematics, more specifically functional analysis and operator theory, the notion of unbounded operator provides an abstract framework for dealing with differential operators, unbounded observables in quantum mechanics, and other cases.
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Undergraduate Texts in Mathematics
Undergraduate Texts in Mathematics (UTM) is a series of undergraduate-level textbooks in mathematics published by Springer-Verlag.
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Uniform distribution (continuous)
In probability theory and statistics, the continuous uniform distribution or rectangular distribution is a family of symmetric probability distributions such that for each member of the family, all intervals of the same length on the distribution's support are equally probable.
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Unit doublet
In mathematics, the unit doublet is the derivative of the Dirac delta function.
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Unit sphere
In mathematics, a unit sphere is the set of points of distance 1 from a fixed central point, where a generalized concept of distance may be used; a closed unit ball is the set of points of distance less than or equal to 1 from a fixed central point.
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Vague topology
In mathematics, particularly in the area of functional analysis and topological vector spaces, the vague topology is an example of the weak-* topology which arises in the study of measures on locally compact Hausdorff spaces.
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Vanish at infinity
In mathematics, a function on a normed vector space is said to vanish at infinity if For example, the function defined on the real line vanishes at infinity.
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Variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean.
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Wave
In physics, a wave is a disturbance that transfers energy through matter or space, with little or no associated mass transport.
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Wave equation
The wave equation is an important second-order linear partial differential equation for the description of waves—as they occur in classical physics—such as mechanical waves (e.g. water waves, sound waves and seismic waves) or light waves.
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Wave function
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system.
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Wave propagation
Wave propagation is any of the ways in which waves travel.
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Weak topology
In mathematics, weak topology is an alternative term for certain initial topologies, often on topological vector spaces or spaces of linear operators, for instance on a Hilbert space.
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Wigner semicircle distribution
The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution supported on the interval the graph of whose probability density function f is a semicircle of radius R centered at (0, 0) and then suitably normalized (so that it is really a semi-ellipse): for −R ≤ x ≤ R, and f(x).
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William Thomson, 1st Baron Kelvin
William Thomson, 1st Baron Kelvin, (26 June 1824 – 17 December 1907) was a Scots-Irish mathematical physicist and engineer who was born in Belfast in 1824.
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Wrapped distribution
In probability theory and directional statistics, a wrapped probability distribution is a continuous probability distribution that describes data points that lie on a unit ''n''-sphere.
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Zero of a function
In mathematics, a zero, also sometimes called a root, of a real-, complex- or generally vector-valued function f is a member x of the domain of f such that f(x) vanishes at x; that is, x is a solution of the equation f(x).
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Construction of Dirac delta function, Delta Function, Delta dirac, Delta distribution, Delta function, Delta measure, Delta pulse, Delta-function, Derac delta function, Dirac Delta Function, Dirac Delta function, Dirac Function, Dirac delta, Dirac delta distribution, Dirac delta functional, Dirac delta functions, Dirac delta-function, Dirac distribution, Dirac function, Dirac pulse, Dirac's delta, Dirac's delta function, Dirac's delta measure, Dirac's function, Impulse function, Nascent delta function, Sampling property, Unit impulse, Unit impulse function, Unity impulse, Unity impulse function.
References
[1] https://en.wikipedia.org/wiki/Dirac_delta_function
