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Screened Poisson Equation
In physics, the screened Poisson equation is a Poisson equation, which arises in (for example) the Klein–Gordon equation, electric field screening in Plasma (physics), plasmas, and nonlocal granular fluidity in granular flow. Statement of the equation The equation is \left[ \Delta - \lambda^2 \right] u(\mathbf) = - f(\mathbf), where \Delta is the Laplace operator, ''λ'' is a constant that expresses the "screening", ''f'' is an arbitrary function of position (known as the "source function") and ''u'' is the function to be determined. In the homogeneous case (''f''=0), the screened Poisson equation is the same as the time-independent Klein–Gordon equation. In the inhomogeneous case, the screened Poisson equation is very similar to the Helmholtz equation#Inhomogeneous Helmholtz equation, inhomogeneous Helmholtz equation, the only difference being the sign within the brackets. Electrostatics In electric-field screening, screened Poisson equation for the electric potential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physics
Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succession of events." It is one of the most fundamental scientific disciplines. "Physics is one of the most fundamental of the sciences. Scientists of all disciplines use the ideas of physics, including chemists who study the structure of molecules, paleontologists who try to reconstruct how dinosaurs walked, and climatologists who study how human activities affect the atmosphere and oceans. Physics is also the foundation of all engineering and technology. No engineer could design a flat-screen TV, an interplanetary spacecraft, or even a better mousetrap without first understanding the basic laws of physics. (...) You will come to see physics as a towering achievement of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function (mathematics), function as input then outputs another function that describes the extent to which various Frequency, frequencies are present in the original function. The output of the transform is a complex number, complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the Operation (mathematics), mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical Chord (music), chord into the sound intensity, intensities of its constituent Pitch (music), pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the #Uncertainty principle, uncerta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yukawa Interaction
In particle physics, Yukawa's interaction or Yukawa coupling, named after Hideki Yukawa, is an interaction between particles according to the Yukawa potential. Specifically, it is between a scalar field (or pseudoscalar field) \ \phi\ and a Dirac field \ \psi\ of the type The Yukawa interaction was developed to model the strong force between hadrons. A Yukawa interaction is thus used to describe the nuclear force between nucleons mediated by pions (which are pseudoscalar mesons). A Yukawa interaction is also used in the Standard Model to describe the coupling between the Higgs field and massless quark and lepton fields (i.e., the fundamental fermion particles). Through spontaneous symmetry breaking, these fermions acquire a mass proportional to the vacuum expectation value of the Higgs field. This Higgs-fermion coupling was first described by Steven Weinberg in 1967 to model lepton masses. Classical potential If two fermions interact through a Yukawa interaction mediate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Laplace Distribution
In the mathematical theory of probability, multivariate Laplace distributions are extensions of the Laplace distribution and the asymmetric Laplace distribution to multiple variables. The marginal distributions of symmetric multivariate Laplace distribution variables are Laplace distributions. The marginal distributions of asymmetric multivariate Laplace distribution variables are asymmetric Laplace distributions. Symmetric multivariate Laplace distribution A typical characterization of the symmetric multivariate Laplace distribution has the characteristic function: : \varphi(t;\boldsymbol\mu,\boldsymbol\Sigma) = \frac, where \boldsymbol\mu is the vector of means for each variable and \boldsymbol\Sigma is the covariance matrix. Unlike the multivariate normal distribution, even if the covariance matrix has zero covariance and correlation the variables are not independent. The symmetric multivariate Laplace distribution is elliptical. Probability density function If \bold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Probability Density Function
In probability theory, a probability density function (PDF), density function, or density of an absolutely continuous random variable, is a Function (mathematics), 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 be equal to that sample. Probability density is the probability per unit length, in other words, while the ''absolute likelihood'' for a continuous random variable to take on any particular value is 0 (since there is an infinite set of possible values to begin with), the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would be close to one sample compared to the other sample. More precisely, the PDF is used to specify the probability of the random variable falling ''within ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, named after Friedrich Bessel who was the first to systematically study them in 1824, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, which represents the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when solving the Helmholtz equation in spherical coordinates. Applications Bessel's equation arises when finding separa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Domain
In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time series. While a time-domain graph shows how a signal changes over time, a frequency-domain graph shows how the signal is distributed within different frequency bands over a range of frequencies. A complex valued frequency-domain representation consists of both the magnitude and the phase of a set of sinusoids (or other basis waveforms) at the frequency components of the signal. Although it is common to refer to the magnitude portion (the real valued frequency-domain) as the frequency response of a signal, the phase portion is required to uniquely define the signal. A given function or signal can be converted between the time and frequency domains with a pair of mathematical operators called transforms. An example is the Fourier transfo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Polar Coordinates
In mathematics, the polar coordinate system specifies a given point (mathematics), point in a plane (mathematics), plane by using a distance and an angle as its two coordinate system, coordinates. These are *the point's distance from a reference point called the ''pole'', and *the point's direction from the pole relative to the direction of the ''polar axis'', a ray (geometry), ray drawn from the pole. The distance from the pole is called the ''radial coordinate'', ''radial distance'' or simply ''radius'', and the angle is called the ''angular coordinate'', ''polar angle'', or ''azimuth''. The pole is analogous to the origin in a Cartesian coordinate system. Polar coordinates are most appropriate in any context where the phenomenon being considered is inherently tied to direction and length from a center point in a plane, such as spirals. Planar physical systems with bodies moving around a central point, or phenomena originating from a central point, are often simpler and more in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Fourier Transform
In mathematics, the Fourier inversion theorem says that for many types of functions it is possible to recover a function from its Fourier transform. Intuitively it may be viewed as the statement that if we know all frequency#Frequency_of_waves, frequency and phase (waves), phase information about a wave then we may reconstruct the original wave precisely. The theorem says that if we have a function f:\R \to \Complex satisfying certain conditions, and we use the Fourier transform#Other conventions, convention for the Fourier transform that :(\mathcalf)(\xi):=\int_ e^ \, f(y)\,dy, then :f(x)=\int_ e^ \, (\mathcalf)(\xi)\,d\xi. In other words, the theorem says that :f(x)=\iint_ e^ \, f(y)\,dy\,d\xi. This last equation is called the Fourier integral theorem. Another way to state the theorem is that if R is the flip operator i.e. (Rf)(x) := f(-x), then :\mathcal^=\mathcalR=R\mathcal. The theorem holds if both f and its Fourier transform are absolutely integrable function, ab ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Larmor Radius
In physics, cyclotron motion, also known as gyromotion, refers to the circular motion exhibited by charged particles in a uniform magnetic field. The circular trajectory of a particle in cyclotron motion is characterized by an angular frequency referred to as the cyclotron frequency or gyrofrequency and a radius referred to as the cyclotron radius, gyroradius, or Larmor radius. For a particle with charge q and mass m initially moving with speed v_\perp perpendicular to the direction of a uniform magnetic field B, the cyclotron radius is: r_ = \frac and the cyclotron frequency is: \omega_ = \frac. An external oscillating field matching the cyclotron frequency, \omega=\omega_c, will accelerate the particles, a phenomenon known as cyclotron resonance. This resonance is the basis for many scientific and engineering uses of cyclotron motion. In quantum mechanical systems, the energies of cyclotron orbits are quantized into discrete Landau levels, which contribute to Landau diamagnet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yukawa Potential
Yukawa (written: 湯川) is a Japanese surname, but is also applied to proper nouns. People * Diana Yukawa (born 1985), Anglo-Japanese solo violinist. She has had two solo albums with BMG Japan, one of which opened to #1 * Hideki Yukawa (1907–1981), Japanese theoretical physicist and the first Japanese Nobel laureate * Kazuyuki Yukawa (1949–2025), Japanese politician * Morio Yukawa (1908–1988), Japanese economist and diplomat * Tsutomu Yukawa (1911–1942), Japanese aikidoka * Yasutoshi Yukawa (1941–2014), Japanese linguist Fictional characters Places * Yukawa Institute for Theoretical Physics, a research institute in the field of theoretical physics, attached to Kyoto University in Japan * Yukawa Station, a train station in Nachikatsuura, Higashimuro District, Wakayama Prefecture, Japan Other * Yukawa interaction, named after Hideki Yukawa, is an interaction between a scalar field φ and a Dirac field Ψ of a particular type * Yukawa potential, a potential of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
