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Limiting Amplitude Principle
In mathematics, the limiting amplitude principle is a concept from operator theory and scattering theory used for choosing a particular solution to the Helmholtz equation. The choice is made by considering a particular time-dependent problem of the forced oscillations due to the action of a periodic force. The principle was introduced by Andrey Nikolayevich Tikhonov and Alexander Andreevich Samarskii. It is closely related to the limiting absorption principle (1905) and the Sommerfeld radiation condition (1912). The terminology -- both the limiting absorption principle and the limiting amplitude principle -- was introduced by Aleksei Sveshnikov. Formulation To find which solution to the Helmholz equation with nonzero right-hand side :\Delta v(x)+k^2 v(x)=-F(x),\quad x\in\R^3, with some fixed k>0, corresponds to the outgoing waves, one considers the wave equation with the source term, :(\Delta-\partial_t^2)u(x,t)=-F(x)e^,\quad t\ge 0, \quad x\in\R^3, with zero initial data u(x,0) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear operators or closed operators, and consideration may be given to nonlinear operators. The study, which depends heavily on the topology of function spaces, is a branch of functional analysis. If a collection of operators forms an algebra over a field, then it is an operator algebra. The description of operator algebras is part of operator theory. Single operator theory Single operator theory deals with the properties and classification of operators, considered one at a time. For example, the classification of normal operators in terms of their spectra falls into this category. Spectrum of operators The spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides cond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scattering Theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a rainbow. Scattering also includes the interaction of billiard balls on a table, the Rutherford scattering (or angle change) of alpha particles by gold nuclei, the Bragg scattering (or diffraction) of electrons and X-rays by a cluster of atoms, and the inelastic scattering of a fission fragment as it traverses a thin foil. More precisely, scattering consists of the study of how solutions of partial differential equations, propagating freely "in the distant past", come together and interact with one another or with a boundary condition, and then propagate away "to the distant future". The direct scattering problem is the problem of determining the distribution of scattered radiation/particle flux basing ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helmholtz Equation
In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenvalue, and is the (eigen)function. When the equation is applied to waves, is known as the wave number. The Helmholtz equation has a variety of applications in physics, including the wave equation and the diffusion equation, and it has uses in other sciences. Motivation and uses The Helmholtz equation often arises in the study of physical problems involving partial differential equations (PDEs) in both space and time. The Helmholtz equation, which represents a time-independent form of the wave equation, results from applying the technique of separation of variables to reduce the complexity of the analysis. For example, consider the wave equation \left(\nabla^2-\frac\frac\right) u(\mathbf,t)=0. Separation of variables begins by assumi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Forced Oscillations
Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum and alternating current. Oscillations can be used in physics to approximate complex interactions, such as those between atoms. Oscillations occur not only in mechanical systems but also in dynamic systems in virtually every area of science: for example the beating of the human heart (for circulation), business cycles in economics, predator–prey population cycles in ecology, geothermal geysers in geology, vibration of strings in guitar and other string instruments, periodic firing of nerve cells in the brain, and the periodic swelling of Cepheid variable stars in astronomy. The term ''vibration'' is precisely used to describe a mechanical oscillation. Oscillation, especially rapid oscillation, may be an undesirable phenomenon in pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Nikolayevich Tikhonov
Andrey Nikolayevich Tikhonov (russian: Андре́й Никола́евич Ти́хонов; October 17, 1906 – October 7, 1993) was a leading Soviet Russian mathematician and geophysicist known for important contributions to topology, functional analysis, mathematical physics, and ill-posed problems. He was also one of the inventors of the magnetotellurics method in geophysics. Other transliterations of his surname include "Tychonoff", "Tychonov", "Tihonov", "Tichonov." Biography Born in Gzhatsk, he studied at the Moscow State University where he received a Ph.D. in 1927 under the direction of Pavel Sergeevich Alexandrov. In 1933 he was appointed as a professor at Moscow State University. He became a corresponding member of the USSR Academy of Sciences on 29 January 1939 and a full member of the USSR Academy of Sciences on 1 July 1966. Research work Tikhonov worked in a number of different fields in mathematics. He made important contributions to topology, functiona ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Andreevich Samarskii
Alexander Andreevich Samarskii (Александр Андреевич Самарский, 19 February 1919, Amvrosiivka, metropolitan Donetsk, Yekaterinoslav Governorate – 11 February 2008, Moscow) was a Soviet and Russian mathematician and academician (USSR Academy of Sciences, Russian Academy of Sciences), specializing in mathematical physics, applied mathematics, numerical analysis, mathematical modeling, finite difference methods. Education and career Born in Amvrosiivka, Yekaterinoslav Governorate, Russian Empire (now, Donetsk Oblast, Ukraine). Samarskii studied from 1936 at Moscow State University, interrupted from 1941 to 1944 by voluntary military service in WW II — he was severely wounded in the Battle of Moscow. In 1948 he received his Russian candidate degree (Ph.D.). At the same time, he worked with Andrey Nikolayevich Tikhonov on mathematical modeling of nuclear weapon explosions and electromagnetic fields in waveguides. In the 1950s Samarskii worked on finite d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Absorption Principle
In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum. The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the L^2 space), but in certain weighted spaces (usually L^2_s, see below), has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing complex parameter into the Helmholtz equation (\Delta+k^2)u(x)=-F(x) for selecting a particular solution. This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire. It is closely related to the Sommerfeld radiation condition and the limiting amplitude principle (1948). The terminology – both the limiting absorption pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sommerfeld Radiation Condition
In applied mathematics, the Sommerfeld radiation condition is a concept from theory of differential equations and scattering theory used for choosing a particular solution to the Helmholtz equation. It was introduced by Arnold Sommerfeld in 1912 and is closely related to the limiting absorption principle (1905) and the limiting amplitude principle (1948). Formulation Arnold Sommerfeld defined the condition of radiation for a scalar field satisfying the Helmholtz equation as : "the sources must be sources, not sinks of energy. The energy which is radiated from the sources must scatter to infinity; no energy may be radiated from infinity into ... the field."A. Sommerfeld, ''Partial Differential Equations in Physics'', Academic Press, New York, New York, 1949. Mathematically, consider the inhomogeneous Helmholtz equation : (\nabla^2 + k^2) u = -f \text \mathbb R^n where n=2, 3 is the dimension of the space, f is a given function with compact support representing a boun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksei Sveshnikov
Aleksei Georgievich Sveshnikov (russian: link=y, Алексей Георгиевич Свешников, 19 November 1924 – 4 July 2022) was a Russian mathematical physicist. Biography Born in Saratov as the son of Georgy Nikolaevich Sveshnikov and Vera Konstantinovna Sveshnikova (''née'' Snitko), A. G. Sveshnikov graduated from a Moscow high school in 1941. As an artillery soldier and platoon commander in WWII, he was in April 1945 seriously wounded on the 4th Ukrainian Front. He was awarded the Order of the Red Star (1945), the Order of the Patriotic War of the 1st Degree (1995), the Medal for Victory over Germany (1945) and many jubilee medals. After demobilization in 1945, he entered the Faculty of Physics of Moscow State University, from which he graduated in 1950. After graduating from the university, he worked at the Physics Department of Moscow State University. There in 1953 he received his Candidate of Sciences degree (PhD) with thesis Принцип излуче ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The (two-way) wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields — as they occur in classical physics — such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. Single mechanical or electromagnetic waves propagating in a pre-defined direction can also be described with the first-order one-way wave equation which is much easier to solve and also valid for inhomogenious media. Introduction The (two-way) wave equation is a second-order partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable repres ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limiting Absorption Principle
In mathematics, the limiting absorption principle (LAP) is a concept from operator theory and scattering theory that consists of choosing the "correct" resolvent of a linear operator at the essential spectrum based on the behavior of the resolvent near the essential spectrum. The term is often used to indicate that the resolvent, when considered not in the original space (which is usually the L^2 space), but in certain weighted spaces (usually L^2_s, see below), has a limit as the spectral parameter approaches the essential spectrum. This concept developed from the idea of introducing complex parameter into the Helmholtz equation (\Delta+k^2)u(x)=-F(x) for selecting a particular solution. This idea is credited to Vladimir Ignatowski, who was considering the propagation and absorption of the electromagnetic waves in a wire. It is closely related to the Sommerfeld radiation condition and the limiting amplitude principle (1948). The terminology – both the limiting absorption pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operators
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
