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List Of Things Named After Vladimir Arnold
A list of things named after Vladimir Arnold, a Russian and Soviet mathematician. *Arnold–Givental conjecture *Arnold's cat map *Arnold's rouble problem *Arnold's spectral sequence * Arnold's stability theorem *Arnold conjecture *Arnold diffusion * Arnold invariants *Arnold tongue * Arnold web *Arnold–Beltrami–Childress flow *Kolmogorov–Arnold–Moser theorem *Kolmogorov–Arnold representation theorem *Liouville–Arnold theorem In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with ''n'' degrees of freedom, there are also ''n'' independent, Poisson commuting first integrals of motion, and the energy level set ... Other *The 10031 Vladarnolda, minor planet. * The '' Arnold Mathematical Journal'', published for the first time in 2015, is named after him.. References {{Reflist Arnold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, russian: link=no, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, algebra, catastrophe theory, topology, algebraic geometry, symplectic geometry, differential equations, classical mechanics, hydrodynamics and singularity theory, including posing the ADE classification problem, since his first main result—the solution of Hilbert's thirteenth problem in 1957 at the age of 19. He co-founded two new branches of mathematics—KAM theory, and topological Galois theory (this, with his student Askold Khovanskii). Arnold was also known as a popularizer of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Tongue
In mathematics, particularly in dynamical systems, Arnold tongues (named after Vladimir Arnold) Section 12 in page 78 has a figure showing Arnold tongues. are a pictorial phenomenon that occur when visualizing how the rotation number of a dynamical system, or other related invariant property thereof, changes according to two or more of its parameters. The regions of constant rotation number have been observed, for some dynamical systems, to form geometric shapes that resemble tongues, in which case they are called Arnold tongues. Arnold tongues are observed in a large variety of natural phenomena that involve oscillating quantities, such as concentration of enzymes and substrates in biological processes and cardiac electric waves. Sometimes the frequency of oscillation depends on, or is constrained (i.e., ''phase-locked'' or ''mode-locked'', in some contexts) based on some quantity, and it is often of interest to study this relation. For instance, the outset of a tumor trigger ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
10031 Vladarnolda
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Liouville–Arnold Theorem
In dynamical systems theory, the Liouville–Arnold theorem states that if, in a Hamiltonian dynamical system with ''n'' degrees of freedom, there are also ''n'' independent, Poisson commuting first integrals of motion, and the energy level set is compact, then there exists a canonical transformation to action-angle coordinates in which the transformed Hamiltonian is dependent only upon the action coordinates and the angle coordinates evolve linearly in time. Thus the equations of motion for the system can be solved in quadratures if the level simultaneous set conditions can be separated. The theorem is named after Joseph Liouville and Vladimir Arnold.J. Liouville, « Note sur l'intégration des équations différentielles de la Dynamique, présentée au Bureau des Longitudes le 29 juin 1853 », '' JMPA'', 1855, pdf/ref> History The theorem was proven in its original form by Liouville in 1853 for functions on \mathbb^ with canonical symplectic structure. It was generalized to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov–Arnold Representation Theorem
In real analysis and approximation theory, the Kolmogorov-Arnold representation theorem (or superposition theorem) states that every multivariate continuous function can be represented as a superposition of the two-argument addition and continuous functions of one variable. It solved a more constrained, yet more general form of Hilbert's thirteenth problem. The works of Vladimir Arnold and Andrey Kolmogorov established that if ''f'' is a multivariate continuous function, then ''f'' can be written as a finite composition of continuous functions of a single variable and the binary operation of addition. More specifically, : f(\mathbf x) = f(x_1,\ldots ,x_n) = \sum_^ \Phi_\left(\sum_^ \phi_(x_)\right) . There are proofs with specific constructions. In a sense, they showed that the only true multivariate function is the sum, since every other function can be written using univariate functions and summing.Persi Diaconis and Mehrdad Shahshahani, ''On Linear Functions of Linear Combinat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kolmogorov–Arnold–Moser Theorem
The Kolmogorov–Arnold–Moser (KAM) theorem is a result in dynamical systems about the persistence of quasiperiodic motions under small perturbations. The theorem partly resolves the small-divisor problem that arises in the perturbation theory of classical mechanics. The problem is whether or not a small perturbation of a conservative dynamical system results in a lasting quasiperiodic orbit. The original breakthrough to this problem was given by Andrey Kolmogorov in 1954. This was rigorously proved and extended by Jürgen Moser in 1962 (for smooth twist maps) and Vladimir Arnold in 1963 (for analytic Hamiltonian systems), and the general result is known as the KAM theorem. Arnold originally thought that this theorem could apply to the motions of the Solar System or other instances of the -body problem, but it turned out to work only for the three-body problem because of a degeneracy in his formulation of the problem for larger numbers of bodies. Later, Gabriella Pinzari sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold–Beltrami–Childress Flow
The Arnold–Beltrami–Childress (ABC) flow or Gromeka–Arnold–Beltrami–Childress (GABC) flow is a three-dimensional incompressible velocity field which is an exact solution of Euler's equation. Its representation in Cartesian coordinates is the following: : \dot = A \sin z + C \cos y, : \dot = B \sin x + A \cos z, : \dot = C \sin y + B \cos x, where (\dot,\dot,\dot) is the material derivative of the Lagrangian motion of a fluid parcel located at (x(t),y(t),z(t)). It is notable as a simple example of a fluid flow that can have chaotic trajectories. It is named after Vladimir Arnold, Eugenio Beltrami, and Stephen Childress. Ippolit S. Gromeka's (1881) name has been historically neglected, though much of the discussion has been done by him first.Zermelo, Ernst. Ernst Zermelo-Collected Works/Gesammelte Werke: Volume I/Band I-Set Theory, Miscellanea/Mengenlehre, Varia. Vol. 21. Springer Science & Business Media, 2010. See also *Beltrami flow References * V. I. Arnold. "S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Web
Arnold may refer to: People * Arnold (given name), a masculine given name * Arnold (surname), a German and English surname Places Australia * Arnold, Victoria, a small town in the Australian state of Victoria Canada * Arnold, Nova Scotia United Kingdom * Arnold, East Riding of Yorkshire * Arnold, Nottinghamshire United States * Arnold, California, in Calaveras County * Arnold, Carroll County, Illinois * Arnold, Morgan County, Illinois * Arnold, Iowa * Arnold, Kansas * Arnold, Maryland * Arnold, Mendocino County, California * Arnold, Michigan * Arnold, Minnesota * Arnold, Missouri * Arnold, Nebraska * Arnold, Ohio * Arnold, Pennsylvania * Arnold, Texas * Arnold, Brooke County, West Virginia * Arnold, Lewis County, West Virginia * Arnold, Wisconsin * Arnold Arboretum of Harvard University, Massachusetts * Arnold Township, Custer County, Nebraska Other uses * Arnold (automobile), a short-lived English car * Arnold of Manchester, a former English coachbuilder * Arnol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold–Givental Conjecture
The Arnold–Givental conjecture, named after Vladimir Arnold and Alexander Givental, is a statement on Lagrangian submanifolds. It gives a lower bound in terms of the Betti numbers of a Lagrangian submanifold on the number of intersection points of with another Lagrangian submanifold which is obtained from by Hamiltonian isotopy, and which intersects transversally. Statement Let (M, \omega) be a compact 2n-dimensional symplectic manifold. An anti-symplectic involution is a diffeomorphism \tau: M \to M such that \tau^* \omega = -\omega. The fixed point set L \subset M of \tau is necessarily a Lagrangian submanifold. Let H_t\in C^\infty(M), 0 \leq t \leq 1 be a smooth family of Hamiltonian functions on M which generates a 1-parameter family of Hamiltonian diffeomorphisms \varphi_t: M \to M. The Arnold–Givental conjecture says, suppose \varphi_1(L) intersects transversely with L, then \# (\varphi_1(L) \cap L) \geq \sum_^n H_*(L; _2). Status The Arnold–Givental conjecture ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Diffusion
In applied mathematics, Arnold diffusion is the phenomenon of instability of integrable Hamiltonian systems. The phenomenon is named after Vladimir Arnold who was the first to publish a result in the field in 1964. More precisely, Arnold diffusion refers to results asserting the existence of solutions to nearly integrable Hamiltonian systems that exhibit a significant change in the action variables. Arnold diffusion describes the diffusion of trajectories due to the ergodic theorem in a portion of phase space unbound by any constraints (''i.e.'' unbounded by Lagrangian tori arising from constants of motion) in Hamiltonian systems. It occurs in systems with more than ''N''=2 degrees of freedom, since the ''N''-dimensional invariant tori do not separate the 2''N''-1 dimensional phase space any more. Thus, an arbitrarily small perturbation may cause a number of trajectories to wander pseudo-randomly through the whole portion of phase space left by the destroyed tori. Background an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Statement Let (M, \omega) be a compact symplectic manifold. For any smooth function H: M \to , the symplectic form \omega induces a Hamiltonian vector field X_H on M, defined by the identity \omega( X_H, \cdot) = dH. The function H is called a Hamiltonian function. Suppose there is a 1-parameter family of Hamiltonian functions H_t: M \to , 0 \leq t \leq 1, inducing a 1-parameter family of Hamiltonian vector fields X_ on M. The family of vector fields integrates to a 1-parameter family of diffeomorphisms \varphi_t: M \to M. Each individual of \varphi_t is a Hamiltonian diffeomorphism of M. The Arnold conjecture says that for each Hamiltonian diffeomorphism of M, it possesses at least as many fixed points as a smooth function on M possesses critical points. Nondegenerate Hamiltonian and weak Arnold conjecture A H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
