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Vladimir Arnold
Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to several areas, including geometrical theory of dynamical systems, algebra, catastrophe theory, topology, real algebraic geometry, symplectic geometry, differential equations, classical mechanics, differential-geometric approach to hydrodynamics, geometric analysis and singularity theory, including posing the ADE classification problem. His first main result was the solution of Hilbert's thirteenth problem in 1957 when he was 19. He co-founded three new branches of mathematics: topological Galois theory (with his student Askold Khovanskii), symplectic topology and KAM theory. Arnold was also a populariser of mathematics. Through his lectures, seminars, and as the author of several textbooks (such as '' Mathematical Methods of Clas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odessa
ODESSA is an American codename (from the German language, German: ''Organisation der ehemaligen SS-Angehörigen'', meaning: Organization of Former SS Members) coined in 1946 to cover Ratlines (World War II aftermath), Nazi underground escape-plans made at the end of World War II by a group of ''SS'' officers with the aim of facilitating secret escape routes, and any directly ensuing arrangements. The concept of the existence of an actual ODESSA organisation has circulated widely in fictional Spy fiction, spy novels and movies, including Frederick Forsyth's best-selling 1972 thriller ''The Odessa File''. The escape-routes have become known as "Ratlines (World War II), ratlines". Known goals of elements within the ''SS'' included allowing ''SS'' members to escape to Argentina or to the Middle East under false passports. Although an unknown number of wanted Nazis and war criminals escaped Germany and often Europe, most experts deny that an organisation called ODESSA ever existed. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Varchenko
Alexander Nikolaevich Varchenko (, born February 6, 1949) is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics. Education and career From 1964 to 1966 Varchenko studied at the Moscow Kolmogorov boarding school No. 18 for gifted high school students, where Andrey Kolmogorov and Ya. A. Smorodinsky were lecturing mathematics and physics. Varchenko graduated from Moscow State University in 1971. He was a student of Vladimir Arnold. Varchenko defended his Ph.D. thesis ''Theorems on Topological Equisingularity of Families of Algebraic Sets and Maps'' in 1974 and Doctor of Science thesis ''Asymptotics of Integrals and Algebro-Geometric Invariants of Critical Points of Functions'' in 1982. From 1974 to 1984 he was a research scientist at the Moscow State University, in 1985–1990 a professor at the Gubkin Institute of Gas and Oil, and since 1991 he has been the Ernest Eliel Professor at the University of North Carolina at Chapel Hill. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold–Givental Conjecture
The Arnold conjecture, named after mathematician Vladimir Arnold, is a mathematical conjecture in the field of symplectic geometry, a branch of differential geometry. Strong Arnold conjecture Let (M, \omega) be a closed (compact without boundary) 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 formula :\omega( X_H, \cdot) = dH. The function H is called a Hamiltonian function. Suppose there is a smooth 1-parameter family of Hamiltonian functions H_t \in C^\infty(M), t \in ,1/math>. This family induces 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 \varphi_t is a called a Hamiltonian diffeomorphism of M. The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M is greater than or equal to the number of critical ... [...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. 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. It is notable as a simple example of a fluid flow that can have chaotic trajectories. 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 In fluid dynamics, a fluid parcel, also known as a fluid element or material element, is an infinitesimal volume of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel rema ... locate ... [...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 triggers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold's Spectral Sequence
In mathematics, Arnold's spectral sequence (also spelled Arnol'd) is a spectral sequence used in singularity theory and normal form theory as an efficient computational tool for reducing a function to canonical form near critical points. It was introduced by Vladimir Arnold Vladimir Igorevich Arnold (or Arnol'd; , ; 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. He is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, and contributed to s ... in 1975.Majid Gazor, Pei Yu,Spectral sequences and parametric normal forms, ''Journal of Differential Equations'' 252 (2012) no. 2, 1003–1031. Definition References Spectral sequences Singularity theory {{Algebra-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold's Rouble Problem
The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a Square (geometry), square or a Rectangle, rectangular napkin can increase its perimeter. The problem is known under several names, including the Margulis napkin problem, suggesting it is due to Grigory Margulis, and the Arnold's rouble problem referring to Vladimir Arnold and the folding of a Russian ruble bank note. It is the first problem listed by Arnold in his book ''Arnold's Problems'', where he calls it the rumpled dollar problem. Some versions of the problem were solved by Robert J. Lang, Svetlana Krat, Alexey S. Tarasov, and Ivan Yaschenko. One form of the problem remains open. Formulations There are several way to define the notion of Paper folding, folding, giving different interpretations. By convention, the napkin is always a unit Square (geometry), square. Folding along a straight line Considering the folding as a reflection along a line that re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold's Problems
''Arnold's Problems'' is a book edited by Soviet mathematician Vladimir Arnold, containing 861 mathematical problems from many different Mathematics#Areas_of_mathematics, areas of mathematics. The book was based on Arnold's seminars at Moscow State University. The problems were created over his decades-long career, and are sorted chronologically (from the period 1956–2003). It was published in Russian as ''Задачи Арнольда'' in 2000, and in a translated and revised English edition in 2004 (printed by Springer-Verlag). The book is divided into two parts: formulations of the problems, and comments upon them by 59 mathematicians. This is the largest part of the book. There are also long outlines for programs of research. Notable problems The problems in ''Arnold's Problems'' are each numbered with a year and a sequence number within the year. They include: *1956–1, the napkin folding problem, on whether a paper rectangle can be folded to a shape with larger perimeter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Invariants
In mathematics, particularly in topology and knot theory, Arnold invariants are Knot invariant, invariants introduced by Vladimir Arnold in 1994Arnold, V. I. (1994). ''Topological Invariants of Plane Curves and Caustics''. University Lecture Series, Vol. 5, American Mathematical Society. for studying the topology and geometry of plane curve, plane curves. The three main invariants—J^+, J^-, and St—provide ways to classify and understand how curves can be deformed while preserving certain properties.Mai, Alexander (2022). "Introduction to Arnold's J+-Invariant". arXiv:2210.00871. Background The fundamental context for Arnold invariants comes from the Whitney–Graustein theorem, which states that any two immersion (mathematics), immersed loops (smooth curves in the plane) with the same rotation number can be deformation (mathematics), deformed into each other through a series of continuous function, continuous Transformation (function), transformations.Whitney, H. (1937). "On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Diffusion
In applied mathematics, Arnold diffusion is the phenomenon of instability of nearly-integrable systems, 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 action-angle variables, 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 ... [...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. Strong Arnold conjecture Let (M, \omega) be a closed (compact without boundary) 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 formula :\omega( X_H, \cdot) = dH. The function H is called a Hamiltonian function. Suppose there is a smooth 1-parameter family of Hamiltonian functions H_t \in C^\infty(M), t \in [0,1]. This family induces 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 \varphi_t is a called a Hamiltonian diffeomorphism of M. The strong Arnold conjecture states that the number of fixed points of a Hamiltonian diffeomorphism of M is greater than or equal to the number of critical point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold's Cat Map
In mathematics, Arnold's cat map is a chaos theory, chaotic map from the torus into itself, named after Vladimir Arnold, who demonstrated its effects in the 1960s using an image of a cat, hence the name. It is a simple and pedagogical example for hyperbolic toral automorphisms. Thinking of the torus \mathbb^2 as the Quotient space (topology), quotient space \mathbb^2/\mathbb^2, Arnold's cat map is the transformation \Gamma : \mathbb^2 \to \mathbb^2 given by the formula :\Gamma (x,y) = (2x+y,x+y) \bmod 1. Equivalently, in matrix (mathematics), matrix notation, this is :\Gamma \left( \begin x \\ y \end \right) = \begin 2 & 1 \\ 1 & 1 \end \begin x \\ y \end \bmod 1 = \begin 1 & 1 \\ 0 & 1 \end \begin 1 & 0 \\ 1 & 1 \end \begin x \\ y \end \bmod 1. That is, with a unit equal to the width of the square image, the image is Shear mapping, sheared one unit up, then two units to the right, and all that lies outside that unit square is shifted back by the unit until it is within the sq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
