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V. I. 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 famous ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Odessa
Odesa (also spelled Odessa) is the third most populous city and municipality in Ukraine and a major seaport and transport hub located in the south-west of the country, on the northwestern shore of the Black Sea. The city is also the administrative centre of the Odesa Raion and Odesa Oblast, as well as a multiethnic cultural centre. As of January 2021 Odesa's population was approximately In classical antiquity a large Greek settlement existed at its location. The first chronicle mention of the Slavic settlement-port of Kotsiubijiv, which was part of the Grand Duchy of Lithuania, dates back to 1415, when a ship was sent from here to Constantinople by sea. After a period of Lithuanian Grand Duchy control, the port and its surroundings became part of the domain of the Ottomans in 1529, under the name Hacibey, and remained there until the empire's defeat in the Russo-Turkish War of 1792. In 1794, the modern city of Odesa was founded by a decree of the Russian empress Catherine t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Anatolyevich Vassiliev
Victor Anatolyevich Vassiliev or Vasilyev ( ru , Виктор Анатольевич Васильев; born April 10, 1956), is a Soviet and Russian mathematician. He is best known for his discovery of the Vassiliev invariants in knot theory (also known as finite type invariants), which subsume many previously discovered polynomial knot invariants such as the Jones polynomial. He also works on singularity theory, topology, computational complexity theory, integral geometry, symplectic geometry, partial differential equations (geometry of wavefronts), complex analysis, combinatorics, and Picard–Lefschetz theory. Biography Vassiliev studied at the Faculty of Mathematics and Mechanics at the Lomonosov University in Moscow until 1981. From 1981 to 1987 he was Senior Researcher at the Documents and Archives Research Institute, Moscow and a part-time mathematics teacher at Specialized Mathematical School No. 57, Moscow. In 1982 he defended his Kandidat nauk thesis under Vladimir A ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert's Thirteenth Problem
Hilbert's thirteenth problem is one of the 23 Hilbert problems set out in a celebrated list compiled in 1900 by David Hilbert. It entails proving whether a solution exists for all 7th-degree equations using algebraic (variant: continuous) functions of two arguments. It was first presented in the context of nomography, and in particular "nomographic construction" — a process whereby a function of several variables is constructed using functions of two variables. The variant for continuous functions was resolved affirmatively in 1957 by Vladimir Arnold when he proved the Kolmogorov–Arnold representation theorem, but the variant for algebraic functions remains unresolved. Introduction William Rowan Hamilton showed in 1836 that every seventh-degree equation can be reduced via radicals to the form x^7 + ax^3 + bx^2 + cx + 1 = 0. Regarding this equation, Hilbert asked whether its solution, ''x'', considered as a function of the three variables ''a'', ''b'' and ''c'', can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gudkov's Conjecture
In real algebraic geometry, Gudkov's conjecture, also called Gudkov’s congruence, (named after Dmitry Gudkov) was a conjecture, and is now a theorem, which states that an M-curve of even degree 2d obeys the congruence : p - n \equiv d^2\, (\!\bmod 8), where p is the number of positive ovals and n the number of negative ovals of the M-curve. (Here, the term M-curve stands for "maximal curve"; it means a smooth algebraic curve over the reals whose genus is k-1, where k is the number of maximal components of the curve.) The theorem was proved by the combined works of Vladimir Arnold and Vladimir Rokhlin. See also *Hilbert's sixteenth problem *Tropical geometry In mathematics, tropical geometry is the study of polynomials and their geometric properties when addition is replaced with minimization and multiplication is replaced with ordinary addition: : x \oplus y = \min\, : x \otimes y = x + y. So f ... References {{reflist Conjectures that have been proved Theorems in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gömböc
The Gömböc ( ) is the first known physical example of a class of convex three-dimensional homogeneous bodies, called mono-monostatic, which, when resting on a flat surface have just one stable and one unstable point of equilibrium. The existence of this class was conjectured by the Russian mathematician Vladimir Arnold in 1995 and proven in 2006 by the Hungarian scientists Gábor Domokos and Péter Várkonyi by constructing at first a mathematical example and subsequently a physical example. Mono-monostatic shapes exist in countless varieties, most of which are close to a sphere, with a stringent shape tolerance (about one part in a thousand). Gömböc is the first mono-monostatic shape which has been constructed physically. It has a sharpened top, as shown in the photo. Its shape helped to explain the body structure of some tortoises in relation to their ability to return to an equilibrium position after being placed upside down. Copies of the Gömböc have been donate ... [...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–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 In fluid dynamics, Beltrami flow ... [...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 (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–A ... in 1975.Majid Gazor, Pei Yu,Spectral sequences and parametric normal forms, ''Journal of Differential Equations'' 252 (2012) no. 2, 1003–1031. Definition References {{Algebra-stub Spectral sequences Singularity theory ... [...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 or a 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. 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 folding, giving different interpretations. By convention, the napkin is always a unit square. Folding along a straight line Considering the folding as a reflection along a line that reflects all the layers of the napkin, the perimeter is always non-increasing, thus never exceeding 4. By considering more general foldings that possibly reflect only a single layer of ... [...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 and ... [...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] |
