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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dynamical System
In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models that describe the swinging of a clock pendulum, fluid dynamics, the flow of water in a pipe, the Brownian motion, random motion of particles in the air, and population dynamics, the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real number, real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a Set (mathematics), set, without the need of a Differentiability, smooth space-time structure defined on it. At any given time, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gabriella Pinzari
Gabriella Pinzari is an Italian mathematician known for her research on the -body problem. Research Pinzari's research on the -body problem has been described as "the most natural way to apply" the Kolmogorov–Arnold–Moser theorem to the problem. The original work of Vladimir Arnold on this theorem attempted to use it to show the stability of the Solar System or similar systems of planetary orbits, but this worked only for the three-body problem because of a degeneracy in Arnold's mathematical framework. Pinzari showed how to eliminate this problem, and extended the solution to larger numbers of bodies, by developing "a rotation-invariant version of the KAM theory". Education and career Pinzari earned master's degrees in both physics and mathematics from Sapienza University of Rome, in 1990 and 1996 respectively. She completed her doctorate in 2009 at Roma Tre University under the supervision of Luigi Chierchia. She joined the faculty of the University of Naples Federico II ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ergodic Theory
Ergodic theory is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, "statistical properties" refers to properties which are expressed through the behavior of time averages of various functions along trajectories of dynamical systems. The notion of deterministic dynamical systems assumes that the equations determining the dynamics do not contain any random perturbations, noise, etc. Thus, the statistics with which we are concerned are properties of the dynamics. Ergodic theory, like probability theory, is based on general notions of measure theory. Its initial development was motivated by problems of statistical physics. A central concern of ergodic theory is the behavior of a dynamical system when it is allowed to run for a long time. The first result in this direction is the Poincaré recurrence theorem, which claims that almost all points in any subset of the phase space eventua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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] [Amazon] |
Stability Of The Solar System
The stability of the Solar System is a subject of much inquiry in astronomy. Though the planets have historically been stable as observed, and will be in the "short" term, their weak gravitational effects on one another can add up in ways that are not predictable by any simple means. For this reason (among others), the Solar System is ''chaotic'' in the technical sense defined by mathematical chaos theory, and that chaotic behavior degrades even the most precise long-term numerical or analytic models for the orbital motion in the Solar System, so they cannot be valid beyond more than a few tens of millions of years into the past or future – about 1% its present age. The Solar System is stable on the time-scale of the existence of humans, and far beyond, given that it is unlikely any of the planets will collide with each other or be ejected from the system in the next few billion years, and that Earth's orbit will be relatively stable. Since Newton's law of gravitation (16 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Michael Herman (mathematician)
Michael Robert Herman (6 November 1942 – 2 November 2000) was a French American mathematician. He was one of the leading experts on the theory of dynamical systems. Born in New York City, he was educated in France. He was a student at École polytechnique before being one of the first members of the Centre de Mathématiques created there by Laurent Schwartz. In 1976 he earned his PhD at the Paris-Sud 11 University, under supervision of Harold Rosenberg. He introduced Herman rings in 1979. Herman received the Salem Prize in 1976. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1978 in Helsinki and the ICM in 1998 in Berlin. Among his students was Jean-Christophe Yoccoz, 1994 Fields Medalist. See also * Almost Mathieu operator *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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Ian C
Ian or Iain is a name of Scottish Gaelic origin, which is derived from the Hebrew given name (Yohanan, ') and corresponds to the English name John. The spelling Ian is an Anglicization of the Scottish Gaelic forename ''Iain''. This name is a popular name in Scotland, where it originated, as well as in other English-speaking countries. The name has fallen out of the top 100 male baby names in the United Kingdom, having peaked in popularity as one of the top 10 names throughout the 1960s. In 1900, Ian ranked as the 180th most popular male baby name in England and Wales. , the name has been in the top 100 in the United States every year since 1982, peaking at 65 in 2003. Other Gaelic forms of the name "John" include " Seonaidh" ("Johnny" from Lowland Scots), "Seon" (from English), "Seathan", and "Seán" and "Eoin" (from Irish). The Welsh equivalent is Ioan, the Cornish counterpart is Yowan and the Breton equivalent is Yann. Notable people named Ian Given name * Ian Agol (b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned this ternary construction only in passing, as an example of a perfect set that is nowhere dense. More generally, in topology, a Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). The Cantor set is naturally homeomorphic to the countable product ^ of the discrete two-point space \underline 2 . By a theorem of L. E. J. Brouwer, this is equivalent to being perfect, nonempty, compac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Quasiperiodic
Quasiperiodicity is the property of a system that displays irregular periodicity. Periodic behavior is defined as recurring at regular intervals, such as "every 24 hours". Quasiperiodic behavior is almost but not quite periodic. The term used to denote oscillations that appear to follow a regular pattern but which do not have a fixed period. The term thus used does not have a precise definition and should not be confused with more strictly defined mathematical concepts such as an almost periodic function or a quasiperiodic function. Climatology Climate oscillations that appear to follow a regular pattern but which do not have a fixed period are called ''quasiperiodic''.''The meteorological glossary: 2d ed.'' 1930. Meteorological Office, Great Britain. "Certain phenomena which recur more or less regularly but without the exactness of truly periodic phenomena are termed quasi-periodic." Within a dynamical system such as the ocean-atmosphere system, oscillations may occur regularly ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses include ring toruses, horn toruses, and spindle toruses. A ring torus is sometimes colloquially referred to as a donut or doughnut. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a torus of revolution, also known as a ring torus. If the axis of revolution is tangent to the circle, the surface is a horn torus. If the axis of revolution passes twice through the circle, the surface is a Lemon (geometry), spindle torus (or ''self-crossing torus'' or ''self-intersecting torus''). If the axis of revolution passes through the center of the circle, the surface is a degenerate torus, a double-covered sphere. If the revolved curve is not a circle, the surface is called a ''toroid'', as in a square toroid. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Initial Condition
In mathematics and particularly in dynamic systems, an initial condition, in some contexts called a seed value, is a value of an evolving variable at some point in time designated as the initial time (typically denoted ''t'' = 0). For a system of order ''k'' (the number of time lags in discrete time, or the order of the largest derivative in continuous time) and dimension ''n'' (that is, with ''n'' different evolving variables, which together can be denoted by an ''n''-dimensional coordinate vector), generally ''nk'' initial conditions are needed in order to trace the system's variables forward through time. In both differential equations in continuous time and difference equations in discrete time, initial conditions affect the value of the dynamic variables (state variables) at any future time. In continuous time, the problem of finding a closed form solution for the state variables as a function of time and of the initial conditions is called the initial valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Doughnut
A doughnut or donut () is a type of pastry made from leavened fried dough. It is popular in many countries and is prepared in various forms as a sweet snack that can be homemade or purchased in bakeries, supermarkets, food stalls, and franchised specialty vendors. ''Doughnut'' is the traditional spelling, while ''donut'' is the simplified version; the terms are used interchangeably. Doughnuts are usually deep fried from a flour dough, but other types of batters can also be used. Various toppings and flavors are used for different types, such as sugar, chocolate or maple glazing. Doughnuts may also include water, leavening, eggs, milk, sugar, oil, shortening, and natural or artificial flavors. The two most common types are the ring doughnut and the filled doughnut, which is injected with fruit preserves (the jelly doughnut), cream, custard, or other sweet fillings. Small pieces of dough are sometimes cooked as doughnut holes. Once fried, doughnuts may be glazed with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
