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Chaotic Scattering
Chaotic scattering is a branch of chaos theory dealing with scattering systems displaying a strong sensitivity to initial conditions. In a classical scattering system there will be one or more ''impact parameters'', ''b'', in which a particle is sent into the scatterer. This gives rise to one or more exit parameters, ''y'', as the particle exits towards infinity. While the particle is traversing the system, there may also be a ''delay time'', ''T''—the time it takes for the particle to exit the system—in addition to the distance travelled, ''s'', which in certain systems, i.e., "billiard-like" systems in which the particle undergoes lossless collisions with ''hard'', fixed objects, the two will be equivalent—see below. In a chaotic scattering system, a minute change in the impact parameter, may give rise to a very large change in the exit parameters. Gaspard–Rice system An excellent example system is the "Gaspard–Rice" (GR) scattering system —also known simply as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures ( length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge. Far-reaching generalizations (such as spectral measures and projection-valued measures) of measure are widely used in quantum physics and physics in general. The intuition behind this concept dates back to ancient Greece, when Archimedes tried to calculate the area of a circle. But it was not until the late 19th and early 20th centuries that measure theory became a branch of mathematics. The foundations of modern measure theory were laid in the works of Émile Borel, Henri Lebesgue, Nikolai Luzin, Johann Radon, Const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lakes Of Wada
In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of ''one'' of the lakes, the other two lakes' boundaries also contain that point. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems. The lakes of Wada were introduced by , who credited the discovery to Takeo Wada. His construction is similar to the construction by of an indecomposable continuum, and in fact it is possible for the common boundary of the three sets to be an indecomposable continuum. Construction of the lakes of Wada The Lakes of Wada are formed by starting with a closed unit square of dry land, and then digging 3 lakes according to the following rule: *On day ''n'' = 1, 2, 3,... extend lake ''n'' mod 3 (=0, 1, 2) so that i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaplan–Yorke Conjecture
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest \lambda_1\geq\lambda_2\geq\dots\geq\lambda_n, let ''j'' be the largest index for which : \sum_^j \lambda_i \geqslant 0 and : \sum_^ \lambda_i < 0. Then the conjecture is that the dimension of the attractor is : This idea is used for the definition of the Lyapunov dimension. Examples Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to estimate the fractal dimension and the Hausdorff dimension of the corresponding attractor. * The Hénon map with parameters ''a'' = 1.4 and ''b'' = 0.3 has the ordered Lyapunov exponents and . In this case, we find ''j''&nb ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Dimension
In information theory, information dimension is an information measure for random vectors in Euclidean space, based on the normalized entropy of finely quantized versions of the random vectors. This concept was first introduced by Alfréd Rényi in 1959. Simply speaking, it is a measure of the fractal dimension of a probability distribution. It characterizes the growth rate of the Shannon entropy given by successively finer discretizations of the space. In 2010, Wu and Verdú gave an operational characterization of Rényi information dimension as the fundamental limit of almost lossless data compression for analog sources under various regularity constraints of the encoder/decoder. Definition and Properties The entropy of a discrete random variable Z is :\mathbb_0(Z)=\sum_P_Z(z)\log_2\frac where P_Z(z) is the probability measure of Z when Z=z, and the supp(P_Z) denotes a set \. Let X be an arbitrary real-valued random variable. Given a positive integer m, we create a new ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector \delta \mathbf_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by : , \delta\mathbf(t) , \approx e^ , \delta \mathbf_0 , where \lambda is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GR Uncertainty Frac
GR may refer to: Arts, entertainment, and media Film and television * ''Golmaal Returns'', a 2008 Bollywood film * ''Generator Rex'', an animated TV series * Guilty Remnant, a cult-like organization portrayed in '' The Leftovers'', an HBO television series Gaming * Game Revolution, a video game review web site * ''GeneRally'', a racing game * GamesRadar, a website owned by Future Publishing * ''Ghost Recon'', a video game series * * ''Ghost Reveries'', a 2005 album by Opeth Companies, groups, and organizations * Aurigny Air Services (IATA airline designator) * Gemini Air Cargo (IATA airline designator) * Globalise Resistance, a UK anti-capitalist group * Gonnema Regiment, an infantry regiment of the South African Army * Goodrich Corporation, an aerospace manufacturer in Charlotte, North Carolina, United States * Greetings & Readings, an independent bookseller * Toyota Gazoo Racing, Toyota's racing division Places * Garden Reach, a neighbourhood of Indian city Kolkata * Giurgiu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unstable Manifold
In mathematics, and in particular the study of dynamical systems, the idea of ''stable and unstable sets'' or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor. In the case of hyperbolic dynamics, the corresponding notion is that of the hyperbolic set. Physical example The gravitational tidal forces acting on the rings of Saturn provide an easy-to-visualize physical example. The tidal forces flatten the ring into the equatorial plane, even as they stretch it out in the radial direction. Imagining the rings to be sand or gravel particles ("dust") in orbit around Saturn, the tidal forces are such that any perturbations that push particles above or below the equatorial plane results in that particle feeling a restoring force, pushing it back into the plane. Particles effectively oscillate in a harmonic well, damped by collisions. The stable direction is perpendicular to the ring. The unstable di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
GR Basins2
GR may refer to: Arts, entertainment, and media Film and television * ''Golmaal Returns'', a 2008 Bollywood film * ''Generator Rex'', an animated TV series * Guilty Remnant, a cult-like organization portrayed in '' The Leftovers'', an HBO television series Gaming * Game Revolution, a video game review web site * ''GeneRally'', a racing game * GamesRadar, a website owned by Future Publishing * ''Ghost Recon'', a video game series * * ''Ghost Reveries'', a 2005 album by Opeth Companies, groups, and organizations * Aurigny Air Services (IATA airline designator) * Gemini Air Cargo (IATA airline designator) * Globalise Resistance, a UK anti-capitalist group * Gonnema Regiment, an infantry regiment of the South African Army * Goodrich Corporation, an aerospace manufacturer in Charlotte, North Carolina, United States * Greetings & Readings, an independent bookseller * Toyota Gazoo Racing, Toyota's racing division Places * Garden Reach, a neighbourhood of Indian city Kolkata * Giurgiu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncertainty Exponent
In mathematics, the uncertainty exponent is a method of measuring the fractal dimension of a basin boundary. In a chaotic scattering system, the invariant (mathematics)#Invariant set, invariant set of the system is usually not directly accessible because it is non-attracting and typically of measure (mathematics), measure zero. Therefore, the only way to infer the presence of members and to measure the properties of the invariant set is through the basin of attraction, basins of attraction. Note that in a scattering system, basins of attraction are not limit cycles therefore do not constitute members of the invariant set. Suppose we start with a random trajectory and perturb it by a small amount, \epsilon, in a random direction. If the new trajectory ends up in a different basin from the old one, then it is called ''epsilon uncertain''. If we take a large number of such trajectories, then the fraction of them that are epsilon uncertain is the ''uncertainty fraction'', f(\epsilo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uncountable Set
In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers. Characterizations There are many equivalent characterizations of uncountability. A set ''X'' is uncountable if and only if any of the following conditions hold: * There is no injective function (hence no bijection) from ''X'' to the set of natural numbers. * ''X'' is nonempty and for every ω-sequence of elements of ''X'', there exists at least one element of X not included in it. That is, ''X'' is nonempty and there is no surjective function from the natural numbers to ''X''. * The cardinality of ''X'' is neither finite nor equal to \aleph_0 (aleph-null, the cardinality of the natural numbers). * The set ''X'' has cardinality strictly greater than \aleph_0. The first three ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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