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Conservative System
In mathematics, a conservative system is a dynamical system which stands in contrast to a dissipative system. Roughly speaking, such systems have no friction or other mechanism to dissipate the dynamics, and thus, their phase space does not shrink over time. Precisely speaking, they are those dynamical systems that have a null wandering set: under time evolution, no portion of the phase space ever "wanders away", never to be returned to or revisited. Alternately, conservative systems are those to which the Poincaré recurrence theorem applies. An important special case of conservative systems are the measure-preserving dynamical systems. Informal introduction Informally, dynamical systems describe the time evolution of the phase space of some mechanical system. Commonly, such evolution is given by some differential equations, or quite often in terms of discrete time steps. However, in the present case, instead of focusing on the time evolution of discrete points, one shifts attenti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
KMS State
In the statistical mechanics of quantum mechanical systems and quantum field theory, the properties of a system in thermal equilibrium can be described by a mathematical object called a Kubo–Martin– Schwinger state or, more commonly, a KMS state: a state satisfying the KMS condition. introduced the condition, used it to define thermodynamic Green's functions, and used the condition to define equilibrium states and called it the KMS condition. Overview The simplest case to study is that of a finite-dimensional Hilbert space, in which one does not encounter complications like phase transitions or spontaneous symmetry breaking. The density matrix of a thermal state is given by :\rho_=\frac=\frac where ''H'' is the Hamiltonian operator and ''N'' is the particle number operator (or charge operator, if we wish to be more general) and :Z(\beta,\mu)\ \stackrel\ \mathrm\left e^ \right/math> is the partition function. We assume that ''N'' commutes with ''H,'' or in oth ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Every
In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to the concept of measure zero, and is analogous to the notion of ''almost surely'' in probability theory. More specifically, a property holds almost everywhere if it holds for all elements in a set except a subset of measure zero, or equivalently, if the set of elements for which the property holds is conull. In cases where the measure is not complete, it is sufficient that the set be contained within a set of measure zero. When discussing sets of real numbers, the Lebesgue measure is usually assumed unless otherwise stated. The term ''almost everywhere'' is abbreviated ''a.e.''; in older literature ''p.p.'' is used, to stand for the equivalent French language phrase ''presque partout''. A set with full measure is one whose complemen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conull Set
In measure theory, a conull set is a set whose complement is null, i.e., the measure of the complement is zero. For example, the set of irrational numbers is a conull subset of the real line with Lebesgue measure. A property that is true of the elements of a conull set is said to be true almost everywhere In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ..... Sep. 62for an example of this usage. References Measure theory {{Settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Set
In mathematical analysis, a null set N \subset \mathbb is a measurable set that has measure zero. This can be characterized as a set that can be covered by a countable union of intervals of arbitrarily small total length. The notion of null set should not be confused with the empty set as defined in set theory. Although the empty set has Lebesgue measure zero, there are also non-empty sets which are null. For example, any non-empty countable set of real numbers has Lebesgue measure zero and therefore is null. More generally, on a given measure space M = (X, \Sigma, \mu) a null set is a set S\in\Sigma such that \mu(S) = 0. Example Every finite or countably infinite subset of the real numbers is a null set. For example, the set of natural numbers and the set of rational numbers are both countably infinite and therefore are null sets when considered as subsets of the real numbers. The Cantor set is an example of an uncountable null set. Definition Suppose A is a subs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic Decomposition Theorem
In mathematics, ergodicity expresses the idea that a point of a moving system, either a dynamical system or a stochastic process, will eventually visit all parts of the space that the system moves in, in a uniform and random sense. This implies that the average behavior of the system can be deduced from the trajectory of a "typical" point. Equivalently, a sufficiently large collection of random samples from a process can represent the average statistical properties of the entire process. Ergodicity is a property of the system; it is a statement that the system cannot be reduced or factored into smaller components. Ergodic theory is the study of systems possessing ergodicity. Ergodic systems occur in a broad range of systems in physics and in geometry. This can be roughly understood to be due to a common phenomenon: the motion of particles, that is, geodesics on a hyperbolic manifold are divergent; when that manifold is compact, that is, of finite size, those orbits return to t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Process
In probability and statistics, a Bernoulli process (named after Jacob Bernoulli) is a finite or infinite sequence of binary random variables, so it is a discrete-time stochastic process that takes only two values, canonically 0 and 1. The component Bernoulli variables ''X''''i'' are identically distributed and independent. Prosaically, a Bernoulli process is a repeated coin flipping, possibly with an unfair coin (but with consistent unfairness). Every variable ''X''''i'' in the sequence is associated with a Bernoulli trial or experiment. They all have the same Bernoulli distribution. Much of what can be said about the Bernoulli process can also be generalized to more than two outcomes (such as the process for a six-sided die); this generalization is known as the Bernoulli scheme. The problem of determining the process, given only a limited sample of Bernoulli trials, may be called the problem of checking whether a coin is fair. Definition A Bernoulli process is a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic System
Ergodic theory ( Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuba Libre
Rum and Coke, or the Cuba libre ( , ; literally "Free Cuba"), is a highball cocktail consisting of cola, rum, and in many recipes lime juice on ice. Traditionally, the cola ingredient is Coca-Cola ("Coke") and the alcohol is a light rum such as Bacardi; however, the drink may be made with various types of rums and cola brands, and lime juice may or may not be included. The cocktail originated in the early 20th century in Cuba, after the country won independence in the Spanish–American War. It subsequently became popular across Cuba, the United States, and other countries. Its simple recipe and inexpensive, ubiquitous ingredients have made it one of the world's most-popular alcoholic drinks. Drink critics often consider the drink mediocre, but it has been noted for its historical significance. History The drink was created in Cuba in the early 1900s, but its exact origins are not certain. It became popular shortly after 1900, when bottled Coca-Cola was first imported into C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, industrial mixing, ''etc''. The concept appears in ergodic theory—the study of stochastic processes and measure-preserving dynamical systems. Several different definitions for mixing exist, including ''strong mixing'', ''weak mixing'' and ''topological mixing'', with the last not requiring a measure to be defined. Some of the different definitions of mixing can be arranged in a hierarchical order; thus, strong mixing implies weak mixing. Furthermore, weak mixing (and thus also strong mixing) implies ergodicity: that is, every system that is weakly mixing is also ergodic (and so one says that mixing is a "stronger" notion than ergodicity). Informal explanation The mathematical definition of mixing aims to capture the ordinary every-day process of mixing, such as mixing paints, drinks ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Decomposition
In mathematics, the Hopf decomposition, named after Eberhard Hopf, gives a canonical decomposition of a measure space (''X'', μ) with respect to an invertible non-singular transformation ''T'':''X''→''X'', i.e. a transformation which with its inverse is measurable and carries null sets onto null sets. Up to null sets, ''X'' can be written as a disjoint union ''C'' ∐ ''D'' of ''T''-invariant sets where the action of ''T'' on ''C'' is conservative and the action of ''T'' on ''D'' is dissipative. Thus, if τ is the automorphism of ''A'' = L∞(''X'') induced by ''T'', there is a unique τ-invariant projection ''p'' in ''A'' such that ''pA'' is conservative and ''(I–p)A'' is dissipative. Definitions *Wandering sets and dissipative actions. A measurable subset ''W'' of ''X'' is ''wandering'' if its characteristic function ''q'' = χ''W'' in ''A'' = L∞(''X'') satisfies ''q''τ''n''(''q'') = 0 for all ''n''; thus, up to null sets, the translates ''T''''n''(''W'') are pairwise ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
