Wandering setIn dynamical systems and ergodic theory, the concept of a wandering set formalizes a certain idea of movement and mixing. When a dynamical system has a wandering set of non-zero measure, then the system is a dissipative system. This is the opposite of a conservative system, to which the Poincaré recurrence theorem applies. Intuitively, the connection between wandering sets and dissipation is easily understood: if a portion of the phase space "wanders away" during normal time-evolution of the system, and is never visited again, then the system is dissipative.
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: e.g. mixing paint, mixing drinks, industrial mixing. 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.
Horseshoe mapIn the mathematics of chaos theory, a horseshoe map is any member of a class of chaotic maps of the square into itself. It is a core example in the study of dynamical systems. The map was introduced by Stephen Smale while studying the behavior of the orbits of the van der Pol oscillator. The action of the map is defined geometrically by squishing the square, then stretching the result into a long strip, and finally folding the strip into the shape of a horseshoe. Most points eventually leave the square under the action of the map.
Mittag-Leffler functionIn mathematics, the Mittag-Leffler function is a special function, a complex function which depends on two complex parameters and . It may be defined by the following series when the real part of is strictly positive: where is the gamma function. When , it is abbreviated as . For , the series above equals the Taylor expansion of the geometric series and consequently . In the case and are real and positive, the series converges for all values of the argument , so the Mittag-Leffler function is an entire function.
Rössler attractorThe Rössler attractor ˈrɒslər is the attractor for the Rössler system, a system of three non-linear ordinary differential equations originally studied by Otto Rössler in the 1970s. These differential equations define a continuous-time dynamical system that exhibits chaotic dynamics associated with the fractal properties of the attractor. Rössler interpreted it as a formalization of a taffy-pulling machine. Some properties of the Rössler system can be deduced via linear methods such as eigenvectors, but the main features of the system require non-linear methods such as Poincaré maps and bifurcation diagrams.
Universal setIn set theory, a universal set is a set which contains all objects, including itself. In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set. Many set theories do not allow for the existence of a universal set. There are several different arguments for its non-existence, based on different choices of axioms for set theory. In Zermelo–Fraenkel set theory, the axiom of regularity and axiom of pairing prevent any set from containing itself.
Totally bounded spaceIn topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “size” (where the meaning of “size” depends on the structure of the ambient space). The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
Conservative systemIn 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.
Fuzzy setIn mathematics, fuzzy sets (a.k.a. uncertain sets) are sets whose elements have degrees of membership. Fuzzy sets were introduced independently by Lotfi A. Zadeh in 1965 as an extension of the classical notion of set. At the same time, defined a more general kind of structure called an L-relation, which he studied in an abstract algebraic context. Fuzzy relations, which are now used throughout fuzzy mathematics and have applications in areas such as linguistics , decision-making , and clustering , are special cases of L-relations when L is the unit interval [0, 1].
Locally compact spaceIn topology and related branches of mathematics, a topological space is called locally compact if, roughly speaking, each small portion of the space looks like a small portion of a compact space. More precisely, it is a topological space in which every point has a compact neighborhood. In mathematical analysis locally compact spaces that are Hausdorff are of particular interest; they are abbreviated as LCH spaces. Let X be a topological space. Most commonly X is called locally compact if every point x of X has a compact neighbourhood, i.
Quadratic irrational numberIn mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients.
EndomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space V is a linear map f: V → V, and an endomorphism of a group G is a group homomorphism f: G → G. In general, we can talk about endomorphisms in any . In the , endomorphisms are functions from a set S to itself. In any category, the composition of any two endomorphisms of X is again an endomorphism of X.
