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Interval Exchange Transformation
In mathematics, an interval exchange transformation is a kind of dynamical system that generalises circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. They arise naturally in the study of polygonal billiards and in area-preserving flows. Formal definition Let n > 0 and let \pi be a permutation on 1, \dots, n. Consider a vector \lambda = (\lambda_1, \dots, \lambda_n) of positive real numbers (the widths of the subintervals), satisfying :\sum_^n \lambda_i = 1. Define a map T_: ,1rightarrow ,1 called the interval exchange transformation associated with the pair (\pi,\lambda) as follows. For 1 \leq i \leq n let :a_i = \sum_ \lambda_j \quad \text \quad a'_i = \sum_ \lambda_. Then for x \in ,1/math>, define : T_(x) = x - a_i + a'_i if x lies in the subinterval translation, and it rearranges these subintervals so that the subinterval at position i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interval Exchange
In mathematics, an interval exchange transformation is a kind of dynamical system that generalises Irrational rotation, circle rotation. The phase space consists of the unit interval, and the transformation acts by cutting the interval into several subintervals, and then permuting these subintervals. They arise naturally in the study of Dynamical billiards, polygonal billiards and in area-preserving flows. Formal definition Let n > 0 and let \pi be a permutation on 1, \dots, n. Consider a Vector (geometric), vector \lambda = (\lambda_1, \dots, \lambda_n) of positive real numbers (the widths of the subintervals), satisfying :\sum_^n \lambda_i = 1. Define a map T_:[0,1]\rightarrow [0,1], called the interval exchange transformation associated with the pair (\pi,\lambda) as follows. For 1 \leq i \leq n let :a_i = \sum_ \lambda_j \quad \text \quad a'_i = \sum_ \lambda_. Then for x \in [0,1], define : T_(x) = x - a_i + a'_i if x lies in the subinterval [a_i,a_i+\lambda_i). Thus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational
Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. The term is used, usually pejoratively, to describe thinking and actions that are, or appear to be, less useful, or more illogical than other more rational alternatives. Irrational behaviors of individuals include taking offense or becoming angry about a situation that has not yet occurred, expressing emotions exaggeratedly (such as crying hysterically), maintaining unrealistic expectations, engaging in irresponsible conduct such as problem intoxication, disorganization, and falling victim to confidence tricks. People with a mental illness like schizophrenia may exhibit irrational paranoia. These more contemporary normative conceptions of what constitutes a manifestation of irrationality are difficult to demonstrate empirically because it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Odometer Thrice Iterated
Dyadic describes the interaction between two things, and may refer to: *Dyad (sociology), interaction between a pair of individuals **The dyadic variation of Democratic peace theory *Dyadic counterpoint, the voice-against-voice conception of polyphony *People who are not intersex (see also endosex) Mathematics *Dyadic, a relation or function having an arity of two in logic, mathematics, and computer science *Dyadic decomposition, a concept in Littlewood–Paley theory *Dyadic distribution, a type of probability distribution *Dyadic fraction, a mathematical group related to dyadic rationals * Dyadic solenoid, a type of dyadic fraction *Dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to , 1)^\infty : x \mapsto (x_0, x_1, x_2, ... *Dyadics, tensor math (including dyadic products) *A synonym for binary relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Odometer, Twice Iterated
Dyadic describes the interaction between two things, and may refer to: *Dyad (sociology), interaction between a pair of individuals **The dyadic variation of Democratic peace theory *Dyadic counterpoint, the voice-against-voice conception of polyphony *People who are not intersex (see also endosex) Mathematics *Dyadic, a relation or function having an arity of two in logic, mathematics, and computer science *Dyadic decomposition, a concept in Littlewood–Paley theory *Dyadic distribution, a type of probability distribution *Dyadic fraction, a mathematical group related to dyadic rationals * Dyadic solenoid, a type of dyadic fraction *Dyadic transformation The dyadic transformation (also known as the dyadic map, bit shift map, 2''x'' mod 1 map, Bernoulli map, doubling map or sawtooth map) is the mapping (i.e., recurrence relation) : T: , 1) \to , 1)^\infty : x \mapsto (x_0, x_1, x_2, ... *Dyadics, tensor math (including dyadic products) *A synonym for binary relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dyadic Odometer
In mathematics, a Markov odometer is a certain type of topological dynamical system. It plays a fundamental role in ergodic theory and especially in orbit theory of dynamical systems, since a theorem of H. Dye asserts that every ergodic nonsingular transformation is orbit-equivalent to a Markov odometer. The basic example of such system is the "nonsingular odometer", which is an additive topological group defined on the product space of discrete spaces, induced by addition defined as x \mapsto x+\underline, where \underline:=(1,0,0,\dots). This group can be endowed with the structure of a dynamical system; the result is a conservative dynamical system. The general form, which is called "Markov odometer", can be constructed through Bratteli–Vershik diagram to define ''Bratteli–Vershik compactum'' space together with a corresponding transformation. Nonsingular odometers Several kinds of non-singular odometers may be defined. These are sometimes referred to as adding machin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Entropy
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing holes, opening holes, tearing, gluing, or passing through itself. A topological space is a set endowed with a structure, called a ''topology'', which allows defining continuous deformation of subspaces, and, more generally, all kinds of continuity. Euclidean spaces, and, more generally, metric spaces are examples of a topological space, as any distance or metric defines a topology. The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connect ... [...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] |
Invariant (mathematics)
In mathematics, an invariant is a property of a mathematical object (or a class of mathematical objects) which remains unchanged after operations or transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used. For example, the area of a triangle is an invariant with respect to isometries of the Euclidean plane. The phrases "invariant under" and "invariant to" a transformation are both used. More generally, an invariant with respect to an equivalence relation is a property that is constant on each equivalence class. Invariants are used in diverse areas of mathematics such as geometry, topology, algebra and discrete mathematics. Some important classes of transformations are defined by an invariant they leave unchanged. For example, conformal maps are defined as transformations of the plane that preserve angles. The discovery of invariants is an important ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ergodic
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 the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost All
In mathematics, the term "almost all" means "all but a negligible amount". More precisely, if X is a set, "almost all elements of X" means "all elements of X but those in a negligible subset of X". The meaning of "negligible" depends on the mathematical context; for instance, it can mean finite, countable, or null. In contrast, "almost no" means "a negligible amount"; that is, "almost no elements of X" means "a negligible amount of elements of X". Meanings in different areas of mathematics Prevalent meaning Throughout mathematics, "almost all" is sometimes used to mean "all (elements of an infinite set) but finitely many". This use occurs in philosophy as well. Similarly, "almost all" can mean "all (elements of an uncountable set) but countably many". Examples: * Almost all positive integers are greater than 1012. * Almost all prime numbers are odd (2 is the only exception). * Almost all polyhedra are irregular (as there are only nine exceptions: the five platonic solids and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Howard Masur
Howard Alan Masur is an American mathematician who works on topology, geometry and combinatorial group theory. Biography Masur was an invited speaker at the 1994 International Congress of Mathematicians in Zürich. and is a fellow of the American Mathematical Society. Along with Yair Minsky, Masur is one of the pioneers of the study of curve complex geometry. He also contributed to the understanding of the convergence of geodesic rays in Teichmüller theory. Masur was a Ph.D. student of Albert Marden at the University of Minnesota-Minneapolis. Awards and recognitions The Hubbard–Masur theorem is named after Masur and John H. Hubbard. In 2009, a conference of mathematicians honored Masur's 60th birthday in France France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of overseas regions and territories in the Americas and the Atlantic, Pacific and Indian Oceans. Its metropolitan area .... Se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of Mathematics
The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as the founding editor-in-chief. It was "intended to afford a medium for the presentation and analysis of any and all questions of interest or importance in pure and applied Mathematics, embracing especially all new and interesting discoveries in theoretical and practical astronomy, mechanical philosophy, and engineering". It was published in Des Moines, Iowa, and was the earliest American mathematics journal to be published continuously for more than a year or two. This incarnation of the journal ceased publication after its tenth year, in 1883, giving as an explanation Hendricks' declining health, but Hendricks made arrangements to have it taken over by new management, and it was continued from March 1884 as the ''Annals of Mathematics''. The n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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