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Schröder's Equation
Schröder's equation, named after Ernst Schröder, is a functional equation with one independent variable: given the function , find the function such that Schröder's equation is an eigenvalue equation for the composition operator that sends a function to . If is a fixed point of , meaning , then either (or ) or . Thus, provided that is finite and does not vanish or diverge, the eigenvalue is given by . Functional significance For , if is analytic on the unit disk, fixes , and , then Gabriel Koenigs showed in 1884 that there is an analytic (non-trivial) satisfying Schröder's equation. This is one of the first steps in a long line of theorems fruitful for understanding composition operators on analytic function spaces, cf. Koenigs function. Equations such as Schröder's are suitable to encoding self-similarity, and have thus been extensively utilized in studies of nonlinear dynamics (often referred to colloquially as chaos theory). It is also used in studies of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ernst Schroeder
Ernst is both a surname and a given name, the German, Dutch, and Scandinavian form of Ernest. Notable people with the name include: Surname * Adolf Ernst (1832–1899) German botanist known by the author abbreviation "Ernst" * Anton Ernst (1975-) South African Film Producer * Alice Henson Ernst (1880-1980), American writer and historian * Britta Ernst (born 1961), German politician * Cornelia Ernst, German politician * Edzard Ernst, German-British Professor of Complementary Medicine * Emil Ernst, astronomer * Ernie Ernst (1924/25–2013), former District Judge in Walker County, Texas * Eugen Ernst (1864–1954), German politician * Fabian Ernst, German soccer player * Gustav Ernst, Austrian writer * Heinrich Wilhelm Ernst, Moravian violinist and composer * Jim Ernst, Canadian politician * Jimmy Ernst, American painter, son of Max Ernst * Joni Ernst, U.S. Senator from Iowa * K.S. Ernst, American visual poet * Karl Friedrich Paul Ernst, German writer (1866–1933) * Ken Ernst, U.S. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaston Julia
Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French Algerian mathematician who devised the formula for the Julia set. His works were popularized by French mathematician Benoit Mandelbrot; the Julia and Mandelbrot fractals are closely related. He founded, independently with Pierre Fatou, the modern theory of holomorphic dynamics. Military service Julia was born in the Algerian town of Sidi Bel Abbes, at the time governed by the French. During his youth, he had an interest in mathematics and music. His studies were interrupted at the age of 21, when France became involved in World War I and Julia was conscripted to serve with the army. During an attack he suffered a severe injury, losing his nose. His many operations to remedy the situation were all unsuccessful, and for the rest of his life he resigned himself to wearing a leather strap around the area where his nose had been. Career in mathematics Julia gained attention for his mathematical work at the age of 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semigroup
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative internal binary operation on it. The binary operation of a semigroup is most often denoted multiplicatively: ''x''·''y'', or simply ''xy'', denotes the result of applying the semigroup operation to the ordered pair . Associativity is formally expressed as that for all ''x'', ''y'' and ''z'' in the semigroup. Semigroups may be considered a special case of magmas, where the operation is associative, or as a generalization of groups, without requiring the existence of an identity element or inverses. The closure axiom is implied by the definition of a binary operation on a set. Some authors thus omit it and specify three axioms for a group and only one axiom (associativity) for a semigroup. As in the case of groups or magmas, the semigroup operation need not be commutative, so ''x''·''y'' is not necessarily equal to ''y''·''x''; a well-known example of an operation that is as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuous Group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two structures together and consequently they are not independent from each other. Topological groups have been studied extensively in the period of 1925 to 1940. Haar and Weil (respectively in 1933 and 1940) showed that the integrals and Fourier series are special cases of a very wide class of topological groups. Topological groups, along with continuous group actions, are used to study continuous symmetries, which have many applications, for example, in physics. In functional analysis, every topological vector space is an additive topological group with the additional property that scalar multiplication is continuous; consequently, many results from the theory of topological groups can be applied to functional analysis. Formal definitio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iterated Function
In mathematics, an iterated function is a function (that is, a function from some set to itself) which is obtained by composing another function with itself a certain number of times. The process of repeatedly applying the same function is called iteration. In this process, starting from some initial object, the result of applying a given function is fed again in the function as input, and this process is repeated. For example on the image on the right: :with the circle‑shaped symbol of function composition. Iterated functions are objects of study in computer science, fractals, dynamical systems, mathematics and renormalization group physics. Definition The formal definition of an iterated function on a set ''X'' follows. Let be a set and be a function. Defining as the ''n''-th iterate of (a notation introduced by Hans Heinrich Bürmann and John Frederick William Herschel), where ''n'' is a non-negative integer, by: f^0 ~ \stackrel ~ \operatorname_ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
One-parameter Group
In mathematics, a one-parameter group or one-parameter subgroup usually means a continuous group homomorphism :\varphi : \mathbb \rightarrow G from the real line \mathbb (as an additive group) to some other topological group G. If \varphi is injective then \varphi(\mathbb), the image, will be a subgroup of G that is isomorphic to \mathbb as an additive group. One-parameter groups were introduced by Sophus Lie in 1893 to define infinitesimal transformations. According to Lie, an ''infinitesimal transformation'' is an infinitely small transformation of the one-parameter group that it generates. It is these infinitesimal transformations that generate a Lie algebra that is used to describe a Lie group of any dimension. The action of a one-parameter group on a set is known as a flow. A smooth vector field on a manifold, at a point, induces a ''local flow'' - a one parameter group of local diffeomorphisms, sending points along integral curves of the vector field. The local flow of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Conjugacy
In mathematics, two functions are said to be topologically conjugate if there exists a homeomorphism that will conjugate the one into the other. Topological conjugacy, and related-but-distinct of flows, are important in the study of iterated functions and more generally dynamical systems, since, if the dynamics of one iterative function can be determined, then that for a topologically conjugate function follows trivially. To illustrate this directly: suppose that f and g are iterated functions, and there exists a homeomorphism h such that :g = h^ \circ f \circ h, so that f and g are topologically conjugate. Then one must have :g^n = h^ \circ f^n \circ h, and so the iterated systems are topologically conjugate as well. Here, \circ denotes function composition. Definition f\colon X \to X, g\colon Y \to Y, and h\colon Y \to X are continuous functions on topological spaces, X and Y. f being topologically semiconjugate to g means, by definition, that h is a surjection such t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a flow may be viewed as a continuous motion of points over time. More formally, a flow is a group action of the real numbers on a set. The idea of a vector flow, that is, the flow determined by a vector field, occurs in the areas of differential topology, Riemannian geometry and Lie groups. Specific examples of vector flows include the geodesic flow, the Hamiltonian flow, the Ricci flow, the mean curvature flow, and Anosov flows. Flows may also be defined for systems of random variables and stochastic processes, and occur in the study of ergodic dynamical systems. The most celebrated of these is perhaps the Bernoulli flow. Formal definition A flow on a set is a group action of the additive group of real numbers on . More explicitl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orbit (dynamics)
In mathematics, specifically in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. It can be understood as the subset of phase space covered by the trajectory of the dynamical system under a particular set of initial conditions, as the system evolves. As a phase space trajectory is uniquely determined for any given set of phase space coordinates, it is not possible for different orbits to intersect in phase space, therefore the set of all orbits of a dynamical system is a partition of the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems. For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces. Definition Given a dynamical system (''T'', ''M'', Φ) with ''T'' a group, ''M'' a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Phase-space Orbit Of-Logistic Map
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space. For mechanical systems, the phase space usually consists of all possible values of position and momentum variables. It is the outer product of direct space and reciprocal space. The concept of phase space was developed in the late 19th century by Ludwig Boltzmann, Henri Poincaré, and Josiah Willard Gibbs. Introduction In a phase space, every degree of freedom or parameter of the system is represented as an axis of a multidimensional space; a one-dimensional system is called a phase line, while a two-dimensional system is called a phase plane. For every possible state of the system or allowed combination of values of the system's parameters, a point is included in the multidimensional space. The system's evolving state over time traces a path (a phase-space trajectory for the system) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carleman Matrix
In mathematics, a Carleman matrix is a matrix used to convert function composition into matrix multiplication. It is often used in iteration theory to find the continuous iteration of functions which cannot be iterated by pattern recognition alone. Other uses of Carleman matrices occur in the theory of probability generating functions, and Markov chains. Definition The Carleman matrix of an infinitely differentiable function f(x) is defined as: :M = \frac\left frac (f(x))^j \right ~, so as to satisfy the (Taylor series) equation: :(f(x))^j = \sum_^ M x^k. For instance, the computation of f(x) by :f(x) = \sum_^ M x^k. ~ simply amounts to the dot-product of row 1 of M with a column vector \left ,x,x^2,x^3,...\right\tau. The entries of M /math> in the next row give the 2nd power of f(x): :f(x)^2 = \sum_^ M x^k ~, and also, in order to have the zeroth power of f(x) in M /math>, we adopt the row 0 containing zeros everywhere except the first position, such that :f(x)^0 = 1 = \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Series
In mathematics, an asymptotic expansion, asymptotic series or Poincaré expansion (after Henri Poincaré) is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular, often infinite, point. Investigations by revealed that the divergent part of an asymptotic expansion is latently meaningful, i.e. contains information about the exact value of the expanded function. The most common type of asymptotic expansion is a power series in either positive or negative powers. Methods of generating such expansions include the Euler–Maclaurin summation formula and integral transforms such as the Laplace and Mellin transforms. Repeated integration by parts will often lead to an asymptotic expansion. Since a '' convergent'' Taylor series fits the definition of asymptotic expansion as well, the phrase "asymptotic series" usually implies a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
