|
Classification Of Fatou Components
In mathematics, Fatou components are components of the Fatou set. They were named after Pierre Fatou. Rational case If f is a rational function :f = \frac defined in the extended complex plane, and if it is a nonlinear function (degree > 1) : d(f) = \max(\deg(P),\, \deg(Q))\geq 2, then for a periodic component U of the Fatou set, exactly one of the following holds: # U contains an attracting periodic point # U is parabolic # U is a Siegel disc: a simply connected Fatou component on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of the unit disc onto itself by an irrational rotation angle. # U is a Herman ring: a double connected Fatou component (an annulus) on which ''f''(''z'') is analytically conjugate to a Euclidean rotation of a round annulus, again by an irrational rotation angle. File:Julia-set_N_z3-1.png, Julia set (white) and Fatou set (dark red/green/blue) for f: z\mapsto z-\frac(z) with g: z \mapsto z^3-1 in the complex plane. Basilica Juli ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Domain (mathematical Analysis)
In mathematical analysis, a domain or region is a non-empty connected open set in a topological space, in particular any non-empty connected open subset of the real coordinate space or the complex coordinate space . This is a different concept than the domain of a function, though it is often used for that purpose, for example in partial differential equations and Sobolev spaces. The basic idea of a connected subset of a space dates from the 19th century, but precise definitions vary slightly from generation to generation, author to author, and edition to edition, as concepts developed and terms were translated between German, French, and English works. In English, some authors use the term ''domain'', some use the term ''region'', some use both terms interchangeably, and some define the two terms slightly differently; some avoid ambiguity by sticking with a phrase such as ''non-empty connected open subset''. One common convention is to define a ''domain'' as a connected open se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Dynamics
Complex dynamics is the study of dynamical systems defined by Iterated function, iteration of functions on complex number spaces. Complex analytic dynamics is the study of the dynamics of specifically analytic functions. Techniques *General **Montel's theorem **Poincaré metric **Schwarz lemma **Riemann mapping theorem **Carathéodory's theorem (conformal mapping) **Böttcher's equation *Combinatorics, Combinatorial ** Hubbard trees ** Spider algorithm ** Tuning **Lamination (topology), Laminations **Cantor function, Devil's Staircase algorithm (Cantor function) **Orbit portraits **Jean-Christophe Yoccoz, Yoccoz puzzles Parts * Holomorphic dynamics (dynamics of holomorphic functions) ** in one complex variable ** in several complex variables * Conformal dynamics unites holomorphic dynamics in one complex variable with differentiable dynamics in one real variable. See also *Arithmetic dynamics *Chaos theory *Complex analysis *Complex quadratic polynomial *Fatou set *Infinite co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Complex Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Limit Sets
Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 song by Paenda; see Austria in the Eurovision Song Contest 2019 * ''Limits'' (collection), a collection of short stories and essays by Larry Niven * The Limit, a Dutch band *The Limit, an episode from ''The Amazing World of Gumball'' Mathematics * Limit (mathematics), the value that a function or sequence "approaches" as the input or index approaches some value ** Limit of a function ***(ε,_δ)-definition of limit, formal definition of the mathematical notion of limit ** Limit of a sequence ** One-sided limit, either of the two limits of a function as a specified point is approached from below or from above * Limit of a net * Limit point, in topological spaces * Limit (category theory) ** Direct limit ** Inverse limit Other uses * Limit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractals
In mathematics, a fractal is a geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller scales is called self-similarity, also known as expanding symmetry or unfolding symmetry; if this replication is exactly the same at every scale, as in the Menger sponge, the shape is called Affine geometry, affine self-similar. Fractal geometry lies within the mathematical branch of measure theory. One way that fractals are different from finite geometric figures is how they Scaling (geometry), scale. Doubling the edge lengths of a filled polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the conventional dimension of the filled polygon). Likewise, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alan F
Alan may refer to: People *Alan (surname), an English and Turkish surname *Alan (given name), an English given name ** List of people with given name Alan ''Following are people commonly referred to solely by "Alan" or by a homonymous name.'' * Alan (Chinese singer) (born 1987), female Chinese singer of Tibetan ethnicity, active in both China and Japan * Alan (Mexican singer) (born 1973), Mexican singer and actor *Alan (wrestler) (born 1975), a.k.a. Gato Eveready, who wrestles in Asistencia Asesoría y Administración * Alan (footballer, born 1979) (Alan Osório da Costa Silva), Brazilian footballer * Alan (footballer, born 1998) (Alan Cardoso de Andrade), Brazilian footballer * Alan I, King of Brittany (died 907), "the Great" * Alan II, Duke of Brittany (c. 900–952) *Alan III, Duke of Brittany(997–1040) * Alan IV, Duke of Brittany (c. 1063–1119), a.k.a. Alan Fergant ("the Younger" in Breton language) * Alan of Tewkesbury, 12th century abbott * Alan of Lynn (c. 1348–1423), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lennart Carleson
Lennart Axel Edvard Carleson (born 18 March 1928) is a Swedish mathematician, known as a leader in the field of harmonic analysis. One of his most noted accomplishments is his proof of Lusin's conjecture. He was awarded the Abel Prize in 2006 for "his profound and seminal contributions to harmonic analysis and the theory of smooth dynamical systems." Life He was a student of Arne Beurling and received his Ph.D. from Uppsala University in 1950. He did his post-doctoral work at Harvard University where he met and discussed Fourier series and their convergence with Antoni Zygmund and Raphaël Salem who were there in 1950 and 1951. He is a professor emeritus at Uppsala University, the Royal Institute of Technology in Stockholm, and the University of California, Los Angeles, and has served as director of the Mittag-Leffler Institute in Djursholm outside Stockholm 1968–1984. Between 1978 and 1982 he served as president of the International Mathematical Union. Carleson married Butte ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fritz John
Fritz John (14 June 1910 – 10 February 1994) was a German-born mathematician specialising in partial differential equations and ill-posed problems. His early work was on the Radon transform and he is remembered for John's equation. He was a 1984 MacArthur Fellow. Life and career John was born in Berlin, Imperial Germany, the son of Hedwig (née Bürgel) and Hermann Jacobson-John. He studied mathematics from 1929 to 1933 in Göttingen where he was influenced by Richard Courant, among others. Following Hitler's rise to power in 1933 "non-aryans" were being expelled from teaching posts, and John, being half Jewish, emigrated from Germany to England. John published his first paper in 1934 on Morse theory. He was awarded his doctorate in 1934 with a thesis entitled ''Determining a function from its integrals over certain manifolds'' from Göttingen. With Richard Courant's assistance he spent a year at St John's College, Cambridge. During this time he published papers on the Radon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Montel's Theorem
In complex analysis, an area of mathematics, Montel's theorem refers to one of two theorems about families of holomorphic functions. These are named after French mathematician Paul Montel, and give conditions under which a family of holomorphic functions is normal. Locally uniformly bounded families are normal The first, and simpler, version of the theorem states that a family of holomorphic functions defined on an open subset of the complex numbers is normal if and only if it is locally uniformly bounded. This theorem has the following formally stronger corollary. Suppose that \mathcal is a family of meromorphic functions on an open set D. If z_0\in D is such that \mathcal is not normal at z_0, and U\subset D is a neighborhood of z_0, then \bigcup_f(U) is dense in the complex plane. Functions omitting two values The stronger version of Montel's Theorem (occasionally referred to as the Fundamental Normality Test) states that a family of holomorphic functions, all of which omit t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
No-wandering-domain Theorem
In mathematics, the no-wandering-domain theorem is a result on dynamical systems, proven by Dennis Sullivan in 1985. The theorem states that a rational map ''f'' : Ĉ → Ĉ with deg(''f'') ≥ 2 does not have a wandering domain, where Ĉ denotes the Riemann sphere. More precisely, for every component ''U'' in the Fatou set of ''f'', the sequence :U,f(U),f(f(U)),\dots,f^n(U), \dots will eventually become periodic. Here, ''f'' ''n'' denotes the ''n''-fold iteration of ''f'', that is, :f^n = \underbrace_n . The theorem does not hold for arbitrary maps; for example, the transcendental map f(z)=z+2\pi\sin(z) has wandering domains. However, the result can be generalized to many situations where the functions naturally belong to a finite-dimensional parameter space, most notably to transcendental entire and meromorphic functions with a finite number of singular values. References * Lennart Carleson and Theodore W. Gamelin, ''Complex Dynamics'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wandering Set
In 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. The language of wandering sets can be used to give a precise, mathematical definition to the concept of a dissipative system. The notion of wandering sets in phase space was introduced by Birkhoff in 1927. Wandering points A common, discrete-time definition of wandering sets starts with a map f:X\to X of a topological space ''X''. A point x\in X is said to be a wandering point if there is a neighbourhood ''U'' of ''x'' and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.PNG "Click for more on -> Theorems In Complex Analysis")
