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Siegel Disk
Siegel disc is a connected component in the Fatou set where the dynamics is analytically conjugate to an irrational rotation. Description Given a holomorphic endomorphism f:S\to S on a Riemann surface S we consider the dynamical system generated by the iterates of f denoted by f^n=f\circ\stackrel\circ f. We then call the orbit \mathcal^+(z_0) of z_0 as the set of forward iterates of z_0. We are interested in the asymptotic behavior of the orbits in S (which will usually be \mathbb, the complex plane or \mathbb=\mathbb\cup\, the Riemann sphere), and we call S the phase plane or ''dynamical plane''. One possible asymptotic behavior for a point z_0 is to be a fixed point, or in general a ''periodic point''. In this last case f^p(z_0)=z_0 where p is the period and p=1 means z_0 is a fixed point. We can then define the ''multiplier'' of the orbit as \rho=(f^p)'(z_0) and this enables us to classify periodic orbits as ''attracting'' if , \rho, 1 and indifferent if \rho=1. Indifferent p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Space
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is ''locally connected'', which neither implies nor follows from connectedness. Formal definition A topological space X is said to be if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice. For a topologi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Golden Ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( or \phi) denotes the golden ratio. The constant \varphi satisfies the quadratic equation \varphi^2 = \varphi + 1 and is an irrational number with a value of The golden ratio was called the extreme and mean ratio by Euclid, and the divine proportion by Luca Pacioli, and also goes by several other names. Mathematicians have studied the golden ratio's properties since antiquity. It is the ratio of a regular pentagon's diagonal to its side and thus appears in the construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of \varphi—may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has been used to analyze the proportions of natural object ... [...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] |
Herman Ring
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou componentJohn Milnor''Dynamics in one complex variable'' Third Edition, Annals of Mathematics Studies, 160, Princeton Univ. Press, Princeton, NJ, 2006. where the rational function is conformally conjugate to an irrational rotation of the standard annulus. Formal definition Namely if ''ƒ'' possesses a Herman ring ''U'' with period ''p'', then there exists a conformal mapping :\phi:U\rightarrow\ and an irrational number \theta, such that :\phi\circ f^\circ\phi^(\zeta)=e^\zeta. So the dynamics on the Herman ring is simple. Name It was introduced by, and later named after, Michael Herman (1979) who first found and constructed this type of Fatou component. Function * Polynomials do not have Herman rings. * Rational functions can have Herman rings. According to the result of Shishikura, if a rational function ''ƒ'' possesses a Herman ring, then the degree of ''ƒ'' is at least ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douady Rabbit
The Douady rabbit is any of various particular filled Julia sets associated with the parameter near the center period 3 buds of Mandelbrot set for complex quadratic map. It is named after French mathematician Adrien Douady. Formula The rabbit is generated by iterating the Mandelbrot set map z=z^2+c on the complex plane with c fixed to lie in the period three bulb off the main cardiod and z ranging over the plane. The pixels in the image are then colored to show whether for a particular value of z the iteration converged or diverged Variants Twisted rabbit or rabbits with twisted ears = is the composition of the “rabbit” polynomial with n-th powers of the Dehn twists about its ears. Corabbit is symmetrical image of rabbit. Here parameter c \approx -0.1226 -0.7449i It is one of 2 other polynomials inducing the same permutation of their post-critical set are the rabbit. 3D The julia set has no direct analog in 3D 4D Quaternion julia set with parameters c = −0,123 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is an independent publisher with close connections to Princeton University. Its mission is to disseminate scholarship within academia and society at large. The press was founded by Whitney Darrow, with the financial support of Charles Scribner, as a printing press to serve the Princeton community in 1905. Its distinctive building was constructed in 1911 on William Street in Princeton. Its first book was a new 1912 edition of John Witherspoon's ''Lectures on Moral Philosophy.'' History Princeton University Press was founded in 1905 by a recent Princeton graduate, Whitney Darrow, with financial support from another Princetonian, Charles Scribner II. Darrow and Scribner purchased the equipment and assumed the operations of two already existing local publishers, that of the ''Princeton Alumni Weekly'' and the Princeton Press. The new press printed both local newspapers, university documents, '' The Daily Princetonian'', and later added book publishing to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brjuno Number
In mathematics, a Brjuno number (sometimes spelled Bruno or Bryuno) is a special type of irrational number named for Russian mathematician Alexander Bruno, who introduced them in . Formal definition An irrational number \alpha is called a Brjuno number when the infinite sum :B(\alpha) = \sum_^\infty \frac converges to a finite number. Here: * q_n is the denominator of the th convergent \tfrac of the continued fraction expansion of \alpha. * B is a Brjuno function Importance The Brjuno numbers are important in the one–dimensional analytic small divisors problems. Bruno improved the diophantine condition in Siegel's Theorem, showed that germs of holomorphic functions with linear part e^ are linearizable if \alpha is a Brjuno number. showed in 1987 that this condition is also necessary, and for quadratic polynomials is necessary and sufficient. Properties Intuitively, these numbers do not have many large "jumps" in the sequence of convergents, in which the denominat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexander Bruno
Alexander Dmitrievich Bruno (russian: Александр Дмитриевич Брюно) (26 June 1940, Moscow) is a Russian people, Russian mathematician who has made contributions to the Normal form (bifurcation theory), normal forms theory. Bruno developed a new level of mathematical analysis and called it "power geometry". He also applied it to the solution of several problems in mathematics, mechanics, celestial mechanics, and hydrodynamics. The Brjuno numbers were introduced by him in 1971, and are named after him. Bruno won third prize at the Moscow Mathematical Olympiade in 1956 and first prize in 1957. He studied at Moscow State University, where he won second prizes for student papers in 1960 and 1961, and earned a master's degree there in 1962.Bio Keldysh Institute of Applied Mathematics, Retrieved 2015-05-04. He completed a doct ... [...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] |
Diophantine Approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria. The first problem was to know how well a real number can be approximated by rational numbers. For this problem, a rational number ''a''/''b'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''a''/''b'' and ''α'' may not decrease if ''a''/''b'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of continued fractions. Knowing the "best" approximations of a given number, the main problem of the field is to find sharp upper and lower bounds of the above difference, expressed as a function of the denominator. It appears that these bounds depend on the nature of the real numbers to be approximated: the lower bound for the approximation of a rational number by another rational number is larger than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Component (analysis)
In topology and other branches of mathematics, a topological space ''X'' is locally connected if every point admits a neighbourhood basis consisting entirely of open, connected sets. Background Throughout the history of topology, connectedness and compactness have been two of the most widely studied topological properties. Indeed, the study of these properties even among subsets of Euclidean space, and the recognition of their independence from the particular form of the Euclidean metric, played a large role in clarifying the notion of a topological property and thus a topological space. However, whereas the structure of ''compact'' subsets of Euclidean space was understood quite early on via the Heine–Borel theorem, ''connected'' subsets of \R^n (for ''n'' > 1) proved to be much more complicated. Indeed, while any compact Hausdorff space is locally compact, a connected space—and even a connected subset of the Euclidean plane—need not be locally connected (see below). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
