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Siegel Disc
A Siegel disc or Siegel disk 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 , \ ... [...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 (set theory), union of two or more disjoint set, disjoint Empty set, non-empty open (topology), open subsets. Connectedness is one of the principal topological properties that distinguish topological spaces. A subset of a topological space X is a if it is a connected space when viewed as a Subspace topology, subspace of X. Some related but stronger conditions are #Path connectedness, path connected, Simply connected space, simply connected, and N-connected space, n-connected. Another related notion is Locally connected space, 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. So ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carl Ludwig Siegel
Carl Ludwig Siegel (31 December 1896 – 4 April 1981) was a German mathematician specialising in analytic number theory. He is known for, amongst other things, his contributions to the Thue–Siegel–Roth theorem in Diophantine approximation, Siegel's method, Siegel's lemma and the Siegel mass formula for quadratic forms. He has been named one of the most important mathematicians of the 20th century.Pérez, R. A. (2011''A brief but historic article of Siegel'' NAMS 58(4), 558–566. André Weil, without hesitation, named Siegel as the greatest mathematician of the first half of the 20th century. Atle Selberg said of Siegel and his work: Biography Siegel was born in Berlin, where he enrolled at the Humboldt University in Berlin in 1915 as a student in mathematics, astronomy, and physics. Amongst his teachers were Max Planck and Ferdinand Georg Frobenius, whose influence made the young Siegel abandon astronomy and turn towards number theory instead. His best-known student ... [...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
A Douady rabbit is a fractal derived from the filled Julia set, Julia set of the function f_c(z) = z^2+c, when Parameter (computer programming), parameter c is near the center of one of the Mandelbrot set#Main cardioid and period bulbs, period three bulbs of the Mandelbrot set for a complex quadratic map. It is named after France, French mathematician Adrien Douady. Background The Douady rabbit is generated by iterating the Mandelbrot set, Mandelbrot set map z_=z_n^2+c on the complex plane, where parameter c is fixed to lie in one of the two period three bulb off the main cardioid and z ranging over the plane. The resulting image can be colored by corresponding each pixel with a starting value z_0 and calculating the amount of iteration, iterations required before the value of z_n escapes a bounded region, after which it will diverge toward infinity. It can also be described using the ''Complex quadratic polynomial#Forms, logistic map form'' of the complex quadratic map, specific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Princeton University Press
Princeton University Press is a nonprofit publisher closely affiliated with 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 Princeton graduate and manager of the ''Alumni Weekly'', Whitney Darrow. It began as Princeton Alumni Press, a small printing house which published the ''Princeton Alumni Weekly''. The press received financial support from Princeton alumnus, 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 Princ ... [...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 convergent series, converges to a finite number. Here: * q_n is the denominator of the th Continued_fraction#Infinite_continued_fractions_and_convergents, convergent \tfrac of the simple continued fraction, continued fraction expansion of \alpha. * B is a #Brjuno_function, Brjuno function Examples Consider the golden ratio : :\phi = \frac = 1+\cfrac. Then the ''n''th convergent \frac can be found via the recurrence relation: :\begin p_n = p_ + p_ & \text p_0=1,p_1=2, \\ q_n = q_ + q_ & \text q_0=q_1=1. \end It is easy to see that q_ |
Alexander Bruno
Alexander Dmitrievich Bruno () (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 doctorate from Moldova State University, Kishinev State Univers ... [...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 m ... [...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 ''p''/''q'' is a "good" approximation of a real number ''α'' if the absolute value of the difference between ''p''/''q'' and ''α'' may not decrease if ''p''/''q'' is replaced by another rational number with a smaller denominator. This problem was solved during the 18th century by means of simple 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 i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers 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 case of irrational numbers, the decimal expansion does not terminate, nor end ... [...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 of open connected sets. As a stronger notion, the space ''X'' is locally path connected if every point admits a neighbourhood basis consisting of open path 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 ... [...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 summation, sum to the larger of the two quantities. Expressed algebraically, for quantities and with , is in a golden ratio to if \frac = \frac = \varphi, where the Greek letter Phi (letter), phi ( or ) denotes the golden ratio. The constant satisfies the quadratic equation 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; it also goes by 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 Straightedge and compass construction, construction of the dodecahedron and icosahedron. A golden rectangle—that is, a rectangle with an aspect ratio of —may be cut into a square and a smaller rectangle with the same aspect ratio. The golden ratio has bee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
