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Lucjan Böttcher
{{Infobox scientist , name = Lucjan Emil Böttcher , native_name = , native_name_lang = , image = , image_size = , alt = , caption = , birth_date = , birth_place = Warsaw, Congress Poland, Russian Empire , death_date = , death_place = Lwów, Poland , resting_place = , resting_place_coordinates = , other_names = , residence = , citizenship = Poland , nationality = Polish , fields = , workplaces = , alma_mater = Lwów Polytechnic School, University of Leipzig , thesis_title = Beiträge zu der Theorie der Iterationsrechnung , thesis_url = , thesis_year = 1898 , doctoral_advisor = Sophus Lie , academic_advisors = , doctoral_students = , notable_students = , known_for = Böttcher's equation , author_abbrev_bot = , author_abbrev_zoo = , influences = , influence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Warsaw
Warsaw ( pl, Warszawa, ), officially the Capital City of Warsaw,, abbreviation: ''m.st. Warszawa'' is the capital and largest city of Poland. The metropolis stands on the River Vistula in east-central Poland, and its population is officially estimated at 1.86 million residents within a greater metropolitan area of 3.1 million residents, which makes Warsaw the 7th most-populous city in the European Union. The city area measures and comprises 18 districts, while the metropolitan area covers . Warsaw is an Alpha global city, a major cultural, political and economic hub, and the country's seat of government. Warsaw traces its origins to a small fishing town in Masovia. The city rose to prominence in the late 16th century, when Sigismund III decided to move the Polish capital and his royal court from Kraków. Warsaw served as the de facto capital of the Polish–Lithuanian Commonwealth until 1795, and subsequently as the seat of Napoleon's Duchy of Warsaw. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Habilitation
Habilitation is the highest university degree, or the procedure by which it is achieved, in many European countries. The candidate fulfills a university's set criteria of excellence in research, teaching and further education, usually including a dissertation. The degree, abbreviated "Dr. habil." (Doctor habilitatus) or "PD" (for "Privatdozent"), is a qualification for professorship in those countries. The conferral is usually accompanied by a lecture to a colloquium as well as a public inaugural lecture. History and etymology The term ''habilitation'' is derived from the Medieval Latin , meaning "to make suitable, to fit", from Classical Latin "fit, proper, skillful". The degree developed in Germany in the seventeenth century (). Initially, habilitation was synonymous with "doctoral qualification". The term became synonymous with "post-doctoral qualification" in Germany in the 19th century "when holding a doctorate seemed no longer sufficient to guarantee a proficient transfer o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Samuel Lattès
Samuel Lattès (21 February 1873 (Nice) – 5 July 1918) was a French mathematician. From 1892 to 1895 he studied at the École Normale Superieure. After this he was a teacher in Algiers, Dijon and Nice. After a promotion to Paris in 1906 he moved first to Montpellier in 1908 and then to Besançon, before he took up a professorship at the University of Toulouse in 1911. He died of typhus in 1918. Today Lattès is best known for his work on complex sets, particularly for examples of rational functions including the Riemann sphere in its Julia set. Today these examples are described as Lattès maps or Lattès examples.Für eine moderne Darstellung der Lattèsschen Beispiele und neuere Ergebnisse dazu siehe: John Milnor, ''On Lattès maps''. In ''Dynamics on the Riemann sphere'', European Mathematical Society, Zürich, 2006, S. 9–43 See also *Pierre Fatou *Gaston Julia *Lattès map In mathematics, a Lattès map is a rational map ''f'' = Θ''L''Θ−1 from the complex ... [...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] |
Pierre Fatou
Pierre Joseph Louis Fatou (28 February 1878 – 9 August 1929) was a French mathematician and astronomer. He is known for major contributions to several branches of analysis. The Fatou lemma and the Fatou set are named after him. Biography Pierre Fatou's parents were Prosper Ernest Fatou (1832-1891) and Louise Eulalie Courbet (1844-1911), both of whom were in the military. Pierre's family would have liked for him to enter the military as well, but his health was not sufficiently good for him to pursue a military career. Fatou entered the École Normale Supérieure in Paris in 1898 to study mathematics and graduated in 1901 when he was appointed an intern (''stagiaire'') in the Paris Observatory. Fatou was promoted to assistant astronomer in 1904 and to astronomer (''astronome titulaire'') in 1928. He worked in this observatory until his death. Fatou was awarded the Becquerel prize in 1918; he was a knight of the Legion of Honour (1923). He was the president of the French ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holomorphic Dynamics
Complex dynamics is the study of dynamical systems defined by 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 *Combinatorial ** Hubbard trees ** Spider algorithm ** Tuning ** Laminations ** Devil's Staircase algorithm (Cantor function) **Orbit portraits ** 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 compositions of analytic functions *Julia set *Mandelbrot set *Symbolic dynamics Notes Refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Function
In the mathematical field of complex analysis, elliptic functions are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse. Important elliptic functions are Jacobi elliptic functions and the Weierstrass \wp-function. Further development of this theory led to hyperelliptic functions and modular forms. Definition A meromorphic function is called an elliptic function, if there are two \mathbb- linear independent complex numbers \omega_1,\omega_2\in\mathbb such that : f(z + \omega_1) = f(z) and f(z + \omega_2) = f(z), \quad \forall z\in\mathbb. So elliptic functions have two periods and are therefore also called ''doubly periodic''. Period lattice and fundamental domain Iff is an elliptic function with periods \omega_1,\omega_2 it also holds that : f(z+\gamma)=f(z) for every linear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julia Set
In the context of complex dynamics, a branch of mathematics, the Julia set and the Fatou set are two complementary sets (Julia "laces" and Fatou "dusts") defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values. Thus the behavior of the function on the Fatou set is "regular", while on the Julia set its behavior is "chaotic". The Julia set of a function is commonly denoted \operatorname(f), and the Fatou set is denoted \operatorname(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century. Formal definition Let f(z) be a non-constant holomorphic function from the Riemann sphere on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fatou Components
Fatou may refer to: People * Aminata Fatou Diallo (born 1995), French footballer * Cecilia Fatou-Berre (1901 – 1989), religious sister * Fatou Baldeh (born 1983), Gambian women's rights activist * Fatou Bensouda, Gambian lawyer and former Chief Prosecutor of the International Criminal Court * Fatou Bintou Fall (born 1981), Senegalese athlete * Fatou Camara (journalist), Gambian journalist * Fatou Coulibaly (born 1987), Ivorian footballer * Fatou Diagne (born 1996), Senegalese basketball player * Fatou Diatta, Senegalese rapper and activist * Fatou Dieng (athlete) (born 1983), Mauritanian sprint athlete * Fatou Dieng (basketball) (born 1983), Senegalese basketball player * Fatou Diome, French-Senegalese writer * Fatou Dioup (born 1994), Mauritanian footballer * Fatou Jagne, Gambian jurist * Fatou Jallow, Gambian model * Fatou Jaw-Manneh, Gambian journalist * Fatou Kandé Senghor (born 1971), Senegalese film director * Fatou Kanteh (born 1997), Gambian footballer * Fatou Keïta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Sphere
In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value \infty for infinity. With the Riemann model, the point \infty is near to very large numbers, just as the point 0 is near to very small numbers. The extended complex numbers are useful in complex analysis because they allow for division by zero in some circumstances, in a way that makes expressions such as 1/0=\infty well-behaved. For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, the Riemann sphere is the prototypical example of a Riemann surface, and is one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Mapping
In mathematics, in particular the subfield of algebraic geometry, a rational map or rational mapping is a kind of partial function between algebraic varieties. This article uses the convention that varieties are irreducible. Definition Formal definition Formally, a rational map f \colon V \to W between two varieties is an equivalence class of pairs (f_U, U) in which f_U is a morphism of varieties from a non-empty open set U\subset V to W, and two such pairs (f_U, U) and (_, U') are considered equivalent if f_U and _ coincide on the intersection U \cap U' (this is, in particular, vacuously true if the intersection is empty, but since V is assumed irreducible, this is impossible). The proof that this defines an equivalence relation relies on the following lemma: * If two morphisms of varieties are equal on some non-empty open set, then they are equal. f is said to be birational if there exists a rational map g \colon W \to V which is its inverse, where the composition is taken i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Calculus
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along with Derivative, differentiation, integration is a fundamental, essential operation of calculus,Integral calculus is a very well established mathematical discipline for which there are many sources. See and , for example. and serves as a tool to solve problems in mathematics and physics involving the area of an arbitrary shape, the length of a curve, and the volume of a solid, among others. The integrals enumerated here are those termed definite integrals, which can be interpreted as the signed area of the region in the plane that is bounded by the Graph of a function, graph of a given function between two points in the real line. Conventionally, areas above the horizontal axis of the plane are posi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
