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Mandelbrot Set
The Mandelbrot set () is a two-dimensional set (mathematics), set that is defined in the complex plane as the complex numbers c for which the function f_c(z)=z^2+c does not Stability theory, diverge to infinity when Iteration, iterated starting at z=0, i.e., for which the sequence f_c(0), f_c(f_c(0)), etc., remains bounded in absolute value. This set was first defined and drawn by Robert W. Brooks and Peter Matelski in 1978, as part of a study of Kleinian groups. Afterwards, in 1980, Benoit Mandelbrot obtained high-quality visualizations of the set while working at IBM's Thomas J. Watson Research Center in Yorktown Heights, New York. Images of the Mandelbrot set exhibit an infinitely complicated Boundary (topology), boundary that reveals progressively ever-finer Recursion, recursive detail at increasing magnifications; mathematically, the boundary of the Mandelbrot set is a ''fractal curve''. The "style" of this recursive detail depends on the region of the set boundary being ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mandel Zoom 00 Mandelbrot Set
Mandel is a surname (and occasional given name) that occurs in multiple cultures and languages. It is a Dutch, German and Jewish surname, meaning "almond", from the Middle High German and Middle Dutch ''mandel''.''Dictionary of American Family Names''"Mandel Family History" Oxford University Press, 2013. Retrieved on 18 January 2016. Mandel can be a locational surname, from places called Mandel, such as Mandel, Germany. Mandel may also be a Dutch language, Dutch surname, from the Middle Dutch ''mandele'', meaning a number of sheaves of harvested wheat. Notable people *Alon Mandel (born 1988), Israeli swimmer *Babaloo Mandel (born 1949), American screenwriter *David Mandel (born 1970), American television producer and writer *Edgar Mandel (born 1928), German actor *Eli Mandel (1922–1992), Canadian writer *Emily St. John Mandel (born 1979), Canadian novelist *Naum Korzhavin, Emmanuil Mandel (1925–2018), Russian poet *Ernest Mandel (1923–1995), Belgian politician, professor an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Fractal Curve
A fractal curve is, loosely, a mathematical curve (mathematics), curve whose shape retains the same general pattern of Pathological (mathematics), irregularity, regardless of how high it is magnified, that is, its graph takes the form of a fractal. In general, fractal curves are nowhere rectifiable curves — that is, they do not have finite Curve length, length — and every subarc longer than a single Point (geometry), point has infinite length. A famous example is the boundary of the Mandelbrot set. Fractal curves in nature Fractal curves and fractal patterns are widespread, in nature, found in such places as broccoli, snowflakes, feet of geckos, frost crystals, and Lightning strike, lightning bolts. See also Romanesco broccoli, dendrite (crystal), dendrite crystal, Patterns_in_nature#Trees,_fractals, trees, fractals, Hofstadter's butterfly, Lichtenberg figure, and self-organized criticality. Dimensions of a fractal curve Most of us are used to mathematical curves having ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Parameter Space
The parameter space is the space of all possible parameter values that define a particular mathematical model. It is also sometimes called weight space, and is often a subset of finite-dimensional Euclidean space. In statistics, parameter spaces are particularly useful for describing parametric families of probability distributions. They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator. Sometimes, parameters are analyzed to view how they affect their statistical model. In that context, they can be viewed as inputs of a function, in which case the technical term for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
New York (state)
New York, also called New York State, is a U.S. state, state in the northeastern United States. Bordered by New England to the east, Canada to the north, and Pennsylvania and New Jersey to the south, its territory extends into both the Atlantic Ocean and the Great Lakes. New York is the List of U.S. states and territories by population, fourth-most populous state in the United States, with nearly 20 million residents, and the List of U.S. states and territories by area, 27th-largest state by area, with a total area of . New York has Geography of New York (state), a varied geography. The southeastern part of the state, known as Downstate New York, Downstate, encompasses New York City, the List of U.S. cities by population, most populous city in the United States; Long Island, with approximately 40% of the state's population, the nation's most populous island; and the cities, suburbs, and wealthy enclaves of the lower Hudson Valley. These areas are the center of the expansive New ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Bernard Maskit
Bernard (Bernie) Maskit (27 May 1935 – 15 March 2024) was an American mathematician who worked on Kleinian groups, low dimensional geometry and topology, and related topics. Life and Work Maskit studied for both his bachelors and doctoral degrees at New York University, earning his Ph.D. in 1964 under the supervision of Lipman Bers,with a thesis entitled ''On Klein's Combination Theorem''. After postdoctoral studies at the Institute for Advanced Study, he held an assistant professorship at the Massachusetts Institute of Technology from 1965 to 1972. He then moved to the mathematics department at Stony Brook University, where he retired in 2008 and was then a professor emeritus until his death. In 2012, he became one of the inaugural fellows of the American Mathematical Society. Maskit’s main area of mathematical expertise was the study of Kleinian groups acting on low dimensional hyperbolic spaces, where he made fundamental contributions. His works include thplanarity theore ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaston Julia
Gaston Maurice Julia (3 February 1893 – 19 March 1978) was a French mathematician who devised the formula for the Julia set. His works were popularized by 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 25, in 1918, when his 199-pag ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 mathematical analysis, 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 Henri Becquerel, Becquerel prize in 1918; he was a knight of the Legion of Honour (1923). ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Complex Dynamics
Complex dynamics, or holomorphic dynamics, is the study of dynamical systems obtained by Iterated function, iterating a complex analytic mapping. This article focuses on the case of algebraic dynamics, where a polynomial or rational function is iterated. In geometric terms, that amounts to iterating a mapping from some algebraic variety to itself. The related theory of arithmetic dynamics studies iteration over the rational numbers or the p-adic numbers instead of the complex numbers. Dynamics in complex dimension 1 A simple example that shows some of the main issues in complex dynamics is the mapping f(z)=z^2 from the complex numbers C to itself. It is helpful to view this as a map from the complex projective line \mathbf^1 to itself, by adding a point \infty to the complex numbers. (\mathbf^1 has the advantage of being compact space, compact.) The basic question is: given a point z in \mathbf^1, how does its ''orbit'' (or ''forward orbit'') :z,\; f(z)=z^2,\; f(f(z))=z^4, f(f(f(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mandel
Mandel is a surname (and occasional given name) that occurs in multiple cultures and languages. It is a Dutch, German and Jewish surname, meaning "almond", from the Middle High German and Middle Dutch ''mandel''.''Dictionary of American Family Names''"Mandel Family History" Oxford University Press, 2013. Retrieved on 18 January 2016. Mandel can be a locational surname, from places called Mandel, such as Mandel, Germany. Mandel may also be a Dutch surname, from the Middle Dutch ''mandele'', meaning a number of sheaves of harvested wheat. Notable people * Alon Mandel (born 1988), Israeli swimmer *Babaloo Mandel (born 1949), American screenwriter *David Mandel (born 1970), American television producer and writer * Edgar Mandel (born 1928), German actor *Eli Mandel (1922–1992), Canadian writer * Emily St. John Mandel (born 1979), Canadian novelist * Emmanuil Mandel (1925–2018), Russian poet *Ernest Mandel (1923–1995), Belgian politician, professor and writer *Frank Mandel, Ame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Julia Set
In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under iterated function, repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small Perturbation theory, 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 "chaos theory, 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. Form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Image Coordinate
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the ''number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate of a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
