Lyapunov Fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, ''r'', periodically switches between two values ''A'' and ''B''. A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent \lambda) in the ''a''−''b'' plane for given periodic sequences of ''a'' and ''b''. In the images, yellow corresponds to \lambda 0 (chaos). Lyapunov fractals were discovered in the late 1980s by the Germano-Chilean physicist Mario Markus from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991. Properties Lyapunov fractals are generally drawn for values of ''A'' and ''B'' in the interval ,4/math>. For larger values, the interval ,1is ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Physicist
A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate causes of Phenomenon, phenomena, and usually frame their understanding in mathematical terms. They work across a wide range of Physics#Research fields, research fields, spanning all length scales: from atom, sub-atomic and particle physics, through biological physics, to physical cosmology, cosmological length scales encompassing the universe as a whole. The field generally includes two types of physicists: Experimental physics, experimental physicists who specialize in the observation of natural phenomena and the development and analysis of experiments, and Theoretical physics, theoretical physicists who specialize in mathematical modeling of physical systems to rationalize, explain and predict natural phenomena. Physicists can apply their k ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Algorithm
In mathematics and computer science, an algorithm () is a finite sequence of Rigour#Mathematics, mathematically rigorous instructions, typically used to solve a class of specific Computational problem, problems or to perform a computation. Algorithms are used as specifications for performing calculations and data processing. More advanced algorithms can use Conditional (computer programming), conditionals to divert the code execution through various routes (referred to as automated decision-making) and deduce valid inferences (referred to as automated reasoning). In contrast, a Heuristic (computer science), heuristic is an approach to solving problems without well-defined correct or optimal results.David A. Grossman, Ophir Frieder, ''Information Retrieval: Algorithms and Heuristics'', 2nd edition, 2004, For example, although social media recommender systems are commonly called "algorithms", they actually rely on heuristics as there is no truly "correct" recommendation. As an e ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Critical Point (mathematics)
In mathematics, a critical point is the argument of a function where the function derivative is zero (or undefined, as specified below). The value of the function at a critical point is a . More specifically, when dealing with functions of a real variable, a critical point is a point in the domain of the function where the function derivative is equal to zero (also known as a ''stationary point'') or where the function is not differentiable. Similarly, when dealing with complex variables, a critical point is a point in the function's domain where its derivative is equal to zero (or the function is not ''holomorphic''). Likewise, for a function of several real variables, a critical point is a value in its domain where the gradient norm is equal to zero (or undefined). This sort of definition extends to differentiable maps between and a critical point being, in this case, a point where the rank of the Jacobian matrix is not maximal. It extends further to differentiable ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Prize-winners being featured since its inception. In print since 1845, it is the oldest continuously published magazine in the United States. ''Scientific American'' is owned by Springer Nature, which is a subsidiary of Holtzbrinck Publishing Group. History ''Scientific American'' was founded by inventor and publisher Rufus Porter (painter), Rufus Porter in 1845 as a four-page weekly newspaper. The first issue of the large-format New York City newspaper was released on August 28, 1845. Throughout its early years, much emphasis was placed on reports of what was going on at the United States Patent and Trademark Office, U.S. Patent Office. It also reported on a broad range of inventions including perpetual motion machines, an 1860 devi ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Recreational Mathematics
Recreational mathematics is mathematics carried out for recreation (entertainment) rather than as a strictly research-and-application-based professional activity or as a part of a student's formal education. Although it is not necessarily limited to being an endeavor for amateurs, many topics in this field require no knowledge of advanced mathematics. Recreational mathematics involves mathematical puzzles and games, often appealing to children and untrained adults and inspiring their further study of the subject. The Mathematical Association of America (MAA) includes recreational mathematics as one of its seventeen Special Interest Groups, commenting: Mathematical competitions (such as those sponsored by mathematical associations) are also categorized under recreational mathematics. Topics Some of the more well-known topics in recreational mathematics are Rubik's Cubes, magic squares, fractals, logic puzzles and mathematical chess problems, but this area of mathemati ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Science Popularization
Popular science (also called pop-science or popsci) is an interpretation of science intended for a general audience. While science journalism focuses on recent scientific developments, popular science is more broad ranging. It may be written by professional science journalists or by scientists themselves. It is presented in many forms, including books, film and television documentaries, magazine articles, and web pages. History Before the modern specialization and professionalization of science, there was often little distinction between "science" and "popular science", and works intended to share scientific knowledge with a general reader existed as far back as Greek and Roman antiquity. Without these popular works, much of the scientific knowledge of the era might have been lost. For example, none of the original works of the Greek astronomer Eudoxus (4th century BC) have survived, but his contributions were largely preserved due to the didactic poem ''Phenomena'' written ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Max Planck Institute Of Molecular Physiology
The Max Planck Institute of Molecular Physiology () is located in Dortmund, next to the Technical University of Dortmund. It is one of 80 institutes in the Max Planck Society (Max Planck Gesellschaft). Origins The Institute was originally founded by Max Rubner as the ''Kaiser-Wilhelm Institut für Arbeitsphysiologie'', part of the Kaiser Wilhelm Institute in 1913. During the First World War the Institute focussed on nutrition and diet, and how ersatz substitutes could be found for particular foodstuffs which were becoming scarce. They worked alongside the Kriegsernährungsamt (War Nutrition Office) to try out different food surrogates both for the military and the civilian population. They also developed aptitude tests for different types of work, such as evaluating distance perception for military drivers, pilots and railway workers. This was further developed to provide a means for selecting artillery officers. Departments Mechanistic Cell Biology The Department of Mechan ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Mario Markus (physicist)
Mario Markus (born 29 July 1944) is a German-Chilean physicist who worked in the Max Planck Institute for Molecular Physiology in Dortmund. In addition to his scientific work, Markus has exhibited his own computer graphics, as well as written novels, poetry, and translated Spanish poems. Biography In 1970 he obtained his physics diploma under Konrad Tamm at the Institute for Applied Physics at the University of Heidelberg with a thesis on the Pinch-Effekt in plasmas made up of electrons and holes in semiconductors. From March 1970 to 1973 he conducted research for his doctorate on instabilities in plasmas at this institute. At the beginning of 1973 he received his doctorate (Dr. rer. nat.). After working as an assistant at the Institute for Theoretical Physics at the University of Heidelberg, he moved to the Max Planck Institute for Biophysics in Frankfurt am Main in April 1974. From January 1975 he was a research associate at the former Max Planck Institute (MPI) for Nutri ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lyapunov Exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectory, trajectories. Quantitatively, two trajectories in phase space with initial separation vector \boldsymbol_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by , \boldsymbol(t) , \approx e^ , \boldsymbol_0 , where \lambda is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaos theory, chaotic (provided some other conditions are met, e.g., ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Lyapunov Fractal Segment
Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include: * Alexey Lyapunov (1911–1973), Russian mathematician * Aleksandr Lyapunov (1857–1918), son of Mikhail (1820–1868), Russian mathematician and mechanician, after whom the following are named: ** Lyapunov dimension ** Lyapunov equation ** Lyapunov exponent ** Lyapunov function ** Lyapunov fractal ** Lyapunov stability ** Lyapunov's central limit theorem ** Lyapunov time ** Lyapunov vector ** Lyapunov (crater) * Boris Lyapunov (1862–1943), son of Mikhail (1820–1868), Russian expert in Slavic studies * Mikhail Lyapunov (1820–1868), Russian astronomer * Mikhail Nikolaevich Lyapunov (1848–1909), Russian military officer and lawyer * Prokopy Lyapunov (d. 1611), Russian statesman * Sergei Lyapunov (1859–1924), son of Mikhail (1820–1868), Russian composer * Zakhary Lyapunov (d. after 1612), R ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |
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Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (Алекса́ндр Миха́йлович Ляпуно́в, – 3 November 1918) was a Russian mathematician, mechanician and physicist. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory. Biography Early life Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer employed by the Demidov Lyceum. His brother, Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (now Ulyanovsk Oblast). After the death of his father in 1868, Aleksandr Lyapunov was educated by his uncle R. M. Sechenov, brother of the physio ... [...More Info...]       [...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]   |