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Symbolic Dynamics
In mathematics, symbolic dynamics is the study of dynamical systems defined on a discrete space consisting of infinite sequences of abstract symbols. The evolution of the dynamical system is defined as a simple shift of the sequence. Because of their explicit, discrete nature, such systems are often relatively easy to characterize and understand. They form a key tool for studying topological or smooth dynamical systems, because in many important cases it is possible to reduce the dynamics of a more general dynamical system to a symbolic system. To do so, a Markov partition is used to provide a finite cover for the smooth system; each set of the cover is associated with a single symbol, and the sequences of symbols result as a trajectory of the system moves from one covering set to another. History The idea goes back to Jacques Hadamard's 1898 paper on the geodesics on surfaces of negative curvature. It was applied by Marston Morse in 1921 to the construction of a nonperi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Levinson
Norman Levinson (August 11, 1912 in Lynn, Massachusetts – October 10, 1975 in Boston) was an American mathematician. Some of his major contributions were in the study of Fourier transforms, complex analysis, non-linear differential equations, number theory, and signal processing. He worked closely with Norbert Wiener in his early career. He joined the faculty of the Massachusetts Institute of Technology in 1937. In 1954, he was awarded the Bôcher Memorial Prize of the American Mathematical Society and in 1971 the Chauvenet Prize (after winning in 1970 the Lester R. Ford Award) of the Mathematical Association of America for his paper ''A Motivated Account of an Elementary Proof of the Prime Number Theorem''. In 1974 he published a paper proving that more than a third of the zeros of the Riemann zeta function lie on the critical line, a result later improved to two fifths by Conrey. He received both his bachelor's degree and his master's degree in electrical engineering ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gibbs Measure
In physics and mathematics, the Gibbs measure, named after Josiah Willard Gibbs, is a probability measure frequently seen in many problems of probability theory and statistical mechanics. It is a generalization of the canonical ensemble to infinite systems. The canonical ensemble gives the probability of the system ''X'' being in state ''x'' (equivalently, of the random variable ''X'' having value ''x'') as :P(X=x) = \frac \exp ( - \beta E(x)). Here, is a function from the space of states to the real numbers; in physics applications, is interpreted as the energy of the configuration ''x''. The parameter is a free parameter; in physics, it is the inverse temperature. The normalizing constant is the partition function. However, in infinite systems, the total energy is no longer a finite number and cannot be used in the traditional construction of the probability distribution of a canonical ensemble. Traditional approaches in statistical physics studied the limit of intensive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yakov Sinai
Yakov Grigorevich Sinai (; born September 21, 1935) is a Russian–American mathematician known for his work on dynamical systems. He contributed to the modern metric theory of dynamical systems and connected the world of deterministic (dynamical) systems with the world of probabilistic (stochastic) systems. He has also worked on mathematical physics and probability theory. His efforts have provided the groundwork for advances in the physical sciences. Sinai has won several awards, including the Nemmers Prize, the Wolf Prize in Mathematics and the Abel Prize. He has served as professor of mathematics at Princeton University since 1993 and holds the position of Senior Researcher at the Landau Institute for Theoretical Physics in Moscow, Russia. Biography Yakov Grigorevich Sinai was born into a Russian Jewish academic family on September 21, 1935, in Moscow, Soviet Union (now Russia). His parents, Nadezda Kagan and Gregory Sinai, were both microbiologists. His grandfat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Anosov Diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold ''M'' is a certain type of mapping, from ''M'' to itself, with rather clearly marked local directions of "expansion" and "contraction". Anosov systems are a special case of Axiom A systems. Anosov diffeomorphisms were introduced by Dmitri Anosov, Dmitri Victorovich Anosov, who proved that their behaviour was in an appropriate sense ''generic'' (when they exist at all). Overview Three closely related definitions must be distinguished: * If a differentiable map (mathematics), map ''f'' on ''M'' has a Hyperbolic set, hyperbolic structure on the tangent bundle, then it is called an Anosov map. Examples include the Bernoulli map, and Arnold's cat map. * If the map is a diffeomorphism, then it is called an Anosov diffeomorphism. * If a flow (mathematics), flow on a manifold splits the tangent bundle into three invariant subbundles, with one subbundle that is exponen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The National Academy Of Sciences Of The United States Of America
''Proceedings of the National Academy of Sciences of the United States of America'' (often abbreviated ''PNAS'' or ''PNAS USA'') is a peer-reviewed multidisciplinary scientific journal. It is the official journal of the National Academy of Sciences, published since 1915, and publishes original research, scientific reviews, commentaries, and letters. According to ''Journal Citation Reports'', the journal has a 2022 impact factor of 9.4. ''PNAS'' is the second most cited scientific journal, with more than 1.9 million cumulative citations from 2008 to 2018. In the past, ''PNAS'' has been described variously as "prestigious", "sedate", "renowned" and "high impact". ''PNAS'' is a delayed open-access journal, with an embargo period of six months that can be bypassed for an author fee ( hybrid open access). Since September 2017, open access articles are published under a Creative Commons license. Since January 2019, ''PNAS'' has been online-only, although print issues are available ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Benjamin Weiss
Benjamin Weiss (; born 1941) is an American-Israeli mathematician known for his contributions to ergodic theory, topological dynamics, probability theory, game theory, and descriptive set theory. Biography Benjamin ("Benjy") Weiss was born in New York City. In 1962 he received B.A. from Yeshiva University and M.A. from the Graduate School of Science, Yeshiva University. In 1965, he received his Ph.D. from Princeton under the supervision of William Feller. Academic career Between 1965 and 1967, Weiss worked at the IBM Research. In 1967, he joined the faculty of the Hebrew University of Jerusalem; and since 1990 occupied the Miriam and Julius Vinik Chair in Mathematics (Emeritus since 2009). Weiss held visiting positions at Stanford, MSRI, and IBM Research Center. Weiss published over 180 papers in ergodic theory, topological dynamics, orbit equivalence, probability, information theory, game theory, descriptive set theory; with notable contributions including introduction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Roy Adler
Roy Lee Adler (February 22, 1931 – July 26, 2016) was an American mathematician. Adler earned his Ph.D. in 1961 from Yale University under the supervision of Shizuo Kakutani (''On some algebraic aspects of measure preserving transformations''). He then worked as a mathematician for IBM at the Thomas J. Watson Research Center. Adler studies dynamical systems, ergodic theory, symbolic and topological dynamics and coding theory. The road coloring problem that was solved by Avraham Trakhtman in 2007 came from him, along with L. W. Goodwyn and Benjamin Weiss. He was a fellow of the American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, .... A paper was written on his work and the impact of his work by Bruce Kitchens and others. Writings *With Brian M ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Toral Automorphism
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Hyperbolic may refer to: * of or pertaining to a hyperbola, a type of smooth curve lying in a plane in mathematics ** Hyperbolic geometry, a non-Euclidean geometry ** Hyperbolic functions, analogues of ordinary trigonometric functions, defined using the hyperbola * of or pertaining to hyperbole, the use of exaggeration as a rhetorical device or figure of speech * ''Hyperbolic'' (album), by Pnau, 2024 See also * Exaggeration * Hyperboloid In geometry, a hyperboloid of revolution, sometimes called a circular hyperboloid, is the surface generated by rotating a hyperbola around one of its principal axes. A hyperboloid is the surface obtained from a hyperboloid of revolution by def ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Information Theory
Information theory is the mathematical study of the quantification (science), quantification, Data storage, storage, and telecommunications, communication of information. The field was established and formalized by Claude Shannon in the 1940s, though early contributions were made in the 1920s through the works of Harry Nyquist and Ralph Hartley. It is at the intersection of electronic engineering, mathematics, statistics, computer science, Neuroscience, neurobiology, physics, and electrical engineering. A key measure in information theory is information entropy, entropy. Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a Fair coin, fair coin flip (which has two equally likely outcomes) provides less information (lower entropy, less uncertainty) than identifying the outcome from a roll of a dice, die (which has six equally likely outcomes). Some other important measu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A Mathematical Theory Of Communication
"A Mathematical Theory of Communication" is an article by mathematician Claude E. Shannon published in '' Bell System Technical Journal'' in 1948. It was renamed ''The Mathematical Theory of Communication'' in the 1949 book of the same name, a small but significant title change after realizing the generality of this work. It has tens of thousands of citations, being one of the most influential and cited scientific papers of all time, as it gave rise to the field of information theory, with ''Scientific American'' referring to the paper as the "Magna Carta of the Information Age", while the electrical engineer Robert G. Gallager called the paper a "blueprint for the digital era". Historian James Gleick rated the paper as the most important development of 1948, placing the transistor second in the same time period, with Gleick emphasizing that the paper by Shannon was "even more profound and more fundamental" than the transistor. It is also noted that "as did relativity and qua ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shift Of Finite Type
In mathematics, subshifts of finite type are used to model dynamical systems, and in particular are the objects of study in symbolic dynamics and ergodic theory. They also describe the set of all possible sequences executed by a finite-state machine. The most widely studied shift spaces are the subshifts of finite type. Motivating examples A (one-sided) shift of finite type is the set of all sequences, infinite on one end only, that can be made up of the letters A, B, C, like AAA\cdots, ABAB\cdots, \dots. A (two-sided) shift of finite type is similar, but consists of sequences that are infinite on both ends. A subshift can be defined by a directed graph on these letters, such as the graph A \to B \to C \to A. It consists of sequences whose transitions between consecutive letters are only those allowed by the graph. For this example, the subshift consists of only three one-sided sequences: ABCABC\cdots, BCABCA\cdots, CABCAB\cdots. Similarly, the two-sided subshift described by th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
