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Takens' Theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a \nu-dimensional manifold M. The dynamics is given by a smooth map :f: M \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank (linear Algebra)
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental result in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James A
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tank En ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George Sugihara
George Sugihara (born in Tokyo, Japan) is currently a professor of biological oceanography in the Physical Oceanography Research Division at the Scripps Institution of Oceanography, where he is the inaugural holder of the McQuown Chair in Natural Science. Sugihara is a theoretical biologist who works across a variety of fields ranging from ecology and landscape ecology, to epidemiology, to genetics, to geoscience and atmospheric science, to quantitative finance and economics. His work involves inductive theoretical approaches to understanding nature from observational data. The general approach is different from most theory and involves minimalist inductive theory – Inductive data-driven explorations of nature using minimal assumptions. The aim is to avoid inevitable assumptions of deductive first-principle models and produce an understanding that passes the validation test of out-of-sample prediction. His initial work on fisheries as complex, chaotic systems led to work on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricardo Mañé
Ricardo Mañé Ramirez ( Montevideo, 14 January 1948 – Montevideo, 9 March 1995) was a Uruguayan mathematician, known for his contributions to dynamical systems and ergodic theory. He was a doctoral student of Jacob Palis at IMPA. He was an invited speaker at the International Congresses of Mathematicians of 1983 and 1994http://www2.profmat-sbm.org.br/sitesbm/socios_honorarios.asp and is a recipient of the 1994 TWAS Prize. Selected publications *Expansive diffeomorphisms, Proceedings of the Symposium on Dynamical Systems (University of Warwick, 1974) ''Lecture Notes in Mathematics'' Vol. 468 pp. 162–174, Springer-Verlag, 1975. *Persistent manifolds are normally hyperbolic, ''Transactions of the American Mathematical Society'', Vol. 246, (Dec., 1978), pp. 261–283. *On the dimension of the compact invariant sets of certain non-linear maps, Springer, ''Lectures Notes in Mathematics'' Vol. 898 (1981) 230–242. *An ergodic closing lemma, ''Annals of Mathematics'' S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James Doyne Farmer
J. Doyne Farmer (born 22 June 1952) is an American complex systems scientist and entrepreneur with interests in chaos theory, complexity and econophysics. He is Baillie Gifford Professor of Mathematics at Oxford University, where he is also Director of the Complexity Economics at the Institute for New Economic Thinking at the Oxford Martin School. Additionally he is an external professor at the Santa Fe Institute. His current research is on complexity economics, focusing on systemic risk in financial markets and technological progress. During his career he has made important contributions to complex systems, Chaos theory, chaos, artificial life, Mathematical and theoretical biology, theoretical biology, Time series, time series forecasting and econophysics. He co-founded Prediction Company, one of the first companies to do fully automated quantitative trading. While a graduate student he led a group that called itself Eudaemons, Eudaemonic Enterprises and built the first wea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James P
James is a common English language surname and given name: *James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, York, James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * James (2005 film), ''James'' (2005 film), a Bollywood film * James (2008 film), ''James'' (2008 film), an Irish short film * James (2022 film), ''James'' (2022 film), an Indian Kannada ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Norman Packard
Norman Harry Packard (born 1954 in Billings, Montana) is a chaos theory physicist and one of the founders of the Prediction Company and ProtoLife. He is an alumnus of Reed College and the University of California, Santa Cruz. Packard is known for his contributions to chaos theory, complex systems, and artificial life. He coined the phrase "the edge of chaos". Biography Between 1976 and 1981, Packard formed the Dynamical Systems Collective at UC Santa Cruz with fellow physics graduate students, Rob Shaw, Doyne Farmer, and James Crutchfield. The collective was best known for its work in probing chaotic systems for signs of order. Around the same time, he worked with Doyne Farmer and other friends in Santa Cruz, California to form the Eudaemons collective , to develop a strategy for beating the roulette wheel using a toe-operated computer. The computer could, in theory, predict in what area a roulette ball would land on a wheel, giving the player a significant statistical advan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonlinear Dimensionality Reduction
Nonlinear dimensionality reduction, also known as manifold learning, refers to various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with the goal of either visualizing the data in the low-dimensional space, or learning the mapping (either from the high-dimensional space to the low-dimensional embedding or vice versa) itself. The techniques described below can be understood as generalizations of linear decomposition methods used for dimensionality reduction, such as singular value decomposition and principal component analysis. Applications of NLDR Consider a dataset represented as a matrix (or a database table), such that each row represents a set of attributes (or features or dimensions) that describe a particular instance of something. If the number of attributes is large, then the space of unique possible rows is exponentially large. Thus, the larger the dimensionality, the more difficult it becomes to sample the space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Embedding Theorem
In mathematics, particularly in differential topology, there are two Whitney embedding theorems, named after Hassler Whitney: *The strong Whitney embedding theorem states that any differentiable manifold, smooth real numbers, real -dimension (mathematics), dimensional manifold (required also to be Hausdorff space, Hausdorff and second-countable) can be smooth map, smoothly embedding, embedded in the real coordinate space, real -space (), if . This is the best linear bound on the smallest-dimensional Euclidean space that all -dimensional manifolds embed in, as the real projective spaces of dimension cannot be embedded into real -space if is a power of two (as can be seen from a characteristic class argument, also due to Whitney). *The weak Whitney embedding theorem states that any continuous function from an -dimensional manifold to an -dimensional manifold may be approximated by a smooth embedding provided . Whitney similarly proved that such a map could be approximated by an imm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
