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Almost Mathieu Operator
In mathematical physics, the almost Mathieu operator arises in the study of the quantum Hall effect. It is given by : ^_\omega un) = u(n+1) + u(n-1) + 2 \lambda \cos(2\pi (\omega + n\alpha)) u(n), \, acting as a self-adjoint operator on the Hilbert space \ell^2(\mathbb). Here \alpha,\omega \in\mathbb, \lambda > 0 are parameters. In pure mathematics, its importance comes from the fact of being one of the best-understood examples of an ergodic Schrödinger operator. For example, three problems (now all solved) of Barry Simon's fifteen problems about Schrödinger operators "for the twenty-first century" featured the almost Mathieu operator. In physics, the almost Mathieu operators can be used to study metal to insulator transitions like in the Aubry–André model. For \lambda = 1, the almost Mathieu operator is sometimes called Harper's equation. The spectral type If \alpha is a rational number, then H^_\omega is a periodic operator and by Floquet theory its spectrum is purely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications In Mathematical Physics
''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, statistical mechanics and quantum field theory, and in operator algebras, quantum information and relativity. History Rudolf Haag conceived this journal with Res Jost, and Haag became the Founding Chief Editor. The first issue of ''Communications in Mathematical Physics'' appeared in 1965. Haag guided the journal for the next eight years. Then Klaus Hepp succeeded him for three years, followed by James Glimm, for another three years. Arthur Jaffe began as chief editor in 1979 and served for 21 years. Michael Aizenman became the fifth chief editor in the year 2000 and served in this role until 2012. The current editor-in-chief is Horng-Tzer Yau. Archives Articles from 1965 to 1997 are available in electronic form free of charge, via Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hofstadter's Butterfly
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps n the graphform a very striking pattern somewhat resembling a butterfly." The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect and the theory of topological quantum numbers. History The first mathematical description of electrons on a 2D lattice, acted on by a perpendicular homogeneous magnetic field, was studied by Rudolf Peierls and his student R. G. Harper in the 1950s. Hofstadter first described the structure in 1976 in an article on the energy levels of Bloch electrons in perpendicular magnetic fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Douglas Hofstadter
Douglas Richard Hofstadter (born February 15, 1945) is an American scholar of cognitive science, physics, and comparative literature whose research includes concepts such as the sense of self in relation to the external world, consciousness, analogy-making, artistic creation, literary translation, and discovery in mathematics and physics. His 1979 book '' Gödel, Escher, Bach: An Eternal Golden Braid'' won both the Pulitzer Prize for general nonfiction"General Nonfiction" . ''Past winners and finalists by category''. The Pulitzer Prizes. Retrieved March 17, 2012. and a (at that time called The American Book Award) for Science. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides with the standard measure of length, area, or volume. In general, it is also called ''n''-dimensional volume, ''n''-volume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set ''A'' is here denoted by ''λ''(''A''). Henri Lebesgue described this measure in the year 1901, followed the next year by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq\mathbb, the Lebesgue oute ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Functional Analysis
The ''Journal of Functional Analysis'' is a mathematics journal published by Elsevier. Founded by Paul Malliavin, Ralph S. Phillips, and Irving Segal, its editors-in-chief are Daniel W. Stroock, Stefaan Vaes, and Cedric Villani. It is covered in databases including Scopus, the Science Citation Index, and the SCImago Journal Rank The SCImago Journal Rank (SJR) indicator is a measure of the prestige of scholarly journals that accounts for both the number of citations received by a journal and the prestige of the journals where the citations come from. Rationale Citati ... service. References {{Mathematics-journal-stub Elsevier academic journals Mathematics journals Publications established in 1967 Semi-monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artur Avila
Artur Avila Cordeiro de Melo (born 29 June 1979) is a Brazilian and naturalized French mathematician working primarily in the fields of dynamical systems and spectral theory. He is one of the winners of the 2014 Fields Medal, being the first Latin American and lusophone to win such an award. He has been a researcher at both the IMPA and the CNRS (working a half-year in each one). He has been a professor at the University of Zurich since September 2018. Biography At the age of 16, Avila won a gold medal at the 1995 International Mathematical Olympiad and received a scholarship for the Instituto Nacional de Matemática Pura e Aplicada (IMPA) to start a M.S. degree while still attending high school in Colégio de São Bento and Colégio Santo Agostinho in Rio de Janeiro. He completed his M.S. degree in 1997. Later he enrolled in the Federal University of Rio de Janeiro (UFRJ), earning his B.S in mathematics. At the age of 19, Avila began writing his doctoral thesis on the the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cantor Set
In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Through consideration of this set, Cantor and others helped lay the foundations of modern point-set topology. The most common construction is the Cantor ternary set, built by removing the middle third of a line segment and then repeating the process with the remaining shorter segments. Cantor mentioned the ternary construction only in passing, as an example of a more general idea, that of a perfect set that is nowhere dense. More generally, in topology, ''a'' Cantor space is a topological space homeomorphic to the Cantor ternary set (equipped with its subspace topology). By a theorem of Brouwer, this is equivalent to being perfect nonempty, compact metrizable and zero dimensional. Construction and formula of the ternary set The Cantor tern ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hofstadter's Butterfly
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps n the graphform a very striking pattern somewhat resembling a butterfly." The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect and the theory of topological quantum numbers. History The first mathematical description of electrons on a 2D lattice, acted on by a perpendicular homogeneous magnetic field, was studied by Rudolf Peierls and his student R. G. Harper in the 1950s. Hofstadter first described the structure in 1976 in an article on the energy levels of Bloch electrons in perpendicular magnetic fiel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Statistical Physics
The ''Journal of Statistical Physics'' is a biweekly publication containing both original and review papers, including book reviews. All areas of statistical physics as well as related fields concerned with collective phenomena in physical systems are covered. The ''Journal of Statistical Physics'' has an impact factor of 1.243 (2019). The journal was established by Howard Reiss. Joel L. Lebowitz is the honorary editor. In the period 1969-1979 the journal published about 65 articles per year, while in the 1980-2016 period approximately 220 articles per year. In total, as to 2017, more than 9000 articles have appeared on this journal. According to Web of Science as of July 2017 the 10 most cited articles which have appeared on this journal are: # Tsallis, C, Possible generalization of Boltzmann-Gibbs statistics, J. Stat. Phys., vol. 52(1-2), 479-487, (1988). Times Cited: 4,245 # Feigenbaum, MJ, Quantitative universality for a class of non-linear transformations, J. Stat. Phys., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Svetlana Jitomirskaya
Svetlana Yakovlevna Jitomirskaya (born June 4, 1966) is a Soviet-born American mathematician working on dynamical systems and mathematical physics. She is a distinguished professor of mathematics at the University of California, Irvine. She is best known for solving the ten martini problem along with mathematician Artur Avila. Education and career Jitomirskaya was born and grew up in Kharkiv. Both her mother, Valentina Borok, and her father, Yakov Zhitomirskii, were professors of mathematics. Her undergraduate studies were at Moscow State University, where she was a student of, among others, Vladimir Arnold and Yakov Sinai. She obtained her Ph.D. from Moscow State University in 1991 under the supervision of Yakov Sinai. She joined the mathematics department at the University of California, Irvine in 1991 as a lecturer, and she became an assistant professor there in 1994 and a full professor in 2000. Honors In 2005, she was awarded the Ruth Lyttle Satter Prize in Mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean Bourgain
Jean, Baron Bourgain (; – ) was a Belgian mathematician. He was awarded the Fields Medal in 1994 in recognition of his work on several core topics of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, ergodic theory and nonlinear partial differential equations from mathematical physics. Biography Bourgain received his PhD from the Vrije Universiteit Brussel in 1977. He was a faculty member at the University of Illinois, Urbana-Champaign and, from 1985 until 1995, professor at Institut des Hautes Études Scientifiques at Bures-sur-Yvette in France, at the Institute for Advanced Study in Princeton, New Jersey from 1994 until 2018. He was an editor for the ''Annals of Mathematics''. From 2012 to 2014, he was a visiting scholar at UC Berkeley. His research work included several areas of mathematical analysis such as the geometry of Banach spaces, harmonic analysis, analytic number theory, combinatorics, ergodic theory, partial differential equ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
