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Weyl Law
In mathematics, especially spectral theory, Weyl's law describes the asymptotic behavior of eigenvalues of the Laplace–Beltrami operator. This description was discovered in 1911 (in the d=2,3 case) by Hermann Weyl for eigenvalues for the Laplace–Beltrami operator acting on functions that vanish at the boundary of a bounded domain \Omega \subset \mathbb^d. In particular, he proved that the number, N(\lambda), of Dirichlet eigenvalues (counting their multiplicities) less than or equal to \lambda satisfies : \lim_ \frac = (2\pi)^ \omega_d \mathrm(\Omega) where \omega_d is a volume of the unit ball in \mathbb^d. In 1912 he provided a new proof based on variational methods. Generalizations The Weyl law has been extended to more general domains and operators. For the Schrödinger operator : H=-h^2 \Delta + V(x) it was extended to : N(E,h)\sim (2\pi h)^ \int _ dx d\xi as E tending to +\infty or to a bottom of essential spectrum and/or h\to +0. Here N(E,h) is the number of ei ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard Courant
Richard Courant (January 8, 1888 – January 27, 1972) was a German American mathematician. He is best known by the general public for the book '' What is Mathematics?'', co-written with Herbert Robbins. His research focused on the areas of real analysis, mathematical physics, the calculus of variations and partial differential equations. He wrote textbooks widely used by generations of students of physics and mathematics. He is also known for founding the institute now bearing his name. Life and career Courant was born in Lublinitz, in the Prussian Province of Silesia. His parents were Siegmund Courant and Martha Courant ''née'' Freund of Oels. Edith Stein was Richard's cousin on the paternal side. During his youth his parents moved often, including to Glatz, then to Breslau and in 1905 to Berlin. He stayed in Breslau and entered the university there, then continued his studies at the University of Zürich and the University of Göttingen. He became David Hilbert's a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Ivrii
Victor Ivrii ( rus, Виктор Яковлевич Иврий), (born 1 October 1949) is a Russian, Canadian mathematician who specializes in analysis, microlocal analysis, spectral theory and partial differential equations. He is a professor at the University of Toronto Department of Mathematics. He was an invited speaker at International Congress of Mathematicians, Helsinki—1978 and Berkeley—1986. Education and Degrees He graduated from Physical Mathematical School at Novosibirsk State University in 1965, received his University Diploma (equivalent to MSci) in 1970 and PhD in 1973 in Novosibirsk State University. He defended his Doktor nauk thesis in St. Petersburg Department of Steklov Institute of Mathematics of Russian Academy of Sciences in 1982. Scientific Contributions Weakly hyperbolic equations His first main works were devoted to the well-posedness of the Cauchy problem for weakly hyperbolic equations. In particular he discovered a necessary (later proven ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current managing editors are Camillo De Lellis (Institute for Advanced Study The Institute for Advanced Study (IAS), located in Princeton, New Jersey, in the United States, is an independent center for theoretical research and intellectual inquiry. It has served as the academic home of internationally preeminent schola ..., Princeton) and Jean-Benoît Bost ( University of Paris-Sud). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Guillemin
Victor William Guillemin (born 1937 in Boston) is an American mathematician working in the field of symplectic geometry, who has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics. He is a tenured Professor in the Department of Mathematics at the Massachusetts Institute of Technology. His uncle Ernst Guillemin was a Professor of Electrical Engineering and Computer Science (EECS) at MIT, and his daughter Karen Guillemin is a Professor of Biology at the University of Oregon. Professional career Guillemin received a Ph.D. in mathematics from Harvard University in 1962, after earlier completing his B. A. at Harvard in 1959, as well as an M. A. at the University of Chicago in 1960. His thesis, entitled ''Theory of Finite G-Structures,'' was written under the direction of Shlomo Sternberg. He is the author or co-author of numerous books and monographs, including a widely used textbook on differential topology, written jointly wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hans Duistermaat
Johannes Jisse (Hans) Duistermaat (The Hague, December 20, 1942 – Utrecht, March 19, 2010) was a Dutch mathematician. Biography Duistermaat attended primary school in Jakarta, at the time capital of the Dutch East Indies, where his family moved after the end of World War II. In 1957, a few years after the Indonesian independence, they came back to the Netherlands and Duistermaat completed his high school studies in Vlaardingen. From 1959 to 1965 he studied studied mathematics at Utrecht University, and he obtained his PhD degree at the same institution in 1968, with a thesis on the mathematical structures of thermodynamics entitled "''Energy and Entropy as Real Morphisms for Addition and Order''". His original supervisor was the applied mathematician Günther K. Braun, who passed away one year before the thesis defense, so the official supervision was taken over by geometer Hans Freudenthal. After a postdoctoral stay in Lund (1969–70), Duistermaat returned to the Netherla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Advances In Mathematics
''Advances in Mathematics'' is a peer-reviewed scientific journal covering research on pure mathematics. It was established in 1961 by Gian-Carlo Rota. The journal publishes 18 issues each year, in three volumes. At the origin, the journal aimed at publishing articles addressed to a broader "mathematical community", and not only to mathematicians in the author's field. Herbert Busemann writes, in the preface of the first issue, "The need for expository articles addressing either all mathematicians or only those in somewhat related fields has long been felt, but little has been done outside of the USSR. The serial publication ''Advances in Mathematics'' was created in response to this demand." Abstracting and indexing The journal is abstracted and indexed in: * |
Robert Thomas Seeley
Robert Thomas Seeley (born February 26, 1932, in Bryn Mawr, Pennsylvania, United States–died November 30, 2016, in Newton, Massachusetts) was a mathematician who worked on pseudo differential operators and the heat equation approach to the Atiyah–Singer index theorem. Seeley did his undergraduate studies at Haverford College, and earned his Ph.D. from the Massachusetts Institute of Technology in 1959, under the supervision of Alberto Pedro Calderón. He taught at Harvey Mudd College and then in 1962 joined the faculty of Brandeis University. In 1972 he moved to the University of Massachusetts Boston; he retired as an emeritus professor. U. Mass. Boston Mathematics, retrieved 2016-12-01. In 2012 he became a fellow of the |
Boris Levitan
Boris Levitan (7 June 1914 – 4 April 2004) was a mathematician known in particular for his work on almost periodic functions, and Sturm–Liouville operators, especially, on inverse scattering. Life Boris Levitan was born in Berdyansk (south-eastern Ukraine), and grew up in Kharkiv. He graduated from Kharkov University in 1936; in 1938, he submitted his PhD thesis "''Some Generalization of Almost Periodic Function''" under the supervision of Naum Akhiezer. Then he defended the habilitation thesis "''Theory of Generalized Translation Operators''". He was drafted into the army at the beginning of World War II in 1941, and served until 1944. From 1944 to 1961 he worked at the Dzerzhinsky Military Academy, and from 1961 until about 1992 at Moscow University. In 1992 Levitan emigrated to the United States. During the last years of his life, he worked in the University of Minnesota The University of Minnesota, formally the University of Minnesota, Twin Cities, (UMN ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Essential Spectrum
In mathematics, the essential spectrum of a bounded operator (or, more generally, of a densely defined closed linear operator) is a certain subset of its spectrum, defined by a condition of the type that says, roughly speaking, "fails badly to be invertible". The essential spectrum of self-adjoint operators In formal terms, let ''X'' be a Hilbert space and let ''T'' be a self-adjoint operator on ''X''. Definition The essential spectrum of ''T'', usually denoted σess(''T''), is the set of all complex numbers λ such that :T-\lambda I_X is not a Fredholm operator, where I_X denotes the ''identity operator'' on ''X'', so that I_X(x)=x for all ''x'' in ''X''. (An operator is Fredholm if its kernel and cokernel are finite-dimensional.) Properties The essential spectrum is always closed, and it is a subset of the spectrum. Since ''T'' is self-adjoint, the spectrum is contained on the real axis. The essential spectrum is invariant under compact perturbations. That is, if ''K ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theory
In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of systems of linear equations and their generalizations. The theory is connected to that of analytic functions because the spectral properties of an operator are related to analytic functions of the spectral parameter. Mathematical background The name ''spectral theory'' was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in infinitely many variables. The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics that spectral theory could explain features of atomic spectra was ther ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Microlocal Analysis
In mathematical analysis, microlocal analysis comprises techniques developed from the 1950s onwards based on Fourier transforms related to the study of variable-coefficients-linear and nonlinear partial differential equations. This includes generalized functions, pseudo-differential operators, wave front sets, Fourier integral operators, oscillatory integral operators, and paradifferential operators. The term ''microlocal'' implies localisation not only with respect to location in the space, but also with respect to cotangent space directions at a given point. This gains in importance on manifolds of dimension greater than one. See also *Algebraic analysis Algebraic analysis is an area of mathematics that deals with systems of linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of functions such as hyperfunctions and microfunc ... * Microfunction External linkslecture notes by Richard Melrose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
