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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 an n-ball, volume of the unit ball in \mathbb^d. In 1912 he provided a new proof based on variational methods. Weyl's law can be extended to closed Riemannian manifolds, where another proof can be given using the Minakshisundaram–Pleijel zeta function#Applications, Minakshisundaram–Pleijel zeta function. Generalizations The Weyl law has been extended to more general domains and operators. For ... [...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] |
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 \sigma_(T), is the set of all real numbers \lambda \in \R 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.) The definition of essential spectrum \sigma_(T) will remain unchanged if we allow it to consist of all those complex numbers \lambda \in \C (instead of just real numbers) such that the above condition holds. This is due to the fact that the spectrum of self-adjoint cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Victor Ivrii
Victor Ivrii (), (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 to be sufficient) condition for Cauchy proble ... [...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 (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals 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. He works at the Massachusetts Institute of Technology in the field of symplectic geometry, and he has also made contributions to the fields of microlocal analysis, spectral theory, and mathematical physics. Education and career Guillemin obtained a B.A. at Harvard University in 1959, as well as an M. A. at the University of Chicago in 1960. He received a Ph.D. in mathematics from Harvard University in 1962; his dissertation, entitled ''Theory of Finite G-Structures,'' was written under the direction of Shlomo Sternberg. He worked at Columbia University from 1963 to 1966 and then moved to the Massachusetts Institute of Technology as assistant professor. He became associate professor in 1969 and full professor in 1973. Awards and honors Guillemin was awarded in 1969 a Sloan Research Fellowship, in 1988 a Guggenheim fellowship and in 1996 a Humboldt fellowship. In 1970 he was invited speaker at th ... [...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 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 Netherlands in 1 ... [...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: * [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Thomas Seeley
Robert Thomas Seeley (February 26, 1932 – November 30, 2016) was an American mathematician who worked on pseudo differential operators and the heat equation approach to the Atiyah–Singer index theorem. Life and career Seeley was born in Bryn Mawr, Pennsylvania on February 26, 1932. He 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 who worked on almost periodic functions, Sturm–Liouville operators and inverse scattering. Levitan was born in Berdyansk (southeastern 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. He then defended the habilitation thesis "Theory of Generalized Translation Operators". Levitan 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, he emigrated to the United States. During the last years of his life, he worked for the University of Minnesota The University of Minnesota Twin Cities (historically known as University of Minnesota) is a public university, public Land-grant universit ... [...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 (now in Poland). His parents were Siegmund Courant and Martha Freund of Oels. Edith Stein was Richard's cousin on the maternal 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 assista ... [...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 In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ... greater than one. See also * Algebraic analysis * Microfunction External linkslecture notes by Richard Melrose F ... [...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 operator (mathematics), operators in a variety of mathematical spaces. It is a result of studies of linear algebra and the solutions of System of linear equations, 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 axis theorem, principal axes of an ellipsoid, in an infinite-dimensional setting. The later discovery in quantum mechanics t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
