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Harold Widom
Harold Widom (September 23, 1932 – January 20, 2021) was an American mathematician best known for his contributions to operator theory and random matrices. He was appointed to the Department of Mathematics at the University of California, Santa Cruz in 1968 and became professor emeritus in 1994. Education and research Widom was born in Newark, New Jersey. He studied at Stuyvesant High School, graduating in 1949, and was a member of the school's math team along with his brother Benjamin Widom (1944, 1948). Widom attended City College of New York until 1951, during which he was one of the winners of the William Lowell Putnam Mathematical Competition (1951). At the University of Chicago he obtained an M.S. (1952) and Ph.D., the latter on a thesis ''Embedding of AW*-algebras'' advised by Irving Kaplansky (1955). He taught mathematics at Cornell University (1955–68) where he started his work on Toeplitz and Wiener-Hopf operators, partly inspired by Mark Kac. Widom was appoint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newark, New Jersey
Newark ( , ) is the List of municipalities in New Jersey, most populous City (New Jersey), city in the U.S. state of New Jersey and the county seat, seat of Essex County, New Jersey, Essex County and the second largest city within the New York metropolitan area.New Jersey County Map New Jersey Department of State. Accessed July 10, 2017. The city had a population of 311,549 as of the 2020 United States census, 2020 U.S. census, and was calculated at 307,220 by the Population Estimates Program for 2021, making it List of United States cities by population, the nation's 66th-most populous municipality. [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Master Of Science
A Master of Science ( la, Magisterii Scientiae; abbreviated MS, M.S., MSc, M.Sc., SM, S.M., ScM or Sc.M.) is a master's degree in the field of science awarded by universities in many countries or a person holding such a degree. In contrast to the Master of Arts degree, the Master of Science degree is typically granted for studies in sciences, engineering and medicine and is usually for programs that are more focused on scientific and mathematical subjects; however, different universities have different conventions and may also offer the degree for fields typically considered within the humanities and social sciences. While it ultimately depends upon the specific program, earning a Master of Science degree typically includes writing a thesis. The Master of Science degree was first introduced at the University of Michigan in 1858. One of the first recipients of the degree was De Volson Wood, who was conferred a Master of Science degree at the University of Michigan in 1859. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equations And Operator Theory
''Integral Equations and Operator Theory'' is a journal dedicated to operator theory and its applications to engineering and other mathematical sciences. As some approaches to the study of integral equations (theoretically and numerically) constitute a subfield of operator theory, the journal also deals with the theory of integral equations and hence of differential equations. The journal consists of two sections: a main section consisting of refereed papers and a second consisting of short announcements of important results, open problems, information, etc. It has been published monthly by Springer-Verlag since 1978. The journal is also available online by subscription. The founding editor-in-chief of the journal, in 1978, was Israel Gohberg. Its current editor-in-chief is Christiane Tretter Christiane Tretter (born 28 December 1964) is a German mathematician and mathematical physicist who works as a professor in the Mathematical Institute (MAI) of the University of Bern in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Asymptotic Combinatorics
In combinatorics, the symbolic method is a technique for counting combinatorial objects. It uses the internal structure of the objects to derive formulas for their generating functions. The method is mostly associated with Philippe Flajolet and is detailed in Part A of his book with Robert Sedgewick, ''Analytic Combinatorics'', while the rest of the book explains how to use complex analysis in order to get asymptotic and probabilistic results on the corresponding generating functions. During two centuries, generating functions were popping up via the corresponding recurrences on their coefficients (as can be seen in the seminal works of Bernoulli, Euler, Arthur Cayley, Schröder, Ramanujan, Riordan, Knuth, , etc.). It was then slowly realized that the generating functions were capturing many other facets of the initial discrete combinatorial objects, and that this could be done in a more direct formal way: The recursive nature of some combinatorial structures translates, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fredholm Determinant
In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a trace-class operator. The function is named after the mathematician Erik Ivar Fredholm. Fredholm determinants have had many applications in mathematical physics, the most celebrated example being Gábor Szegő's limit formula, proved in response to a question raised by Lars Onsager and C. N. Yang on the spontaneous magnetization of the Ising model. Definition Let ''H'' be a Hilbert space and ''G'' the set of bounded invertible operators on ''H'' of the form ''I'' + ''T'', where ''T'' is a trace-class operator. ''G'' is a group because (I+T)^ - I = - T(I+T)^, so (''I''+''T'')−1−''I'' is trace class if ''T'' is. It has a natural metric given by , where is the trace-class norm. If ''H'' is a Hilbert space with inner product (\cdot, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Painlevé Transcendents
In mathematics, Painlevé transcendents are solutions to certain nonlinear second-order ordinary differential equations in the complex plane with the Painlevé property (the only movable singularities are poles), but which are not generally solvable in terms of elementary functions. They were discovered by , , , and . History Painlevé transcendents have their origin in the study of special functions, which often arise as solutions of differential equations, as well as in the study of isomonodromic deformations of linear differential equations. One of the most useful classes of special functions are the elliptic functions. They are defined by second order ordinary differential equations whose singularities have the Painlevé property: the only movable singularities are poles. This property is rare in nonlinear equations. Poincaré and L. Fuchs showed that any first order equation with the Painlevé property can be transformed into the Weierstrass elliptic function or the Ric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Random Matrices
In probability theory and mathematical physics, a random matrix is a matrix-valued random variable—that is, a matrix in which some or all elements are random variables. Many important properties of physical systems can be represented mathematically as matrix problems. For example, the thermal conductivity of a lattice can be computed from the dynamical matrix of the particle-particle interactions within the lattice. Applications Physics In nuclear physics, random matrices were introduced by Eugene Wigner to model the nuclei of heavy atoms. Wigner postulated that the spacings between the lines in the spectrum of a heavy atom nucleus should resemble the spacings between the eigenvalues of a random matrix, and should depend only on the symmetry class of the underlying evolution. In solid-state physics, random matrices model the behaviour of large disordered Hamiltonians in the mean-field approximation. In quantum chaos, the Bohigas–Giannoni–Schmit (BGS) conjecture asserts ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tracy–Widom Distribution
The Tracy–Widom distribution is a probability distribution from random matrix theory introduced by . It is the distribution of the normalized largest eigenvalue of a random Hermitian matrix. The distribution is defined as a Fredholm determinant. In practical terms, Tracy–Widom is the crossover function between the two phases of weakly versus strongly coupled components in a system. It also appears in the distribution of the length of the longest increasing subsequence of random permutations, as large-scale statistics in the Kardar-Parisi-Zhang equation, in current fluctuations of the asymmetric simple exclusion process (ASEP) with step initial condition, and in simplified mathematical models of the behavior of the longest common subsequence problem on random inputs. See and for experimental testing (and verifying) that the interface fluctuations of a growing droplet (or substrate) are described by the TW distribution F_2 (or F_1) as predicted by . The distribution ''F''1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Craig Tracy
Craig Arnold Tracy (born September 9, 1945) is an American mathematician, known for his contributions to mathematical physics and probability theory. Born in United Kingdom, he moved as infant to Missouri where he grew up and obtained a B.Sc. in physics from University of Missouri (1967). He studied as a Woodrow Wilson Fellow at the Stony Brook University where he obtained a Ph.D. on the thesis entitled ''Spin-Spin Scale-Functions in the Ising and XY-Models'' (1973) advised by Barry M. McCoy, in which (also jointly with Tai Tsun Wu and Eytan Barouch) he studied Painlevé functions in exactly solvable statistical mechanical models. He then was on the faculty of Dartmouth College (1978–84) before joining University of California, Davis (1984) where he is now a professor. With Harold Widom he worked on the asymptotic analysis of Toeplitz determinants and their various operator theoretic generalizations. This work gave them both the George Pólya and the Norbert Wie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudodifferential Operators
In mathematical analysis a pseudo-differential operator is an extension of the concept of differential operator. Pseudo-differential operators are used extensively in the theory of partial differential equations and quantum field theory, e.g. in mathematical models that include ultrametric pseudo-differential equations in a non-Archimedean space. History The study of pseudo-differential operators began in the mid 1960s with the work of Kohn, Nirenberg, Hörmander, Unterberger and Bokobza. They played an influential role in the second proof of the Atiyah–Singer index theorem via K-theory. Atiyah and Singer thanked Hörmander for assistance with understanding the theory of pseudo-differential operators. Motivation Linear differential operators with constant coefficients Consider a linear differential operator with constant coefficients, : P(D) := \sum_\alpha a_\alpha \, D^\alpha which acts on smooth functions u with compact support in R''n''. This operator can be wr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integral Equations
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; I^1 (u), I^2(u), I^3(u), ..., I^m(u)) = 0where I^i(u) is an integral operator acting on ''u.'' Hence, integral equations may be viewed as the analog to differential equations where instead of the equation involving derivatives, the equation contains integrals. A direct comparison can be seen with the mathematical form of the general integral equation above with the general form of a differential equation which may be expressed as follows:f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n) ; D^1 (u), D^2(u), D^3(u), ..., D^m(u)) = 0where D^i(u) may be viewed as a differential operator of order ''i''. Due to this close connection between differential and integral equations, one can often convert between the two. For example, one method of solvi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mark Kac
Mark Kac ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, the idea of understanding the extent to which the spectrum allows one to read back the geometry. (In the end, the answer was "no", in general.) Biography He was born to a Polish-Jewish family; their town, Kremenets ( Polish: "Krzemieniec"), changed hands from the Russian Empire (by then Soviet Ukraine) to Poland after the Peace of Riga, when Kac was a child.Obituary in ''Rochester Democrat & Chronicle'', 11 November 1984 Kac completed his Ph.D. in mathematics at the Polish University of L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
