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Mikio Sato
is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto. Education Sato studied at the University of Tokyo and then did graduate study in physics as a student of Shin'ichiro Tomonaga. Since 1970, Sato has been professor at the Research Institute for Mathematical Sciences attached to Kyoto University. His disciples include Masaki Kashiwara, Takahiro Kawai, Tetsuji Miwa, and Michio Jimbo, who have been called the "Sato School". Research Sato is known for his innovative work in a number of fields, such as prehomogeneous vector spaces and Bernstein–Sato polynomials; and particularly for his hyperfunction theory. This theory initially appeared as an extension of the ideas of distribution theory; it was soon connected to the local cohomology theory of Grothendieck, for which it was an independent realization in terms of sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kyoto University
, mottoeng = Freedom of academic culture , established = , type = Public (National) , endowment = ¥ 316 billion (2.4 billion USD) , faculty = 3,480 (Teaching Staff) , administrative_staff = 3,978 (Total Staff) , students = 22,615 , president = Nagahiro Minato , city = Kyoto , state = Kyoto , country = Japan , coor = , undergrad = 13,038 , postgrad = 9,308 , campus = Urban,, , colors = Dark blue , nickname = Kyodai , mascot = None , free_label = Athletics , free = 48 varsity teams , affiliations = Kansai Big Six, ASAIHL , logo = , website www.kyoto-u.ac.jp , or , is a public research university located in Kyoto, Japan. Founded in 1897, it is one of the former Imperial Universities and the second oldest university in Japan. KyotoU is consistently ranked amongst the top two in Japan, the top ten in Asia, and the world's top fifty institutions of higher education. Founded upon the principles of its motto, “ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Michio Jimbo
is a Japanese mathematician working in mathematical physics and is a professor of mathematics at Rikkyo University. He is a grandson of the linguist . Career After graduating from the University of Tokyo in 1974, he studied under Mikio Sato at the Research Institute for Mathematical Sciences in Kyoto University. He has made important contributions to mathematical physics, including (independently of Vladimir Drinfeld) the initial development of the study of quantum groups, the development of the theory of \tau-functions for the KP ( Kadomtsev–Petviashvili) integrable hierarchy, and other related integrable hierarchies , E. Date, M. Jimbo, M. Kashiwara and T. Miwa, "Operator approach to the Kadomtsev-Petviashvili equation III". ''J. Phys. Soc. Jap.'' 50 (11): 3806–3812 (1981). doi:10.1143/JPSJ.50.3806. M. Jimbo and T. Miwa, "Solitons and infinite-dimensional Lie algebras", ''Publ. Res. Inst. Math. Sci.'', 19(3):943–1001 (1983). and development of the theory of isomonodr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassmannian
In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than . When is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension k(n-k). The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general. Notations for the Grassmannian vary between authors; notations include , , , or to denote the Grassmannian of -dimensional subspaces of an -dimensional vector space . Motivation By giving a collection of subspaces of some vector space a topological st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium. (Dispersive effects are a property of certain systems where the speed of a wave depends on its frequency.) Solitons are the solutions of a widespread class of weakly nonlinear dispersive partial differential equations describing physical systems. The soliton phenomenon was first described in 1834 by John Scott Russell (1808–1882) who observed a solitary wave in the Union Canal in Scotland. He reproduced the phenomenon in a wave tank and named it the " Wave of Translation". Definition A single, consensus definition of a soliton is difficult to find. ascribe three properties to solitons: # They are of permanent form; # They are localized within a region; # They can interact with other solitons, and emerge from the collision unchan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Holonomic System
In classical mechanics, holonomic constraints are relations between the position variables (and possibly time) that can be expressed in the following form: :f(u_1, u_2, u_3,\ldots, u_n, t) = 0 where \ are the ''n'' generalized coordinates that describe the system. For example, the motion of a particle constrained to lie on the surface of a sphere is subject to a holonomic constraint, but if the particle is able to fall off the sphere under the influence of gravity, the constraint becomes non-holonomic. For the first case, the holonomic constraint may be given by the equation :r^2-a^2=0 where r is the distance from the centre of a sphere of radius a, whereas the second non-holonomic case may be given by :r^2 - a^2 \geq 0 Velocity-dependent constraints (also called semi-holonomic constraints) such as :f(u_1,u_2,\ldots,u_n,\dot_1,\dot_2,\ldots,\dot_n,t)=0 are not usually holonomic. Holonomic system In classical mechanics a system may be defined as holonomic if all constraints ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
D-module
In mathematics, a ''D''-module is a module over a ring ''D'' of differential operators. The major interest of such ''D''-modules is as an approach to the theory of linear partial differential equations. Since around 1970, ''D''-module theory has been built up, mainly as a response to the ideas of Mikio Sato on algebraic analysis, and expanding on the work of Sato and Joseph Bernstein on the Bernstein–Sato polynomial. Early major results were the Kashiwara constructibility theorem and Kashiwara index theorem of Masaki Kashiwara. The methods of ''D''-module theory have always been drawn from sheaf theory and other techniques with inspiration from the work of Alexander Grothendieck in algebraic geometry. The approach is global in character, and differs from the functional analysis techniques traditionally used to study differential operators. The strongest results are obtained for over-determined systems ( holonomic systems), and on the characteristic variety cut out by the sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harmonic Analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis). In the past two centuries, it has become a vast subject with applications in areas as diverse as number theory, representation theory, signal processing, quantum mechanics, tidal analysis and neuroscience. The term "harmonics" originated as the Ancient Greek word ''harmonikos'', meaning "skilled in music". In physical eigenvalue problems, it began to mean waves whose frequencies are integer multiples of one another, as are the frequencies of the harmonics of music notes, but the term has been generalized beyond its original meaning. The classical Fourier transform on R''n'' is still an area of ongoing research, particularly concerning Fourier transformation on more general objects such as tempered di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to how is thought of as an unknown number to be solved for in an algebraic equation like . However, it is usually impossible to write down explicit formulas for solutions of partial differential equations. There is, correspondingly, a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity, and stability. Among the many open questions are the ex ... [...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] |
Microfunction
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 microfunctions. Semantically, it is the application of algebraic operations on analytic quantities. As a research programme, it was started by the Japanese mathematician Mikio Sato in 1959. This can be seen as an algebraic geometrization of analysis. It derives its meaning from the fact that the differential operator is right-invertible in several function spaces. It helps in the simplification of the proofs due to an algebraic description of the problem considered. Microfunction Let ''M'' be a real-analytic manifold of dimension ''n'', and let ''X'' be its complexification. The sheaf of microlocal functions on ''M'' is given as :\mathcal^n(\mu_M(\mathcal_X) \otimes \mathcal_) where * \mu_M denotes the microlocalization functor, * \mathcal_ i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could be the ring of continuous functions defined on that open set. Such data is well behaved in that it can be restricted to smaller open sets, and also the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original open set (intuitively, every piece of data is the sum of its parts). The field of mathematics that studies sheaves is called sheaf theory. Sheaves are understood conceptually as general and abstract objects. Their correct definition is rather technical. They are specifically defined as sheaves of sets or as sheaves of rings, for example, depending on the type of data assigned to the open sets. There are also maps (or morphisms) from o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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