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Microlocalization Functor
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 *Microfunction Algebraic analysis is an area of mathematics that deals with systems of Partial differential equation, linear partial differential equations by using sheaf theory and complex analysis to study properties and generalizations of Function (mathematic ... External linkslecture notes by Richard Melrose Fourie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, which will output a function depending on temporal frequency or spatial frequency respectively. That process is also called ''analysis''. An example application would be decomposing the waveform of a musical chord into terms of the intensity of its constituent pitches. The term ''Fourier transform'' refers to both the frequency domain representation and the mathematical operation that associates the frequency domain representation to a function of space or time. The Fourier transform of a function is a complex-valued function representing the complex sinusoids that comprise the original function. For each frequency, the magnitude (absolute value) of the complex value represents the amplitude of a constituent complex sinusoid with that ... [...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 calculus, 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 Numerical methods for partial differential equations, numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematics, 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 a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Function
In mathematics, generalized functions are objects extending the notion of functions. There is more than one recognized theory, for example the theory of distributions. Generalized functions are especially useful in making discontinuous functions more like smooth functions, and describing discrete physical phenomena such as point charges. They are applied extensively, especially in physics and engineering. A common feature of some of the approaches is that they build on operator aspects of everyday, numerical functions. The early history is connected with some ideas on operational calculus, and more contemporary developments in certain directions are closely related to ideas of Mikio Sato, on what he calls algebraic analysis. Important influences on the subject have been the technical requirements of theories of partial differential equations, and group representation theory. Some early history In the mathematics of the nineteenth century, aspects of generalized function theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-differential Operator
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 wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Front Set
In mathematical analysis, more precisely in microlocal analysis, the wave front (set) WF(''f'') characterizes the singularities of a generalized function ''f'', not only in space, but also with respect to its Fourier transform at each point. The term "wave front" was coined by Lars Hörmander around 1970. Introduction In more familiar terms, WF(''f'') tells not only ''where'' the function ''f'' is singular (which is already described by its singular support), but also ''how'' or ''why'' it is singular, by being more exact about the direction in which the singularity occurs. This concept is mostly useful in dimension at least two, since in one dimension there are only two possible directions. The complementary notion of a function being non-singular in a direction is ''microlocal smoothness''. Intuitively, as an example, consider a function ƒ whose singular support is concentrated on a smooth curve in the plane at which the function has a jump discontinuity. In the direction ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Integral Operator
In mathematical analysis, Fourier integral operators have become an important tool in the theory of partial differential equations. The class of Fourier integral operators contains differential operators as well as classical integral operators as special cases. A Fourier integral operator T is given by: :(Tf)(x)=\int_ e^a(x,\xi)\hat(\xi) \, d\xi where \hat f denotes the Fourier transform of f, a(x,\xi) is a standard symbol which is compactly supported in x and \Phi is real valued and homogeneous of degree 1 in \xi. It is also necessary to require that \det \left(\frac\right)\neq 0 on the support of ''a.'' Under these conditions, if ''a'' is of order zero, it is possible to show that T defines a bounded operator from L^ to L^. Examples One motivation for the study of Fourier integral operators is the solution operator for the initial value problem for the wave operator. Indeed, consider the following problem: : \frac\frac(t,x) = \Delta u(t,x) \quad \mathrm \quad (t,x) \in \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oscillatory Integral Operator
In mathematics, in the field of harmonic analysis, an oscillatory integral operator is an integral operator of the form :T_\lambda u(x)=\int_e^ a(x, y) u(y)\,dy, \qquad x\in\R^m, \quad y\in\R^n, where the function ''S''(''x'',''y'') is called the phase of the operator and the function ''a(x,y)'' is called the symbol of the operator. ''λ'' is a parameter. One often considers ''S''(''x'',''y'') to be real-valued and smooth, and ''a''(''x'',''y'') smooth and compactly supported. Usually one is interested in the behavior of ''T''λ for large values of ''λ''. Oscillatory integral operators often appear in many fields of mathematics (analysis, partial differential equations, integral geometry, number theory) and in physics. Properties of oscillatory integral operators have been studied by Elias Stein and his school. Hörmander's theorem The following bound on the ''L''2 → ''L''2 action of oscillatory integral operators (or ''L''2 → ''L''2 operator norm) was obtained by Lars H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cotangent Space
In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T^*_x\!\mathcal M is defined as the dual space of the tangent space at ''x'', T_x\mathcal M, although there are more direct definitions (see below). The elements of the cotangent space are called cotangent vectors or tangent covectors. Properties All cotangent spaces at points on a connected manifold have the same dimension, equal to the dimension of the manifold. All the cotangent spaces of a manifold can be "glued together" (i.e. unioned and endowed with a topology) to form a new differentiable manifold of twice the dimension, the cotangent bundle of the manifold. The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic to each other via many possible isomorphisms. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A Surface (mathematics), surface, such as the Boundary (mathematics), boundary of a Cylinder (geometry), cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the Euclidean plane, plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
