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Propagation Of Singularities Theorem
In microlocal analysis, the propagation of singularities theorem (also called the Duistermaat–Hörmander theorem) is theorem which characterizes the wavefront set of the distributional solution of the partial (pseudo) differential equation :Pu=f for a pseudodifferential operator P on a smooth manifold. It says that the propagation of singularities follows the bicharacteristic flow of the principal symbol of P. The theorem appeared 1972 in a work on ''Fourier integral operators'' by Johannes Jisse Duistermaat and Lars Hörmander and since then there have been many generalizations which are known under the name propagation of singularities. Propagation of singularities theorem We use the following notation: *X is a C^-differentiable manifold, and C^_0(X) is the space of smooth functions u with a compact set K \subset X, such that u \mid = 0. *L^m_(X) denotes the class of pseudodifferential operators of type (\sigma,\delta) with symbol a(x,y,\theta) \in S^m_(X \times X \times ... [...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] |
Wavefront 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 dimensions 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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Distribution (mathematics)
Distributions, also known as Schwartz distributions are a kind of generalized function in mathematical analysis. Distributions make it possible to derivative, differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions (weak solutions) than Solution of a differential equation, classical solutions, or where appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are singular, such as the Dirac delta function, Dirac delta function. A Function (mathematics), function f is normally thought of as on the in the function Domain (function), domain by "sending" a point x in the domain t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Differential Equation
In mathematics, a partial differential equation (PDE) is an equation which involves a multivariable function and one or more of its partial derivatives. The function is often thought of as an "unknown" that solves the equation, similar to how is thought of as an unknown number solving, e.g., an algebraic equation like . However, it is usually impossible to write down explicit formulae 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 existence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudodifferential 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 writt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and returns another function (in the style of a higher-order function in computer science). This article considers mainly linear differential operators, which are the most common type. However, non-linear differential operators also exist, such as the Schwarzian derivative. Definition Given a nonnegative integer ''m'', an order-m linear differential operator is a map P from a function space \mathcal_1 on \mathbb^n to another function space \mathcal_2 that can be written as: P = \sum_a_\alpha(x) D^\alpha\ , where \alpha = (\alpha_1,\alpha_2,\cdots,\alpha_n) is a multi-index of non-negative integers, , \alpha, = \alpha_1 + \alpha_2 + \cdots + \alpha_n, and for each \alpha, a_\alpha(x) is a function on some open domain in ''n''-dimensional space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1972
Within the context of Coordinated Universal Time (UTC) it was the longest year ever, as two leap seconds were added during this 366-day year, an event which has not since been repeated. (If its start and end are defined using Solar time, mean solar time [the legal time scale], its duration was 31622401.141 seconds of Terrestrial Time (or Ephemeris Time), which is slightly shorter than 1908 in science#Astronomy, 1908). Events January * January 1 – Kurt Waldheim becomes Secretary-General of the United Nations. * January 4 – The first scientific hand-held calculator (HP-35) is introduced (price $395). * January 7 – Iberia Airlines Flight 602 crashes into a 462-meter peak on the island of Ibiza; 104 are killed. * January 9 – The RMS Queen Elizabeth, RMS ''Queen Elizabeth'' catches fire and sinks in Hong Kong's Victoria harbor while undergoing conversion to a floating university. * January 10 – Independence leader Sheikh Mujibur Rahman returns to Bangladesh after s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johannes Jisse 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 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lars Hörmander
Lars Valter Hörmander (24 January 1931 – 25 November 2012) was a Swedish mathematician who has been called "the foremost contributor to the modern theory of linear partial differential equations". Hörmander was awarded the Fields Medal in 1962 and the Wolf Prize in 1988. In 2006 he was awarded the Steele Prize for Mathematical Exposition for his four-volume textbook ''Analysis of Linear Partial Differential Operators'', which is considered a foundational work on the subject. Hörmander completed his Ph.D. in 1955 at Lund University. Hörmander then worked at Stockholm University, at Stanford University, and at the Institute for Advanced Study in Princeton, New Jersey. He returned to Lund University as a professor from 1968 until 1996, when he retired with the title of professor emeritus. Biography Education Hörmander was born in Mjällby, a village in Blekinge in southern Sweden where his father was a teacher. Like his older brothers and sisters before him, he attended ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Set
In mathematics, specifically general topology, compactness is a property that seeks to generalize the notion of a closed and bounded subset of Euclidean space. The idea is that a compact space has no "punctures" or "missing endpoints", i.e., it includes all ''limiting values'' of points. For example, the open interval (0,1) would not be compact because it excludes the limiting values of 0 and 1, whereas the closed interval ,1would be compact. Similarly, the space of rational numbers \mathbb is not compact, because it has infinitely many "punctures" corresponding to the irrational numbers, and the space of real numbers \mathbb is not compact either, because it excludes the two limiting values +\infty and -\infty. However, the ''extended'' real number line ''would'' be compact, since it contains both infinities. There are many ways to make this heuristic notion precise. These ways usually agree in a metric space, but may not be equivalent in other topological spaces. One su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
