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Edge-of-the-wedge Theorem
In mathematics, Bogoliubov's edge-of-the-wedge theorem implies that holomorphic functions on two "wedges" with an "edge" in common are analytic continuations of each other provided they both give the same continuous function on the edge. It is used in quantum field theory to construct the analytic continuation of Wightman functions. The formulation and the first proof of the theorem were presented by Nikolay Bogoliubov at the International Conference on Theoretical Physics, Seattle, USA (September, 1956) and also published in the book ''Problems in the Theory of Dispersion Relations''. Further proofs and generalizations of the theorem were given by R. Jost and H. Lehmann (1957), F. Dyson (1958), H. Epstein (1960), and by other researchers. The one-dimensional case Continuous boundary values In one dimension, a simple case of the edge-of-the-wedge theorem can be stated as follows. *Suppose that ''f'' is a continuous complex-valued function on the complex pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fundamental Solution
In mathematics, a fundamental solution for a linear partial differential operator is a formulation in the language of distribution theory of the older idea of a Green's function (although unlike Green's functions, fundamental solutions do not address boundary conditions). In terms of the Dirac delta "function" , a fundamental solution is a solution of the inhomogeneous equation Here is ''a priori'' only assumed to be a distribution. This concept has long been utilized for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis. In the context of functional analysis, fundamental solutions are usually developed via the Fredholm alternative and explored in F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
London
London is the capital and largest city of England and the United Kingdom, with a population of just under 9 million. It stands on the River Thames in south-east England at the head of a estuary down to the North Sea, and has been a major settlement for two millennia. The City of London, its ancient core and financial centre, was founded by the Romans as '' Londinium'' and retains its medieval boundaries.See also: Independent city § National capitals The City of Westminster, to the west of the City of London, has for centuries hosted the national government and parliament. Since the 19th century, the name "London" has also referred to the metropolis around this core, historically split between the counties of Middlesex, Essex, Surrey, Kent, and Hertfordshire, which largely comprises Greater London, governed by the Greater London Authority.The Greater London Authority consists of the Mayor of London and the London Assembly. The London Mayor is distinguished fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boston
Boston (), officially the City of Boston, is the state capital and most populous city of the Commonwealth of Massachusetts, as well as the cultural and financial center of the New England region of the United States. It is the 24th- most populous city in the country. The city boundaries encompass an area of about and a population of 675,647 as of 2020. It is the seat of Suffolk County (although the county government was disbanded on July 1, 1999). The city is the economic and cultural anchor of a substantially larger metropolitan area known as Greater Boston, a metropolitan statistical area (MSA) home to a census-estimated 4.8 million people in 2016 and ranking as the tenth-largest MSA in the country. A broader combined statistical area (CSA), generally corresponding to the commuting area and including Providence, Rhode Island, is home to approximately 8.2 million people, making it the sixth most populous in the United States. Boston is one of the oldest ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dordrecht
Dordrecht (), historically known in English as Dordt (still colloquially used in Dutch, ) or Dort, is a city and municipality in the Western Netherlands, located in the province of South Holland. It is the province's fifth-largest city after Rotterdam, The Hague, Zoetermeer and Leiden, with a population of . The municipality covers the entire Dordrecht Island, also often called ''Het Eiland van Dordt'' ("the Island of Dordt"), bordered by the rivers Oude Maas, Beneden Merwede, Nieuwe Merwede, Hollands Diep, and Dordtsche Kil. Located about 17 km south east of Rotterdam, Dordrecht is the largest and most important city in the Drechtsteden and is also part of the Randstad, the main conurbation in the Netherlands. Dordrecht is the oldest city in Holland and has a rich history and culture. Etymology The name Dordrecht comes from ''Thuredriht'' (circa 1120), ''Thuredrecht'' (circa 1200). The name seems to mean 'thoroughfare'; a ship-canal or -river through which ships were pulle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reading, Massachusetts
Reading ( ) is a town in Middlesex County, Massachusetts, United States, north of central Boston. The population was 25,518 at the 2020 census. History Settlement and American independence Many of the Massachusetts Bay Colony's original settlers arrived from England in the 1630s through the ports of Lynn and Salem. In 1639 some citizens of Lynn petitioned the government of the colony for a "place for an inland plantation". They were initially granted six square miles, followed by an additional four. The first settlement in this grant was at first called "Lynn Village" and was located on the south shore of the "Great Pond", now known as Lake Quannapowitt. On June 10, 1644 the settlement was incorporated as the town of Reading, taking its name from the town of Reading in England. The first church was organized soon after the settlement, and the first parish separated and became the town of "South Reading" in 1812, renaming itself as Wakefield in 1868. Thomas Parker was one of t ... [...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] |
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] |
Hyperfunction
In mathematics, hyperfunctions are generalizations of functions, as a 'jump' from one holomorphic function to another at a boundary, and can be thought of informally as distributions of infinite order. Hyperfunctions were introduced by Mikio Sato in 1958 in Japanese, (1959, 1960 in English), building upon earlier work by Laurent Schwartz, Grothendieck and others. Formulation A hyperfunction on the real line can be conceived of as the 'difference' between one holomorphic function defined on the upper half-plane and another on the lower half-plane. That is, a hyperfunction is specified by a pair (''f'', ''g''), where ''f'' is a holomorphic function on the upper half-plane and ''g'' is a holomorphic function on the lower half-plane. Informally, the hyperfunction is what the difference f -g would be at the real line itself. This difference is not affected by adding the same holomorphic function to both ''f'' and ''g'', so if ''h'' is a holomorphic function on the whole compl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Embedding Theorem
In mathematics, there is in mathematical analysis a class of Sobolev inequalities, relating norms including those of Sobolev spaces. These are used to prove the Sobolev embedding theorem, giving inclusions between certain Sobolev spaces, and the Rellich–Kondrachov theorem showing that under slightly stronger conditions some Sobolev spaces are compactly embedded in others. They are named after Sergei Lvovich Sobolev. Sobolev embedding theorem Let denote the Sobolev space consisting of all real-valued functions on whose first weak derivatives are functions in . Here is a non-negative integer and . The first part of the Sobolev embedding theorem states that if , and are two real numbers such that :\frac-\frac = \frac -\frac, then :W^(\mathbf^n)\subseteq W^(\mathbf^n) and the embedding is continuous. In the special case of and , Sobolev embedding gives :W^(\mathbf^n) \subseteq L^(\mathbf^n) where is the Sobolev conjugate of , given byp. (Note that 1/p^*p.) Thus, an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Space
In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense to make the space complete, i.e. a Banach space. Intuitively, a Sobolev space is a space of functions possessing sufficiently many derivatives for some application domain, such as partial differential equations, and equipped with a norm that measures both the size and regularity of a function. Sobolev spaces are named after the Russian mathematician Sergei Sobolev. Their importance comes from the fact that weak solutions of some important partial differential equations exist in appropriate Sobolev spaces, even when there are no strong solutions in spaces of continuous functions with the derivatives understood in the classical sense. Motivation In this section and throughout the article \Omega is an open subset of \R^n. There are many c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
