|
Fréchet–Kolmogorov Theorem
In functional analysis, the Fréchet–Kolmogorov theorem (the names of Riesz or Weil are sometimes added as well) gives a necessary and sufficient condition for a set of functions to be relatively compact in an ''L''''p'' space. It can be thought of as an ''L''''p'' version of the Arzelà–Ascoli theorem, from which it can be deduced. The theorem is named after Maurice René Fréchet and Andrey Kolmogorov. Statement Let B be a subset of L^p(\mathbb^n) with p\inrelatively compact if and only if the following properties hold: #(Equicontinuous) \lim_\Vert\tau_h f-f\Vert_ = 0 uniformly on B. #(Equitight) \lim_\int_\left, f\^p=0 uniformly on B. The first property can be stated as \forall \varepsilon >0 \, \, \exists \delta >0 such that \Vert\tau_h f-f\Vert_ < \varepsilon \, \, \forall f \in B, \forall h with Usually, the Fréchet–Kolmogorov theorem is formulated with the extra assumption that is bounded (i.e., |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Scientiarum Mathematicarum
''Acta Scientiarum Mathematicarum'' is a Hungarian mathematical journal published by the János Bolyai Mathematical Institute (University of Szeged). It was established by Alfréd Haar and Frigyes Riesz in 1922. The current editor-in-chief is Lajos Molnár. The journal is abstracted and indexed in Scopus and Zentralblatt MATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastru .... References External links * * {{ISSN, 0001-6969 Mathematics journals Academic journals established in 1922 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second-largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rellich–Kondrachov Theorem
In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the ''L''2 theorem and Kondrashov the ''L''''p'' theorem. Statement of the theorem Let Ω ⊆ R''n'' be an open, bounded Lipschitz domain, and let 1 ≤ ''p'' < ''n''. Set :p^ := \frac. Then the Sobolev space ''W''1,''p''(Ω; R) is continuously embedded in the ''L''''p'' space ''L''''p''∗(Ω; R) and is compactly embedded in ''L''''q''(Ω; R) for every 1 ≤ ''q'' < ''p''∗. In symbols, :W^ (\Omega) \hookrightarrow L^ (\Omega) and :W^ (\Omega) \subset \subset L^ (\Omega) \text 1 \leq q 2527916 Zblbr>1180.46001* {{DEFAULTSORT:Rellich-Kondrachov theorem Theorems in mathematical analysis Sobolev spaces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helly's Selection Theorem
In mathematics, Helly's selection theorem (also called the ''Helly selection principle'') states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point. The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures. Statement of the theorem Let (''f''''n'')''n'' ∈ N be a sequence of increasing functions mapping a real interval I into the real line R, and suppose that it is uniformly bounded: there are ''a,b'' ∈ R such that ''a'' ≤ ''f''''n'' ≤ ''b'' for every ''n'' ∈ N. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Space
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 suc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Burgers' Equation
Burgers' equation or Bateman–Burgers equation is a fundamental partial differential equation and convection–diffusion equation occurring in various areas of applied mathematics, such as fluid mechanics, nonlinear acoustics, gas dynamics, and traffic flow. The equation was first introduced by Harry Bateman in 1915 and later studied by Johannes Martinus Burgers in 1948. For a given field u(x,t) and diffusion coefficient (or ''kinematic viscosity'', as in the original fluid mechanical context) \nu, the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system: :\frac + u \frac = \nu\frac. The term u\partial u/\partial x can also be rewritten as \partial(u^2/2)/\partial x. When the diffusion term is absent (i.e. \nu=0), Burgers' equation becomes the inviscid Burgers' equation: :\frac + u \frac = 0, which is a prototype for conservation equations that can develop discontinuities (shock waves). The reason f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Relatively Compact Subspace
In mathematics, a relatively compact subspace (or relatively compact subset, or precompact subset) of a topological space is a subset whose closure is compact. Properties Every subset of a compact topological space is relatively compact (since a closed subset of a compact space is compact). And in an arbitrary topological space every subset of a relatively compact set is relatively compact. Every compact subset of a Hausdorff space is relatively compact. In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is ''not'' necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. Every compact subset of a (possibly non-Hausdorff) topological vector space is complete and relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marcel Riesz
Marcel Riesz ( ; 16 November 1886 – 4 September 1969) was a Hungarian mathematician, known for work on summation methods, potential theory, and other parts of analysis, as well as number theory, partial differential equations, and Clifford algebras. He spent most of his career in Lund, Sweden. Marcel is the younger brother of Frigyes Riesz, who was also an important mathematician and at times they worked together (see F. and M. Riesz theorem). Biography Marcel Riesz was born in Győr, Austria-Hungary. He was the younger brother of the mathematician Frigyes Riesz. In 1904, he won the Loránd Eötvös competition. Upon entering the Budapest University, he also studied in Göttingen, and the academic year 1910-11 he spent in Paris. Earlier, in 1908, he attended the 1908 International Congress of Mathematicians in Rome. There he met Gösta Mittag-Leffler, in three years, Mittag-Leffler would offer Riesz to come to Sweden. Riesz obtained his PhD at Eötvös Loránd Universit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Andrey Kolmogorov
Andrey Nikolaevich Kolmogorov ( rus, Андре́й Никола́евич Колмого́ров, p=ɐnˈdrʲej nʲɪkɐˈlajɪvʲɪtɕ kəlmɐˈɡorəf, a=Ru-Andrey Nikolaevich Kolmogorov.ogg, 25 April 1903 – 20 October 1987) was a Soviet mathematician who played a central role in the creation of modern probability theory. He also contributed to the mathematics of topology, intuitionistic logic, turbulence, classical mechanics, algorithmic information theory and Analysis of algorithms, computational complexity. Biography Early life Andrey Kolmogorov was born in Tambov, about 500 kilometers southeast of Moscow, in 1903. His unmarried mother, Maria Yakovlevna Kolmogorova, died giving birth to him. Andrey was raised by two of his aunts in Tunoshna (near Yaroslavl) at the estate of his grandfather, a well-to-do Russian nobility, nobleman. Little is known about Andrey's father. He was supposedly named Nikolai Matveyevich Katayev and had been an Agronomy, agronomist. Katayev ha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maurice René Fréchet
Maurice may refer to: *Maurice (name), a given name and surname, including a list of people with the name Places * or Mauritius, an island country in the Indian Ocean *Maurice, Iowa, a city * Maurice, Louisiana, a village * Maurice River, a tributary of the Delaware River in New Jersey Other uses * ''Maurice'' (2015 film), a Canadian short drama film * Maurice (horse), a Thoroughbred racehorse * ''Maurice'' (novel), a 1913 novel by E. M. Forster, published in 1971 ** ''Maurice'' (1987 film), a British film based on the novel * ''Maurice'' (Shelley), a children's story by Mary Shelley *Maurice, a character from the Madagascar ''franchise'' *Maurices, an American retail clothing chain *Maurice or Maryse, a type of cooking spatula See also *Church of Saint Maurice (other) * *Maurice Debate, a 1918 debate in the British House of Commons *Maurice Lacroix, Swiss manufacturer of mechanical timepieces, clocks, and watches *Mauricie, Quebec, Canada *Moritz (other) * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
