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Aubin–Lions Lemma
In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution. The result is named after the French mathematicians Jean-Pierre Aubin and Jacques-Louis Lions Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture pr .... In the original proof by Aubin, the spaces ''X''0 and ''X''1 in the statement of the lemma w ... [...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] |
Jacques-Louis Lions
Jacques-Louis Lions (; 3 May 1928 – 17 May 2001) was a French mathematician who made contributions to the theory of partial differential equations and to stochastic control, among other areas. He received the SIAM's John von Neumann Lecture prize in 1986 and numerous other distinctions.Jacques-Louis Lions Casinapioiv.va. Retrieved on 9 May 2016. isces.org Lions is listed as an . Biography After being part of the French Résistance in 1943 ...[...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorems In Functional Analysis
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In the mainstream of mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice, or of a less powerful theory, such as Peano arithmetic. A notable exception is Wiles's proof of Fermat's Last Theorem, which involves the Grothendieck universes whose existence requires the addition of a new axiom to the set theory. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Banach Spaces
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vectors and is complete in the sense that a Cauchy sequence of vectors always converges to a well-defined limit that is within the space. Banach spaces are named after the Polish mathematician Stefan Banach, who introduced this concept and studied it systematically in 1920–1922 along with Hans Hahn and Eduard Helly. Maurice René Fréchet was the first to use the term "Banach space" and Banach in turn then coined the term "Fréchet space." Banach spaces originally grew out of the study of function spaces by Hilbert, Fréchet, and Riesz earlier in the century. Banach spaces play a central role in functional analysis. In other areas of analysis, the spaces under study are often Banach spaces. Definition A Banach space is a complete norme ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lions–Magenes Lemma
In mathematics, the Lions–Magenes lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a criterion for moving a time derivative of a function out of its action (as a functional) on the function itself. Statement of the lemma Let ''X''0, ''X'' and ''X''1 be three Hilbert spaces with ''X''0 ⊆ ''X'' ⊆ ''X''1. Suppose that ''X''0 is continuously embedded in ''X'' and that ''X'' is continuously embedded in ''X''1, and that ''X''1 is the dual space of ''X''0. Denote the norm on ''X'' by , , ⋅ , , ''X'', and denote the action of ''X''1 on ''X''0 by \langle\cdot,\cdot\rangle. Suppose for some T>0 that u \in L^2 (, T X_0) is such that its time derivative \dot \in L^2 (, T X_1). Then u is almost everywhere equal to a function continuous from ,T/math> into X, and moreover the following equality holds in the sense of scalar distributions on (0,T): :\frac\frac \, u\, _X^2 = \langle\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuously Embedded
In mathematics, one normed vector space is said to be continuously embedded in another normed vector space if the inclusion function between them is continuous. In some sense, the two norms are "almost equivalent", even though they are not both defined on the same space. Several of the Sobolev embedding theorems are continuous embedding theorems. Definition Let ''X'' and ''Y'' be two normed vector spaces, with norms , , ·, , ''X'' and , , ·, , ''Y'' respectively, such that ''X'' ⊆ ''Y''. If the inclusion map (identity function) :i : X \hookrightarrow Y : x \mapsto x is continuous, i.e. if there exists a constant ''C'' > 0 such that :\, x \, _Y \leq C \, x \, _X for every ''x'' in ''X'', then ''X'' is said to be continuously embedded in ''Y''. Some authors use the hooked arrow "↪" to denote a continuous embedding, i.e. "''X'' ↪ ''Y''" means "''X'' and ''Y'' are normed spaces with ''X'' continuously embedded in ''Y''". This is a consist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactly Embedded
In mathematics, the notion of being compactly embedded expresses the idea that one set or space is "well contained" inside another. There are versions of this concept appropriate to general topology and functional analysis. Definition (topological spaces) Let (''X'', ''T'') be a topological space, and let ''V'' and ''W'' be subsets of ''X''. We say that ''V'' is compactly embedded in ''W'', and write ''V'' ⊂⊂ ''W'', if * ''V'' ⊆ Cl(''V'') ⊆ Int(''W''), where Cl(''V'') denotes the closure of ''V'', and Int(''W'') denotes the interior of ''W''; and * Cl(''V'') is compact. Definition (normed spaces) Let ''X'' and ''Y'' be two normed vector spaces with norms , , •, , ''X'' and , , •, , ''Y'' respectively, and suppose that ''X'' ⊆ ''Y''. We say that ''X'' is compactly embedded in ''Y'', and write ''X'' ⊂⊂ ''Y'', if * ''X'' is continuously embedded in ''Y''; i.e., there is a constant ''C'' such that , , ''x'', , ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflexive Space
In the area of mathematics known as functional analysis, a reflexive space is a locally convex topological vector space (TVS) for which the canonical evaluation map from X into its bidual (which is the strong dual of the strong dual of X) is an isomorphism of TVSs. Since a normable TVS is reflexive if and only if it is semi-reflexive, every normed space (and so in particular, every Banach space) X is reflexive if and only if the canonical evaluation map from X into its bidual is surjective; in this case the normed space is necessarily also a Banach space. In 1951, R. C. James discovered a Banach space, now known as James' space, that is reflexive but is nevertheless isometrically isomorphic to its bidual (any such isomorphism is thus necessarily the canonical evaluation map). Reflexive spaces play an important role in the general theory of locally convex TVSs and in the theory of Banach spaces in particular. Hilbert spaces are prominent examples of reflexive Banach spaces. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Pierre Aubin , a character from ''JoJo's Bizarre Ad ...
Jean-Pierre or Jean Pierre may refer to: People * Karine Jean-Pierre b.1977, White House Deputy Press Secretary for President Joe Biden 2021- * Jean-Pierre, Count of Montalivet (1766–1823), French statesman and Peer of France * Eugenia Pierre (better known as Jean Pierre, 1944–2002), Trinidadian netballer and parliamentarian Places * Jean-Pierre Bay, on the Gouin Reservoir in Quebec, Canada Arts and entertainment *"Jean Pierre", song by Miles Davis from ''Miles! Miles! Miles!'' * Jean-Pierre, chef on television series ''Metalocalypse'' * Jean-Pierre Delmas, in French animated television series ''Code Lyoko'' * Jean Pierre, a character in ''Fighter's History'' *Jean Pierre Polnareff The ''JoJo's Bizarre Adventure'' manga series features a large cast of characters created by Hirohiko Araki. Spanning several generations, the series is split into eight parts, each following a different descendant of the Joestar family. Parts 7 ... [...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] |
Mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History One of the earliest known mathematicians were Thales of Miletus (c. 624–c.546 BC); he has been hailed as the first true mathematician and the first known individual to whom a mathematical discovery has been attributed. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' Theorem. The number of known mathematicians grew when Pythagoras of Samos (c. 582–c. 507 BC) established the Pythagorean School, whose doctrine it was that mathematics ruled the universe and whose motto was "All is number". It was the Pythagoreans who coined the term "mathematics", and with whom the study of mathematics for its own sake begins. The first woman mathematician recorded by history was Hypati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pacific and Indian Oceans. Its Metropolitan France, metropolitan area extends from the Rhine to the Atlantic Ocean and from the Mediterranean Sea to the English Channel and the North Sea; overseas territories include French Guiana in South America, Saint Pierre and Miquelon in the North Atlantic, the French West Indies, and many islands in Oceania and the Indian Ocean. Due to its several coastal territories, France has the largest exclusive economic zone in the world. France borders Belgium, Luxembourg, Germany, Switzerland, Monaco, Italy, Andorra, and Spain in continental Europe, as well as the Kingdom of the Netherlands, Netherlands, Suriname, and Brazil in the Americas via its overseas territories in French Guiana and Saint Martin (island), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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