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Trudinger's Theorem
In mathematical analysis, Trudinger's theorem or the Trudinger inequality (also sometimes called the Moser–Trudinger inequality) is a result of functional analysis on Sobolev spaces. It is named after Neil Trudinger (and Jürgen Moser). It provides an inequality between a certain Sobolev space norm and an Orlicz space norm of a function. The inequality is a limiting case of Sobolev imbedding and can be stated as the following theorem: Let \Omega be a bounded domain in \mathbb^n satisfying the cone condition. Let mp=n and p>1. Set : A(t)=\exp\left( t^ \right)-1. Then there exists the embedding : W^(\Omega)\hookrightarrow L_A(\Omega) where : L_A(\Omega)=\left\. The space :L_A(\Omega) is an example of an Orlicz space In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the ''L'p'' spaces. Like the ''L'p'' spaces, they are Banach spaces. The spaces are na .... Referenc ... [...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] |
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 (e.g. Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, Topological space#Definition, topology, etc.) and the linear transformation, linear functions defined on these spaces and respecting these structures in a suitable sense. 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 continuous function, continuous, unitary operator, unitary etc. 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 variati ... [...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] |
Neil Trudinger
Neil Sidney Trudinger (born 20 June 1942) is an Australian mathematician, known particularly for his work in the field of nonlinear elliptic partial differential equations. After completing his B.Sc at the University of New England (Australia) in 1962, he continued his graduate studies at Stanford University. He was awarded a Ph.D in 1966 for his thesis "Quasilinear Elliptical Partial Differential Equations in n Variables". After the award of his doctorate from Stanford University, Trudinger became a Courant Instructor at the Courant Institute of Mathematical Sciences of New York University during the academic year 1966–67. He then returned to Australia where he was appointed as a lecturer at Macquarie University in 1967. In 1970, he moved to University of Queensland where he was first appointed as a Reader, then as Professor. In 1973 he moved to the Australian National University. In 2016 he moved to the University of Wollongong, where he is currently appointed as a Distinguis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Moser
Jürgen Kurt Moser (July 4, 1928 – December 17, 1999) was a German-American mathematician, honored for work spanning over four decades, including Hamiltonian dynamical systems and partial differential equations. Life Moser's mother Ilse Strehlke was a niece of the violinist and composer Louis Spohr. His father was the neurologist Kurt E. Moser (July 21, 1895 – June 25, 1982), who was born to the merchant Max Maync (1870–1911) and Clara Moser (1860–1934). The latter descended from 17th century French Huguenot immigrants to Prussia. Jürgen Moser's parents lived in Königsberg, German empire and resettled in Stralsund, East Germany as a result of the second world war. Moser attended the Wilhelmsgymnasium (Königsberg) in his hometown, a high school specializing in mathematics and natural sciences education, from which David Hilbert had graduated in 1880. His older brother Friedrich Robert Ernst (Friedel) Moser (August 31, 1925 – January 14, 1945) served in the German Army ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orlicz Space
In mathematical analysis, and especially in real, harmonic analysis and functional analysis, an Orlicz space is a type of function space which generalizes the ''L''''p'' spaces. Like the ''L''''p'' spaces, they are Banach spaces. The spaces are named for Władysław Orlicz, who was the first to define them in 1932. Besides the ''L''''p'' spaces, a variety of function spaces arising naturally in analysis are Orlicz spaces. One such space ''L'' log+ ''L'', which arises in the study of Hardy–Littlewood maximal functions, consists of measurable functions ''f'' such that the integral :\int_ , f(x), \log^+ , f(x), \,dx < \infty. Here log+ is the of the logarithm. Also included in the class of Orlicz spaces are many of the most important |
Limiting Case (mathematics)
In mathematics, a limiting case of a mathematical object is a special case that arises when one or more components of the object take on their most extreme possible values. For example: * In statistics, the limiting case of the binomial distribution is the Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution. *A circle is a limiting case of various other figures, including the Cartesian oval, the ellipse, the superellipse, and the Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached. *Archimedes calculated an approximate value of π by treating the circle as the limiting case of a regular polygon with 3 × 2''n'' sides, as ''n'' gets large. *In electricity and magnetism, the long wavelength limit is the limiting case when the wavelength is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cone Condition
In mathematics, the cone condition is a property which may be satisfied by a subset of a Euclidean space. Informally, it requires that for each point in the subset a cone with vertex in that point must be contained in the subset itself, and so the subset is "non-flat". Formal definitions An open subset S of a Euclidean space E is said to satisfy the ''weak cone condition'' if, for all \boldsymbol \in S, the cone \boldsymbol + V_ is contained in S. Here V_ represents a cone with vertex in the origin, constant opening, axis given by the vector \boldsymbol(\boldsymbol), and height h \ge 0. S satisfies the ''strong cone condition'' if there exists an open cover In mathematics, and more particularly in set theory, a cover (or covering) of a set X is a collection of subsets of X whose union is all of X. More formally, if C = \lbrace U_\alpha : \alpha \in A \rbrace is an indexed family of subsets U_\alpha\s ... \ of \overline such that for each \boldsymbol \in \overline \cap S_k there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sobolev Spaces
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] |
Inequalities
Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * Spatial inequality, the unequal distribution of income and resources across geographical regions * Income inequality metrics, used to measure income and economic inequality among participants in a particular economy * International inequality, economic differences between countries Healthcare * Health equity, the study of differences in the quality of health and healthcare across different populations Mathematics * Inequality (mathematics), a relation between two values when they are different Social sciences * Educational inequality, the unequal distribution of academic resources to socially excluded communities * Gender inequality, unequal treatment or perceptions of individuals due to their gender * Participation inequality, the pheno ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
