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Lorentz Spaces
In mathematical analysis, Lorentz spaces, introduced by George G. Lorentz in the 1950s,G. Lorentz, "On the theory of spaces Λ", ''Pacific Journal of Mathematics'' 1 (1951), pp. 411-429. are generalisations of the more familiar L^ spaces. The Lorentz spaces are denoted by L^. Like the L^ spaces, they are characterized by a norm (technically a quasinorm) that encodes information about the "size" of a function, just as the L^ norm does. The two basic qualitative notions of "size" of a function are: how tall is the graph of the function, and how spread out is it. The Lorentz norms provide tighter control over both qualities than the L^ norms, by exponentially rescaling the measure in both the range (p) and the domain (q). The Lorentz norms, like the L^ norms, are invariant under arbitrary rearrangements of the values of a function. Definition The Lorentz space on a measure space (X, \mu) is the space of complex-valued measurable functions f on ''X'' such that the following quasi ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequence Space
In functional analysis and related areas of mathematics, a sequence space is a vector space whose elements are infinite sequences of real or complex numbers. Equivalently, it is a function space whose elements are functions from the natural numbers to the field ''K'' of real or complex numbers. The set of all such functions is naturally identified with the set of all possible infinite sequences with elements in ''K'', and can be turned into a vector space under the operations of pointwise addition of functions and pointwise scalar multiplication. All sequence spaces are linear subspaces of this space. Sequence spaces are typically equipped with a norm, or at least the structure of a topological vector space. The most important sequence spaces in analysis are the spaces, consisting of the -power summable sequences, with the ''p''-norm. These are special cases of L''p'' spaces for the counting measure on the set of natural numbers. Other important classes of sequences ... [...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] |
Hardy–Littlewood Inequality
In mathematical analysis, the Hardy–Littlewood inequality, named after G. H. Hardy and John Edensor Littlewood, states that if f and g are nonnegative measurable real functions vanishing at infinity that are defined on n-dimensional Euclidean space \mathbb R^n, then :\int_ f(x)g(x) \, dx \leq \int_ f^*(x)g^*(x) \, dx where f^* and g^* are the symmetric decreasing rearrangement In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. Definition for sets Given a measurable set, A, in \R ...s of f and g, respectively. The decreasing rearrangement f^* of f is defined via the property that for all r >0 the two super-level sets :E_f(r)=\left\ \quad and \quad E_(r)=\left\ have the same volume (n-dimensional Lebesgue measure) and E_(r) is a ball in \mathbb R^n centered at x=0, i.e. it has maximal symmetry. Proof The layer cake represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interpolation Space
In the mathematics, mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a number of data points, obtained by sampling (statistics), sampling or experimentation, which represent the values of a function for a limited number of values of the Dependent and independent variables, independent variable. It is often required to interpolate; that is, estimate the value of that function for an intermediate value of the independent variable. A closely related problem is the function approximation, approximation of a complicated function by a simple function. Suppose the formula for some given function is known, but too complicated to evaluate efficiently. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original. The resulting gai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasinorm
In linear algebra, functional analysis and related areas of mathematics, a quasinorm is similar to a norm in that it satisfies the norm axioms, except that the triangle inequality is replaced by \, x + y\, \leq K(\, x\, + \, y\, ) for some K > 1. Definition A on a vector space X is a real-valued map p on X that satisfies the following conditions: : p \geq 0; : p(s x) = , s, p(x) for all x \in X and all scalars s; there exists a real k \geq 1 such that p(x + y) \leq k (x) + p(y)/math> for all x, y \in X. * If k = 1 then this inequality reduces to the triangle inequality. It is in this sense that this condition generalizes the usual triangle inequality. A is a quasi-seminorm that also satisfies: Positive definite/: if x \in X satisfies p(x) = 0, then x = 0. A pair (X, p) consisting of a vector space X and an associated quasi-seminorm p is called a . If the quasi-seminorm is a quasinorm then it is also called a . Multiplier The infimum of all values of k that satisf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cavalieri's Principle
In geometry, Cavalieri's principle, a modern implementation of the method of indivisibles, named after Bonaventura Cavalieri, is as follows: * 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. * 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross section (geometry), cross-sections of equal area, then the two regions have equal volumes. Today Cavalieri's principle is seen as an early step towards integral calculus, and while it is used in some forms, such as its generalization in Fubini's theorem and layer cake representation, results using Cavalieri's principle can often be shown more directly via integration. In the other direction, Cavalieri's principle gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Decreasing Rearrangement
In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function. Definition for sets Given a measurable set, A, in \R^n, one defines the ''symmetric rearrangement'' of A, called A^*, as the ball centered at the origin, whose volume (Lebesgue measure) is the same as that of the set A. An equivalent definition is A^* = \left\, where \omega_n is the volume of the unit ball and where , A, is the volume of A. Definition for functions The rearrangement of a non-negative, measurable real-valued function f whose level sets f^(y) (for y \in \R_) have finite measure is f^*(x) = \int_0^\infty \mathbb_(x) \, dt, where \mathbb_A denotes the indicator function of the set A. In words, the value of f^*(x) gives the height t for which the radius of the symmetric rearrangement of \ is equal to x. We have the following motivation for this definition. Because the iden ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
George G
George may refer to: Names * George (given name) * George (surname) People * George (singer), American-Canadian singer George Nozuka, known by the mononym George * George Papagheorghe, also known as Jorge / GEØRGE * George, stage name of Giorgio Moroder * George, son of Andrew I of Hungary Places South Africa * George, South Africa, a city ** George Airport United States * George, Iowa, a city * George, Missouri, a ghost town * George, Washington, a city * George County, Mississippi * George Air Force Base, a former U.S. Air Force base located in California Computing * George (algebraic compiler) also known as 'Laning and Zierler system', an algebraic compiler by Laning and Zierler in 1952 * GEORGE (computer), early computer built by Argonne National Laboratory in 1957 * GEORGE (operating system), a range of operating systems (George 1–4) for the ICT 1900 range of computers in the 1960s * GEORGE (programming language), an autocode system invented by Charles Leonard Hamblin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measurable Function
In mathematics, and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in direct analogy to the definition that a continuous function between topological spaces preserves the topological structure: the preimage of any open set is open. In real analysis, measurable functions are used in the definition of the Lebesgue integral. In probability theory, a measurable function on a probability space is known as a random variable. Formal definition Let (X,\Sigma) and (Y,\Tau) be measurable spaces, meaning that X and Y are sets equipped with respective \sigma-algebras \Sigma and \Tau. A function f:X\to Y is said to be measurable if for every E\in \Tau the pre-image of E under f is in \Sigma; that is, for all E \in \Tau f^(E) := \ \in \Sigma. That is, \sigma (f)\subseteq\Sigma, where \sigma (f) is the σ-algeb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
