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Hanner's Inequalities
In mathematics, Hanner's inequalities are results in the theory of ''L''''p'' spaces. Their proof was published in 1956 by Olof Hanner. They provide a simpler way of proving the uniform convexity of ''L''''p'' spaces for ''p'' ∈ (1, +∞) than the approach proposed by James A. Clarkson in 1936. Statement of the inequalities Let ''f'', ''g'' ∈ ''L''''p''(''E''), where ''E'' is any measure space A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that .... If ''p'' ∈ , 2 then :\, f+g\, _p^p + \, f-g\, _p^p \geq \big( \, f\, _p + \, g\, _p \big)^p + \big, \, f\, _p-\, g\, _p \big, ^p. The substitutions ''F'' = ''f'' + ''g'' and ''G'' = ''f'' − ''g'' yield the second of Hanner's inequalities: : ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lp Space
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Olof Hanner
Olof Hanner (7 December 1922 in Stockholm – 19 September 2015 in Gothenburg) was a Swedish mathematician. Education and career Hanner earned his Ph.D. from Stockholm University in 1952. He was a professor at the University of Gothenburg from 1963 to 1989. Contributions In a 1956 paper, Hanner introduced the Hanner polytopes and the Hanner spaces having these polytopes as their metric balls. Hanner was interested in a Helly property of these shapes, later used to characterize them by : unlike other convex polytopes, it is not possible to find three translated copies of a Hanner polytope that intersect pairwise but do not have a point of common intersection. Subsequently, the Hanner polytopes formed a class of important examples for the Mahler conjecture and for Kalai's 3''d'' conjecture. In another paper from the same year, Hanner proved a set of inequalities related to the uniform convexity of ''L''''p'' spaces, now known as Hanner's inequalities. Other contributions of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniformly Convex Space
In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity was first introduced by James A. Clarkson in 1936. Definition A uniformly convex space is a normed vector space such that, for every 00 such that for any two vectors with \, x\, = 1 and \, y\, = 1, the condition :\, x-y\, \geq\varepsilon implies that: :\left\, \frac\right\, \leq 1-\delta. Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. Properties * The unit sphere can be replaced with the closed unit ball in the definition. Namely, a normed vector space X is uniformly convex if and only if for every 00 so that, for any two vectors x and y in the closed unit ball (i.e. \, x\, \le 1 and \, y\, \le 1 ) with \, x-y\, \ge \varepsilon , one has \left\, \right\, \le 1-\delta (note that, given \varepsilon , the corresponding value ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
James A
James is a common English language surname and given name: * James (name), the typically masculine first name James * James (surname), various people with the last name James James or James City may also refer to: People * King James (other), various kings named James * Saint James (other) * James (musician) * James, brother of Jesus Places Canada * James Bay, a large body of water * James, Ontario United Kingdom * James College, a college of the University of York United States * James, Georgia, an unincorporated community * James, Iowa, an unincorporated community * James City, North Carolina * James City County, Virginia ** James City (Virginia Company) ** James City Shire * James City, Pennsylvania * St. James City, Florida Arts, entertainment, and media * ''James'' (2005 film), a Bollywood film * ''James'' (2008 film), an Irish short film * ''James'' (2022 film), an Indian Kannada-language film * James the Red Engine, a character in ''Thomas the Tan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Measure Space
A measure space is a basic object of measure theory, a branch of mathematics that studies generalized notions of volumes. It contains an underlying set, the subsets of this set that are feasible for measuring (the -algebra) and the method that is used for measuring (the measure). One important example of a measure space is a probability space. A measurable space consists of the first two components without a specific measure. Definition A measure space is a triple (X, \mathcal A, \mu), where * X is a set * \mathcal A is a -algebra on the set X * \mu is a measure on (X, \mathcal) In other words, a measure space consists of a measurable space (X, \mathcal) together with a measure on it. Example Set X = \. The \sigma-algebra on finite sets such as the one above is usually the power set, which is the set of all subsets (of a given set) and is denoted by \wp(\cdot). Sticking with this convention, we set \mathcal = \wp(X) In this simple case, the power set can be writt ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram Rule
In mathematics, the simplest form of the parallelogram law (also called the parallelogram identity) belongs to elementary geometry. It states that the sum of the squares of the lengths of the four sides of a parallelogram equals the sum of the squares of the lengths of the two diagonals. We use these notations for the sides: ''AB'', ''BC'', ''CD'', ''DA''. But since in Euclidean geometry a parallelogram necessarily has opposite sides equal, that is, ''AB'' = ''CD'' and ''BC'' = ''DA'', the law can be stated as 2AB^2 + 2BC^2 = AC^2 + BD^2\, If the parallelogram is a rectangle, the two diagonals are of equal lengths ''AC'' = ''BD'', so 2AB^2 + 2BC^2 = 2AC^2 and the statement reduces to the Pythagorean theorem. For the general quadrilateral with four sides not necessarily equal, AB^2 + BC^2 + CD^2+DA^2 = AC^2+BD^2 + 4x^2, where x is the length of the line segment joining the midpoints of the diagonals. It can be seen from the diagram that x = 0 for a parallelogram, and so the ... [...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 n ... [...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 phe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
