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Loomis–Whitney Inequality
In mathematics, the Loomis–Whitney inequality is a result in geometry, which in its simplest form, allows one to estimate the "size" of a d-dimensional set by the sizes of its (d-1)-dimensional projections. The inequality has applications in incidence geometry, the study of so-called "lattice animals", and other areas. The result is named after the American mathematicians Lynn Harold Loomis and Hassler Whitney, and was published in 1949. Statement of the inequality Fix a dimension d\ge 2 and consider the projections :\pi_ : \mathbb^ \to \mathbb^, :\pi_ : x = (x_, \dots, x_) \mapsto \hat_ = (x_, \dots, x_, x_, \dots, x_). For each 1 ≤ ''j'' ≤ ''d'', let :g_ : \mathbb^ \to [0, + \infty), :g_ \in L^ (\mathbb^). Then the Loomis–Whitney inequality holds: :\left\, \prod_^d g_j \circ \pi_j\right\, _ = \int_ \prod_^ g_ ( \pi_ (x) ) \, \mathrm x \leq \prod_^ \, g_ \, _. Equivalently, taking f_ (x) = g_ (x)^, we have :f_ : \mathbb^ \to [0, + \infty), :f_ \in L^ (\mathbb^) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperplane
In geometry, a hyperplane is a generalization of a two-dimensional plane in three-dimensional space to mathematical spaces of arbitrary dimension. Like a plane in space, a hyperplane is a flat hypersurface, a subspace whose dimension is one less than that of the ambient space. Two lower-dimensional examples of hyperplanes are one-dimensional lines in a plane and zero-dimensional points on a line. Most commonly, the ambient space is -dimensional Euclidean space, in which case the hyperplanes are the -dimensional "flats", each of which separates the space into two half spaces. A reflection across a hyperplane is a kind of motion ( geometric transformation preserving distance between points), and the group of all motions is generated by the reflections. A convex polytope is the intersection of half-spaces. In non-Euclidean geometry, the ambient space might be the -dimensional sphere or hyperbolic space, or more generally a pseudo-Riemannian space form, and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bulletin Of The American Mathematical Society
The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. It also publishes, by invitation only, book reviews and short ''Mathematical Perspectives'' articles. History It began as the ''Bulletin of the New York Mathematical Society'' and underwent a name change when the society became national. The Bulletin's function has changed over the years; its original function was to serve as a research journal for its members. Indexing The Bulletin is indexed in Mathematical Reviews, Science Citation Index, ISI Alerting Services, CompuMath Citation Index, and Current Contents/Physical, Chemical & Earth Sciences. See also *'' Journal of the American Mathematical Society'' *'' Memoirs of the American Mathematical Society'' *'' Notices of the American Mathematical Society'' *'' Proceedings of the Ame ... [...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] |
Oxford University Press
Oxford University Press (OUP) is the publishing house of the University of Oxford. It is the largest university press in the world. Its first book was printed in Oxford in 1478, with the Press officially granted the legal right to print books by decree in 1586. It is the second-oldest university press after Cambridge University Press, which was founded in 1534. It is a department of the University of Oxford. It is governed by a group of 15 academics, the Delegates of the Press, appointed by the Vice Chancellor, vice-chancellor of the University of Oxford. The Delegates of the Press are led by the Secretary to the Delegates, who serves as OUP's chief executive and as its major representative on other university bodies. Oxford University Press has had a similar governance structure since the 17th century. The press is located on Walton Street, Oxford, Walton Street, Oxford, opposite Somerville College, Oxford, Somerville College, in the inner suburb of Jericho, Oxford, Jericho. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John Wiley & Sons, Inc
John Wiley & Sons, Inc., commonly known as Wiley (), is an American multinational publishing company that focuses on academic publishing and instructional materials. The company was founded in 1807 and produces books, journals, and encyclopedias, in print and electronically, as well as online products and services, training materials, and educational materials for undergraduate, graduate, and continuing education students. History The company was established in 1807 when Charles Wiley opened a print shop in Manhattan. The company was the publisher of 19th century American literary figures like James Fenimore Cooper, Washington Irving, Herman Melville, and Edgar Allan Poe, as well as of legal, religious, and other non-fiction titles. The firm took its current name in 1865. Wiley later shifted its focus to scientific, technical, and engineering subject areas, abandoning its literary interests. Wiley's son John (born in Flatbush, New York, October 4, 1808; died in East O ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a linear endomorphism. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brascamp–Lieb Inequality
In mathematics, the Brascamp–Lieb inequality is either of two inequalities. The first is a result in geometry concerning integrable functions on ''n''-dimensional Euclidean space \mathbb^. It generalizes the Loomis–Whitney inequality and Hölder's inequality. The second is a result of probability theory which gives a concentration inequality for log-concave probability distributions. Both are named after Herm Jan Brascamp and Elliott H. Lieb. The geometric inequality Fix natural numbers ''m'' and ''n''. For 1 ≤ ''i'' ≤ ''m'', let ''n''''i'' ∈ N and let ''c''''i'' > 0 so that :\sum_^m c_i n_i = n. Choose non-negative, integrable functions :f_i \in L^1 \left( \mathbb^ ; , + \infty\right) and surjective linear maps :B_i : \mathbb^n \to \mathbb^. Then the following inequality holds: :\int_ \prod_^m f_i \left( B_i x \right)^ \, \mathrm x \leq D^ \prod_^m \left( \int_ f_i (y) \, \mathrm y \right)^, where ''D'' is given by :D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isoperimetric Inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the square of the circumference of a closed curve in the plane and the area of a plane region it encloses, as well as its various generalizations. '' Isoperimetric'' literally means "having the same perimeter". Specifically, the isoperimetric inequality states, for the length ''L'' of a closed curve and the area ''A'' of the planar region that it encloses, that :4\pi A \le L^2, and that equality holds if and only if the curve is a circle. The isoperimetric problem is to determine a plane figure of the largest possible area whose boundary has a specified length. The closely related ''Dido's problem'' asks for a region of the maximal area bounded by a straight line and a curvilinear arc whose endpoints belong to that line. It is named after Dido, the legendary founder and first queen of Carthage. The solution to the isoperimetric problem is given by a circle and was known already in Ancient Greece. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting Measure
In mathematics, specifically measure theory, the counting measure is an intuitive way to put a measure on any set – the "size" of a subset is taken to be the number of elements in the subset if the subset has finitely many elements, and infinity \infty if the subset is infinite. The counting measure can be defined on any measurable space (that is, any set X along with a sigma-algebra) but is mostly used on countable sets. In formal notation, we can turn any set X into a measurable space by taking the power set of X as the sigma-algebra \Sigma; that is, all subsets of X are measurable sets. Then the counting measure \mu on this measurable space (X,\Sigma) is the positive measure \Sigma \to ,+\infty/math> defined by \mu(A) = \begin \vert A \vert & \text A \text\\ +\infty & \text A \text \end for all A\in\Sigma, where \vert A\vert denotes the cardinality of the set A. The counting measure on (X,\Sigma) is σ-finite if and only if the space X is countable In mathematic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indicator Function
In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , then the indicator function of is the function \mathbf_A defined by \mathbf_\!(x) = 1 if x \in A, and \mathbf_\!(x) = 0 otherwise. Other common notations are and \chi_A. The indicator function of is the Iverson bracket of the property of belonging to ; that is, \mathbf_(x) = \left x\in A\ \right For example, the Dirichlet function is the indicator function of the rational numbers as a subset of the real numbers. Definition Given an arbitrary set , the indicator function of a subset of is the function \mathbf_A \colon X \mapsto \ defined by \operatorname\mathbf_A\!( x ) = \begin 1 & \text x \in A \\ 0 & \text x \notin A \,. \end The Iverson bracket provides the equivalent notation \left x\in A\ \right/math> or that can be used instead of \mathbf_\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician who works in the field of geometry is called a ''List of geometers, geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point (geometry), point, line (geometry), line, plane (geometry), plane, distance, angle, surface (mathematics), surface, and curve, as fundamental concepts. Originally developed to model the physical world, geometry has applications in almost all sciences, and also in art, architecture, and other activities that are related to graphics. Geometry also has applications in areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental in Wiles's proof of Fermat's Last Theorem, Wiles's proof of Fermat's ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
