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Order Unit
An order unit is an element of an ordered vector space which can be used to bound all elements from above. In this way (as seen in the first example below) the order unit generalizes the unit element in the reals. According to H. H. Schaefer, "most of the ordered vector spaces occurring in analysis do not have order units." Definition For the ordering cone K \subseteq X in the vector space X, the element e \in K is an order unit (more precisely an K-order unit) if for every x \in X there exists a \lambda_x > 0 such that \lambda_x e - x \in K (that is, x \leq_K \lambda_x e). Equivalent definition The order units of an ordering cone K \subseteq X are those elements in the algebraic interior of K; that is, given by \operatorname(K). Examples Let X = \R be the real numbers and K = \R_+ = \, then the unit element 1 is an . Let X = \R^n and K = \R^n_+ = \left\, then the unit element \vec = (1, \ldots, 1) is an . Each interior point of the positive cone of an ordered topolo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Vector Space
In mathematics, an ordered vector space or partially ordered vector space is a vector space equipped with a partial order that is compatible with the vector space operations. Definition Given a vector space ''X'' over the real numbers R and a preorder ≤ on the set ''X'', the pair is called a preordered vector space and we say that the preorder ≤ is compatible with the vector space structure of ''X'' and call ≤ a vector preorder on ''X'' if for all ''x'', ''y'', ''z'' in ''X'' and ''λ'' in R with the following two axioms are satisfied # implies # implies . If ≤ is a partial order compatible with the vector space structure of ''X'' then is called an ordered vector space and ≤ is called a vector partial order on ''X''. The two axioms imply that translations and positive homotheties are automorphisms of the order structure and the mapping is an isomorphism to the dual order structure. Ordered vector spaces are ordered groups under their additio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as '' The Lancet'', '' Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Examples
Example may refer to: * '' exempli gratia'' (e.g.), usually read out in English as "for example" * .example, reserved as a domain name that may not be installed as a top-level domain of the Internet ** example.com, example.net, example.org, example.edu, second-level domain names reserved for use in documentation as examples * HMS ''Example'' (P165), an Archer-class patrol and training vessel of the Royal Navy Arts * ''The Example'', a 1634 play by James Shirley * ''The Example'' (comics), a 2009 graphic novel by Tom Taylor and Colin Wilson * Example (musician), the British dance musician Elliot John Gleave (born 1982) * ''Example'' (album), a 1995 album by American rock band For Squirrels See also * * Exemplar (other), a prototype or model which others can use to understand a topic better * Exemplum, medieval collections of short stories to be told in sermons * Eixample The Eixample (; ) is a district of Barcelona between the old city (Ciutat Vella) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can be complex numbers or, more generally, elements of any field. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. The terms real vector space and complex vector space are often used to specify the nature of the scalars: real coordinate space or complex coordinate space. Vector spaces generalize Euclidean vectors, which allow modeling of physical quantities, such as forces and velocity, that have not only a magnitude, but also a direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix, which allows computing in vector spaces. This provides a concise and synthetic way for manipulating and studying systems of li ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe was the first president and Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance, due to concerns about competing with the American Journal of Mathematics. The result was the '' Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influential ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Interior
In functional analysis, a branch of mathematics, the algebraic interior or radial kernel of a subset of a vector space is a refinement of the concept of the interior. Definition Assume that A is a subset of a vector space X. The ''algebraic interior'' (or ''radial kernel'') ''of A with respect to X'' is the set of all points at which A is a radial set. A point a_0 \in A is called an of A and A is said to be if for every x \in X there exists a real number t_x > 0 such that for every t \in , t_x a_0 + t x \in A. This last condition can also be written as a_0 + , t_xx \subseteq A where the set a_0 + , t_xx ~:=~ \left\ is the line segment (or closed interval) starting at a_0 and ending at a_0 + t_x x; this line segment is a subset of a_0 + radial points of the set. If M is a linear subspace of X and A \subseteq X then this definition can be generalized to the ''algebraic interior of A with respect to M'' is: \operatorname_M A := \left\. where \operatorname_M A \subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Topological Vector Space
In mathematics, specifically in functional analysis and order theory, an ordered topological vector space, also called an ordered TVS, is a topological vector space (TVS) ''X'' that has a partial order ≤ making it into an ordered vector space whose positive cone C := \left\ is a closed subset of ''X''. Ordered TVS have important applications in spectral theory. Normal cone If ''C'' is a cone in a TVS ''X'' then ''C'' is normal if \mathcal = \left \mathcal \right, where \mathcal is the neighborhood filter at the origin, \left \mathcal \right = \left\, and := \left(U + C\right) \cap \left(U - C\right) is the ''C''-saturated hull of a subset ''U'' of ''X''. If ''C'' is a cone in a TVS ''X'' (over the real or complex numbers), then the following are equivalent: # ''C'' is a normal cone. # For every filter \mathcal in ''X'', if \lim \mathcal = 0 then \lim \left \mathcal \right = 0. # There exists a neighborhood base \mathcal in ''X'' such that B \in \mathcal implies \left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sublinear Functional
In linear algebra, a sublinear function (or functional as is more often used in functional analysis), also called a quasi-seminorm or a Banach functional, on a vector space X is a real-valued function with only some of the properties of a seminorm. Unlike seminorms, a sublinear function does not have to be nonnegative-valued and also does not have to be absolutely homogeneous. Seminorms are themselves abstractions of the more well known notion of norms, where a seminorm has all the defining properties of a norm that it is not required to map non-zero vectors to non-zero values. In functional analysis the name Banach functional is sometimes used, reflecting that they are most commonly used when applying a general formulation of the Hahn–Banach theorem. The notion of a sublinear function was introduced by Stefan Banach when he proved his version of the Hahn-Banach theorem. There is also a different notion in computer science, described below, that also goes by the name "subline ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Archimedean Ordered Vector Space
In mathematics, specifically in order theory, a binary relation \,\leq\, on a vector space X over the real or complex numbers is called Archimedean if for all x \in X, whenever there exists some y \in X such that n x \leq y for all positive integers n, then necessarily x \leq 0. An Archimedean (pre)ordered vector space is a (pre)ordered vector space whose order is Archimedean. A preordered vector space X is called almost Archimedean if for all x \in X, whenever there exists a y \in X such that -n^ y \leq x \leq n^ y for all positive integers n, thenx = 0. Characterizations A preordered vector space (X, \leq) with an order unit u is Archimedean preordered if and only if n x \leq u for all non-negative integers n implies x \leq 0. Properties Let X be an ordered vector space over the reals that is finite-dimensional. Then the order of X is Archimedean if and only if the positive cone of X is closed for the unique topology under which X is a Hausdorff TVS. Order unit norm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Functional
In mathematics, in the field of functional analysis, a Minkowski functional (after Hermann Minkowski) or gauge function is a function that recovers a notion of distance on a linear space. If K is a subset of a real or complex vector space X, then the or of K is defined to be the function p_K : X \to , \infty valued in the extended real numbers, defined by p_K(x) := \inf \ \quad \text x \in X, where the infimum of the empty set is defined to be positive infinity \,\infty\, (which is a real number so that p_K(x) would then be real-valued). The Minkowski function is always non-negative (meaning p_K \geq 0) and p_K(x) is a real number if and only if \ is not empty. This property of being nonnegative stands in contrast to other classes of functions, such as sublinear functions and real linear functionals, that do allow negative values. In functional analysis, K is usually assumed to have properties (such as being absorbing in X, for instance) that will guarantee that for ... [...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] |
