|
Aronszajn
Nachman Aronszajn (26 July 1907 – 5 February 1980) was a Polish American mathematician. Aronszajn's main field of study was mathematical analysis, where he systematically developed the concept of reproducing kernel Hilbert space. He also contributed to mathematical logic. Life An Ashkenazi Jew, Aronszajn received his Ph.D. from the University of Warsaw, in 1930, in Poland. Stefan Mazurkiewicz was his thesis advisor. He also received a Ph.D. from Paris University, in 1935; this time Maurice Fréchet was his thesis advisor. He joined the Oklahoma State University faculty, but moved to the University of Kansas in 1951 with his colleague Ainsley Diamond after Diamond, a Quaker, was fired for refusing to sign a newly instituted loyalty oath.. Aronszajn retired in 1977. He was a Summerfield Distinguished Scholar from 1964 to his death. Work He introduced, together with Prom Panitchpakdi, injective metric spaces under the name of "hyperconvex metric spaces". Together with Ke ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aronszajn Tree
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of height ''κ'' in which all levels have size less than ''κ'' and all branches have height less than ''κ'' (so Aronszajn trees are the same as \aleph_1-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by . A cardinal ''κ'' for which no ''κ''-Aronszajn trees exist is said to have the tree property (sometimes the condition that ''κ'' is regular and uncountable is included). Existence of κ-Aronszajn trees Kőnig's lemma states that \aleph_0-Aronszajn trees do not exist. The existence of Aronszajn trees (=\aleph_1-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aronszajn Trees
In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of height ''κ'' in which all levels have size less than ''κ'' and all branches have height less than ''κ'' (so Aronszajn trees are the same as \aleph_1-Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by . A cardinal ''κ'' for which no ''κ''-Aronszajn trees exist is said to have the tree property (sometimes the condition that ''κ'' is regular and uncountable is included). Existence of κ-Aronszajn trees Kőnig's lemma states that \aleph_0-Aronszajn trees do not exist. The existence of Aronszajn trees (=\aleph_1-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reproducing Kernel Hilbert Space
In functional analysis (a branch of mathematics), a reproducing kernel Hilbert space (RKHS) is a Hilbert space of functions in which point evaluation is a continuous linear functional. Roughly speaking, this means that if two functions f and g in the RKHS are close in norm, i.e., \, f-g\, is small, then f and g are also pointwise close, i.e., , f(x)-g(x), is small for all x. The converse does not need to be true. Informally, this can be shown by looking at the supremum norm: the sequence of functions \sin^n (x) converges pointwise, but do not converge uniformly i.e. do not converge with respect to the supremum norm (note that this is not a counterexample because the supremum norm does not arise from any inner product due to not satisfying the parallelogram law). It is not entirely straightforward to construct a Hilbert space of functions which is not an RKHS. Some examples, however, have been found. Note that ''L''2 spaces are not Hilbert spaces of functions (and hence not RKH ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aronszajn–Smith Theorem
In the field of mathematics known as functional analysis, the invariant subspace problem is a partially unresolved problem asking whether every bounded operator on a complex Banach space sends some non-trivial closed subspace to itself. Many variants of the problem have been solved, by restricting the class of bounded operators considered or by specifying a particular class of Banach spaces. The problem is still open for separable Hilbert spaces (in other words, each example, found so far, of an operator with no non-trivial invariant subspaces is an operator that acts on a Banach space that is not isomorphic to a separable Hilbert space). History The problem seems to have been stated in the mid-1900s after work by Beurling and von Neumann,. who found (but never published) a positive solution for the case of compact operators. It was then posed by Paul Halmos for the case of operators T such that T^2 is compact. This was resolved affirmatively, for the more general class of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aronszajn Line
In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality \aleph_1 which contains no subset order-isomorphic to * \omega_1 with the usual ordering * the reverse of \omega_1 * an uncountable subset of the Real numbers with the usual ordering. Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal ''κ'', a ''κ''-Aronszajn tree is a tree of .... References Order theory {{settheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Injective Metric Space
In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher-dimensional vector spaces. These properties can be defined in two seemingly different ways: hyperconvexity involves the intersection properties of closed balls in the space, while injectivity involves the isometric embeddings of the space into larger spaces. However it is a theorem of that these two different types of definitions are equivalent. Hyperconvexity A metric space X is said to be hyperconvex if it is convex and its closed balls have the binary Helly property. That is: #Any two points x and y can be connected by the isometric image of a line segment of length equal to the distance between the points (i.e. X is a path space). #If F is any family of closed balls _r(p) = \ such that each pair of balls in F meets, then there exists a point x common to all the balls in F. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prom Panitchpakdi
A promenade dance, commonly called a prom, is a dance party for high school students. It may be offered in semi-formal black tie or informal suit for boys, and evening gowns for girls. This event is typically held near the end of the school year. There may be individual junior (11th grade) and senior (12th grade) proms or they may be combined. At a prom, a "prom king" and a "prom queen" may be revealed. These are honorary titles awarded to students elected in a school-wide vote prior to the prom. Other students may be honored with inclusion in a ''prom court''. The selection method for a prom court is similar to that of homecoming queen/princess, king/prince, and court. Inclusion in a prom court may be a reflection of popularity of those students elected and their level of participation in school activities, such as clubs or sports. The prom queen and prom king may be given crowns to wear. Members of the prom court may be given sashes to wear and photographed together. Similar e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quaker
Quakers are people who belong to a historically Protestant Christian set of Christian denomination, denominations known formally as the Religious Society of Friends. Members of these movements ("theFriends") are generally united by a belief in each human's ability to experience Inward light, the light within or see "that of God in every one". Some profess a priesthood of all believers inspired by the First Epistle of Peter. They include those with evangelicalism, evangelical, Holiness movement, holiness, Mainline Protestant, liberal, and Conservative Friends, traditional Quaker understandings of Christianity. There are also Nontheist Quakers, whose spiritual practice does not rely on the existence of God. To differing extents, the Friends avoid creeds and Hierarchical structure, hierarchical structures. In 2017, there were an estimated 377,557 adult Quakers, 49% of them in Africa. Some 89% of Quakers worldwide belong to ''evangelical'' and ''programmed'' branches that hold ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ainsley Diamond
Ainsley (also spelt Ainsleigh) is both a unisex given name and a surname and place name. It is derived from words meaning hermitage and clearing. Notable people with the name include: Given name Men *Ainsley Battles (born 1978), American football player * Ainsley Bennett (born 1954), British sprinter *Ainsley Hall (born 1972), Cayman Islands cricketer *Ainsley Harriott (born 1957), English television chef *Ainsley Maitland-Niles (born 1997), English footballer *Ainsley Waugh (born 1981), Jamaican sprinter *Ainsley Melham (born 1991), Australian actor *Ainsley Iggo (1924–2012), Scottish neurophysiologist * Ainsley Robinson, Canadian wrestler *A. C. de Zoysa (1923–1983), Sri Lankan criminal lawyer Women *Ainsley Earhardt (born 1978), American television news anchor *Ainsley Gardiner, film producer from New Zealand *Ainsley Gotto (1946–2018), Australian businesswoman *Ainsley Howard, English actress * Ainsley Hamill, Scottish singer Surname Ainsleigh *Daniel Ainsleig ... [...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] |
Paris University
, image_name = Coat of arms of the University of Paris.svg , image_size = 150px , caption = Coat of Arms , latin_name = Universitas magistrorum et scholarium Parisiensis , motto = ''Hic et ubique terrarum'' (Latin) , mottoeng = Here and anywhere on Earth , established = Founded: c. 1150Suppressed: 1793Faculties reestablished: 1806University reestablished: 1896Divided: 1970 , type = Corporative then public university , city = Paris , country = France , campus = Urban The University of Paris (french: link=no, Université de Paris), metonymically known as the Sorbonne (), was the leading university in Paris, France, active from 1150 to 1970, with the exception between 1793 and 1806 under the French Revolution. Emerging around 1150 as a corporation associated with the cathedral school of Notre Dame de Paris, it was considered the second-oldest university in Europe. Haskins, C. H.: ''The Rise of Universities'', Henry Holt and Company, 1923, p. 292. Officially chartered i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
