|
Shadow (mathematics)
In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every such hyperreal x, the unique real x_0 infinitely close to it, i.e. x-x_0 is infinitesimal. As such, it is a mathematical implementation of the historical concept of adequality introduced by Pierre de Fermat,Karin Usadi Katz and Mikhail G. Katz (2011) A Burgessian Critique of Nominalistic Tendencies in Contemporary Mathematics and its Historiography. Foundations of Science.Searxiv The authors refer to the Fermat-Robinson standard part. as well as Leibniz's Transcendental law of homogeneity. The standard part function was first defined by Abraham Robinson who used the notation ^x for the standard part of a hyperreal x (see Robinson 1974). This concept plays a key role in defining the concepts of the calculus, such as continuity, the derivati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonstandard Analysis
The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals. Nonstandard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Nonstandard analysis originated in the early 1960s by the mathematician Abraham Robinson. He wrote: ... the idea of infinitely small or ''infinitesimal'' quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the formative stages of the Differential and Integral Calculus. As for the objection ... that the distance between two distinct real numbers cannot be infinitely small, Gottfried Wilhelm Leibniz argued that the theory of infinitesimals implies the introduction of ideal numbers which might be infinitely small or infinitely ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ultrapower
The ultraproduct is a mathematical construction that appears mainly in abstract algebra and mathematical logic, in particular in model theory and set theory. An ultraproduct is a quotient of the direct product of a family of structures. All factors need to have the same signature. The ultrapower is the special case of this construction in which all factors are equal. For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pion ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithmetic operations. It has two major branches, differential calculus and integral calculus; the former concerns instantaneous Rate of change (mathematics), rates of change, and the slopes of curves, while the latter concerns accumulation of quantities, and areas under or between curves. These two branches are related to each other by the fundamental theorem of calculus, and they make use of the fundamental notions of convergence (mathematics), convergence of infinite sequences and Series (mathematics), infinite series to a well-defined limit (mathematics), limit. Infinitesimal calculus was developed independently in the late 17th century by Isaac Newton and Gottfried Wilhelm Leibniz. Later work, including (ε, δ)-definition of limit, codify ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelmus A
"Wilhelmus van Nassouwe", usually known just as "Wilhelmus" ( nl, Het Wilhelmus, italic=no; ; English translation: "The William"), is the national anthem of both the Netherlands and the Kingdom of the Netherlands. It dates back to at least 1572, making it the oldest national anthem in use today, provided that the latter is defined as consisting of both a melody and lyrics. Although "Wilhelmus" was not recognized as the official national anthem until 1932, it has always been popular with parts of the Dutch population and resurfaced on several occasions in the course of Dutch history before gaining its present status. It was also the anthem of the Netherlands Antilles from 1954 to 1964. "Wilhelmus" originated in the Dutch Revolt, the nation's struggle to achieve independence from the Spanish Empire. It tells of the Father of the Nation William of Orange who was stadholder in the Netherlands under the King of Spain. In the first person, as if quoting himself, William speaks to the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graduate Texts In Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag. The books in this series, like the other Springer-Verlag mathematics series, are yellow books of a standard size (with variable numbers of pages). The GTM series is easily identified by a white band at the top of the book. The books in this series tend to be written at a more advanced level than the similar Undergraduate Texts in Mathematics series, although there is a fair amount of overlap between the two series in terms of material covered and difficulty level. List of books #''Introduction to Axiomatic Set Theory'', Gaisi Takeuti, Wilson M. Zaring (1982, 2nd ed., ) #''Measure and Category – A Survey of the Analogies between Topological and Measure Spaces'', John C. Oxtoby (1980, 2nd ed., ) #''Topological Vector Spaces'', H. H. Schaefer, M. P. Wolff (1999, 2nd ed., ) #''A Course in Homological Algebra'', Peter Hilton, Urs Stammbac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Robert Goldblatt
__notoc__ Robert Ian Goldblatt (born 1949) is a mathematical logician who is Emeritus Professor in the School of Mathematics and Statistics at Victoria University, Wellington, New Zealand. His most popular books are ''Logics of Time and Computation'' and ''Topoi: the Categorial Analysis of Logic''. He has also written a graduate level textbook on hyperreal numbers which is an introduction to nonstandard analysis. He has been Coordinating Editor of The ''Journal of Symbolic Logic'' and a Managing Editor of ''Studia Logica''. He was elected Fellow and Councillor of the Royal Society of New Zealand, President of the New Zealand Mathematical Society, and represented New Zealand to the International Mathematical Union. In 2012 he was awarded theJones Medalfor lifetime achievement in mathematics. Books and handbook chapters * 1979: ''Topoi: The Categorial Analysis of Logic'', North-Holland. Revised edition 1984. Dover Publications edition 2006Internet edition Project Euclid. :: Benjami ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
An Infinitesimal Approach
An, AN, aN, or an may refer to: Businesses and organizations * Airlinair (IATA airline code AN) * Alleanza Nazionale, a former political party in Italy * AnimeNEXT, an annual anime convention located in New Jersey * Anime North, a Canadian anime convention * Ansett Australia, a major Australian airline group that is now defunct (IATA designator AN) * Apalachicola Northern Railroad (reporting mark AN) 1903–2002 ** AN Railway, a successor company, 2002– * Aryan Nations, a white supremacist religious organization * Australian National Railways Commission, an Australian rail operator from 1975 until 1987 * Antonov, a Ukrainian (formerly Soviet) aircraft manufacturing and services company, as a model prefix Entertainment and media * Antv, an Indonesian television network * ''Astronomische Nachrichten'', or ''Astronomical Notes'', an international astronomy journal * ''Avisa Nordland'', a Norwegian newspaper * ''Sweet Bean'' (あん), a 2015 Japanese film also known as ''An'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonstandard Calculus
In mathematics, nonstandard calculus is the modern application of infinitesimals, in the sense of nonstandard analysis, to infinitesimal calculus. It provides a rigorous justification for some arguments in calculus that were previously considered merely heuristic. Non-rigorous calculations with infinitesimals were widely used before Karl Weierstrass sought to replace them with the (ε, δ)-definition of limit starting in the 1870s. (See history of calculus.) For almost one hundred years thereafter, mathematicians such as Richard Courant viewed infinitesimals as being naive and vague or meaningless. Contrary to such views, Abraham Robinson showed in 1960 that infinitesimals are precise, clear, and meaningful, building upon work by Edwin Hewitt and Jerzy Łoś. According to Howard Keisler, "Robinson solved a three hundred year old problem by giving a precise treatment of infinitesimals. Robinson's achievement will probably rank as one of the major mathematical advances of the tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Adequality
Adequality is a technique developed by Pierre de Fermat in his treatise ''Methodus ad disquirendam maximam et minimam''''METHOD FOR THE STUDY OF MAXIMA AND MINIMA'' English translation of Fermat's treatise ''Methodus ad disquirendam maximam et minimam''. (a treatise circulated in France c. 1636) to calculate of functions, s to curves, |
Microcontinuity
In nonstandard analysis, a discipline within classical mathematics, microcontinuity (or ''S''-continuity) of an internal function ''f'' at a point ''a'' is defined as follows: :for all ''x'' infinitely close to ''a'', the value ''f''(''x'') is infinitely close to ''f''(''a''). Here ''x'' runs through the domain of ''f''. In formulas, this can be expressed as follows: :if x\approx a then f(x)\approx f(a). For a function ''f'' defined on \mathbb, the definition can be expressed in terms of the halo as follows: ''f'' is microcontinuous at c\in\mathbb if and only if f(hal(c))\subseteq hal(f(c)), where the natural extension of ''f'' to the hyperreals is still denoted ''f''. Alternatively, the property of microcontinuity at ''c'' can be expressed by stating that the composition \text\circ f is constant on the halo of ''c'', where "st" is the standard part function. History The modern property of continuity of a function was first defined by Bolzano in 1817. However, Bolzano's work wa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halo (mathematics)
In nonstandard analysis, a monad (also called halo) is the set of points infinitesimally close to a given point. Given a hyperreal number ''x'' in R∗, the monad of ''x'' is the set :\text(x)=\. If ''x'' is finite (limited), the unique real number in the monad of ''x'' is called the standard part of ''x''. See also *Infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ... Notes References H. Jerome Keisler: Foundations of Infinitesimal Calculus, available for downloading Nonstandard analysis {{mathanalysis-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperfinite Set
In nonstandard analysis, a branch of mathematics, a hyperfinite set or *-finite set is a type of internal set. An internal set ''H'' of internal cardinality ''g'' ∈ *N (the hypernaturals) is hyperfinite if and only if there exists an internal bijection between ''G'' = and ''H''. Hyperfinite sets share the properties of finite sets: A hyperfinite set has minimal and maximal elements, and a hyperfinite union of a hyperfinite collection of hyperfinite sets may be derived. The sum of the elements of any hyperfinite subset of *R always exists, leading to the possibility of well-defined integration. Hyperfinite sets can be used to approximate other sets. If a hyperfinite set approximates an interval, it is called a ''near interval'' with respect to that interval. Consider a hyperfinite set K = with a hypernatural ''n''. ''K'' is a near interval for 'a'',''b''if ''k''1 = ''a'' and ''k''''n'' = ''b'', and if the difference between successive elements of ''K'' is infinitesimal. Phrased ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
