|
Standard Part Function
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, 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 derivative, and the integral, in nonstandard analysis. The latter theory is a rigorous formalization of calculations with infinitesimals. The standard part of ''x'' is sometimes referred to as its shadow. Definition Nonstandard analysis ... [...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 (ε, δ)-definition of limit, limits 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 inf ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Calculus
Calculus is the mathematics, 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. Originally called infinitesimal calculus or "the calculus of infinitesimals", 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. 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. It is the "mathematical backbone" for dealing with problems where variables change with time or another reference variable. Infinitesimal calculus was formulated separately ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wilhelmus A
"Wilhelmus van Nassouwe", known simply as "Wilhelmus", or written with the article as "Het Wilhelmus", is the national anthem of both the Netherlands and its sovereign state, 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. The name is derived from its first word, which is the Latinization of the Germanic name Wilhelm (the English version is William), used for official records during that time in Dutch speaking lands or when considering the entire first line which is how the anthem is also referred to, "Wilhelmus van Nassouwe ... [...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. 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 twentieth century." Histor ... [...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'' (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (''parisotēs'') to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as ''adaequalitas''. Paul Tannery's French translation of Fermat's Latin treatises on maxima and minima used the words ''adéquation'' and ''adégaler''. Fermat's method Fermat used ''adequality'' first to find max ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Halo (mathematics)
In nonstandard analysis, a monad or also a 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''. References 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. Phra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Internal Set
In mathematical logic, in particular in model theory and nonstandard analysis, an internal set is a set that is a member of a model. The concept of internal sets is a tool in formulating the transfer principle, which concerns the logical relation between the properties of the real numbers R, and the properties of a larger field denoted *R called the hyperreal numbers. The field *R includes, in particular, infinitesimal ("infinitely small") numbers, providing a rigorous mathematical justification for their use. Roughly speaking, the idea is to express analysis over R in a suitable language of mathematical logic, and then point out that this language applies equally well to *R. This turns out to be possible because at the set-theoretic level, the propositions in such a language are interpreted to apply only to internal sets rather than to all sets (note that the term "language" is used in a loose sense in the above). Edward Nelson's internal set theory is an axiomatic app ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dedekind Cut
In mathematics, Dedekind cuts, named after German mathematician Richard Dedekind (but previously considered by Joseph Bertrand), are а method of construction of the real numbers from the rational numbers. A Dedekind cut is a partition of a set, partition of the rational numbers into two Set (mathematics) , sets ''A'' and ''B'', such that each element of ''A'' is less than every element of ''B'', and ''A'' contains no greatest element. The set ''B'' may or may not have a smallest element among the rationals. If ''B'' has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique irrational number which, loosely speaking, fills the "gap" between ''A'' and ''B''. In other words, ''A'' contains every rational number less than the cut, and ''B'' contains every rational number greater than or equal to the cut. An irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
