Halo (mathematics)

	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]
	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]
	Numberphile ''Numberphile'' is an Educational entertainment, educational YouTube channel featuring videos that explore topics from a variety of fields of mathematics. In the early days of the channel, each video focused on a specific number, but the channel has since expanded its scope, featuring videos on more advanced mathematical concepts such as Fermat's Last Theorem, the Riemann hypothesis and Kruskal's tree theorem. The videos are produced by Brady Haran, a former BBC video journalist and creator of Periodic Videos, Sixty Symbols, and several other YouTube channels. Videos on the channel feature several university professors, maths communicators and famous mathematicians. In 2018, Haran released a spin-off (media), spin-off audio podcast titled ''The Numberphile Podcast''. YouTube channel The ''Numberphile'' YouTube channel was started on 15 September 2011. Most videos consist of Haran interviewing an expert on a number, mathematical theorem or other mathematical concept. The expert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Hyperreal Number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer $n$ . Similarly, $x$ is said to be infinitesimal if, and only if, $, x, <1/n$ for all positive integers $n$ . The term "hyper-real" was introduced by Edwin Hewitt in 1948. The hyperreal numbers satisfy the transfer principle, a rigorous version of Leibniz's heuristic [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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]
	University Of Wisconsin Press The University of Wisconsin Press (sometimes abbreviated as UW Press) is a Non-profit organization, non-profit university press publishing Peer review, peer-reviewed books and journals. It publishes work by scholars from the global academic community; works of fiction, memoir and poetry under its imprint, Terrace Books; and serves the citizens of Wisconsin by publishing important books about Wisconsin, the Upper Midwest, and the Great Lakes region (North America), Great Lakes region. UW Press annually awards the Brittingham Prize in Poetry, the Felix Pollak Prize in Poetry, and The Four Lakes Prize in Poetry. The press was founded in 1936 in Madison, Wisconsin, Madison and is one of more than 120 member presses in the Association of University Presses. The Journals Division was established in 1965. The press employs approximately 25 full and part-time staff, produces 40 to 60 new books a year, and publishes 13 journals. It also distributes books and some annual journals for sele ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]