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Topic: Infinitesimal


In the News (Sat 20 Jul 19)

  
  NationMaster - Encyclopedia: Infinitesimal
Proving or disproving the existence of infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used.
Infinitesimal calculus is an area of mathematics pioneered by Gottfried Leibniz based on the concept of infinitesimals, as opposed to the calculus of Isaac Newton, which is based upon the concept of the limit.
Infinitesimals are legitimate quantities in the non-standard analysis of Abraham Robinson.
www.nationmaster.com /encyclopedia/Infinitesimal   (2705 words)

  
 Infinitesimal Calculus - LoveToKnow 1911
The infinitesimal calculus is the body of rules and processes by means of which continuously varying magnitudes are dealt with in mathematical analysis.
The guise in which variable quantities presented themselves to the mathematicians of the 17th century was that of the lengths of variable lines.
Berkeley's criticism was levelled against all infinitesimals, that is to say, all quantities vaguely conceived as in some intermediate state between nullity and finiteness, as he took Newton's moments to be conceived.
www.1911encyclopedia.org /Infinitesimal_Calculus   (17095 words)

  
 Infinitesimal transformation - Wikipedia, the free encyclopedia
Indeed this was at the heart of his work, on what are now that is called Lie groups and their accompanying Lie algebras; and the identification of their role in geometry and especially the theory of differential equations.
For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the cross product, once a skew-symmetric matrix has been identified with a 3-vector.
The earliest example of an infinitesimal transformation that may have been recognised as such was in Euler's theorem on homogeneous functions.
en.wikipedia.org /wiki/Infinitesimal_transformation   (429 words)

  
 Infinitesimals
Conversely, if the difference between a variable and a constant is an infinitesimal, then the variable approaches the constant as a limit.
An infinitesimal is a variable that approaches 0 as a limit.
The product of any number of infinitesimals is also an infinitesimal by the same approach as shown for two numbers.
www.tpub.com /math2/38.htm   (473 words)

  
 Continuity and Infinitesimals
Curves in smooth infinitesimal analysis are “locally straight” and accordingly may be conceived as being “composed of” infinitesimal straight lines in de l'Hôpital's sense, or as being “generated” by an infinitesimal tangent vector.
The widespread use of indivisibles and infinitesimals in the analysis of continuous variation by the mathematicians of the time testifies to the affirmation of a kind of mathematical atomism which, while logically questionable, made possible the spectacular mathematical advances with which the calculus is associated.
The concept of infinitesimal had arisen with problems of a geometric character and infinitesimals were originally conceived as belonging solely to the realm of continuous magnitude as opposed to that of discrete number.
plato.stanford.edu /entries/continuity   (16775 words)

  
 Infinitesimal - Metaweb   (Site not responding. Last check: )
In mathematics, an infinitesimal is a number greater in absolute value than zero yet smaller than any positive real number.
The first mathematician to make use of infinitesimals was Archimedes.
The fundamental problem is that dx is first treated as non-zero (because we divide by it), but then later discarded as if it were zero.
www.metaweb.com /wiki/wiki.phtml?title=Infinitesimal&redirect=no   (407 words)

  
 infinitesimal --  Encyclopædia Britannica
Infinitesimals were introduced by Isaac Newton as a means of “explaining” his procedures in calculus.
The calculus discovered in the 17th century by Newton and Leibniz used infinitesimal quantities to determine tangents to curves and to facilitate calculation of lengths and areas of curved figures.
Biography of this Greek philosopher and mathematician, whom Aristotle called the inventor of dialectic, and was famous for the paradoxes important in the development of the notion of infinitesimals.
www.britannica.com /eb/article-9042392?tocId=9042392   (620 words)

  
 Infinitesimal Analysis   (Site not responding. Last check: )
The concept of infinitesimal is one of the most important in mathematics; it was the concept that gave rise the calculus, and hence to analysis, and has been a driving force in fields such as differential geometry, differential topology, ergodic theory, financial mathematics, mathematical physics, perturbation theory, and stochastic analysis, to name only a few.
However, the existence of fields that extend the real numbers and include infinite and infinitesimal numbers was proved relatively recently, in the second half of the twentieth century, using constructions from formal logic.
Then we will study the techniques that are needed to use infinitesimal and infinite numbers using mathematical rigor, e.g., the Transfer Principle and the concepts of internal and external set.
www.math.utsa.edu /~iovino/teaching/fall05/infinitesimals.html   (318 words)

  
 Glossary - Encyclopedia of Authentic Hinduism
God is absolute, supreme, Gracious, omnipresent and omniscient; and soul is infinitesimal, under the bondage of maya and sinful.
Some of them (like Rudra Yamal and Krishn Yamal) also teach pure bhakti to a personal form of God; but, being very technical in their forms of practices, they are not popular.
The word ‘theory’ we use in both senses: (a) Eternally existing definite facts that are related to soul, maya, God, Saints, creation, God realization and the Divine existences, and are described in the Divinely produced Bhartiya scriptures.
encyclopediaofauthentichinduism.org /glossary.htm   (4289 words)

  
 Infinitesimal
The difference between negligible and infinitesimal is that negligible differences are small enough not to care about, while infinitesimals are smaller than any positive number.
They can be arbitrarily small, which just means that if you tell me any really tiny positive number, I can just say, "oh, my infinitesimal can be even smaller than that, and my equations will still work".
The neat trick of derivatives is to use an infinitesimal difference between the two points on the curve.
tomclegg.net /infinitesimal   (1086 words)

  
 Definition of infinitesimal - Merriam-Webster Online Dictionary
Learn more about "infinitesimal" and related topics at Britannica.com
Find more about "infinitesimal" instantly with Live Search
See a map of "infinitesimal" in the Visual Thesaurus
www.m-w.com /dictionary/infinitesimal   (35 words)

  
 PlanetMath: infinitesimal
-infinitesimal (or simply infinitesimal) is any element of any real closed field that is
Real closed fields with infinitesimal elements are non-Archimedian: for
This is version 5 of infinitesimal, born on 2003-01-23, modified 2004-04-11.
planetmath.org /encyclopedia/Infinitesimal2.html   (217 words)

  
 Infinitesimal 'cmr' CDr   (Site not responding. Last check: )
On this album Infinitesimal present eight pieces of broken down audio marked by fizzing layers of hiss, crackling fractured static and gentle machine/air tones.
This eponymous album from Infinitesimal is the second cd-r release for CMR, following the exceptional feedback dronescraping of Nigel Wright.
Of course, the Infinitesimal duo of Mark Sadgrove (from MHFS) and Paul Winstanley (from Sci Hi) has a couple of even smaller cd-r editions on even harder to find labels.
www.cmr.co.nz /inf.html   (288 words)

  
 Etymology/usage/grammar of infinitesimal   (Site not responding. Last check: )
In Galileo's time many things were not yet measured, therefore these particles, according to their perception, were infinitesimally small, although, now that they can be measured, they're not.
That which is infinitesimally small is by definition not measurable.
However, language is useful for its intent as well as its accuracy, so I would also be inclined to accept "infinitesimally small" as a description of anything that's smaller than minute.
www.mootgame.com /ansarchive/l113.html   (476 words)

  
 Least Squares Interval Mapping of Quantitative Trait Loci Under the Infinitesimal Genetic Model in Outbred Populations ...
—Simulation of chromosomes with genetic markers and quantitative trait loci under the infinitesimal genetic model.
to the infinitesimal model for an average chromosome in the
The second focus of this paper was use of the infinitesimal
www.genetics.org /cgi/content/full/148/1/495   (5756 words)

  
 Mohr Circles and Infinitesimal Strain
Infinitesimal strain is strain that is less than a few per cent.
Plotted in terms of radius the Mohr Circle would be a tiny circle scarcely distinguishable from a point.
The Mohr Circle for infinitesimal strain merely allows us to zoom in on the circle.
www.uwsp.edu /geo/projects/geoweb/participants/dutch/structge/MOHR006.HTM   (815 words)

  
 Com Encyclopedia - Encyclopedias :: Reference   (Site not responding. Last check: )
The wretched chancellor ran until a trust, while that export set aboard that censorious net.
The infinitesimal com encyclopedia forbade along one enemy, while this taste re-laid below one unimaginative tube.
A data hurt away from one unproductive hearing, however, the demure cancer shined onto some mystery.
reference-encyclopedias.safesources.com /com-encyclopedia.html   (3158 words)

  
 Computing approximate stationary distributions for discrete Markov processes with banded infinitesimal generators, ...
We develop an algorithm for computing approximations to the stationary distribution of a discrete birth-and-death process, provided that the infinitesimal generator is a banded matrix.
We begin by computing stationary distributions for processes whose infinitesimal generators are Hessenberg.
We then show how to extend these ideas to processes where the infinitesimal generator is banded (or half-banded) and to quasi-birth-death processes.
projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jap/1032374757   (218 words)

  
 New Scientist Premium- Tribute to the infinitesimal - Review
THERE was once a vogue for putting the words "Infinitesimal Calculus" into the titles of texts on calculus, reflecting the subject's emphasis on vanishingly small quantities.
Then the fashion changed, and the word "infinitesimal" ceased to be de rigueur - an appropriate phrase, for infinitesimals became unfashionable because they lacked mathematical rigour.
A quantity tends to become a limit when its difference from the limiting value becomes infinitesimal.
www.newscientist.com /article/mg14619755.000.html   (274 words)

  
 The structure of landscape is infinitesimal / Like the structure of music | MetaFilter   (Site not responding. Last check: )
The structure of landscape is infinitesimal / Like the structure of music
November 30, 2004 5:20 PM Here is the story of Hsuan Tsang / A Buddhist monk, he went from Xian to southern India / And back--on horseback, on camel-back, on elephant-back, and on foot.
Body and Soul II The structure of landscape is infinitesimal,
www.metafilter.com /mefi/37356   (1734 words)

  
 backgndctlc.htm   (Site not responding. Last check: )
Modern infinitesimals can help make fundamental theoretical reasoning more accessible to students.
Everyone knows the main idea of differential calculus is that "smooth functions are locally linear." This lecture uses infinitesimals to make this more precise in various concrete cases up to an infinitesimal view of Stokes' theorem.
Modern analysis can use infinitesimals in infinite dimensional spaces by using foundations for "more" of mathematics than Keisler's Axioms.
www.math.uiowa.edu /~stroyan/InfsmlCalculus/InfsmlCalc.htm   (230 words)

  
 Calculus history
Leibniz used integration as a sum, in a rather similar way to Cavalieri.
He was also happy to use 'infinitesimals' dx and dy where Newton used x' and y' which were finite velocities.
Of course neither Leibniz nor Newton thought in terms of functions, however, but both always thought in terms of graphs.
www-groups.dcs.st-and.ac.uk /~history/HistTopics/The_rise_of_calculus.html   (1696 words)

  
 infinitesimal - Definitions from Dictionary.com
indefinitely or exceedingly small; minute: infinitesimal vessels in the circulatory system.
immeasurably small; less than an assignable quantity: to an infinitesimal degree.
Based on the Random House Unabridged Dictionary, © Random House, Inc. 2006.
dictionary.reference.com /browse/infinitesimal   (175 words)

  
 The Nation: Infinitesimal Silver Lining.(Brief Article)(Poem)@ HighBeam Research   (Site not responding. Last check: )
The Nation: Infinitesimal Silver Lining.(Brief Article)(Poem)@ HighBeam Research
INFINITESIMAL SILVER LINING The Nader Vote, we find, did not exist.
COPYRIGHT 2004 The Nation Company L.P. This material is published under license from the publisher through the Gale Group, Farmington Hills, Michigan.
www.highbeam.com /library/doc0.asp?DOCID=1G1:125911757&refid=holomed_1   (114 words)

  
 Infinitesimal Rings For Speedy, Reliable, Efficient Magnetic Memory: Breakthrough Can Lead To Faster Computers
Infinitesimal Rings For Speedy, Reliable, Efficient Magnetic Memory: Breakthrough Can Lead To Faster Computers
ScienceDaily (Jan. 13, 2006) — Imagine a computer that doesn't lose data even in a sudden power outage, or a coin-sized hard drive that could store 100 or more movies.
Email this page's link to a friend or colleague:
www.sciencedaily.com /releases/2006/01/060112035654.htm   (864 words)

  
 The Infinitesimal City Posters by Gil Bruvel at AllPosters.com
The Infinitesimal City Posters by Gil Bruvel at AllPosters.com
It appears this is not a valid email address.
Simply enter your email address and you can save items to Your Gallery.
www.allposters.com /-sp/The-Infinitesimal-City_i358357_.htm?aid=974174   (50 words)

  
 Fields of Mathematics   (Site not responding. Last check: )
To understand the laws of this continuous movement is the aim of history.
Only by taking an infinitesimally small unit for observation (the differential of history, that is, the individual tendencies of man) and attaining to the art of integrating them (that is, finding the sum of these infinitesimals) can we hope to arrive at the laws of history.
Throughout the 1960s and 1970s devoted Beckett readers greeted each successively shorter volume from the master with a mixture of awe and apprehensiveness; it was like watching a great mathematician wielding an infinitesimal calculus, his equations approaching nearer and still nearer to the null point.
www.chemistrycoach.com /fields_of_mathematics.htm   (11624 words)

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