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	CAUCHY, A.L.(1789-1857)
		Cauchy was born in Paris in 1789 and recceived his early education from his father.
		His name is net by the student of calculus in the so-called Cauchy root test and Cauchy ratio test for convergence or divergence of a series of positive terms, and and in the Cauchy ratio test of two given series.
		Cauchy's work exhibits great attention to rigor, and as such was largely reaponsible for inspiring other mathematicians to attempt the banishment of blind formal manipulation and of intuitive proofs from analysis.
	library.thinkquest.org /22584/temh3041.htm (493 words)

	Net (mathematics) - Wikipedia, the free encyclopedia
		In topology and related areas of mathematics a net or Moore-Smith sequence is a generalization of a sequence, intended to unify the various notions of limit and generalize them to arbitrary topological spaces.
		Nets generalize this concept by weakening the order relation on the index set to that of a directed set.
		This may be useful to guide the intuition since the notion of limit of a net is very similar to that of limit of a sequence.
	en.wikipedia.org /wiki/Net_(mathematics) (1168 words)

	Cauchy sequence: Definition and Links by Encyclopedian.com
		Let (x n) be an arbitrary Cauchy sequence in E; let F n be the closure of the set {x k : k >= n} in E...point of a Cauchy sequence is a limit point (x n); hence any Cauchy sequence in E...
		In mathematical analysis, a Cauchy sequence is a sequence whose terms become arbitrarily close to each other as the sequence progresses.
		A metric space in which every Cauchy sequence has a limit is called complete.
	www.encyclopedian.com /ca/Cauchy-net.html (535 words)

	Encyclopaedia Britannica entry (Site not responding. Last check: 2007-10-25)
		Cauchy became a military engineer and in 1810 went to Cherbourg to work on the harbours and fortifications for Napoleon's English invasion fleet.
		Cauchy returned to Paris in 1813, and Lagrange and Laplace persuaded him to devote himself entirely to mathematics.
		Cauchy made substantial contributions to the theory of numbers and wrote three important papers on error theory.
	www.aam314.vzz.net /EB/Cauchy.html (455 words)

	Cauchy sequence - Wikipedia, the free encyclopedia
		In constructive mathematics, Cauchy sequences often must be given with a modulus of Cauchy convergence to be useful.
		The converse (that every Cauchy sequence has a modulus) follows from the well-ordering property of the natural numbers (let α(k) be the smallest possible N in the definition of Cauchy sequence, taking r to be 1 / k).
		Any Cauchy sequence with a modulus of Cauchy convergence is equivalent (in the sense used to form the completion of a metric space) to a regular Cauchy sequence; this can be proved without using any form of the axiom of choice.
	en.wikipedia.org /wiki/Cauchy_sequence (1060 words)

	Navier-Stokes Equations: Cauchy's Integral and the Distortion of Vorticity (Site not responding. Last check: 2007-10-25)
		Equation (Vi2) is known as Cauchy's integral of the vorticity transport equation and provides us with an exact, explicit solution for the variation of the vorticity of a material blob.
		As mentioned in the introduction to our discussion of vorticity, one of the motivations for examining vorticity dynamics is to determine the conditions under which it vanishes.
		Later Cauchy (1815) provided the exact integral (Vi2a) for the case of constant density motions from which the result at the left follows immediately.
	www.navier-stokes.net /nsvci.htm (918 words)

	Cauchy sequence - Article from FactBug.org - the fast Wikipedia mirror site
		In mathematical analysis, a Cauchy sequence, named after Augustin Cauchy, is a sequence whose elements become close as the sequence progresses.
		Cauchy sequences require the notion of distance so they can only be defined in a metric space.
		They are of interest because in a complete space, all such sequences converge to a limit, and one can test for "Cauchiness" without knowing the value of the limit (if it exists), in contrast to the definition of convergence.
	www.factbug.org /cgi-bin/a.cgi?a=6085 (598 words)

	Analytic Functions, The Magnus Effect, and Wings
		From examining the streamlines around the cylinder, it might seem surprising that any net lift is produced, because the air upstream of the cylinder is pulled upward along mirror images of the downward paths of air downstream of the cylinder.
		Well, in fact there is a net downward change in the momentum of the air, which we can see by considering a circular "control volume" of radius R surrounding the cylinder.
		Of course, this assumes the circulation of the flow extends indefinitely, whereas in fact the circulation flow breaks down at sufficient distances, and the organized pressure gradients are gradually dissipated through exchanges of momentum between neighboring parts of the fluid.
	www.mathpages.com /home/kmath258/kmath258.htm (3185 words)

	Cauchy distribution (Site not responding. Last check: 2007-10-25)
		As a probability distribution, it is known as the Cauchy distribution while among physicists it is known as the Lorentz distribution or the Breit-Wigner distribution.
		The Cauchy distribution is often cited as an example of a distribution which has no mean, variance or higher moments defined, although its mode and median are well defined and are both equal to x
		The Cauchy distribution is an infinitely divisible probability distribution.
	cauchy-distribution.iqnaut.net (632 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Cauchy's Residue Theorem is helpful to compute difficult definite integrals.
		Therefore, we'll choose an example of a sequence in the the field of rationals (a notoriously incomplete space, as was first glimpsed by a disciple of Pythagoras about 2500 years ago).
		However, it is more or less universally understood that the Cauchy principal value is used whenever needed, and some authors don't bother to insist on this with special typography.
	home.att.net /~numericana/answer/analysis.htm (4095 words)

	[No title] (Site not responding. Last check: 2007-10-25)
		The posterior moments of parameters specifying distributions are minimum mean square error Bayesian estimators for the corresponding moments of those parameters, and as such are ubiquitous in the Bayesian approach to statistical inference of distributions.
		The cauchy distribution is most notable for its wide tails, decided absence of high-order moments, and non-existence of less-than-data dimension sufficient statistics.
		In this paper the posterior moments of the position parameter of the Cauchy distribution are found in closed form.
	www.realtime.net /~drwolf/pages/momcauchy98.html (154 words)

	Sketching the History of Hypercomplex Numbers (Site not responding. Last check: 2007-10-25)
		The memoirs of Augustin-Louis Cauchy (1789-1857) give the first clear theory of functions of a complex variable.
		Cauchy publishes Memoire sur les integrales definies, prises entre des limites imaginaires.
		Cauchy shows that an analytic function of a complex variable can be expanded about a point in a power series in the neighborhood of the singularity.
	history.hyperjeff.net /hypercomplex.html (1939 words)

	PlanetMath: filter
		Also, the two kinds of limit that one sees in elementary real analysis - the limit of a sequence at infinity, and the limit of a function at a point - are both special cases of the limit of a filter: the
		The notion of a Cauchy sequence can be extended with no difficulty to any uniform space (but not just a topological space), getting what is called a Cauchy filter; any convergent filter on a uniform space is a Cauchy filter, and if the converse holds then we say that the uniform space is complete.
		Cross-references: complete, convergent, uniform space, Cauchy sequence, Fréchet filter, function, infinity, sequence, real, limit, Tychonoff's theorem, cluster point, accumulation point, meets, converge, neighbourhoods, topological space, point, proper subset, finite, axiom, intersection, subsets
	planetmath.org /encyclopedia/filter.html (443 words)

	Chaos and Fractals in Financial Markets, Part 4, by J. Orlin Grabbe
		Henry’s net winnings in dollars, then, are the total number of heads minus the total number of tails.
		Note that at –3, for example, the probability density of the Cauchy distribution is g(-3) =.0318, while for the normal distribution, the value is f(-3) =.0044.
		Similarly, for the Cauchy distribution the standard deviation (or the variance, which is the square of the standard deviation) doesn’t exist.
	www.aci.net /kalliste/Chaos4.htm (2704 words)

	Cauchy sequences and completeness of reals. -- Mathology.net (Site not responding. Last check: 2007-10-25)
		By theorem 4 every Cauchy sequence is bounded.
		By theorem 6 we already know that a Cauchy sequence is convergent.
		But this is the definition of Cauchy sequence.
	www.mathology.net /mathology/vis_articolo.asp?id=44&lang=ita (311 words)

	Building convex polytopes
		Equivalently, a polyhedral metric is obtained by gluing together edges of a simple polygon together in equallength pairs, so that the resulting 2complex complex is homeomorphic to a sphere; the metric is convex if the sum of the angles incident to each vertex is at most 2pi.
		It is open whether every polytope can be unfolded into a simple net, that is, one that does not overlap itself, by cutting along edges; however, even nonsimple nets can be used to define polyhedral metrics.
		As with Cauchy's Rigidity Theorem, there are numerical approximation algorithms, but I'm looking for a purely combinatorial algorithm.
	compgeom.cs.uiuc.edu /~jeffe/open/makepoly.html (1455 words)

	N-body problem, and Cauchy Sequences - GameDev.Net Discussion Forums
		In Euclidian space, a sequence is a Cauchy sequence iff its converging.
		I hear the real numbers are constructed by Cauchy sequences, and I can see why.
		Indeed, real numbers can be constructed from rational numbers using Cauchy sequences.
	www.gamedev.net /community/forums/topic.asp?topic_id=374716 (643 words)

	Hausdorff space (Site not responding. Last check: 2007-10-25)
		Limits of sequences, nets, and filters (when they exist) are unique in Hausdorff spaces.
		Similarly, a space is preregular iff all of the limits of a given net (or filter) are topologically indistinguishable.
		Specifically, a space is complete iff every Cauchy net has at least one limit, while a space is Hausdorff iff every Cauchy net has at most one limit (since only Cauchy nets can have limits in the first place).
	hausdorff-space.kiwiki.homeip.net (918 words)

	Navier-Stokes Equations: Cauchy's Hypothesis and Theorem (Site not responding. Last check: 2007-10-25)
		The key to further simplifications is Cauchy's Hypothesis and Theorem which permit us to replace t and Q by the stress tensor T and the heat flux vector q, respectively.
		Result (5) is known as Cauchy's Theorem for the existence of the stress tensor and the heat flux vector.
		The ability to represent the force and heat flow at a surface independently of the orientation of the surface will permit us to write our governing equations in a local form ultimately leading to a relatively simple set of governing equations.
	www.navier-stokes.net /nsch.htm (376 words)

	Newran - random number generator library
		Generates random numbers from a standard Cauchy distribution.
		Cauchy C; for (int i=0; i<100; i++) cout << C.Next() << "\n";
		This class has been augmented to ensure only one copy of the arrays generated by the constructor exist at any given time.
	www.robertnz.net /nr02doc.htm (2816 words)

	[No title]
		] The Cauchy data for a hyperbolic partial differential equation consist of the value of the field and its time derivative on some spacelike surface.
		] A net whose members are elements of a topological vector space and which satisfies the condition that for any neighborhood of the origin of the space there is an element a of the directed system that indexes the net such that if b and c are also members of this directed system and b
		is bounded on an interval (a,b) except in the neighborhood of a point c, the Cauchy principal value of
	www.accessscience.com /Dictionary/C/C12/DictC12.html (1783 words)

	Mathematics Other Homework Help
		Problem: State the definition of a Cauchy sequence.
		Also, by negating and definition, state the definition of a sequence to be not Cauchy.
		An investment project is expected to generate earnings before taxes of $60,000 per year.
	www.brainmass.com /homeworkhelp/math/other/pg96 (310 words)

	TenSen.net (Site not responding. Last check: 2007-10-25)
		So the analogy is, the flux capacitor is to time travel as the Cauchy-Goursat Theorem is to complex analysis.
		It's true, just go ahead and try to do something in complex analysis which does not either depend on this theorem or is motivated by it.
		So now this is the second mathematical result involving Cauchy that I have found that relates directly to American cinema.
	www.tensen.net (368 words)

	[No title]
		Next, it is shown that over all closed bounded intervals in any monotone incomplete ordered field, there are continuous not uniformly continuous unbounded functions whose ranges are not closed, and continuous 1-1 functions which map every interior point to an interior point (of the image) but are not open.
		These are achieved using appropriate nets cofinal in gaps or coinitial in their complements.
		In our third main theorem, an ordered field is constructed which has parametrically definable regular gaps but no $\emptyset$-definable divergent Cauchy functions (while we show that, in either of the two cases where parameters are or are not allowed, any definable divergent Cauchy function gives rise to a definable regular gap).
	www.emis.ams.org /proceedings/TopoSym2001/20.tex (1041 words)

	ORDINAL REAL NUMBERS 1
		This is the third paper of a series of five papers that have as goal the definition of topological complete linearly ordered fields (continuous numbers) that include the real numbers and are obtained from the ordinal numbers in a method analogous to the way that Cauchy derived the real numbers from the natural numbers.
		It is actually the same ideas that lead to the process of construction of the real numbers from the natural numbers through fundamental (Cauchy) sequenses.
		The characteristic of the (strong) Cauchy completion of a linearly ordered field F,is the same with that of the field F. Proof.
	www.softlab.ntua.gr /~kyritsis/PapersInMaths/InfinityandStochastics/OR1.htm (3795 words)

	Smyth Completeness In Terms Of Nets: The General Case (ResearchIndex)
		Smyth completeness is the appropriate notion of completeness for quasi-uniform spaces carrying an additional topology to serve as domains of computation [2, 3].
		The goal of this paper is to provide a better understanding of Smyth completeness by giving a characterization in terms of nets.
		We develop the notion of computational Cauchy net and an appropriate notion of strong convergence to get the result that a space is Smyth complete if and only if every computational Cauchy net...
	citeseer.ist.psu.edu /254038.html (318 words)

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