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Topic: Infinitely divisible

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	Infinite divisibility - Wikipedia, the free encyclopedia
		By contrast, the ring of integers is not infinitely divisible.
		The Poisson distributions, the normal distributions, and the gamma distributions are infinitely divisible probability distributions.
		This concept of infinite divisibility of probability distributions was introduced in 1929 by Bruno de Finetti.
	en.wikipedia.org /wiki/Infinite_divisibility (1274 words)

	The Internet Classics Archive \| Physics by Aristotle
		Moreover, it is plain that everything continuous is divisible into divisibles that are infinitely divisible: for if it were divisible into indivisibles, we should have an indivisible in contact with an indivisible, since the extremities of things that are continuous with one another are one and are in contact.
		That infinite time will not be occupied in passing over BE is evident if the time be taken as limited in one direction: for as the part will be passed over in less time than the whole, the time occupied in traversing this part must be finite, the limit in one direction being given.
		And we now see that in most cases the fact that all the terms are divisible or infinite is a direct consequence of the fact that the thing that changes is divisible or infinite: for the attributes 'divisible' and 'infinite' belong in the first instance to the thing that changes.
	classics.mit.edu /Aristotle/physics.6.vi.html (7134 words)

	Atomism and Infinite Divisibility - Chapter 4 - Aristotle on Infinite Divisibility
		Infinite divisibility had already long been a topic of philosophical discussion before The Philosopher came on the scene.
		(Infinite is an attribute predicated of divisibility and divisibility is an attribute predicated of magnitude.) Once the infinite is predicated of processes, and limited to the process of addition and division at that, the subjects of the processes come under examination.
		Since his purpose was not to clarify infinite divisibility itself but only to clarify it in relation to its use in accounting for coming to be, alteration, growth, and undergoing the contrary of these, he stopped short of unearthing the contradiction.
	www.xenodochy.org /rekphd/chapter4.html (8923 words)

	Paradoxes Resolved, Origins Illuminated - Formal Logic and Scientific Method
		If assuming both finite or infinite divisibility in turn leads to contradictions, then it is a matter of personal preference since there is no way to scientifically distinguish them.
		Again, all the arguments leading to infinite divisibility appear to arise from a particular geometry, which carries with it a set of assumptions.
		My reference to infinite divisibility leading to an universe that is completely occupied by matter from the infinitely large to the infinitesimal comes straight from Dr. Flandern's Meta Model, so if you have problems with it, you should take it up with him.
	www.metaresearch.org /msgboard/topic.asp?TOPIC_ID=478&whichpage=2 (5033 words)

	Atomism and Infinite Divisibility - Chapter 1 - Preliminary Notions
		Division into smaller portions is most often achieved by a process of repeated or successive division.
		Since division is a process that is applied to something, an immediate dichotomy concerning the question is possible.
		It is from the former that the name 'infinite divisibility' derives and is given to the view opposing atomism.
	www.xenodochy.org /rekphd/chapter1.html (3926 words)

	zeno
		In other words, to get a refined sense of what it is to be infinitely divisible is to solve these paradoxes.
		There are, then, two ways for a magnitude to contain an infinity of points: to be infinitely long (that would certainly do it) or to be finitely long but infinitely divisible.
		But what seems conceptually muddled about this is that for something to be infinitely divisible is not for there to be a point at which we have an infinite number of finite intervals.
	www.smcm.edu /users/mstaber/zeno.htm (1969 words)

	Infinitely divisible representations of Lie algebras; by R. F. Streater; non-commutative probability; non-abelian ... (Site not responding. Last check: 2007-10-13)
		We relate the classification of infinitely divisible representations to the existence of a one-cocycle for the Lie algebra, related to the method of Araki by taking the group elements to be close to the identity.
		In lectures at MaPhySto, Aarhus, I explain that cocycles of the Lie algebra R are one-to-one to the class of infinitely divisible measures defined by Kolmogorov, whereas for the Lie group R, the cocycles are one-to-one with the larger class of Levy measures.
		It is shown that any infinitely divisible representation of a Lie algebra (or of the enveloping algebra) is associated with a conditionally positive semi-definite form on the algebra.
	www.mth.kcl.ac.uk /~streater/divireps.html (357 words)

	Paradoxes Resolved, Origins Illuminated - Planck limits
		They are the mathematical tools by which we observe and measure the real and tangible, and they are infinite by postulate, and by the lack of any intelligible meaning to finiteness for dimensions.
		The meaning of "infinite" for dimensions is simple: We cannot name any finite limit to any dimension, however large, and be assured that we will never find an example of a real, tangible substance or property that will exceed the limit.
		If infinite divisibility is possible then construction of an encoder with infinite holes should be possible in theory and such an encoder would prohibit motion.
	www.metaresearch.org /msgboard/topic.asp?TOPIC_ID=193 (6723 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		The class of infinitely divisible distributions is shown to provide useful formulations for problems involving heavy-tailed distributions and for problems involving convolutions.
		Secondly, a generalized, parametric theory of multivariate statistical analysis based on infinitely divisible distributions is outlined.
		Because the infinitely divisible class includes the normal family, this theory is more general than that based on multivariate normal distributions.
	www.uic.edu /classes/idsc/ids594/my_abstracts/abstr18.html (119 words)

	Review: Beyond Space and Time
		It is the non-physical Sector 3 that constitutes the ethical human realm an infinite continuous whole of natural existence in distinction from the quantized or finitely divisible physical realms, from which infinity is excluded.
		When counting an infinite whole, Thomas Paine and Georg Cantor have taught us to identify the proper part to establish that the whole is infinite and not finite.
		The total human worth of the humankind, the dead, the presently living, and the yet to be born, is infinitely greater than the total value of all the commodities presently in the global market.
	www.reciprocalsystem.com /isus/articles/revbst.htm (883 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		If space is infinitely divisible, so is time, as can be seen from the nature of motion, so if time is not infinitely divisible, neither is space.
		A contrary demonstration is not a mere difficulty, as would be claimed by the defenders of infinite divisibility.
		A further consequence is that all pretended demonstartions of infinite divisiblity are sophistical, because they would have to prove that mathematical points are impossible.
	www-philosophy.ucdavis.edu /mattey/phi174/TB1P2S2.HTM (448 words)

	On the Infinite Divisibility and Composition of One-Dimensional Continua: an Anc
		On the Infinite Divisibility and Composition of One-Dimensional Continua: an Anc
		On the Infinite Divisibility and Composition of One-Dimensional Continua: an Ancient and Medieval Prespective
		Division everywhere is the concept of division used by
	www.vivboard.net /doc/n0052.htm (4363 words)

	Zeno's race course, part 2
		Then the time required for all of the infinite sequence of tasks would be only twice that required for the first.
		Suppose now that the lamp is off, and I succeed in pressing the button an infinite number of times, perhaps making one jab in one minute, another jab in the next half minute, and so on, according to Russell’s recipe.
		This is because G is the limit point of the infinite sequence of Z-points.
	faculty.washington.edu /smcohen/320/zeno3.htm (1736 words)

	Continuity and Infinitesimals
		In attacking infinite divisibility the atomists were at the same time mounting a claim that the continuous is ultimately reducible to the discrete, whether it be at the physical, theoretical, or perceptual level.
		In that case, the Thesis implies that the sequence of divisions is finite; the Antithesis, that it is infinite.
		Cantor's analysis of the continuum in terms of infinite point sets led to his theory of transfinite numbers and to the eventual freeing of the concept of set from its geometric origins as a collection of points, so paving the way for the emergence of the concept of general abstract set central to today's mathematics.
	plato.stanford.edu /entries/continuity (16772 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		Measurement of dependence in the infinitely divisible class of multivariate distributions, based on developments in probability theory for that class, is discussed.
		When the infinitely divisible variables contain no normal component (in particular, when they are discrete), the cumulant of order (2,2) can be used as a measure of pairwise dependence; when a normal component is present, the appropriate measure also involves the covariance.
		A method of testing normality against infinitely divisible alternatives is given.
	www.uic.edu /classes/idsc/ids594/my_abstracts/abstr20.html (110 words)

	Wilmott Forums - is beta distribution infinitely divisible? (Site not responding. Last check: 2007-10-13)
		The Beta-distribution lives on the interval [0,1] so it is impossible to be infinitely divisible.
		Any distribution on a bounded interval can not be infinitely divisible, other examples are the uniform and the Binomial one.
		be infinitely decomposable, leading to the concept of a triangular array, defining infinite divisibility: A random variable X is said to be infinitely divisible if there exists a triangular array X11 ; X21, X22; X31, X32, X33;.
	www.wilmott.com /messageview.cfm?catid=4&threadid=7601 (222 words)

	A Treatise of Human Nature, by David Hume (chapter9) (Site not responding. Last check: 2007-10-13)
		The infinite divisibility of space implies that of time, as is evident from the nature of motion.
		It is likewise certain that this idea, as conceived by the imagination, though divisible into parts or inferior ideas, is not infinitely divisible, nor consists of an infinite number of parts: For that exceeds the comprehension of our limited capacities.
		These consequences we may carry one step farther, and conclude that all the pretended demonstrations for the infinite divisibility of extension are equally sophistical; since it is certain these demonstrations cannot be just without proving the impossibility of mathematical points; which it is an evident absurdity to pretend to.
	etext.library.adelaide.edu.au /h/hume/david/h92t/chapter9.html (1383 words)

	Handout—Class 5—July 24th (Site not responding. Last check: 2007-10-13)
		The axiom in geometry that any finite extension is infinitely divisible generates paradoxes and difficulties.
		So, if a finite extension is infinitely divisible then I must perceive infinite parts in my idea of the finite extension I am considering — that is, I must perceive infinite parts in it.
		So, infinite parts are not contained in the extension — finite extension is not infinitely divisible.
	ruccs.rutgers.edu /~rafaeua/18thcentury/h5.htm (1115 words)

	SikhSpectrum.com Monthly. Time Reality and Religion
		here is, in a sense, a correspondence between the idea of infinite divisibility of time of classical materialism and its notion of infinite velocity of cosmic action.
		As infinite time and infinite space are both devoid of "content", so their being the aspects of Brahman would not make the latter a determinate Being.
		Thus the ultimate reality, having infinite time and infinite space as its aspects, retains its abstractness as well as its timelessness, that is, its unchangeability.
	www.sikhspectrum.com /052003/time_jasbir.htm (6288 words)

	Current Commutation Relations, Continuous Tensor Products and Infinitely Divisible Group Representations, by R. F. ...
		The theory can be easily reformulated in terms of infinitely divisible positive-definite functions on groups, namely, the expectation of the unitary operators representing the group in the cyclic vector (known as the characteristic function of the cyclic representation), which is always a positive semi-definite continuous function on the group.
		The concept of infinite divisibility in probability was introduced by di Finetti, and the classification of these was achieved by Lévy.
		He avoided the use of the term infinite divisibility, and analysed the more general problem of all factorisable representations of current algebra.
	www.mth.kcl.ac.uk /~streater/infidivi.html (631 words)

	Volume 22, Number 1, 1996 (Site not responding. Last check: 2007-10-13)
		A description of all random variables \nu admitting an analog of the Gaussian distribution under \nu-summation, that is, the summation of a random number \nu of random terms, is given.
		The \nu-infinitely divisible distributions are described for these \nu-summations and finite estimates of the approximation of \nu-sum distributions with the help of \nu-accompanying infinitely divisible distributions are given.
		The results include, in particular, the description of geometrically infinitely divisible and geometrically stable distributions as well as their domains of attraction.
	www.math.bas.bg /~serdica/n4_96.html (664 words)

	Another trick question! (Site not responding. Last check: 2007-10-13)
		Just because you can prove you've got an infinite number of time intervals to run through, doesn't meant that the sum of these (the total time) is infinite.
		Aristotle's response to that one in particular was that there are two senses in which something can be infinite: in divisibility, or in extent.
		In a finite length of time someone can come into contact with things infinitely divisible and so a finite length of time can be taken to cover a finite length.
	www.physicsforums.com /showthread.php?p=235032 (1082 words)

	Re: Infinitely divisible particle (was Congregating bosons/fermions) (Site not responding. Last check: 2007-10-13)
		Similarly one might have > > infinitely many fermions with infinitely many sublattices or just one > > field describing infinitely many fermions interacting in a strange > > way.
		By macroscopically identical I mean that the expectation values of the Dirac bispinor components and the expectation values of scalar field are identical on all four sublattices.
		Prev by thread: Re: Infinitely divisible particle (was Congregating bosons/fermions)
	www.lns.cornell.edu /spr/2002-07/msg0043079.html (444 words)

	RAND \| Research Memoranda \| Infinitely Divisible Distributions, Conditions for Independence, and Central Limit Theorems.
		Infinitely Divisible Distributions, Conditions for Independence, and Central Limit Theorems.
		If a random variable is the sum of a large number of small, statistically independent components, it can under certain conditions be approximated by the normal distribution.
		When these conditions cannot be obtained, the random variable can be approximated by a member of a broader class of infinitely divisible (I.D.) random variables.
	www.rand.org /pubs/research_memoranda/RM6042 (301 words)

	[No title] (Site not responding. Last check: 2007-10-13)
		infinitely divisible and lots of others are not.
		cumulant is negative fails to be infinitely divisible.
		Is it within the literature on infinite divisibility, being
	mathforum.org /kb/plaintext.jspa?messageID=3742766 (309 words)

	Abstracts (Site not responding. Last check: 2007-10-13)
		Continuity of stochastic integrals with respect to infinitely divisible random measures, (with J. Rosinski).
		Sufficient conditions for boundedness and continuity are obtained for stochastically continuous infinitely divisible processes, without Gaussian component, $ \{Y(t), t\in T\}$, where $T$ is a compact metric space or pseudo-metric space.
		The methods developed are also used to show that certain infinitely divisible random fields are bounded.
	home.earthlink.net /~mbmarcus/abstract.html (458 words)

	Chaper 2, Section 1 (Site not responding. Last check: 2007-10-13)
		Most of these philosophers, including Plato and Aristotle, believed that matter was infinitely divisible.
		Despite Democritus's proposition, however, the erroneous idea that matter could be divided infinitely was widely believed until the early nineteenth century.
		As these scientists discovered patterns of reactivity that seemed inconsistent with the idea of infinitely divisible matter, Democritus's notion of atoms reemerged.
	cwx.prenhall.com /bookbind/pubbooks/blb/chapter2/medialib/blb0201.html (627 words)

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