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Topic: Summability criterion

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	PlanetMath: Cesàro summability
		Cesàro summability is a generalized convergence criterion for infinite series.
		Cesàro summability is a generalization of the usual definition of the limit of an infinite series.
		This is version 3 of Cesàro summability, born on 2002-10-29, modified 2002-10-29.
	planetmath.org /encyclopedia/CesaroSummability.html (157 words)

	PlanetMath: Abel summability
		Abel summability is a generalized convergence criterion for power series.
		The standard example of a divergent series that is nonetheless Abel summable is the alternating series
		This is version 3 of Abel summability, born on 2002-10-30, modified 2006-09-30.
	planetmath.org /encyclopedia/AbelianTheorem.html (388 words)

	an informal discussion of proof issues
		Criterion which are only satisfied by Condorcet: (2) Condorcet Criterion (3) Generalized Condorcet Criterion Quorum rule does not satisfy these criterion.
		It would be interesting to fully define the cases where these criterion are satisfied even with supermajority and the cases where these criterion are not satisfied.
		(I) Summability Criterion Supermajority satisfies this criterion, using a mechanism analogous to that which was used for quorum: add additional rows and columns to the array to record supermajority requirements, and add 0 to this part of the array for each vote.
	lists.debian.org /debian-vote/2002/12/msg00059.html (595 words)

	Summability criterion - Electowiki (Site not responding. Last check: 2007-11-06)
		Each vote should map onto a summable array, where the summation operation is associative and commutative, and the winner should be determined from the array sum for all votes cast.
		Under first-order summable methods like plurality or Approval voting, the votes at any level (precinct, ward, county, etc.) can be compressed into a list of ten numbers.
		In first-order summable election systems, adding new ballots to the count (say, ballots that were found after the initial count, or late absentee ballots, or ballots that were initially ruled invalid) is as simple as "summing" the original result with the newly-found ballots.
	wiki.electorama.com /wiki/Summability_criterion (734 words)

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	30.1 Introduction
		First you may be able to assign a value to such a series even if it fails to converge, by using the notion of summability.
		It is said to be C1 summable if the sequence of averages of its first k partial sums converges.
		summability: the sequence of averages of partial sums of a series defines a series which has them as its partial sums; if this series is C
	ocw.mit.edu /ans7870/18/18.013a/textbook/HTML/chapter30/section01.html (507 words)

	NSDL Metadata Record -- Ces\`aro summability
		Ces\`aro summability is a generalized convergence criterion for infinite series.
		Ces\`aro summability is a generalization of the usual definition of the limit of an infinite series.
		The standard example of a divergent series, that is nonetheless Ces\`aro summable is...
	nsdl.org /mr/1031640 (113 words)

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	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		Thus, the investigation of the convergence of singular integrals bears on questions of convergence and summability of trigonometric Fourier series; Fourier series in orthogonal polynomials, and also expansions in general orthogonal systems.
		Lebesgue established a criterion for the convergence of singular integrals in the case of continuous functions
		Since the conditions of Lebesgue and Faddeev are difficult to verify for concrete singular integrals, a whole series of papers was devoted to the search for effective sufficient conditions for the convergence of singular integrals, both at isolated points and uniformly.
	eom.springer.de /S/s085570.htm (345 words)

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	Borda count - Article from FactBug.org - the fast Wikipedia mirror site
		The Borda count satisfies the monotonicity criterion, the consistency criterion, the summability criterion, and the participation criterion.
		It does not satisfy the Condorcet criterion, the weak defensive strategy criterion, or various related criteria.
		As compared with the Borda count, however, Instant Borda Runoff does not satisfy the Summability criterion, the Monotonicity criterion, the Participation criterion, and the Consistency criterion, while it does satisfy the Majority criterion, and the Independence of Clone Candidates criterion.
	www.factbug.org /cgi-bin/a.cgi?a=4155 (1602 words)

	MIT OpenCourseWare, Department Template
		The j-th partial sum of a series is defined to be the sum of its first j terms.
		A sequence is said to converge to value A if, for any positive criterion c, there is an n such that all the terms in the sequence after the n
		You can go further by defining C2 and more generally Ck summability: The sequence of averages of partial sums of a series defines a series which has them as its partial sums; if this series is C1 summable the original series is said to be C2 summable.
	www.cs.cityu.edu.hk /~luoyan/mirror/mit/ocw.mit.edu/18/18.013a/f01/required-readings/chapter30/section01.html (603 words)

	Re: an informal discussion of proof issues
		> > Criterion which are only satisfied by Condorcet: > (2) Condorcet Criterion > (3) Generalized Condorcet Criterion > > Quorum rule does not satisfy these criterion.
		> > (8) Favorite Betrayal Criterion [Approval only] > > The quorum portion of the rules satisfies this criterion, the proof is > essentially the same as for monotonicity.
		However, since the underlying > system does not satisfy this criterion the point is moot.
	lists.debian.org /debian-vote/2002/12/msg00087.html (329 words)

	Search Results for Series
		Da Cunha develops a criterion for the convergence of a series which he uses to define the exponential function in a rather modern way, and from these develops the binomial series.
		Another paper published in the same year On the summability (C) of allied series and the existence of (CP) extends the conditions for the Cesaro summability of Fourier-Lebesgue series and of their conjugates given by Bosanquet in 1937 to the case of functions integrable in the Cesaro-Perron sense.
		Her doctoral dissertation was On the summability of trigonometric series with localized properties - on Fourier constants and convergence factors of double Fourier series.
	www-groups.dcs.st-and.ac.uk /~history/Search/historysearch.cgi?SUGGESTION=Series&CONTEXT=1 (16352 words)

	Condorcet Criterion (Site not responding. Last check: 2007-11-06)
		The Condorcet criterion for a voting system is that it chooses the Condorcet winner when one exists.
		Any method conforming to the Condorcet criterion is known as a Condorcet method.
		Non-ranking methods such as plurality voting and approval voting cannot comply with the Condorcet criterion because they do not allow each voter to fully specify their preferences.
	read-and-go.hopto.org /Voting-system-criteria/Condorcet-Criterion.html (196 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-06)
		The first convergence criterion for Fourier series was obtained by P.G.L. Dirichlet in 1829.
		This Lebesgue criterion is stronger than all those given above and stronger than the de la Vallée-Poussin criterion and the Young criterion.
		A convergence criterion of another type is given by the Hardy–Littlewood theorem (1932): The Fourier series of a function
	eom.springer.de /f/f041090.htm (1838 words)

	Voting system (Site not responding. Last check: 2007-11-06)
		Monotonicity criterion - Is it impossible to cause a choice to lose by ranking it higher, or win by ranking it lower?
		Participation criterion - Is it always better to vote honestly than to not vote?
		Summability criterion - Can the vote-tallying process be distributed across many locations?
	voting-system.iqnaut.net (2064 words)

	[EM] River method -- updated summary
		Summability (depends on the pairwise matrix only except for rare tiebreaking situations.
		The final result is easily justified by showing the resulting tree-like diagram of n-1 arrows which shows that any majority argument against the winner is in conflict with stronger arguments of the same kind.
		The main consequence of this difference is that the River diagrams are much simpler (tree-like) and can thus show the process and reasons for the final decisions easily and efficiently.
	lists.electorama.com /pipermail/election-methods-electorama.com/2004-October/013971.html (3493 words)

	A lower estimate for the Lebesgue constants of linear means of Laguerre expansions (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		In this paper, Kalnei's problem is considered in the setting of Laguerre series on weighted L 1 -spaces; the admitted scale of weights contains in particular the appropriate "natural weights" occurring in transplantation and convolution.
		2 aro summability of Laguerre expansions and norm estimates wi..
		1 aro summability and operator norms for Laguerre expansions (context) - Gorlich, Markett et al.
	citeseer.ist.psu.edu /122573.html (350 words)

	Talk:Summability criterion - Electowiki (Site not responding. Last check: 2007-11-06)
		Is DSC really second-order summable, or is this an error?
		This means (assuming tie-break rules that don't create added complication) that recording, for every set of 2 or more candidates, the number of votes giving top place within the set to each candidate in the set, will suffice.
		This is a summable array of n(2^(n-1)-1) numbers, still failing the criterion but enough to make the statement that "the votes cannot be compressed by summing" incorrect.
	wiki.electorama.com /wiki/index.php?title=Talk:Summability_criterion&printable=yes (283 words)

	[No title]
		\] It is known that this method of summability cannot be included by any matrix method, but for bounded sequences it is included by the Ces\'{a}ro matrix method $C_{1}$.
		In this paper these results are extended by comparing statistical convergence with the intersection of a collection $\cal T$ of summability matrices, each of which is somewhat like $C_{1}$.
		It is shown that a bounded sequence is statistically convergent if and only if it is summable by every matrix in $\cal T$.
	www.csse.monash.edu.au /mirrors/bibliography/Techreports/kent-state-university.unique (3670 words)

	Lecture outline for Fourier Analysis
		Convergence of summability operators in homogeneous Banach spaces.
		Additional examples of summability kernels: de la Vallee Poussin kernel, Poisson kernel, Abel means & radial limits of harmonic functions.
		Mon (9/30): Lebesgue criterion for pointwise convergence of Fejer sums.
	www.math.sc.edu /~sharpley/math750_f02/lectures.html (688 words)

	mp_arc 05-36 (Site not responding. Last check: 2007-11-06)
		Summation of divergent series and Borel summability for strongly dissipative equations with periodic or quasi-periodic forcing terms (401K, postscript) Jan 28, 05
		We consider a class of second order ordinary differential equations describing one-dimensional systems with a quasi-periodic analytic forcing term and in the presence of damping.
		We show the existence of a summation criterion of the series in this case also, but, however, this can not be interpreted as Borel summability.
	www.ma.utexas.edu /mp_arc-bin/mpa?yn=05-36 (190 words)

	Gintautas Dzemyda,
		The papers for the book were selected because they relate to these topics and also satisfy the criterion of theoretical soundness combined with practical applicability.
		Several summability methods of d-dimensional trigonometric-, Walsh-, spline-, and Ciesielski-Fourier series and Fourier transforms as well as the d-dimensional dyadic derivative are investigated.
		Audience: This book will be useful for researchers as well as for graduate or postgraduate students whose work involves Fourier analysis, approximations and expansions, sequences, series, summability, probability theory, stochastic processes, several complex variables, and analytic spaces.
	www.yurinsha.com /348/p4.htm (709 words)

	Talk:Instant-runoff voting/archive1 - Enc.yclopedia.info rich with information, resources and facts (Site not responding. Last check: 2007-11-06)
		Summability criterion, making them easier to audit" because it is simply not true that the Condorcet criterion implies the
		All your examples (Condorcet criterion, Monoticity criterion, Participation criterion) are concepts with a rigid definition in a formal system (i.e., they are mathematics).
		I think that perhaps invulnerability to burying should be a criterion by itself; I don't think that later-no-help and later-no-harm say exactly the same thing.
	enc.yclopedia.info /info.php?title=Talk:Instant-runoff_voting/archive1 (11229 words)

	Citations: Separability of internal representations in multilayer perceptrons with application to learning - Takahashi, ... (Site not responding. Last check: 2007-11-06)
		It is then very important to have a natural termination criterion for the output layer learning algorithm so that inseparability can be detected efficiently (Grossman et al.
		They start from the notion of summability of boolean functions [16] and ncyclicity can be viewed as a reinterpretation of n summability.
		They start from the notion of summability of boolean functions [15] and n cyclicity can be viewed as a reinterpretation of n summability.
	citeseer.ist.psu.edu /context/603049/0 (1218 words)

	AMS Journals :: Print and Electronic
		A limit-point criterion for a class of Sturm-Liouville operators defined in ${L^p}$ spaces.
		Necessary and sufficient conditions for absolute summability of the trace formulas for certain one dimensional Schrödinger operators.
		A criterion for correct solvability of the Sturm-Liouville equation in the space $L_{p}(R)$.
	www.mathaware.org /joursearch/servlet/DoSearch?f1=msc&v1=47E05 (152 words)

	[EM] Instant Runoff Normalized Ratings
		This is the most important criterion to me.
		Thus an additional X > Y ballot cannot elect Y over X. Summability: No. Because this is an instant runoff method, all votes must be centrally collected and run together.
		It is not even representable as a candidates-factorial size array as IRV could be.
	lists.electorama.com /pipermail/election-methods-electorama.com/2004-May/012885.html (606 words)

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