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Topic: Degenerate case

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	Degenerate (mathematics) - Wikipedia
		In mathematics, a degenerate case is a limiting case in which a class of object changes its nature so as to belong to another, usually simpler, class.
		A point is a degenerate case of a circle, namely one with radius 0.
		The circle is a degenerate form of an ellipse, namely one with eccentricity 0.
	wikipedia.findthelinks.com /de/Degenerate_case.html (81 words)

	General position - Wikipedia, the free encyclopedia
		In geometry, general position for a set of points, or other configuration, means the general case situation, as opposed to some more special or coincidental cases that are possible.
		This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating exactly certain theorems and when writing computer programs.
		In some contexts, e.g., when discussing Voronoi tessellations and Delaunay triangulations in the plane, a stricter definition is used: a set of points in the plane is then said to be in general position only if no three of them lie on the same straight line and no four lie on the same circle.
	en.wikipedia.org /wiki/General_position (360 words)

	Expert About de:Degenerate
		Degenerate gas is a particular configuration, usually reached at high densities, of a gas composed of subatomic particles with half-integral intrinsic angular momentum (spin).
		For example, the point is a degenerate case of the circle as the radius approaches 0, and the circle is a degenerate form of an ellipse as the eccentricity approaches 0.
		Degenerate art (from the German: entartete Kunst) is a term that became notorious during the Nazi rule of Germany to refer to any art reflecting values or aesthetics contrary to the Aryan ones.
	www.expertsite.biz /dir/de/degenerate.htm (1332 words)

	Degeneracy (mathematics) - Enpsychlopedia (Site not responding. Last check: 2007-11-05)
		A point is a degenerate circle, namely one with radius 0.
		Also it is a degenerate form of a rectangle, if this has a side of length 0.
		Thus, there may be as few as one degenerate "side" or as many as $n$ (in which case $R$ reduces to a singleton point).
	www.enpsychlopedia.com /psypsych/Mathematical_degeneracy (253 words)

	[No title]
		In some cases -- that is, for some classes of linear parts -- the standard normal form is unique (trivial) and thus standard theory gives a completely defined answer; in some other cases, the standard NF is not unique, and PRF theory is able to improve the classification provided by the standard theory.
		In cases {\bf S1,N1} the PRF is trivial (i.e.
		Thus, the PRF in case {\bf (db)} is $$ \^W \ = \ Y_0 \ + \ b_\nu Y_\nu \ + \ \sum_{k=\mu}^\infty \=a_k X_k \.
	www.ma.utexas.edu /mp_arc/papers/01-17 (6430 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-05)
		That is called a "degenerate" case, because some feature has been lost, allowing the thing we are looking at to "degenerate into" something simpler.
		A degenerate case is sort of like the boundary of a region.
		Because a cylinder can be thought of as a degenerate cone, we can treat parallel lines as if they were a special case of the hyperbola.
	mathforum.org /library/drmath/view/54756.html (1298 words)

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	The Case for Elliptic Curve Cryptography
		This case will be based on both the relative security offered by elliptic curves and first generation public key systems and the relative performance of these algorithms.
		Similar to both RSA and Diffie-Hellman, the first years of analysis yielded some degenerate cases for elliptic curve parameters that one should avoid.
		In the elliptic curve case, there is actually one additional bit that needs to be transmitted in each direction which allows the recovery of both the x and y coordinates of an elliptic curve point.
	www.nsa.gov /ia/industry/crypto_elliptic_curve.cfm (1818 words)

	Odd and Degenerate Types
		The general case is handled by a recent result of Jeff Burdges (article in preparation as of August 2002) showing that any nontrivial solvable signalizer functor gives rise to a nilpotent one.
		If G is degenerate, this can be simplified: G is either algebraic, or has proper 2-generated core, or has Prufer 2-rank at most 2.
		In the degenerate case, a simple K*-group is minimal, and in particular if H is the connected component of an involution then O(H)=H>1.
	www.rci.rutgers.edu /~cherlin/FMR/oddtype.html (603 words)

	[No title]
		This amplitude is a function of the areas of the 10 faces of the 4-simplex, and Barrett and Williams have shown that one contribution to its asymptotics comes from the Regge action for all non-degenerate 4-simplices with the specified face areas.
		In Section 3 we analyse the contribution of `degenerate 4-simplices' to the integral for the 10j symbols.
		The way degenerate geometries govern the asymptotics of the 10j symbols raises the possibility that in the limit of large spins, the Barrett-Crane model reduces to a theory of degenerate metrics.
	jdc.math.uwo.ca /papers/asymp.txt (3445 words)

	The Coupled Two Level Eigensystem Problem (Site not responding. Last check: 2007-11-05)
		For degenerate unperturbed systems, the effect of the perturbation is to break the degeneracy.
		A plot of the split in the eigenvalues induced by the coupling between the unperturbed states shows that the splitting of non-degenerate states asymptotically approaches the degenerate case.
		Note that the symmetry of the state is not simply determined by the sign of the root used, but involves the sign of the coupling term as well.
	128.42.6.89 /faculty/wong/Derivations/TwoLevel (443 words)

	Degenerate Total Necessary What's What A Consumer Could Crave Next Degenerate (Site not responding. Last check: 2007-11-05)
		The degenerate ethics of the American WASP by Chris M., Sept. 5, 2005 [Re Race not an issue in rescue efforts, says Rice] "Secretary of State Condoleezza Rice said, "Nobody, especially the president...
		DEGENERATE GAS A degenerate gas is one that is super-compressed and very dense.
		Non-degenerate and Degenerate Semiconductors A semiconductor is said to be 'non-degenerate' if only a small number of donor and/or acceptor atoms have been introduced into it, such that the...
	peaceart.serapeace.com /degenerate (888 words)

	Introduction
		In the degenerate case, it is necessary to use prior information about the source signals to perform demixing, because of the ill-posed nature of the inverse mathematical problem.
		Even in the case of dissonant or deliberately random music, pitches are in general discretized to the 12-tone Western scale, leading to overlap of some harmonics.
		However, in the cases of musical trios or four speaker examples, the two-source assumption is of great benefit.
	ccrma-www.stanford.edu /~asmaster/BTSM/master/node1.html (519 words)

	Goals and Scenarios
		Initialize the bool variable for the three possible solution cases: extremely degenerate, degenerate and regular case.
		3.5 If the equation is a degenerate case, the output interface can show that there is an unique solution and the value of the root.
		In each case, there is a corresponding root computing unit.
	www.engr.umd.edu /~masoud/HomePage/ENSE622/sections.htm (1169 words)

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		A degenerate range is like a text editor caret (insertion point) ¡Xit does not actually select any characters.
		The end points of a degenerate range are adjacent.
		If the range is degenerate, passing a large positive number causes the text to the right of the range to be searched.
	www.siteexperts.com /forums/getMessage.asp?m_id=77810 (534 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-11-05)
		Date: 03/04/98 at 16:35:26 From: Craig Kiczek Subject: Conic Sections I am aware that there are a number of degenerate cases for the graphs of equations in conic form.
		As I have been told, one of these cases is two parallel lines.
		The degenerate cases arise if you slice (a) through the vertex parallel to the bases...this gives a point; (b) through the vertex tangent to the cones...this gives a line; (c) through the vertex but intersecting both cones...this gives two intersecting lines.
	mathforum.org /library/drmath/view/54458.html (425 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		The case of my primary interest is in dimension 3.
		case - a genus 3 curve in the class (m1,m2,m3), is counted in a paper of
		degenerate case, namely curves in the class (m1,m2,m3) with genus greater
	www.mathforum.com /kb/plaintext.jspa?messageID=4178332 (239 words)

	Define degenerate - Definition of degenerate from Free-Online-Dictionary.org (Site not responding. Last check: 2007-11-05)
		of degenerare to degenerate, cause to degenerate, fr.
		Homer never tires of sneering at "men who live in these degenerate days," which is perhaps why they suffered him to beg his bread -- a marked instance of returning good for evil, by the way, for if they had forbidden him he would certainly have starved.
		Degenerate CMS-03 Comment In the degenerate case where there are no signers, the EncapsulatedContentInfo value being "signed" is...
	www.free-online-dictionary.org /define-degenerate.html (632 words)

	Calculating a rotation matrix based on location/target (Site not responding. Last check: 2007-11-05)
		This document will also emulate the specific nature of 3DS cameras, when it comes to the degenerate case (input vector points directly up).
		There is a degenerate case when the [delta y] of the direction vector is anything but zero.
		TESTING Many people claim to have perfectly working code, and I have found that in over 50% of them, this was not the case.
	www.whisqu.se /per/docs/math5.htm (268 words)

	Topology (Site not responding. Last check: 2007-11-05)
		Any degenerate case is very near a non degenerate case.
		To handle degenerate cases we make all of these intersection observations on fudged coordinates.
		These observations describe non degenerate polygon pairs that are within floating point precision of the delivered polygon pairs.
	www.cap-lore.com /MathPhys/IP/Logic.html (446 words)

	Journal of Lie Theory, Vol. 13, No. 1, pp. 279--289, 2003 (Site not responding. Last check: 2007-11-05)
		Abstract: This is an investigation of local (analytic or formal) integrating factors near certain degenerate stationary points of plane analytic vector fields.
		The method is to use blow-ups and then apply known results from the non-degenerate case.
		The main result is that in general there is no formal integrating factor which is algebraic over the quotient field of the formal power series ring.
	www.emis.de /journals/JLT/vol.13_no.1/18.html (85 words)

	Re: Degenerate CMS-03 Comment (Site not responding. Last check: 2007-11-05)
		If an S/MIME v3 agent needs to send a degenerate signedData to a S/MIME v2 legacy agent, then the inclusion of the id-absent-content OID may cause problems.
		In >this case, the content type within the EncapsulatedContentInfo value being >"signed" should be data (as defined in section 4), and the content field of >the EncapsulatedContentInfo value should be omitted." > >Recommend that this should be changed to: "In the degenerate case where >there are no signers, the EncapsulatedContentInfo value being "signed" is >irrelevant.
		In this case, the EncapsulatedContentInfo eContentType MUST be >set to id-absent-content and the eContent field of the >EncapsulatedContentInfo value MUST be omitted." > >Recommend that Russ should define an id-absent-content OID.
	www.imc.org /ietf-smime/archive1/msg01283.html (174 words)

	Degeneracy (mathematics) - the free encyclopedia (Site not responding. Last check: 2007-11-05)
		The line is a degenerate form of a
		A hyperbola can degenerate into two lines crossing at a point, through a family of hyperbolas having those lines as common asymptotes.
		Another usage of the word comes in eigenproblems: a degenerate
	www.encyclopedia-of-knowledge.com /default.asp?t=Mathematical_degeneracy (105 words)

	General position Info - Bored Net - Boredom (Site not responding. Last check: 2007-11-05)
		A set of points in the d-dimensional Euclidean space is said to be in general position, if no d + 1 of them lie in a (d-1)-dimensional plane.
		See the article about affine transformation for more.
		If d + 1 points are in a (d-1)-dimensional plane, it is called degenerate case or degenerate configuration.
	www.borednet.com /e/n/encyclopedia/g/ge/general_position.html (274 words)

	NationMaster - Encyclopedia: Mathematical degeneracy (Site not responding. Last check: 2007-11-05)
		Updated 576 days 19 hours 50 minutes ago.
		Degeneracy (mathematics) In mathematics, a degenerate case is a limiting case in which a class of object changes its nat...
		Mathematics Mathematics is commonly defined as the study of Philosophy of mathematics.
	www.nationmaster.com /encyclopedia/Mathematical-degeneracy (345 words)

	Degenerate case
		In this last case -only having one center of projection- depth can not be observed.
		In the absence of noise the detection of these degenerate cases would not be too hard.
		In this case the fundamental matrix (corresponding to a 3D scene and more than a pure rotation), a general homography (corresponding to a planar scene) and a rotation-induced homography are computed.
	www.cs.unc.edu /~marc/tutorial/node58.html (293 words)

	Stonewall 25: Case 3
		This case includes the earliest writings in English about homosexuality, mostly male homosexuality.
		Symonds's friend and literary executor, Horatio Brown, bought up and destroyed all copies of this edition (except for a few already distributed) and persuaded Ellis to drop Symonds's name from the title page and remove other references to his collaboration in the project.
		He rejected current theories that "inverts"' were degenerate, perverted, or sick; challenged stereotypes of homosexual behavior and appearance; insisted on the ubiquity of homosexuality across time, cultures, and class; and emphasized the association of homosexuality with intellectual and artistic distinction.
	www.columbia.edu /cu/lweb/eresources/exhibitions/sw25/case3.html (946 words)

	The perturbation of a degenerate state
		Assume that this eigenvalue is g-fold degenerate, i=1,2,...,g.
		is degenerate then the corresponding eigenvector is still not uniquely defined.
		Therefore, if first-order perturbation theory removes the degeneracy of a set of degenerate eigenvalues, we can proceed to find higher-order corrections to the now uniquely defined eigenvalues just like in the non degenerate case.
	electron6.phys.utk.edu /qm2/modules/m8/degenerate.htm (211 words)

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