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Topic: Removable singularities

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Laurent expansion

	Singularity -- from Wolfram MathWorld (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		Singularities are extremely important in complex analysis, where they characterize the possible behaviors of analytic functions.
		A logarithmic singularity is a singularity of an analytic function whose main
		Removable singularities are singularities for which it is possible to assign a
	mathworld.wolfram.com.cob-web.org:8888 /Singularity.html (135 words)

	PlanetMath: potential theory
		To the extent that it is possible to draw a distinction between these two fields, the difference is more one of emphasis than subject matter and rests on the following distinction -- potential theory focuses on the properties of the functions as opposed to the properties of the equation.
		In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwartz's theorem, Morera's theorem, the Casorati-Weierstrass theorem, Laurent series, and the classification of singularities as removable, poles and essential) generalize to results on harmonic functions in any dimension.
		As alluded to in the last section, one can classify the isolated singularities of harmonic functions as removable singularities, poles, and essential singularities.
	planetmath.org /encyclopedia/PotentialTheory.html (1101 words)

	Removable Singularities for Lu = Ψ(u) and Orlicz Capacities (ResearchIndex)
		Removable Singularities for Lu = Ψ(u) and Orlicz Capacities (ResearchIndex)
		Removable Singularities for Lu = Ψ(u) and Orlicz Capacities
		A compact set R d is called a removable singularity for the equation Lu = u if this equation has no nontrivial solution in R d n.
	citeseer.ist.psu.edu /352779.html (482 words)

	Defining a Singularity - SciForums.com
		You need to realise that in a mathematical sense, a singularity is a point at which some physical quantity seems to become unphysical, usually by becoming infinite.
		Some singularities in the mathematics of physical theories turn out to be removable, so that they are not a problem for the physics.
		Other singularities, such as the singularity at the centre of a fl hole, or the singularity at the moment the universe began, are non-removable.
	www.sciforums.com /showthread.php?t=39720 (4456 words)

	Pole and Singularity
		Removable singularities and Liouville-type property of analytic multivalued functions.
		Removable singularities for analytic functions of Zygmund class.
		On the removable singularities of functions of several complex variables.
	math.fullerton.edu /mathews/c2003/PoleBib/Links/PoleBib_lnk_3.html (383 words)

	Removable singularity - Wikipedia, the free encyclopedia
		In complex analysis, a removable singularity of a function is a point at which the function is not defined (a singularity) but at which the function can be so defined that it is continuous at the singularity.
		for z ≠ 0 has a removable singularity at z = 0: we can define f(0) = 1 and the resulting function will be continuous and even infinitely differentiable (a consequence of L'Hôpital's rule).
		Riemann's theorem on removable singularities states that a singularity a of a holomorphic function f is removable if and only if there exists a neighborhood of a on which f is bounded.
	en.wikipedia.org /wiki/Removable_singularity (167 words)

	AMCA: Removable singularities for Hardy spaces of analytic functions by Anders Bjoern
		This is a conformally invariant definition which is the same as the usual one for the unit disc.
		A relatively closed subset E of a domain \Omega is a weakly removable singularity for H
		Weak removability is in fact independent of the domain \Omega, and if E is compact, then the two types of removability coincide.
	at.yorku.ca /c/a/h/s/27.htm (213 words)

	Springer Online Reference Works
		The situation may be described in other words by saying that "the set E is set, removable for a class of functionsremovable for the class K" or that "E is a null-set for the class K" , briefly:
		On removable sets for different classes of analytic functions of one complex variable and related unsolved problems see [3], [4], [6], [9].
		The problem of removable sets can also be posed for harmonic, subharmonic and other functions.
	eom.springer.de /r/r081230.htm (441 words)

	Czechoslovak Mathematical Journal, Vol. 56, No. 1, pp. 179-227, 2006 (Site not responding. Last check: 2007-09-10)
		The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathop BMO$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g.
		In the case when weak and strong removability are the same for all sets, in particular if $\meas$ is absolutely continuous with respect to the Lebesgue measure $\Leb$, we are able to say more than in the general case.
		When $\dd\mu= w\dd m$ and $w$ is a Muckenhoupt $A_p$ weight, $1removable singularities are characterized as the null sets of the weighted Sobolev space capacity with respect to the dual exponent $p'=p/(p-1)$ and the dual weight $w'=w^{1/(1-p)}$.
	cmj.math.cas.cz /cmj56-1/12.html (321 words)

	Removable Hard Drive Case (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		Removable media 1: '''Removable media''' are transportable disk drivedrives 3: e much larger amounts of data from other types of removable disks, like CD burner s or Zip disk s.
		Brass instrument 46: Most brass instruments are fitted with a removable mouthpiece.
		Fairchild Channel F 24: ajor changes were in design, the controllers were removable from the base unit instead of being wired directl
	www.relativeaccess.com.cob-web.org:8888 /File/1635-Removable.Hard.Drive.Case.Html (318 words)

	Amazon.com: "removable singularities": Key Phrase page (Site not responding. Last check: 2007-09-10)
		Chapter 1 Removable Singularities 1.1 Bochner's Theorems 1.1.1 Sheaf of solutions Suppose that P(x, D) = 1:1,,I
		Key Phrases in this book: removability results, nondecreasing real valued function, nonlinear elliptic equations, porous media equation, nonnegative solution, quasilinear elliptic equations, semilinear elliptic equations, quasilinear equations, isolated singularities, radial solutions, removable singularities, spherical average (See more)
		in Section 2.7 we study an alternative definition of capacity, which ap- pears naturally in connection with the study of removable singularities of solutions of partial differential equations.
	www.amazon.com /phrase/removable-singularities (430 words)

	On Removable Singularities for CR Functions in Higher Codimension (ResearchIndex)
		On Removable Singularities for CR Functions in Higher Codimension (ResearchIndex)
		On Removable Singularities for CR Functions in Higher Codimension (1996)
		0.3: Removable Singularities for Lu = Ψ(u) and Orlicz Capacities - Kuznetsov
	citeseer.ist.psu.edu /merker96removable.html (266 words)

	Session 13 \\ {\bf Singularities and Residues }
		A branch point is a special type of singularity such that if a circuit is described around it, the function after the circuit assumes a different value.
		Once and if all branch points have been suitably removed by cutting the only remaining singularities are points wher df/dz does not exist.
		Moreover, if the function is suitably redefined at z = a so as to eliminate the singularity, the point is a removable singularity.
	www.rh.edu /~ernesto/C_S2000/mes/Notes/mes13.html (1136 words)

	Atlas: Removable singularities for Hardy spaces of analytic functions by Anders Bjoern (Site not responding. Last check: 2007-09-10)
		Atlas: Removable singularities for Hardy spaces of analytic functions by Anders Bjoern
		Since our Hardy spaces are conformally invariant also the removable singularities are and Hasumi (1978) proved that different p give different removable singlarities.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cahs-27.
	atlas-conferences.com /c/a/h/s/27.htm (215 words)

	Computational and Applied Mathematics Group (CAM)
		There was a huge progress in understanding the geometric nature of removable singularities for some subtle classes of analytic and harmonic functions.
		This progress culminated in the solution of several problems of Vitushkin from the 50's.
		A 100 year old problem of Painlevé asking for ``geometric" description of removable sets for bounded analytic functions was solved by Xavier Tolsa.
	cam.ucsd.edu /abstracts/volberg.html (133 words)

	Removable singularities for subharmonic functions., Stephen J. Gardiner
		Removable singularities for subharmonic functions., Stephen J. Gardiner
		[9] R. Kaufman and J.-M. Wu, Removable singularitiesfor analytic or subharmonic functions, Ark. Mat, 18 (1980), 107-16.
		[12] A. O'Farrell, The I-reduction for removable singularities, and the negative Holder spaces, Proc.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.pjm/1102645043 (210 words)

	Amazon.com: "removable singularities theorem": Key Phrase page (Site not responding. Last check: 2007-09-10)
		See all pages with references to removable singularities theorem.
		There is a natural analogue for holomorphic functions of several variables of Riemann's removable singularities theorem, but much more is true:...
		The Removable Singularities Theorem We consider a homogeneous function F of degree k in R':...
	www.amazon.com /phrase/removable-singularities-theorem (487 words)

	Eduardo Santillan - Topological properties of removable singularities for analytic functions
		In his doctoral tesis, Shiffman proved a partial characterisation of the removable singularities by using the Hausdorff measure.
		Moreover, in 1994, professors Chirca, Stout and Lupacciolu showed more results which are completely independet of those of Shiffman; they used the cover dimension.
		The actual challenge is to improve that results to characterise the removable singularities by using just topological properties.
	www.cms.math.ca /Events/winter98/w98-abs/node182.e (116 words)

	Smith: Removable singularities for the Yang-Mills-Higgs equations in two dimensions
		Smith: Removable singularities for the Yang-Mills-Higgs equations in two dimensions
		Smith, P. Removable singularities for the Yang-Mills-Higgs equations in two dimensions.
		, Removable Singularities for the Yang-Mills Higgs equation in 2 dimensions,
	www.numdam.org /numdam-bin/item?id=AIHPC_1990__7_6_561_0 (131 words)

	Publications of Sergei Kuznetsov
		Removable Lateral Singularities of Semilinear Parabolic PDE's and Besov Capacities, J.
		Polar boundary sets for superdiffusions and removable lateral singularities for nonlinear parabolic PDE's, Communications on Pure and Applied Mathematics, 51 (1998), no. 3, pp 303-340.
		Superdiffusions and removable singularities for quasilinear partial differential equations (with E. Dynkin), Comm.
	spot.colorado.edu /~sek/skpp.html (1343 words)

	Penelope Smith
		Errata for that paper (*ed below) has been submitted to JMAA and is undergoing review.
		"Removable Singularities for the Fermionic Yang-Mills-Higgs Equations in Two Dimensions", in preparation
		"Removable singularities for the Yang-Mills-Higgs equations in two dimensions.
	comet.lehman.cuny.edu /sormani/others/smith.html (320 words)

	Refraction Between Moving Media (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		(which are just the speeds of light in the moving medium) generally correspond to removable singularities, because both the numerator and denominator of the expression for sin(
		It isn’t clear what, if any, optical effects would appear at these two removable singularities.
		The other two distinguished speeds represent the onset of total internal reflection if their values fall in the range from -1 to +1.
	www.mathpages.com.cob-web.org:8888 /rr/s2-08/2-08.htm (1236 words)

	Topics and HW (Site not responding. Last check: 2007-09-10)
		Click here for the program of Exam 2.
		Removable points and principal part of Laurent's expansion.
		Continue: removable points and principal part of Laurent's expansion; essential points and principal part of Laurent's series expansion.
	www.math.uiuc.edu /~inik/teaching/class2/main/topics.html (762 words)

	Argument Principle and Rouche's Theorem
		We now derive two results based on Cauchy's residue theorem.
		They have important practical applications and pertain only to functions all of whose isolated singularities are poles.
		By definition, meromorphic functions have no essential singularities.
	math.fullerton.edu /mathews/c2003/RoucheTheoremMod.html (560 words)

	Applied Mathematics (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		Björn, A., Removable singularities for Hardy spaces, Matematiska institutionen, Uppsala universitet, 25 April 1995.
		Björn, A., Different types of removable singularities for Hardy spaces of analytic functions, Matematiska institutionen, KTH, 26 April 1995.
		Kozlov, V.: Description of singularities of solutions to the Lamé and Stokes systems in non-smooth domains.
	math.liu.se.cob-web.org:8888 /organisation/actrep/ar94_95/node77.html (270 words)

	Duzaar, Fuchs: On removable singularities of p-harmonic maps
		Duzaar, Fuchs: On removable singularities of p-harmonic maps
		Duzaar, F. Fuchs, M. On removable singularities of p-harmonic maps.
		, On Removable Singularities of Stationary Harmonic Maps, J.
	www.numdam.org /numdam-bin/item?id=AIHPC_1990__7_5_385_0 (265 words)

	Mathematics-Online lexicon: Removable Singularities (Site not responding. Last check: 2007-09-10)
		One has to verify that the limit from the right coincides with the limit from the left.
		Then the function has a removable singularity at the point
		Usually one mark this removable singularity with an empty box in the sketch.
	www.mathematics-online.org /inhalt/aussage/aussage499 (66 words)

	Geometry and Topology Seminar (via CobWeb/3.1 planetlab2.cs.unc.edu) (Site not responding. Last check: 2007-09-10)
		Although the seminar will end promptly at 5:30, it will not be unusual for the speaker to talk for several sessions.
		In this talk I will cover some recent results and techniques that should have a big impact on the classical theory of complete embedded minimal surfaces in the next decade.
		As this is a GANG seminar, my talk will one in which there is a dialogue between the speaker and the audience, so feel free to interrupt me at any time.
	www.math.umass.edu.cob-web.org:8888 /~sullivan/geotopF05.html (776 words)

	U.S. Pregrant 20050007177 - Method and circuit for perturbing removable singularities in coupled translinear loops (Site not responding. Last check: 2007-09-10)
		Method and circuit for perturbing removable singularities in coupled translinear loops
		A Trafton-Hastings clamp circuit (36) is connected to generate a piecewise-polynomial-continuous current IY, the value of which becomes undefined when current IX=0 due to a removable singularity in the transfer equation at this point.
		A current mirror (38) comprising a plurality of transistors (M1, M2, M3) is coupled to the Trafton-Hastings clamp circuit (36), and operates to add additional currents in transistors Q3 and Q5 to IX, when the Trafton-Hastings clamp transistor (Q7) conducts, so as to perturb the removable singularity in the transfer equation into the left half-plane.
	cxp.paterra.com /uspregrant20050007177.html (205 words)

	Sibner: Removable singularities of Yang-Mills fields in $R^3$
		Sibner: Removable singularities of Yang-Mills fields in $R^3$
		Sibner, L. Removable singularities of Yang-Mills fields in $R^3$.
		: Removable singularities of coupled Yang-Mills fields in R3, Comm.
	math-doc.ujf-grenoble.fr /numdam-bin/item?id=CM_1984__53_1_91_0 (65 words)

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