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Topic: Singular point

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	Singular point of an algebraic variety - Wikipedia, the free encyclopedia
		In mathematics, a singular point of an algebraic variety V is a point P that is 'special' (so, singular), in the geometric sense that V is not locally flat there.
		A general algebraic variety V being defined by several polynomials, or in algebraic terms an ideal of polynomials, the condition on a point P to be a singular point of V is that none of those polynomials have a non-zero linear (degree 1) term, when written in terms of variables
		Points of V that are not singular are non-singular.
	en.wikipedia.org /wiki/Singular_point_of_an_algebraic_variety (239 words)

	Mathematical singularity - Wikipedia, the free encyclopedia
		In mathematics, a singularity is in general a point at which a given mathematical object is not defined, or a point of an exceptional set where it fails to be well-behaved in some particular way, such as differentiability.
		The point a is a removable singularity of f if there exists a holomorphic function g defined on all of U such that f(z) = g(z) for all z in U − {a}.
		In algebraic geometry and commutative algebra, a singularity is a prime ideal whose localization is not a regular local ring (alternately a scheme (mathematics) with a stalk that is not a regular local ring).
	en.wikipedia.org /wiki/Mathematical_singularity (461 words)

	Tangents, Normals, Continuity and Taylor's Theorem (Site not responding. Last check: 2007-07-09)
		The gradient of a curve at a point is the same as the gradient of the tangent to the curve at that point.
		A singular point of a curve is any point at which the curve does not have a unique tangent.
		A singular point of a curve can occur where the curve is pointed, where it crosses itself, or where it branches in two directions.
	www.geocities.com /mathfair2002/school/calc/calc0.htm (693 words)

	[No title] (Site not responding. Last check: 2007-07-09)
		Given a point in the plane or a value of the potential V, it is difficult to predict the behavior of the corresponding equipotentials.
		Equipotentials are plane curves; the separation lines are given by those curves that have singular points (if an equipotential has no singular point, it is a "regular" one, not a limiting curve).
		The singular points of a curve are those for which all partial derivatives become zero simultaneously; but, since the partial derivatives of the potential V are the components of the electric field E, singular points are those for which E=0:
	alumnus.caltech.edu /~muresan/projects/esfields/field_223.html (544 words)

	Mathematics 535 (Fall 2002) Information (Site not responding. Last check: 2007-07-09)
		Note that the singular points of a cubic surface are intersections of the cubic with quadrics given by taking the various derivatives.
		Hence if 3 singular points are on a line, the whole line is in the cubic and in any quadrics containing the three points, hence the entire line is singular.
		Similarly the line joining 2 singular points on a cubic surface is entirely contained in the surface, since a line hitting a cubic surface 4 times is wholly in the surface.
	www.math.rutgers.edu /courses/535/535-f02/Movie4.html (405 words)

	Find Singularity Utility
		The purpose of the Find Singularity utility is to compute the coordinates and other information about a singular point on a level curve on a plane.
		Planar singularities on a curve are special because the partial derivatives of the function which defines the curve vanish at that point.
		The purpose of the Singularity algorithm is to locate the singular point or points on a level set curve.
	www.geom.uiuc.edu /~streed/pisces/docs/sing.html (672 words)

	Classical structure of conifolds
		When this is true, the generic point of that locus will locally be an intersection of k hypersurfaces, meeting transversally.
		However, in the example which we will study in detail there are k=16 singular points on the conifold which impose only 15 conditions on the parameters.
		This is clearly a special property of the particular collection of 16 points which we are considering.
	www.cgtp.duke.edu /~drm/condensation/node2.html (530 words)

	Continuation and the Parameter IPROB.
		The singular point labeled ``A'' in the figure on the left is a limit (turning) point and those labeled ``B'' in the figure on the right are bifurcation points (this figure corresponds to the special case of a linear eigenvalue problem).
		At the conclusion of this iteration, some tests are made to determine if the point is a bifurcation point, a limit point, or a regular point.
		The algorithms in PLTMG were designed to handle only simple limit and bifurcation points, although on occasion we have observed them to work on higher degree singular points as well.
	www.ima.umn.edu /~bank/guide/node27.html (916 words)

	Point of View
		After they adopt their points of view, good writers are careful to consistently use the same points of view all the way through the essay--unless they have a good reason to change those points of view.
		This point of view is extremely handy to those writers who are in the process of narrating stories that concern themselves.
		The third person plural point of view is the point of view most commonly encountered in a formal piece of writing such as a research paper, for example.
	www.taft.cc.ca.us /newTC/Academic/LiberalArts/OWL/POVIEW.HTML (610 words)

	[No title] (Site not responding. Last check: 2007-07-09)
		# # Maple is used to determine if a point is singular or regular (part 2).
		Determine the nature of all singular points of the # given differential equation (which has at least one nonpolynomial # coefficient).
		\dx / \dx / # T-238/15b: Singular points of DE.
	calclab.math.tamu.edu /docs/math308/series/ab-sinpt2-R4.txt (299 words)

	[No title]
		EX: y" = exy, every point x ((is a regular point x5y" = y, every point x except for x = 0 and x = (is a regular point If p(x), q(x), or r(x) is not analytic at x = xo, the point x = xo is said to be a singular point.
		Consider a second order homogeneous linear equation y” + p(x)y’ + q(x)y = 0 (4.1-1) Regular singular point: The point x = xo is called a regular singular point of (4.1-1) if not both of p(x), q(x) are analytic but both (x — xo)p(x) and (x — xo)2q(x) are analytic in the neighborhood of xo.
		Irregular singular point: The point x = xo is called an irregular singular point of (4.1-1) if it is neither a regular point nor a regular singular point.
	www.csupomona.edu /~tknguyen/egr509/Notes/Chapter4-1.doc (382 words)

	Singular Curves and Surfaces (Site not responding. Last check: 2007-07-09)
		Note that unlike the circle, which is smooth, the folium has a singular point at the origin where the curve crosses itself.
		Equivalently, a point is singular if the polynomial and all of its first-order partial derivatives vanish at the point.
		by first converting it to an equivalent equation having rational solutions, then using the techniques of this section [the singular point is not at the origin in this case].
	www.mast.queensu.ca /~reidl/ColemanEllis/CE2.html (211 words)

	[No title] (Site not responding. Last check: 2007-07-09)
		\ dx / \ dx / -------------------------------------------------------------------------------- # T-238/15b: Singular points of DE.
		> P(x)=0; solve(", x); others:=nPi; x sin(x) = 0 0, 0 others := n Pi -------------------------------------------------------------------------------- # T-238/15c: Nature of singular* point: x0 = 0 is an irregular singular # point, since the series expansion about x0 = 0 is a Laurent series, not # a Taylor series.
		> (x-0)Q(x)/P(x);\ irregular:=series(", x=0); 3 ------ sin(x) -1 3 4 irregular := 3 x + 1/2 x + 7/120 x + O(x) -------------------------------------------------------------------------------- # T-238/15d: Nature of singular* point: x0 = Pi is a regular singular # point of the DE since the relevant expressions below have Taylor # series expansions.
	calclab.math.tamu.edu /docs/math308/series/ab-sinpt2-R3.txt (272 words)

	Chapter 4.2
		After that, the image obtained must be transformed by the reflection R with the reflection line containing the singular point O, and finally, by the rotation S.
		The form of a fundamental region of conformal symmetry groups is defined by the invariance of all the points of inversion circles and reflection lines.
		According to the criterion of maximal constructional and visual simplicity, constructions of conformal symmetry rosettes are mostly based on the use of circles and lines as homologous elements of conformal symmetry transformations.
	www.emis.de /monographs/jablan/chap42.htm (7801 words)

	powerseriesII.htm
		is a singular point for this differential equation.
		We can now formulate the type of singular points which series methods can handle.
		is a regular singular point and the indicial equation is given by
	germain.umemat.maine.edu /faculty/bray/Archive_notes/powerseriesII.htm (671 words)

	TAM 310 - Spring 97
		A function is analytic at a point if it can be expanded in a Taylor Series about the point or centered at the point.
		If either function fails to have a Taylor Series at a point, the point is called a Singular Point, or Singularity of the ODE.
		The power series expansion of the solution of an ODE about an Ordinary Point converges in the circle (remember the Complex Plane?) centered at the Ordinary Point of radius just large enough to reach the closest Singular Point.
	www.tam.cornell.edu /courses/310Sp97/L14Feb/lec14feb.html (226 words)

	Amazon.com: Generalized Cauchy-Riemann Systems with a Singular Point: Books (Site not responding. Last check: 2007-07-09)
		Usmanov, Zafar Dzuraevich Usmanov "The basic aim of this chapter is to establish a connection between the solutions of the equations by means of a linear integral equation with..." (more)
		A theory of generalized Cauchy-Riemann systems with polar singularities of order not less than one is presented and its application to study of infinitesimal bending of surfaces having positive curvature and an isolated flat point is given.
		Presents a theory of generalized Cauchy-Riemann systems with polar singularities of order not less than one and gives its application to the study of infinitesimal bendings of surfaces having positive curvature and an isolated flat point.
	www.amazon.com /exec/obidos/tg/detail/-/0582292808?v=glance (590 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		Energy Citations Database (ECD) Document #6891177 - Matched uniform approximations for a singular boundary point and an interior turning point
		Availability information may be found in the Availability, Publisher, Research Organization, Resource Relation and/or Author (affiliation information) fields and/or via the "Full-text Availability" link.
		Matched uniform approximations for a singular boundary point and an interior turning point
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=6891177 (99 words)

	Definition of singular - Merriam-Webster Online Dictionary
		singular adventure>
		For More Information on "singular" go to Britannica.com
		Get the Top 10 Search Results for "singular"
	www.m-w.com /cgi-bin/dictionary?singular (194 words)

	Frobenius Series Solution of a D. E.
		is a singular point of (1) and that
		They will have Maclaurin series expansions with radius of convergence
		This method is attributed to the german mathemematican
	math.fullerton.edu /mathews/n2003/FrobeniusSeriesMod.html (373 words)

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