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Topic: Singularity theory

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Isolated singularity

Essential singularity

Mathematical singularity

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Algebraic geometry

Monodromy

Submersion

Proof that holomorphic functions are analytic

Hassler Whitney

Jacobian matrix

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Connected space

Complex projective space

Power series

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	Singularity theory - Wikipedia, the free encyclopedia
		In mathematics, singularity theory is the study of the failure of manifold structure.
		In singularity theory the general phenomenon of points and sets of singularities is studied, as part of the concept that manifolds (spaces without singularities) may acquire special, singular points by a number of routes.
		Such singularities in algebraic geometry are the easiest in principle to study, since they are defined by polynomial equations and therefore in terms of a coordinate system.
	en.wikipedia.org /wiki/Singularity_theory (1082 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.
		See singularity theory for general discussion of the geometric theory, which only covers some aspects.
		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)

	Singularity - Wikipedia, the free encyclopedia
		mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved.
		gravitational singularity (physics) - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum, namely fl holes, white holes and worm holes.
		technological singularity - a theoretical point in the development of a scientific civilization at which technological progress accelerates into infinity or beyond prediction.
	en.wikipedia.org /wiki/Singularity (169 words)

	Singularity theory (Site not responding. Last check: 2007-07-20)
		In mathematics, singularity theory is the study of thefailure of manifold structure.
		That implies the breakdown of parametrization of points; it is prominent in cosmology,where a gravitational singularity, at which the gravitational field is strong enough to change the verystructure of space-time, is identified with a fl hole.
		Such singularities in algebraic geometry are the easiest inprinciple to study, since they are defined by polynomialequations and therefore in terms of a coordinate system.
	www.therfcc.org /singularity-theory-3596.html (1008 words)

	Technological singularity - Wikipedia, the free encyclopedia
		In future studies, a technological singularity (also referred to as just the Singularity) is a predicted future event when technological progress and societal change accelerate due to the advent of superhuman intelligence, changing our environment beyond the ability of pre-Singularity humans to comprehend or reliably predict.
		Devoid of the next level of technical infusion, the Singularity is never approached, and human society collapses in a deflationary wait for the Big Bang or in a variant of intellectual Cargo Cultism.
		Singularity themes are common in cyberpunk novels, one of the most famous examples being the recursively self-improving AI Neuromancer from William Gibson's novel of the same name.
	en.wikipedia.org /wiki/Technological_singularity (3556 words)

	Reflection Groups and Singularity Theory (Site not responding. Last check: 2007-07-20)
		Finite subgroups in SU(2), Weyl groups and Kleinian singularities.
		Discriminant, monodromy group, intersection form, period map and Gauss-Manin connection of an isolated singularity.
		Singularity theory and isomonodromy deformations of differential operators with rational coefficients.
	www.sissa.it /fm/course/rgst.html (45 words)

	Singularity Theory
		The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis and geometry, or phenomena from physics or some other science, on parameters.
		The primary purpose of the ASI was the introduction of new developments in singularity theory to a broader audience, thus establishing new contacts and advancing a broad front of research.
		Bogaevsky and Ishikawa studied singularities of Legendre mappings, in particular a generalisation of Mather’s theory to the Legendrian case.
	www.newton.cam.ac.uk /reports/0001/sgt.html (2778 words)

	INI Programme SGT
		As a consequence Singularity Theory lies at the crossroads of the paths connecting the most important areas of applications of mathematics with its most abstract parts.
		The main goal in most problems of singularity theory is to understand the dependence of some objects of analysis and geometry, or physics, or from some other science on parameters.
		In one sense singularity theory can be viewed as the modern equivalent of the differential calculus, and this explains its central position and wide applicability.
	www.newton.cam.ac.uk /programmes/SGT/sgt.html (457 words)

	Mathematical singularity - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-07-20)
		Suppose U is an open subset of C, a is an element of U and f is a holomorphic function defined on U {a}.
		Singularities and Groups in Bifurcation Theory : Volume 2 (Applied Mathematical Sciences)
		Singularity Theory I (Encyclopaedia of Mathematical Sciences, 6)
	encyclopedia.worldsearch.com /mathematical_singularity.htm (452 words)

	Singularity theory - The Jiggies Reference Guide (Site not responding. Last check: 2007-07-20)
		Projection is one way, very obvious in visual terms when three-dimensional objects are projected into two dimensions (for example in one of our eye); in looking at classical statuary the folds of drapery are amongst the most obvious features.
		The foundational work on smoothsingularities is formulated as the construction of equivalence relations on singular points, and germ (mathematics)s.
		Technically this involves group actions of Lie groups on spaces of jet; in less abstract terms Taylor expansions are examined up to change of variable, pinning down singularities with enough derivatives.
	www.jiggies.com /reference/Singularity_theory (913 words)

	Singularity Theory Yields Elegant Description of Polarized Sunlight - January, 2005 (Site not responding. Last check: 2007-07-20)
		Singularity theory offers a "beautiful" approach to explaining the polarization patterns observed in the sunlit sky.
		To test their theory, the scientists compared the intensity patterns it generated with those observed in polarization photographs taken with a digital camera and fish-eye lens.
		Although multiple-scattering theory can account for several additional atmospheric effects that influence the polarization patterns in the sunlit sky, such as haze and surface reflection, the elegance of singularity theory wins the beauty contest that is used to decide among otherwise similarly successful, competing scientific theories.
	www.photonics.com /spectra/tech/XQ/ASP/techid.1676/placement.HomeIndex/QX/read.htm (535 words)

	Mathematical singularity (Site not responding. Last check: 2007-07-20)
		In mathematics, a singularity is in general a point at whicha given mathematical object is not defined or lacks some "nice" property, such as differentiability.
		the point a is a removable singularity off if there exists a holomorphic function g defined on all of U such thatf(z)=g(z) for all z in U-{a}.
		the point a is an essential singularity off if it is neither a removable singularity nor a pole.
	www.therfcc.org /mathematical-singularity-69174.html (229 words)

	Mathematical singularity (Site not responding. Last check: 2007-07-20)
		In mathematics a singularity is in general a point at a given mathematical object is not defined lacks some "nice" property 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 }.
		This book is concerned primarily with a geometrical and in places, a topological approach to spacetime, leading to a full proof of one of Hawking's singularity theorems.The first part introduces the geometry of Minkowski Spacetime as..
	www.freeglossary.com /Mathematical_singularity (435 words)

	Arlie O. Pettters (Site not responding. Last check: 2007-07-20)
		The theory applies to any finite number of deflector planes and highlights the distinctions between single and multiple plane lensing.
		Among the lensing topics discussed are multiple quasars, giant luminous arcs, Einstein rings, the detection of dark matter and planets with lensing, time delays and the age of the universe (Hubble's constant), and microlensing of stars and quasars.
		Other mathematical topics discussed are Morse theory, Whitney singularity theory, Thom catastrophe theory, Mather stability theory, Arnold singularity theory, and the Euler characteristic via projectivized rotation numbers.
	www.math.duke.edu /~petters/apabsbook.html (250 words)

	CCSD thèses-EN-ligne: Some applications of singularity theory to the geometry of curves and surfaces
		Some applications of singularity theory to the geometry of curves and surfaces.
		We study the singularities occurring on the duals of symmetry sets and their generic transitions in 1-parameter families of curves $ gamma$.
		A first step to define an analogous theory to study the local rotational symmetry in the plane is given.
	tel.ccsd.cnrs.fr /documents/archives0/00/00/19/16/index_fr.html (737 words)

	Singularity Theory Encyclopedia Article, Definition, History, Biography (Site not responding. Last check: 2007-07-20)
		Looking For singularity theory - Find singularity theory and more at Lycos Search.
		Find singularity theory - Your relevant result is a click away!
		Look for singularity theory - Find singularity theory at one of the best sites the Internet has to offer!
	www.folkartmuseum.com /encyclopedia/Singularity_theory (1227 words)

	Real and Complex Singularities (Site not responding. Last check: 2007-07-20)
		The boundaries of singularity theory are broad and vague, connecting the most important applications of mathematics and science with more abstract areas.
		Optics, robotics, computer vision, Hamiltonian mechanics, bifurcation theory and differential equations are among the variety of topics that benefit from developments in the theory.
		With singularity theory encompassing more and more applications, Real and Complex Singularities provides insight into the future of this expanding field.
	www.ramex.com /title.asp?id=6362 (403 words)

	Singularity Theory - Cambridge University Press
		Singularity theory is a broad subject with vague boundaries.
		This volume consists of two dozen articles from some of the best known figures in singularity theory, and it presents an up-to-date survey of research in this area.
		Singular phenomena in kinematics P. Donelan and C. Gibson; 22.
	www.cambridge.org /catalogue/print.asp?isbn=0521658888&print=y (409 words)

	The geometry of resonance tongues: a singularity theory approach (Site not responding. Last check: 2007-07-20)
		Resonance tongues and tongue boundaries are studied for nondegenerate and (certain) degenerate Hopf bifurcations of maps using singularity theory methods of normal forms and universal unfoldings.
		We recover the standard theory of tongues (the nondegenerate case) in a straightforward way and we find certain surprises in the tongue boundary structure when degeneracies are present.
		For example, the tongue boundaries at degenerate singularities in weak resonance are much blunter than expected from the nondegenerate theory.
	www.math.uh.edu /~mg/reprints/abstracts/01BG.html (92 words)

	Universidade de Lisboa
		We intend to pursue the study of the singularities of Legendrian varieties.
		Roughly speaking a solution of an ODE on the Riemann sphere is a special function if the sheaf of solutions of the ODE is a rigid local system (cf K 96 book at Princeton UP).
		Neto and Silva intend to pursue the study of holonomic D-modules near a singular point of the characteristic variety.
	ptmat.lmc.fc.ul.pt /membros/memb-l1-pt.html (116 words)

	Reprint List by Keyword: Singularity theory
		The geometry of resonance tongues: a singularity theory approach.
		A singularity theory analysis of the thermal chainbranching model.
		A singularity theory approach to steady state bifurcation theory.
	www.math.uh.edu /~mg/reprints/key.Singularity_theory.html (180 words)

	Singularity (Site not responding. Last check: 2007-07-20)
		mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways
		Singularities of Differentiable Maps, Vol II: Monodromy and Asymptotic Integrals (Monographs in Mathematics)
		Resolution of Singularities of Embedded Algebraic Surfaces (Springer Monographs in Mathematics)
	www.freeglossary.com /Singularities (212 words)

	The Inverted Pendulum: A Singularity Theory Approach - Broer, Hoveijn, Noort, Vegter (ResearchIndex) (Site not responding. Last check: 2007-07-20)
		The qualitative dynamics of the Poincar'e map corresponding to the central periodic solution is studied via an approximating integrable normal form.
		At bifurcation points we construct local universal models in the appropriate symmetry context, using equivariant singularity theory.
		Broer, H.W., Hoveijn, I., Noort, M. van and Vegter, G.: The inverted pendulum: a singularity theory approach, Journ.
	citeseer.ist.psu.edu /broer98inverted.html (647 words)

	Singularity Fun Theory (Site not responding. Last check: 2007-07-20)
		Singularity Fun Theory is in the early stages of development, so please don't expect a full mathematical analysis.
		In either case, the solution from the perspective of Singularity Fun Theory is the same; make the autopotent design decision to exempt low-level introspection from sphexishness (that is, from the internal perception of sphexishness that gives rise to boredom).
		The key requirement for local altruism (that is, altruism toward a loved one) is that the loved one greatly desires something that he/she/ve would not otherwise be able to obtain.
	www.sysopmind.com /essays/funtheory.html (3306 words)

	Singularity Theory and Bifurcation Theory (Site not responding. Last check: 2007-07-20)
		Catastrophe theory and equations of state: conditions for a butterfly singularity, Math.
		Progress in catastrophe theory, Report of Conference on Dynamic Nonlinear Theory and General Urban/Regional Systems, Washington DC 1981, Univ. of Kansas 1982, 367-376.
		Overview of catastrophe theory, in Catastrophe Theory: State of the Art and Potential Applications, F.C.Johnson and R.C.Lacher, ONR Report NR 170-938 (1983) 9-14.
	members.aol.com /istewjoat/RESEARCH/ressing.html (267 words)

	Singularity theory bei eLexi - das Onlinelexikon (Site not responding. Last check: 2007-07-20)
		\nIn Ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law.
		In economics, the theory of comparative advantage explains why it can be beneficial for two countries to trade, even though one of them may be able to produce every kind of item more cheaply than the
		The "Colonel Bogey March" is one of the most successful marches ever published.
	www.elexi.de /en/s/si/singularity_theory.html (1156 words)

	Singularity Theory and Phantom Edges in Scale Space
		The process of detecting edges in a one-dimensional signal by finding the zeros of the second derivative of the signal can be interpreted as the process of detecting the critical points of a general class of contrast functions that are applied to the signal.
		The points in the scale space at which these events occur are seen to be singular points of the contrast function.
		Using ideas from singularity, or catastrophy theory, the scale map contours near these singular points are found to be either vertical or parabolic.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/trans/tp/&toc=comp/trans/tp/1988/05/i5toc.xml&DOI=10.1109/34.6782 (525 words)

	Citebase - Chiral Rings, Singularity Theory and Electric-Magnetic Duality
		Theories with gauge-mediated supersymmetry breaking provide an interesting alternative to the scenario in which the soft terms of the low-energy fields are induced by gravity.
		Recently N. Seiberg has shown that certain N=1 supersymmetric Yang -- Mills theories have the property that their infrared physics can be equivalently described by {\it two different} gauge theories, with the strong coupling region of one corresponding to the weak coupling region of the other.
		Classically, the singular points on the moduli space are associated with extra massless particles.
	citebase.eprints.org /cgi-bin/citations?archiveID=oai:arXiv.org:hep-th/9510222 (1145 words)

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