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Topic: Mathematical singularity

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	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.
		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).
		In linear algebra a square matrix is said to be singular when it is not invertible, that is when its determinant is zero.
	en.wikipedia.org /wiki/Mathematical_singularity (599 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 (178 words)

	SANKHYAKHARIKA (Site not responding. Last check: 2007-11-06)
		Not withstanding the power of mathematics, scientific analysis must also be unequivocally bound to such rigorous human logic, that, it is deemed to be of an axiomatic nature and it should only be supported by mathematical calculations to maintain precision and logical continuity.
		The singular element in the midst of an infinite sea of such components cannot at all have the freedom to move, transfer or translocate in any manner whatsoever in a relative way or in other words each component is always at rest relative to each other.
		Any mathematical description of phenomenon in physics etc. is based on movement or displacement and therefore a definition of singularity reflects the state of its movement, whereas the holographic Purusha cannot move in any way.
	chennai.sancharnet.in /gopsvas/SankEinstein.htm (9181 words)

	Singular
		In complex analysis, a singular point of a function f is a point such that f is analytic in its neighbourhood but not at it.
		A measure or distribution is called singular if its support has zero Lebesgue measure.
		SINGULAR is also the name of a computer algebra system.
	www.brainyencyclopedia.com /encyclopedia/s/si/singular.html (200 words)

	The Singularity (Site not responding. Last check: 2007-11-06)
		The "Singularity" was a term originally coined by Vernor Vinge, a mathematician and science-fiction author, to describe the breakdown in our predictions of the future that occurs when some form of greater-than-human intelligence comes into play - the moment when human intelligence, the source of technology, is improved by that technology.
		The term "Singularity" was coined by Vinge in analogy with the singularity at a center of a fl hole, where our current understanding of physics breaks down; beyond the point where transhuman intelligence exists, our model of the future breaks down.
		Vernor Vinge's original definition of the Singularity is probably the most important, since it captures the essential idea of a change that is different in kind from any that humanity has previously experienced.
	www.singinst.org /intro/singularity.html (518 words)

	CAUSE AND EFFECT AND THE FIRST CAUSE.
		An example of simple mathematical singularity is the point when x = 1 in 1/(x-1) It simply means that at that point the value of x is indefinite.
		In fact the singularities in elementary particle theories have been removed by normalization process in Physics so that we could still handle the Physics with laws that we are familiar with.
		Singularity is therefore the greatest argument for the existence of God beyond this physical universe.
	www.acns.com /~mm9n/cause/cause.htm (2062 words)

	Research Groups
		SINGULAR is a system to compute with ideals and modules over commutative and non-commutative polynomial rings and over localizations of the commutative polynominal ring.
		SINGULAR is used a great deal all over the world and is one of the most efficient systems for computations with polynomials, as has been recognized by being awarded the 2004 Richard D. Jenks Memorial Prize for Excellence in Software Engineering for Computer Algebra.
		Singularity theory is a relatively new branch of mathematics.
	www.mathematik.uni-kl.de /grad_school/navigator/research_groups/research_groups.html (1872 words)

	INI Programme SGT
		Singularities arise naturally in a huge number of different areas of mathematics and science.
		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 spite of its fundamental character, and the central position it now occupies in mathematics, singularity theory is a surprisingly young subject.
	www.newton.cam.ac.uk /programmes/SGT/sgt.html (457 words)

	Singularity \| MetaFilter (Site not responding. Last check: 2007-11-06)
		The third fallacy is that singularity adocates consider technological advancement in isolation, and not techonolgy as part of an ecological relationship with other economic factors.
		Consciousness singularity - a state of consciousness in which the level of concentration and control of mental chaotic tendencies is so strong that the usual linear registration of events breaks down.
		The crux of the singularity is the recursion of intelligence.
	www.metafilter.com /mefi/41801 (6624 words)

	A Comment on the Singularity
		A "tech Singularity" is a projected brief interval of time in which the practical innovation rate becomes so fast as to seem essentially instantaneous to humans.
		The Singularity will probably be brought about by the convergence of a large number of interrelated technologies, a process that Kurzweil has described very well.
		On the other hand, the Singularity may be a multifurcation point, so that a small nudge in one direction or another can have a big effect on what path is taken.
	home.mchsi.com /~deering9/bensing.html (553 words)

	BBC - h2g2 - The Vingean Singularity
		Although the Singularity is sometimes referred to as the brainchild of Vinge, it was briefly mentioned in the 1950s by John von Neumann, as Raymond Kurzweil points out in his excellent book précis The Singularity is Near.
		This is a mathematical singularity, corresponding to the Singularity.
		The term 'singularity' (lower case 's') refers to singularities in general, such as mathematical singularities and the centre of a fl hole.
	www.bbc.co.uk /h2g2/guide/A569522 (1892 words)

	Singularity Math Trialogue
		Ungoverned AI progress will resemble a singularity as far as human scope and time scale are concerned, and result in a new regime with properties that are not simply an extrapolation of what happened before the "singularity".
		RAY: Yes, I understand that, and I use the term "singularity" in this sense, that is a "rupture" in the fabric of human history caused by the cycle of strong AI (nonbiological intelligence) having access to its own source code and improving it.
		However, I am not definining the technological "singularity" as a mathematical singularity, but (like Vinge) using the word as a metaphor for an event with an event horizon beyond which it is difficult to penetrate (analogously with a fl hole).
	www.kurzweilai.net /articles/art0151.html?m=1 (3546 words)

	The Socio-technological Singularity (Site not responding. Last check: 2007-11-06)
		In mathematics, the point where the value of an otherwise finite and continuous function becomes infinite is called a "singularity".
		Since traditional mathematical operations, like differentiation, integration and extrapolation, are based on continuity, they cannot be applied to this singular point.
		The inside of a "fl hole" is an example of a singularity in the geometry of space.
	pespmc1.vub.ac.be /SINGULAR.html (1142 words)

	Singularities in Mathematical Models (Site not responding. Last check: 2007-11-06)
		The term `singularity' is used in applied mathematics to indicate that a conventional way of modelling a certain physical process mathematically leads to consequences which for some reasons cannot be accepted.
		If a solution exists but some physical arguments indicate a `singularity', one has to find out whether it appeared due to simplifications and/or additional assumptions made in the process of obtaining the solution (see [1] for a review of such cases) or whether it is inherent in the very formulation of the problem.
		A standard remedy for the singularity is to use `microscopic cutoffs' to stop the solution on its way to the singularity which is inherent in the very formulation of the problem.
	web.mat.bham.ac.uk /Y.D.Shikhmurzaev/singul.htm (1712 words)

	What is the Technological Singularity?
		The Technological Singularity, or simply "Singularity," is a multi-faceted concept in futurism with several overlapping and sometimes conflicting definitions.
		The term "Technological Singularity" was coined by analogy to the singularity in the center of a fl hole, where the forces of nature become so intense and unpredictable that our ability to calculate the behavior of matter in these circumstances drops to zero.
		However, this notion of the singularity is not the same as Vinge intended; referring to the emergence of superhuman intelligence, along with superhuman thinking speeds.
	www.wisegeek.com /what-is-the-technological-singularity.htm (677 words)

	Paradoxes Resolved, Origins Illuminated - Black holes
		They were based on an extra mathematical singularity in the spacetime metric that occurred because Einstein truncated an expansion in potential to first order, seeing no point in retaining second-order effects that were far beyond any conceivable means of detection at that time.
		The first singularity is at the origin when distance goes to zero, and can be avoided by the finite size of masses.
		This second singularity corresponds to what we now call the "event horizon" and cannot be eliminated in the present form of the equations.
	www.metaresearch.org /msgboard/topic.asp?TOPIC_ID=406 (2590 words)

	Paths to the Singularity
		This not actually a mathematical singularity, but the rate of progress will become so fast over such a short amount of time that there is no effective difference.
		I believe that the singularity will occur when a technology-based entity comes into existence that is modestly more intelligent than the most intelligent human, with "intelligence" defined as "technical creativity." Since this entity is technology-based and technically creative, it can immediately increase its own capabilities in a rapid feedback loop.
		However, by the time of the singularity, the network will be an essential element of nearly all economic activity, so the shutdown will create havoc and is likely to result in civil unrest or warfare.
	www.ciar.org /~ttk/techsing/techsing.paths.html (2863 words)

	Light Cone Dynamics
		In this, T0 is "present" or the mathematical singularity, and t1 is the time of the event when we started the evaluation.
		But, it appears that actually it’s not necessary, but mathematically it is. This is because it still is required (remember we still have to do the outside observer, so we have to take everything in account).
		But, in order to keep them correct, mathematically speaking, we have to assume that some physical change happens when we are at this point on the evolution of t+.
	www.lightcones.blogspot.com (4360 words)

	Atheism Awareness (Site not responding. Last check: 2007-11-06)
		He had reached a singularity (note the often misuse of this word and huge misunderstanding of what a singularity is and is not).
		A singularity (referring to the "Big Bang") is not the astrophysics singularity (point in space-time where gravitational forces cause matter to have infinite density and infinitesimal volume, and space and time become infinitely distorted).
		It is the mathematical singularity (point where the derivative does not exist for a given function of a random variable, but every neighborhood of which contains points for which the derivative exists).
	www.alabamaatheist.org /awareness/questions/origins.htm (965 words)

	fUSION Anomaly. Singularity
		This is a singular event in the sense that it happens once in human history.
		The singularity is a mathematical paradox because it is infinitely massive yet infinitely small.
		chakra which relates to the singularity is located between the navel and the pelvic bone.
	fusionanomaly.net /singularity.html (2303 words)

	Edge: THE EVOLUTION OF COOKING
		So I became a student of technology trends, and have developed mathematical models about how technology evolves in different areas like computers, electronics in general, communication storage devices, biological technologies like genetic scanning, reverse engineering of the human brain, miniaturization, the size of technology, and the pace of paradigm shifts.
		I've now refined these mathematical models, and have begun to really examine what the 21st century would be like.
		We actually have very detailed mathematical models of several dozen regions of the human brain and how they work, and have recreated their methodologies using conventional computation.
	www.edge.org /3rd_culture/kurzweil_singularity/kurzweil_singularity_p2.html (1242 words)

	Wikinfo \| Gravitational singularity
		In a rotating fl hole, the singularity occurs on a ring, and is called a "ring singularity".
		However, in 1991 Shapiro and Teukolsky performed computer simulations of a rotating plane of dust which indicated that general relativity allows for naked singularities.
		It is generally assumed that a theory of quantum gravity - a theory that unifies general relativity with quantum mechanics - will provide a better description of what actually occurs where general relativity predicts a singularity.
	www.wikinfo.org /wiki.php?title=Gravitational_singularity (428 words)

	WISDOM AND INSPIRATION FROM OUR VISITORS
		A mathematical singularity is, by its very nature, something we have no concept of.
		That is mathematical fact and has been done via such systems as topology, chaos theory, perturbation theory, modern algebra and the realm of probability.
		You can study mathematics and physics to the point of understanding the language it is written in, then using that language to operate in those dimensions.
	www.eclecticchurch.org /WISDOM.htm (1312 words)

	Singularity Formation in Free Surface Flows (Site not responding. Last check: 2007-11-06)
		To a physicist they are also appealing because their mathematical formulation is in many senses unique: it has to self-consistently describe the flow inside each of the fluids (water, air) as well as the motion of their interface.
		One peculiarity is due to a discontinuous change in the physical properties of the fluids at the interface, which shows up as a mathematical singularity in the equations of fluid motion.
		Such changes are usually associated with the appearance of an additionional singularity - in the curvature of the interface.
	cns.physics.gatech.edu /~roman/singularity.html (175 words)

	Paradoxes Resolved, Origins Illuminated
		[quote]GR and related abstract mathematical concepts obviously go against the common sense of the masses...[/quote] Much of my own recent writing has the theme that GR has more than one physical interpretation (long recognized), and that only the so-called "geometric interpretation" violates common sense by requiring that space curves.
		GR and related abstract mathematical concepts obviously go against the common sense of the masses but is there a higher level of common sense where these are more or less self-evident truths the majority of people fail to comprehend simply because they haven't reached that kind of mental sophistication yet?
		GR and related abstract mathematical concepts obviously go against the common sense of the masses...
	www.metaresearch.org /msgboard/post.asp?method=ReplyQuote&REPLY_ID=3095&TOPIC_ID=227&FORUM_ID=9 (1669 words)

	Science a GoGo Forums: Singularity in Real Life? (Site not responding. Last check: 2007-11-06)
		The major issue is not that a mathematical singularity exists in a theory, but the physical meaning of that singularity.
		You cannot enrich mathematics in relation with nature, because while the former is abstract, our knowledge of the latter is empirical, obeyng rules that are not mathematical in essence.
		What matters is only the meaning of the mathematical symbols, and physics assumes its right to use math as it sees fit, and within limits that are not mathematical in nature.
	www.scienceagogo.com /cgi-bin/ultimatebb.cgi?ubb=get_topic;f=1;t=000310;p=2 (4475 words)

	Amazon.com: Singularity Theory and Gravitational Lensing: Books: Arlie O. Petters,Harold Levine,Joachim Wambsganss (Site not responding. Last check: 2007-11-06)
		This monograph, unique in the literature, is the first to develop a mathematical theory of gravitational 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), microlensing of stars and quasars.
		Mathematical topics discussed: Morse theory, Whitney singularity theory, Thom catastrophe theory, Mather stability theory, Arnold singularity theory, and the Euler characteristic via projectivized rotation numbers.
	www.amazon.com /exec/obidos/tg/detail/-/0817636684?v=glance (621 words)

	MMJ: Vol.1 (2001), N.4. ``Fundamental Mathematics Today'' Conference
		The main goal of the Conference was to present to the current students and graduates of the University a panorama of modern mathematics, stressing on the areas well established at the University.
		A symplectic reflection algebra is the algebra of deformations of a pair (a symplectic space, a group of its symplectomorphisms generated by reflections).
		Vershik (Peterburg Division of Mathematical Institute RAS) Metric classification of functions of several arguments, random matrix and space of Polish spaces, The space of degenerate random tensors is closely related to the classification problem for functions of several arguments.
	www.ams.org /distribution/mmj/vol1-4-2001/fundmathtoday.html (1898 words)

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