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Topic: Kolmogorov space

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Preregular space

Topological space

Hausdorff space

Affine space

Base (topology)

Open mapping theorem

Axiom of dependent choice

Vitali set

Adjoint functors

General topology

Free object

Curve

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Isomorphism

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	Regular space
		X is a regular space iff, given any closed set F and any point x that does not belong to F, there are a neighbourhood U of x and a neighbourhood V of F that are disjoint.
		On the other hand, spaces that are regular but not completely regular, or preregular but not regular, are usually constructed only to provide counterexamples to conjectures, showing the boundaries of possible theorems.
		Suppose that A is a set in a topological space X and f is a continuous function from A to a regular space Y.
	www.ebroadcast.com.au /lookup/encyclopedia/t3/T3_space.html (917 words)

	Kolmogorov space: Definition and Links by Encyclopedian.com
		...Kolmogorov space Kolmogorov space In topology and related branches of...the T 0 spaces or Kolmogorov spaces form a broad class of well behaved topological spaces.The T...well behaved topological spaces.The T 0 condition is one of the separation axioms.
		This constructs a quotient space of the original seminormed vector space, and this quotient is a normed vector space.
		From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it's a vector space, and it has a seminorm, and these define a pseudometric and a uniform structure that are compatible with the topology.
	www.encyclopedian.com /ko/Kolmogorov-space.html (1015 words)

	Kolmogorov biography
		Kolmogorov's mother had been on a journey from the Crimea back to her home in Tunoshna near Yaroslavl and it was in the home of his maternal grandfather in Tunoshna that Kolmogorov spent his youth.
		Kolmogorov was appointed a professor at Moscow University in 1931.
		If Kolmogorov made a major contribution to Hilbert's sixth problem, he completely solved Hilbert's Thirteenth Problem in 1957 when he showed that Hilbert was wrong in asking for a proof that there exist continuous functions of three variables which could not be represented by continuous functions of two variables.
	www-groups.dcs.st-and.ac.uk /~history/Biographies/Kolmogorov.html (2094 words)

	Andrey Kolmogorov - Wikipedia, the free encyclopedia
		Andrey Nikolaevich Kolmogorov (Russian: Андре́й Никола́евич Колмого́ров) (April 25, 1903 - October 20, 1987) was a Soviet mathematician who made major advances in the fields of probability theory and topology.
		He was a founder of algorithmic complexity theory which is often referred to as simply Kolmogorov complexity theory.
		The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography.
	en.wikipedia.org /wiki/Kolmogorov (273 words)

	Britain.tv Wikipedia - Space
		Another way to frame this is to ask, "Can space itself be measured, or is space part of the measurement system?"?title=The same debate applies also to time, and an important formulation in both areas was given by Immanuel Kant.
		Schopenhauer, in the preface to his On the Will in Nature, stated that "space is the condition of the possibility of juxtaposition."?title=This is in accordance with Kant's understanding of space as a form in the mind of an observing subject.
		In physics, time and mass are fundamental quantities and space is defined via time â” as the distance light travels in exactly 1/299 792 458 second.
	www.britain.tv /wikipedia.php?title=Space (1459 words)

	Kolmogorov space - Biocrawler (Site not responding. Last check: 2007-11-05)
		From the point of view of topology, the seminormed vector space that we started with has a lot of extra structure; for example, it is a vector space, and it has a seminorm, and these define a pseudometric and a uniform structure that are compatible with the topology.
		And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics) generally want to study.
		The Legacy of Andrei Nikolaevich Kolmogorov (http://www.kolmogorov.com/) Curriculum Vitae and Biography.
	www.biocrawler.com /encyclopedia/T0 (1287 words)

	[No title]
		Kolmogorov complexity is a modern notion of randomness dealing with the quantity of information in individual objects; that is, pointwise randomness rather than average randomness as produced by a random source.
		It was proposed by A.N. Kolmogorov in 1965 to quantify the randomness of individual objects in an objective and absolute manner.
		`Kolmogorov' complexity was earlier proposed by Ray Solomonoff in a Zator Technical Report in 1960 and in a long paper in Information and Control in 1964.
	homepages.cwi.nl /~paulv/kolmogorov.html (824 words)

	Probability - Kolmogorov
		Definition: If (S,S,m) is a measure space, and m(S) = 1 then (i) m is a probability measure, and (ii) (S,S,m) is a probability triple.
		S is called the sample space; a point in S is called a sample point.
		The (first 3) Kolmogorov axioms and the Bayesian axioms can be derived from each other (NO!), and we sometimes blur the distinction.
	www.quantum.bowmain.com /probability.htm (533 words)

	The Kolmogorov turbulence theory in the light of six-dimensional Navier-Stokes' equation (Site not responding. Last check: 2007-11-05)
		The classical turbulence theory by Kolmogorov is reconsidered using Navier-Stokes' equation generalized to 6D physical-plus-eddy space.
		Strong pseudo-singularity is shown to reveal itself along the boundary `ridge' line separating the dissipation and inertial sub-ranges surrounding the origin of the eddy space.
		It is supported by the observation that the universal power spectrum calculated rediscovers the Kolmogorov's -5/3 power law as independent of the dimensional approach.
	www.ca-homes.com /science/tsuge-8 (107 words)

	Chronology for 1950 to 1960
		Serre uses spectral sequences to the study of the relations between the homology groups of fibre, total space and base space in a fibration.
		of a space and to prove important results on the homotopy groups of spheres.
		Kolmogorov solves "Hilbert's Thirteenth Problem" on continuous functions of three variables which cannot be represented by continuous functions of two variables.
	www-groups.dcs.st-and.ac.uk /~history/Chronology/1950_1960.html (231 words)

	Symmetric Kolmogorov phase space
		on the embedding space we notice that it is invariant under the reflection r.
		As expected, a Galerkin projection onto those three K-L eigenmodes, augmented by a fourth mode representing the basic Kolmogorov flow 67#67, generates an ODE which has, for 68#68 a periodic orbit with reflection symmetry.
		However, in order to resolve the homoclinic behavior properly in a Galerkin projection we found we would need many more modes making it a system that is almost as complicated to analyze as the original PDE simulations.
	math.la.asu.edu /~dieter/papers/K-L-symmetry/sym_kol_rev/node5.html (691 words)

	Kolmogorov Turbulence Theory in 6D Navier-Stokes Equation
		Its success is the finding that the dissipation region in the sense of Kolmogorov has the shape like a 'worm' with the radius of the Kolmogorov length aligned randomly streamwise.
		In this paper the classical Kolmogorov turbulence theory is reconsidered in the light of the 6D Navier-Stokes equation.
		It is one of the new findings that have been hidden behind the classical dimensional analysis, yet is totally consistent with the Kolmogorov theory, leading, as an example, to the -5/3 law for the power spectrum.
	www.ca-homes.com /science/tsuge-8/abstract.htm (388 words)

	Kolmogorov space - Wikipedia, the free encyclopedia
		spaces or Kolmogorov spaces, named after Andrey Kolmogorov, form a broad class of well-behaved topological spaces.
		Topological spaces X and Y are Kolmogorov equivalent when their Kolmogorov quotients are homeomorphic.
		Many properties of topological spaces are preserved by this equivalence; that is, if X and Y are Kolmogorov equivalent, then X has such a property if and only if Y does.
	en.wikipedia.org /wiki/Kolmogorov_space (1202 words)

	Kolmogorov Complexity, Complexity Cores, and the Distribution of Hardness (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Problems that are complete for exponential space are provably intractable and known to be exceedingly complex in several technical respects.
		However, every problem decidable in exponential space is efficiently reducible to every complete problem, so each complete problem must have a highly organized structure.
		The authors have recently exploited this fact to prove that complete problems are, in two respects, unusually simple for problems in expontential space.
	citeseer.ist.psu.edu /197299.html (803 words)

	FAH Excerpt: Separation (Site not responding. Last check: 2007-11-05)
		The two entries in each row of the chart are closely related: a space satisfies the condition in the left column if and only if the space is Kolmogorov and satisfies the condition in the right column in the same row.
		spaces, but the abstract theory can be developed more clearly if we classify properties according to the various axioms in the chart.
		space'', and ``completely regular Hausdorff space'' are used interchangeably in the literature; they all describe the same thing.
	www.math.vanderbilt.edu /~schectex/ccc/excerpts/separat.html (481 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		-space or a Kolmogorov space, if for any two distinct points of the space there exists an open set containing one of the points but not containing the other.
		It is precisely in these spaces that the closure of the union of any family of sets is the same as the union of the closures of these sets.
		Thus, the study of discrete spaces is equivalent to that of partially ordered sets.
	eom.springer.de /K/k055670.htm (279 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		] A topological space where, for each pair of points, at least one has a neighborhood not containing the other.
		] A topological space where, for each pair of distinct points, each one has a neighborhood not containing the other.
		] A region of space bounded by a tubular surface consisting of the lines of force which pass through a given closed curve.
	www.accessscience.com /Dictionary/T/T29/DictT29.html (2557 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		For a Markov process with a countable set of states, the transition function is completely determined by the transition probabilities
		In the case of a finite number of states, equations (3) and (4) hold, provided that the limits in (5) exist.
		Besides in a probabilistic context, Kolmogorov equations of course also occur as equations modelling real diffusions, such as the diffusion of molecules of a substance through porous material or the spread of some property through a biological structured population.
	eom.springer.de /K/k055700.htm (267 words)

	The Unity of Mathematics
		In historical terms, this result precedes Gelfand's theorem and is the foundation for it-he starts with a general commutative Banach algebra and reconstructs a space from it-thus establishing in what sense that the space to algebra correspondence is surjective, and hence by the aforementioned theorem, bi-unique.
		Stone's paper is the true starting point of these ideas, but this paper of Kolmogorov and Gelfand is the second landmark on the path that led to Grothendieck's concept of a scheme(with Gelfand's representation theorem probably as the third).
		As an aside, this paper was not Kolmogorov's first foray into topological algebra-earlier he conjectured the possibility of a classification of locally compact fields, a problem which was solved by Pontryagin.
	log24.com /log03/0902.htm (1986 words)

	Tenure-track Faculty
		The subject deals with (possibly infinite-dimensional) vector spaces with symmetry (manifested by ``actions'' of groups); through these linear spaces, the geometric, analytic, and algebraic properties of the groups are explored.
		His main interest is intersection theory on the moduli space of curves, with specific attention to the classes that show up in enumerative geometry problems.
		She is also interested in undergraduate research, in mathematics competitions at the high school and college levels, and in the preparation of mathematics graduate students as teachers of mathematics.
	www.math.okstate.edu /undergrad/handbook/node76.html (1435 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		We apply recent results on extracting randomness from independent sources to ``extract'' Kolmogorov complexity.
		This result holds for both classical and space-bounded Kolmogorov complexity.
		We use the extraction procedure for space-bounded complexity to establish zero-one laws for polynomial-space strong dimension.
	www.cs.iastate.edu /~pavan/papers/kex.html (126 words)

	New 'metal sandwich' may break superconductor record, theory suggests \| Commercial Space Watch
		He said that once a new superconductive material is identified, scientists typically can manipulate the substance -- twisting it or doping it with other elements ñ to create related structures that might have even more appealing properties.
		Curtarolo and Kolmogorov decided it was time to try something else.
		Additional calculations identified the binary alloy lithium monoboride as a promising candidate that might be both structurally stable and superconductive at temperatures that exceed those of the current binary alloy record-holder.
	www.comspacewatch.com /news/viewpr.html?pid=19790 (1154 words)

	Citations: About Realization of Sets in 3-dimensional Space - Kolmogorov, Bardzin (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		Kolmogorov and Y. Bardzin, About Realization of Sets in 3dimensional Space, Problems in Cybernetics, 1967, pp.
		In this paper we present an algorithm for producing 3 D orthogonal drawings of graphs of maximum degree six, and a second algorithm that produces 3 D orthogonal drawings of graphs of....
		Kolmogorov and Y. Bardzin, About Realization of Sets in 3-dimensional Space, Problems in Cybernetics, 1967, pp.
	citeseer.ist.psu.edu /context/794046/0 (401 words)

	Solar Eruptive Phenomena: Reconciling Observations and Models I - SPA-Solar and Heliospheric Physics [SH]
		Following the catastrophe, on the other hand, magnetic reconnection plays an essential role in assisting the loss of equilibrium to smoothly develop a plausible eruption by converting the magnetic energy previously stored in the system into heat and kinetic energy at a reasonable rate.
		In this presentation we will briefly introduce the fundamental properties of the catastrophe models of solar eruptions, display the expected observational consequences, and discuss what consequences can be directly deduced from the model calculations and what needs to be further investigated via improving our existing models on the basis of more rigorous calculations.
		We examine the relationship between the acceleration of the flux rope and the thermal energy release rate in a loss-of-equilibrium model of coronal mass ejections.
	www.agu.org /meetings/fm05/fm05-sessions/fm05_SH11C.html (1563 words)

	An Introduction to Kolmogorov Complexity and Its Applications
		Kolmogorov complexity is a central concept and a powerful tool in the understanding of the quantitative nature of information and its processing and transmission.
		The basic concepts of Kolmogorov complexity should be understood by any technically educated person, and they should be studied by all computer scientists."
		Together with Ming Li they pioneered applications of Kolmogorov complexity and co-authored ``An Introduction to Kolmogorov Complexity and its Applications,'' Springer-Verlag, New York, 1993 (2nd Edition 1997), parts of which have been translated into Chinese, Russian and Japanese.
	www.tcs.hut.fi /~pkaski/kca (454 words)

	Enumerations of the Kolmogorov function , Richard Beigel, Harry Buhrman, Peter Fejer, Lance Fortnow, Piotr Grabowski, ...
		For any given constant k, the Kolmogorov function is k-enumerable relative to an oracle A if and only if A is at least as hard as the halting problem.
		Although every 2-enumerator for C is Turing hard for K, we show that reductions must depend on the specific choice of the 2-enumerator and there is no bound on the quantity of their queries.
		PSPACE is the class of all sets A for which there is a polynomially bounded function g such that there is a polynomial time Turing reduction which reduces A to every strong 2-enumerator for g.
	projecteuclid.org /Dienst/UI/1.0/Summarize/euclid.jsl/1146620156 (594 words)

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