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	Kolmogorov space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		More generally, functions are identified in the construction of the quotient space precisely when they are topologically indistinguishable in the original seminormed space.
		Topological spaces X and Y are Kolmogorov equivalent iff their Kolmogorov quotients are homeomorphic.
		The nice thing about Kolmogorov equivalence is that 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 iff Y does.
	www.absoluteastronomy.com /encyclopedia/k/ko/kolmogorov_space.htm (1511 words)

	Hausdorff space - Wikipedia, the free encyclopedia
		A topological space is preregular if and only if its Kolmogorov quotient is Hausdorff.
		Pseudometric spaces typically are not Hausdorff, but they are preregular, and their use in analysis is usually only in the construction of Hausdorff gauge spaces.
		In contrast, non-preregular spaces are encountered much more frequently in abstract algebra and algebraic geometry, in particular as the Zariski topology on an algebraic variety or the spectrum of a ring.
	en.wikipedia.org /wiki/Separated_uniformity (1227 words)

	20th WCP: Coherence and Epistemic Rationality
		Betting quotients are then defined in the usual way in terms of odds, so that if a person's subjectively fair odds on h are q, then her subjectively fair betting quotient for h is q/(q +1).
		Since a set of betting quotients which is bound to lose must confer a advantage to one side, they cannot all be fair.
		She then believes a proposition p, namely that the fair betting quotient is x, yet p is logically false and so she has committed a logical error.
	www.bu.edu /wcp/Papers/TKno/TKnoVine.htm (3166 words)

	Kolmogorov space - Wikipedia, the free encyclopedia
		And it is a Hilbert space that mathematicians (and physicists, in quantum mechanics) generally want to study.
		(R) usually denotes the Kolmogorov quotient, the set of equivalence classes of square integrable functions which differ on sets of measure zero, rather than simply the vector space of square integrable functions which the notation suggests.
		The Legacy of Andrei Nikolaevich Kolmogorov Curriculum Vitae and Biography.
	en.wikipedia.org /wiki/Kolmogorov_quotient (1253 words)

	A Conversation with Joe Doob
		Kolmogorov's 1933 monograph on the foundations of (mathematical) probability appeared just when I was desperately trying to find out what the subject was all about.
		Kolmogorov defined a random variable as a measurable function on a probability measure space.
		I had already noted that the quotient of a positive superharmonic function divided by a positive harmonic function satisfied an average inequality like that of a superharmonic function, with an averaging measure depending on the denominator function.
	www.dartmouth.edu /~chance/Doob/conversation.html (7051 words)

	Hausdorff space - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-10-21)
		They also arise in the model theory of intuitionistic logic: every complete Heyting algebra is the algebra of open sets of some topological space, but this space need not be preregular, much less Hausdorff.
		In fact, a quotient space of a Hausdorff space X is itself Hausdorff if and only if the kernel of the quotient map is closed as a subset of the Cartesian product X × X.
		Compact preregular spaces are normal, meaning that they satisfy Urysohn's lemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
	xahlee.org /_p/wiki/Hausdorff_space.html (828 words)

	Hausdorff space -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-21)
		A topological space is preregular if and only if its (Click link for more info and facts about Kolmogorov quotient) Kolmogorov quotient is Hausdorff.
		Almost all spaces encountered in (The abstract separation of a whole into its constituent parts in order to study the parts and their relations) analysis are Hausdorff; most importantly, the (Any rational or irrational number) real numbers are a Hausdorff space.
		Let f : X → Y be a (Click link for more info and facts about quotient map) quotient map with X a compact Hausdorff space.
	www.absoluteastronomy.com /encyclopedia/H/Ha/Hausdorff_space.htm (1626 words)

	Conceptual Proofs of L log L Criteria for Mean Behavior of Branching Processes - Lyons, Pemantle, Peres (ResearchIndex) (Site not responding. Last check: 2007-10-21)
		The Kesten-Stigum Theorem is a fundamental criterion for the rate of growth of a supercritical branching process, showing that an L log L condition is decisive.
		In critical and subcritical cases, results of Kolmogorov and later authors give the rate of decay of the probability that the process survives at least n generations.
		The multi type case has an additional difficulty not present in the single type case: namely, the convergence of the quotient in (1) is no...
	citeseer.ist.psu.edu /lyons95conceptual.html (768 words)

	Mathematics of Computer Algebra and Analysis - MOCAA (Site not responding. Last check: 2007-10-21)
		This main tool is an adaption of the theory of Minimum Description Length which is closely related to the Kolmogorov complexity.
		This theory justifies the use of various "magic constants" in Maple's simplifier for selecting when one representation of an algebraic formula is "simpler" than another.
		Monagan and R. Pearce have developed a package for computing in quotient rings and have developed an algorithm for simplifying fractions to lowest terms when the quotient ring is an integral domain.
	www.cecm.sfu.ca /~pborwein/MITACS/AchievementsSummary.htm (897 words)

	Kolmogorov space : Kolmogorov equivalence (Site not responding. Last check: 2007-10-21)
		This quotient space is the Kolmogorov quotient of
		The nice thing about Kolmogorov equivalence is that many properties of topological spaces are preserved by this equivalence; that is, if
		Botcher's mattress, with an expression that was in the House that afternoon in announcing his opposition policy.
	www.city-search.org /ko/kolmogorov-equivalence.html (1166 words)

	Fuzzy Sets and Systems. (Site not responding. Last check: 2007-10-21)
		M.A. Samhan, Fuzzy quotient algebras and fuzzy factor congruences, Fuzzy Sets and Systems 73 (2) (1995) pp.
		Mashinchi, M. Mukaidono, Generalized fuzzy quotient subgroups, Fuzzy Sets and Systems 74 (2) (1995) pp.
		Sang Yeol Joo, Yun Kyong Kim, Kolmogorov's strong law of large numbers for fuzzy random variables, Fuzzy Sets and Systems 120 (3) (2001) pp.
	www.elsevier.com /cdweb/journals/01650114/viewer.htt?viewtype=keywords&rangeselected=94 (883 words)

	Information
		The standard, and widely accepted, formal axiomatization of probability theory is due to Kolmogorov.
		One philosophical issue concerns just what is meant by "degree of belief." It's easy enough to see intuitively what this is, but not easy to define it in a non-circular way.
		There are some theorems (known as "Dutch Book' theorems) which show that a person will accept a set of bets in which he is guaranteed to lose money (a so-called "Dutch Book') iff his method for calculating the probabilities of compound statements does not conform to the standard probability axioms.
	serendip.brynmawr.edu /local/scisoc/information/baker.html (7635 words)

	Hausdorff space (Site not responding. Last check: 2007-10-21)
		They also arise in the modeltheory of intuitionistic logic : every complete Heytingalgebra is the algebra of open sets of some topological space, but this spaceneed not be preregular, much less Hausdorff.
		In fact, a quotient space of a Hausdorff spaceX is itself Hausdorff if and only if the kernel of thequotient map is closed as a subset of the Cartesian product X × X.
		Compact preregular spaces are normal, meaning that they satisfy Urysohn'slemma and the Tietze extension theorem and have partitions of unity subordinate to locally finite open covers.
	www.therfcc.org /hausdorff-space-35857.html (799 words)

	Architecture Taxonomy
		The addi quadruple assigns the sum of its two integer operands to the result.
		The divi quadruple assigns the quotient of the first integer operand over the second operand to the result field.
		The divf quadruple assigns the quotient of the first floating point operand over the second operand to the result.
	www.cs.pdx.edu /~cmay/class_notes/cs321/ho/quads.html (2176 words)

	CASE Mathematics
		Topics include homomorphisms and quotient structures, the theory of polynomials, canonical forms for linear transformations and the principal axis theorem.
		This course is required of all students majoring in mathematics.
		Eigenvalues and eigenvectors: the QR algorithm, Rayleigh quotient, inverse iteration.
	www.cwru.edu /artsci/math/courses.htm (2161 words)

	[No title]
		Vector Spaces : Definitions, subspaces, quotient spaces, linear transformations, direct sum and its universal property.
		Module : Definition, sub and quotient modules, homomorphism, isomorphism theorems, irreducible modules, decomposition of a module over a Euclidian Ring.
		Krishnan) :} Discrete parameter stochastic processes, Kolmogorov's consistency theorem; strong and weak stationarity; autocovariance and autocorrelation.
	members.rediff.com /arnabc/course.htm (2107 words)

	[No title] (Site not responding. Last check: 2007-10-21)
		, or Kolmogorov, if any two distinct points in X are topologically distinguishable.
		is preregular), in a fairly precise sense; see Kolmogorov quotient for more information.
		, and you go from the left side to the right side by removing that requirment, using the Kolmogorov quotient operation.
	www.online-encyclopedia.info /encyclopedia/s/se/separation_axiom.html (1450 words)

	Results overview
		Table 11: Overview of the percentage of explained variance in the standardized EGG parametrization measured for the entire vowel durations (N=5244).
		The effects of gender, vowel type, stress and tenseness are investigated for Open Quotient I(OQI), Open Quotient II(OQII), Speed Quotient (SQ),closing (ECA) and opening (SOA) slopes and duration closed phase duration (CT), opening (SOT) and closing duration (ECT).
		In the following sections these results will be discussed for each variable and the specific impact of the individual factors will be analysed as well.
	www.ims.uni-stuttgart.de /phonetik/EGG/pagel6.htm (662 words)

	Carnap, Rudolf [Internet Encyclopedia of Philosophy]
		In the early 1930s, the Russian mathematician Andrei Nikolaevich Kolmogorov (1903-1987) formulated the first axiomatic system for probability.
		A special case is the theory that the probability is a fair betting quotient - this interpretation was supported by Carnap.
		Example: suppose you bet that the score would be five; you bet a dollar and, if you win, you will receive six dollars: this is a fair bet.
	www.iep.utm.edu /c/carnap.htm (5674 words)

	course_descriptions (Site not responding. Last check: 2007-10-21)
		The course will be an introduction to some of the modern topics in information coding and processing in the context of signal quantization and compression.
		Linear parabolic PDE's and their relations with stochastic differential equations: the forward and backward Kolmogorov equation, exit times, fundamental solutions, boundary value problems, maximum principle.
		Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus.
	www.math.nyu.edu /degree/course_descriptions.html (7943 words)

	[No title]
		Estimating the mean value and the covariance function of stationary random processes.
		Chapman- Kolmogorov equations of discrete and continuous parameter processes.
		Students are assumed to interpret computed results of a statistical analysis of natural and artifical data with a penetrating criticism.
	www.cs.elte.hu /probability/common/oktatas.a.html (909 words)

	Toward Real-world Models of Trust
		The second one is that if we formulate these concepts into metric funtions that obey the 4 properties, then it is irrelevant which one we use.
		The structure of x, while a scalar, is that of an operator that represents a process (see the specific definition of process in [Ger97].
		This operator can be shown to obey the properties of a quotient ring in Mathematics, also called a skew-field.
	www.safevote.com /papers/trustdef.htm (19140 words)

	Abstracts of Joseph Y. Halpern's Publications
		It is established that CTL has the small model property by showing that any satisfiable CTL formula is satisfiable in a small finite model obtained from the small ``pseudomodel'' resulting from the Fischer-Ladner quotient construction.
		Then an exponential-time algorithm is given for deciding satisfiability in CTL, and the axiomatization of UB given by Ben-Ari, Manna, and Pnueli is extended to a complete axiomatization for CTL.
		Belief functions are mathematical objects defined to satisfy three axioms that look somewhat similar to the Kolmogorov axioms defining probability functions.
	www.cs.cornell.edu /home/halpern/abstract.html (17982 words)

	Mathematics Colloquium and Seminars (Site not responding. Last check: 2007-10-21)
		One of the entries has connections to the famous "circle problem" in number theory.
		Another involves a divergent continued fraction with three distinct limit points for the partial quotients.
		A third is an unusual identity for the deriviative of a quotient of certain hypergeometric series that converge for z=1 only, i.e., not for z<1 or z>1.
	www.maths.mq.edu.au /talks/colloquium.html (2107 words)

	A Gentle Introduction to Functional Analysis by Dzung Minh Ha (Site not responding. Last check: 2007-10-21)
		2.6 Kolmogorov's theorem (Minkowski functionals, semi-norms, Kolgomorov's criterion for normability)
		2.10 Linear functionals and hyperplanes (includes quotient spaces)
		Dual spaces of closed subsapces and quotient spaces
	www.matrixeditions.com /functionalanalysis.html (561 words)

	Incoming Calendar Events
		As appliactions, we will discuss some global behaviors of the periodic Kolmogorov two species reaction-diffusion systems and, if time permits, a periodic competitor- competitor-mutualist parabolic system.
		To solve the above two problems, we first derive the Kolmogorov forward equations for the state probabilities of the processes.
		Then we transform these equations to equations for the Laplace transforms and represent them in matrix form.
	www.math.yorku.ca /incoming_cal.html (17159 words)

	C++0x Standard Library wishlist (Site not responding. Last check: 2007-10-21)
		Paper: N1668 = 04-0108, "A Proposal to add Mathematical Functions for Statistics to the C++ Standard Library", Paul Bristow
		More math functions for statistics: Student's t function, Kolmogorov statistic, incomplete beta function, digamma function, and so on.
		A mechanism for accessing integer operations that every processor has: addition with carry, full-width multiplication, and division with remainder and quotient.
	lafstern.org /matt/wishlist.html (2418 words)

	NAG Fortran Library, Mark 19 : Library Contents (Site not responding. Last check: 2007-10-21)
		Compute quotient of two real scalars, with overflow flag
		Apply complex similarity rotation to 2 by 2 Hermitian matrix
		Compute quotient of two complex scalars, with overflow flag
	www.nag.com /numeric/fl/manual19/html/genint/libconts_fl19.html (5702 words)

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