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Topic: Fixed point lemma for normal functions

	Encyclopedia: Fixed point theorem
		For example, the cosine function is continuous in [-1,1] and maps it into [-1, 1], and thus should have a fixed point.
		Every lambda expression has a fixed point, and a fixed point combinator is a "function" which takes as input a lambda expression and produces as output a fixed point of that expression.
		Every closure operator on a poset has many fixed points; these are the "closed elements" with respect to the closure operator, and they are the main reason the closure operator was defined in the first place.
	www.nationmaster.com /encyclopedia/Fixed-point-theorem (459 words)

	Fixed point - Wikipedia, the free encyclopedia
		Fixed point has many meanings in science, most of them mathematical.
		Fixed point — a number x that makes f(x) = x.
		Fixed-point arithmetic — manner of doing arithmetic on computers: a fixed number of decimal (or binary) digits is kept after the decimal point, any remaining digits are rounded.
	en.wikipedia.org /wiki/Fixed_point (117 words)

	Normal function (Site not responding. Last check: 2007-10-22)
		In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff it is continuous (with respect to the order topology) and strictly mononotically increasing.
		A simple normal function is given by f(α) = 1 + α; note however that f(α) = α + 1 is not normal.
		Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof.
	www.sciencedaily.com /encyclopedia/normal_function (359 words)

	Station Information - Fixed-point lemma for normal functions
		The fixed-point lemma for normal functions is a basic result in axiomatic set theory; it states that any normal function has arbitrarily large fixed points.
		Let f : Ord → Ord be a normal function.
		If this was not the case, we could choose a minimal α with f(α) < α; then, since f is normal and thus monotone, f(f(α)) < f(α), which is a contradiction to α being minimal.
	www.stationinformation.com /encyclopedia/f/fi/fixed_point_lemma_for_normal_functions.html (304 words)

	CS620 fixed on Tue Mar 25 12:07:52 PST 2003
		We will show that some of these functionals, when repeatedly applied to a function generate a series that tends toward a definite limit and that this is a particular fixed point of the functional.
		The denotational semantics is in terms of the fixed points of continuous functionals.
		Polaris or the "pole star" is close to the visual fixed point in the night sky.
	www.csci.csusb.edu /dick/cs620/fixed.html (2288 words)

	Fix Your Credit (Site not responding. Last check: 2007-10-22)
		A fixative is a liquid, similar to varnish, which is usually sprayed over a finished piece of artwork to better preserve it and prevent smudging.
		Admittedly, there seems little to separate fixed 3D from its precursor, the graphic adventure game (Monkey Island, Sam and Max etc.), but whereas the latter overlays 2D characters over a 2D background, fixed 3D is at least 3D overlaid on 2D, and often onto 3D.
		Fixed access: In personal communications service (PCS), terminal access to a network in which there is a set relationship between a terminal and the access interface.
	www.wwwtln.com /finance/79/fix-your-credit.html (591 words)

	[No title]
		Assuming that the size of a generation of a population depends solely on the size of the previous generation and may thus be expressed as a function of it, questions concerning the further development of the population reduce to iteration of this function.
		From this point of view, the iteration theory of entire functions and of meromorphic functions with one pole which is an omitted value is quite different from that of general meromorphic functions, which have at least two poles or only one pole which is not omitted.
		Clearly, this attracting fixed point of $f^{-n_j-1}$ is a repelling periodic point of $f$ of period $n_j+1$.
	www.ams.org /journals/bull/pre-1996-data/199329-2/Bergweiler (9297 words)

	Talk Abstracts
		Fixed points and periodic points of holomorphic and quasiregular maps
		Random iteration is a generalization of standard polynomial iteration in complex dynamics where instead of looking at the iterates of a fixed polynomial one allows the functions one considers to vary at each stage of the iterative process.
		We show that if f is the class B (that is, the class of functions for which the singularities of the inverse function are bounded), then the Julia set of f has Hausdorff dimension strictly greater than one.
	www.maths.warwick.ac.uk /~lasse/dec_04_meeting/abstracts.html (2248 words)

	Articles - Fixed-point theorem (Site not responding. Last check: 2007-10-22)
		In mathematics, a fixed-point theorem is a result saying that a function F will have at least one fixed point, under some conditions on F that can be stated in general terms.
		By contrast, the Brouwer fixed point theorem is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, but it doesn't describe how to find the fixed point (See also Sperner's lemma).
		For example, the cosine function is continuous in [-1,1] and maps it into [-1, 1], and thus must have a fixed point.
	www.poncier.com /articles/Fixed-point_theorem (420 words)

	FUNCTION - Online Information article about FUNCTION (Site not responding. Last check: 2007-10-22)
		The rule for a function of a function is not stated explicitly but is illustrated by 'examples in which new variables are introduced, in much the same way as in Newton's Methodus fluxionum.
		In connexion with the problem of maxima and minima, it is noted that the differential of y is positive or negative according as y increases or decreases when x increases, and the discrimination of maxima from minima depends upon the sign of ddy, the differential of dy.
		A tangent is defined as a line joining two " infinitely " near points of a curve, and the " infinitely " small distances (e.g., the distance between the feet of the ordinates of such points) are said to be expressible by means of the differentials (e.g., dx).
	encyclopedia.jrank.org /FRA_GAE/FUNCTION.html (8019 words)

	Fixed-point lemma for normal functions - Wikipedia, the free encyclopedia
		The fixed-point lemma for normal functions is a basic result in axiomatic set theory; it states that any normal function has arbitrarily large fixed points and can often be used to construct ordinal numbers with interesting properties.
		Let f : Ord → Ord be a normal function.
		is normal (see aleph number); thus, there exists an ordinal Θ such that Θ = א
	en.wikipedia.org /wiki/Fixed-point_lemma_for_normal_functions (187 words)

	Representation of Propositional Expert Systems as Partial Functions
		Lemma 6.2: For a given EDB, the evaluation procedure selects an acyclic subset of IDB clauses, which are the only clauses to generate tuples.
		From a practical point of view, the key to the algorithm developed from this theorem is in the method of choosing an attribute in part 1 of the induction step.
		Unfortunately, this method is of limited utility, since K is constructed in conjunctive normal form, and the process of conversion from conjunctive to disjunctive normal forms is exponential in the number of conjuncts.
	www.itee.uq.edu.au /~colomb/Papers/PartialFunctions.html (10395 words)

	Julia set (Site not responding. Last check: 2007-10-22)
		While infinity is a point attractor, the second attractor may be either a point attractor or a periodic cycle.
		(A point attractor is essentially a periodic cycle of period 1.) The exact shape of the basin of attraction to this second attractor depends on c.
		Instead of using an "escape time" method to find the points that do not belong to the Julia set, the "random game" method uses the reverse formula to track the convergence of a single point towards the edge of the Julia set.
	www.worldhistory.com /wiki/J/Julia-set.htm (1393 words)

	Fixed point theorems in infinite-dimensional spaces (Site not responding. Last check: 2007-10-22)
		In mathematics, a number of fixed point theorems in infinite-dimensional spaces generalise the Brouwer fixed point theorem.
		One way in which fixed-point theorems of this kind have had a larger influence on mathematics as a whole has been that one approach is to try to carry over methods of algebraic topology, first proved for finite simplicial complexes, to spaces of infinite dimension.
		The Schauder fixed point theorem states, in one version, that if C is a nonempty closed convex subset of a Banach space V and f is a continuous map from C to C whose image is countably compact, then f has a fixed point.
	www.kiwipedia.com /en/schauder-fixed-point-theorem.html (291 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Function ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is wildly non-injective, and function ", Cell[BoxData[ \(TraditionalForm\`h\)]], " is injective, and therefore bijective.
		\n\n\ ", StyleBox["Examples:", FontWeight->"Bold"], " \nThe inverse of the logarithm function is the exponential function.
		\nThe inverse of the successor function on the integers is the \ predecessor function (subtract 1).
	www.cs.cmu.edu /afs/andrew.cmu.edu/course/15/354/www/NBooks/Functions.nb (9924 words)

	[No title]
		Weinstein: $\omega$ symplectic, $d\theta=\omega$, $\phi$ an exhausting function; $\theta$ is associated to a "compressing" form.
		Lemma: Suppose $D$ has normal crossings (locally, $D=\{z_1 \dots z_k=0\}$ in holomorphic coordinates, $z=(z_1,\dots,z_n)$ (i.e., if $D$ is smooth).
		Clearly, $\tilde \phi$ has the same critical points as $\phi$; thus, with the new function, $M$ is of finite type iff it was originally.
	www.math.uchicago.edu /~msmukler/nov02 (538 words)

	[No title] (Site not responding. Last check: 2007-10-22)
		Let $s\in\Sigma$ be the image of the unique closed point of $\Spec(W)$ by $\Spec(W)\to\Sigma$, $B_1$ be the completion of the local ring $\mathcal{O}_{\Sigma, s}$, and $W_1$ be a valuation ring of the quotient field of $B_1$ such that $\dim_kW_1=0$ and $W_1$ dominates $W$.
		The field $k(x_1, x_2, \ldots, x_r)$ is isomorphic to the rational function field over $k$ with $r$ variables, and there is a valuation ring $\overline{V}_1$ of $k(x_1, x_2, \ldots, x_r)$ such that $k\subset\overline{V}_1$ and $\dim_k\overline{V}_1=0$.
		Fixing the numbers of $y_1, y_2,\ldots, y_{j-1}$ and exchanging the numbers $y_j, y_{j+1},\ldots, y_n$, we can assume that $y_j\in \q'_j\cap Z'$.
	home.imf.au.dk /esn/preprints/139 (8944 words)

	Symbolic Asymptotics:Multiseries of Inverse Functions
		From the point of view of symbolic computation, it is important that we have finite expressions for these coefficient functions and for any coefficients in their expansions etc., in order that zero-equivalence tests can be made.
		Lemma 1 Let f and g be two elements of a Hardy field such that g does not tend to a non-zero finite constant then f=o(g) (resp.
		Lemma 2 Let h be an element of a Hardy field with h not asymptotic to a non-zero constant.
	algo.inria.fr /papers/html/SaSh99/inv.html (4856 words)

	Earliest Known Uses of Some of the Words of Mathematics (N)
		Normal correlation appears in W. Sheppard, "On the application of the theory of error to cases of normal distribution and normal correlation," Phil.
		Normal law is found in Francis Galton’s "Results Derived from the Natality Table of Korosi by Employing the Method of Contours or Isogens," Proceedings of the Royal Society, 55, (1894), p.
		Normal variate was in wide use in the 1930s and is found in Joseph Pepper's "Studies in the Theory of Sampling," Biometrika, 21, (1929), p.
	members.aol.com /jeff570/n.html (5360 words)

	Comp432 - Functional Programming References Light Version (Site not responding. Last check: 2007-10-22)
		Function composition allows other functions to be joined together to make larger (more powerful) functions.
		In contrast to normal order evaluation, and used by most imperative languages, is applicative order evaluation (call-by-value)- evaluate the arguments to a function before evaluating the function.
		Functions can't take types as arguments, nor (apart from the use of type classes) have types which depend on values supplied to the function.
	www.mcs.vuw.ac.nz /~db/comp432_lite.html?type=bibtex& (2493 words)

	\Large Normal Distribution \\ \large characterizations with applications (Site not responding. Last check: 2007-10-22)
		We have managed to avoid functional equations for non-differentiable functions; in many proofs in the literature lack of differentiability is a major technical difficulty.
		Fall as it may, its deviation from the mark is error, and the probability of that error is the unknown function of its square, ie.
		First, normality can frequently be recognized from the conditional moments of linear combinations of independent random variables; we illustrate this by a simple proof of the well known fact that the independence of the sample mean and the sample variance characterizes normal populations, and by the proof of the central limit theorem.
	math.uc.edu /~brycw/probab/charakt/charakt.htm (5891 words)

	Ernest Lane (ResearchIndex) (Site not responding. Last check: 2007-10-22)
		An interpolation lemma due to Katetov is simplified in order to make it more general and easier to use.
		Several results are established to illustrate applications of this lemma, including a characterization of monotonically normal spaces and a topology-free insertion theorem.
		1 A theorem of insertion and extension of functions for normal..
	citeseer.ist.psu.edu /104739.html (341 words)

	Bounds For Fixed Point Free Elements In A Transitive Group and Applications to Curves over Finite Fields - Guralnick, ... (Site not responding. Last check: 2007-10-22)
		Let A be a finite group with G a normal subgroup with A=G cyclic generated by xG.
		Guralnick and D. Wan, Bounds for fixed point free elements in a transitive group and applications to curves over finite fields, Israel J. Math.
		3 the number of fixed point free elements in a permutation gro..
	citeseer.ist.psu.edu /12049.html (727 words)

	Recursion Theory (Math/Phil 291A) Schedule (Site not responding. Last check: 2007-10-22)
		Equivalence of Recursive functions and Turing machine-computable functions.
		If we can show that a given set is recursive, then we know there is some (possibly inefficient) algorithm to determine whether or not a given input is in the set.
		Fixed Point Theorem and another equivalent definition of the set of partially recursive functions.
	www.math.lsa.umich.edu /~mvd/rec/schedule.html (389 words)

	Normal function - InfoSearchPoint.com (Site not responding. Last check: 2007-10-22)
		In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) iff it is continuous and strictly mononotically increasing.
		That is, a function is normal iff the following two conditions hold:
		It is easy to show that normal functions have arbitrarily large fixed points; see Fixed-point lemma for normal functions for a proof.
	www.infosearchpoint.com /display/Normal_function (118 words)

	2003-05 Dedman College Graduate Catalog - - SMU
		Numerical solution of linear and nonlinear equations, interpolation and approximation of functions, numerical integration, floating-point arithmetic, and the numerical solution of initial value problems in ordinary differential equations.
		Approximation of a real continuous function by an approximating function depending on a finite number of parameters.
		Green's functions for the heat, wave, and Laplace equations.
	www.smu.edu /catalogs/graduate/dedman/mathematics.asp (1302 words)

	University College Dublin Statistics Department
		Point and interval estimation using a single sample.
		Instead the units STAT 2205, STAT 2206, STAT 2207 and STAT 2221 may be taken with permission of the Department of Statistics.
		Define and use simple commutation functions suitable for valuing pension fund benefits and contributions.
	www.ucd.ie /statdept/schb.html (861 words)

	Encyclopedia: Fixed-point lemma for normal functions (Site not responding. Last check: 2007-10-22)
		Updated 260 days 9 hours 25 minutes ago.
		Other descriptions of Fixed-point lemma for normal functions
		Click for other authoritative sources for this topic (summarised at Factbites.com).
	www.nationmaster.com /encyclopedia/Fixed_point-lemma-for-normal-functions (213 words)

	Lambda Calculus and Types: Synopsis
		Terms, free and bound variables, alpha-conversion, substitution, variable convention, contexts, the formal theory lambda beta, the n rule, fixed point combinators, brief mention of other lambda-theories.
		Church numerals, definability of total recursive functions (with extension to partial functions).
		Consequences: no fixed point combinators, poor definability power.
	web.comlab.ox.ac.uk /oucl/courses/topics04-05/lcat/synopsis.html (218 words)

	[No title]
		It also % introduces the required assumption for arbitrary precision % IEEE-854 floating point numbers.
		"" (SKOSIMP) (("" (LEMMA* "value_positive") (("" (INST?) (("" (ASSERT) NIL))))))) (fp_sqrt_TCC2
		"" (SKOSIMP) (("" (REWRITE "zero_times3") (("" (LEMMA* "expt_nonzero") (("" (INST?) (("" (ASSERT) NIL))))))))) (overflow_TCC1
	shemesh.larc.nasa.gov /fm/ftp/larc/Floating_Point/dump.2.29.96 (1426 words)

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