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Topic: Characteristic function

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Central limit theorem

Orthogonal functions

Normal distribution

Fourier transform

Laplace transform

Mellin transform

Chirplet

Wavelet

Epilepsy

	Characteristic function - Wikipedia, the free encyclopedia
		The indicator function of a subset A of a set B is the function with domain B, whose value is 1 at each point in A and 0 at each point that is in B but not in A.
		Characteristic functions are used in the most frequently seen proof of the central limit theorem.
		The characteristic function is closely related to the Fourier transform: the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f.
	en.wikipedia.org /wiki/Characteristic_function (403 words)

	Characteristic function (Site not responding. Last check: 2007-10-07)
		The indicator function of a subset A of a set B is the function with domain B whose value is 1 at each in A and 0 at each point that in B but not in A.
		Characteristic functions are used in the most seen proof of the central limit theorem.
		The characteristic function is closely related to Fourier transform : the characteristic function of a distribution density function f is proportional to the inverse Fourier of f.
	www.freeglossary.com /Characteristic_function (526 words)

	Characteristic function (Site not responding. Last check: 2007-10-07)
		The characteristic function is closely related to the Fourier transform : the characteristic function of a distribution with density function f is proportional to the inverse Fourier transform of f''.
		The Function Pointer Tutorials Introduction to C and C++ function pointers, callbacks and functors by Lars Haendel.
		Function Definitions in APL Example that the derivative of the integral returns the original function.
	www.serebella.com /encyclopedia/article-Characteristic_function.html (715 words)

	PlanetMath: characteristic function (Site not responding. Last check: 2007-10-07)
		It is also true that each characteristic function determines a unique distribution; hence the name, since it characterizes the distribution function (see property 6.)
		Other important result related to characteristic functions is the Paul Lévy continuity theorem.
		This is version 2 of characteristic function, born on 2002-12-10, modified 2004-02-04.
	www.planetmath.org /encyclopedia/CharacteristicFunction2.html (211 words)

	The Inversion of a Characteristic Function (Site not responding. Last check: 2007-10-07)
		One of the interesting and important properties of the transformation that generates the characteristic function is that if it is applied a second time it generates the original function; i.e., the characteristic function of a characteristic function is essentially the original function.
		Note the factor of (1/2π) and the difference in the sign of the argument of the exponential function, -ωz instead of +ωz.
		Below is the case for the the normal distribution, which happens to have the same functional form as the original distribution.
	www.sjsu.edu /faculty/watkins/inverse.htm (195 words)

	Indicator function - Wikipedia, the free encyclopedia
		In the mathematical subfield of set theory, the indicator function, or characteristic function, is a function defined on a set X which is used to indicate membership of an element in a subset A of X.
		The term "characteristic function" has an unrelated meaning in probability theory; see characteristic function.
		As suggested by the previous example, the indicator function is a useful notational device in combinatorics.
	en.wikipedia.org /wiki/Indicator_function (301 words)

	Characteristic function - Term Explanation on IndexSuche.Com (Site not responding. Last check: 2007-10-07)
		The indicator_function of a subset ''A'' of a set ''B'' is the function with domain ''B'', whose value is 1 at each point in ''A'' and 0 at each point that is in ''B'' but not in ''A''.
		Given a characteristic function φ, it is possible to reconstruct the corresponding cumulative probability distribution function ''F'': :F_X(y) - F_X(x) = \lim_{\tau \to +\infty} \frac{1} {2\pi} \int_{-\tau}^{+\tau} \frac{e^{-itx} - e^{-ity}} {it}\, \varphi_X(t)\, dt.
		The characteristic function is closely related to the Fourier_transform: the characteristic function of a distribution with density function ''f'' is proportional to the inverse Fourier_transform of ''f''.
	www.indexsuche.com /Characteristic_function.html (358 words)

	Addition of Vectors and Matrices
		Function is a correspondence f between elements of a space X and those of a space Y such that any element x of X has a unique corresponding element y of Y which is denoted y=f(x).
		Thus there is a 1-1 correspondence between subsets of X and their characteristic functions.
		It must also be noted that functions become interesting when the spaces X and Y are topological.
	www.cut-the-knot.com /do_you_know/add_vec.shtml (956 words)

	ipedia.com: Characteristic Article (Site not responding. Last check: 2007-10-07)
		In mathematics, the characteristic of a ring R with identity element 1 R is defined to be the smallest positive integer n such that n 1 R = 0.
		The characteristic of the ring R may be equivalently defined as the unique natural number n such that nZ is the kernel of the unique ring homomorphism from Z to R which sends 1 to 1
		And yet another equivalent definition: the characteristic of R is the unique natural number n such that R contains a subring isomorphic to the factor ring Z/nZ.
	www.ipedia.com /characteristic.html (524 words)

	Central limit theorem - Wikipedia, the free encyclopedia
		For a theorem of such fundamental importance to statistics and applied probability, the central limit theorem has a remarkably simple proof using characteristic functions.
		But, this limit is just the characteristic function of a standard normal distribution, N(0,1), and the central limit theorem follows from the Lévy continuity theorem, which confirms that the convergence of characteristic functions implies convergence in distribution.
		Thus the central limit theorem can be interpreted as a statement about the properties of density functions under convolution: the convolution of a number of density functions tends to the normal density as the number of density functions increases without bound, under the conditions stated above.
	www.wikipedia.org /wiki/Central_limit_theorem (1010 words)

	Characteristic Function (Site not responding. Last check: 2007-10-07)
		Harrell talked of the indigenous materials, the design deriving from the function required for that climate, and the fact that...
		The curvature of the corrected output characteristic of Figure 2 (red trace) is intrinsic...
		of, pertaining to, or characteristic of the British playwright Harold Pinter...
	www.wikiverse.org /characteristic-function (450 words)

	Method and apparatus for generating representation of an image from a transparency - Patent 5375000
		The signals obtained are modified with a function (11,10,8) constituting the inverse of the film characteristic function to compensate.
		The application of an inverse film characteristic function to the image removes the effects of the film so that a more accurate representation of the original image is obtained.
		Once the effect of the film characteristic is removed, it is possible to apply any chosen film characteristic by application of a new film characteristic function to the compensated output.
	www.freepatentsonline.com /5375000.html (1735 words)

	Characteristic function - Open Encyclopedia (Site not responding. Last check: 2007-10-07)
		The last form is valid only when f--the probability density function--exists.
		More than that, there is a bijection between cumulative probability functions and characteristic functions.
		In other words, two probability distributions never share the same characteristic function.
	www.open-encyclopedia.com /Characteristic_function (369 words)

	PHIL 3340: Review for Exam 1
		a function relates two sets: it is from a set A to a set B. (So each of the ordered pairs "in" the function has a first member from A and a second member from B.) We say f: A -> B to indicate that the function f is from A to B. domain, range
		(The characteristic function of a k-place relation is a function of k arguments.)
		a relation is recursive iff its characteristic function is recursive.
	www.trinity.edu /cbrown/metalogic/exam1review.html (760 words)

	Indicator function (Site not responding. Last check: 2007-10-07)
		In mathematics, the indicator function (sometimes also called characteristic function) of a subset ''A of a set X is a function from X into {0,1} defined as follows: :
		The term characteristic function is potentially confusing because it is also used to denote a quite different concept that is also prevalent in probability theory; see characteristic function.
		The indicator function is a basic tool in probability theory : if X is a probability space with probability measure P and A is a measurable set, then I
	www.serebella.com /encyclopedia/article-Indicator_function.html (537 words)

	Functions
		Furthermore, there are numerous terms (vector, sequence, measure, length, volume, etc.) that are functions in disguise and their functional ancestry is seldom mentioned.
		Two functions with the same domain and region but defined by different rules, will be distinguished by different function names.
		Function is a correspondence f between elements of a space X and those of a space Y such that any element x of X has a unique corresponding element y of Y which is denoted
	www.cut-the-knot.com /do_you_know/function.shtml (448 words)

	The Characteristic Function of a Probability Distribution
		The expected value of the function exp(iωz) is called the characteristic function for the probability distribution p(z), where ω is parameter that can have any real value and i is the square root of -1.
		The connection between the moments of a probability distribution and its characteristic function is seen from taking the derivaative of the characteristic function with respect to the parameter ω.
		Thus for a stable distribution the graph of the logarithm of the real component of the log-characteristic function as a function of the logarithm of ω is a straight line, the slope of which is the stability index of the distribution, α.
	www.sjsu.edu /faculty/watkins/charact.htm (1272 words)

	V-function and shapley for 2 examples (Site not responding. Last check: 2007-10-07)
		The characteristic function is one of the basic concepts in cooperative game theory.
		The characteristic function is a function v that takes as its argument any sub-set of the players of a game say set T out of the entire set of players N. the function v(T) = $40 says that if the players in get together as a coalition they can guarantee themselves as a group $40.
		The normalized Characteristic function treats the amounts that the players can guarantee for themselves as if they were the disagreement point in a bargaining game.
	www.economics.laurentian.ca /Strategic_Think.27/Modules/Course_Schedule.98/Characteristic%20function/V-function_examples.htm (1298 words)

	Developments in Characteristic Function Theory:0195205774:Luckas, E.:eCampus.com
		Characteristic functions were originally introduced as tools for the study of limit theorems of probability theory.
		Later it became clear that the theory of characteristic functions is also of intrinsic mathematical interest.
		The present monograph is intended to cover the results of significant advancement in the theory.
	www.ecampus.com /bk_detail.asp?isbn=0195205774 (47 words)

	Möbius function and characteristic polynomial for subspace arrangements embedded in B_{n}.-abstract (Site not responding. Last check: 2007-10-07)
		Möbius function and characteristic polynomial for subspace arrangements embedded in B_{n}.
		They give formulas for the exponential generating functions for the M\"obius function and for the characteristic polynomial for a class of subspace arrangements embedded in $\cal B_n$.
		As a corollary the exponential generating function for the Euler characteristic is also calculated.
	www.mai.liu.se /~svlin/artiklar/bn.abs.html (110 words)

	R: Operating Characteristic Function (Site not responding. Last check: 2007-10-07)
		is a generic function which calls the proper function depending on the type of `qcc' object.
		are passed to the specific function depending on the type of chart.
		The function invisibly returns a matrix or a vector of beta values, the probability of type II error.
	www.maths.lth.se /help/R/.R/library/qcc/html/oc.curves.html (183 words)

	Semicontinuity (Site not responding. Last check: 2007-10-07)
		In mathematical analysis, semicontinuity is a property of real-valued functions that is weaker than continuity.
		The characteristic function of an open set is lower semicontinuous.
		The characteristic function of a closed set is upper semicontinuous.
	www.wikiverse.org /semicontinuity (102 words)

	Hamilton's Papers on Geometrical Optics
		A key feature of Hamilton's approach is the use of a characteristic function, V, with the property that, in an isotropic medium, the light rays are perpendicular to the level surfaces of V.
		This auxiliary function W is a function of the direction cosines of the final ray.
		He discusses the relationship between his theory of the characteristic function and the undulatory theory of light, and in particular applies his theory to the study of the propagation of light in biaxal crystals such as arragonite.
	www.maths.tcd.ie /pub/HistMath/People/Hamilton/Optics.html (1502 words)

	An Inequality for a Multidimensional Characteristic Function
		Let $\xi $ be a vector-valued random variable in $\mathbf{R}^s$ and a corresponding density function $p_\xi(x)$ be "close" to the "standard" normal density.
		Under this condition an inequality for a characteristic function is proved.
		The inequality obtained is of interest for the problem of a lower estimator of the rate of convergence in the local limit theorem for densities.
	epubs.siam.org /sam-bin/dbq/article/97811 (161 words)

	Citations: person Games in Partition Function Form - Thrall, Lucas (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		Authors mostly use the characteristic function form, that besides its merits, lacks externalities: ignores the side effects of coalition formation.
		define the partition function [11, pp509 511] an extension of the characteristic function that assigns a value to a coalition given the whole partition and hence accounts for externalities too.
		The core is determined by the inspection of the characteristic function.
	citeseer.ist.psu.edu /context/1110941/0 (913 words)

	Estimation Based on the Empirical Characteristic Function (ResearchIndex) (Site not responding. Last check: 2007-10-07)
		As an alternative the characteristic function is used.
		Certain distributions whose characteristic functions can be expressed in terms of jtj ff are such examples.
		Tailweight properties are first examined; it is shown that these laws are Paretian, their tail index ff being one of the parameters defining these laws.
	citeseer.ist.psu.edu /77896.html (409 words)

	Error Bounds for Numerical Inversion of a Probability Characteristic Function
		Error Bounds for Numerical Inversion of a Probability Characteristic Function: SIAM Journal on Numerical Analysis Vol.
		Given the characteristic function of a continuous real random variable, it is possible to construct a finite Fourier series that approximates the density over any specified finite interval.
		Explicit error bounds can be obtained directly from the characteristic function, without prior knowledge of either the distribution or density, making these series useful for numerical inversion.
	epubs.siam.org /sam-bin/dbq/article/31085 (188 words)

	Ultra-Formal Verification of a Result in a System of Combinatory Logic
		A level-2 object is thought of as a function from level-1 objects to level-1 objects, and a level-1 object is thought of as a function from level-0 objects to level-0 objects.
		The next two assert the existence of a pairing function (which is not a combinator) and projections (which are combinators).
		Because the pairing function is surjective, anything that can be said about pairs can be recast into an equivalent statement about the projections, and vice-versa.
	www.cityauditorphilwood.com /warren/allone.html (8234 words)

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