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Topic: Chebyshev

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	Chebyshev biography
		Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".
		Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal.
		In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal.
	www-history.mcs.st-andrews.ac.uk /Biographies/Chebyshev.html (3026 words)

	Chebyshev Filters
		Chebyshev filters are used to separate one band of frequencies from another.
		The primary attribute of Chebyshev filters is their speed, typically more than an order of magnitude faster than the windowed-sinc.
		This chapter presents the information needed to use Chebyshev filters without wading through a mire of advanced mathematics.
	www.dspguide.com /ch20.htm (116 words)

	PlanetMath: Chebyshev equation
		Chebyshev's equation is the second order linear differential equation
		These polynomials are, up to multiplication by a constant, the Chebyshev polynomials.
		This is version 3 of Chebyshev equation, born on 2002-11-21, modified 2002-11-21.
	planetmath.org /encyclopedia/ChebyshevEquation.html (167 words)

	Chebyshev
		In 1847 Chebyshev was appointed to the University of St Petersburg.
		Chebyshev also came close to proving the prime number theorem, proving that if pi (n)log n)/n had a limit as n-> infinity then that limit is 1.
		Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling.
	library.wolfram.com /examples/quintic/people/Chebyshev.html (246 words)

	One tailed version of Chebyshev's inequality - by Henry Bottomley
		Chebyshev's Inequalities: one of his own and one from 'Probability and Random Processes', by Grimmett and Stirzaker, published by Oxford Science Publications, ISBN 0 19 853665 8.
		Pafnuty Lvovich Chebyshev was a notable Russian mathematician, who was born on 16 May 1821 and died on 8 December 1894.
		Stijn van Dongen has compiled a list of 33 western European verions of his surname, of which Chebyshev seems to be the most popular in English, Tchebycheff in French, Cebysev in Spanish, and Tchebyscheff in German.
	www.btinternet.com /~se16/hgb/cheb.htm (1002 words)

	Chebyshev polynomials
		Chebyshev's polynomial of the second kind can be defined in terms of the differential equation
		Formulas for Chebyshev polynomials of the second kind, from
		A plot of the Chebyshev polynomials of the second kind as functions of
	www.unc.edu /~wjt/ChebyshevPolynomials2.htm (342 words)

	Chebyshev Series Approximations for the Bessel Function Y n (z) of Complex Argument - Belward, Zhang (ResearchIndex)
		Results of numerical experiments are presented to demonstrate the computed accuracy by using the Chebyshev series approximation.
		Advantages and disadvantages of the Chebyshev series approximation compared with other polynomial approximation methods, e.g., the tau-method approximations, are discussed.
		The Chebyshev coefficients of J 0 (z) and Y 0 (z) are taken from [22] and
	citeseer.ist.psu.edu /zhang96chebyshev.html (478 words)

	Chebyshev/Butterworth Filter Design
		As no corresponding derivation was found for the Chebyshev filter, the number of required branches is estimated to be the next odd number less than that calculated for the Butterworth design.
		The normalized Butterworth filter component values are calculated in statement 270; the Chebyshev values are calculated in statements 300 to 330.
		These values are scaled to the desired frequency and impedance in the subroutine at statements 340 to 390; and the subroutine at statements 400 to 510 rounds these values to 3 significant digits and prints them in the most common units.
	www.qsl.net /kp4md/filter.htm (823 words)

	Chebyshev
		In 1847 Chebyshev was appointed to the University of St Petersburg.
		Chebyshev was also interested in mechanics and studied the problems involved in converting rotary motion into rectilinear motion by mechanical coupling.
		About Chebyshev's attitude towards applications, see an interesting remark by Clive J. Grant and a quotation by Chebyshev.
	www.mathsoc.spb.ru /pantheon/chebyshe/index.html (265 words)

	Pafnuty Chebyshev Summary
		In general, Chebyshev was looking for derivations of the leading results of probability by methods that could not be faulted for rigor, but which were not dependent on mathematical ideas that seemed out of proportion to the depth of the subject.
		Chebyshev was able to get a decent approximation for the number of prime numbers less than a fixed number compared to known functions of that fixed number, but he did not prove that there was a limiting value.
		Chebyshev is known for his work in the field of probability and statistics.
	www.bookrags.com /Pafnuty_Chebyshev (1783 words)

	Chebyshev's inequality
		However, the bounds provided by Chebyshev's inequality cannot, in general (remaining sound for variables of arbitrary distribution), be improved upon.
		The theorem can be useful despite these loose bounds because it applies to random variables of any distribution, and because these bounds can be calculated knowing no more about the distribution than the mean and variance.
		Chebyshev's inequality is used for proving the weak law of large numbers.
	www.xasa.com /wiki/en/wikipedia/c/ch/chebyshev_s_inequality.html (237 words)

	Chebyshev Interpolation
		Using the CP applet, observe how the extrema of the Chebyshev polynomials are not evenly distributed and how they cluster around the boundary.
		Chebyshev pseudospectral methods, which are based on the interpolating Chebyshev approximation (12), are well established as powerful methods for the numerical solution of PDEs with sufficiently smooth solutions.
		In these cases, the Chebyshev pseudospectral method produces approximations that are contaminated with Gibbs oscillations and suffer from the corresponding loss of spectral accuracy, just like the Chebyshev interpolation methods that the pseudospectral methods are based on.
	www.mathdl.org /images/upload_library/4/vol6/Sarra/Chebyshev.html (2857 words)

	The Chebyshev UCL Proposal
		This means the Chebyshev UCL tends to be much lower than the Land UCL when computed for lognormal distributions, implying that the Chebyshev UCL achieves much less than 95% coverage when the underlying distribution is truly lognormal (or close to it.) It is the opposite of conservative.
		The Chebyshev 95% UCL of the mean as described in Singh et al.
		This is due partly to the use of the Chebyshev UCL whenever the hypothesis of Normality is rejected, which happens with greater frequency as sigma increases.
	www.quantdec.com /envstats/notes/class_12/ucl.htm (3360 words)

	Biographies
		Chebyshev was a prolific mathematician, making contributions to number theory, probability and integration.
		In addition to his interest in mathematics, Chebyshev was interested in mechanical systems and their properties.
		In 1867 Chebyshev published a paper "On Mean Values" in which he formulated the inequality known today by his name.
	tulsagrad.ou.edu /statistics/biographies/Chebyshev.htm (352 words)

	PlanetMath: Chebyshev polynomial
		The Chebyshev polynomials of first kind are defined by the simple formula
		Related are the Chebyshev polynomials of the second kind that are defined as
		This is version 7 of Chebyshev polynomial, born on 2002-02-19, modified 2005-04-28.
	planetmath.org /encyclopedia/ChebyshevPolynomial.html (140 words)

	Chebyshev's Inequalities
		Chebyshev's inequality and its descendants allow you to place an upper bound on the probability that some random variable is >= a set value, given only the mean and variance of that variable.
		The main use for Chebyshev's inequality is in proving theorems, such as laws of large numbers.
		It would be nice to use Chebyshev's inequalities as a defence against variables in real life where outliers are expected, but in such cases the variance of the variable in question may not be known - and judging the error of estimates of variance will probably involve you in distributional assumptions anyway.
	www.mcdowella.demon.co.uk /Chebyshev.html (767 words)

	Statistical Measures of Data - CHEBYSHEV'S THEOREM (Site not responding. Last check: )
		The Russian mathematician P. Chebyshev (1821- 1894) discovered that the fraction of observations falling between two distinct values, whose differences from the mean have the same absolute value, is related to the variance of the population.
		Using the concept of z scores, we can restate Chebyshev's Theorem to say that for any population or sample, the proportion of all observations, whose z score has an absolute value less than or equal to k, is no less than
		1, Chebyshev's Theorem provides a lower bound to the proportion of measurements that are within a certain number of standard deviations from the mean.
	library.thinkquest.org /10030/3smodcm.htm (279 words)

	Determination of optimal Chebyshev-expanded hydrophobic discrimination function for globular proteins
		The scoring function is expanded using the Chebyshev polynomials, for which the coefficients are determined by minimizing the Z-score of native structures in the ensembles of alternate conformations.
		In fact, the Chebyshev approximating polynomial is very nearly the same as the minimax polynomial.
		The second is that the Chebyshev approximation achieves a very low error for relatively few terms, thus leaving us fewer parameters to deal with.
	www.research.ibm.com /journal/rd/453/fain.html (3467 words)

	GNU Scientific Library -- Reference Manual - Chebyshev Approximations
		A Chebyshev approximation is a truncation of the series f(x) = \sum c_n T_n(x), where the Chebyshev polynomials T_n(x) = \cos(n \arccos x) provide an orthogonal basis of polynomials on the interval [-1,1] with the weight function 1 / \sqrt{1-x^2}.
		The computation of the Chebyshev approximation is an O(n^2) process, and requires n function evaluations.
		For smooth functions the Chebyshev approximation converges extremely rapidly and errors would not be visible.
	linux.duke.edu /~mstenner/free-docs/gsl-ref-1.1/gsl-ref_28.html (434 words)

	Chebyshev's bias
		Chebyshev’s Bias deserves to be much better known than it is, though, so to get the word out, I’m going to blog it, right here.
		The situation resembles those “first past the post” election systems, where a nationwide majority of 51 percent can give your party a landslide in terms of parliamentary seats; or a foot race with very well-matched runners, in which one runner manages to stay slightly ahead for most of the race, and gets all the glory.
		The wonderful and amazing result got by Rubinstein and Sarnak is this: Take a Chebyshev bias; for illustration, I'll take the bias to remainder 3 when you divide a prime by 4.
	olimu.com /Notes/ChebyshevsBias.htm (1564 words)

	Chebyshev's bias
		Chebyshev’s Bias deserves to be much better known than it is, though, so to get the word out, I’m going to blog it, right here.
		The situation resembles those “first past the post” election systems, where a nationwide majority of 51 percent can give your party a landslide in terms of parliamentary seats; or a foot race with very well-matched runners, in which one runner manages to stay slightly ahead for most of the race, and gets all the glory.
		The wonderful and amazing result got by Rubinstein and Sarnak is this: Take a Chebyshev bias; for illustration, I'll take the bias to remainder 3 when you divide a prime by 4.
	www.olimu.com /Notes/ChebyshevsBias.htm (1564 words)

	Ephemerides of the Largest Asteroids
		The disadvantage is that the values produced by the Chebyshev polynomials is not exactly the same as that of the tabulated ephemeris; however, the difference between the tabulated and the Chebyshev representations can be made arbitrarily small.
		The coefficients of the Chebyshev polynomials were calculated using the algorithm detailed in (Newhall 1989) which is also used for storage of the JPL DE series planetary ephemerides.
		The ephemeris used to generate the Chebyshev polynomials was produced by using the initial parameters as determined from the fit to data and cutting them off at the formal uncertainty in the parameters.
	aa.usno.navy.mil /hilton/ephemerides/README.html (1608 words)

	Support vector machine with orthogonal Chebyshev kernel (Site not responding. Last check: )
		An orthogonal Chebyshev kernel function for Support Vector Machine (SVM) is proposed based on extensive research about the properties of kernel functions.
		Chebyshev polynomials are firstly constructed through Chebyshev formulae.
		As Chebyshev polynomial has the best uniform proximity and its orthogonality promises the minimum data redundancy in feature space, it is possible to represent the data with less support vectors.
	csdl2.computer.org /persagen/DLAbsToc.jsp?resourcePath=/dl/proceedings/&toc=comp/proceedings/icpr/2006/2521/02/2521toc.xml&DOI=10.1109/ICPR.2006.1096 (196 words)

	Chebyshev polynomial and the Pascal Triangle
		He is known for his work in the field of probability and statistics.This article refers to what are commonly known as Chebyshev polynomials of the first kind.
		Chebyshev polynomials of the first kind are very important in numerical approximation.
		It is obvious that Pascal`s Triangle structure is built in these relations, which certainly indicates the existing connection between the numbers of Pascal`s Triangle and Chebyshev polynomials of the second kind.
	milan.milanovic.org /math/english/fibo/fibo6.html (230 words)

	Chebyshev Distance
		Chebyshev distance is also called Maximum value distance.
		The Chebyshev Distance between point A and B is
		Chebyshev distance is a special case of Minkowski distance with
	people.revoledu.com /kardi/tutorial/Similarity/ChebyshevDistance.html (70 words)

	História - Biografias: Pafnuty Lvvich Chebyshev
		A prova deste resultado foi completada somente dois anos depois da morte de Chebyshev por Hadamard e independentemente por Vallèe Poussin.
		Chebyshev estava interessado, também, em mecânica e estudou os problemas envolvidos na conversão do movimento rotativo em movimento retilíneo através de junção mecânica.
		ROY, R. The work of Chebyshev on orthogonal polynomials, in Topics in polynomials of one and several variables and their applications.
	www.pucrs.br /famat/statweb/historia/daestatistica/biografias/Chebyshev.htm (344 words)

	MATHnetBASE: Mathematics Online (Site not responding. Last check: )
		Chebyshev polynomials crop up in virtually every area of numerical analysis, and they hold particular importance in recent advances in subjects such as orthogonal polynomials, polynomial approximation, numerical integration, and spectral methods.
		It includes rigorous yet down-to-earth coverage of the theory along with an in-depth look at the properties of all four kinds of Chebyshev polynomials-properties that lead to a range of results in areas such as approximation, series expansions, interpolation, quadrature, and integral equations.
		Far from being an esoteric subject, Chebyshev polynomials lead one on a journey through all areas of numerical analysis.
	www.mathnetbase.com /ejournals/books/book_summary/summary.asp?id=915 (210 words)

	Analog Filter Design Demystified - Maxim/Dallas
		It is superior for applications in which the passband includes only one frequency of interest (e.g., the derivation of a sine wave from a square wave, by filtering out the harmonics).
		Similar to the Chebyshev response, it has ripple in the passband and severe rolloff at the expense of ripple in the stopband.
		A 5th-order, 1dB-ripple Chebyshev lowpass filter is constructed from two non-identical 2nd-order sections and an output RC network.
	www.maxim-ic.com /appnotes.cfm/appnote_number/1795 (2295 words)

	Chebyshev's Equation (Site not responding. Last check: )
		Chebyshev polynomials are found at integer values of n.
		Press the mouse down on the slider knob and drag the mouse back and forth, or click the mouse in the slider channel at the desired value for the parameter.
		[Chebyshev Function] brings up sliders to set values of the parameters R and delta as well as n.
	www.math.wayne.edu /xde/Media/JavaTools/funcchby.html (80 words)

	Chebyshev Filters by Nuhertz Technologies
		The Chebyshev Type I Filter is the filter type that results in the sharpest pass band cut off and contains the largest group delay.
		A standard Chebyshev Type I Filter's pass band attenuation is defined to be the same value as the pass band ripple amplitude.
		Below are examples of Chebyshev Type I low pass, high pass, band pass and band stop filters and the low pass step response.
	www.filter-solutions.com /chevy1.html (180 words)

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