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Topic: Exponential sum

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	Exponential sum - Wikipedia, the free encyclopedia (Site not responding. Last check: 2007-11-05)
		A basic distinction is between a complete exponential sum, which is typically a sum over all residue classes modulo some integer N (or more general finite ring), and an incomplete exponential sum where the range of summation is restricted by some inequality.
		Examples of complete exponential sums are Gauss sums and Kloosterman sums; these are in some sense finite field or finite ring analogues of the gamma function and some sort of Bessel function, respectively, and have many 'structural' properties.
		An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by Gauss).
	en.wikipedia.org /wiki/Exponential_sum (614 words)

	Exponential sum -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-05)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, an exponential sum may be a finite (The sum of a series of trigonometric expressions; used in the analysis of periodic functions) Fourier series (i.e.
		An example of an incomplete sum is the partial sum of the quadratic Gauss sum (indeed, the case investigated by (A unit of magnetic flux density equal to 1 maxwell per square centimeter) Gauss).
		One of the most general types of exponential sum is the Weyl sum, with exponents 2πif(n) where f is a fairly general real-valued (Click link for more info and facts about smooth function) smooth function.
	www.absoluteastronomy.com /encyclopedia/e/ex/exponential_sum.htm (740 words)

	exponential-sum whorls (Site not responding. Last check: 2007-11-05)
		The first "few" partial sums of this series are plotted in the complex plane.
		Count the whorls and estimate the length of the sum.
		To generate similar pictures of the sums of e(f(n)) for much larger values of f(n), compute the fractional parts of f(n) recursively using power-series approximations to the first or (if necessary) higher finite differences of f - which after all is also a key ingredient in our analytic estimates on such exponential sums.
	www.math.harvard.edu /~elkies/M259.98/whorls.html (136 words)

	Numerical methods
		This remarkable convergence rate for smooth, exponentially decaying integrands follows from the observation that by multiplying the terms near the boundary by appropriate factors, it is possible to obtain algorithms of higher and higher order in the step size [14].
		Exponentially decaying integrands are impervious to any such coefficients, and so converge faster than any power of the step size.
		This means it is advisable to group them in size (using the argument of the exponential) as they are summed, then combine the groups from smallest to largest to minimize roundoff error.
	www.stats.bris.ac.uk /~macpd/tln/node3.html (489 words)

	314_report (Site not responding. Last check: 2007-11-05)
		The algorithm used in the first summing program was very simple, it preformed the sum directly by multiplying -x on the top and taking a factorial of the step number on the bottom.
		After calculating the sum the program would save the data in a comma-separated variable file (*.csv) which I could open directly with Microsoft Excel to analyze the data.
		After calculating the sum, the program also calculated the absolute error, assuming that the math.h exp() function was exact.
	w3.physics.uiuc.edu /lindenle/314/314_re~1.htm (504 words)

	Exponential Sums for (ResearchIndex)
		Identities between exponential sums including Kloosterman sums were studied by many researchers (see e.g.
		0.4: Hyper-Kloosterman Sums and Estimation of Exponential Sums of..
		Hyper-Kloosterman Sums and Estimation of Exponential Sums of..
	citeseer.ist.psu.edu /199284.html (377 words)

	LMS Proceedings Abstract, paper PLMS 1495 (Site not responding. Last check: 2007-11-05)
		Exponential sums and the Riemann zeta function V
		A Van der Corput exponential sum is $S = \Sigma \exp (2 \pi i f(m))$, where $m$ has size $M$, the function $f(x)$ has size $T$ and $\alpha = (\log M) / \log T
		Bounds for sums of type $S$ lead to bounds for integer points close to curves, and another branching iteration.
	www.lms.ac.uk /publications/proceedings/abstracts/p1495a.html (166 words)

	Algorithms for fitting two classes of exponential sums to empirical data (Site not responding. Last check: 2007-11-05)
		When fitting an exponential sum model to data and estimating the parameters determining its shape, often a least squares criterion is used.
		For exponential sums this is simple to apply as the sum of a set of equidistant data points is a geometrical sum.
		The interpolation in single points of a classical exponential sum is equivalent to the classical Prony method and thus generalized interpolation can be viewed as a generalization of the Prony method.
	www.math.kth.se /optsyst/seminar/peterssonII.html (316 words)

	Fourier Analysis of Time Series
		Sums can be done with matrix multiplication, as describes in Appendix I.
		The exponential sum is the sum of a geometric progression, which can be summed as done in Appendix G. Thus, we have
		Thus, the sum appears to be of the same form as before, but there are half as many terms with a different coefficient for the
	people.uncw.edu /hermanr/signals/Notes/Signals.htm (1799 words)

	A Review of the Parameter Estimation Problem of Fitting Positive Exponential Sums to Empirical Data - Petersson, ...
		Abstract: Exponential sum models are used frequently: In heat diffusion, diffussion of chemical compounds, time series in medicine, economics, physical sciences and technology.
		There are many different ways of estimating parameters in exponential sums and model a fit criterion, which gives a valid result from the fit.
		3 Decomposition of multicomponent exponential decays by spectr..
	citeseer.ist.psu.edu /petersson97review.html (1341 words)

	[No title]
		Exponential sums have important applications in many areas of mathematics, including Diophantine equations and Goppa codes.
		The principal aim of this research is to study properties of exponential sums over finite fields, particularly those properties that are reflected in the L-function of the exponential sum and hence have a cohomological interpretation.
		A key component of the project will be to study the p-adic differential equation that describes the variation of p-adic cohomology in a parametrized family of exponential sums.
	www.osu-ours.okstate.edu /report93/as/math.html (1764 words)

	Radiative Transfer Model (Site not responding. Last check: 2007-11-05)
		In order to achieve this accuracy, the transmission function of molecular band absorption should be expressed as a sum of exponential terms for each gas species.
		More than a few thousand terms of the combined exponential functions may be needed for a single narrow wave number interval, thus requiring the computing power provided by high-performance parallel systems.
		It is suitable for cases where the total number of the exponential terms is almost the same for each interval.
	beowulf.gsfc.nasa.gov /ESS/annual.reports/ess95contents/app.gci.wan.html (372 words)

	Correlated k--Distribution Research (Site not responding. Last check: 2007-11-05)
		The essence of both the CKD and ESFT approaches is that both reduce the expression for the transmission of a spectral band to a sum of a number of exponential-in-path transmission functions.
		The disadvantage is that the number of exponential transmission functions in the summation tend to be large for transmissions that change by orders of magnitude typical of paths associated with troposphere--stratosphere radiative transfer.
		The total number of radiative transfer calculations for a given water vapor only atmospheric column is the sum of the second column in the table (50 for the infrared spectrum and 42 for the solar spectrum).
	www.agu.org /revgeophys/stephe01/node9.html (310 words)

	Oxford University Press
		It is an excellent and important work for all mathematicians who deal with exponential sums and lattice point theory.
		In analytic number theory a large number of problems can be "reduced" to problems involving the estimation of exponential sums in one or several variables.
		The audience for the book will be mathematics graduate students and faculties with a research interest in analytic theory; more specifically, those with an interest in exponential sum methods.
	www.oup.com /ca/isbn/0-19-853466-3 (343 words)

	A Uniquely Identifiable Model (Site not responding. Last check: 2007-11-05)
		The function describing the observations, in this case a "sum" of exponential (there is just one) is (d/V1)exp(-k(0,1)t).
		So, the parameters that can be unequivocally determined based on the fit of a single exponential function to the data are (d/V1) and -k(0,1).
		If one were writing this as a standard sum of exponentials, it would be written Aexp(-at) where A and a can be uniquely estimated from the monoexponentially decaying data; here A and a are the observational parameters.
	depts.washington.edu /rfpk/training/tutorials/modeling/part3/05.html (193 words)

	Multiple Channels Mediate Calcium Leakage in the A7r5 Smooth Muscle-Derived Cell Line -- Obejero-Paz et al. 75 (3): ...
		sum of the open states within the burst was divided by the sum
		The mean burst length was calculated by fitting exponential functions to the burst time distribution, or by averaging the burst lengths when few bursts were observed.
		Single Lorentzian functions and the sum of multiple Lorentzians are depicted as dashed and solid lines, respectively.
	www.biophysj.org /cgi/content/full/75/3/1271 (6952 words)

	An Introduction to the Theory of Newton Polygons for L-functions of Exponential Sums, by Daqing Wan (Site not responding. Last check: 2007-11-05)
		An Introduction to the Theory of Newton Polygons for L-functions of Exponential Sums, by Daqing Wan
		The aim is to give an elementary and self-contained introduction to the theory of Newton polygons (namely, the $p$-adic Riemann hypothesis) for L-functions of exponential sums over a finite field.
		They are also enough for a number of further applications such as Mazur's conjecture for a generic hypersurface and a weaker form of the more general Adolphson-Sperber conjecture for a generic exponential sum.
	www.math.uiuc.edu /Algebraic-Number-Theory/0178 (296 words)

	CodeGuru Forums - Fitting a Sum of Exponentials to Numerical Data
		A finite sum of exponential functions may be expressed by a linear combination of powers of the independent variable and by successive integrals of the sum.
		This is proved for the general case and the connection between the parameters in the sum and the coefficients in the linear combination is highlighted.
		The fitting of exponential functions to a given data- set is therefore reduced to a multilinear approximation procedure.
	www.codeguru.com /forum/showthread.php?referrerid=177422&t=342027 (355 words)

	Journal of the American Mathematical Society (Site not responding. Last check: 2007-11-05)
		Abstract: The aim of this paper is to extend recent work of S. Konyagin and the author on Gauss sum estimates for large degree to the case of `sparse' polynomials.
		J. Bourgain, Estimates on exponential sums related to the Diffie-Hellman distributions, to appear in GAFA.
		L.J. Mordell, On a sum analogous to a Gauss sum, Quart.
	80-www.ams.org.library.uor.edu /jams/2005-18-02/S0894-0347-05-00476-5/home.html (411 words)

	Research Interests (Site not responding. Last check: 2007-11-05)
		By an exponential sum we mean a finite sum whose summands are all found on the complex unit circle.
		The goal is to find upper and lower bounds on these sums, a task to which mathematicians have applied considerable intellectual firepower over the years.
		Bounds on Exponential Sums and the Polynomial Waring's Problem Mod p (with T. Cochrane and C. Pinner); to appear in the Journal of the London Mathematical Society.
	www.math.ksu.edu /~jasonr/Research.html (1123 words)

	REU 2002 Robinson
		They computed the generalized exponential sum $F^(i^)$ associated with their polynomial directly using $p$-adic analysis.
		Thomas Wright wrote the preprint \lq\lq A stationary phase formula for generalized exponential sums." In this paper, Wright gives a formula for computing generalized exponential sums that is analogous to Igusa's stationary phase formula for local zeta functions.
		Abstract: This paper gives a formula for Generalized Exponential sums which depends entirely upon the singularities mod $p$, similar to the purpose of the Stationary Phase Formula for Igusa Local Zeta Functions.
	www.mtholyoke.edu /~robinson/reu/reu02/reu02.htm (736 words)

	Pub 1 (Site not responding. Last check: 2007-11-05)
		The discrete ordinate radiative transfer code was used, along with the exponential sum fitting of transmission of water vapor, ozone, carbon dioxide, carbon monoxide, methane, and nitrous oxide.
		The spectral data base for the exponential sum fitting was obtained from the LOWTRAN7 transmittances.
		Observed vertical profiles of temperature, water vapor and ozone, obtained with conventional radiosondes and with Raman lidar, were used as inputs to the calculations.
	www.atmos.umd.edu /~laszlo/pub1.html (221 words)

	Wolfram Research, Inc.
		This exponential generating function is simply the Taylor series for the exponential.
		Here is the exponential generating function of the sequence of squares of integers.
		This computes the exponential power sum for a series that has a different form when
	documents.wolfram.com /v3/AddOns/DscM_RSolve-.html (769 words)

	Initial Values for Two Classes of Exponential Sum Least Squares Fitting Problems - Petersson, Holmstrom (ResearchIndex)
		Abstract: The authors have earlier developed new initial value algorithms to least squares fitting of two classes of exponential sum models by generalized interpolation (GI).
		In this report the class f (t) = P p i=1 (a i t + c i) exp (\Gammab i t) and its subclass, when all c i = 0, are treated.
		Initial Values for a Class of Exponential Sum Least..
	citeseer.ist.psu.edu /367981.html (478 words)

	Cardiff University School of Mathematics Home Page (Site not responding. Last check: 2007-11-05)
		It turns out that there are excellent reasons why sieve methods alone cannot solve these problems, but they give partial information on these and many other problems where the `deeper' methods of analytic number theory, such as exponential sums will not work.
		For example pairs of consecutive odd numbers which are either prime or very hard to factorise do keep on occurring.
		Sieve methods can be purely combinatorial like the "sieve of Eratosthenes", or partly combinatorial and partly analytic: finding an inequality by rearranging a sum of squares which must be positive - even the modulus squared of an exponential sum in the so-called "large sieve".
	www.cf.ac.uk /maths/numbertheory/sieves.html (174 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		Returns the ith segment of length number of exponential segments whose values sum to sum.
		The exponential curve is defind by power, where 01 yields convex slopes.
		The type of the value returned normally depends on the type of the arguments specified to the function.
	ccrma.stanford.edu /CCRMA/Courses/AlgoComp/cm/doc/dict/explseg.fun.html (73 words)

	Super-greenhouse Gases - Example Analysis
		The k value chosen was the one that resulted in the smallest error (the calculated error is the sum of the absolute difference between the theoretical and experimental data).
		If the curve could not be fitted using a one term exponential (i.e.
		It is important to note that these solutions are most likely not unique, however, this is not important for this analysis since the objective of the exponential sum fits is to simply find a function that represents the data.
	www.users.globalnet.co.uk /~mfogg/Sample.html (518 words)

	The Mathematical Institute Eprints Archive - An estimate for Heilbronn's exponential sum
		The Mathematical Institute Eprints Archive - An estimate for Heilbronn's exponential sum
		Heath-Brown, D.R. An estimate for Heilbronn's exponential sum.
		No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without prior permission of the copyright owner.
	eprints.maths.ox.ac.uk /archive/00000157 (83 words)

	Additive and Combinatorial Number Theory (L24) (Site not responding. Last check: 2007-11-05)
		An exponential sum is a sum such as
		Such expressions may look peculiar at first, and indeed there is no hope of evaluating them with a simple formula, but it can be extremely useful to obtain upper bounds for their modulus.
		Because of the hybrid nature of the material, this course could be classified equally well as combinatorics or number theory, and would also be a good complement to courses in harmonic analysis (in particular the course entitled Restriction and Kakeya Phenomena).
	www.maths.cam.ac.uk /CASM/courses/02-03/descriptions/node34.html (507 words)

	Physics Help and Math Help - Physics Forums - Path integrals and the sum of surfaces, is this general?
		The original surface would then be the result of a sum of surfaces.
		The expansion (if it exists) will prescribe a surface, a piece of which is the same as a piece of the surface prescribed by the original function; it is possible that they may only have a single point in common!
		The surfaces prescribed by the partial sums may or may not eventually look like the surface prescribed by the expansion.
	www.physicsforums.com /printthread.php?t=9396&pp=40 (1599 words)

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