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	Encyclopedia: Diophantine equation (Site not responding. Last check: 2007-11-07)
		The word Diophantine refers to the Greek mathematician of the third century A.D., Diophantus of Alexandria, who made a study of such equations and was one of the first mathematicians to introduce symbolism into algebra.
		The depth of the study of general Diophantine equations is shown by the characterisation of Diophantine sets as recursively enumerable.
		The field of Diophantine approximation deals with the cases of Diophantine inequalities: variables are still supposed to be integral, but some coefficients may be irrational numbers, and the equality sign is replaced by upper and lower bounds.
	www.nationmaster.com /encyclopedia/Diophantine-equation (1071 words)

	Diophantine approximation (Site not responding. Last check: 2007-11-07)
		The smallness of the distance from the real number to be approximated and the rational number that approximates it is a crude measure of how good the approximation is. A subtler measure considers how good the approximation is by comparison to the size of the denominator.
		This showed that Diophantine appproximation results were closely related to the general problem of cancellation in exponential sums, which occurs all over analytic number theory in the bounding of error terms.
		Diophantine approximation; Nevanlinna theory, especially as related to diophantine approximation; Arakelov theory.
	www.serebella.com /encyclopedia/article-Diophantine_approximation.html (523 words)

	Diophantine approximation - Wikipedia, the free encyclopedia
		In number theory, the field of Diophantine approximation, named after Diophantus of Alexandria, deals with the approximation of real numbers by rational numbers.
		The smallness of the distance (in an absolute value sense) from the real number to be approximated and the rational number that approximates it is a crude measure of how good the approximation is. A subtler measure considers how good the approximation is by comparison to the size of the denominator.
		This showed that Diophantine approximation results were closely related to the general problem of cancellation in exponential sums, which occurs all over analytic number theory in the bounding of error terms.
	en.wikipedia.org /wiki/Diophantine_approximation (404 words)

	Diophantine approximation - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-07)
		There are still simply-stated unsolved problems remaining in Diophantine approximation, for example Littlewood's conjecture.
		Diophantine Approximations and Diophantine Equations (Lecture Notes in Mathematics, Vol.
		Diophantine Approximations and Transcendental Numbers: Proceedings of the Colloquium Held at C.R.M., Luminy June 18-22/Appoximations Diophantiennes
	encyclopedia.worldsearch.com /diophantine_approximation.htm (425 words)

	Maurice Dodson's Home page
		Simultaneous Diophantine approximation on the circle and Hausdorff dimension (with H. Dickinson), Math.
		Diophantine approximation and a lower bound for Hausdorff dimension (with B P Rynne and J A G Vickers), Mathematika 37 (1990), 59-73.
		Metric inhomogeneous Diophantine approximation with integral polynomials (with V.I.Bernik and H.Dickinson)
	www-users.york.ac.uk /~mmd1/Publications.html (1123 words)

	Diophantine Approximation (Site not responding. Last check: 2007-11-07)
		Diophantine Approximation is the branch of number theory dealing with problems such as whether certain given numbers are rational or irrational, or algebraic or transcendental, and how well given numbers can be approximated by rational or algebraic numbers.
		The purpose of the workshop was to discuss new developments in Diophantine approximation and its applications.
		In the lectures, a very broad range of Diophantine approximation was covered: irrationality and transcendence problems, approximation of real numbers by algebraic numbers, applications to Diophantine equations, elliptic curves and Diophantine geometry, connections with Nevanlinna theory, transcendence results for Drin’feld modules, and computational aspects of Diophantine approximation.
	www.lc.leidenuniv.nl /lc/web/2003/20030728/report.html (291 words)

	Diophantine approximation (Site not responding. Last check: 2007-11-07)
		The smallness of the distance (in absolute value sense) from the real number to approximated and the rational number that approximates is a crude measure of how good approximation is. A subtler measure considers how the approximation is by comparison to the of the denominator.
		This showed that Diophantine results were closely related to the general of cancellation in exponential sums which occurs over analytic number theory in the bounding of error terms.
		Diophantine Approximation: Lectures Given at the C.I.M.E. Summer School Held in Cetraro, Italy, June 28-July 6, 2000 (Lecture Notes in Mathematics (Springer-Verlag), 1819.)
	www.freeglossary.com /Diophantine_approximation (465 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		The phenomenon of ``small divisors'' is closely related to Diophantine approximation and it is perhaps no coincidence that Dirichlet, Kronecker and Siegel all worked on small divisor problems.
		In particular, Sprind zuk's solution of Mahler's problem and recent work in Diophantine approximation and Hausdorff dimension on manifolds [4] are of use in the problem of lower dimensional invariant tori and of averaging in multifrequency systems ([2], [5]).
		It is possible to extend the KAM results to consider the degenerate case of m-dimensional elliptic invariant tori in a Hamiltonian system with m+n degrees of freedom and obtain a formula for the Hausdorff dimension of the exceptional set of frequencies which fail to satisfy the relevant Diophantine approximation condition [6].
	www.maths.soton.ac.uk /staff/Vickers/dynamics.html (512 words)

	Simon Kristensen's homepage (Site not responding. Last check: 2007-11-07)
		Diophantine criteria occur naturally in the theory of partial differential equations as well as in ergodic theory and other branches of dynamics.
		Conversely, Diophantine approximation draws on various methods from dynamics, functional analysis and other branches of mathematics.
		A branch of Diophantine approximation in which I have been particularly interested is the so-called metric theory of Diophantine approximation.
	www.maths.ed.ac.uk /~simonk/research.html (194 words)

	Powell's Books - Diophantine Approximation on Linear Algebraic Groups: Transcendence Properties of the Exponential ...
		The value of Waldschmidt's new monograph for the further development of the subject cannot be overestimated.
		Diophantine approximation is an important part of number theory, which studies integers and the relations between them.
		The theory of transcendental numbers is closely related to the study of Diophantine approximation.
	www.powells.com /cgi-bin/biblio?inkey=4-3540667857-4 (427 words)

	Department of Computer Science
		Given real numbers $\al_1,\dots,\al_n$, a simultaneous diophantine $\eps$-approximation is a sequence of integers $P_1,\dots,P_n,Q$ such that $Q>0$ and for all $j\in\{1,\dots,n\}$, $Q\al_j-P_j\leq\eps$.
		A simultaneous diophantine approximation is said to exclude the prime $p$ if $Q$ is not divisible by $p$.
		We generalize the result to simultaneous diophantine $\eps$-approximations excluding several primes.
	www.cs.uchicago.edu /research/publications/techreports/TR-2001-25 (253 words)

	11J: Diophantine approximation, transcendental number theory (Site not responding. Last check: 2007-11-07)
		Lang, Serge: "Introduction to Diophantine approximations", Springer-Verlag, New York, 1995.
		Quality of approximations of an irrational by rationals.
		Maximizing a sum of sines (of different periods) (really a question of approximating a number by rationals).
	www.math.niu.edu /~rusin/known-math/index/11JXX.html (343 words)

	Amazon.com: Books: Diophantine Analysis (Site not responding. Last check: 2007-11-07)
		Simon (Series Editor), M.M. Dodson "Diophantine approximation is a more quantitative and general study of the density of the rationals Q in the reals R while a smooth manifold is..." (more)
		This volume explores Diophantine approximation on smooth manifolds embedded in Euclidean space, developing a coherent body of theory comparable to that of classical Diophantine approximation.
		Diophantine approximation is a more quantitative and general study of the density of the rationals Q in the reals R while a smooth manifold is locally diffeomorphic to Euclidean space.
	www.amazon.com /exec/obidos/tg/detail/-/0521432758?v=glance (563 words)

	Citations: Factorization of univariate integer polynomials by diophantine approximation and an . 4 improved basis ... (Site not responding. Last check: 2007-11-07)
		Citations: Factorization of univariate integer polynomials by diophantine approximation and an.
		His concept of semi reduction approximates the length of the shortest lattice vector up to a factor 2 whereas we get close to a factor (4=3) n=2.
		Sch"onhage, A., "Factorization of univariate integer polynomials by diophantine approximation and an.
	citeseer.ist.psu.edu /context/353304/0 (1532 words)

	A Note On Metric Inhomogeneous Diophantine Approximation (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		3 A metrical theorem in Diophantine approximation (context) - Schmidt - 1960
		2 Diophantine approximation and a lower bound for Hausdorff di..
		2 Diophantine approximation and Hausdorff dimension (context) - Baker, Schmidt - 1970
	citeseer.ist.psu.edu /90184.html (341 words)

	Simon Kristensen's homepage (Site not responding. Last check: 2007-11-07)
		Diophantine approximation and the solubility of the Schrödinger equation, Phys.
		Metric Diophantine approximation with respect to planar distance functions, to appear in The Proceedings of the Institute of Mathematics NAN Belarus, (
		Approximating numbers with missing digits by algebraic numbers, to appear in Proc.
	www.maths.ed.ac.uk /~simonk/papers.html (304 words)

	[No title]
		Carlitz: Diophantine approximation in fields of characteristic p.
		Ito: On Legendre's theorem related to diophantine approximations.
		3595 Wolfgang Schmidt: Diophantine approximations and diophantine equations.
	felix.unife.it /Root/d-Mathematics/d-Number-theory/b-Diophantine-approximation (412 words)

	Diophantine geometry in characteristic p, survey update. (Site not responding. Last check: 2007-11-07)
		Updates on "Diophantine geometry in characteristic p: a survey" by José Felipe Voloch, which appeared in: Arithmetic Geometry, F. Catanese, ed., Symposia Mathematica XXXVII, Cambridge Univ. Press, 1997, pp.
		Of course, diophantine approximation in positive characteristic gives a different result.
		For other results on diophantine approximation on abelian varieties see T. Scanlon, "The abc theorem for commutative algebraic groups in characteristic p", IMRN, (1997), 17, pp 881-898.
	www.ma.utexas.edu /users/voloch/survey.html (474 words)

	Diophantine approximation exponents and continued fractions for algebraic power series, by Dinesh S. Thakur (Site not responding. Last check: 2007-11-07)
		Diophantine approximation exponents and continued fractions for algebraic power series, by Dinesh S. Thakur
		This is a corrected and improved version of the preprint 147.
		For each rational number not less than 2, we provide an explicit family of continued fractions of algebraic power series in finite characterisic (together with the algebraic equations they satisfy) which have that rational number as its diophantine approximation exponent.
	www.math.uiuc.edu /Algebraic-Number-Theory/0156 (112 words)

	Atlas: The critical determinant of the double paraboloid and simultaneous Diophantine approximation by Werner Georg ... (Site not responding. Last check: 2007-11-07)
		Atlas: The critical determinant of the double paraboloid and simultaneous Diophantine approximation by Werner Georg Nowak
		A combination of these ideas yields improved bounds for the simultaneous Diophantine approximation constants (in the sense of Hurwitz' theorem), with respect to the Euclidean norm, for dimension 3, 4, 5.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacf-43.
	atlas-conferences.com /cgi-bin/abstract/cacf-43 (186 words)

	Cryptology ePrint Archive (Site not responding. Last check: 2007-11-07)
		Exploiting the inherent probabilism of quantum computation we are able to substitute the continued fraction algorithm to find a certain unknown fraction by a simultaneous Diophantine approximation.
		While the continued fraction algorithm is able to find a Diophantine approximation to a single known fraction with a denominator greater than $N^2$, our simultaneous Diophantine approximation method computes in polynomial time unusually good approximations to known fractions with a denominator of size $N^{1+\varepsilon}$, where $\varepsilon$ is allowed to be an arbitrarily small positive constant.
		As these unusually good approximations are almost unique we are able to recover an unknown denominator using fewer qubits in the quantum part of our algorithm.
	eprint.iacr.org /2000/036 (246 words)

	Diophantine approximation
		Measure theory and metric Diophantine approximation: Diophantine approximation on manifolds.
		Metrical Diophantine approximation, harmonic analysis and applications to signal processing, chaos and biology.
		Analytic number theory: application of sieve methods, Diophantine approximation, metric number theory.
	www.omniknow.com /common/wiki.php?in=en&term=Diophantine_approximation (792 words)

	Diophantine approximation and deformation, by M. Kim, D. Thakur, J. F. Voloch (Site not responding. Last check: 2007-11-07)
		Diophantine approximation and deformation, by M. Kim, D. Thakur, J. Voloch
		We associate certain curves over function fields to given algebraic power series and show that bounds on the rank of Kodaira-Spencer map of this curves imply bounds on the exponents of the power series, with more generic curves giving lower exponents.
		We also prove a hierarchy of exponent bounds for approximation by algebraic quantities of bounded degree.
	www.math.uiuc.edu /Algebraic-Number-Theory/0191 (121 words)

	Diophantine approximation and analytic number theory (Site not responding. Last check: 2007-11-07)
		The objective of this workshop is to gather together researchers with expertise in both Diophantine approximation and analytic number theory in an environment that fosters the presentation and sharing of the latest ideas in both fields.
		The participants named below have been chosen either as experts in analytic number theory whose work involves problems in Diophantine approximation, or as experts in Diophantine approximation whose methods also lend themselves to the resolution of open questions in analytic number theory.
		A reduction of a problem in one area of number theory (and indeed in many other mathematical fields as well) often involves a very simply stated question in the other area, which can seem difficult to resolve if one is not well-versed in the techniques of the second area.
	www.pims.math.ca /birs/workshops/2004/04w5507 (286 words)

	Simultaneous Diophantine Approximation with Excluded Primes (ResearchIndex) (Site not responding. Last check: 2007-11-07)
		, # n, a simultaneous diophantine #-approximation is a sequence of integers P 1,.
		36 An Introduction to Diophantine Approximations (context) - Cassels - 1957
		Simultaneous Diophantine Approximation with Excluded Primes - Stefankovic (2001)
	citeseer.ist.psu.edu /462833.html (248 words)

	CWI Tract (Site not responding. Last check: 2007-11-07)
		These bounds are often so large that it is practically impossible to use enumeration up to the upper bound, in order to determine all the solutions.
		Computational Diophantine approximation theory of linear forms is developed to deal with this problem.
		This method is applied to several types of Diophantine equations.
	www.cwi.nl /publications/Abstracts_tracts/tr-65.html (168 words)

	Publisher description for Library of Congress control number 00266475 (Site not responding. Last check: 2007-11-07)
		This book is concerned with Diophantine approximation on smooth manifolds embedded in Euclidean space, and its aim is to develop a coherent body of theory comparable with that which already exists for classical Diophantine approximation.
		A wide range of techniques from the number theory arsenal are used to obtain the upper and lower bounds required, and this is an indication of the difficulty of some of the questions considered.
		The authors go on to consider briefly the p-adic case, and they conclude with a chapter on some applications of metric Diophantine approximation.
	www.loc.gov /catdir/description/cam021/00266475.html (202 words)

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