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Topic: Darboux integral

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	Riemann integral - Wikipedia, the free encyclopedia
		In a branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann (pronounced REE mahn), was the first rigorous definition of the integral of a function on an interval.
		The definition of the Lebesgue integral is not obviously a generalization of the Riemann integral, but it is not hard to prove that every Riemann-integrable function is Lebesgue-integrable and that the values of the two integrals agree whenever they are both defined.
		An integral which is in fact a direct generalization of the Riemann integral is the Henstock-Kurzweil integral.
	en.wikipedia.org /wiki/Riemann_integral (2320 words)

	Integral - Wikipédia
		An integral which can only be evaluated by considering it as the limit of integrals on successively larger and larger integrals is called an improper integral.
		Improper integrals usually turn up when the range of the function is infinite or, in the case of the Riemann integral, when the domain is infinite.
		The Lebesgue integral was created by Henri Lebesgue to integrate a wider class of functions and to prove very strong theorems about interchanging limits and integrals.
	su.wikipedia.org /wiki/Integral (1270 words)

	Darboux integral - Wikipedia, the free encyclopedia
		In real analysis, a branch of mathematics, the Darboux integral is one possible definition of the integral of a function.
		Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal.
		Darboux integrals are named after their discoverer: Gaston Darboux.
	en.wikipedia.org /wiki/Darboux_integral (302 words)

	Integral (Site not responding. Last check: 2007-10-08)
		Intuitively, the integral of a continuous, positive real-valued function f of one real variable x between a left endpoint a and a right endpoint b represents the area bounded by the lines x=a, x=b, the x-axis, and the curve defined by the graph of f.
		Improper integrals usually turn up when the range of the function to be integrated is infinite or, in the case of the Riemann integral, when the domain of the function is infinite.
		The Riemann-Stieltjes integral, an extension of the Riemann integral.
	www.worldhistory.com /wiki/I/Integral.htm (1457 words)

	Darboux (Site not responding. Last check: 2007-10-08)
		Darboux was appointed to the Collège de France for the academic year 1866-1867, then he taught at the Lycée Louis le Grand (where Galois was educated) between 1867 and 1872.
		In 1875 he gave his way of looking at the Riemann integral, defining upper and lower sums and defining a function to be integrable if the difference between the upper and lower sums tends to zero as the mesh size gets smaller.
		Darboux is known for a wider range of mathematics to that described above.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Darboux.html (776 words)

	Integral
		The integral value of a real number x is defined to be the largest integer which is less than or equal to x; it is often denoted by ⌊x⌋ and also called the floor function.
		In the integral calculus, the integral of a function is informally defined as the size of the area delimited by the x axis and the graph of the function.
		Both the Riemann and the Lebesgue integral are approaches to integration which seek to measure the area under the curve, and the overall schema in both cases is the same.
	www.wordlookup.net /in/integral.html (883 words)

	Darboux, Jean Gaston (Site not responding. Last check: 2007-10-08)
		In defining the Riemann integral 1879, he derived the Darboux sums and used the Darboux integrals.
		Darboux was born in Nîmes and educated at the Ecole Normale Supérieure and at the Sorbonne in Paris, where he became a professor 1873 and assistant to mathematician Joseph Liouville.
		Important among Darboux's other work were his papers on the theory of integrations, proof of the existence of integrals of continuous functions, and the theory of analytical functions.
	www.cartage.org.lb /en/themes/Biographies/MainBiographies/D/Darboux/1.html (157 words)

	PlanetMath: proof of Ruffa's formula for continuous functions
		is a Darboux upper sum for the integral
		(Darboux's definition of the integral may be thought of as a modern incarnation of the ancient method of exhaustation.) Hence
		This is version 3 of proof of Ruffa's formula for continuous functions, born on 2005-01-11, modified 2005-02-26.
	planetmath.org /encyclopedia/proofofruffasformulaforcontinuousfunctions.html (154 words)

	v6n2
		In the article, K. Petr's formula of single integral is generalized to that of double integral, some important special cases and estimates of its remainder are established.
		In this paper we obtain some new nonlinear integral inequality of Gronwall type involving functions of two independent variables which can be used in the analysis of the behavior of the solutions of some partial differential equations.
		Integral representations for generalised Mathieu series are obtained which recapture the Mathieu series as a special case.
	rgmia.vu.edu.au /v6n2.html (834 words)

	Analysis, Convergence, Series, Complex Analysis - Numericana
		Uniform convergence does imply that the integral of the (uniform) limit is the limit of the integrals.
		Therefore, the contribution of the outer semicircle to the contour integral tends to zero as the radius tends to infinity.
		The integral along the right part is exactly the integral we are asked to compute, whereas the left part contributes i times that quantity.
	home.att.net /~numericana/answer/analysis.htm (3995 words)

	Suzko (Site not responding. Last check: 2007-10-08)
		The first- and second-order matrix Darboux transformations, factorization, supersymmetry, chains of transformations are studied.
		An interrelation is established between the differential and integral transformation operators.
		It is shown that in a particular case the second-order integral transformations turn into expressions for matrix solutions and potentials obtained by the inverse problem with degenerate integral kernels.
	www.imath.kiev.ua /~appmath/Abstracts2005/Suzko.html (140 words)

	Science Fair Projects - Darboux integral
		The upper Darboux sum of f with respect to x
		The lower Darboux sum of f with respect to
		, then we say that f is Darboux-integrable and set ∫f to be the common value of the upper and lower Darboux integrals.
	www.all-science-fair-projects.com /science_fair_projects_encyclopedia/Darboux_integral (450 words)

	Riemann Integration In Rn - (Site not responding. Last check: 2007-10-08)
		His integral integrates any function which is Riemann integrable, and also has good...
		Dina matematik, integral ngarupakeun konsep umum ngeunaan daerah tina gambar nu teratur ka wewengkon nu diwatesan ku fungsi.
		Differential and Integral Calculus II Grade: An examination at the end of the semester.
	integration.fabug.com /index.php?k=riemann-integration-in-rn (793 words)

	Polynomial inverse integrating factors of quadratic differential systems
		In lots of cases the expression of a first integral of the system is not easy, but on the other side the so-called inverse integrating factors have, usually, an easier expression.
		Due to the extension and, in some cases, the complexity of the classification of the polynomial inverse integrating factors, in this work we present all the polynomial inverse integrating factors of six of the ten normal forms.
		In the case of quadratic systems having a rational first integral, we also compute all the critical remarkable values and their critical remarkable invariant algebraic curves.
	www.icmc.usp.br /~getds/abst/ferragut.html (319 words)

	PlanetMath:
		Darboux coordinates (=Darboux's Theorem (symplectic geometry)) owned by bwebste
		derivation of a definite integral formula using the method of exhaustion owned by ruffa
		direct integral (in direct integral of Hilbert spaces) owned by rspuzio
	planetmath.org /encyclopedia/D (1501 words)

	Volume 23, Number 2, 1997
		In this paper the classical Kirchhoff case of motion of a rigid body in an infinite ideal fluid is considered.
		Then for the corresponding Hamiltonian system on the zero integral level, the KAM theory conditions are checked.
		This is achieved by use of the coherent state path integral for the coupled systems of the cyclotron and the guiding center motion.
	www.math.bas.bg /serdica/n2_97.html (776 words)

	Physique Mathématique 2001
		Resume: A connection is made between: separation of variables in the spectral Darboux coordinates naturally associated to isospectral Lax matrix flows on finite dimensional Poisson subspaces of loop algebras in the rational R-matrix setting and: separation of variables in the "Nijenhuis coordinates" associated with multi-Hamiltonian systems.
		This is extended to multi-Hamiltonian structures related to trigonometric and elliptic R-matrices, and to the quadratic (Sklyanin) brackets on loop groups, viewed as Poisson Lie groups.
		The large n limit of that integral can be computed, and the results involve elliptic functions even in the one cut-case (because there is another "ghost" cut).
	www.crm.umontreal.ca /act/seminars/Archives/PhysMath2001_Sch1.html (1357 words)

	Jean-Gaston Darboux -- Encyclopædia Britannica
		French mathematician who made important contributions to geometry and analysis and after whom the Darboux integral is named.
		After acting as an assistant in mathematical physics (1866–67) at the Collège de France, Paris, Darboux taught at the Lycée Louis le Grand (1867–72), the École Normale Supérieure (1872–73), and the Sorbonne (1873–90).
		More results on "Jean-Gaston Darboux" when you join.
	www.britannica.com /eb/article-9028755 (647 words)

	MATH 275 001 - Review Notes for Test #3 -- 11/12/04
		Evaluating such integrals by means of a parametrization.
		For 15.1: 2 Upper and Lower Darboux sums for the partition of the region caused by slicing horizontally with y = 1 and vertically with x = 1.
		For 15.1: 4 Upper and Lower Darboux sums for the partition of the region caused by slicing horizontally with y = 2 and vertically with x = 1.
	math.boisestate.edu /~kerr/275fa04/review_suggestions_3.html (1258 words)

	MATH 252 EXAM 2 INFORMATION (Site not responding. Last check: 2007-10-08)
		Definition of of the Darboux-Stieltjes Integral, the ``jump'' sum, the upper and lower sums.
		The restricted linearity of the Riemann-Stieltjes integral in the ``dF''
		Be able to evaluate the Stieltjes integral for particular example.
	www.math.wvu.edu /~sherm/m252/exam2.html (421 words)

	Integrability of Bounded Total Functions
		All these results have been obtained by Darboux's theorem in our previous article [10].
		In addition, we have proved the first mean value theorem to Riemann integral.
		The definition of the Riemann definite integral and some related lemmas.
	www.mizar.org /JFM/Vol12/integra4.html (177 words)

	MATH411 (Site not responding. Last check: 2007-10-08)
		Advanced Calculus is critical for those students who wish to go on to advanced mathematics.
		In this course, a theoretical foundation of analysis is presented suitable for students who have completed the standard elementary analysis courses (Differential and Integral Calculus).
		- prove that the Riemann integral and the Darboux integral are equivalent
	magisw.addu.edu.ph /syllabi/nsdiv/math411.htm (1053 words)

	Class Work in Math 424 - Spring 2005 (Site not responding. Last check: 2007-10-08)
		Class 31 (W 4/6): continue with integration; additivity of the integral.
		Class 29 (F 4/1): continue with integration; existence of the integral of continuous functions.
		Class 28 (W 3/30): continue with integration; linearity and positivity of the integral.
	www.math.uiuc.edu /~henson/Math424/Spring2005/inclass.html (422 words)

	Differential and Integral Calculus II (Site not responding. Last check: 2007-10-08)
		Properties of integrable functions and of the integral.
		Improper Riemann integrals; conditional and absolute convergence, some simple tests for convergence.
		Fichtengolz, G.M. Course in Differential and Integral Calculus,1,2
	www.math.tau.ac.il /~leviatan/calc2.html (162 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		Andrzej J. Maciejewski, Maria Przybylska Darboux Polynomials and First Integrals of Natural Polynomial Hamiltonian Systems (348K, postscript) ABSTRACT.
		We show that for a natural polynomial Hamiltonian system the existence of a single Darboux polynomial (a partial polynomial first integral) is equivalent to the existence of an additional first integral functionally independent with the Hamiltonian function.
		Moreover, we show that, in a case when the degree of potential is odd, the system does not admit any proper Darboux polynomial, i.e., the only Darboux polynomials are first integrals.
	www.ma.utexas.edu /mp_arc/a/04-142 (86 words)

	Energy Citations Database (ECD) - Energy and Energy-Related Bibliographic Citations
		The Wigner phase-space distribution function provides the basis for Moyal{close_quote}s deformation quantization alternative to the more conventional Hilbert space and path integral quantizations.
		The general features of time-independent Wigner functions are explored here, including the functional ({open_quotes}star{close_quotes}) eigenvalue equations they satisfy; their projective orthogonality spectral properties; their Darboux ({open_quotes}supersymmetric{close_quotes}) isospectral potential recursions; and their canonical transformations.
		These features are illustrated explicitly through simple solvable potentials: the harmonic oscillator, the linear potential, the P{umlt o}schl-Teller potential, and the Liouville potential.{copyright}{ital 1998}{ital The American Physical Society}
	www.osti.gov /energycitations/product.biblio.jsp?osti_id=638834 (218 words)

	Cumulative Index
		Necessary and Sufficient Conditions for Weighted Orlicz Class Inequalities for Maximal Functions and Singular Integrals.
		Gogoladze L. On Estimating Integral Moduli of Continuity of Functions of Several Variables in Terms of Fourier Coefficients, 6 (1999), No. 4, 307-322
		Tevzadze T. Estimation of the Integral Modulus of Smoothness of an Even Function of Several Variables with Quasiconvex Fourier Coefficients, 3 (1996), No. 4, 379-400
	www.maths.tcd.ie /EMIS/journals/GMJ/gmjetin.htm (7795 words)

	CI - I
		On the Solvability of a Spatial Problem of Darboux Type for the Wave Equation, 2 (1995), No. 4, 385--394
		On the Solvability of a Darboux Type Non-Characteristic Spatial Problem for the Wave Equation, 3 (1996), No. 1, 53--68
		Boundary Integral Equations of Plane Elasticity in Domains with Peaks.
	www.heldermann.de /GMJ/gmjcumui.htm (5388 words)

	wikipedia:requested articlmathematics (Site not responding. Last check: 2007-10-08)
		Bézout's lemma - Complemented lattice - Discrete valuation - Formal parameter - Gaussian periods - Ideal supporting algebra - Integral ideal - Inverse operation - Invertible - Kummer extension - Kummer theory - Real closed field - Trivial representation -
		Algebraic torus - Formal group law - Geometric invariant theory - Group scheme - Hodge cycle - Hyperelliptic curve - Jacobian variety - Poncelet-Steiner theorem - Semistable elliptic curve - Singularity theory - Theta-function
		Arithmetic harmonic mean - Asymptotic series - Darboux integral - Denjoy integral - Geometric harmonic mean - Henstock-Kurzweil-Stieltjes integral - Multivariable calculus - Perron integral - Poisson summation formula - Surface integral -
	www.yourencyclopedia.net /Wikipedia:Requested_articles/mathematics.html (872 words)

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