Sturm's theorem - Factbites

Sturm's theorem - Factbites
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Topic: Sturm's theorem

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	Dictionary of Meaning www.mauspfeil.net
		Sturm's theorem is a basic result for proving the existence of real zeroes of functions.
		In 1829 he discovered the theorem, regarding the determination of the number of real roots of a Sturm's theorem numerical equation included between given limits, which bears his name, and in the following year he was appointed professor of mathematics at the College Rollin.
		'''Jacques Charles François Sturm''' (September 29, 1803 - December 15, 1855), France French mathematician, of Germany German extraction, was born in Geneva.
	www.mauspfeil.net /Jacques_Charles_Fran%E7ois_Sturm.html

	PlanetMath: Sturm's theorem
		This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.
		This is version 3 of Sturm's theorem, born on 2004-07-22, modified 2004-07-22.
		For a proof, see Wolpert, N., ``Proof of Sturm's Theorem''
	planetmath.org /encyclopedia/SturmsTheorem.html

	sturm.refs
		Since Budan's theorem doesn't handle all cases, Sturm's theorem is more powerful.
		On the other hand, Budan's theorem doesn't require as much computation as Sturm's theorem does.
		Constructive Aspects of the Fundamental Theorem of Algebra.
	www.math.niu.edu /~rusin/known-math/97/sturm.refs (247 words)

	Jacques Charles François Sturm -- Facts, Info, and Encyclopedia article
		(Click link for more info and facts about Sturm's theorem) Sturm's theorem is a basic result for proving the existence of real zeroes of functions.
		Jacques Charles François Sturm -- Facts, Info, and Encyclopedia article
		Originally tutor to the son of (French romantic writer (1766-1817)) Madame de Staël, he resolved, with his school-fellow Colladon, to try his fortune in (The capital and largest city of France; and international center of culture and commerce) Paris, and obtained employment on the Bulletin universel.
	www.absoluteastronomy.com /encyclopedia/J/Ja/Jacques_Charles_Fran%E7ois_Sturm.htm (247 words)

	Sturm, Charles-Francois -- Encyclopædia Britannica
		French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations.
		in full Jacques-Charles-François Sturm French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations.
		Charles-François Sturm, pencil sketch by Daniel Colladon, 1822; in the Academy of Sciences, …
	www.britannica.com /eb/article?tocId=9070049 (247 words)

	September 29 - Today in Science History
		Sturm provided a complete solution to the problem with his theorem which first appeared in Mémoire sur la résolution des équations numériques (1829; "Treatise on Numerical Equations").
		French mathematician whose work resulted in Sturm's theorem, an important contribution to the theory of equations.
		With Swiss engineer Daniel Colladon, he made the first accurate determination of the velocity of sound in water (1826) and a year later wrote a prizewinning essay on compressible fluids.
	www.todayinsci.com /9/9_29.htm (247 words)

	Mathematics
		First order equations: separable, homogeneous and exact equations: integrating factors, linear, Bernoulli's, Riccati's, Lagrange's and Clairaut's equations, existence and uniqueness theorems for equations and systems, linear equations and linear sytems, Sturm's separation and comparison theorems, solutions by power series, regular singular points, Bessel's equation, Sturm-Liouville systems.
		First order differential equations: linear, separable, exact, integrating factors, homogeneous equations, existence and uniqueness theorem (without proof), linear differential equations of order n, systems of linear differential equations, solution of differential equations by power series, Bessel's equation.
		Existence and uniqueness theorems, dependence on initial values and parameters: self-adjoint problems on finite intervals, oscillation and comparison theorems, singular self-adjoint problems, two dimensional autonomous systems and the Poincare-Bendixon theory.
	www.math.technion.ac.il /department/courses/sub010.html (247 words)

	IngentaConnect Paley-Wiener-Type Theorems for a Class of Integral Transforms
		Keywords: Paley– Wiener theorem; singular Sturm– Liouville problems; Fourier transform; Hankel transform; Weber transform; Jacobi transform; Kontorovich– Lebedev transform
		Unlike the classical Paley–Wiener theorem, our theorems use real variable techniques and do not require analytic continuation to the complex plane.
		The class of integral transformations considered is related to singular Sturm–Liouville boundary-value problems on a half line and on the whole line.
	www.ingentaconnect.com /content/ap/ay/2002/00000266/00000001/art07740 (197 words)

	PlanetMath: Sturm's theorem
		This is version 3 of Sturm's theorem, born on 2004-07-22, modified 2004-07-22.
		This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.
		The total number of distinct real roots will depend only on the leading terms of the Sturm sequence polynomials.
	planetmath.org /encyclopedia/SturmsTheorem.html (222 words)

	Sturm
		Sturm's theorem is a basic result for proving t...
		Sturm Charles Francois Sturm was a French-Swiss matematician.
		Sturm und Drang Sturm und Drang (literally: "storm and stress") was a mainly literary protest movement in Friedrich Maxi...
	www.brainyencyclopedia.com /topics/sturm.html (222 words)

	PlanetMath: Sturm's theorem
		This root-counting theorem was produced by the French mathematician Jacques Sturm in 1829.
		This is version 3 of Sturm's theorem, born on 2004-07-22, modified 2004-07-22.
		For a proof, see Wolpert, N., ``Proof of Sturm's Theorem''
	planetmath.org /encyclopedia/SturmsTheorem.html (222 words)

	calc.txt
		Initial Statements ------------------ Jacques Calmet opened the panel session by arguing that algebraic topology is a field in which automated theorem proving could be successfully applied and that Jesus Aransay a PhD student currently in Karlsruhe is indeed working on this problem.
		Thomas Sturm presented his point calling himself "somewhat of an outsider to the Calculemus community".
		Subject of the Panel Discussion: Challenging Mathematical Problems (for details see also http://www.ags.uni-sb.de/~calculemus2002/panel/) Panel Members: James H. Davenport, Jvrg Siekmann, Jacques Calmet, Thomas Sturm, Alain Colmerauer, Claude Kirchner Chair: Jacques Calmet Format: 5 minute opening remarks for each panel member, followed by an hour of general discussion, open to the floor.
	4c.ucc.ie /~tw/ar/calc.txt (222 words)

	Volume 6 Abstracts
		By use of the Sturm-Desargues involution theorem it is proved that with any such quadrangle an infinite number of butterfly points is associated which are located on an equilateral hyperbola.
		The butterfly theorem and some of its generalizations deal with a specific point related to a quadrangle inscribed into a circle.
		Finally an infinite number of quadrangles sharing the same butterfly curve is presented.
	www.heldermann.de /JGG/jggabs06.htm (222 words)

	Graduate School, Temple University
		Existence and uniqueness theorems, continuous and smooth dependence on parameters, linear differential equations, asymptotic behavior of solutions, isolated singularities, nonlinear equations, Sturm-Liouville problems, numerical solution of ODEs.
		Dirichlet's theorem on primes in an arithmetic progression, the prime number theorem, algebraic number fields.
		An introduction to the ideas and techniques of number theory, elementary, analytic, and algebraic.
	www.temple.edu /gradcourses/cst/gsc_d01305.htm (903 words)

	Abstracts
		Really satisfactory results in inverse spectral theory have so far only been obtained for the simplest Sturm-Liouville equation $-u''+qu=\lambda u$ and some closely related equations.
		Inverse spectral theory should therefore attempt to reconstruct the equivalence class of the equation from spectral data, two equations being equivalent if they can be transformed into each other unitarily via a Liouville transformation.
		This seems to be a difficult problem to attack in full generality, and a first step might be to show that the potential $q$ is uniquely determined by appropriate spectral data, provided the coefficients $p$ and $w$ are known.
	www.cs.cf.ac.uk /Gregynog99/Bennewitz.html (254 words)

	Routh-Hurwitz stability criterion - Wikipedia, the free encyclopedia
		Using the Routh-Hurwitz theorem, we can replace the condition on p and q by a condition on the generalized Sturm chain, which will give in turn a condition on the coefficients of f.
		By the fundamental theorem of algebra, each polynomial of degree n must have n roots in the complex plane (i.e., for an f with no roots on the imaginary line, p+q=n).
		Routh, E. J., A Treatise on the Stability of a Given State of Motion.
	en.wikipedia.org /wiki/Routh-Hurwitz_stability_criterion (608 words)

	Charles François Sturm
		Sturm's theorem is a basic result for proving the existence of real zeroes of functions.
		Jacques Charles-François Sturm ( 1803 – 1855) was a French - Swiss mathematician and mathematical physicist.
	www.encyclopedia-1.com /c/ch/charles_francois_sturm.html (608 words)

	Edward Routh - Wikipedia, the free encyclopedia
		Central tenets of modern control systems theory relies upon the Routh stability criterion, an application of Sturm's Theorem to evaluate Cauchy indices through the use of the Euclidiean algorithm.
		Besides his intensive work in teaching, which had a persistent effect on the presentation of mathematical physics, he contributed original research, for example the Routh-Hurwitz theorem.
		Edward John Routh (1831-1907) was a British mathematician, noted as the outstanding coach of students preparing for the Mathematical Tripos examination of the University of Cambridge in its heyday in the middle of the nineteenth century.
	en.wikipedia.org /wiki/Edward_John_Routh (143 words)

	Francois
		Sturm's theorem is a basic result for proving t...
		Charles François Sturm Jacques Charles-François Sturm (Joseph Liouville.
		Francois Gevaert Francois Auguste Gevaert was born at Huysse, near Jesuit's church.
	www.brainyencyclopedia.com /topics/francois.html (143 words)

	Harmonic Analysis In Hypercomplex Systems; Author: Berezansky, Yurij M.; Author: Berezanskii, Iu M.; Hardback; Book
		Topics covered include Fourier transforms, the Plancherel theorem, the Peter-Weyl theorem, representation theory, duality, Gelfand pairs, Sturm-Liouville operators, and Lie theory.
		Topics covered include Fourier transforms, the Plancherel theorem, the Peter-Weyl theorem, representation theory, duality and others.
		Berezansky and S. Krein in the 1950s and are a generalisationof the notion of a hypergroup (a family of generalised shift operators) which was introduced in the 1970s.The book gives a state-of-the-art account of hypercomplex systems theory.
	www.netstoreusa.com /mabooks/079/0792350294.shtml (294 words)

	appliedmath.html
		Hilbert spaces: Basic geometry, orthogonality, bases, projections, and examples; Bessel’s inequality and the Parseval Theorem; the Riesz Representation Theorem; compact and Hilbert-Schmidt operators; spectral theory for compact, self-adjoint and normal operators; Sturm-Liouville Theory.
		Differential Calculus in Banach Spaces and Calculus of Variations: The Fréchet derivative; the Chain Rule and Mean Value Theorems; Banach’s Contraction Mapping Theorem and Newton’s Method; Inverse and Implicit Function Theorems, and applications to nonlinear functional equations; extremum problems, Lagrange multipliers, and problems with constraints; the Euler-Lagrange equation.
		The Fourier Transform and Sobolev Spaces: The Schwartz space and tempered distributions; the Fourier transform; the Plancherel Theorem; convolutions; fundamental solutions of PDE’s; Sobolev spaces; Imbedding Theorems; the Trace Theorems for H
	www.ma.utexas.edu /Prelims-Syllabi/appliedmath.html (325 words)

	MSC.names
		Sturm's theorem about the roots of equations, which, as he informed me with his own lips, stared him in the face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums".
		(There's another topologist A.H. Stone: "metrizable implies paracompact") Sturm is Jacques Charles-Francois (worked with Liouville), not geometer Rudolf nor Johannes Sturm, 1507-1589; Jacques is author of Sturm sequences for solvability of polynomials.
		J. C-F Sturm (Liouville's friend & colleague) did indeed discover Sturm sequences.
	www.math.niu.edu /~rusin/known-math/98/MSC.names (10614 words)

	Master of Science in Electrical and Computer Engineering
		Existence and uniqueness theorems; continuation of solutions; continuous dependence and stability, Lyapunovs direct method; differential inequalities and their applications; boundary-value problems and Sturm-Liouville theory.
		Algebra II Galois theory, solvability of equations by radicals, separable extensions, normal basis theorem, norm and trace, cyclic and cyclotomic extensions, Kummer extensions.
		Review of functional spaces and embedding theorems; existence and regularity of solutions of boundary-value problems for second-order elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundary-value problems for second-order parabolic and hyperbolic equations.
	eng.ku.edu.tr /grad/gradscience/math.html (10614 words)

	Department of Mathematics, University of Strathclyde
		Closed Operators: Graphs and closed operators; Closed Graph Theorem and Bounded Inverse Theorem.
		Statement of the Spectral Theorem on infinite-dimensional spaces.
		Applications will be centred mainly on Sturm-Liouville type problems.
	www.maths.strath.ac.uk /ungrad/classes/921.htm (267 words)

	Master of Science in Electrical and Computer Engineering
		Existence and uniqueness theorems; continuation of solutions; continuous dependence and stability, Lyapunovs direct method; differential inequalities and their applications; boundary-value problems and Sturm-Liouville theory.
		Review of functional spaces and embedding theorems; existence and regularity of solutions of boundary-value problems for second-order elliptic equations; maximum principles for elliptic and parabolic equations; comparison theorems; existence, uniqueness and regularity theorems for solutions of initial boundary-value problems for second-order parabolic and hyperbolic equations.
		First order equations, method of characteristics; the Cauchy-Kovalevskaya theorem; Laplace's equation: potential theory and Greens's function, properties of harmonic functions, the Dirichlet problem on a ball; heat equation: the Cauchy problem, initial boundary-value problem, the maximum principle; wave equation: the Cauchy problem, the domain of dependence, initial boundary-value problem.
	www.eng.ku.edu.tr /grad/gradscience/math.html (1537 words)

	mingarelli-tex
		However, {\it there is no Sturm oscillation theorem} (or even its extension by Haupt and Richardson, cf., \cite{abm}) for the eigenfunctions as we show presently.
		Thus, in an interesting connection with operator theory, degenerate Sturm-Liouville operators give rise to self-adjoint operators on a Kre\v{i}n space whose spectrum is all of $\mathbb{C}$ (see \cite{alm}, Section 4] for more details about this connection.
		For this reason among others (likely of a physical nature), Sturm restricted himself to cases where the leading term $p(x) >0$ in the interval under consideration (and then also to cases where the weight $r(x) >0$).
	www.univie.ac.at /EMIS/journals/EJDE/Monographs/Monographs/Volumes/2004/130/mingarelli-tex (2573 words)

	Hermite
		Sturm and Cauchy gave a good report on this memoir in 1851 but a priority dispute with Liouville seems to have prevented its publication.
		Hermite may have still been an undergraduate but it is likely that his ideas from around 1843 helped Liouville to his important 1844 results which include the result now known as Liouville 's theorem.
		Also like Galois he was attracted by the problem of solving algebraic equations and one of the two papers attempted to show that the quintic cannot be solved in radicals.
	www-groups.dcs.st-and.ac.uk /~history/Mathematicians/Hermite.html (2573 words)

	PUC-RIO - Ensino e Pesquisa - Coordenação Central de Cooperação Internacional - Centers
		Theorems of existence and uniqueness of solutions.Continuous dependence on initial values.Classical Sturm-Liouville problems,fundamental solutions and Green functions.
		Integration: idea of integral, fundamental theorem of calculus, indefinite integral, techniques of integration, area between curves.
		Integration of differential forms, partitions of unity; Stokes's theorem, closed and exact forms; square summable functions.
	www.puc-rio.br /ensinopesq/ccci/deptos/mat.html (2573 words)

	Department of Mathematics, University of Strathclyde
		Statement of the Spectral Theorem on infinite-dimensional spaces.
		The Spectral Theorem: Spectral Theorem on finite-dimensional spaces.
		Closed Operators: Graphs and closed operators; Closed Graph Theorem and Bounded Inverse Theorem.
	www.maths.strath.ac.uk /ungrad/classes/921.htm (267 words)

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