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Topic: Schwarzian derivative

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	Schwarzian derivative - Wikipedia, the free encyclopedia
		In mathematics, the Schwarzian derivative is a certain operator that is invariant under all linear fractional transformations.
		The Schwarzian derivative of a function of one complex variable f is defined by
		The Schwarzian derivative has a curious interplay with second-order linear ordinary differential equations.
	www.wikipedia.org /wiki/Schwarzian_derivative (419 words)

	Encyclopedia: Derivative (Site not responding. Last check: 2007-10-08)
		If the second derivative is positive at a critical point, that point is a local minimum; if negative, it is a local maximum; if zero, it may or may not be a local minimum or local maximum.
		The common thread is that the derivative at a point serves as a linear approximation of the function at that point.
		Perhaps the most natural situation is that of functions between differentiable manifolds; the derivative at a certain point then becomes a linear transformation between the corresponding tangent spaces and the derivative function becomes a map between the tangent bundles.
	www.nationmaster.com /encyclopedia/Derivative (5362 words)

	Derivative (Site not responding. Last check: 2007-10-08)
		In mathematics, the derivative of a function is one of the two central concepts of calculus.
		The derivative of a function at a point measures the rate at which the function's value changes as the function's argument changes.
		Derivatives are defined by taking the limit of the slope of secant lines as they approach a tangent line.
	hallencyclopedia.com /Derivative (2293 words)

	Connection (mathematics) - Wikipedia, the free encyclopedia
		In differential geometry, a connection (also connexion) or covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold.
		That is, partial derivatives are not an intrinsic notion on a manifold: a connection 'fixes up' the concept and permits discussion in geometric terms.
		There is also a concept of projective connection; the most commonly-met form of this is the Schwarzian derivative in complex analysis.
	www.wikipedia.org /wiki/Connection%2B(vector%2Bbundle) (493 words)

	Read about Derivative at WorldVillage Encyclopedia. Research Derivative and learn about Derivative here! (Site not responding. Last check: 2007-10-08)
		As it turns out, the derivative is an extremely versatile concept which can be viewed in many different ways.
		physics is the concept of the "time derivative" — the rate of change over time — which is required for the precise definition of several important concepts.
		second derivative test provide ways to determine if the critical points are maxima, minima or neither.
	encyclopedia.worldvillage.com /s/b/Derivative (2031 words)

	Encyclopedia: Schwarzian derivative (Site not responding. Last check: 2007-10-08)
		The Schwarzian derivative of a function of one complex variable f is defined by Complex analysis is the branch of mathematics investigating holomorphic functions, i.
		The Schwarzian derivative of a linear fractional transformation Möbius transformations should not be confused with the Möbius transform.
		In mathematics, the Virasoro group is a central extension of the orientation-preserving diffeomorphism group of the circle.
	www.nationmaster.com /encyclopedia/Schwarzian-derivative (831 words)

	Schwarzian derivative -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		In (A science (or group of related sciences) dealing with the logic of quantity and shape and arrangement) mathematics, the Schwarzian derivative of a function of one (additional info and facts about complex variable) complex variable is defined by
		If we follow a function by a fractional linear transformation then the composition has the same Schwarzian derivative as.
		On the other hand the Schwarzian derivative of, where is again fractional linear, is given by the
	www.absoluteastronomy.com /encyclopedia/s/sc/schwarzian_derivative.htm (150 words)

	Derivative Quotient Rule (Site not responding. Last check: 2007-10-08)
		In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist.
		derivative derivative Qualitatively the derivative is a of the change of a function in a small around a specified point.
		Quotient Rule In Words: The derivative of a quotient is the derivative of the top times the bottom, minus the top times the derivative of the bottom, all over the...
	riskmgmt.biz /mysite/economics%20TE2.2/derivative-quotient-rule.html (1348 words)

	Georgia Tech Geometry and Topology Seminar (Site not responding. Last check: 2007-10-08)
		Abstract: The classical Schwarzian derivative measures the failure of a diffeomorphism of the line to be a linear fractional transformation.
		A contact Schwarzian derivative measures the failure of a contactomorphism to be an element of the linear symplectic group acting by linear fractional transformations.
		The classical Schwarzian derivative will be reviewed, and there will be described the construction of the contact Schwarzian.
	www.math.gatech.edu /%7Efox/seminar/seminar.html (954 words)

	04-03 (Site not responding. Last check: 2007-10-08)
		The main aim in the present article is to give sufficient conditions for a locally univalent meromorphic function in the unit disk to have specific geometric properties such as starlikeness and convexity in terms of the Schwarzian derivative.
		To this end, we establish estimates of fundamental solutions to an ODE of the form 2y''+φ y=0 in the unit disk, where φ is an analytic function satisfying a given growth condition.
		As by-products, growth and distortion estimates are derived for a locally univalent strongly normalized analytic function f in the unit disk with a prescribed growth of the Schwarzian derivative.
	www.cajpn.org /complex/pp04/0403.html (107 words)

	ipedia.com: Connection (mathematics) Article (Site not responding. Last check: 2007-10-08)
		In differential geometry, connection or covariant derivative is a way of specifying a derivative of a vector field along an other vector field on a manifold.
		In differential geometry, connection (spelt also as connexion) or covariant derivative is a way of specifying a derivative of a vector field along an other vector field on a manifold.
		This family is called parallel displacement along the curve and it gives an equivalent description of connection (which in case of Levi-Civita connection on a Riemannian manifold is called parallel transport).
	www.ipedia.com /connection__mathematics_.html (524 words)

	[No title]
		We realized that the obstruction to the existence of projective structure invariant under the perturbed action is its Schwarzian cocycle, and smooth rigidity is equivalent to this cocycle being a coboundary.
		\section{The Schwarzian cocycle.} Let $x(t)$ and $y(t)$ be two $C^3$ curves in $S^1$ which differ by a transformation $PSL(2,\Bbb{R})$: \begin{equation} \label{eq-1} y(t)=\frac{ax(t)+b}{cx(t)+d} \end{equation} where $ad-bc=1.$ To find a differential invariant, let us eliminate constants $a$ and $b$ from the relations resulting from the differentiation of (~\ref{eq-1}).
		\end{equation} The Schwarzian derivative was already known to Lagrange (\cite{lagrange}) and Klein (\cite{klein}) and plays a key role in the theory of functions of one complex variable.
	www.ma.utexas.edu /mp_arc/e/96-171.latex (1388 words)

	Derivative--from Eric Weisstein's World of Mathematics (Site not responding. Last check: 2007-10-08)
		The derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have.
		A 3-D generalization of the derivative to an arbitrary direction is known as the directional derivative.
		In general, derivatives are mathematical objects which exist between smooth functions on manifolds.
	ciencias.unizar.es /~mdg/2003/10eduunivesq/laboratorio99/tercera%20parte/eric/Derivative.html (304 words)

	Application Of A Derivative (Site not responding. Last check: 2007-10-08)
		Formulae are derived for the configuration interaction (CI) energy gradient where the CI is based upon a two-configuration self-consistent-field (TCSCF) reference wave function.^A derivation of the analytic CI energy second derivative is...
		Possio's subsonic derivative theory and its application to flexural-torsional wing flutter part 1 Possio's derivative theory for an infinite aerofoil moving at subsonic speeds Possio's subsonic derivative theory and its application to flexural-...
		Directional derivative of the increasing rearrangement mapping and application to a queer differential equation in plasma physics Directional derivative of the increasing rearrangement mapping and application to a queer differential equation...
	riskmgmt.biz /mysite/economics%20TE2.2/application-of-a-derivative.html (1390 words)

	Registration & Records - Course Catalog
		Algebra review, functions, graphs, limits, derivatives, integrals, logarithmic and exponential functions, functions of several variables, applications in management, applications in biological and social sciences.
		Functions, graphs, limits, derivatives, rules of differentiation, definite integrals, fundamental theorem of calculus, applications of derivatives and integrals.
		Operations with complex numbers, derivatives, analytic functions, integrals, definitions and properties of elementary functions, multivalued functions, power series, residue theory and applications, conformal mapping.
	www2.acs.ncsu.edu /reg_records/crs_cat/MA.html (5337 words)

	Schwarzian derivative - Enpsychlopedia (Site not responding. Last check: 2007-10-08)
		The Schwarzian derivative of a function of one complex variable $f$ is defined by
		If we follow a function $f$ by a fractional linear transformation $g$ then the composition $g\circ f$ has the same Schwarzian derivative as $f$ .
		On the other hand the Schwarzian derivative of $f\circ g$ , where $g$ is again fractional linear, is given by the remarkable chain-like rule
	www.grohol.com /psypsych/Schwarzian_derivative (524 words)

	Math Department Course Offerings (Site not responding. Last check: 2007-10-08)
		This course covers functions, the derivative with applications, techniques of differentiation, the exponential and logarithmic functions with applications, and an introduction to the definite integral.
		Topics covered are the laws of probability, probability distributions, moments, the central limit theorem, confidence intervals, tests of hypotheses, correlation and regression, statistical quality control, and reliability and life testing.
		Topics covered include dynamical systems, hyperbolicity, symbolic dynamics, topological conjugacy, chaos, structural stability, Sarkovski’s Theorem, the Schwarzian derivative, bifurcation theory, maps of the circle, Mose-Smale diffeomorphisms, homoclinic points and bifurcations, period-doubling, kneading theory, genealogy of periodic units.
	www.msoe.edu /math/courses.html (1215 words)

	Schwarzian derivative (Site not responding. Last check: 2007-10-08)
		In mathematics, the Schwarzian derivative of a function of one complex variable
		The Schwarzian derivative of a linear fractional transformation :
		The Schwarzian derivative can also be defined as the following limit :
	www.keywordmage.net /sc/schwarzian-derivative.html (99 words)

	Colloquium and Seminar
		The consequences of his theory have been derived in a wide range of flows ranging from simple flows on the plane to multi-layer quasi-geostrophic flows on the rotating sphere.
		In this talk we prove in mathematical terms that Kraichnan's theory is equivalent to the well-known Gaussian model of theoretical physicists, and thus suffers, as all Gaussian models do, from a low temperature catastrophe where the partition function is not defined.
		This completes the classification of the extremal domains for the Schwarzian in all three classical geometries hence answering the question first posed in the 50's as to how far one can distort a disk under a convex map in Euclidean, spherical and hyperbolic geometries.
	www.math.wm.edu /%7Eilya/colloquia-2004-2005.html (860 words)

	On a theorem of Nehari and quasidiscs (ResearchIndex) (Site not responding. Last check: 2007-10-08)
		Let f be a locally injective analytic map of the unit disc D and let ff; zg be its Schwarzian derivative.
		We use the classical connection between Schwarzian derivative and second order linear equations to show that, for a particular class of functions p, the image f(D) is a quasidisc.
		The analysis centers on the differential equation y 00 + py = 0 and a finiteness condition of a positive solution y.
	citeseer.ist.psu.edu /chuaqui93theorem.html (236 words)

	Derivatives Portal
		Derivatives Portal is an online professional source of relevant documentation in the field of derivatives and risk.
		This portal was initiated by IMC Derivatives Foundation, a Dutch non-profit organisation and has the mission to promote the use and knowledge of derivatives among institutional investors and academic institutions.
		We provide short descriptions and links to the most frequently read articles, books, journals, papers, newspapers/magazines, websites and upcoming events around the field of derivatives trading and risk management.
	www.derivativesportal.org (268 words)

	The Schwarzian Derivative and Measured Laminations on Riemann Surfaces (Site not responding. Last check: 2007-10-08)
		The Schwarzian Derivative and Measured Laminations on Riemann Surfaces
		A holomorphic quadratic differential on a hyperbolic Riemann surface has an associated measured foliation, which can then be straightened to yield a measured geodesic lamination.
		Our main tools include a decomposition of the Schwarzian derivative of a projective structure using the Osgood-Stowe Schwarzian tensor, and analytic estimates for the Thurston metric of a
	hypatia.rice.edu /~ddumas/work/schwarzian/abstract_html (157 words)

	CMFT 5 (2005), 1--17 (Site not responding. Last check: 2007-10-08)
		It has been previously shown that the number of sign changes of arg(w(t))' on the omitted arc is finite.
		Here, we derive upper bounds for that number in terms of both the spherical and Schwarzian derivatives.
		Support point, omitted arc, wiggle point, waver point, spherical derivative, Schwarzian derivative, level set.
	www.heldermann.de /CMF/CMF05/CMF051/cmf05001.htm (133 words)

	[No title] (Site not responding. Last check: 2007-10-08)
		\documentclass[amssymb,mathfont]{amsart} \pagestyle{empty} \title{Math 295 problems 12} \author{Igor Kriz} \begin{document} \maketitle \vspace{10mm} {\bf 1.} \noindent Using the axioms for $\ln x$ and theorems proven in class, compute: $$\lim_{x\rightarrow 0}\frac{a^{x}-1}{x}\;\;(a>0).$$ \vspace{10mm} {\bf 2.} \noindent Compute the derivative of $$\root^{3}\of{2e^{x}-2^{x}+1}+(\ln x)^{5}.$$ \vspace{10mm} {\bf 3.} \noindent Prove that $a$ is a double root of a polynomial $f(x)$ (i.e.
		$f(x)=(x-a)^{2}g(x)$ for some polynomial $g(x)$) if and only if $a$ is the root of both $f$ and its derivative $f^{\prime}$.
		\vspace{10mm} {\bf 4.} \noindent If $f$ is three times differentiable and $f'(x)\neq 0$, the {\em Schwarzian derivative} of $f$ at $x$ is defined to be $$\mathcal{D}f(x)=\frac{f^{\prime\prime\prime}(x)}{f^{\prime}(x)}- \frac{3}{2}\left(\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}\right)^{2}.$$ (a) Show that $$\mathcal{D}(f\circ g)=[\mathcal{D}f\circ g].(g^{\prime})^{2}+\mathcal{D}g.$$ (b) Show that if $f(x)=\frac{ax+b}{cx+d}$, with $ad-bc\neq 0$, then $\mathcal{D}f=0$.
	www.math.lsa.umich.edu /~ikriz/29512.tex (147 words)

	Hirotaka Tamanoi
		He is also interested in generalized Schwarzian derivatives in several variables and Moebius invariant differential operators.
		Tamanoi: Higher Schwarzian operators and combinatorics of the Schwarzian derivative.
		(3) H. Tamanoi: Higher Schwarzian operators and combinatorics of Schwarzian derivative.
	www.math.ucsc.edu /Faculty/Tamanoi.html (552 words)

	Amazon.co.uk: Projective Differential Geometry Old and New: From the Schwarzian Derivative to the Cohomology of ... (Site not responding. Last check: 2007-10-08)
		The book opens by discussing the Schwarzian derivative and its connection to the Virasoro algebra.
		Related topics include differential operators, the cohomology of the group of diffeomorphisms of the circle, and the classical four-vertex theorem.
		A final chapter considers various versions of multi-dimensional Schwarzian derivative.
	amazon.co.uk /exec/obidos/ASIN/0521831865 (435 words)

	Papers
		The techniques depend on the nonlinear $L^2$ theory for the Schwarzian derivative developed earlier by the authors in the paper '$L^2$ estimates, harmonic measure and the Schwarzian derivative'.
		Our next result is an $L^p$ estimate relating the derivative of a conformal mapping to its Schwarzian derivative.
		The proofs in this paper are simplified somewhat by our later paper $L^2$ estimates, harmonic measure and the Schwarzian derivative.
	www.math.sunysb.edu /~bishop/papers/papers.html (5965 words)

	Abstracts
		In the simplest case, there is a point c at which the map has no derivative (it has two one-sided derivatives).
		In the simplest case there is a surface G in phase space along which the map has no derivative (or has two one-sided derivatives).
		An expression giving a lower bound for the average lifetime of a chaotic transient is derived and shown to agree well with numerical experiments.
	www-chaos.umd.edu /publications/abstracts.html (4733 words)

	Symbolic dynamics (Site not responding. Last check: 2007-10-08)
		More generally one can derive period-km orbit from K and M orbits with periods k and m.
		Copy m times the symbolic dynamics of K and replace each of the (m-1) C's (exept the first C) by one after another of the (m-1) symbols of M, interchanging L and R if the number of L in K is odd.
		By combining the kneading theory with an additional property of the quadratic map (namely that it has a negative Schwarzian derivative), we obtain a detailed description of how periodic orbits arise as c (and hence the kneading sequence) increases.
	www.people.nnov.ru /fractal/MSet/Symbolic.htm (745 words)

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