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Topic: Frobenius method

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Power series method

Ordinary differential equation

Frobenius theorem

Proof that holomorphic functions are analytic

Georg Frobenius

Analytic function

Frobenius norm

Group theory

Ferdinand Georg Frobenius

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	Ferdinand Georg Frobenius Summary
		Frobenius only remained in the position for a year when he decided to relocate to Zürich, Switzerland to teach mathematics at the Eidgenössische Polytechnikum (Federal Polytechnic) there.
		Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory.
		Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin.
	www.bookrags.com /Ferdinand_Georg_Frobenius (597 words)

	Ferdinand Georg Frobenius - Wikipedia, the free encyclopedia
		Ferdinand Georg Frobenius (October 26, 1849 - August 3, 1917) was a German mathematician, best-known for his contributions to the theory of differential equations and to group theory.
		Frobenius was born in Charlottenburg, a suburb of Berlin, and was educated at the University of Berlin.
		Group theory was one of Frobenius' principal interests in the second half of his career.
	en.wikipedia.org /wiki/Ferdinand_Georg_Frobenius (229 words)

	[No title]
		This family of iterative methods is described in section 3.3 and statements on the convergence of such iterative methods (with and without preconditioning) are given.
		Krylov subspace methods are particularly interesting for the implementation on vector and parallel computers, since their computational complexity is dominated by matrix-vector multiplications, which are efficiently vectorizable and parallelizable.
		For practical purposes, the projection method on the initial projection pattern should be practical in the sense of definition 4.3, because this guarantees, that the final projection method is practical as well.
	www.ubka.uni-karlsruhe.de /vvv/1999/mathematik/4/4.text (10424 words)

	Dom::GaloisField -- finite fields (Site not responding. Last check: 2007-09-21)
		Method matrixRepresentation: isomorphism to the algebra generated by the companion matrix
		This method returns a randomly chosen primitive element of the field.
		This method chooses a random normal element simply by choosing random elements until a normal one is found.
	www.sciface.com /STATIC/DOC25/de/Dom/GaloisF.shtml (455 words)

	Leo Frobenius Summary
		His method involved the notion that individual elements of culture should be investigated according to their placement within the organic whole of which they are parts.
		Frobenius went on twelve research expeditions to various parts of Africa to document the lives of tribal peoples.
		In 1925 the institute was removed to Frankfurt, where Frobenius received an honorary lectureship in the department of ethnology and cultural studies at the university.
	www.bookrags.com /Leo_Frobenius (1193 words)

	Civil & Environmental Engineering - course descriptions (Site not responding. Last check: 2007-09-21)
		Investigation of the finite element method as a numerical technique for solving linear ordinary and partial differential equations, using rod and beam theory, heat conduction, elastostatics and dynamics, and advective/diffusive transport as sample systems.
		Presents an overview of advanced nomenical methods for the treatment of engineering problems such as brittle and ductile failure and solid-liquid phase transformations in pure substances.
		Analytical methods for arbitrary discontinuities and interfaces are reviewed, with particular attention to the derivation of jump conditions.
	www.cee.duke.edu /grads/courses.php (2396 words)

	Frobenius Series Solution of a D. E.
		This method is attributed to the german mathemematican
		Use Frobenius series to solve the D. Solution 2.
		Use Frobenius series to solve the D. A solution is known to be the celebrated
	math.fullerton.edu /mathews/n2003/FrobeniusSeriesMod.html (373 words)

	[No title] (Site not responding. Last check: 2007-09-21)
		This method returns an n times n-matrix over the ground field, where n is the degree of the field over its ground field.
		This method implements the unique isomorphism between the field F[X]/ and the algebra of all linear combinations of powers of the companion matrix that maps X to the companion matrix.
		This method tests whether the field elements in the list l constitute a basis of the field, viewed as vector space over the ground field.
	www.sciface.com /STATIC/DOC30/eng/Dom_GaloisF.html (760 words)

	Department of Mathematics, University of Strathclyde
		Secod-order linear equations with constant coefficients: auxiliary equation, complementary function, particular integrals; variation of parameter method; generalisation to higher-order.
		Series solutions: Maclaurin/Taylor series solutions at an ordinary point; Frobenius method at a regular singular point; Legendre's equation and Legendre polynomials; Bessel's equation and Bessel functions.
		Method of Assessment: 2 hour Degree Examination in May/June with August resit.
	www.maths.strath.ac.uk /ungrad/classes/731.htm (291 words)

	The spectrum of the Frobenius-Perron operator and its discretization for circle diffeomorphisms (ResearchIndex) (Site not responding. Last check: 2007-09-21)
		Abstract: To investigate the dynamical behaviour of a discrete dynamical system given by a map f, it is nowadays a standard method to look at the discretization of the Frobenius--Perron operator P f w.
		20 Geometrical Methods in the Theory of Ordinary Differential E..
		9 An adaptive method for the approximation of the generalized..
	citeseer.ist.psu.edu /hotzel01spectrum.html (454 words)

	syllabus (Site not responding. Last check: 2007-09-21)
		If f=cos(t), the equation can be solved using the method of undetermined coeff.s; periodic solutions exist only if there is no "resonance".
		You might get stuck somewhere (in which case the method is useless), or it might work out.
		The method itself is suggested by a change of variables, and it takes a few pages to figure out what the answer is, but the final result is a nice clean, easy, formula for the solution: see formula 13 page 233.
	www.math.gatech.edu /~pacini/4581/4581_syllabus.html (2531 words)

	M (Site not responding. Last check: 2007-09-21)
		The scattering experiment, relationship of the scattering cross-section to the wave function, the scattering amplitude, method of partial waves, expansion of a plane wave in terms of partial waves, zero energy scattering, the scattering length, scattering by a square well potential, effective range, resonance scattering (No derivation).
		Four probe method for the measurement of resistivity of a sample.
		Quinke’s method for the susceptibility of a liquid.
	www.mlsu.org /syll/mscp.html (2098 words)

	PHYS 533 - Mathematical Methods of Physics II (Site not responding. Last check: 2007-09-21)
		It studies some of the mathematical methods used in those courses more thoroughly and prepares for graduate courses in classical and quantum physics.
		Frobenius method of solving SOLDEs in general and carried out in some explicit examples, e.g., Bessel equation, Legendre equation, hypergeometric equation.
		Second linearly independent solution to SOLDEs that cannot be found by Frobenius method.
	www.phas.ucalgary.ca /phasweb/courses/undergrad/533.php (348 words)

	frobenius.html (Site not responding. Last check: 2007-09-21)
		Section 11.5 (the method of Frobenius) will NOT be on the final exam, but here is an explanation of this technique.
		This method is nearly the same as the one shown in 11.1 and 11.2, the only difference is that instead of guessing y =
		We know to use this alternative method, whenever x = 0 is a regular singular point of y'' + P(x)y' + Q(x)y = 0.
	www.central.edu /homepages/lintont/maple/frobenius/frobenius.html (1519 words)

	128_hw3 (Site not responding. Last check: 2007-09-21)
		You will have at least one question where you need to find the Frobenius solution and the second solution using the Frobenius Method.
		You may have a few questions on this topic with very limited scope, that is you will not have to solve the entire problem but only deal with intermediate issues.
		You will have some questions where you use the numerical methods we have and will learn to find the solution to a given ode at a point using a few step sizes.
	zimmer.csufresno.edu /~childebr/181_test_3_study_fall_2003.htm (199 words)

	Numerical Analysis
		Approximation of a class of differential equations using a generalization of the Frobenius method.
		An extrapolation method for a Volterra integral equation with weakly singular kernel.
		Numerical methods and asymptotic error expansions for the Emden-Fowler equations.
	www.math.ist.utl.pt /~numerica/research/volt.html (320 words)

	Course Descriptions
		The method of separation of variables is applied to the three standard second order PDE’s, Laplace’s equation, the diffusion equation and the wave equation.
		To investigate the solution of partial differential equations which occur in mathematical physics by the method of separation of variables in a number of different geometries and to become familiar with the special functions that arise from this method.
		The special cases of the Frobenius method (equal roots, and roots differing by an integer).
	www.ma.man.ac.uk /DeptWeb/UGCourses/Syllabus/MT3612.html (382 words)

	A New Method for Approximate Solution of One-Dimensional Schrödinger Equations -- from Mathematica Information Center
		A general method for approximate solution of one-dimensional Schrödinger equations with a wide range of square-integrable potentials is described.
		This allows the Schrödinger equation to be solved by the Frobenius method.
		In the absence of super-computing power the input requirement of a large number of significant figures was handled by an algebraic computing package, for illustrative purposes.
	library.wolfram.com /infocenter/Articles/876 (122 words)

	School of Physics Spring 2003--Physics 6125 Mathematical Methods of Physics II
		School of Physics Spring 2003--Physics 6125 Mathematical Methods of Physics II Admissions
		Phase integral method (Stokes again) (so that you can read QM by Landau and Lifshitz).
		Solitons and the inverse scattering transform method, Path integrals (II),.
	www.physics.gatech.edu /academics/Classes/spring2003/6125/syllabus.html (153 words)

	IBM 7090/94 Console Operation
		The console displayed the current status of the most important machine registers (via small incandescent lights -- one for each bit) and also allowed the operator/programmer to modify, on the fly, various registers and machine control variables by flipping switches etc. Complete programs, written in binary, could be directly entered through the control console.
		This obviously laborious and slow method could be used to build boot strap loaders, simple maintenance and test programs, and the like (considering the typical cost of the machine -- approximately $100/hour (perhaps $750 in 2001 dollars), this method was used as little as possible).
		The control console for the 7094 differed in minor respects from that used for the 7090 -- primarily in the addition of extra index register displays for the 7094 and an additional switch that would command the 7094 to operate as a 7090 with respect to multiple indexing.
	www.frobenius.com /console-details.htm (401 words)

	Cryptology ePrint Archive
		At PKC 2004, Avanzi, Ciet, and Sica combined Frobenius operations with one point halving to compute scalar multiplications on Koblitz curves using on average 14\% less group additions than with the usual $\tau$-and-add method without increasing memory usage.
		The second result of this paper is an improvement over their expansion, that is simpler to compute, and optimal in a suitable sense, i.e.\ it has minimal Hamming weight among all $\tau$-adic expansions with digits $\{0,\pm1\}$ that allow one halving to be inserted in the corresponding scalar multiplication algorithm.
		The resulting scalar multiplication requires on average 25\% less group operations than the Frobenius method, and is thus 12.5\% faster than the previous known combination.
	eprint.iacr.org /2005/225 (269 words)

	MAS2008 (Site not responding. Last check: 2007-09-21)
		The main aim of this module is the study of differential equations and their solution, particularly partial differential equations.
		Subject knowledge and skills: By the end of the module the student will have a basic working knowledge of mathematical methods required to solve a wide range of problems in both science and engineering.
		Assessment methods: Coursework (25%) and an examination paper in May/June (75%).
	www.maths.ex.ac.uk /official/modules01/MAS2008.html (449 words)

	Çankaya University
		Propositions; methods of proving, sets; functions and relations; equivalence relations; partially ordered sets; totally ordered sets; order preserving functions; order isomorphism; lattices; well-ordered sets; axiom of choice and its equivalences; algebraic structures.
		Roundoff errors, algorithms and convergence, bisection method, fixed point iteration, Newton’s method; error analysis, accelerating convergence; interpolation and Lagrange polynomial, divided differences; cubic splines; numerical differentiation, Richardson’s extrapolation; numerical integration, trapezoid, Simpson’s and Boole’s rules; Romberg integration, adaptive quadrature; Gaussian quadrature; multiple integrals.
		Ordinary differential equations and initial value problems; Euler’s method; higher order Taylor methods; Runge-Kutta methods; error control, systems of ordinary differential equations and higher order equations; linear systems of equations; operations of linear algebra; Gaussian elimination; pivoting strategies, LU factorization; eigenvalues; iterative methods of Gauss-Seidel and Jacobi.
	www.cankaya.edu.tr /eng/fakulte/dersicerik.php?no=22 (2147 words)

	Summa: Symbolic Sums Manipulation -- from Mathematica Information Center
		This is useful to perform basic operation such as simplification, reindexing, and term by term differentiation on sums and series.
		The package was originally designed as an aid in solving ordinary differential equations via the power method.
		Power.nb : Contains a few simple examples of the application of symbolic sum manipulation to the resolution of ODEs with the power series method and the Frobenius method.
	library.wolfram.com /infocenter/MathSource/3336 (205 words)

	[No title] (Site not responding. Last check: 2007-09-21)
		This is also a review of series solutions for ordinary differential equations, and the method of Frobenius.
		Page 171: If x_0 is a regular singular point then there is a Frobenius series solution.
		Method of Frobenius: find two values of the exponent r.
	www.math.vt.edu /people/renardyy/class_home/feb28-mar3.html (276 words)

	[No title]
		First we have to develop a method, Frobenius method of power series solutions to solve this equation.
		Pages 211-217 describe the Frobenius method for power series solution of differential equations that can applied in limited circumstances when coefficients in the differential equation are not analytic.
		This is an application of the Frobenius method to the solution of a differential equation.
	www.csun.edu /~lcaretto/me501a/week08.doc (458 words)

	Session 2 \\{\bf Series Solutions of ODE's}
		One starts by assuming that the solution can be expressed by a simple power series then substitutes the series into the differential equation and thus determines the required values of the coefficients in the power series.
		In its implementation due to Frobenius the method allows the determination of solutions to a large number of differential equations of great importance in applications.
		If x is complex the solution has a pole at the origin when the exponent is a negative integer and a branch point when the exponent is nonintegral.
	www.rh.edu /~ernesto/F2003/MES/Notes/Notes_html/s02.html (1362 words)

	Frobenius method with imaginary powers Text - Physics Forums Library
		What's more, the two complex solutions I got turned out to be conjugates of eachother, so adding them gets one real solution, and subtracting them gets a pure imaginary which can then be multiplied by i to get another, linearly independent real solution.
		02-02-2005, 08:36 PM Using StatusX's method, I worked the equation from scratch and obtained a different answer than he (generator function is different).
		Using Mathematica, I calculated the first 50 terms of the series, calculated the first and second derivatives and back-substituted into the ODE in the range (0.1,10).
	www.physicsforums.com /archive/index.php/t-61952.html (2290 words)

	Applied Mathematics 3161
		Here we will learn the powerful method of using power series to solve ODEs, solve systems of linear differential equations, and investigate some numerical methods for computing approximate solutions of ODEs.
		Differential equations are used to describe anything that moves in a continuous manner, so there will be many applications seen in this course, from the oscillations of an aging spring to the spread of pollution through a systems of waterways.
		Solving ODEs by Power Series: Method of Frobenius.
	www.cs.mun.ca /~foster/am3161.html (235 words)

	Home Page of Gerald Teschl
		Then we establish the Frobenius method for linear equations in the complex domain and investigates Sturm-Liouville type boundary value problems including oscillation theory.
		Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.
		Beginning with the basics for iterated interval maps and ending with the Smale-Birkhoff theorem and the Melnikov method for homoclinic orbits.
	www.mat.univie.ac.at /~gerald/ftp/book-ode/index.html (193 words)

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