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Topic: Formal power series

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	PlanetMath: formal power series
		Formal power series allow one to employ much of the analytical machinery of power series in settings which don't have natural notions of convergence.
		This latter series is guaranteed to converge in
		This is version 8 of formal power series, born on 2002-07-01, modified 2005-02-13.
	planetmath.org /encyclopedia/FormalPowerSeries.html (760 words)

	PlanetMath: formal power series
		Formal power series allow one to employ much of the analytical machinery of power series in settings which don't have natural notions of convergence.
		Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain.
		This latter series is guaranteed to converge in
	www.planetmath.org /encyclopedia/FormalPowerSeries.html (765 words)

	Kids.Net.Au - Encyclopedia > Formal power series
		Formal power series are devices in mathematics that make it possible to employ much of the analytical machinery of power series in settings that do not have natural notions of "convergence".
		In mathematical analysis, every convergent power series defines a function with values in the real or complex numbers.
		Formal power series can also be interpreted as functions, but one has to be careful with the domain and codomain.
	www.kids.net.au /encyclopedia-wiki/fo/Formal_power_series (1454 words)

	Series (mathematics) Summary
		The idea of an infinite series expansion of a function was first conceived in India by Madhava in the 14th century, who also developed the concepts of the power series, the Taylor series, the Maclaurin series, rational approximations of infinite series, and infinite continued fractions.
		Series for the expansion of sines and cosines, of multiple arcs in powers of the sine and cosine of the arc had been treated by Jakob Bernoulli (1702) and his brother Johann Bernoulli (1701) and still earlier by Viète.
		Formal power series are also used in combinatorics to describe and study sequences that are otherwise difficult to handle; this is the method of generating functions.
	www.bookrags.com /Series_(mathematics) (3434 words)

	In focus - Power chords - - Music Seek
		Specifically, the term refers to the powerful and distinct sonic effect caused by the combination of two pitch classes separated by the interval of a perfect fifth (or its inversion, a perfect fourth) when subjected to a degree of distortion, usually through adequate amplification or other electronic processing, (e.g., a fuzz box).
		The respective overtone series of the remaining perfect intervals match to a very high degree and combine to produce the distinct and stable 'raw power' sound that is the 'trademark' of power chords.
		The criticism sometimes levelled at the use of consecutive power chords, i.e., that they violate an important rule of harmony, is largely based on a misunderstanding of the rule.
	www.musicseek.net /infocus-powerchords.htm (915 words)

	ChapterZero » Blog Archive » An Inversion Theorem for Formal Power Series
		Another name for a formal power series is a generating function, because it is associated with, or generates the sequence
		This entry was posted on Sunday, January 2nd, 2005 at 6:16 pm and is filed under Mathematics.
		2 Responses to “An Inversion Theorem for Formal Power Series”
	www.tangentspace.net /cz/archives/2005/01/an-inversion-theorem-for-formal-power-series (222 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		A formal power series is an infinite sequence of numbers (a_0, a_1, a_2,...), which we choose to write as a_0 + a_1 x + a_2 x^2 +...
		to be the formal power series (a_0 + b_0) + (a_1 + b_1) x + (a_2 + b_2) x^2 +...
		to be the formal power series (a_0 b_0) + (a_0 b_1 + a_1 b_0) x + (a_0 b_2 + a_1 b_1 + a_2 b_0) x^2 +...
	www.math.wisc.edu /~propp/475/Apr04.doc (1139 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-11-05)
		The concept of a formal group law, and thus of a formal Lie group, can be generalized to the case of arbitrary commutative ground rings (see [2], [5]).
		The theory of formal groups over fields can be generalized to the case of arbitrary formal ground schemes [7].
		This is the covariant classification of commutative formal groups in contrast with the earlier contravariant classification of commutative formal groups over perfect fields by Dieudonné modules.
	eom.springer.de /F/f040820.htm (889 words)

	[No title]
		Operational definition A “formal power series” is an infinite sequence of numbers (a_0, a_1, a_2,...), which we operate on by certain formal rules that are easier to remember (and whose motivation is clearer) if we represent the sequence by the symbol a_0 + a_1 x + a_2 x^2 +...
		For instance, the “formal derivative” of the sequence (a_0, a_1, a_2, a_3,...) is defined as (a_1, 2 a_2, 3 a_3,...).
		With this notion of convergence, the ring of formal power series becomes a “topological ring”, once we have proved the following theorem: Theorem: The operations of addition and multiplication respect the notion of convergence.
	www.math.harvard.edu /~propp/192/09-18.doc (1696 words)

	Math Forum Discussions
		I too was talking only about power series centered at the origin.
		that a power series that converges in a neighborhood of 0 defines an
		series representation; is this representation convergent near the origin?
	www.mathforum.org /kb/thread.jspa?threadID=65925&messageID=278602 (575 words)

	[No title]
		But such a series is invertible in R (trivially in the first case, by a standard theorem in the second case).
		This is not quite folklore, I think, at least in the convergent power series case, where it is a theorem in a famous paper of W. Rueckert about 1939, where an extensive study of local analytic algebras was made.
		Whether the formal power series case is folklore, I do not know; at least it appears in the Zariski and Samuel book on commutative algebra, and a nice proof is due to H. Sagres, J. Reine Angew.
	www.math.niu.edu /~rusin/known-math/00_incoming/weier_prep (1579 words)

	Citations: Approximation de s'eries formelles par des s'eries rationnelles - Hespel (ResearchIndex) (Site not responding. Last check: 2007-11-05)
		....series f : Sigma K, we define the rationality measure R f (n) to be the minimum possible rank of any recognizable (rational) series g such that (f; w) g; w) for all w with jwj n.
		6.1] proves that a formal power series is rational iff it is recognizable.
		In this case the dimension of the smallest possible matrix representation (the dimension of the square matrix fl) is an invariant called the rank of the rational series.
	citeseer.comp.nus.edu.sg /context/84752/0 (692 words)

	Math Forum Discussions - Re: Formal composition of formal power series
		Math Forum Discussions - Re: Formal composition of formal power series
		Bruce Reznik wrote on iterates of formal power series:
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	www.mathforum.com /kb/thread.jspa?forumID=253&threadID=567620&messageID=1693644 (175 words)

	Overview of the powseries Package
		The primary resulting limitation to remember is that you can neither create power series with multivariate monomials, nor create the isomorphic equivalent which is a formal power series with a coefficient ring that is itself based on a (different) ring of formal power series.
		Perform the natural term-by-term differentiation with respect to the variable that the power series is based on.
		For instance, as an ODE the Riccati equation has a formal power series solution which is of no use in an analytic context.
	web.mit.edu /maple/www/pkg-powseries.html (1211 words)

	Formal Power Series & Linear Systems of Meromorphic Ordinary at Quantum Books
		Simple Ordinary Differential Equations may have solutions in terms of power series whose coefficients grow at such a rate that the series has a radius of convergence equal to zero.
		In fact, every linear meromorphic system has a formal solution of a certain form, which can be relatively easily computed, but which generally involves such power series diverging everywhere.
		In this book the author presents the classical theory of meromorphic systems of ODE in the new light shed upon it by the recent achievements in the theory of summability of formal power series.
	www.quantumbooks.com /p/SPR06Y/0387986901 (114 words)

	MTH 207 Lab Lesson 19 - Taylor Series
		The general theory of formal power series is beyond the scope of this course (see Stewart Chapter 10), however a few comments are in order.
		The problem with Formal Power Series is that we can't guarentee convergence of the series for a given value of
		Note that in this case we have equality between the function and the power series.
	www.sfu.ca /~rpyke/macm316/maple/less19.htm (440 words)

	On the Transcendence of Formal Power Series (Site not responding. Last check: 2007-11-05)
		In this perspective, f(z)=exp(z) is transcendental ``because'' its growth is too fast at infinity, a fact incompatible with the fact that an algebraic function is locally described by a Puiseux series (i.e., a series involving fractional powers).
		The resulting series is algebraic, since a theorem of Furstenberg states that algebraic functions over finite fields are closed under Hadamard (termwise) products.
		-- Transcendence of binomial and Lucas's formal power series.
	algo.inria.fr /seminars/sem97-98/allouche.html (1118 words)

	Formal power series - Definition, explanation
		Then R((G)) is the ring of formal power series on G; because of the condition that the indexing set be well-ordered the product is well-defined, and we of course assume that two elements which differ by zero are the same.
		This theory is due to Hans Hahn, who also showed that one obtains subfields when the number of (non-zero) terms is bounded by some fixed infinite cardinality.
		Bell series are used to study the properties of multiplicative arithmetic functions
	www.calsky.com /lexikon/en/txt/f/fo/formal_power_series.php (1738 words)

	AddALL.com - Formal Power Series and Linear Systems of Meromorphic Ordinary Differentialequations
		AddALL.com - Formal Power Series and Linear Systems of Meromorphic Ordinary Differentialequations
		Formal Power Series and Linear Systems of Meromorphic Ordinary Differentialequations
		If you cannot find this book in our new and in print search, be sure to try our used and out of print search too!
	www.addall.com /detail/0387986901.html (70 words)

	19th International Conference on Formal Power Series and Algebraic Combinatorics (Site not responding. Last check: 2007-11-05)
		19th International Conference on Formal Power Series and Algebraic Combinatorics
		Name: 19th International Conference on Formal Power Series and Algebraic Combinatorics
		Submission of papers: Authors are invited to submit extended abstracts of at most twelve pages by November 19, 2006.
	www.ams.org /mathcal/info/2007_jul2-6_tianjin.html (113 words)

	Series Solutions and Frobenius Method
		Power series solutions around a singular point of the system of hypergeometric differential equations of type (3,6) by use of special values of 3F2.
		A necessary condition for a power series to be a formal solution of a singular linear differential equation of order k.
		Estimate of the radius of convergence of power series in a small parameter which represent periodic solutions of systems of differential equations.
	math.fullerton.edu /mathews/n2003/frobeniusdiffeqns/FrobeniusSeriesBib/Links/FrobeniusSeriesBib_lnk_3.html (2042 words)

	Amazon.com: "Power Series Approach": Key Phrase page (Site not responding. Last check: 2007-11-05)
		Another procedure can be based on the construction of a specially well suited power series approach around a chosen point providing a polynomial approximation to a piece of the solution curve within a given algebraic error.
		we present the formal power series approach and analytic theory of ordinary generating functions,...
		power series approach is only valid for circuits consisting of memoryless elements or circuits with memory elements but operated at sufficiently low frequencies...
	www.amazon.com /phrase/Power-Series-Approach (520 words)

	[No title]
		Some functors from commutative rings to groups 1.1 A formal diffeomorphism of the line, with coefficients in a commutative ring A, is an element g of the ring A[[x]] of formal power series with coefficients * *in A, such that g(0) = 0 and g0(0) is a unit.
		(g0O g1)(x) = g0(g1(x)) of formal power series makes this* * set into a monoid with e(x) = x as identity element, and it is an exercise in induc* tion to show that such an invertible power* series [ie with leading coefficient a uni* *t] possesses a composition inverse in G(A).
		This is natura* lly isomorphic to the set of ring homomorphisms from a polynomial algebra on gener- ators {wk, k > 0} to A; the group structure endows this representing ring with *the structure of a (commutative and cocommutative) Hopf algebra.
	hopf.math.purdue.edu /Morava/Virasoro.txt (1454 words)

	PrintThisPage
		Thus Q(x) is the formal reciprocal of P(x) as a power series.
		Observe that this is pure formal algebra: no questions of analytical convergence are involved at all.
		P(x) was taken to 550 terms and Q(x) produced as the Taylor series of P(x)^(-1).
	www.mathsoft.com /printThisPage.aspx?1148 (249 words)

	Citations: The calculus of formal power series for diffeomorphisms and vector fields: [Preprint - Feng (ResearchIndex)
		Feng K. The calculus of formal power series for diffeomorphisms and vector fields: [Preprint].
		The crucial fact is that any symplectic algorithm possesses its own formal integrals of motion depending on the step size parameter and approximating the original integrals of motion (with the same order of accuracy as that of the method itself) and being invariant under the algorithm
		It can also be proved that the majority of invariant tori of integrable systems are preserved under symplectic algorithms; resulting in a new version of the celebrated K. theorems and the theoretical infinite time tracking capability of the symplectic algorithms [6] although practically....
	citeseer.ist.psu.edu /context/1848273/0 (648 words)

	Volume 24 "Formal Power Series and Algebraic Combinatorics" Louis J. Billera, Curtis Greene, Rodica Simion, Richard ...
		Volume 24 "Formal Power Series and Algebraic Combinatorics" Louis J. Billera, Curtis Greene, Rodica Simion, Richard P.Stanley, Eds.
		This volume is devoted to the invited lectures presented at the Sixth International Conference on Formal Power Series and Algebraic Combinatorics/ Series Formelles et Combinatoire Algebrique, held at DIMACS during May 23-27, 1994, and organized by the editors.
		Originally planned as a small workshop to highlight the vitality of current research in algebraic combinatorics, it was later merged into the FPSAC/SFCA series of annual conferences, becoming the 1994 meeting of this series.
	dimacs.rutgers.edu /Volumes/Vol24.html (319 words)

	WWU Math Department - Colloquium
		Abstract: Formal power series have useful applications in many branches of mathematics.
		In combinatorics, they occur as generating functions, facilitate solving different counting problems, and are helpful in finding closed forms for recurrence relations.
		Prior to the publication of the paper Formal Power Series by Ivan Niven in 1969, a theory of formal power series was simply assumed by many mathematicians.
	www.wwu.edu /depts/math/colloquium/c_050406.html (125 words)

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