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Topic: Abelian integral

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Analytic function

	Path integral formulation Encyclopedia
		The path integral formulation of quantum mechanics is a description of quantum theory which generalizes the action principle of classical mechanics.
		However, the path integral formulation is also extremely important in direct application to quantum field theory, in which the "paths" or histories being considered are not the motions of a single particle, but the possible time evolutions of a field over all space.
		In one philosophical interpretation of quantum mechanics, the "sum over histories" interpretation, the path integral is taken to be fundamental and reality is viewed as a single indistinguishable "class" of paths which all share the same events.
	www.hallencyclopedia.com /topic/Path_integral_formulation.html (3321 words)

	Abelian - Wikipedia, the free encyclopedia
		An abelian extension is a field extension for which the associated Galois group is abelian.
		An abelian von Neumann algebra is a von Neumann algebra of operators on a Hilbert space in which all elements commute.
		An abelian integral is a function related to the indefinite integral of a differential of the first kind.
	en.wikipedia.org /wiki/Abelian (255 words)

	Abelian integral - Wikipedia, the free encyclopedia
		In mathematics, an abelian integral in Riemann surface theory is a function related to the indefinite integral of a differential of the first kind.
		Logically speaking, therefore, an abelian integral should be a function such as f.
		This is a natural step in the theory of integration to the case of integrals involving algebraic functions √A, where A is a polynomial of degree > 4.
	en.wikipedia.org /wiki/Abelian_integral (285 words)

	Introduction to "Taming Wild Extensions" of Lindsay N. Childs
		The observation of {Ch87} for abelian extensions and {CM94} in general, was that for wild Galois extensions, if the associated order \frak A is a Hopf order in KG, then S is free of rank one over \frak A. Noether's theorem is the case where \frak A = RG.
		This generalized Noether's theorem was perhaps the first general integral normal basis theorem for wildly ramified Galois extensions of local fields.
		Mazur {Mz70} noted that describing all the possible abelian group schemes over the valuation ring of a local field "is a delicate matter", citing {TO70} as evidence.
	math.albany.edu:8000 /~lc802/mono.html (1829 words)

	Algebraic groups (Site not responding. Last check: 2007-10-12)
		For the purposes of algebraic geometry over the complex numbers, an abelian variety is a complex torus (a torus of real dimension 2n that is a complex manifold) that is also a projective algebraic variety of dimension n, i.e.
		An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods, equivalently, it is a function in the function field of an abelian variety.
		For example there was much interest in the case of hyperelliptic integrals that may be expressed in terms of elliptic integrals: this comes down to asking that J is a product of elliptic curves, up to a finite-to-one mapping (called an isogeny of abelian varieties).
	read-and-go.hopto.org /Algebraic-groups (436 words)

	Publications
		``Duality in Noetherian integral domains'', Rocky Mountain Journal of Mathematics 29 (2), (1999), 519-529.
		``Numerical Invariants for a class of Butler Groups'', with W. Ullery and C. Vinsonhaler, Proceedings of Abelian Groups and Modules, Contemporary Mathematics, Vol 171, 159-172, (1995).
		``Hyper-types of Torsion-Free Abelian Groups of Finite Rank'', with C. Vinsonhaler and W. Wickless, Australian Math.
	www.auburn.edu /~goetehp/pubs.html (986 words)

	The cyclicity problem for non-generic quadratic Hamiltonian systems (Site not responding. Last check: 2007-10-12)
		Limit cycles bifurcating from an annulus filled by regular Hamiltonian cycles (or in a neighborhood of a non-degenerate center), are in general directly controlled by the zeros of the associated Abelian integrals.
		This is now well understood for the quadratic Hamiltonian systems : the Abelian integrals have no more than 2 zeros in their whole domain of definition, which may be the union of two intervals.
		In a recent work in collaboration with Chengzhi Li, we prove that the maximal number of limit cycles which bifurcate from the union of the 2-saddle-loops with respect to quadratic perturbations is two, in the reversible direction.
	www.icmc.usp.br /~getds/abst/roussarie.html (306 words)

	Creation
		Given an integral ideal I belonging to the maximal order of a number field, the ray class group modulo I is the quotient of the subgroup generated by the ideals coprime to I by the subgroup generated by the principal ideals generated by elements congruent to 1 modulo I and T if present.
		The ray class group is returned as an abelian group A, together with a mapping between A and a set of representatives for the ray classes.
		Let I be an integral ideal of an absolute maximal order and let T be a set of real places given by an increasing sequence containing integers i, 1 <= i <= r_1 where r_1 is the number of real zeros of the defining polynomial of the field.
	www.umich.edu /~gpcc/scs/magma/text661.htm (1839 words)

	Abelian Integrals
		Although her results were considered to be of little real importance at the time, Weierstrass still felt that the paper, demonstrated proof of a high level of mathematical competence.
		By doing some work with this corollary she reduced the question of degeneracy to a question of whether or not there are Abelian integrals of first kind associated with f(x, y) = 0 which reduce to elliptic integrals.
		This purely algebraic condition is far easier to work with than the transcendental ones given by Weierstrass and is the main achievement of this paper which was otherwise of little interest.
	www-groups.dcs.st-and.ac.uk /~history/Projects/Ellison/Chapters/Ch6.html (447 words)

	Earliest Known Uses of Some of the Words of Mathematics (A) (Site not responding. Last check: 2007-10-12)
		ABELIAN EQUATION is named for a kind of equation treated by Niels Henrik Abel (1802-1829) in his "Mémoire sur une classe particulière d'équations résoluble algébriquement" (1829) Oeuvres Complètes, 1, 478-514.
		Abelian function appears in the title "Zur Theorie der Abelschen Functionen" by Karl Weierstrass (1815-1897) in Crelle's Journal, 47 (1854) reprinted in Werke I, p.
		An early use of "Abelian" to refer to commutative groups is H. Weber, "Beweis des Satzes, dass jede eigentlich primitive quadratische Form unendlich viele Primzahlen darzustellen fähig ist," Mathematische Annalen, 20 (1882), 301--329.
	www.members.aol.com /jeff570/a.html (7127 words)

	Mary Roman (Site not responding. Last check: 2007-10-12)
		Weierstrass was really into Abelian integrals and worked to solve these equations and to simplify them.
		Abelian Integrals "…are functions defined using certain types of definite integrals; in some sense they are generalizations of the familiar trigonometric functions" (Koblitz 242).
		The partial differential equation she had was from the earlier works of her friend, Karl Weierstrass, to which he simplified the complex and made it clearer for others to understand for the future in mathematics concerning the motion or reflection of objects.
	www.sienahts.edu /~mroman/math.html (394 words)

	[No title]
		Jacobi based his theory on the integral u =int(0 to \varphi {d\varphi}\over \sqrt{1-k^2\sin^2\varphi}}, with a parameter k, between 0 and 1, called the modulus of the elliptic integral.
		Abelian integrals are defined like elliptic integrals by u = \int(0 to v) R(t,\sqrt{f(t)})dt = I(v) where R(x,y) is a rational function of x and y, except that the function f is of a very general type which includes all polynomials.
		In fact during her stay in Berlin she produced three outstanding papers; on differential equations, on Abelian integrals and on Saturn's rings, and managed to obtain a doctorate from the University of Gottingen.
	math.nist.gov /opsf/personal/weierstrass.html (3054 words)

	CompleteAbelian integrals as rational envelopes - Yakovenko (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		The integral of the form over a continuous family of closed curves on the level sets fH = tg can be extended to a complete Abelian integral, a multivalued analytic function I! (t) of a point t 2 C varying over the set of regular values of the polynomial.
		We prove that this function may be represented as a linear combination of a certain family of analytic multivalued...
		Abelian integral can be represented as a linear combination of a finite number of integrals of some 1 forms # k with coe#cients from C(t)
	citeseer.ist.psu.edu /136523.html (363 words)

	[No title] (Site not responding. Last check: 2007-10-12)
		Show the number of elements of order $\ell$ in a finite Abelian group $G$ is $\ell^r-1$ where $r$ is the number of cyclic groups of $\ell$-power order in any decomposition of $G$ into a product of groups of prime power order.
		Show every Abelian group is isomorphic to a subgroup of the group of units of some commtive ring.\smk 13.
		Let $G$ be an Abelian group and $\Z[G]$ be the ring made out of $\stt{\sum_{g\in G}a_gg\colon g\in G}$ with coordinatewise addition and multiplication ``$\cdot$'' such that $mg\cdot nh =mngh$.
	math.berkeley.edu /~coleman/Courses/Sp03/Fin-II (518 words)

	AMERICAN MATHEMATICAL MONTHLY -MAY 2001
		Such a formula was proved in 1879 by E. Lundberg for some "sines" and "cosines" arising from the inversion of an Abelian integral.
		A convergent integral containing a parameter was differentiated under the integral sign with respect to the parameter without justification.
		This yielded a divergent integral that is listed even today in standard integral tables as converging.
	www.maa.org /pubs/monthly_may01_toc.html (623 words)

	ANGEO - Abstracts (Site not responding. Last check: 2007-10-12)
		The retrieval is based on the Abelian integral inversion of the atmospheric bending angle profile into the refractivity index profile.
		The problem of the upper boundary condition of the Abelian integral is described by examples.
		The retrieved temperature profiles are compared with corresponding profiles which have already been calculated by scientists of UCAR and Jet Propulsion Laboratory (JPL), using Abelian integral inversion too.
	www.copernicus.org /EGU/annales/15/4/443.htm?FrameEngine=false (303 words)

	Tesla Secondary Simulation Project
		The model has been applied to the task of mapping out the performance of secondary resonators in small signal CW operation [vsd], from which semi-empirical formulae have been obtained for the resonant frequencies and effective inductances [formulae].
		The integral equations turn out to have the same mathematical form as those of the elementary lumped approximation, except that the components of the lumped circuit are replaced by the corresponding integral operators.
		These lead to an eigenequation for the normal modes, the solution of which provides the basis functions from which the time domain response of the resonator is computed.
	www.abelian.demon.co.uk /tssp (1153 words)

	Rushanan, Joseph John (1986-05-22) Topics in integral matrices and Abelian group codes. ...
		We consider the SNF of a matrix [...] to be the ratio of two [...]-modules-a finitely generated abelian group; this is called the Smith group of A. The Smith group provides a unified setting to present both new and old results.
		In particular, the multiplicities of integer eigenvalues are shown to relate to the multiplicities in the type of the Smith group.
		We take abelian group codes to be ideals in the group ring [...], where [...] is a finite abelian group of odd order [...] and [...] is a finite field with characteristic relatively prime to [...].
	etd.caltech.edu /etd/available/etd-05022003-113743 (356 words)

	Citebase - On the multiplicity of the hyperelliptic integrals (Site not responding. Last check: 2007-10-12)
		The reasoning goes as follows: we consider the analytic curve parameterized by the integrals along δ(t) of the n ``Petrov'' forms of H (polynomial 1-forms that freely generate the module of relative cohomology of H), and interpret the multiplicity of I(t) as the order of contact of γ(t) and a linear hyperplane of \textbf C
		Using the Picard-Fuchs system satisfied by γ(t), we establish an algebraic identity involving the wronskian determinant of the integrals of the original form ω along a basis of the homology of the generic fiber of H. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of I(t).
		Citation coverage and analysis is incomplete and hit coverage and analysis is both incomplete and noisy.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0312323 (603 words)

	Index (riemann package API documentation)
		Creates function which delivers the largest word length of all element in section of the the subset which is used for the approximation of integral of first kind.
		Creates function which delivers the smallest word length of all element in section of the the subset which is used for the approximation of integral of first kind.
		Creates function counting the number of elements in subset which is used for the approximation of the integral of first kind.
	www-sfb288.math.tu-berlin.de /~jtem/riemann/api/index-all.html (1823 words)

	Atlas: Surace area and capacity of n-dimensional ellipsoids by Garry Tee (Site not responding. Last check: 2007-10-12)
		The surface area of a general n-dimensional ellipsoid is represented as an abelian integral, which can readily be evaluated numerically.
		If there are only 2 values for the semi-axes then the area is expressed as an elliptic integral, which reduces in most cases to elementary functions.
		The capacity of a general n-dimensional ellipsoid is represented as a hyperelliptic integral, which can readily be evaluated numerically.
	atlas-conferences.com /cgi-bin/abstract/caog-51 (145 words)

	[ref] 17 Cyclotomic Numbers (Site not responding. Last check: 2007-10-12)
		Since the underlying basis of the external representation of cyclotomics is an integral basis (see Integral Bases for Abelian Number Fields), the subring of cyclotomic integers in a cyclotomic field is formed by those cyclotomics for which the external representation is a list of integers.
		For example, square roots of integers are cyclotomic integers (see ATLAS irrationalities), any root of unity is a cyclotomic integer, character values are always cyclotomic integers, but all rationals which are not integers are not cyclotomic integers.
		A special integral basis of cyclotomic fields is chosen that admits to obtain easily a conversion of arbitrary sums of roots of unity into the basis, as well as to reduce a cyclotomic represented w.r.t.
	www.math.psu.edu /local_doc/gap4/htm/ref/CHAP017.htm (1627 words)

	Réunion Toulouse 2004
		This means that every bifurcating limit cycle is related to a zero of Abelian integral.
		This diagram cannot be deduced from the bifurcation diagram for Abelian integral zeros.
		As an application we give a "natural" integral representation of the invariants of Martinet and Ramis associated to resonant foliations, yielding many explicit examples of nonconjugated foliations.
	www.smc.math.ca /Reunions/Toulouse2004/abs/ss7.html (1135 words)

	Citebase - Grope cobordism of classical knots (Site not responding. Last check: 2007-10-12)
		The Kontsevich integral of a knot is a graph-valued invariant which (when graded by the Vassiliev degree of graphs) is characterized by a universal property; namely it is a universal Vassiliev invariant of knots.
		We use the 2-loop term of the Kontsevich integral to show that there are (many) knots with trivial Alexander polynomial which don't have a Seifert surface whose genus equals the rank of the Seifert form.
		In 1999, Rozansky conjectured the existence of a rational presentation of the Kontsevich integral of a knot.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/0012118 (1322 words)

	Re: Geometric quantization
		I get the vague feeling that somewhere >along the way you did something like an "integral over a loop" of a function >taking values in an abelian group: > > integral_C f(x) dx, f(x) in G, G an abelian group > >where C is a curve in the base space of your fiber bundle.
		It's true that you can integrate functions taking values in an abelian group much more easily than you can functions taking values in a nonabelian, because when you integrate, you are basically adding a bunch of things up, and abelianness means you don't have to worry about what order to add them up in.
		Physicists have tried to generalize gauge theory to situations where you take a kind of holonomy of something along a higher-dimensional surface, but for the most part they have failed miserably EXCEPT when the gauge group is *abelian*, since a higher- dimensional surface doesn't provide you with an ordering.
	www.lns.cornell.edu /spr/2000-05/msg0025136.html (1052 words)

	Bulletin of the American Mathematical Society
		II [Singularities of differentiable maps], Monodromiya i asimptotiki integralov [Monodromy and the asymptotic behavior of integrals], ``Nauka'', Moscow, 1984.
		Ilyashenko, S. Yakovenko, Double exponential estimate for the number of zeros of complete abelian integrals and rational envelopes of linear ordinary differential equations with an irreducible monodromy group, Invent.
		A. Varchenko, Estimation of the number of zeros of an abelian integral depending on a parameter, and limit cycles (Russian), Funktsional.
	www.ams.org /bull/2002-39-03/S0273-0979-02-00946-1/home.html (2658 words)

	Redundant Picard-Fuchs System For Abelian Integrals (ResearchIndex) (Site not responding. Last check: 2007-10-12)
		We derive an explicit system of Picard--Fuchs differential equations satisfied by Abelian integrals of monomial forms and majorize its coefficients.
		2 Linear estimate for the number of zeros of Abelian integrals..
		1 Estimation of the number of zeros of an Abelian integral dep..
	citeseer.ist.psu.edu /novikov00redundant.html (405 words)

	Re: Categorified Gauge Theory \| The String Coffee Table
		The algebra I wrote down on spr is the most general Lie algebra with an abelian ideal; an ideal means that [J,e]- e and [e,e] - e, and abelian that [e,e] = 0 (I cant find tilde on the Spanish keyboard).
		I guess that we indeed agree what it means for H to take values in an abelian ideal of some algebra and for g to act on this algebra and hence on that ideal by automorphisms.
		Once I wrote down some H^2(G,K) for G a current algebra (it’s the same thing as finding abelian extensions), by restriction from Askar Dzhumadildaev’s classification for G an algebra of vector fields, but I don’t know about H^3.
	golem.ph.utexas.edu /string/archives/000487.html (1512 words)

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