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	Holomorphic function - Wikipedia, the free encyclopedia
		Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C that are complex-differentiable at every point.
		A function that is holomorphic on the whole complex plane is called an entire function.
		Close to points with non-zero derivative, holomorphic functions are conformal in the sense that they preserve angles and the shape (but not size) of small figures.
	en.wikipedia.org /wiki/Holomorphic_function (887 words)

	Talk:Holomorphic function - Wikipedia, the free encyclopedia
		The inverse trigonometric functions likewise have seams and are holomorphic everywhere except the seams.
		Anyway it is a theorem that a holomorphic function is a complex analytic function and vice versa and the article reflects this.
		That a holomorphic function is complex analytic is a very profound theorem using intimate properties of the complex number field, and it has dramatic consequences (like every polynomial with complex entries has complex roots).
	en.wikipedia.org /wiki/Talk:Holomorphic_function (2726 words)

	PlanetMath: normal family (Site not responding. Last check: 2007-11-07)
		This definition is often used in complex analysis for spaces of holomorphic functions.
		This is similar to the holomorphic case, but instead of using the standard metric for convergence we must use the spherical metric.
		In more modern language, one would give a metric on the space of continuous (holomorphic) functions that corresponds to convergence on compact subsets and then you'd say ``precompact set of functions'' in such a metric space instead of saying ``normal family of continuous (holomorphic) functions''.
	planetmath.org /encyclopedia/NormalFamily.html (270 words)

	[No title]
		The holomorph* ic K - groups will be defined to be the homotopy groups K-qhol(X) = ssq(Khol(X)): A variant of this construction was first incidentally introduced in [12], and s ubsequently developed in [16] where one obtains various connective spectra associated to an * algebraic variety X, using spaces of algebraic cycles.
		Finally we describe a necessary conditions for the holomorphic K - theory of * a smooth variety to be Bott periodic (i.e Khol(X) ~=Khol(X)[1_b]) in terms of the Hodge *filtration of its cohomology.
		The Chern character for holomorphic K - theory In this section we study the Chern character for holomorphic K - theory that * *was defined by the authors in [6].
	hopf.math.purdue.edu /CohenR-Lima-Filho/holo-k-th.txt (9457 words)

	Holomorphic function (Site not responding. Last check: 2007-11-07)
		The term analytic function is used interchangeably with "holomorphic function", although note that the former term has several other meanings.
		A function that is holomorphic on the whole complex plane is called entire.
		All polynomial functions with complex coefficients are holomorphic on C, and so are the trigonometric functions and the exponential function.
	www.portaljuice.com /holomorphic_function.html (521 words)

	Holomorphic K-theory, algebraic co-cycles, and loop groups, by Ralph L. Cohen and Paulo Lima-Filho (Site not responding. Last check: 2007-11-07)
		In this paper we study the ``holomorphic K-theory" of a projective variety.
		This theory is defined in terms of the homotopy type of spaces of holomorphic maps from the variety to Grassmannians and loop groups.
		Using the Chern character studied by the authors in a companion paper, we show that there is a rational isomorphism between holomorphic K-theory and the appropriate "morphic cohomology", defined by Lawson and Friedlander.
	www.math.uiuc.edu /K-theory/0380 (229 words)

	Citebase - Holomorphic curvature of Finsler metrics and complex geodesics
		Holomorphic curvature of Finsler metrics and complex geodesics
		In this paper we address the question whether, given a smooth complex Finsler metric on a complex manifold, it is possible to give purely differential geometric properties of the metric ensuring the existence of such a fibration in complex geodesics of the manifold.
		Finally, we show that a complex Finsler metric of constant holomorphic sectional curvature -4 satisfying the given simmetry condition on the curvature is necessarily the Kobayashi metric.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:math/9207201 (453 words)

	[No title] (Site not responding. Last check: 2007-11-07)
		One of the central problems of holomorphic mapping theory is to determine whether a biholomorphic mapping will extend smoothly (or real analytically) to the boundary.
		But invariant metrics also have applications to the study of the boundary behavior of holomorphic functions in Hardy classes (work of Krantz), to the study of domains with non-compact automorphism groups, and to general questions of geometric function theory.
		Holomorphic mappings are fundamental invariant objects in complex function theory.
	www.aimath.org /projects/mappings/holo_maps.html (662 words)

	Abelian fibred holomorphic symplectic manifolds (Site not responding. Last check: 2007-11-07)
		Irreducible holomorphic symplectic manifolds are higher dimensional generalizations of K3 surfaces.
		The purpose of this article is to describe a framework for understanding irreducible holomorphic symplectic manifolds, which hopefully will lead towards some kind of classification.
		There is evidence to suggest that if the 2n-dimensional irreducible holomorphic symplectic manifold X admits a non-trivial fibration, then the fibres must be abelian varieties and the base must be P^n (a large part of this has been proved by Matsushita).
	www.math.sunysb.edu /~sawon/abelianHK.shtml (575 words)

	9.2 Holomorphic/anti-holomorphic spin-angular momenta
		Since in Minkowski spacetime the restriction of the two constant spinor fields to any 2-surface are constant, and hence holomorphic and anti-holomorphic at the same time, both the holomorphic and anti-holomorphic spin-angular momenta are vanishing.
		Both the holomorphic and anti-holomorphic spin-angular momenta were calculated for small spheres [360].
		In general the anti-holomorphic and the holomorphic spin-angular momenta are diverging near the future null infinity of Einstein-Maxwell spacetimes as
	www.univie.ac.at /EMIS/journals/LRG/Articles/lrr-2004-4/articlesu21.html (410 words)

	AIF : Tome 31 fascicle 4 -- 1981
		The study of the finely holomorphic functions is bases here on the Beppo Levi functions, made precise in the sense of Deny.
		In addition it is shown that every finely holomorphic function is uniquely determined by its Taylor series, and that this series represents the function finely locally in an asymptotic sense.
		Furthermore, a finely holomorphic function has at most countably many zeros.
	annalif.ujf-grenoble.fr /Vol31/E314_3/E314_3.html (115 words)

	Almost Everywhere Holomorphic Functions Exercises (Site not responding. Last check: 2007-11-07)
		holomorphic function on D^1 which equals f a.e.
		Assume that f_n are holomorphic on a domain and that they converge to f
		Assume that f_n are holomorphic on a domain and that they converge
	www.groupsrv.com /science/about81191.html (849 words)

	On holomorphic maps with only fold singularities, Yoshifumi Ando
		Let $f:N\to P$ be a holomorphic map between $n$-dimensional complex manifolds which has only fold singularities.
		Such a map is called a holomorphic fold map.
		By using this result we prove that if the tangent bundles $TN$ and $TP$ are equipped with $\SU(n)$-structures in addition, then a holomorphic fold map $f$ canonically determines the homotopy class of an $\SU(n+1)$-bundle map of $TN\oplus\theta_N$ to $TP\oplus\theta_P$, where $\theta_N$ and $\theta_P$ are the trivial line bundles.
	projecteuclid.org /Dienst/UI/1.0/Display/euclid.nmj/1114631659 (127 words)

	DC MetaData for:Holomorphic vector bundles on non-algebraic surfaces
		The existence problem for holomorphic structures on vector bundles over non-algebraic surfaces is in general still open.
		We solve this problem in the case of rank 2 vector bundles over K3 surfaces and in the case of vector bundles of arbitrary rank over all known surfaces of class VII.
		Our methods, which are based on Donaldson theory and deformation theory, can be used to solve the existence problem of holomorphic vector bundles on further classes of non-algebraic surfaces.
	www.mathematik.uni-osnabrueck.de /preprints/shadow/calg0111.rdf.html (86 words)

	PlanetMath: holomorphic (Site not responding. Last check: 2007-11-07)
		This is version 8 of holomorphic, born on 2001-12-28, modified 2004-10-04.
		Object id is 1146, canonical name is Holomorphic.
		(Several complex variables and analytic spaces :: Holomorphic functions of several complex variables :: Holomorphic functions)
	planetmath.org /encyclopedia/Holomorphic.html (61 words)

	Citebase - Holomorphic Currents and Duality in N=1 Supersymmetric Theories
		Holomorphic Currents and Duality in N=1 Supersymmetric Theories
		The holomorphic currents act as vector fields on the chiral ring.
		In the context of electric-magnetic duality, the algebra generated by the holomorphic currents in the electric theory is isomorphic to the one on the magnetic side.
	www.citebase.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0309125 (674 words)

	Holomorphic function (Site not responding. Last check: 2007-11-07)
		Holomorphic functions are the central object of study of complex analysis; they are functions defined on an open subset of the complex number plane C with values in C which are complex differentiable at every point.
		The logarithm function is holomorphic on the set { z : z isn't a non-positive real number}.
		The Taylor series may converge on a larger disk; for instance, the Taylor series for the logarithm converges on every disk that doesn't contain 0, even in the vicinity of the negative real line.
	www.explainthis.info /ho/holomorphic-function.html (717 words)

	ARCC Workshop: Holomorphic curves in contact geometry (Site not responding. Last check: 2007-11-07)
		This workshop, sponsored by AIM and the NSF, will be devoted to the development of holomorphic curve techniques in contact geometry and topology.
		The advent of holomorphic curve techniques in contact topology, as exemplified in Symplectic Field Theory (SFT), and asymptotically holomorphic curve techniques, in the spirit of Donaldson, has allowed one to use a diverse set of geometric, analytic and topological tools when studying contact structures.
		The workshop will bring together researchers in contact and symplectic geometry, dynamics, low-dimensional topology and physics, who will explore connections between the various approaches people have taken to using holomorphic curves, develop new holomorphic curve techniques and apply them to various questions in contact and symplectic geometry and low-dimensional topology.
	www.aimath.org /ARCC/workshops/contactgeom2.html (325 words)

	Hsiung (Site not responding. Last check: 2007-11-07)
		Holomorphic sectional and bisectional curvatures of almost Hermitian manifolds
		Friedland and Hsiung \cite{1} proved an analogue of F. Schur's theorem concerning the holomorphic sectional curvature of some almost Hermitian manifolds called almost Hermitian $L$-manifolds, of which K\"ahlerian manifolds are special ones.
		In this paper we shall further extend the above work of Hsiung and Xiong by studying the general sectional, the holomorphic sectional and the holomorphic bisectional curvatures of almost Hermitian manifolds of all classes, together with some relationship among the three types of sectional curvatures.
	www.ma.kagu.sut.ac.jp /~sutjmath/31/Hsiung.html (123 words)

	Citebase - Holomorphic matrix models
		Authors: Lazaroiu, C. This is a study of holomorphic matrix models, the matrix models which underlie the conjecture of Dijkgraaf and Vafa.
		I also show that planar solutions of the holomorphic model probe the entire moduli space of the associated algebraic curve.
		In this case, use of the holomorphic model is crucial, since the Hermitian approach and its attending regularization would lead to a singular algebraic curve, thus contradicting the requirements of the conjecture.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:hep-th/0303008 (1478 words)

	Holomorphic Dynamics -- S. Morosawa T. Ueda Y. Nishimura M. Taniguchi
		This is a comprehensive introduction to holomorphic dynamics, that is the dynamics induced by the iteration of various analytic maps in complex number spaces.
		This has been the focus of much attention in recent years, with, for example, the discovery of the Mandelbrot set, and work on chaotic behavior of quadratic maps.
		The treatment is mathematically unified, emphasizing the substantial role played by classical complex analysis in understanding holomorphic dynamics as well as giving an up-to-date coverage of the modern theory." "The book will be welcomed by graduate students and professionals in pure mathematics and science who seek a reasonably self-contained introduction to this exciting area.
	www.frontlist.com /detail/0521662583 (136 words)

	Lisa Jeffrey - Holomorphic bundles and the Verlinde formula
		The moduli space M(n,d) of semistable holomorphic bundles of (coprime) rank n and degree d over a closed Riemann surface of genus g is a smooth Kahler manifold.
		We show how to use the Riemann-Roch formula and formulas for the intersection numbers in the cohomology of M(n,d) (proved in joint work with F. Kirwan) to establish the Verlinde formula, which is a formula for the dimension of the space of holomorphic sections of a line bundle over M(n,d).
		We also show how to extend our proof to prove a more general version of the Verlinde formula which gives the dimension of the space of holomorphic sections of line bundles over more general moduli spaces related to M(n,d).
	www.cms.math.ca /Events/summer98/s98-abs/node84.e (149 words)

	ipedia.com: Taniyama-Shimura theorem Article (Site not responding. Last check: 2007-11-07)
		The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions...
		The Taniyama-Shimura theorem establishes an important connection between elliptic curves, which are objects from algebraic geometry, and modular forms, which are certain periodic holomorphic functions investigated in number theory.
		Despite the name, which was a carry over from the Taniyama-Shimura Conjecture, the theorem is the work of Andrew Wiles, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor.
	www.ipedia.com /taniyama_shimura_theorem.html (534 words)

	Holomorphic Spaces (Site not responding. Last check: 2007-11-07)
		The term holomorphic spaces is short for spaces of holomorphic functions.
		This book is a collection of expository articles arising from MSRI’s fall 1995 program on holomorphic spaces.
		The remaining articles, while more specialized, are nevertheless designed in varying degrees to be accessible to the nonexpert; some are minicourses in themselves.
	www.axler.net /Holomorphic.html (98 words)

	Riemann surfaces and holomorphic vector bundles (Site not responding. Last check: 2007-11-07)
		After we have seen a couple of nice applications of these theorems involving certain special points on Riemann surfaces, we will go on to the study of holomorphic differentials.
		We shall then discuss holomorphic line bundles and their relations to divisors and to the Jacobian.
		Finally, we will embark on the study of Semi-Stable holomorphic vector bundles over Riemann surfaces of genus greater than or equal to 2.
	www.imf.au.dk /da/uddannelse/beskrivelser/older/F1996-F1998/node97.html (319 words)

	Girbau, Nicolau: On deformations of holomorphic foliations
		as a holomorphic foliation and as a transversely holomorphic foliation respectively.
		is the sheaf of germs of holomorphic vector fields tangent to
		KALKA, Holomorphic Foliations and Deformations of the Hopf foliation, Pacific J. of Math., 112 (
	www.numdam.org /numdam-bin/item?id=AIF_1989__39_2_417_0 (301 words)

	Fields Institute - Thematic Program 2005-06
		The purpose is two-fold: to survey the recent achievements and outline new directions of research on the one hand; and to foster the interaction between various branches of the dynamics and mathematical physics communities working in renormalization.
		The first half of the program will concentrate on the physical aspects of renormalization: statistical physics, conformal field theory, the underlying Hopf-algebraic structure, renormalization in PDE. The second half will have the dynamical flavour: holomorphic and smooth dynamics; KAM theory and renormalization of Hamiltonian flows; conformal invariance and universality in 2D stochastic processes.
		We also plan to touch on the recent breakthrough in the Geometrization Program based on the dynamics of the Ricci Flow and its connection to the concept of renormalization.
	www.fields.utoronto.ca /programs/scientific/05-06/holodynamics (558 words)

	Gauge Theory Seminar - Abstracts
		We formulate a general statement of estimated transversality for jets of such sections, and describe topological applications such as the existence of "good" linear systems from which interesting topological invariants of symplectic manifolds may be derived.
		The instanton Floer cohomology of a Riemann surface times a circle was shown by Dostoglou and Salamon to be isomorphic as a vector space to the cohomology of the moduli space of stable bundles on the Riemann surface.
		There is a ring structure on this Floer homology given by the cobordism obtained as the product of the surface and a `pair of pants'; Munoz has shown this to be isomorphic to the quantum cohomology of the moduli space of stable bundles on the Riemann surface.
	www.mpim-bonn.mpg.de /html/services/activities/gauge_abstracts.html (8367 words)

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