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Integral complexa: teorema de Cauchy, fórmula integral de Cauchy e aplicações / Complex integral: Cauchy's theorem, Cuchy integral formula and applicationsOliveira, Saulo Henrique de 29 April 2015 (has links)
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Previous issue date: 2015-04-29 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This work ... / Este trabalho ...
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Sobre folheações projetivas sem soluções algébricasPenao, Giovanna Arelis Baldeón 30 May 2018 (has links)
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Previous issue date: 2018-05-30 / O objetivo deste trabalho é estudar um método, apresentado em [6], que nos permite
determinar se uma folheação no plano projetivo possui ou não soluções algébricas, usando
apenas métodos de computação algébrica. Mais especificamente usando bases de Gröbner.
Com este método é possível procurar por outros exemplos de folheações sem soluções
algébricas. / The aim of this work is to present a method, given by S. C. Coutinho and Bruno F. M.
Ribeiro in [6], to check whether certain holomorphic foliations on the complex projective
plane have algebraic solutions, using only methods of algebraic computing or more precisely,
using Gröbner bases. This algorithm is then used to produce examples of foliations without
algebraic solutions.
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Some Descriptions Of The Envelopes Of Holomorphy Of Domains in CnGupta, Purvi 03 1900 (has links) (PDF)
It is well known that there exist domains Ω in Cn,n ≥ 2, such that all holomorphic functions in Ω continue analytically beyond the boundary. We wish to study this remarkable phenomenon. The first chapter seeks to motivate this theme by offering some well-known extension results on domains in Cn having many symmetries. One important result, in this regard, is Hartogs’ theorem on the extension of functions holomorphic in a certain neighbourhood of (D x {0} U (∂D x D), D being the open unit disc in C. To understand the nature of analytic continuation in greater detail, in Chapter 2, we make rigorous the notions of ‘extensions’ and ‘envelopes of holomorphy’ of a domain. For this, we use methods similar to those used in univariate complex analysis to construct the natural domains of definitions of functions like the logarithm. Further, to comprehend the geometry of these abstractly-defined extensions, we again try to deal with some explicit domains in Cn; but this time we allow our domains to have fewer symmetries. The subject of Chapter 3 is a folk result generalizing Hartogs’ theorem to the extension of functions holomorphic in a neighbourhood of S U (∂D x D), where S is the graph of a D-valued function Φ, continuous in D and holomorphic in D. This problem can be posed in higher dimensions and we give its proof in this generality. In Chapter 4, we study Chirka and Rosay’s proof of Chirka’s generalization (in C2) of the above-mentioned result. Here, Φ is merely a continuous function from D to itself. Chapter 5 — a departure from our theme of Hartogs-Chirka type of configurations — is a summary of the key ideas behind a ‘non-standard’ proof of the so-called Hartogs phenomenon (i.e., holomorphic functions in any connected neighbourhood of the boundary of a domain Ω Cn , n ≥ 2, extend to the whole of Ω). This proof, given by Merker and Porten, uses tools from Morse theory to tame the boundary geometry of Ω, hence making it possible to use analytic discs to achieve analytic continuation locally. We return to Chirka’s extension theorem, but this time in higher dimensions, in Chapter 6. We see one possible generalization (by Bharali) of this result involving Φ where is a subclass of C (D; Dn), n ≥ 2. Finally, in Chapter 7, we consider Hartogs-Chirka type configurations involving graphs of multifunctions given by “Weierstrass pseudopolynomials”. We will consider pseudopolynomials with coefficients belonging to two different subclasses of C(D; C), and show that functions holomorphic around these new configurations extend holomorphically to D2 .
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The Role Of Potential Theory In Complex DynamicsBandyopadhyay, Choiti 05 1900 (has links) (PDF)
Potential theory is the name given to the broad field of analysis encompassing such topics as harmonic and subharmonic functions, the Dirichlet problem, Green’s functions, potentials and capacity. In this text, our main goal will be to gain a deeper understanding towards complex dynamics, the study of dynamical systems defined by the iteration of analytic functions, using the tools and techniques of potential theory. We will restrict ourselves to holomorphic polynomials in C.
At first, we will discuss briefly about harmonic and subharmonic functions. In course, potential theory will repay its debt to complex analysis in the form of some beautiful applications regarding the Julia sets (defined in Chapter 8) of a certain family of polynomials, or a single one.
We will be able to provide an explicit formula for computing the capacity of a Julia set, which in some sense, gives us a finer measurement of the set. In turn, this provides us with a sharp estimate for the diameter of the Julia set. Further if we pick any point w from the Julia set, then the inverse images q−n(w) span the whole Julia set. In fact, the point-mass measures with support at the discrete set consisting of roots of the polynomial, (qn-w) will eventually converge to the equilibrium measure of the Julia set, in the weak*-sense. This provides us with a very effective insight into the analytic structure of the set.
Hausdorff dimension is one of the most effective notions of fractal dimension in use. With the help of potential theory and some ergodic theory, we can show that for a certain holomorphic family of polynomials varying over a simply connected domain D, one can gain nice control over how the Hausdorff dimensions of the respective Julia sets change with the parameter λ in D.
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Infinitely Divisible Metrics, Curvature Inequalities And Curvature FormulaeKeshari, Dinesh Kumar 07 1900 (has links) (PDF)
The curvature of a contraction T in the Cowen-Douglas class is bounded above by the
curvature of the backward shift operator. However, in general, an operator satisfying the curvature inequality need not be contractive. In this thesis, we characterize a slightly smaller class of contractions using a stronger form of the curvature inequality. Along the way, we find conditions on the metric of the holomorphic Hermitian vector bundle E corresponding to the operator T in the Cowen-Douglas class which ensures negative definiteness of the curvature function. We obtain a generalization for commuting tuples
of operators in the Cowen-Douglas class.
Secondly, we obtain an explicit formula for the curvature of the jet bundle of the Hermitian holomorphic bundle E f on a planar domain Ω. Here Ef is assumed to be a pull-back of the tautological bundle on gr(n, H ) by a nondegenerate holomorphic map f :Ω →Gr (n, H ).
Clearly, finding relationships amongs the complex geometric invariants inherent in the
short exact sequence
0 → Jk(Ef ) → Jk+1(Ef ) →J k+1(Ef )/ Jk(Ef ) → 0
is an important problem, whereJk(Ef ) represents the k-th order jet bundle. It is known that the Chern classes of these bundles must satisfy
c(Jk+1(Ef )) = c(Jk(Ef )) c(Jk+1(Ef )/ Jk(Ef )).
We obtain a refinement of this formula:
trace Idnxn ( KJk(Ef )) - trace Idnxn ( KJk-1(Ef ))= KJk(Ef )/ Jk-1(Ef )(z).
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Trace de Dixmier d'opérateurs de Hankel / Dixmier trace of Hankel operatorsTytgat, Romaric 02 December 2013 (has links)
Nous nous intéressons aux opérateurs de Hankel $H_{bar{f}}$ de symbole anti holomorphe $bar{f}$ et regardons l'espace de Dixmier $mathcal{D}^{p}$ associé ($pgeq1$), c'est à dire l'ensemble des $f$ tel que $|H_{bar{f}}|^{p}$ soit dans l'idéal de Macaev $mathcal{S}^{+}_{1}$. Notre approche est de voir l'espace de Dixmier comme une certaine limite des classes de Schatten. Quand $f in mathcal{D}^{p}$, nous étudions $Tr_{omega}(|$H_{bar{f}}$|^{p})$ la trace de Dixmier de $|H_{bar{f}}|^{p}$. Nous redémontrons certains résultats classiques quand $f$ est holomorphe sur le disque alors que nous donnons de nouveaux résultats quand $f$ est entière. Nous utilisons notre méthode pour étudier l'espace de Dixmier du petit opérateur de Hankel, des opérateurs de Toeplitz $T_{varphi}$ ($varphi$ définie sur le disque ou sur le plan complexe tout entier) ainsi que pour l'opérateur de composition. / We study Hankel operators $H_{bar{f}}$ with anti holomorphic symbol $bar{f}$ and we are interested to the Dixmier space $mathcal{D}^{p}$ ($pgeq1$), the set of functions $f$ such that $|H_{bar{f}}|^{p} in mathcal{S}^{+}_{1}$ the Macaev ideal. We look Dixmier space as a limit of Schatten class. When $f in mathcal{D}^{p}$, we study $Tr_{omega}(|$H_{bar{f}}$|^{p})$ the Dixmier trace of $|H_{bar{f}}|^{p}$. We have different results when $f$ is an entire or a holomorphic function of the unit disk in the complex plan. We study also the Dixmier space of the little Hankel operator, Toeplitz operator and composition operator.
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D-bar and Dirac Type Operators on Classical and Quantum DomainsMcBride, Matthew Scott 29 August 2012 (has links)
Indiana University-Purdue University Indianapolis (IUPUI) / I study d-bar and Dirac operators on classical and quantum domains subject to the APS boundary conditions, APS like boundary conditions, and other types of global boundary conditions. Moreover, the inverse or inverse modulo compact operators to these operators are computed. These inverses/parametrices are also shown to be bounded and are also shown to be compact, if possible. Also the index of some of the d-bar operators are computed when it doesn't have trivial index. Finally a certain type of limit statement can be said between the classical and quantum d-bar operators on specialized complex domains.
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Operators on wighted spaces of holomorphic functionsBeltrán Meneu, María José 24 March 2014 (has links)
The Ph.D. Thesis ¿Operators on weighted spaces of holomorphic functions¿ presented
here treats different areas of functional analysis such as spaces of holomorphic
functions, infinite dimensional holomorphy and dynamics of operators.
After a first chapter that introduces the notation, definitions and the basic results
we will use throughout the thesis, the text is divided into two parts. A first one,
consisting of Chapters 1 and 2, focused on a study of weighted (LB)-spaces of entire
functions on Banach spaces, and a second one, corresponding to Chapters 3 and
4, where we consider differentiation and integration operators acting on different
classes of weighted spaces of entire functions to study its dynamical behaviour. In
what follows, we give a brief description of the different chapters:
In Chapter 1, given a decreasing sequence of continuous radial weights on a Banach
space X, we consider the weighted inductive limits of spaces of entire functions
VH(X) and VH0(X). Weighted spaces of holomorphic functions appear naturally
in the study of growth conditions of holomorphic functions and have been investigated
by many authors since the work of Williams in 1967, Rubel and Shields
in 1970 and Shields and Williams in 1971. We determine conditions on the family
of weights to ensure that the corresponding weighted space is an algebra or
has polynomial Schauder decompositions. We study Hörmander algebras of entire
functions defined on a Banach space and we give a description of them in terms of
sequence spaces. We also focus on algebra homomorphisms between these spaces
and obtain a Banach-Stone type theorem for a particular decreasing family of
weights. Finally, we study the spectra of these weighted algebras, endowing them
with an analytic structure, and we prove that each function f ¿ VH(X) extends
naturally to an analytic function defined on the spectrum. Given an algebra homomorphism,
we also investigate how the mapping induced between the spectra
acts on the corresponding analytic structures and we show how in this setting
composition operators have a different behavior from that for holomorphic functions
of bounded type. This research is related to recent work by Carando, García,
Maestre and Sevilla-Peris. The results included in this chapter are published by
Beltrán in [14]. Chapter 2 is devoted to study the predual of VH(X) in order to linearize this space
of entire functions. We apply Mujica¿s completeness theorem for (LB)-spaces to
find a predual and to prove that VH(X) is regular and complete. We also study
conditions to ensure that the equality VH0(X) = VH(X) holds. At this point,
we will see some differences between the finite and the infinite dimensional cases.
Finally, we give conditions which ensure that a function f defined in a subset
A of X, with values in another Banach space E, and admitting certain weak
extensions in a space of holomorphic functions can be holomorphically extended
in the corresponding space of vector-valued functions. Most of the results obtained
have been published by the author in [13].
The rest of the thesis is devoted to study the dynamical behaviour of the following
three operators on weighted spaces of entire functions: the differentiation operator
Df(z) = f (z), the integration operator Jf(z) = z
0 f(¿)d¿ and the Hardy
operator Hf(z) = 1
z z
0 f(¿)d¿, z ¿ C.
In Chapter 3 we focus on the dynamics of these operators on a wide class of
weighted Banach spaces of entire functions defined by means of integrals and
supremum norms: the weighted spaces of entire functions Bp,q(v), 1 ¿ p ¿ ¿,
and 1 ¿ q ¿ ¿. For q = ¿ they are known as generalized weighted Bergman
spaces of entire functions, denoted by Hv(C) and H0
v (C) if, in addition, p = ¿.
We analyze when they are hypercyclic, chaotic, power bounded, mean ergodic
or uniformly mean ergodic; thus complementing also work by Bonet and Ricker
about mean ergodic multiplication operators. Moreover, for weights satisfying
some conditions, we estimate the norm of the operators and study their spectrum.
Special emphasis is made on exponential weights. The content of this chapter is
published in [17] and [15].
For differential operators ¿(D) : Bp,q(v) ¿ Bp,q(v), whenever D : Bp,q(v) ¿
Bp,q(v) is continuous and ¿ is an entire function, we study hypercyclicity and
chaos. The chapter ends with an example provided by A. Peris of a hypercyclic
and uniformly mean ergodic operator. To our knowledge, this is the first example
of an operator with these two properties. We thank him for giving us permission
to include it in our thesis.
The last chapter is devoted to the study of the dynamics of the differentiation and
the integration operators on weighted inductive and projective limits of spaces of
entire functions. We give sufficient conditions so that D and J are continuous on
these spaces and we characterize when the differentiation operator is hypercyclic,
topologically mixing or chaotic on projective limits. Finally, the dynamics of these
operators is investigated in the Hörmander algebras Ap(C) and A0
p(C). The results
concerning this topic are included by Bonet, Fernández and the author in [16]. / Beltrán Meneu, MJ. (2014). Operators on wighted spaces of holomorphic functions [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/36578 / Premios Extraordinarios de tesis doctorales
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Harmonicity in Slice Analysis: Almansi decomposition and Fueter theorem for several hypercomplex variablesBinosi, Giulio 10 June 2024 (has links)
The work is situated within the theory of slice analysis, a generalization of complex analysis for hypercomplex numbers, considering function of both quaternionic and Clifford variables, in both one and several variables.
%We first characterize some partial slice sets of
The primary focus of the thesis is on the harmonic and polyharmonic properties of slice regular functions. We derive explicit formulas for the iteration of the Laplacian on slice regular functions, proving that their degree of harmonicity increases with the dimension of the algebra. Consequently, we present Almansi-type decompositions for slice functions in several variables. Additionally, using the harmonic properties of the partial spherical derivatives and their connection with the Dirac operator in Clifford analysis, we achieve a generalization of the Fueter and Fueter-Sce theorems in the several variables context. Finally, we establish that regular polynomials of sufficiently low degree are the unique slice regular functions in the kernel of the iteration of the Laplacian, whose power is less than Sce index.
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The differential geometry of the fibres of an almost contract metric submersionTshikunguila, Tshikuna-Matamba 10 1900 (has links)
Almost contact metric submersions constitute a class of Riemannian submersions whose
total space is an almost contact metric manifold. Regarding the base space, two types
are studied. Submersions of type I are those whose base space is an almost contact
metric manifold while, when the base space is an almost Hermitian manifold, then the
submersion is said to be of type II.
After recalling the known notions and fundamental properties to be used in the
sequel, relationships between the structure of the fibres with that of the total space
are established. When the fibres are almost Hermitian manifolds, which occur in the
case of a type I submersions, we determine the classes of submersions whose fibres
are Kählerian, almost Kählerian, nearly Kählerian, quasi Kählerian, locally conformal
(almost) Kählerian, Gi-manifolds and so on. This can be viewed as a classification of
submersions of type I based upon the structure of the fibres.
Concerning the fibres of a type II submersions, which are almost contact metric
manifolds, we discuss how they inherit the structure of the total space.
Considering the curvature property on the total space, we determine its corresponding
on the fibres in the case of a type I submersions. For instance, the cosymplectic
curvature property on the total space corresponds to the Kähler identity on the fibres.
Similar results are obtained for Sasakian and Kenmotsu curvature properties.
After producing the classes of submersions with minimal, superminimal or umbilical
fibres, their impacts on the total or the base space are established. The minimality of
the fibres facilitates the transference of the structure from the total to the base space.
Similarly, the superminimality of the fibres facilitates the transference of the structure
from the base to the total space. Also, it is shown to be a way to study the integrability
of the horizontal distribution.
Totally contact umbilicity of the fibres leads to the asymptotic directions on the total
space.
Submersions of contact CR-submanifolds of quasi-K-cosymplectic and
quasi-Kenmotsu manifolds are studied. Certain distributions of the under consideration
submersions induce the CR-product on the total space. / Mathematical Sciences / D. Phil. (Mathematics)
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