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dc.contributor Ortega Cerdà, Joaquim
dc.creator Arraz Almirall, Alexis
dc.date 2018-10-08T07:59:53Z
dc.date 2018-10-08T07:59:53Z
dc.date 2018-06-27
dc.date.accessioned 2024-12-16T10:26:48Z
dc.date.available 2024-12-16T10:26:48Z
dc.identifier http://hdl.handle.net/2445/125123
dc.identifier.uri http://fima-docencia.ub.edu:8080/xmlui/handle/123456789/21543
dc.description Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2018, Director: Joaquim Ortega Cerdà
dc.description [en] In this project we deal with random analytic functions. Here we specifically use Gaussian analytic functions. Without technicalities, a GAF $f$ (for short) is a random holomorphic function on a region of $\mathbb{C}$ such that $( f ( z 1 ) , ..., f ( z n ))$ is a random vector with normal distribution. One way to generate them is using linear combinations of holomorphic functions whose coefficients are Gaussian random variables in $\mathbb{C}$ (or in $\mathbb{R}$ in special cases). For finding the zero set of a GAF we work on four isometric - invariant Hilbert spaces of analytic functions: the Fock space in $\mathbb{C}$, the finite space of polynomials in $\mathbb{S}^2$, the weighted Bergman space in $\mathbb{D}$ and the Paley - Wiener space. The first intensity determines the average of the distribution of the zero set of a GAF, and the Edelman - Kostlan formula gives an explicit expression of it. A result of uniqueness, called Calabi’s Rigidity, concludes that the first intensity determines the distribution of the zero set of a GAF. At the end, some examples made in C++ and gnuplot clarify the theory in these Hilbert spaces.
dc.format 81 p.
dc.format application/pdf
dc.language eng
dc.rights cc-by-nc-nd (c) Alexis Arraz Almirall, 2018
dc.rights http://creativecommons.org/licenses/by-nc-nd/3.0/es/
dc.rights info:eu-repo/semantics/openAccess
dc.source Treballs Finals de Grau (TFG) - Matemàtiques
dc.subject Funcions de variables complexes
dc.subject Teoria geomètrica de funcions
dc.subject Processos puntuals
dc.subject Treballs de fi de grau
dc.subject Functions of complex variables
dc.subject Geometric function theory
dc.subject Point processes
dc.subject Bachelor's theses
dc.title Zeros of random analytic functions
dc.type info:eu-repo/semantics/bachelorThesis


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