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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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