Search results for "Dirichlet'n ongelma"

showing 2 items of 2 documents

Harmoniset funktiot kompleksialueessa ja konformikuvaukset

2014

Tämän tutkielman tarkoituksena on syventää tietoja kompleksianalyysistä tutustumalla harmonisiin funktioihin ja konformikuvauksiin. Funktioita, jotka toteuttavat Laplacen yhtälön, kutsutaan harmonisiksi funktioiksi. Harmonisten funktioiden määrittämiseen voidaan käyttää Cauchy-Riemannin yhtälöitä. Harmoniset funktioit ovat yhteydessä analyyttisiin funktioihin, sillä harmonisten funktioiden avulla voidaan selittää analyyttisten kuvausten teoriaa ja päinvastoin. Tämän tutkielman kannalta tärkeimpiä analyyttisiä kuvauksia ovat injektiiviset kuvaukset, jotka tunnetaan myös konformikuvauksina. Konformikuvaukset ovat alueiden välisiä kuvauksia, jotka säilyttävät kulmien suuruuden ja suunnan ja jo…

konformikuvausLaplacen yhtälöfunktioteorialineaarinen rationaalikuvausPoissonin integrointikaavaharmoninen funktioanalyyttinen funktioDirichlet'n ongelmayhtälötCauchy-Riemannin yhtälötSchwarz-Christoffelin kaavaanalyyttiset funktiotfunktiot

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Stokastinen reunasäännöllisyys ja häiritty köydenvetopeli

2014

Tämän tutkielman tarkoituksena on tutkia Dirichlet'n ongelmaa todennäköisyysteorian ja peliteorian näkökulmasta. Osoittautuu, että käytössä olevan alueen reuna vaikuttaa siihen, onko olemassa Dirichlet'n ongelman ratkaisevaa funktiota, joka saavuttaa oikeat reuna-arvot jatkuvasti. Tutkielmassa määritellään joukon reunasäännöllisyys jatkuva-aikaisen satunnaisliikkeen avulla. Käyttökelpoinen ja riittävä ehto reunapisteen säännöllisyydelle on Poincarén kartioehto. Tässä työssä näytetään myös Wienerin kriteerio, joka on riittävä ja välttämätön ehto reunapisteen säännöllisyydelle. Tutkielmassa käsitellään myös p-Laplacen yhtälöä käyttäen apuna häirittyä köydenvetopeliä (tug-of-war with noise). H…

todennäköisyysteoriap-harmonious funktiotstokastinen reunasäännöllisyysBrownin liikeDirichlet'n ongelmahäiritty köydenvetopelip-harmoniset funktiotWienerin kriteeriofunktiotstokastiset prosessit

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PEPhysical Sciences and Engineering
- PE2Fundamental Constituents of Matter
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  - PE11_12Semi-conducting and magnetic materials engineering
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LSLife Sciences
- LS1Molecules of Life: Biological Mechanisms, Structures and Functions
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- LS2Integrative Biology: from Genes and Genomes to Systems
  - LS2_1Genetics
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  - LS4_1Organ and tissue physiology and pathophysiology
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- LS3Cell Biology, Development, Stem Cells and Regeneration
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SHSocial Sciences and Humanities
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