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No. of bitstreams: 1 santos_sp_me_sjrp.pdf: 1777749 bytes, checksum: 414449c29a5a2c7be6d6e5dc4eb7fbe0 (MD5) Previous issue date: 2018-02-08 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Seja S uma superf´ıcie regular em R3 sem pontos parab´olicos. O conjunto focal de S ´e o lugar geom´etrico dos centros das esferas que possuem contato degenerado com S em cada ponto. Tal contato ´e medido pelas singularidades da fam´ılia de func¸˜oes distaˆncia ao quadrado D associada a` S. O conjunto focal ´e uma superf´ıcie, por´em n˜ao necessariamente regular, e pode tamb´em ser visto como o conjunto bifurca¸ca˜o da fam´ılia D. A t´ecnica de associar uma variedade singular X(S) a uma subvariedade suave S do espa¸co euclidiano e descobrir alguns aspectos da geometria de S a partir daqueles de X(S) esta´ na essˆencia das aplica¸co˜es da Teoria das Singularidades `a Geometria Diferencial. Neste trabalho, estudamos os modelos, a menos de difeomorfismos, para o conjunto focal de superf´ıcies imersas em R3 gen´ericas, reunimos os principais resultados sobre a geometria da superf´ıcie focal encontrados na literatura e os apresentamos de forma mais explicativa e com uma linguagem moderna. Al´em disso, mostramos que a superf´ıcie focal pode ser parametrizada por uma frente de onda e utilizamos resultados conhecidos para tais aplica¸co˜es no estudo da geometria da superf´ıcie focal. / Let S be a immersed surface in R3 without parabolic points. The focal set of S is the locus of the centres of spheres that have a degenerate contact with S in each point. This contact is measured by singularities of the family of distance squared function D associated with S. The focal set is a surface, but is not necessarily regular, and it can also be seen as the bifurcation set of the family D. The approach of associating a singular variety X(S) to a smooth submanifold S in an Euclidian space and recover some aspects of the geometry of S from that of X(S) is at the essence of applications of singularity theory to the Differential Geometry. In this work, we study models, unless diffeomorphism, of focal set of the immersed generics surfaces in R3. We have also gathered some results about the geometry of the focal set of the literature and we present them in a more explanatory way and in a modern notation. Furthermore, we show that the focal surface can be parametrized by a wave front and use the known results of such applications in the study of the focal set. / Fapesp: 2016/21226-5

In this study, I introduce the alternative-dependent focal Luce model (ADFLM), a random choice model generalizing the well-known Luce model (1959). In the ADFLM, focal alternatives are chosen more frequently relative to their utilities. I identify utilities, focal sets, and the magnitude of focal biases from choice data. Additionally, I axiomatically characterize the ADFLM by weakening the independence of irrelevant alternatives (IIA) axiom. This model can explain the well-known behavioral phenomena, the attraction and compromise effects. Furthermore, I also study the seller's profit maximization problem in the ADFLM. I also study an asymmetric dynamic patent race with a deadline under complete information. In my model, two firms decide whether to invest in RandD. The patent arrives randomly according to a Poisson process, and the large firm has a higher hazard rate than the small firm. I find the unique sub-game perfect Nash equilibrium strategy for this game. At the equilibrium, the large firm will stay longer in the race, while the small firm will quit earlier. The large firm's optimal stopping time is not affected by the competition, while the small firm's stopping time is reduced. Additionally, I find that companies will remain longer in the race if the investigation cost is lower, the winning premium is higher, the deadline is extended further, and the hazard rate is more prominent. Moreover, the market becomes more efficient with the competition since the patent is easier to realize. / Doctor of Philosophy / In this research, I study the consumer's behavior when individuals have limited cannot or do not give the same attention to each alternative available to them. In my study, I characterize the alternative-dependent focal Luce model (ADFLM), a consumer behavior model. Moreover, I solve the seller's profit maximization problem when the consumer's behavior follows the ADFLM. Meanwhile, I also study a dynamic patent race problem that occurs when firms compete for a patent with a deadline. If no firms achieve the patent, the stopping time (when the firm quits the patent race) of the large firm's (with a higher success rate every period) is not affected by the introduction of the small firm. However, the small firm quits earlier when the large firm is introduced. The competition between the two companies increases the overall probability of receiving a patent.