Thesis (Ph. D.)--Columbia University, 1917. / Vita.

Thesis (Ph. D.)--University of Illinois, 1911. / Vita. "Reprinted from 'Proceedings of the London mathematical society' series 2, vol. 11, nos. 1140-1142, pages 185-224."

Proeschrift--Amsterdam. / "Stellingen": [3] p. laid in.

Thesis (Ph. D.)--University of Illinois, 1911. / Vita. "Reprinted from 'Proceedings of the London mathematical society' series 2, vol. 11, nos. 1140-1142, pages 185-224."

Texte remanié: Thèse doctorat--Mathématiques--Paris, 1878. N°: 408. / Thèses présentées à la Faculté des sciences de Paris. N° 408.

Diss.--Paris. / Part of the Cornell Digital Library Math Collection.

The well-known Riemann-Hurwitz formula for Riemann surfaces (or the corresponding formulas of the same name for curves/function fields) is used in genus computations. In 1979, Yûji Kida proved a strikingly analogous formula in [Kid80] for p-extensions of CM-fields (p an odd prime) which is similarly used to compute Iwasawa λ -invariants. However, the relationship between Kida’s formula and the statement for surfaces is not entirely clear since the proofs are of a very different flavor. Also, there were a few hypotheses for Kida’s result which were not fully satisfying; for example, Kida’s formula requires CM-fields rather than more general number fields and excludes the prime p = 2. Around a year after Kida’s result was published, Kenkichi Iwasawa used Galois cohomology in [Iwa81] to establish a more general formula (about representations) that did not exclude the prime p = 2 nor need the CM-field assumption. Moreover, Kida’s formula follows as a corollary from Iwasawa’s formula. We’ll prove a slight generalization of Iwasawa’s formula and use this to give a new proof of a result of Kida in [Kid79] and Ferrero in [Fer80] which computes λ-invariants in imaginary quadratic extensions for the prime p = 2. We go on to produce special generalizations of Iwasawa’s formula in the case of cyclic p-extensions; these formulas can be realized as statements about Q(p)-representations, and, in the cases of degree p or p², about p-adic integral representations. One upshot of these formulas is a vanishing criterion for λ-invariants which generalizes a result of Takashi Fukuda et al. in [FKOT97]. Other applications include new congruences and inequalities for λ-invariants that cannot be gleaned from Iwasawa’s formula. Lastly, we give a scheme theoretic approach to produce a general formula for finite, separable morphisms of Dedekind schemes which simultaneously encompasses the classical Riemann-Hurwitz formula and Iwasawa’s formula.

Student Number : 9701675D - MSc dissertation - School of Mathematics - Faculty of Science / The role of invariants in obtaining exact solutions of di#11;erential equations is reviewed. The examples considered are nonlinear evolution type equations like the Fisher and Fitzhugh-Nagumo equations. Finally, we look directly at an equation (formulated) governing some non-Newtonian uid in a rotating system.

No description available.

One of the restrictions used in all of the works done on phylogenetic invariants for group based models has been that the group be abelian. In my thesis, I aim to generalize the method of invariants for group-based models of DNA sequence evolution to include nonabelian groups. By using a nonabelian group to act one the nucleotides, one could capture the structure of the symmetric model for DNA sequence evolution. If successful, this line of research would unify the two separated strands of active research in the area today: Allman and Rhodes’s invariants for the symmetric model and Strumfels and Sullivant’s toric ideals of phylogenetic invariants. Furthermore, I want to look at the statistical properties of polynomial invariants to get a better understanding of how they behave when used with real, “noisy” data.