Let K be a number field with ring of integers O. Consider the set of n x n alternating matrices with fixed rank, r
The principal ideas behind the proof were first introduced in a paper of Katznelson where the problem of counting matrices is reduced to one of counting lattice points. Because our matrices have entries in O rather than being restricted to the set of rational integers, the lattices of Katznelson are replaced in the present work with O-modules. The generalization from the rational numbers to K renders the standard tools of lattice-theory less directly applicable, and we rely on the Minkowski map and novel arguments to, at various turns, reduce to the lattice case, or abstract results from lattices to O-modules. Ancillary results in our work include a new formula for the discriminant of a torsion-free O-module in terms of its pseudo-basis and a novel structure theorem about the set of alternating matrices whose rows lie in a specified O-module.
One goal of this thesis is to prove theorems that elucidate the Cohen-Lenstra-Martinet conjectures for the distributions of class groups of number fields, and further the understanding of their implications. We start by giving a simpler statement of the conjectures. We show that the probabilities that arise are inversely proportional to the number of automorphismsof structures slightly larger than the class groups. We find the moments of the Cohen-Lenstra-Martinet distributions and prove that the distributions are determined by their moments. In order to apply these conjectures to class groups of non-Galois fields, we prove a new theorem on the capitulation kernel (of ideal classes that become trivial in a larger field) to relate the class groups of non-Galois fields to the class groups of Galois fields. We then construct an integral model of the Hecke algebra of a finite group, show that it acts naturally on class groups of non-Galois fields, and prove that the Cohen-Lenstra-Martinet conjectures predict a distribution for class groups of non-Galois fields that involves the inverse of the number of automorphisms of the class group as a Hecke-module. The Cohen-Lenstra-Martinet Heuristics give a prediction for the distribution for the p-Sylow subgroups of the class groups of random Γ-number fields when p ∤ |Γ|. In this thesis, we prove several results on the distributions of the class groups for some p||Γ|, and show that the behaviour is qualitatively different than the predicted behaviour when p ∤ |Γ|. We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the bad part of the class group. For general number fields, our result is conditional on a natural conjecture on field counting. For abelian or D4 fields, our result is unconditional.
The challenges posed by climate change in national parks and other protected areas demand creative approaches, new ideas, and experiments that are beyond the capacity of any single park or agency staff. Research fellowships provide a critical way that the National Park Service (NPS) and its partners can address the agency’s needs to address climate change adaptation challenges. At least 30 such programs support stewardship-relevant science in national parks. Some national programs and initiatives at Acadia National Park in Maine, Rocky Mountain National Park in Colorado, and Sequoia and Kings Canyon National Parks in California serve as examples of how researchers in these programs are informing restoration, relocation, vegetation and fire management, and resource protection activities; documenting change that has already occurred; providing baseline data on biodiversity; and conducting novel experiments. Successful fellowship programs have strong engagement of resource managers, emphasize communication with management and public audiences, and incorporate ongoing support and evaluation. As a result
of these successes, NPS and partners are working to expand and strengthen the sustainability and effectiveness of research grants and fellowships.
Cookie SettingseScholarship uses cookies to ensure you have the best experience on our website. You can manage which cookies you want us to use.Our Privacy Statement includes more details on the cookies we use and how we protect your privacy.