Determinants Associated to Zeta Matrices of Posets

We consider the matrix ${\frak Z}_P=Z_P+Z_P^t$, where the entries of $Z_P$ are the values of the zeta function of the finite poset $P$. We give a combinatorial interpretation of the determinant of ${\frak Z}_P$ and establish a recursive formula for this determinant in the case in which $P$ is a boolean algebra.Comment: 14 pages, AMS-Te

Marine Biodiversity in the Caribbean: Regional Estimates and Distribution Patterns

This paper provides an analysis of the distribution patterns of marine biodiversity and summarizes the major activities of the Census of Marine Life program in the Caribbean region. The coastal Caribbean region is a large marine ecosystem (LME) characterized by coral reefs, mangroves, and seagrasses, but including other environments, such as sandy beaches and rocky shores. These tropical ecosystems incorporate a high diversity of associated flora and fauna, and the nations that border the Caribbean collectively encompass a major global marine biodiversity hot spot. We analyze the state of knowledge of marine biodiversity based on the geographic distribution of georeferenced species records and regional taxonomic lists. A total of 12,046 marine species are reported in this paper for the Caribbean region. These include representatives from 31 animal phyla, two plant phyla, one group of Chromista, and three groups of Protoctista. Sampling effort has been greatest in shallow, nearshore waters, where there is relatively good coverage of species records; offshore and deep environments have been less studied. Additionally, we found that the currently accepted classification of marine ecoregions of the Caribbean did not apply for the benthic distributions of five relatively well known taxonomic groups. Coastal species richness tends to concentrate along the Antillean arc (Cuba to the southernmost Antilles) and the northern coast of South America (Venezuela – Colombia), while no pattern can be observed in the deep sea with the available data. Several factors make it impossible to determine the extent to which these distribution patterns accurately reflect the true situation for marine biodiversity in general: (1) highly localized concentrations of collecting effort and a lack of collecting in many areas and ecosystems, (2) high variability among collecting methods, (3) limited taxonomic expertise for many groups, and (4) differing levels of activity in the study of different taxa

Hypergraphs and automorphic forms

grantor: University of TorontoIn the late '80s, A. Lubotzky, R. Phillips and P. Samak [22] used Deligne's proof of Ramanujan's conjecture and the Jacquet-Langlands correspondence for cuspidal representations of 'GL'(2) to construct a class of Ramanujan graphs which are the best known expander graphs. Their graphs are Cayley graphs of the group 'PSL' &parl0;2,Z/qZ&parr0; or 'PGL' &parl0;2,Z/qZ&parr0; . Unfortunately, this strategy cannot be applied for groups in general because, in the general case, there is no equivalent of the Jacquet-Langlands correspondence. However, J. Rogawski has completely classified the representations of the unitary group in three variables in [25]. In this thesis we consider a form of 'U'(3) which at a place p is isomorphic to 'GL'3 &parl0;Qp&parr0; . We study a finite disconnected union of finite quotients of the building attached to the group 'GL'3 &parl0;Qp&parr0; . We view this object as a hypergraph and, using the classification of automorphic representations of the group 'U'(3) and Deligne's Theorem, we give an estimation of the spectrum of the adjacency matrix of its underlying graph. We show that the underlying graph is an expander graph with good expansion coefficient.Ph.D