Geometry of Integrable Lattice Equations and their Reductions

Modern research into discrete integrable systems has provided new insights into a wide variety of fields, including generalisations of special functions, orthogonal polynomials and dynamical systems theory. In this thesis, we extend one of the most productive insights in this area to higher dimensions. In particular, we show how to apply ideas from resolution of singularities and birational geometry to discrete systems in higher dimensions. The most widely studied setting for these ideas lies in spaces of dimension two. By blowing up at certain points to resolve singularities found in maps on surfaces, new surfaces are constructed on which the map becomes an isomorphism, a so-called space of initial conditions. This has led to new developments in the field, including the discovery of new examples of integrable maps by Sakai with solutions that have unexpectedly rich properties. On the other hand, this geometric approach has never been applied to integrable partial difference equations (often called lattice equations), which share other properties with the maps in dimension two. In this thesis, we overcome this gap. In particular, we examine spaces of initial conditions for integrable lattice equations, which are members of the equations classified by Adler et al, known as ABS equations. By explicitly calculating the induced map on their resolved initial value spaces, we find transformations to new lattice equations, and hence find novel reductions to discrete Painlevé equations. We also show that an equation arising from the geometry of ABS equations is satisfied by the coefficients of a cluster algebra associated with a form of the discrete mKdV

Opening Weekend: The First-Year Experience

Opening Weekend, at Bowling Green State University, is the four-day antecedent to the start of the Fall Semester in August. University sponsored programming during Opening Weekend provides academic support, social opportunities, and leadership opportunities for students. Through an analysis of the strengths, weaknesses, opportunities, and threats of Opening Weekend, this project stands to recommend and identify areas of growth for the program that will benefit incoming students to the university

Seismic detection of transient changes beneath Black Rapids Glacier, Alaska

Thesis (Ph.D.) University of Alaska Fairbanks, 1998To gain new insight into the mechanisms of basal motion, I have demonstrated the feasibility of an active seismic technique to measure temporal changes in basal conditions on sub-hourly time-scales. Using changes observed in the summer of 1993 on Black Rapids Glacier, I have determined part of the basal morphology and the mechanisms of seismic change there. One region of the glacier's bed was monitored daily using seismic reflections, for a period of 45 days. The majority of these reflections were nearly identical. However, the englacial drainage of two ice-marginal lakes and one supra-glacial pothole upglacier of the study site each caused significant anomalies in the daily reflections, as well as dramatic increases in basal motion. Two of these seismic anomalies were nearly identical despite the fact that their drainage events occurred at different locations. Further, these two seismic anomalies were followed by records identical to the non-anomalous state, showing that the changes were seismically reversible. In one of these events, two records taken 36 minutes apart revealed that the transition between the anomalous and normal states occurred completely within this short interval. Reflection arrival times during the anomalies require that a basal layer at least 5 m thick was either created or changed in situ. Reflection amplitudes indicate that such a layer could be either water or a basal till, but water layers of such thickness are not physically reasonable. Published values of wave speeds and densities of till are then compared to those constrained by the observed reflection coefficients. Only a decrease in till saturation can produce the observed changes in reflection amplitudes in the time required. Because the transition from anomalous to normal states can occur in as little as 36 minutes, any mechanisms involving the diffusion of water through a thick till layer are ruled out, such as a change in porosity or pore-water (or effective) pressure. We therefore interpret the cause of the seismic anomalies as due to a temporary decrease in saturation, and propose that such a change may occur quickly and reversibly following a lake drainage by a redistribution of the overburden pressure

Higher Education Product Baskets: Degree Offering Distributions and the Financial Strength of Colleges and Universities

This paper evaluates the relationship between the distributions of degrees offered by a college and the financial strength of that institution. While no causal relationship is established, the findings generally show that the more spe-cialized an institution is, the more net wealth it is likely to hold. Additional evidence points to how this effect differs de-pending on the degrees themselves: High concentrations of STEM fields, for example, tend to benefit the home college's financial position.This research highlights the importance of the considerations by which university systems balance the types of insti-tutions in their network. It adds to the small but growing research into higher education finance. Finally, it advocates for an understanding of public institutions as policy platforms. By paying attention to the implementers of public poli-cies, those policies might have more sustainable impacts

Intrinsic electrophysiological properties of entorhinal cortex stellate cells and their contribution to grid cell firing fields

The medial entorhinal cortex (MEC) is an increasingly important focus for investigation of mechanisms for spatial representation. Grid cells found in layer II of the MEC are likely to be stellate cells, which form a major projection to the dentate gyrus. Entorhinal stellate cells are distinguished by distinct intrinsic electrophysiological properties, but how these properties contribute to representation of space is not yet clear. Here, we review the ionic conductances, synaptic, and excitable properties of stellate cells, and examine their implications for models of grid firing fields. We discuss why existing data are inconsistent with models of grid fields that require stellate cells to generate periodic oscillations. An alternative possibility is that the intrinsic electrophysiological properties of stellate cells are tuned specifically to control integration of synaptic input. We highlight recent evidence that the dorsal-ventral organization of synaptic integration by stellate cells, through differences in currents mediated by HCN and leak potassium channels, influences the corresponding organization of grid fields. Because accurate cellular data will be important for distinguishing mechanisms for generation of grid fields, we introduce new data comparing properties measured with whole-cell and perforated patch-clamp recordings. We find that clustered patterns of action potential firing and the action potential after-hyperpolarization (AHP) are particularly sensitive to recording condition. Nevertheless, with both methods, these properties, resting membrane properties and resonance follow a dorsal-ventral organization. Further investigation of the molecular basis for synaptic integration by stellate cells will be important for understanding mechanisms for generation of grid fields

Census 2020: The effect of a census undercount in Pierce County

CAUR is partnering with GTCF to provide data for a Census 2020 campaign in Pierce County. The purpose of this campaign is to provide materials and information about the importance of a complete and accurate census count. The research materials provided by CAUR will be used to provide information for a public education video and materials to be distributed to community leaders as they encourage constituents to actively participate in the upcoming 2020 Census. Census data has a wide variety of applications, both public/government and in private industry. CAUR worked closely with GTCF to identify four different use cases for census data that help convey the importance of having a complete and accurate count. These use cases represent aspects of day-to-day life for Pierce County residents that might be changed if the census count is inaccurate. For example, we present details on how bus routes might be altered if are using inaccurate census data, due to significant undercounts, potentially missing entire neighborhoods where bus service would be needed. We also evaluate the difference in federal funding to Pierce County if 10% of residents do not complete their census forms. The examples we use here provide insight into the issues that arise from both inaccurate and incomplete census data. In some of our use cases, we simulate an inaccurate census count in key areas. For example, we simulate the differences of siting a new health clinic if some houses incorrectly report the number of children in their homes. In other cases we examine the repercussions of failing to complete the census entirely. Federal grants are sometimes based on the number of residents in an area, and if the census count does not include all residents, then there is a proportional drop in grant funding. It is important to relay the message that the census is important, and that it should be completed honestly and accurately

Measure Transport with Kernel Stein Discrepancy

Measure transport underpins several recent algorithms for posterior approximation in the Bayesian context, wherein a transport map is sought to minimise the Kullback--Leibler divergence (KLD) from the posterior to the approximation. The KLD is a strong mode of convergence, requiring absolute continuity of measures and placing restrictions on which transport maps can be permitted. Here we propose to minimise a kernel Stein discrepancy (KSD) instead, requiring only that the set of transport maps is dense in an $L^2$ sense and demonstrating how this condition can be validated. The consistency of the associated posterior approximation is established and empirical results suggest that KSD is competitive and more flexible alternative to KLD for measure transport