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Ph. D. Thesis / In order to compute more realistic production plans and schedules, techniques using nonlinear programming (NLP) and mixed-integer nonlinear programming (MINLP) have gathered a lot of attention from the industry and academy. Efficient solution of these problems to a proven ε-global optimality remains a challenge due to their combinatorial, nonconvex, and large dimensionality attributes. The key contributions of this work are: 1) the generalization of the inventory pinch decomposition method to scheduling problems, and 2) the development of a deterministic global optimization method. An inventory pinch is a point at which the cumulative total demand touches its corresponding concave envelope. The inventory pinch points delineate time intervals where a single fixed set of operating conditions is most likely to be feasible and close to the optimum. The inventory pinch method decomposes the original problem in three different levels. The first one deals with the nonlinearities, while subsequent levels involve only linear terms by fixing part of the solution from previous levels. In this heuristic method, infeasibilities (detected via positive value of slack variables) are eliminated by adding at the first level new period boundaries at the point in time where infeasibilities are detected. The global optimization algorithm presented in this work utilizes both piecewise McCormick (PMCR) and Normalized Multiparametric Disaggregation (NMDT), and employs a dynamic partitioning strategy to refine the estimates of the global optimum. Another key element is the parallelized bound tightening procedure. Case studies include gasoline blend planning and scheduling, and refinery planning. Both inventory pinch method and the global optimization algorithm show promising results and their performance is either better or on par with other published techniques and commercial solvers, as exhibited in a number of test cases solved during the course of this work. / Thesis / Doctor of Philosophy (PhD) / Optimal planning and scheduling of production systems are two very important tasks in industrial practice. Their objective is to ensure optimal utilization of raw materials and equipment to reduce production costs. In order to compute realistic production plans and schedules, it is often necessary to replace simplified linear models with nonlinear ones including discrete decisions (e.g., “yes/no”, “on/off”). To compute a global optimal solution for this type of problems in reasonable time is a challenge due to their intrinsic nonlinear and combinatorial nature. The main goal of this thesis is the development of efficient algorithms to solve large-scale planning and scheduling problems. The key contributions of this work are the development of: i) a heuristic technique to compute near-optimal solutions rapidly, and ii) a deterministic global optimization algorithm. Both approaches showed results and performances better or equal to those obtained by commercial software and previously published methods.

Habitat degradation and hunting pose the most proximate threats to many primate species, while climate change is expected to exacerbate these threats (habitat and climate change combined henceforth as "global change") and present new challenges. Madagascar's lemurs are earth's most endangered primates, placing added urgency to their conservation in the face of global change. My dissertation focused on the critically endangered golden-crowned sifaka (Propithecus tattersalli; hereafter, "sifaka") which is endemic to fragmented forests across a gradient of dry, moderate, and wet forest types in northeastern Madagascar. I surveyed sifakas across their global range and investigated factors affecting their densities. I explored sifaka diets across different forest types and evaluated if nutritional factors influenced sifaka densities. Lastly, I investigated sifaka range-wide genetic diversity and conducted a connectivity analysis to prioritize corridor-restoration and other potential conservation efforts. Sifaka densities varied widely across forest fragments (6.8 (SE = 2.0-22.8) to 78.1 (SE = 53.1-114.8) sifakas/km²) and populations have declined by as much as 30-43% in 10 years, from ~18,000 to 10,222-12,631 individuals (95% CI: 8,230-15,966). Tree cutting, normalized difference vegetation index (NDVI) during the wet season, and Simpson's diversity index (1-D) predicted sifaka densities range-wide. Sifakas consumed over 101 plant species and spent 27.1% of their active time feeding on buds, flowers, fruits, seeds, and young and mature leaves. Feeding effort and plant part consumption varied by season, forest type, and sex. Minerals in sifaka food items (Mg (β = 0.62, SE = 0.19) and K (β = 0.58, SE = 0.20)) and wet season NDVI (β = 0.43, SE = 0.20) predicted sifaka densities. Genetic measures across forest fragments indicated that sifaka populations are becoming more isolated (moderate FIS values: mean = 0.27, range = 0.11-0.60; high M-ratios: mean = 0.59, range = 0.49-0.82; low overall effective population size: Ne = 139.8-144 sifakas). FST comparisons between fragments (mean = 0.12, range = 0.01-0.30) supported previous findings that sifakas still moved across the fragmented landscape. Further validation of these genetic results is needed. I identified critical corridors that conservation managers could protect and/or expand via active reforestation to ensure the continued existence of this critically-endangered lemur. / Doctor of Philosophy / Worldwide, many species of primates are threatened with extinction due to habitat degradation, hunting, and climate change (habitat and climate combined threats, henceforth, "global change"). These threats work at different time scales, with hunting being the most immediate and climate change likely to have its fullest impact experienced from the present to a longer time frame. Lemurs are a type of primate found only on Madagascar, an island experiencing rapid global change, which puts lemurs at a heightened risk of extinction. My dissertation research focused on the critically endangered golden-crowned sifaka (Propithecus tattersalli; hereafter, "sifaka"), a species of lemur found only in a few isolated forests across a dry to wet gradient in northeastern Madagascar. To better understand their extinction risk, I conducted surveys to estimate the number of sifakas remaining and investigated several factors that might determine how many sifakas can live in one place. I then explored how sifaka diets varied depending on the forest type that they inhabit and tested whether nutrients in their food might determine sifaka numbers. Lastly, I calculated sifaka genetic diversity to assess their ability to adapt to new environmental conditions and to determine whether sifakas can move across the landscape to find new mates and to potentially colonize new areas of habitat. Sifaka densities varied widely across their range (6.8-78.1 sifakas/km² ). Only 10,222-12,631 sifakas remain, which is 30-43% less than the range of estimates obtained 10 years ago (~18,000 sifakas). Tree cutting, normalized difference vegetation index (NDVI; a measure of plant health or "greenness" obtained from satellite data), and a tree species diversity index were useful measures to predict sifaka densities. Sifakas ate different plant parts (buds, flowers, fruits, seeds, and leaves) from over 101 plant species. The amount of time they spent eating each day varied by the time of year, forest type, and sex. On average, they spent a quarter of their day eating. Magnesium and potassium concentrations in sifaka food items also were useful nutrition-related measures to predict sifaka densities. Genetic analyses suggested that sifaka populations are becoming more isolated and inbred, meaning sifakas are breeding with other sifakas to which they are closely related. However, it appears that sifakas still can move between forest patches to find new mates and to potentially colonize new areas, if such areas are created. Further validation of these genetic results is needed. I also identified critical areas that will be important to protect and reforest to ensure that movements between populations can continue.

We will study various quantities related to connectivity of a graph. To this end, we look at Nordhaus-Gaddum type problems, which are problems where the same quantity is studied for a graph $G$ and its complement $G^c$ at the same time. For a graph $G$ on $n$ vertices with normalized Laplacian eigenvalues $0 = \lambda_1(G) \leq \lambda_2(G) \leq \cdots \leq \lambda_n(G)$ and graph complement $G^c$, we prove that \begin{equation*} \max\{\lambda_2(G),\lambda_2(G^c)\}\geq \frac{2}{n^2}. \end{equation*} We do this by way of lower bounding $\max\{i(G), i(G^c)\}$ and $\max\{h(G), h(G^c)\}$ where $i(G)$ and $h(G)$ denote the isoperimetric number and Cheeger constant of $G$, respectively. We also discuss some related Nordhaus-Gaddum questions.

This dissertation thesis deals with the application of the concept of information distance to the social network data analysis. We consider this data as recorded acts of social action. As such, they express certain attitudes, values and intentions. We introduce a formula for calculating the Normalized Social Distance and, based on the series of case studies, we prove the usefulness and validity of this approach. The application of formal mathematical and computer science techniques to massive data records of human action in social network environments is enabled by the change brought by new media and the associated technological advancement. This change is accompanied by a gradual transition of research methods in the humanities, referred to as the onset of digital humanities. This approach is characterized by the application of quantitative methods in the field of humanities and the discovery of new data areas useful for analyses. In case of social media data, the differentiation between quantitative and qualitative methods is no longer valid. A good example is also this thesis, in which information theory specifically combines the methods of a traditional social network analysis and the Goffman's frame analysis of human action. Keywords Information distance, Normalized Social Distance, Kolmogorov...

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Doctor of Philosophy / Department of Biological & Agricultural Engineering / Stacy L. Hutchinson / The Soil Conservation Service-Curve Number (SCS-CN) method is widely used to estimate direct runoff from rainfall events; however, the method does not account for the dynamic rainfall-runoff relationship. This study used back-calculated curve numbers (CNs) and Normalized Difference Vegetation Index (NDVI) to develop NDVI-based CNs (CN[subscript]NDV) using four small northeastern Kansas grassland watersheds with average areas of 1 km² and twelve years (2001–2012) of daily precipitation and runoff data. Analysis indicated that the CN[subscript]NDVI model improved runoff predictions compared to the SCS-CN method. The CN[subscript]NDVI also showed greater variability in CNs, especially during growing season, thereby increasing the model’s ability to estimate relatively accurate runoff from rainfall events since most rainfall occurs during the growing season. The CN[subscript]NDVI model was applied to small, disturbed grassland watersheds to assess the model’s ability to detect land cover change impact for military maneuver damage and large, diverse land use/cover watersheds to assess the impact of scaling up the model. CN[subscript]NDVI application was assessed using a paired watershed study at Fort Riley, Kansas. Paired watersheds were identified through k-means and hierarchical-agglomerative clustering techniques. At the large watershed scale, Daymet precipitation was used to estimate runoff, which was compared to direct runoff extracted from stream flow at gauging points for Chapman (grassland dominated) and Upper Delaware (agriculture dominated) watersheds. In large, diverse watersheds, CN[subscript]NDVI performed better in moderate and overall flow years. Overall, CN[subscript]NDVI more accurately simulated runoff compared to SCS-CN results: The calibrated model increased by 0.91 for every unit increase in observed flow (r = 0.83), while standard CN-based flow increased by 0.506 for every unit increase in observed flow (r = 0.404). Therefore, CN[subscript]NDVI could help identify land use/cover changes and disturbances and spatiotemporal changes in runoff at various scales. CN[subscript]NDVI could also be used to accurately estimate runoff from precipitation events in order to instigate more timely land management decisions.