<p> The object of this dissertation is to investigate the properties, associated boundary conditions and generalized resolvents of symmetric ordinary differential operators associated with formally self-adjoint nth order ordinary differential expressions with complex coefficients. </p> <p> While symmetric differential operators with equal deficiency indices have been studied in some detail, expecially the particular case when the underlying differential expression has real coefficients, little research has been done on the properties of symmetric differential operators with unequal deficiency indices which are associated with a differential expression with complex coefficients. </p> <p> By extending the symmetric differential operators with unequal deficiency indices to suitable operators with equal deficiency indices in larger Hilbert spaces and introducing a new type of boundary conditions to these extensions, we obtain important information about the original symmetric differential operators with unequal deficiency indices. We are able to generate some well-known theorems of I. M. Glazman (1950) and E. A. Coddington (1954) dealing with the characterization of self-adjoint extensions of symmetric operators in terms of boundary conditions, the relation between the deficiency indices of operators on the whole real line and on the half-line, and the resolvent of self-adjoint extensions, from the theory of symmetric, in particular real, differential operators with equal deficiency indices. We also generalize the result of W. N. Everitt (1959) concerning the number of integrable-square solutions of differential equations with one particular and one singular end-point to the case in which both end-points are singular. Finally, under certain assumptions, we extend some of the fundamental results of K. Kodaira (1950) based upon the methods of algebraic geometry, concerning Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expansion with real coefficients to the case of Green's functions and the minimal symmetric differential operator associated with an even-order formally self-adjoint ordinary differential expression with complex coefficients. </p> / Thesis / Doctor of Philosophy (PhD)

Nitrogen (N) is the most critical fertilizer applied nutrient for supporting plant growth. It is a critical part of photosynthesis as a component of chlorophyl, hence it is a key indicator of plant health. In recent years, rapid development of multispectral sensing technology and machine learning (ML) methods make it possible to estimate leaf chemical components such as N for predicting yield spatially and temporally. The objectives of this study were to compare the relationships between canopy reflectance and corn (Zea mays L.) leaf N concentration acquired by two multispectral sensors: red-edge multispectral camera mounted on the Unmanned Aerial Vehicle (UAV) and crop circle ACS-430. Four fertilizer N rates were applied, ranging from deficient to excessivein order to have a broad rangein plant N status. Spectral information was collected at different phenological stages of corn to calculate vegetation indices (VIs) for each stage. Moreover, leaf samples were taken simultaneously to determine N concentration. Different ML methods (Multi-Layer Perceptron (MLP), Support Vector Machines (SVMs), Random Forest regression, Regularized regression models, and Gradient Boosting) were used to estimate leaf N% from VIs and predict yield from VIs. Random Forest Regression was utilized as a feature selection method to choose the best combination of variables for different stages and to interpret the relationships between VIs and corn leaf N concentration and grain yield. The Canopy Chlorophyll Content Index (SCCCI) and Red-edge Ratio Vegetation Index (RERVI) were selected as the most efficient VIs in leaf N estimation and SCCCI, Red-edge chlorophyll index (CIRE), RERVI, Soil Adjusted Vegetation Index (SAVI), and Normalized Difference Vegetation Index (NDVI) were chosen as the most effective VIs in predicting corn grain yield. The results derived from using a red-edge multispectral camera showed that the SCCCI was the most proper index for predicting yield at most of the phenological stages and Gradient Boosting was the best-fitted model to estimate leaf N% with an 80% coefficient of determination. Using a Crop Circle ACS-430 showed that the Support Vector Regression (SVR) model achieved the best performance measures than other models tested in the prediction of leaf N concentration.

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BACKGROUND. Frailty is a syndrome characterized by a decreased resistance to stressors, leading to increased vulnerability to adverse outcomes, including mortality. Multi-morbidity refers to the presence of two or more chronic diseases, and is associated with increased risk of adverse health outcomes. Most of the literature in frailty is based on older people (65+ years) living in high income countries. OBJECTIVE. To compare the predictive ability of three frailty indices for all-cause and one-year mortality among high- (HIC), middle- (MIC), and low- income country (LIC) participants; and to assess the mortality risk associated with multi-morbidity. METHODS. Using data from the Prospective Urban and Rural Epidemiological (PURE) study, we developed three indices using different definitions of frailty (one phenotypic frailty index; two cumulative deficit indices). All indices were tested for predictive ability for mortality both individually and with multi-morbidity. RESULTS. Prevalence of phenotypic frailty was greatest in LIC (8%), intermediate in MIC (7%), and lowest in HIC (4%). Multi-morbidity was most prevalent in HIC (20%), intermediate in MIC (15%), and lowest in LIC (13%). Increased frailty was associated with greater mortality risk using all frailty indices (e.g. HR (95% CI) of 2.63 (2.35-2.95) for the phenotypically frail relative to the robust). At each frailty level, mortality risk was higher within one year of baseline measurement than afterwards, and increased if it was accompanied by concurrent multi-morbidity (e.g. HR of phenotypic frailty increases from 2.27 (1.96-2.62) to 5.08 (4.34-5.95) if accompanied by multi-morbidity). CONCLUSION. All frailty indices predicted mortality. This study is unique in evaluating the prognostic ability of frailty indices in middle-aged adults across HIC, MIC, and LICs. / Thesis / Master of Science (MSc)

This dissertation contains three related papers. The first paper revisits the concept of integrated rural development and provides examples on how to design balanced development work programs for the Trifinio region, a small rural region shared by 3 Central American countries. Work programs should balance 3 development objectives: economic development, social development and environmental protection. Finding a balance between these 3 competing objectives is difficult. The literature of Sustainable Development recognizes that policy makers often fail to balance objectives while the Integrated Rural Development literature points out the challenges of combining the objectives in a manageable project. We argue that, by focusing on identifying sources of economic friction and by accurately measuring tradeoffs using appropriate tools, we can design sound work programs. We present a toolkit that allows policy makers to identify sources of economic friction, measure their drag on the economy, and prioritize these sources so as to reduce the frictions that slow rural development. The toolkit contains 4 tools to assist in program design and 1 for implementation. GIS and building municipal indices of outcomes, household surveys, conjoint analysis and economic field experiments, are the tools that we have applied to design work programs in the Trifinio. In addition, balanced programs must be multi-dimensional in scope so we propose a tool that focuses on the institutional setup required for successful program execution. Finally we make policy recommendations and suggest additional tools that may also be added to our tool kit. In the second paper we create municipal indices of agricultural value of production, personal consumption and poverty in the Trifinio region of Central America with the objective of using them to guide investment priorities. Our indices synthesize information from the complex economic, social and geographic system of this region. In this respect we depart from established practices of estimating indices—for outcomes such as competitiveness—that select factors and create the index by adding them up. The established practice follows a normative approach because the index results from adding factors that should have an impact on the outcome. In this context the index author does not observe the outcome or the impact of factors; and does not know the functional relationship between factors and outcome. The author assumes all the information to create indices. Our methodology follows a positive approach and departs from the established practice because we estimate the outcome and identify factors that have an impact on it. To do it we use household survey and municipal level data to estimate determinants of agricultural value of production, consumption and poverty for the 45 municipalities in the Trifinio region. We then show how to identify municipalities in greatest need, identify factors of greatest impact on the outcome, and identify complementary activities. In addition we use GIS to develop a method that allows for the "generation" of missing agricultural-related data by extrapolating high quality yet limited information from a subsection of the region to the whole. The data generated has been validated in the field by agriculture experts thus confirming the legitimacy of this innovation. Finally we offer policy recommendations. The final paper presents an economic model of group formation with an application to data collected from an agricultural credit program in western Honduras. We formulate a simple theory of group formation using the concept of centers of gravity to explain why individuals join a group. According to our theory, prospective members join based on the potential benefits and costs of group membership, and based on their perception of social distance between themselves and other group members. Social distance is unobservable by outsiders but known by the individual: if you are in then you know who has blue hair. Thus, we argue that social distance helps explain preferences for group formation. To test our theory we analyze data collected from members and non-members of PRODERT, a program that has helped create 188 "Cajas Rurales" (CRs). Using conjoint analysis we test for differences in preferences between members and non-members for the main attributes of the CR. We find that members and non-members exhibit similar preferences for the attributes of the CR; therefore non-membership is not related to supply factors. Using information gathered by executing field experiments, we estimate a proxy for social distance. We use this proxy to run a group formation equation and find that it explains, along with individual characteristics, participation in the CR. Finally we offer suggestions on how to balance performance and coverage in programs in which beneficiaries decide who joins. Small cohesive groups may show exceptional performance at the cost of low coverage, and the opposite may be true. / Ph. D.

Strong tornadoes have impacted the central Appalachian Mountains multiple times in recent years. The topography of this region leads to unique spatial patterns of tornado damage as the tornado vortices pass over ridges in forested areas, and this damage can be detected with vegetation indices derived from remotely sensed imagery. The objectives of this study were to 1) Classify forest damage from the April 19, 2019 EF-3 tornado in Franklin County, VA using remotely-sensed images, 2) Quantify the spatial patterns of forest damage intensity across the path using derived vegetation indices and terrain variables (primarily slope, aspect, elevation, and exposure), and 3) Use regression models to determine if relationships exist among terrain variables along the and forest damage patterns. I generated EVI and NDII vegetation indices from Sentinel-2 imagery and compared the derived damage to the underlying terrain variables. Results revealed that the two vegetation indices were effective for classifying tornado damage, and discrete damage classes aligned well with NWS EF-scale tornado intensity estimations. ANOVA testing suggested that EF-3 equivalent damage was more likely to occur on downslope topography, leeward of the tornado's direction of travel. OLS and geographically weighted regression (GWR) modeling performed poorly, suggesting that an alternative method may be more suitable for modeling, the scale of assessment was inadequate, or that important predictor variables were not captured. Overall, the intensity of the tornado was clearly modified by terrain interactions, and the remote sensing methodology used was effective for reliably identifying and rating damage in forested areas. / Master of Science / Strong tornadoes have impacted the central Appalachian Mountains multiple times in recent years. The topography of this region leads to unique spatial patterns of tornado damage as the tornado vortices pass over ridges in forested areas, and this damage can be detected with vegetation indices derived from remotely sensed imagery. The objectives of this study were to 1) Classify forest damage from the April 19, 2019 EF-3 tornado in Franklin County, VA using remotely-sensed images, 2) Quantify the spatial patterns of forest damage intensity across the path using derived vegetation indices and terrain variables (primarily slope, aspect, elevation, and exposure), and 3) Use regression models to determine if relationships exist among terrain variables along the and forest damage patterns. I generated EVI and NDII vegetation indices from Sentinel-2 imagery and compared the derived damage to the underlying terrain variables. Results revealed that the two vegetation indices were effective for classifying tornado damage, and discrete damage classes aligned well with NWS EF-scale tornado intensity estimations. ANOVA testing suggested that EF-3 equivalent damage was more likely to occur on downslope topography, leeward of the tornado's direction of travel. OLS and geographically weighted regression (GWR) modeling performed poorly, suggesting that an alternative method may be more suitable for modeling, the scale of assessment was inadequate, or that important predictor variables were not captured. Overall, the intensity of the tornado was clearly modified by terrain interactions, and the remote sensing methodology used was effective for reliably identifying and rating damage in forested areas.

A recent trend in education has been toward formative assessments to enable teachers, parents, and administrators assist students succeed. Cognitive diagnostic modeling (CDM) has the potential to provide valuable information for stakeholders to assist students identify their skill deficiency in specific academic subjects. Cognitive diagnosis models are mainly viewed as a family of latent class confirmatory probabilistic models. These models allow the mapping of students' skill profiles/academic ability. Using a complex simulation studies, the methodological issues in one of the existing cognitive models, referred to as compensatory reparameterized unified model (CRUM) under the log-linear model family of CDM, was investigated. In order for practitioners to implement these models, their item parameter recovery and examinees' classifications need to be studied in detail. A series of complex simulated data were generated for investigation with the following designs: three attributes with seven items, three attributes with thirty five items, four attributes with fifteen items, and five attributes with thirty one items. Each dataset was generated with observations of: 50, 100, 500, 1,000, 5,000, and 10,000 examinees. The first manuscript is the report of the investigation of how accurately CRUM could recover item parameters and classify examinees under true QMattrix specification and various research designs. The results suggested that the test length with regards to number of attributes and sample size affects the item parameter recovery and examinees classification accuracy. The second manuscript is the report of the investigation of the sensitivity of relative fit indices in detecting misfit for over- and opposite-Q-Matrix misspecifications. The relative fit indices under investigation were Akaike information criterion (AIC), Bayesian information criterion (BIC), and sample size adjusted Bayesian information criterion (ssaBIC). The results suggested that the CRUM can be a robust model given the consideration to the observation number and item/attribute combinations. The findings of this dissertation fill some of the existing gaps in the methodological issues regarding cognitive models' applicability and generalizability. It helps practitioners design tests in CDM framework in order to attain reliable and valid results. / Ph. D.