ScholarWorks@UMassAmherst
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Item A Descriptive Approach to the Class NP(1997-02)Descriptive coraplexity is the study of the expressive power of logical languages. There exists a close relationship between the expressive power of a logical language and the computational complexity of the properties captured by such a language. R. Fagin proved that every sentence in second-order existential logic expresses a problem that can be decided by a non-deterministic polynomial time Turing machine: SOƎ = NP [9]. With this fact as our starting point, we study the sentences that express properties that are complete for the class NP via first-order projections (fops). This type of reductions arises naturally in descriptive complexity, and it is known that all problems complete for NP via fops are FO-isomorphic [1]. Our study of SOƎ sentences includes a normal form for sentences that describe NP-complete properties, the development of syntactic tools for proving problems NP-complete via fops, the definition of syntactic families of problems that have a similar syntactic structure, the study of the approximability of the problems in the syntactic families that we define, and a descriptive version of the POP theorem. This dissertation is organized in seven chapters. In Chapter 1, we present an overview of our research area and results. In Chapter 2, we review the concepts and definitions of descriptive complexity. In Chapter 3 we describe a normal form for SOB sentences that define a NP-complete problems via fops. In Chapter 4 we present syntactic tools to prove problems NP-complete via fops and use them to prove that a large number of the known NP-complete problems remain complete via fops. Among these tools, we define families of problems with similar syntactic structure. In Chapter 5, we study the approximation properties of some of the families defined in Chapter 4. In Chapter 6, we prove a descriptive version of the PCP theorem. This descriptive version implies both the PCP theorem in its computational version and Fagin's theorem. We close this work with Chapter 7 presenting our conclusions and suggestions for future research motivated by this dissertation.Item Bending-Driven Patterning of Objects in Lipid Membranes: Colloidal Assembly in Elastic1 2D Fluids(University of Massachusetts Amherst, 2024-07)Item Primate Bone Histomorphology and its Relationship to Biomechanical Forces(1994)Observations of nonhuman primate long bone microanatomy were undertaken with the explicit goal of recording intra- and interspecific variation in the percentage of osteonal bone. One-hundred and eight specimens with age, sex, body size, and positional behavior data representing 7 species (Galago seneqalensis, Otolemur crassicaudatus, Macaca fascicularis, Macaca mulatta, Macaca arctoides, Erythrocebus patas, and Cercopithecus aethiops) were examined. Humeral and femoral shafts were examined histologically at the proximal, midshaft, and distal cross sections. Furthermore, the humeral and femoral midshaft sections were histologically examined at the anterior, posterior, medial, and lateral quadrants. Each cross section and quadrant were read specifically for percentages of osteonal bone. Data on cortical bone area and cortical thickness were also recorded for each section. Descriptive statistics, analysis of variance, and Model II regression (maximum likelihood estimation) were used to test for intra- and interspecific variation in osteonal bone for this population of primates. In general, several points concerning primate skeletal microanatomy have been revealed: 1) humeral and femoral microanatomy do not mirror each other in osteonal pattern, suggesting that different factors affect the bone microstructure of these limb bones, 2) primate species with different positional behavior express different distributions of osteonal bone and these differences reflect body size and mechanical usage, 3) after sexual maturation age does not affect intraspecific variations in the percentage of osteonal bone, 4) osteon area scales positively allometrically on body size (cortical area) suggesting that the rate of osteon area increases faster than the rate of increase in body size among primate species. Other conclusions have been drawn from this study, the most important point of which is that mechanical loading plays a large role in stimulating secondary osteon formation and that osteonal bone distribution reflects positional behavior and body size. This has profound implications for how future primate histological research can be conducted. The results of this study also have implications for the application of histological analysis to extinct primate specimens. Overall, this information will give primatologists additional important tools to be used in examining skeletal form as it relates to function in both living and extant primate species.Item Code and Sample Data for "Mesoscale Polymer Ribbon Arrays: High Aspect Ratio Surface Structures and their Digital Reconstruction"(2024-07-11)Mesoscale polymer ribbon arrays are high aspect ratio surface structures that exhibit unique three-dimensional morphologies in aqueous environments. To better correlate the impact of processing and aging on the morphology of these and similar high-aspect ratio surface structures, we have developed a MATLAB code capable of extracting quantitative morphological descriptors from 3D confocal microscopy data. This code was used to produce the results outlined in “Mesoscale Polymer Arrays: High Aspect Ratio Surface Structures and Their Digital Reconstruction.” In addition to this code, we have included instructions for data processing and four test datasets. This process first skeletonizes .tif stacks obtained via confocal microscopy. It then builds a model of individual ribbon positions in 3D space. Finally, it calculates key quantitative morphological descriptors such as the radius of curvature and end-to-end distance of these high aspect ratio structures.
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