We investigate the steady state solutions that can exist for a two dimensional swarm of biological organisms, which have pairwise social interaction forces. The three steady states we investigate using a continuum model are a ribbon migrating swarm, a circular migrating swarm, and a milling swarm. We solve these numerically by reformulating the integral equation that arises from the continuum model as an energy minimization problem. For the ribbon migrating solution, we are able to determine an analytic solution from Carleman's equation which arises after an asymptotic expansion of the social interaction potential. Using this technique we are able to show the existence of a square root singularity that emerges at the boundary of the compactly supported swarm. The analytic solution agrees with the numerical solution for certain parameter values in the social interaction potential. We then demonstrate the existence of solutions for a migrating and milling circular swarm which contain a square root singularity. The milling swarm looks similar to the infinite ribbon, so we are able to use an asymptotic expansion of the potential to obtain an analytic solution in this case as well. The singularities in the density of the swarm suggest that the Morse potential should not be used for modeling biological swarming.

This dissertation was to study computational theory and methods for ?nding multiple saddle critical points in Banach spaces. Two local minimax methods were developed for this purpose. One was for unconstrained cases and the other was for constrained cases. First, two local minmax characterization of saddle critical points in Banach spaces were established. Based on these two local minmax characterizations, two local minimax algorithms were designed. Their ?ow charts were presented. Then convergence analysis of the algorithms were carried out. Under certain assumptions, a subsequence convergence and a point-to-set convergence were obtained. Furthermore, a relation between the convergence rates of the functional value sequence and corresponding gradient sequence was derived. Techniques to implement the algorithms were discussed. In numerical experiments, those techniques have been successfully implemented to solve for multiple solutions of several quasilinear elliptic boundary value problems and multiple eigenpairs of the well known nonlinear p-Laplacian operator. Numerical solutions were presented by their pro?les for visualization. Several interesting phenomena of the solutions of quasilinear elliptic boundary value problems and the eigenpairs of the p-Laplacian operator have been observed and are open for further investigation. As a generalization of the above results, nonsmooth critical points were considered for locally Lipschitz continuous functionals. A local minmax characterization of nonsmooth saddle critical points was also established. To establish its version in Banach spaces, a new notion, pseudo-generalized-gradient has to be introduced. Based on the characterization, a local minimax algorithm for ?nding multiple nonsmooth saddle critical points was proposed for further study.

With the advent of large scale genome sequencing efforts along with more sophisticated methods of genetic mapping, a number of loci have been identified that are associated with human diseases. Intriguingly, many genes identified in these loci remain uncharacterized. Although current annotation can provide a prediction of putative function of some of these proteins at a biochemical level, understanding their cellular roles require analysis at a single gene level. Bioinformatic analysis carried out in the laboratory during studies on cyclic nucleotide metabolism in mycobacteria identified putative Class III cyclic nucleotide phosphodiesterases (Class III cNMP PDEs) from the non-redundant database of proteins. One of the proteins identified was the Rv0805 gene product from Mycobacterium tuberculosis. Detailed biochemical characterization of this protein revealed that Rv0805 is indeed a phosphodiesterase (PDE) and could hydrolyze 3’, 5’-cyclic adenosine monophosphate (cAMP) as well as 3’, 5’-cyclic guanosine monophosphate (cGMP). Structural analysis of Rv0805 revealed a metallophosphoesterase (MPE) like fold and presence of two metal atoms at the binuclear metal centre of the protein. Moreover, overexpression of Rv0805 in E. coli and M. smegmatis reduced intracellular cAMP levels indicating that it possesses cAMP PDE activity in vivo. The majority of proteins identified in this bioinformatic analysis were of bacterial or archaebacterial in origin but it was interesting to find some mammalian proteins, since, till date, no Class III cNMP PDE has been found in higher eukaryotes. Interestingly, two genes were identified in the human genome. These genes, 239FB and 239AB, are expressed in the fetal brain and adult brain, respectively and have been annotated as metallophosphoesterases but there has been no biochemical or functional characterization of these proteins. The 239FB gene is present between the FSHB and PAX6 genes on chromosome 11. This gene locus is present within a deletion interval (11p13-14) that is associated with the mental retardation phenotype of WAGR syndrome (Wilms’ tumor, aniridia, genitourinary anomalies, mental retardation). Inspection of available sequenced mammalian genomes indicated a shared synteny of the genes in the WAGR locus, highlighting it’s evolutionary conservation. Most interestingly, nucleotide sequences within the WAGR locus (which include the 5 genes WT1, PAX6, RCN1, ELP4 and 239FB) are amongst the 481 ultra conserved regions of the human genome. Moreover, 239FB is one of only 24 instances where an ortholog of an ultra-conserved element could be partially traced back by sequence similarity in lower eukaryotes such as Ciona intestinalis, Drosophila melanogaster, or Caenorhabditis elegans. Although the function of the 239FB protein is unknown so far, the distinctive expression of the gene in the fetal brain and the presence of an “ancient conserved region” in this gene suggest that this gene may be vital for the development of the nervous system. The work carried out in this thesis has attempted to understand the physiological functions of the 239FB/AB gene family. Amino acid sequence comparison revealed two amino acids changes between the human and rat proteins indicating the extra-ordinary sequence conservation of these proteins. Therefore, to characterize the biochemical properties of 239FB and 239AB proteins, rat proteins were used as model enzymes. Reverse transcription-PCR analysis of RNA prepared from the fetal and adult rat brains as well as Western blot analysis on cytosolic fractions of rat brains from various developmental stages indicated that 239FB is predominantly expressed in fetal brain. Detailed biochemical analyses of the rat 239FB and 239AB proteins were performed which showed that they possess metallophosphodiesterase activity. 239FB showed activity only in the presence of Mn2+ and Co2+ as the added metal cofactors. Surprisingly, the Km for Mn2+ of 239FB was found to be 1.5 mM, which is nearly 60-fold higher than that of its mycobacterial ortholog, Rv0805. A systematic mutational analysis was performed to characterize the residues that are involved in binding either one or both the metals found in the catalytic site of 239FB. Although 239FB shares some of the residues that have been shown to be essential for metal binding and catalytic activity with other MPEs including Rv0805, there are some differences as well. One histidine residue that has been conserved in other MPEs and has been shown to be important for metal binding is replaced by glycine (Gly-252) in 239FB. To study the consequence of replacing the glycine with a histidine in 239FB, a 239FBGly252His mutant protein was generated and characterized. Interestingly, the single mutation of Gly-252 to a histidine residue not only increased the affinity of the protein for metals but increased catalytic activity as well with various phosphodiesters. Moreover, 239FBGly252His mutant protein showed significant activity with cAMP and cGMP which were not hydrolysed by wild type 239FB. Interestingly, in the 239AB protein, histidine 284 is present at a position equivalent to Gly-252 in the 239FB protein. Biochemical characterization of 239AB showed 2’, 3’-cAMP hydrolyzing activity similar to 239FBGly252His mutant protein. A rat 239FB protein with a mutation (His67Arg) corresponding to a single nucleotide polymorphism seen in human 239FB, led to complete inactivation of the protein. The occurrence of this SNP at a very low frequency and only as a heterozygous condition suggests that a complete loss-of-function mutation of 239FB in human populations cannot be tolerated. To gain insights into the function of 239FB in its physiological milieu, yeast two-hybrid screening was performed with 239FB using human fetal brain cDNA library. Dipeptidyl peptidase III, a zinc dependent metallopeptidase, was found as an interacting partner of 239FB in this analysis and the functional consequences of this interaction would be an interesting area of study in future. While a number of metallophosphoesterases have been characterized biochemically and structurally, their biological role(s) and in vivo substrate(s) remain elusive. In order to elucidate the physiological role of 239FB/AB family, the ortholog of 239FB/AB in D. melanogaster was characterized. Sequence comparison of Drosophila ortholog with both the mammalian proteins indicated that it may be an ortholog of 239AB and hence, it was named as d239AB. Enhancer-promoter analysis with a putative promoter region of the d239AB indicated the expression of d239AB in the mushroom bodies in brain and in enterocytes in mid gut. Characterization of a Drosophila line, BS#16242, with a piggybac element inserted in the intron of d239AB showed disruption of d239AB expression. This suggested that BS#16242 line can serve as a d239AB knockout line and hence, was selected for further phenotypic characterization to unravel the physiological roles of d239AB. Though, BS#16242 flies did not show any developmental defects, a severe reduction in the fecundity of these files was observed. Further analysis revealed defective ovulation as a probable reason for reduced fecundity of these flies. In addition to compromised fecundity, BS#16242 flies showed a significant reduction in the life span of male as well as female flies. Moreover, these flies showed less resistance to thermal stress and desiccation. Most interestingly, all these phenotypes were rescued upon neuronal expression of the d239AB transgene in BS#16242 flies indicating that neuronal function of d239AB is important for diverse physiological processes. The phenotypes observed in BS#16242 flies mimic the physiological state under increased insulin signaling, such as decrease in life span, and susceptibility to various stress conditions suggesting that d239AB could play a role in the insulin signaling pathway. Interestingly, overexpression of d239AB transgene in neurons reduced cAMP levels in the brains of Drosophila, indicating that the protein may have cAMP phosphodiesterase activity in vivo. This is the first analysis of the presence of a Class III phosphodiesterase in eukaryotes. Thus, d239AB mediated regulation of cAMP levels in a particular subsets of cells, such as neurons, could also be one of the molecular mechanisms responsible for reduced fecundity and longevity of BS#16242 flies. Interacting partners of d239AB were inspected in the Drosophila interactome (built on protein-protein interactions identified using a yeast two-hybrid approach). Strikingly, most of the d239AB interacting proteins were involved either in transcriptional or translational regulation indicating that d239AB could be involved in the regulation of expression of genes involved in diverse physiological processes. This could explain why disruption of d239AB led to various physiological defects such as reduced fecundity, decreased life span and compromised fitness. In summary, studies described in this thesis suggest that 239FB and 239AB proteins are the first Class III cyclic nucleotide phosphodiesterases reported in eukaryotes. Results shown here suggest the critical role of their ortholog in the physiology of Drosophila. Further genetic manipulation in D. melanogaster and other organisms which harbor orthologs of the 239FB/AB gene could throw light on the diverse biological roles of these enzymes in humans.

Diffusion is a phenomenon in which particles move from regions of higher density to regions of lower density. Many physical systems, in fields as diverse as plant biology and finance, are known to involve diffusion phenomena. Typically, diffusion systems are modeled by partial differential equations (PDEs), which include certain parameters. These parameters characterize a given diffusion system. Therefore, for both modeling and simulation of a diffusion system, one has to either know or determine these parameters. Moreover, as PDEs are infinite order dynamic systems, for computational purposes one has to approximate them by a finite order model. In this thesis, we investigate these two issues of model reduction and parameter estimation by considering certain specific cases of heat diffusion systems. We first address model reduction by considering two specific cases of heat diffusion systems. The first case is a one-dimensional heat diffusion across a homogeneous wall, and the second case is a two-dimensional heat diffusion across a homogeneous rectangular plate. In the one-dimensional case we construct finite order approximations by using some well known PDE solvers and evaluate their effectiveness in approximating the true system. We also construct certain other alternative approximations for the one-dimensional diffusion system by exploiting the different modal structures inherently present in it. For the two-dimensional heat diffusion system, we construct finite order approximations first using the standard finite difference approximation (FD) scheme, and then refine the FD approximation by using its asymptotic limit. As for parameter estimation, we consider the same one-dimensional heat diffusion system, as in model reduction. We estimate the parameters involved, first using the standard batch estimation technique. The convergence of the estimates are investigated both numerically and theoretically. We also estimate the parameters of the one-dimensional heat diffusion system recursively, initially by adopting the standard recursive prediction error method (RPEM), and later by using two different recursive algorithms devised in the frequency domain. The convergence of the frequency domain recursive estimates is also investigated.

In this thesis, the relationship between several high-latitude phenomena and the ionospheric conductance in both hemispheres is studied theoretically and experimentally. </p>Theoretically, the high-latitude electrodynamics is studied by considering currents in the magnetosphere-ionosphere system resulting from the ionospheric sheet current redistribution between the conjugate ionospheres. It is shown that strong flow between the conjugate ionospheres, the interhemispheric currents (IHC), can be set up if the conductance distribution is asymmetric in the conjugate ionospheric regions. Such conditions are typical for solstices owing to the differences in the solar illumination. Analytical and numerical modeling shows that IHCs can appear in the regions of strong conductance gradient, more specifically around the solar terminator line, and that the intensity of the IHCs can be comparable to the intensity of the well known Region 1/Region 2 currents. The effect of IHC excitation on observable magnetic perturbations on the ground is investigated. It is shown that in the vicinity of the solar terminator line, the pattern of magnetic perturbation can be such that an apparent equivalent current vortex can be detected. In addition, strong conductance gradients are shown to affect significantly the quality of the ionospheric plasma flow estimates from the ground-based magnetometer data. </p>Experimentally, the effect of the nightside ionospheric conductance on occurrence of substorms, global storm sudden commencement and radar auroras is investigated. To characterize substorm occurrence, new parameters, the derivatives of the classical AE and AO indices, are introduced. It is shown that the seasonal and diurnal variations of these parameters are controlled by the total nightside ionospheric conductance in the conjugate regions. The substorm onsets preferentially occur at low levels of the total conductance, which is consistent with the idea of the substorm triggering through the magnetosphere-ionosphere feedback instability. It is hypothesized that the total conductance affects the global storm onsets as well. To check this idea, the 33-year sudden storm commencement (SSC) data are considered. The semiannual, annual, semidiurnal, and diurnal variations in the SSC occurrence rate are found to be significant and these components exhibit a strong relationship with the total conductance of the high-latitude ionospheres. Finally, the SuperDARN midnight echo occurrence is shown to correlate, for some radars, with the total conductance minima and presumably with electric field maxima, which is consistent with general expectation that the F-region irregularities occur preferentially during times of enhanced electric fields. The gradients of the high-latitude conductance can also lead to significant errors in the plasma convection estimates from the ground-based magnetometers, and to investigate this effect a statistical assessment of the difference between the true plasma convection (SuperDARN) and the magnetometer-inferred equivalent convection direction is performed. The largest differences are found for the transition region between the dark and sunlit ionospheres and in the midnight sector where strong conductance gradients are expected due to particle precipitation. Consideration of regular conductance gradients due to solar illumination improves the agreement between the radar and magnetometer data. Finally, an attempt is made to demonstrate the effects of conductance upon the properties of traveling convection vortices (TCVs). Joint SuperDARN and magnetometer data reveal that there is resemblance between the magnetometer and radar inferred TCV images on a scale of thousands of kilometers. However, on a smaller scale of hundreds of kilometers, significant differences are observed.

Many optimal stochastic control problems in finance can be formulated in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this thesis, a general framework for solutions of HJB PDEs in finance is developed, with application to asset allocation. The numerical scheme has the following properties: it is unconditionally stable; convergence to the viscosity solution is guaranteed; there are no restrictions on the underlying stochastic process; it can be easily extended to include features as needed such as uncertain volatility and transaction costs; and central differencing is used as much as possible so that use of a locally second order method is maximized. In this thesis, continuous time mean variance type strategies for dynamic asset allocation problems are studied. Three mean variance type strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, are investigated. The numerical method can handle various constraints on the control policy. The following cases are studied: allowing bankruptcy (unconstrained case), no bankruptcy, and bounded control. In some special cases where analytic solutions are available, the numerical results agree with the analytic solutions. These three mean variance type strategies are compared. For the allowing bankruptcy case, analytic solutions exist for all strategies. However, when additional constraints are applied to the control policy, analytic solutions do not exist for all strategies. After realistic constraints are applied, the efficient frontiers for all three strategies are very similar. However, the investment policies are quite different. These results show that, in deciding which objective function is appropriate for a given economic problem, it is not sufficient to simply examine the efficient frontiers. Instead, the actual investment policies need to be studied in order to determine if a particular strategy is applicable to specific investment problem.

Many optimal stochastic control problems in finance can be formulated in the form of Hamilton-Jacobi-Bellman (HJB) partial differential equations (PDEs). In this thesis, a general framework for solutions of HJB PDEs in finance is developed, with application to asset allocation. The numerical scheme has the following properties: it is unconditionally stable; convergence to the viscosity solution is guaranteed; there are no restrictions on the underlying stochastic process; it can be easily extended to include features as needed such as uncertain volatility and transaction costs; and central differencing is used as much as possible so that use of a locally second order method is maximized. In this thesis, continuous time mean variance type strategies for dynamic asset allocation problems are studied. Three mean variance type strategies: pre-commitment mean variance, time-consistent mean variance, and mean quadratic variation, are investigated. The numerical method can handle various constraints on the control policy. The following cases are studied: allowing bankruptcy (unconstrained case), no bankruptcy, and bounded control. In some special cases where analytic solutions are available, the numerical results agree with the analytic solutions. These three mean variance type strategies are compared. For the allowing bankruptcy case, analytic solutions exist for all strategies. However, when additional constraints are applied to the control policy, analytic solutions do not exist for all strategies. After realistic constraints are applied, the efficient frontiers for all three strategies are very similar. However, the investment policies are quite different. These results show that, in deciding which objective function is appropriate for a given economic problem, it is not sufficient to simply examine the efficient frontiers. Instead, the actual investment policies need to be studied in order to determine if a particular strategy is applicable to specific investment problem.

In this thesis, the relationship between several high-latitude phenomena and the ionospheric conductance in both hemispheres is studied theoretically and experimentally. </p>Theoretically, the high-latitude electrodynamics is studied by considering currents in the magnetosphere-ionosphere system resulting from the ionospheric sheet current redistribution between the conjugate ionospheres. It is shown that strong flow between the conjugate ionospheres, the interhemispheric currents (IHC), can be set up if the conductance distribution is asymmetric in the conjugate ionospheric regions. Such conditions are typical for solstices owing to the differences in the solar illumination. Analytical and numerical modeling shows that IHCs can appear in the regions of strong conductance gradient, more specifically around the solar terminator line, and that the intensity of the IHCs can be comparable to the intensity of the well known Region 1/Region 2 currents. The effect of IHC excitation on observable magnetic perturbations on the ground is investigated. It is shown that in the vicinity of the solar terminator line, the pattern of magnetic perturbation can be such that an apparent equivalent current vortex can be detected. In addition, strong conductance gradients are shown to affect significantly the quality of the ionospheric plasma flow estimates from the ground-based magnetometer data. </p>Experimentally, the effect of the nightside ionospheric conductance on occurrence of substorms, global storm sudden commencement and radar auroras is investigated. To characterize substorm occurrence, new parameters, the derivatives of the classical AE and AO indices, are introduced. It is shown that the seasonal and diurnal variations of these parameters are controlled by the total nightside ionospheric conductance in the conjugate regions. The substorm onsets preferentially occur at low levels of the total conductance, which is consistent with the idea of the substorm triggering through the magnetosphere-ionosphere feedback instability. It is hypothesized that the total conductance affects the global storm onsets as well. To check this idea, the 33-year sudden storm commencement (SSC) data are considered. The semiannual, annual, semidiurnal, and diurnal variations in the SSC occurrence rate are found to be significant and these components exhibit a strong relationship with the total conductance of the high-latitude ionospheres. Finally, the SuperDARN midnight echo occurrence is shown to correlate, for some radars, with the total conductance minima and presumably with electric field maxima, which is consistent with general expectation that the F-region irregularities occur preferentially during times of enhanced electric fields. The gradients of the high-latitude conductance can also lead to significant errors in the plasma convection estimates from the ground-based magnetometers, and to investigate this effect a statistical assessment of the difference between the true plasma convection (SuperDARN) and the magnetometer-inferred equivalent convection direction is performed. The largest differences are found for the transition region between the dark and sunlit ionospheres and in the midnight sector where strong conductance gradients are expected due to particle precipitation. Consideration of regular conductance gradients due to solar illumination improves the agreement between the radar and magnetometer data. Finally, an attempt is made to demonstrate the effects of conductance upon the properties of traveling convection vortices (TCVs). Joint SuperDARN and magnetometer data reveal that there is resemblance between the magnetometer and radar inferred TCV images on a scale of thousands of kilometers. However, on a smaller scale of hundreds of kilometers, significant differences are observed.

Pulsed Detonation Engines (PDEs) have the potential to revolutionize fight by better utilizing the chemical energy content of reactive fuel/air mixtures over conventional combustion processes. Combustion by a super-sonic detonation wave results in a significant increase in pressure in addition to an increase in temperature. In order to harness this pressure increase and achieve a high power density, it is desirable to operate PDEs at high frequency. The process of detonation initiation impacts operating frequency by dictating the length of the chamber and contributing to the overall cycle time. Therefore a key challenge in the development of a practical PDEs is the requirement to rapidly initiate a detonation in hydrocarbon-air mixtures. This thesis evaluates the influence of spark energy and airflow velocity on this challenging initiation process. The influence of spark energy, number of sparks and airflow velocity on Deflagration-to-Detonation Transition (DDT) was studied during cyclic operation of a small-scale PDE at the General Electric Global Research Center. Experiments were conducted in a 50 mm square transitioning to cylindrical channel PDE with optical access operating with stoichiometric ethylene-air mixture. Total spark energy was varied from 250 mJ to 4 J and was distributed between one and four spark plugs located in the same axial location. Initial flame acceleration was imaged using high-speed shadowgraph and was characterized by the time to reach 20 cm from the spark plug. Measurements of detonation wave velocity and emergence time, the time it takes the detonation wave to exit the tube, was measured using dynamic pressure transducers and ionization probes. It was found that the flame front spread was faster at higher spark energies and with more spark locations. Initial flame acceleration was 16% faster for the 4-spark, 4 J case when compared to the baseline 1-spark, 1 J case. When looking at the effect of airflow on the influence of spark energy, it was found that airflow had a larger effect on emergence time at high energies, versus energies less than 1 J. Finally, for a selected case of 0.25 J spark energy and 4 sparks, the velocity of the fuel-air mixture during fill was found to have a varying influence on detonation initiation and emergence time.

The prevalent view in computer vision since Marr is that visual perception is a data-driven bottom-up process. In this view, image data is processed in a feed-forward fashion where a sequence of independent visual modules transforms simple low-level cues into more complex abstract perceptual units. Over the years, a variety of techniques has been developed using this paradigm. Yet an important realization is that low-level visual cues are generally so ambiguous that they could make purely bottom-up methods quite unsuccessful. These ambiguities cannot be resolved without taking account of high-level contextual information. In this thesis, we explore different ways of enriching early and mid-level computer vision modules with a capacity to extract and use contextual knowledge. Mainly, we integrate low-level image features with contextual information within uni&amp / #64257 / ed formulations where bottom-up and top-down processing take place simultaneously.