# Department of Mathematics

## Pages

2-stratifold Spines Of Closed 3-manifolds
2-stratifolds are a generalization of 2-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed 3-manifolds that have a 2-stratifold as a spine.
An Efficient And Long-time Accurate Third-order Algorithm For The Stokes-darcy System
A third-order in time numerical IMEX-type algorithm for the Stokes-Darcy system for flows in fluid saturated karst aquifers is proposed and analyzed. A novel third-order Adams-Moulton scheme is used for the discretization of the dissipative term whereas a third-order explicit Adams-Bashforth scheme is used for the time discretization of the interface term that couples the Stokes and Darcy components. The scheme is efficient in the sense that one needs to solve, at each time step, decoupled Stokes and Darcy problems. Therefore, legacy Stokes and Darcy solvers can be applied in parallel. The scheme is also unconditionally stable and, with a mild time-step restriction, long-time accurate in the sense that the error is bounded uniformly in time. Numerical experiments are used to illustrate the theoretical results. To the authors' knowledge, the novel algorithm is the first third-order accurate numerical scheme for the Stokes-Darcy system possessing its favorable efficiency, stability, and accuracy properties., Keywords: beavers, coupling fluid-flow, domain decomposition methods, explicit, implicit, interface boundary-condition, joseph, model, porous-media flow, surface water flows, Publication Note: The publisher's version of record is available at https://doi.org/10.1007/s00211-015-0789-3
Analytical Results on the Role of Flexibility in Flapping Propulsion
Wing or fin flexibility can dramatically affect the performance of flying and swimming animals. Both laboratory experiments and numerical simulations have been used to study these effects, but analytical results are notably lacking. Here, we develop small-amplitude theory to model a flapping wing that pitches passively due to a combination of wing compliance, inertia and fluid forces. Remarkably, we obtain a class of exact solutions describing the wing's emergent pitching motions, along with expressions for how thrust and efficiency are modified by compliance. The solutions recover a range of realistic behaviours and shed new light on how flexibility can aid performance, the importance of resonance, and the separate roles played by wing and fluid inertia. The simple robust estimates afforded by our theory may prove valuable even in situations where details of the flapping motion and wing geometry differ., Keywords: propulsion, swimming/flying, vortex shedding, Note: This is the author's accepted manuscript as accepted for publication. Minor changes may have occurred between this and the version of record, which may be found at: http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=9359446&fileId=S0022112014005333, Citation: Moore, M. Nicholas J. (2014). Analytical results on the role of flexibility in flapping propulsion. Journal of Fluid Mechanics, 757, pp 599-612 doi:10.1017/jfm.2014.533
Automatic stage identification of Drosophila egg chamber based on DAPI images
The Drosophila egg chamber, whose development is divided into 14 stages, is a well-established model for developmental biology. However, visual stage determination can be a tedious, subjective and time-consuming task prone to errors. Our study presents an objective, reliable and repeatable automated method for quantifying cell features and classifying egg chamber stages based on DAPI images. The proposed approach is composed of two steps: 1) a feature extraction step and 2) a statistical modeling step. The egg chamber features used are egg chamber size, oocyte size, egg chamber ratio and distribution of follicle cells. Methods for determining the on-site of the polytene stage and centripetal migration are also discussed. The statistical model uses linear and ordinal regression to explore the stage-feature relationships and classify egg chamber stages. Combined with machine learning, our method has great potential to enable discovery of hidden developmental mechanisms., Keywords: endocycle, follicle cell-differentiation, melanogaster, morphogenesis, notch pathway, oogenesis, pattern-formation, polarity, Proliferation, watershed segmentation, Publication Note: The publisher’s version of record is available at http://www.dx.doi.org/10.1038/srep18850
Automatic stage identification of Drosophila egg chamber based on DAPI images.
The Drosophila egg chamber, whose development is divided into 14 stages, is a well-established model for developmental biology. However, visual stage determination can be a tedious, subjective and time-consuming task prone to errors. Our study presents an objective, reliable and repeatable automated method for quantifying cell features and classifying egg chamber stages based on DAPI images. The proposed approach is composed of two steps: 1) a feature extraction step and 2) a statistical modeling step. The egg chamber features used are egg chamber size, oocyte size, egg chamber ratio and distribution of follicle cells. Methods for determining the on-site of the polytene stage and centripetal migration are also discussed. The statistical model uses linear and ordinal regression to explore the stage-feature relationships and classify egg chamber stages. Combined with machine learning, our method has great potential to enable discovery of hidden developmental mechanisms., Grant Number: R01 GM072562, R01GM072562, Publication Note: This NIH-funded author manuscript originally appeared in PubMed Central at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4702167.
Axisymmetric Squirmers In Stokes Fluid With Nonuniform Viscosity
The ciliary locomotion and feeding of an axisymmetric microswimmer in a complex fluid whose viscosity depends on a surrounding nutrient field are investigated numerically in order to extend previous asymptotic results for weak nutrient-viscosity coupling. Numerical simulations capture nonlinearities inherent in the full system that are missed using perturbation-based linearization methods. The microswimmer's ciliary beating is modeled by a slip velocity, i.e., the squirmer model, and body geometry is modeled by spheroids. It is found that swimming speed and feeding are most affected by a nonuniform viscosity environment when the ratio of advection forces to diffusion transport, characterized by the nondimensional Peclet number, is moderate, i.e., Pe = O(5). These changes are correlated to significant increases in the pressure force on the surface of the squirmer, as well as differences in power expenditure and hydrodynamic efficiency compared to the constant-viscosity case. Additionally, the swimming and feeding changes are found to be more significant in oblate spheroids than prolate spheroids, although the shape has a smaller effect on performance than Peclet number or surface stroke. These results suggest that nonlocal effects from viscosity variation are caused by a modification to the pressure force, as opposed to the strain rate. These results should be useful in interpreting experiments where a microswimmer affects a fluid's local rheology., model, driven, algae, aqueous-solutions, flagella, motility, mucin, organisms, propulsion, temperatures, The publisher's version of record is availible at https://doi.org/10.1103/PhysRevFluids.5.063102
CAM Stochastic Volatility Model for Option Pricing
The coupled additive and multiplicative (CAM) noises model is a stochastic volatility model for derivative pricing. Unlike the other stochastic volatility models in the literature, the CAM model uses two Brownian motions, one multiplicative and one additive, to model the volatility process. We provide empirical evidence that suggests a nontrivial relationship between the kurtosis and skewness of asset prices and that the CAM model is able to capture this relationship, whereas the traditional stochastic volatility models cannot. We introduce a control variate method and Monte Carlo estimators for some of the sensitivities (Greeks) of the model. We also derive an approximation for the characteristic function of the model., Publication Note: The publisher’s version of record is available at http://www.dx.doi.org/10.1155/2016/5496945
Ca2+ Effects on ATP Production and Consumption Have Regulatory Roles on Oscillatory Islet Activity.
Pancreatic islets respond to elevated blood glucose by secreting pulses of insulin that parallel oscillations in β-cell metabolism, intracellular Ca(2+) concentration, and bursting electrical activity. The mechanisms that maintain an oscillatory response are not fully understood, yet several models have been proposed. Only some can account for experiments supporting that metabolism is intrinsically oscillatory in β-cells. The dual oscillator model (DOM) implicates glycolysis as the source of oscillatory metabolism. In the companion article, we use recently developed biosensors to confirm that glycolysis is oscillatory and further elucidate the coordination of metabolic and electrical signals in the insulin secretory pathway. In this report, we modify the DOM by incorporating an established link between metabolism and intracellular Ca(2+) to reconcile model predictions with experimental observations from the companion article. With modification, we maintain the distinguishing feature of the DOM, oscillatory glycolysis, but introduce the ability of Ca(2+) influx to reshape glycolytic oscillations by promoting glycolytic efflux. We use the modified model to explain measurements from the companion article and from previously published experiments with islets., Grant Number: K01-DK101683, R01 DK046409, K01 DK101683, , P30 DK020572, R01-DK46409, Publication Note: This NIH-funded author manuscript originally appeared in PubMed Central at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4744176.
Computing Eigenfunctions And Eigenvalues Of Boundary-value Problems With The Orthogonal Spectral Renormalization Method
The spectral renormalization method was introduced in 2005 as an effective way to compute ground states of nonlinear Schrodinger and Gross-Pitaevskii type equations. In this paper, we introduce an orthogonal spectral renormalization (OSR) method to compute ground and excited states (and their respective eigenvalues) of linear and nonlinear eigenvalue problems. The implementation of the algorithm follows four simple steps: (i) reformulate the underlying eigenvalue problem as a fixed-point equation, (ii) introduce a renormalization factor that controls the convergence properties of the iteration, (iii) perform a Gram-Schmidt orthogonalization process in order to prevent the iteration from converging to an unwanted mode, and (iv) compute the solution sought using a fixed-point iteration. The advantages of the OSR scheme over other known methods (such as Newton's and self-consistency) are (i) it allows the flexibility to choose large varieties of initial guesses without diverging, (ii) it is easy to implement especially at higher dimensions, and (iii) it can easily handle problems with complex and random potentials. The OSR method is implemented on benchmark Hermitian linear and nonlinear eigenvalue problems as well as linear and nonlinear non-Hermitian PT-symmetric models., Keywords: states, symmetric quantum-mechanics, Publication Note: The publisher’s version of record is available at https://doi.org/10.1103/PhysRevA.97.032134
Confidence Building Exercise In Data And Identifiability
Mathematical modeling has a long history in the field of cancer therapeutics, and there is increasing recognition that it can help uncover the mechanisms that underlie tumor response to treatment. However, making quantitative predictions with such models often requires parameter estimation from data, raising questions of parameter identifiability and estimability. Even in the case of structural (theoretical) identifiability, imperfect data and the resulting practical unidentifiability of model parameters can make it difficult to infer the desired information, and in some cases, to yield biologically correct inferences and predictions. Here, we examine parameter identifiability and estimability using a case study of two compartmental, ordinary differential equation models of cancer treatment with drugs that are cell cycle-specific (taxol) as well as non-specific (oxaliplatin). We proceed through model building, structural identifiability analysis, parameter estimation, practical identifiability analysis and its biological implications, as well as alternative data collection protocols and experimental designs that render the model identifiable. We use the differential algebra/input-output relationship approach for structural identifiability, and primarily the profile likelihood approach for practical identifiability. Despite the models being structurally identifiable, we show that without consideration of practical identifiability, incorrect cell cycle distributions can be inferred, that would result in suboptimal therapeutic choices. We illustrate the usefulness of estimating practically identifiable combinations (in addition to the more typically considered structurally identifiable combinations) in generating biologically meaningful insights. We also use simulated data to evaluate how the practical identifiability of the model would change under alternative experimental designs. These results highlight the importance of understanding the underlying mechanisms rather than purely using parsimony or information criteria/goodness-of-fit to decide model selection questions. The overall roadmap for identifiability testing laid out here can be used to help provide mechanistic insight into complex biological phenomena, reduce experimental costs, and optimize model-driven experimentation. (C) 2017 Published by Elsevier Ltd., Keywords: systems, cells, Cancer, Chemotherapy model, colorectal-cancer, Compartmental models, global identifiability, Identifiability, mathematical-models, optimal experimental-design, Parameter estimation, parameter identifiability, practical identifiability, profile likelihood, structural identifiability, Publication Note: The publisher's version of record is available at https://doi.org/10.1016/j.jtbi.2017.07.018
Constructing Thin Subgroups Of Sl(n+1, R) Via Bending
We use techniques from convex projective geometry to produce many new examples of thin subgroups of lattices in special linear groups that are isomorphic to the fundamental groups of finite-volume hyperbolic manifolds. More specifically, we show that for a large class of arithmetic lattices in SO(n, 1) it is possible to find infinitely many noncommensurable lattices in SL(n + 1, R) that contain a thin subgroup isomorphic to a finite-index subgroup of the original arithmetic lattice. This class of arithmetic lattices includes all noncocompact arithmetic lattices as well as all cocompact arithmetic lattices when n is even., 1st betti number, surfaces, The publisher's version of record is availible at https://doi.org/10.2140/agt.2020.20.2071
Data Assimilation In Large Prandtl Rayleigh-benard Convection From Thermal Measurements
This work applies a continuous data assimilation scheme-a framework for reconciling sparse and potentially noisy observations to a mathematical model-to Rayleigh-Benard convection at infinite or large Prandtl numbers using only the temperature field as observables. These Prandtl numbers are applicable to the earth's mantle and to gases under high pressure. We rigorously identify conditions that guarantee synchronization between the observed system and the model, then confirm the applicability of these results via numerical simulations. Our numerical experiments show that the analytically derived conditions for synchronization are far from sharp; that is, synchronization often occurs even when sufficient conditions of our theorems are not met. We also develop estimates on the convergence of an infinite Prandtl model to a large (but finite) Prandtl number generated set of observations. Numerical simulations in this hybrid setting indicate that the mathematically rigorous results are accurate, but of practical interest only for extremely large Prandtl numbers., data assimilation, temperature, number, attractors, turbulence, algorithm, equations, velocity, boussinesq system, determining nodes, heat-transport, large Prandtl limit, Rayleigh-Benard convection, The publisher's version of record is availible at https://doi.org/10.1137/19M1248327
Direct Transmission Via Households Informs Models Of Disease And Intervention Dynamics In Cholera
While several basic properties of cholera outbreaks are common to most settings-the pathophysiology of the disease, the waterborne nature of transmission, and others-recent findings suggest that transmission within households may play a larger role in cholera outbreaks than previously appreciated. Important features of cholera outbreaks have long been effectively modeled with mathematical and computational approaches, but little is known about how variation in direct transmission via households may influence epidemic dynamics. In this study, we construct a mathematical model of cholera that incorporates transmission within and between households. We observe that variation in the magnitude of household transmission changes multiple features of disease dynamics, including the severity and duration of outbreaks. Strikingly, we observe that household transmission influences the effectiveness of possible public health interventions (e.g. water treatment, antibiotics, vaccines). We find that vaccine interventions are more effective than water treatment or antibiotic administration when direct household transmission is present. Summarizing, we position these results within the landscape of existing models of cholera, and speculate on its implications for epidemiology and public health., epidemics, vibrio-cholerae, waterborne pathogen, The publisher's version of record is availible at https://doi.org/10.1371/journal.pone.0229837
Dual Detection System for Simultaneous Measurement of Intracellular Fluorescent Markers and Cellular Secretion.
Glucose-stimulated insulin secretion from pancreatic β-cells within islets of Langerhans plays a critical role in maintaining glucose homeostasis. Although this process is essential for maintaining euglycemia, the underlying intracellular mechanisms that control it are still unclear. To allow simultaneous correlation between intracellular signal transduction events and extracellular secretion, an analytical system was developed that integrates fluorescence imaging of intracellular probes with high-speed automated insulin immunoassays. As a demonstration of the system, intracellular [Ca] ([Ca]) was measured by imaging Fura-2 fluorescence simultaneously with insulin secretion from islets exposed to elevated glucose levels. Both [Ca] and insulin were oscillatory during application of 10 mM glucose with temporal and quantitative profiles similar to what has been observed elsewhere. In previous work, sinusoidal glucose levels have been used to test the entrainment of islets while monitoring either [Ca] or insulin levels; using this newly developed system, we show unambiguously that oscillations of both [Ca] and insulin release are entrained to oscillatory glucose levels and that the temporal correlation of these are maintained throughout the experiment. It is expected that the developed analytical system can be expanded to investigate a number of other intracellular messengers in islets or other stimulus-secretion pathways in different cells., Grant Number: R01 DK080714, Publication Note: This NIH-funded author manuscript originally appeared in PubMed Central at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5089909.
Effects Of Nonuniform Viscosity On Ciliary Locomotion
The effect of nonuniform viscosity on the swimming velocity of a free swimmer at zero Reynolds number is examined. Using the generalized reciprocal relation for Stokes flow with nonuniform viscosity, we formulate the locomotion problem in a fluid medium with spatially varying viscosity. Assuming the limit of small variation in the viscosity of the fluid as a result of nonuniform distribution of nutrients around a swimmer, we derive a perturbation model to calculate the changes in the swimming performance of a spherical swimmer as a result of position-dependent viscosity. The swimmer is chosen to be a spherical squirmer with a steady tangential motion on its surface modeling ciliary motion. The nutrient concentration around the body is described by an advection-diffusion equation. The roles of the surface stroke pattern, the specific relationship between the nutrient and viscosity, and the Peclet number of the nutrient in the locomotion velocity of the squirmer are investigated. Our results show that for a pure treadmill stroke, the velocity change is maximum at the limit of zero Peclet number and monotonically decreases toward zero at very high Peclet number. When higher surface stroke modes are present, larger modification in swimming velocity is captured at high Peclet number where two mechanisms of thinning the nutrient boundary layer and appearance of new stagnation points along the surface of squirmer are found to be the primary reasons behind the swimming velocity modifications. It is observed that the presence of nonuniform viscosity allows for optimal swimming speed to be achieved with stroke combinations other than pure treadmill., Keywords: flow, dynamics, driven, organisms, flagella, low-reynolds-number, microorganisms, mucin, propelled oil droplets, self-propulsion, Publication Note: The publisher’s version of record is available at https://doi.org/10.1103/PhysRevFluids.3.043101
Effects of GABAergic Polarity Changes on Episodic Neural Network Activity in Developing Neural Systems.
Early in development, neural systems have primarily excitatory coupling, where even GABAergic synapses are excitatory. Many of these systems exhibit spontaneous episodes of activity that have been characterized through both experimental and computational studies. As development progress the neural system goes through many changes, including synaptic remodeling, intrinsic plasticity in the ion channel expression, and a transformation of GABAergic synapses from excitatory to inhibitory. What effect each of these, and other, changes have on the network behavior is hard to know from experimental studies since they all happen in parallel. One advantage of a computational approach is that one has the ability to study developmental changes in isolation. Here, we examine the effects of GABAergic synapse polarity change on the spontaneous activity of both a mean field and a neural network model that has both glutamatergic and GABAergic coupling, representative of a developing neural network. We find some intuitive behavioral changes as the GABAergic neurons go from excitatory to inhibitory, shared by both models, such as a decrease in the duration of episodes. We also find some paradoxical changes in the activity that are only present in the neural network model. In particular, we find that during early development the inter-episode durations become longer on average, while later in development they become shorter. In addressing this unexpected finding, we uncover a priming effect that is particularly important for a small subset of neurons, called the "intermediate neurons." We characterize these neurons and demonstrate why they are crucial to episode initiation, and why the paradoxical behavioral change result from priming of these neurons. The study illustrates how even arguably the simplest of developmental changes that occurs in neural systems can present non-intuitive behaviors. It also makes predictions about neural network behavioral changes that occur during development that may be observable even in actual neural systems where these changes are convoluted with changes in synaptic connectivity and intrinsic neural plasticity., Keywords: GABAergic neurons, Activity episodes, Developing neural networks, Excitatory-inhibitory balance, Heterogeneity, Publication Note: This NIH-funded author manuscript originally appeared in PubMed Central at https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5649201.
Enriched Galerkin Finite Elements For Coupled Poromechanics With Local Mass Conservation
Robust and efficient discretization methods for coupled poromechanical problems are critical to address a wide range of problems related to civil infrastructure, energy resources, and environmental sustainability. In this work, we propose a new finite element formulation for coupled poromechanical problems that ensures local (element-wise) mass conservation. The proposed formulation draws on the so-called enriched Galerkin method, which augments piecewise constant functions to the classical continuous Galerkin finite element method. These additional degrees of freedom allow us to obtain a locally conservative and nonconforming solution for the pore pressure field. The enriched and continuous Galerkin formulations are compared in several numerical examples ranging from a benchmark consolidation problem to a complex problem that involves plastic deformation due to unsaturated flow in a heterogeneous porous medium. The results of these examples show not only that the proposed method provides local mass conservation, but also that local mass conservation can be crucial to accurate simulation of deformation processes in fluid-infiltrated porous materials. (C) 2018 Elsevier B.V. All rights reserved., Keywords: numerical-solution, Finite element method, computational model, reactive transport, cam-clay plasticity, saturated porous-media, Coupled poromechanics, Enriched Galerkin method, geological storage, hydro-mechanics, land subsidence, Local mass conservation, penalty parameter, scalable algorithms, Publication Note: The publisher’s version of record is available at https://doi.org/10.1016/j.cma.2018.06.022
First Semester in Numerical Analysis with Julia
First Semester in Numerical Analysis with Julia presents the theory and methods, together with the implementation of the algorithms using the Julia programming language (version 1.1.0). The reader is expected to have studied calculus and linear algebra. Some familiarity with a programming language is beneficial, but not required. The programming language Julia will be introduced in the book. Incorporating coding and computing within the main text was my primary objective in writing this book. The simplicity of Julia allows bypassing the pseudocode, and writing a computer code directly after the description of a method. It also minimizes the distraction the presentation of a computer code might cause to the flow of the main narrative. The Julia codes are written without much concern for efficiency; the priority is to write codes that will mimic the derivations presented in the text., Keywords: Numerical analysis, Julia, Publication Note: This publication was made possible by an Alternative Textbook Grant issued by Florida State University Libraries., Preferred Citation: Ökten, G. (2019). First Semester in Numerical Analysis with Julia. doi:
Generalized Mahalanobis Depth In Point Process And Its Application In Neural Coding
In this paper, we propose to generalize the notion of depth in temporal point process observations. The new depth is defined as a weighted product of two probability terms: (1) the number of events in each process, and (2) the center-outward ranking on the event times conditioned on the number of events. In this study, we adopt the Poisson distribution for the first term and the Mahalanobis depth for the second term. We propose an efficient boot-strapping approach to estimate parameters in the defined depth. In the case of Poisson process, the observed events are order statistics where the parameters can be estimated robustly with respect to sample size. We demonstrate the use of the new depth by ranking realizations from a Poisson process. We also test the new method in classification problems using simulations as well as real neural spike train data. It is found that the new framework provides more accurate and robust classifications as compared to commonly used likelihood methods., Keywords: algorithms, Mahalanobis depth, neural coding, Point process, Poisson process, prototypes, spike train, spike-train space, statistics, Publication Note: The publisher's version of record is available at https://doi.org/10.1214/17-AOAS1030
Genome-Wide Association Study Reveals Multiple Loci Influencing Normal Human Facial Morphology
Numerous lines of evidence point to a genetic basis for facial morphology in humans, yet little is known about how specific genetic variants relate to the phenotypic expression of many common facial features. We conducted genome-wide association meta-analyses of 20 quantitative facial measurements derived from the 3D surface images of 3118 healthy individuals of European ancestry belonging to two US cohorts. Analyses were performed on just under one million genotyped SNPs (Illumina OmniExpress+Exome v1.2 array) imputed to the 1000 Genomes reference panel (Phase 3). We observed genome-wide significant associations (p < 5 x 10(-8)) for cranial base width at 14q21.1 and 20q12, intercanthal width at 21p13.3 and Xq13.2, nasal width at 20p11.22, nasal ala length at 14q11.2, and upper facial depth at 11q22.1. Several genes in the associated regions are known to play roles in craniofacial development or in syndromes affecting the face: MAFB, PAX9, MIPOL1, ALX3, HDAC8, and PAX1. We also tested genotype-phenotype associations reported in two previous genome-wide studies and found evidence of replication for nasal ala length and SNPs in CACNA2D3 and PRDM16. These results provide further evidence that common variants in regions harboring genes of known craniofacial function contribute to normal variation in human facial features. Improved understanding of the genes associated with facial morphology in healthy individuals can provide insights into the pathways and mechanisms controlling normal and abnormal facial morphogenesis., Keywords: auriculocondylar syndrome, cleft-lip, craniofacial complex, geometric morphometrics, hormone receptor gene, japanese population, mandibular height, mutant mice, sonic-hedgehog, unaffected relatives, Publication Note: The publisher’s version of record is available at https://doi.org/10.1371/journal.pgen.1006149