Department of Mathematical Sciences Articles
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Item Foredune and Beach Dynamics on the Southern Shores of Lake Michigan during Recent High Water Levels(MDPI, 2022) Kilibarda, Zoran; Kilibarda, Vesna; Mathematical Sciences, School of ScienceFrom 18 January 2013 (175.16 m a.s.l.) to 8 September 2020 (177.82 m a.s.l.), Lake Michigan experienced its fastest and highest rise (2.67 m) since 1860, when instrumental measurements began. Extensive foredunes developed since the last high lake levels began eroding in 1997 at fast rates. This study focuses on coastal morphodynamics along the 800 m coast within the central Indiana Dunes State Park on Lake Michigan’s southern shores during this time. Severe foredune erosion, in terms of total horizontal dune loss and total volume of eroded sand, occurred unevenly over the three-year period, both temporally, during a single storm, a season, a year, or three years, and spatially, in the eastern, central, and western study areas. Late autumn storms accounted for most foredune erosion in 2018 and 2019, when foredune scarps retreated up to 4 and 9 m, respectively. Erosion was highest in the updrift eastern study area, where about 8 m3/m of sand was removed in 2018 and about 19 m3/m of sand was removed in 2019. The lack of shelf ice along the shore, rising lake levels, and convective storms that triggered meteotsunamis changed the foredune erosion pattern in 2020. Erosion became most vigorous in the downdrift central (13 m scarp retreat) and western (11 m scarp retreat) study areas. The average volume of eroded sand (21.5 m3/m) was more than double that of 2019 (8 m3/m), and almost quadruple the 2018 volume (5.5 m3/m). After foredune erosion events, the beach rapidly recovered and maintained its width as the shoreline migrated landward. On many occasions following these severe erosion events the dry portion of the beach aggraded and absorbed significant sand amounts removed from the foredunes. The remaining sand was transferred to the surf zone, where it changed the sand bar morphology and led to their coalescence and flattening.Item Accessibility of the Boundary of the Thurston Set(Taylor & Francis, 2023) Silvestri, Stefano; Pérez, Rodrigo A.; Mathematical Sciences, School of ScienceConsider two objects associated to the Iterated Function System (IFS) {1+𝜆𝑧,−1+𝜆𝑧}: the locus ℳ of parameters 𝜆∈𝔻∖{0} for which the corresponding attractor is connected; and the locus ℳ0 of parameters for which the related attractor contains 0. The set ℳ can also be characterized as the locus of parameters for which the attractor of the IFS {1+𝜆𝑧,𝜆𝑧,−1+𝜆𝑧} contains 𝜆−1. Exploiting the asymptotic similarity of ℳ and ℳ0 with the respective associated attractors, we give sufficient conditions on 𝜆∈∂ℳ or ∂ℳ0 to guarantee it is path accessible from the complement 𝔻∖ℳ.Item On smooth perturbations of Chebyshëv polynomials and ∂¯ -Riemann–Hilbert method(Cambridge University Press, 2023) Yattselev, Maxim L.; Mathematical Sciences, School of ScienceItem Potential measurement error from vessel reflex and multiple light paths in dual-wavelength retinal oximetry(Wiley, 2024-05) Beach, James M.; Shoemaker, Benjamin; Eckert, George J.; Harris, Alon; Siesky, Brent; Arciero, Julia C.; Mathematical Sciences, School of SciencePurpose This study aims to characterize the dependence of measured retinal arterial and venous saturation on vessel diameter and central reflex in retinal oximetry, with an ultimate goal of identifying potential causes and suggesting approaches to improve measurement accuracy. Methods In 10 subjects, oxygen saturation, vessel diameter and optical density are obtained using Oxymap Analyzer software without diameter correction. Diameter dependence of saturation is characterized using linear regression between measured values of saturation and diameter. Occurrences of negative values of vessel optical densities (ODs) associated with central vessel reflex are acquired from Oxymap Analyzer. A conceptual model is used to calculate the ratio of optical densities (ODRs) according to retinal reflectance properties and single and double-pass light transmission across fixed path lengths. Model-predicted values are compared with measured oximetry values at different vessel diameters. Results Venous saturation shows an inverse relationship with vessel diameter (D) across subjects, with a mean slope of −0.180 (SE = 0.022) %/μm (20 < D < 180 μm) and a more rapid saturation increase at small vessel diameters reaching to over 80%. Arterial saturation yields smaller positive and negative slopes in individual subjects, with an average of −0.007 (SE = 0.021) %/μm (20 < D < 200 μm) across all subjects. Measurements where vessel brightness exceeds that of the retinal background result in negative values of optical density, causing an artifactual increase in saturation. Optimization of model reflectance values produces a good fit of the conceptual model to measured ODRs. Conclusion Measurement artefacts in retinal oximetry are caused by strong central vessel reflections, and apparent diameter sensitivity may result from single and double-pass transmission in vessels. Improvement in correction for vessel diameter is indicated for arteries however further study is necessary for venous corrections.Item Modeling and simulation of flow–osteocyte interaction in a lacuno-canalicular network(AIP, 2023-09) Barber, Jared; Manring, Isaac; Boileau, Sophie; Zhu, Luoding; Mathematical Sciences, School of ScienceOsteocytes are bone cells that can sense mechanical cues (stress and strain) and respond by releasing biochemical signals that direct bone remodeling. This process is called mechanotransduction which, in osteocytes, is not well understood yet because in vivo studies have proven difficult due to the complexity and inaccessibility of the flow–osteocyte lacuna-canaliculi system. While in silico studies (modeling and simulation) have become powerful, currently computational studies for the system often omit the fluid–structure interaction (FSI) between the cell and the surrounding fluids. To investigate the role of FSI in osteocyte mechanotransduction, we introduce a two-dimensional coarse-grained yet integrative model for flow–osteocyte interaction in a lacuno-canalicular network. The model uses the lattice Boltzmann immersed boundary framework to incorporate the flexible osteocyte (membrane, cytoskeleton, and cytosol), its processes, the interstitial fluid, and the rigid extracellular matrix that encases the system. One major result of our model is that the stress and strain tend to attain their local maxima near the regions where the processes meet the membrane of the main body.Item Diagnostic Capability of OCTA-Derived Macular Biomarkers for Early to Moderate Primary Open Angle Glaucom(MDPI, 2024-07-18) Verticchio Vercellin, Alice; Harris, Alon; Oddone, Francesco; Carnevale, Carmela; Siesky, Brent A.; Arciero, Julia; Fry, Brendan; Eckert, George; Sidoti, Paul A.; Antman, Gal; Alabi, Denise; Coleman-Belin, Janet C.; Pasquale, Louis R.; Mathematical Sciences, School of ScienceBackground/Objectives: To investigate macular vascular biomarkers for the detection of primary open-angle glaucoma (POAG). Methods: A total of 56 POAG patients and 94 non-glaucomatous controls underwent optical coherence tomography angiography (OCTA) assessment of macular vessel density (VD) in the superficial (SCP), and deep (DCP) capillary plexus, foveal avascular zone (FAZ) area, perimeter, VD, choriocapillaris and outer retina flow area. POAG patients were classified for severity based on the Glaucoma Staging System 2 of Brusini. ANCOVA comparisons adjusted for age, sex, race, hypertension, diabetes, and areas under the receiver operating characteristic curves (AUCs) for POAG/control differentiation were compared using the DeLong method. Results: Global, hemispheric, and quadrant SCP VD was significantly lower in POAG patients in the whole image, parafovea, and perifovea (p < 0.001). No significant differences were found between POAG and controls for DCP VD, FAZ parameters, and the retinal and choriocapillaris flow area (p > 0.05). SCP VD in the whole image and perifovea were significantly lower in POAG patients in stage 2 than stage 0 (p < 0.001). The AUCs of SCP VD in the whole image (0.86) and perifovea (0.84) were significantly higher than the AUCs of all DCP VD (p < 0.05), FAZ parameters (p < 0.001), and retinal (p < 0.001) and choriocapillaris flow areas (p < 0.05). Whole image SCP VD was similar to the AUC of the global retinal nerve fiber layer (RNFL) (AUC = 0.89, p = 0.53) and ganglion cell complex (GCC) thickness (AUC = 0.83, p = 0.42). Conclusions: SCP VD is lower with increasing functional damage in POAG patients. The AUC for SCP VD was similar to RNFL and GCC using clinical diagnosis as the reference standard.Item Evidence for Quasicritical Brain Dynamics(American Physical Society, 2021) Fosque, Leandro J.; Williams-García , Rashid V.; Beggs, John M.; Oritz, Gerardo; Mathematical Sciences, School of ScienceMuch evidence seems to suggest the cortex operates near a critical point, yet a single set of exponents defining its universality class has not been found. In fact, when critical exponents are estimated from data, they widely differ across species, individuals of the same species, and even over time, or depending on stimulus. Interestingly, these exponents still approximately hold to a dynamical scaling relation. Here we show that the theory of quasicriticality, an organizing principle for brain dynamics, can account for this paradoxical situation. As external stimuli drive the cortex, quasicriticality predicts a departure from criticality along a Widom line with exponents that decrease in absolute value, while still holding approximately to a dynamical scaling relation. We use simulations and experimental data to confirm these predictions and describe new ones that could be tested soon.Item Metabolic blood flow regulation in a hybrid model of the human retinal microcirculation(Elsevier, 2023) Albright, Amanda; Fry, Brendan C.; Verticchio, Alice; Siesky, Brent; Harris, Alon; Arciero, Julia; Mathematical Sciences, School of ScienceThe retinal vascular network supplies perfusion to vital visual structures, including retinal ganglion cells responsible for vision. Impairments in retinal blood flow and oxygenation are involved in the progression of many ocular diseases, including glaucoma. In this study, an established theoretical hybrid model of a retinal microvascular network is extended to include the effects of local blood flow regulation on oxygenation. A heterogeneous representation of the arterioles based on confocal microscopy images is combined with a compartmental description of the downstream capillaries and venules. A Green’s function method is used to simulate oxygen transport in the arterioles, and a Krogh cylinder model is applied to the capillary and venular compartments. Acute blood flow regulation is simulated in response to changes in pressure, shear stress, and metabolism. Model results predict that both increased intraocular pressure and impairment of blood flow regulation can cause decreased tissue oxygenation, indicating that both mechanisms represent factors that could lead to impaired oxygenation characteristic of ocular disease. Results also indicate that the metabolic response mechanism reduces the fraction of poorly oxygenated tissue but that the pressure- and shear stress-dependent response mechanisms may hinder the vascular response to changes in oxygenation. Importantly, the heterogeneity of the vascular network demonstrates that traditionally reported average values of tissue oxygen levels hide significant localized defects in tissue oxygenation that may be involved in disease processes, including glaucoma. Ultimately, the model framework presented in this study will facilitate future comparisons to sectorial-specific clinical data to better assess the role of impaired blood flow regulation in ocular disease.Item A System with Two Spare Units, Two Repair Facilities, and Two Types of Repairers(MDPI, 2022-03-08) Andalib, Vahid; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceAssuming exponential lifetime and repair time distributions, we study the limiting availability 𝐴∞ as well as the per unit time-limiting profit 𝜔 of a one-unit system having two identical, cold standby spare units using semi-Markov processes. The failed unit is repaired either by an in-house repairer within an exponential patience time T or by an external expert who works faster but charges more. When there are two repair facilities, we allow the regular repairer to begin repair or to continue repair beyond T if the expert is busy. Two models arise accordingly as the expert repairs one or all failed units during each visit. We show that (1) adding a second spare to a one-unit system already backed by a spare raises 𝐴∞ as well as 𝜔; (2) thereafter, adding a second repair facility improves both criteria further. Finally, we determine whether the expert must repair one or all failed units to maximize these criteria and fulfill the maintenance management objectives better than previously studied models.Item Solutions of diophantine equations as periodic points of p-adic algebraic functions, III(arXiv, 2020) Morton, Patrick; Mathematical Sciences, School of Science