Department of Mathematical Sciences Articles
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Item Symmetries of the Three-Gap Theorem(Taylor & Francis, 2023) Dasgupta, Aneesh; Roeder, Roland; Mathematical Sciences, School of ScienceThe Three-Gap Theorem states that for any 𝛼∈ℝ and 𝑁∈ℕ, the fractional parts of {0𝛼,1𝛼,…,(𝑁−1)𝛼} partition the unit circle into gaps of at most three distinct lengths. It is also of interest to find patterns in how the order of different gap sizes appear as one goes counterclockwise around the circle. This note is devoted to proving a result about symmetries in this ordering.Item Symplectic isotopy on non-minimal ruled surfaces(Springer Nature, 2023) Buse, Olguta; Li, Jun; Mathematical Sciences, School of ScienceWe prove the stability of Symp(X,w) \Diff0(X) for a one-point blow-up of irrational ruled surfaces and study their topological colimit. Non-trivial generators of π0[Symp(X,w) \Diff0(X)] that differ from Lagrangian Dehn twists are detected.Item Mathematical modeling approaches in the study of glaucoma disparities among people of African and European descents(American Scientific, 2013) Guidoboni, Giovanna; Harris, Alon; Arciero, Julia C.; Siesky, Brent A.; Amireskandari, Annahita; Gerber, Austin L.; Huck, Andrew H.; Kim, Nathaniel J.; Cassani, Simone; Carichino, Lucia; Mathematical Sciences, School of ScienceOpen angle glaucoma (OAG) is a severe ocular disease characterized by progressive and irreversible vision loss. While elevated intraocular pressure (IOP) is a well-established risk factor for OAG, the progression of OAG in many cases, despite IOP treatment, suggests that other risk factors must play significant roles in the development of the disease. For example, various structural properties of the eye, ocular blood flow properties, and systemic conditions have been identified as risk factors for OAG. Ethnicity has also been indicated as a relevant factor that affects the incidence and prevalence of OAG; in fact, OAG is the leading cause of blindness among people of African descent. Numerous clinical studies have been designed to examine the possible correlation and causation between OAG and these factors; however, these studies are met with the challenge of isolating the individual role of multiple interconnected factors. Over the last decade, various mathematical modeling approaches have been implemented in combination with clinical studies in order to provide a mechanical and hemodynamical description of the eye in relation to the entire human body and to assess the contribution of single risk factors to the development of OAG. This review provides a summary of the clinical evidence of ocular structural differences, ocular vascular differences and systemic vascular differences among people of African and European descent, describes the mathematical approaches that have been proposed to study ocular mechanics and hemodynamics while discussing how they could be used to investigate the relevance to OAG of racial disparities, and outlines possible new directions of research.Item Ranking and Visualizing Clusters of the US States by Adversity Childhood Experiences(2023) Do, Anh Phuong; Rashid, Mamunur; Sarkar, Jyotirmoy; Rashid, Maieasha Shifa; Mathematical Sciences, School of ScienceAdverse Childhood Experiences (ACEs) refer to traumatic childhood events like emotional, physical, sexual abuse, and other forms of household dysfunction. ACEs are associated with biomarkers for chronic diseases resulting in early mortality and increased morbidity. According to the Centers for Disease Control and Prevention, ACEs are common: Around 61% of adults across 25 US states reported having experienced at least one type of ACE. Ranking and finding clusters of the US states on ACEs provide a better understanding of the situation and helps prevent or reduce the occurrence of ACEs. The paper aims to apply a Multiple Criteria Decision-Making Model called the Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) method and calculate the overall Composite Index‘ to rank the states. Furthermore, the study uses the K-Means Cluster algorithm to identify and visualize clusters of states experiencing similar ACEs. The BRFSS 2019 data set was used for all analyses. The TOPSIS method suggested that Tennessee had the worst status of ACEs (ranked first) and North Dakota performed the best (ranked last). The elbow method determined that four clusters were present out of the 21 states. Many states ranked with the highest ACEs were clustered together: Tennessee, Florida, Pennsylvania, New Mexico, Delaware, Michigan. To better understand the current performance of the US regarding ACEs, it would be best to collect data from all states. Diagnostic studies, such as this study, can create the foundation for addressing and eradicating child maltreatment and ensuring healthy and nurturing childhoods.Item Effects of Glycinergic Inhibition Failure on Respiratory Rhythm and Pattern Generation(Elsevier, 2014) Shevtsova, Natalia A.; Büsselberg, Dietrich; Molkov, Yaroslav I.; Bischoff, Anne M.; Smith, Jeffrey C.; Richter, Diethelm W.; Rybak, Ilya A.; Mathematical Sciences, School of ScienceInhibitory interactions between neurons of the respiratory network are involved in rhythm generation and pattern formation. Using a computational model of brainstem respiratory networks, we investigated the possible effects of suppressing glycinergic inhibition on the activity of different respiratory neuron types. Our study revealed that progressive suppression of glycinergic inhibition affected all neurons of the network and disturbed neural circuits involved in termination of inspiration. Causal was a dysfunction of postinspiratory inhibition targeting inspiratory neurons, which often led to irregular preterm reactivation of these neurons, producing double or multiple short-duration inspiratory bursts. An increasing blockade of glycinergic inhibition led to apneustic inspiratory activity. Similar disturbances of glycinergic inhibition also occur during hypoxia. A clear difference in prolonged hypoxia, however, is that the rhythm terminates in expiratory apnea. The critical function of glycinergic inhibition for normal respiratory rhythm generation and the consequences of its reduction, including in pathological conditions, are discussed.Item A randomized trial comparing weight loss treatment delivered in large versus small groups(Springer Nature, 2014-09-24) Dutton, Gareth R.; Nackers, Lisa M.; Dubyak, Pamela J.; Rushing, Nicole C.; Huynh, Tuong-Vi T.; Tan, Fei; Anton, Stephen D.; Perri, Michael G.; Mathematical Sciences, School of ScienceBackground: Behavioral interventions for obesity are commonly delivered in groups, although the effect of group size on weight loss has not been empirically evaluated. This behavioral weight loss trial compared the 6- and 12-month weight changes associated with interventions delivered in a large group (LG) or small groups (SG). Methods: Obese adults (N = 66; mean age = 50 years; mean BMI = 36.5 kg/m2; 47% African American; 86% women) recruited from a health maintenance organization were randomly assigned to: (1) LG treatment (30 members/group), or (2) SG treatment (12 members/group). Conditions were comparable in frequency and duration of treatment, which included 24 weekly group sessions (months 1-6) followed by six monthly extended care contacts (months 7-12). A mixed effects model with unstructured covariance matrix was applied to analyze the primary outcome of weight change while accounting for baseline weight and dependence among participants' measurements over time. Results: SG participants lost significantly more weight than LG participants at Month 6 (-6.5 vs. -3.2 kg; p = 0.03) and Month 12 (-7.0 vs. -1.7 kg; p < 0.002). SG participants reported better treatment engagement and self-monitoring adherence at Months 6 and 12, ps < 0.04, with adherence fully mediating the relationship between group size and weight loss. Conclusions: Receiving obesity treatment in smaller groups may promote greater weight loss and weight loss maintenance. This effect may be due to improved adherence facilitated by SG interactions. These novel findings suggest that the perceived efficiency of delivering behavioral weight loss treatment to LGs should be balanced against the potentially better outcomes achieved by a SG approach.Item On the independence number of regular graphs of matrix rings(Elsevier, 2024-01) Nica, Bogdan; Mathematical Sciences, School of ScienceConsider a graph on the non-singular matrices over a finite field, in which two distinct non-singular matrices are joined by an edge whenever their sum is singular. We prove an upper bound for the independence number of this graph. As a consequence, we obtain a lower bound for its chromatic number that significantly improves a previous result of Tomon.Item New combinatorial formulae for nested Bethe vectors(arXiv, 2025) Kosmakov, Maksim; Tarasov, Vitaly; Mathematical Sciences, School of ScienceWe give new combinatorial formulae for vector-valued weight functions (off-shell nested Bethe vectors) for the evaluation modules over the Yangian Y (gl_4). The case of Y (gl_n) for an arbitrary n is considered in the forthcoming paper [KT].Item Monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian(Springer Nature, 2024) Tarasov, Vitaly; Varchenko , Alexander; Mathematical Sciences, School of ScienceWe describe the monodromy of the equivariant quantum differential equation of the cotangent bundle of a Grassmannian in terms of the equivariant K-theory algebra of the cotangent bundle. This description is based on the hypergeometric integral representations for solutions of the equivariant quantum differential equation. We identify the space of solutions with the space of the equivariant K-theory algebra of the cotangent bundle. In particular, we show that for any element of the monodromy group, all entries of its matrix in the standard basis of the equivariant K-theory algebra of the cotangent bundle are Laurent polynomials with integer coefficients in the exponentiated equivariant parameters.Item Modulation of thalamocortical relay by basal ganglia in Parkinson’s disease and dystonia(Springer Nature, 2011-07-18) Guo, Yixin; Park, Choongseok; Rong, Min; Worth, Robert M.; Rubchinsky, Leonid L.; Mathematical Sciences, School of Science