Department of Mathematical Sciences Articles
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Item Similar weight loss and maintenance in African American and White women in the Improving Weight Loss (ImWeL) trial(Taylor & Francis, 2021) Kinsey, Amber W.; Gowey, Marissa A.; Tan, Fei; Zhou, Dali; Ard, Jamy; Affuso, Olivia; Dutton, Gareth R.; Mathematical Sciences, School of ScienceObjective: African Americans (AA) are often underrepresented and tend to lose less weight than White participants during the intensive phase of behavioral obesity treatment. Some evidence suggests that AA women experience better maintenance of lost weight than White women, however, additional research on the efficacy of extended care programs (i.e. continued contacts to support the maintenance of lost weight) is necessary to better understand these differences. Methods: The influence of race on initial weight loss, the likelihood of achieving ≥5% weight reduction (i.e. extended care eligibility), the maintenance of lost weight and extended care program efficacy was examined in 269 AA and White women (62.1% AA) participating in a 16-month group-based weight management program. Participants achieving ≥5% weight reduction during the intensive phase (16 weekly sessions) were randomized to a clustered campaign extended care program (12 sessions delivered in three, 4-week clusters) or self-directed control. Results: In adjusted models, race was not associated with initial weight loss (p = 0.22) or the likelihood of achieving extended care eligibility (odds ratio 0.64, 95% CI [0.29, 1.38]). AA and White women lost −7.13 ± 0.39 kg and −7.62 ± 0.43 kg, respectively, during initial treatment. There were no significant differences in weight regain between AA and White women (p = 0.64) after adjusting for covariates. Clustered campaign program participants (AA: −6.74 ± 0.99 kg, White: −6.89 ± 1.10 kg) regained less weight than control (AA: −5.15 ± 0.99 kg, White: −4.37 ± 1.04 kg), equating to a 2.12 kg (p = 0.03) between-group difference after covariate adjustments. Conclusions: Weight changes and extended care eligibility were comparable among all participants. The clustered campaign program was efficacious for AA and White women. The high representation and retention of AA participants may have contributed to these findings.Item Critical and Ictal Phases in Simulated EEG Signals on a Small-World Network(Frontiers Media, 2021-01-08) Nemzer, Louis R.; Cravens, Gary D.; Worth, Robert M.; Motta, Francis; Placzek, Andon; Castro, Victor; Lou, Jennie Q.; Mathematical Sciences, School of ScienceHealthy brain function is marked by neuronal network dynamics at or near the critical phase, which separates regimes of instability and stasis. A failure to remain at this critical point can lead to neurological disorders such as epilepsy, which is associated with pathological synchronization of neuronal oscillations. Using full Hodgkin-Huxley (HH) simulations on a Small-World Network, we are able to generate synthetic electroencephalogram (EEG) signals with intervals corresponding to seizure (ictal) or non-seizure (interictal) states that can occur based on the hyperexcitability of the artificial neurons and the strength and topology of the synaptic connections between them. These interictal simulations can be further classified into scale-free critical phases and disjoint subcritical exponential phases. By changing the HH parameters, we can model seizures due to a variety of causes, including traumatic brain injury (TBI), congenital channelopathies, and idiopathic etiologies, as well as the effects of anticonvulsant drugs. The results of this work may be used to help identify parameters from actual patient EEG or electrocorticographic (ECoG) data associated with ictogenesis, as well as generating simulated data for training machine-learning seizure prediction algorithms.Item On the Hasse invariants of the Tate normal forms E5 and E7(Elsevier, 2021) Morton, Patrick; Mathematical Sciences, School of ScienceA formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic eld K = Q(pl), proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term 1, and a theorem is stated for the number of quartic factors of a speci c form in terms of the class number of Q(p 5l). These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over Fl of the supersingular polynomial ss(5 ) l (X) corresponding to the Fricke group 0 (5). The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E7 for a point of order 7 are determined, which is used to show that the polynomial ss(N ) l (X) for the group 0 (N) has roots in Fl2 , for any prime l 6= N, when N 2 f2; 3; 5; 7g.Item Statistical Inference on Panel Data Models: A Kernel Ridge Regression Method(Taylor & Francis, 2021) Zhao, Shunan; Liu, Ruiqi; Shang, Zuofeng; Mathematical Sciences, School of ScienceWe propose statistical inferential procedures for nonparametric panel data models with interactive fixed effects in a kernel ridge regression framework. Compared with the traditional sieve methods, our method is automatic in the sense that it does not require the choice of basis functions and truncation parameters. The model complexity is controlled by a continuous regularization parameter which can be automatically selected by the generalized cross-validation. Based on the empirical process theory and functional analysis tools, we derive the joint asymptotic distributions for the estimators in the heterogeneous setting. These joint asymptotic results are then used to construct the confidence intervals for the regression means and the prediction intervals for future observations, both being the first provably valid intervals in literature. The marginal asymptotic normality of the functional estimators in a homogeneous setting is also obtained. Our estimators can also be readily modified and applied to other widely used semiparametric models, such as partially linear models. Simulation and real data analyses demonstrate the advantages of our method.Item Predicting program attendance and weight loss in obesity interventions: Do triggering events help?(Sage, 2021) Borgatti, Alena; Tang, Ziting; Tan, Fei; Salvy, Sarah-Jeanne; Dutton, Gareth; Mathematical Sciences, School of ScienceMedical events that “trigger” motivation to lose weight may improve treatment outcomes compared to non-medical or no triggering events. However, previous findings include only long-term successful participants, not those initiating treatment. The current study compared those with medical triggering events or non-medical triggering events to no triggering events on attendance and weight loss during a weight management program. Medical-triggering-event participants lost 1.8 percent less weight (p = 0.03) than no-triggering-event participants. Non-medical-triggering-event participants attended 1.45 more sessions (p = 0.04) and were 1.83 times more likely to complete the program (p = 0.03) than no-triggering-event participants. These findings fail to support the benefit of medical triggering events when beginning treatment for obesity.Item Strategic Cuts With a Cylindrical Knife(Springer, 2023-02) Bhattacharjee, Rishav; Sarkar, Jyotirmoy; Mathematical Sciences, School of ScienceSuppose your father gives you a hollow cylindrical knife as a birthday gift, and your mother buys you a bag of potatoes. When the knife is pressed into a potato and the outer excess is removed, the interior of the knife yields a cylindrical core. By pressing the knife into a potato from several strategically chosen directions, you can construct some solids of intersection such that all faces are identical or one of two distinct shapes.Item Quantum Toroidal Comodule Algebra of Type A(n-1) and Integrals of Motion(Foundation Compositio Mathematica, 2022-07-07) Feigin, Boris; Jimbo, Michio; Mukhin, Evgeny; Mathematical Sciences, School of ScienceWe introduce an algebra Kn which has a structure of a left comodule over the quantum toroidal algebra of type An−1. Algebra Kn is a higher rank generalization of K1, which provides a uniform description of deformed W algebras associated with Lie (super)algebras of types BCD. We show that Kn possesses a family of commutative subalgebras.Item Scrambled Vandermonde convolutions of Gaussian polynomials(Elsevier, 2022-12) Aspenburg, Magnus; Pérez, Rodrigo A.; Mathematical Sciences, School of ScienceIt is well known that Gaussian polynomials (i.e., q-binomials) describe the distribution of the area statistic on monotone paths in a rectangular grid. We introduce two new statistics, corners and c-index; attach "ornaments" to the grid; and re-evaluate these statistics, in order to argue that all scrambled versions of the c-index statistic are equidistributed with area. Our main result is a representation of the generating function for the bi-statistic (c-index; corners) as a two-variable Vandermonde convolution of the original Gaussian polynomial. The proof relies on explicit bijections between differently ornated paths.Item Some Inequalities on Finsler Manifolds with Weighted Ricci Curvature Bounded Below(Springer, 2022-02-07) Cheng, Xinyue; Shen, Zhongmin; Mathematical Sciences, School of ScienceWe establish some important inequalities under a lower weighted Ricci curvature bound on Finsler manifolds. Firstly, we establish a relative volume comparison of Bishop–Gromov type. As one of the applications, we obtain an upper bound for volumes of the Finsler manifolds. Further, when the S-curvature is bounded on the whole manifold, we obtain a theorem of Bonnet–Myers type on Finsler manifolds. Finally, we obtain a sharp Poincaré–Lichnerowicz inequality by using integrated Bochner inequality, from which we obtain a better lower bound for the first eigenvalue on the Finsler manifolds.Item Landau–Ginzburg mirror, quantum differential equations and qKZ difference equations for a partial flag variety(Elsevier, 2023-02) Tarasov, Vitaly; Varchenko, Alexander; Mathematical Sciences, School of ScienceWe consider the system of quantum differential equations for a partial flag variety and construct a basis of solutions in the form of multidimensional hypergeometric functions, that is, we construct a Landau–Ginzburg mirror for that partial flag variety. In our construction, the solutions are labeled by elements of the K-theory algebra of the partial flag variety. To establish these facts we consider the equivariant quantum differential equations for a partial flag variety and introduce a compatible system of difference equations, which we call the qKZ equations. We construct a basis of solutions of the joint system of the equivariant quantum differential equations and qKZ difference equations in the form of multidimensional hypergeometric functions. Then the facts about the non-equivariant quantum differential equations are obtained from the facts about the equivariant quantum differential equations by a suitable limit. Analyzing these constructions we obtain a formula for the fundamental Levelt solution of the quantum differential equations for a partial flag variety.