The Association for Women in Mathematics (AWM) has invited graduate students and recent PhD recipients to give  poster presentations at the 2019 AWM Research Symposium at Rice University in Houston, Texas on April 6-7, 2019.

Posters will be exhibited during the entire symposium in and around the first floor of Duncan Hall.  Poster presenters will be at their posters to discuss their work on Saturday from 2:30-3:15pm.

Anna Aboud, An Efficient Algorithm for Perturbed Data Sets
Abstract: With applications in signal processing and data science, the Kaczmarz algorithm is an iterative method used to reconstruct a vector x in a Hilbert space from the inner products {(x, φ_n)}. To address limitations revealed by large spatial and temporal data sets, we dualize the Kaczmarz algorithm so that the reconstruction of x can be obtained from {(x, φ_n)} by using a second sequence {ψ_n}.

Lale Asik^*, Angela Peace, Dynamics of a Producer-Grazer Model Incorporating the Effects of Phosphorus Loading on Grazer’s Growth
Abstract: Recent work in ecological stoichiometry has indicated that consumer dynamics are not only affected by insufficient food nutrient content (low phosphorus (P): carbon (C) ratio) but also by excess food nutrient content (extremely high P:C ratio). This phenomenon is known as the “stoichiometric knife edge.” While the Peace et al. (2014) model has captured this phenomenon, it does not explicitly track P loading of the aquatic environment. Here, we extend the Peace et al. (2014) model by mechanistically deriving and tracking P loading in order to investigate the growth response of the grazer to the producer of varying P:C ratios. We analyze the dynamics of the system such as boundedness and positivity of the solutions, existence and stability conditions of boundary equilibria. Bifurcation diagram and simulations show that our model behaves qualitatively similar but quantitatively different to the Peace et al. (2014) model. Furthermore, the structure of our model can easily be extended to incorporate seasonal phosphorus loading.

Rhea Palek Bakshi, On the KBSA of the Thickened T-Shirt
Abstract: Skein modules were introduced by Przytycki as a way to extend link polynomials to links in arbitrary 3-manifolds. Since their introduction, skein modules have become central to the theory of 3-manifolds. In 1997, Frohman and Gelca established a product-to-sum formula for the Kauffman bracket skein algebra of the thickened torus. We discover a similar formula for the multiplication of curves in the thickened T-shirt. This is joint work with Mukherjee, Przytycki, Silvero and Wang.

Catherine Berrouet, A Mathematical Model of Anti-Cancer Drug’s IC₅₀ Values in Monolayer and Spheroid Cultures
Abstract:
Traditionally, the monolayer (two-dimensional) cell cultures are used for initial evaluation of the effectiveness of anticancer therapies. In particular, these experiments provide the IC₅₀ curves that determine drug concentration that can inhibit growth of a tumor colony by half when compared to the cells grown with no exposure to the drug. Low IC₅₀ value means that the drug is effective at low concentrations, and thus will show lower systemic toxicity when administered to the patient.  However, in these experiments cells are grown in a monolayer, while in vivo tumors expand as three- dimensional multicellular masses. Therefore, we performed computational studies to compare the  IC₅₀ curves for cells grown as a two-dimensional monolayer and a cross section through a three- dimensional spheroid. Our results identified conditions (drug diffusivity, drug action mechanisms  and cell proliferation capabilities) under which these IC₅₀ curves differ significantly. This will help experimentalists to better determine drug dosage for future in vivo experiments and clinical trials.

Ariel Bowman, Mathematical Modeling of a Network of neurons regarding G1D Transport Deficiency Epilepsy Seizures
Abstract: G1D Transport Deficiency Epilepsy can be identified by the high number of seizures during the infant stage which is usually rapid, irregular eye movement, and small brain size. We want to identify brain activity specific to G1D by using EEG data. An EEG is a test used to monitor the electrical activity within the brain using small electrodes, which measures voltage fluctuations across the scalp. Wilson-Cowan (1972) introduced a model showing the dynamics of a network of neurons consisting of excitatory and inhibitory neurons. Taylor et. al (2014) then adapted the Wilson-Cowan model to epileptic seizures using a thalamo-cortical based theory. Fan et. al (2018) projects that thalamic reticulus nuclei control spike wave discharges specifically in absence seizures. We want to study the EEG patterns to identify the single mechanism that causes G1D epileptic behavior. The goal is to find out how an entirely connected brain network shows the neuronal func- tionality as a unit regarding G1D. Our coupled thalamo-cortical model goes beyond a connection in a logical unidirectional pattern shown by Fan (2018) but in a bidirectional small world pattern more analogous to realistic seizure activity. Using our model, we are able to study stability analysis for equilibrium and periodic behavior, parameter values which cause synchronized activity or more stable activity and identify a synchronization index, and sensitivity analysis regarding parameters that directly affect Spike Wave Dis- charges and other spiking behavior. We will show explicitly how our 32-unit network model is a more accurate picture of G1D and its limitations.

Danielle Brager, Mathematically Investigating Cone Photoreceptor Death in Retinitis Pigmentosa
Abstract: Retinitis Pigmentosa (RP) is a heterogeneous group of inherited degenerative retinal diseases. Typically, those afflicted with RP experience a loss of night vision that progresses to day-light blindness as a result of the death of rod photoreceptors followed by the death of cone photoreceptors. We mathematically model RP based on the experimental work of Léveillard et al. We investigate the progression of RP and the role of RdCVF using stability and bifurcation analysis.

Juliette Bruce, Asymptotic Syzygies for Products of Projective Space
Abstract: I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.

Sutthirut Charoenphon, Vanishing Relaxation Time Dynamics of the Moore-Gibson-Thompson (MGT) Equation Arising in High Frequency Ultrasound
Abstract: The MGT equation is a model describing acoustic wave propagation and arises as a model of high-frequency ultrasound (HFU) waves. The dynamic response of MGT depends on the physical parameters, in particular, the relaxation parameter τ , which accounts for the finite speed of propagation. Since is relatively small, it is important to trace the dynamics with vanishing parameter τ → 0. It is shown that the decay rates for the finite energy are preserved uniformly. The corresponding result provides not only a robust stabilizing mechanism for HFU waves but also leads to a new ”higher energy” stability estimates.

Weiqi Chu, Nonlinear Constitutive Models for Nano-scale Heat Conduction
Abstract: Heat conduction properties of nano mechanical systems have significant impacts on the performances of nano devices. There have been enormous experimental and numerical observations that indicate the breakdown of the conventional model of heat conduction, the Fourier’s Law. The indications include, but not limited to, the size dependence of the heat conductivity, propagation heat pulses behavior, and delay phenomena.
I will present a rigorous approach that leads, from a many-particle description, to a nonlinear, stochastic constitutive relation for the modeling of transient heat conduction processes at nanoscale. The nonlinearity gives rise to traveling waves, which also implies that temperature can propagate with finite speed. By enforcing statistical consistency, in that the statistics of the local energy is consistent with that from an all-atom description, we identify the driving force as well as the model parameters in these generalized constitutive models, such as heat propagation speed, relaxation time, thermal conductivity and its size dependence.

Ngoc Do, Theoretically exact solution of the inverse source problem for the wave equation with spatially and temporally reduced data
Abstract: The inverse source problem for the wave equation arises in several promising emerging modalities of medical imaging. Of special interest here are theoretically exact inversion formulas, explicitly expressing solution of the problem in terms of the measured data. Almost all known formulas of this type require data to be measured on a closed surface completely surrounding the object. This, however, is too restrictive for practical applications. I will present an alternative approach that, under certain restriction on geometry, yields explicit, theoretically exact reconstruction from the data measured on a finite open surface. Numerical simulations illustrating the work of the method will be presented. This is joint work with Leonid Kunyansky

Francesca Gandini, Ideals associated to subspace arrangements
Abstract:
Given a collection of t subspaces in an n-dimensional \(\mathbb{K}\)-vector space W we can associated to them t vanishing ideals in the symmetric algebra \(S(W^∗) = \mathbb{K}[x_1, x_2, \ldots , x_n]\). As a subspace is defined by a set of linear equations, its vanishing ideal is generated by linear forms so it is a linear ideal. Conca and Herzog showed that the Castelnuovo-Mumford regularity of the product of t linear ideals is equal to t. Derksen and Sidman showed that the Castelnuovo-Mumford regularity of the intersection of t linear ideals is at most t and they also showed that similar results hold for a more general class of ideals constructed from linear ideals. We show that analogous results hold when we replace the symmetric algebra S(W∗) with the exterior algebra \((\bigwedge W^∗)\) and work over a field of characteristic 0. To prove these results we rely on the functoriality of free resolutions and construct a functor Ω from the category of polynomial functors to itself. The functor Ω transforms resolutions of ideals in the symmetric algebra to resolutions of ideals in the exterior algebra.

Xiaoqian Gong, Weak Measure-valued Solution to a Nonlinear Hyperbolic Conservation Law Modeling a Highly Re-entrant Manufacturing System
Abstract: We consider an optimal control problem governed by a nonlinear hyperbolic conservation law model of a highly re-entrant semi-conductor manufacturing system. For the case of transition between equilibria, we report progress on the conjectured non-existence of L¹-optimal controls. In the setting of Borel  measures, we formulate a new notion of weak solution to the hyperbolic conservation law with flux boundary condition and non-local velocity and establish the well-posedness of such solution.

Emily L. Johnson*, Austin J. Herrema, Davide Proserpio, Michael C.H. Wu, Josef Kiendl, Ming-Chen Hsu,  Penalty Coupling of Non-Matching Isogeometric Kirchhoff–Love Shell Patches with Application to Complex Structures
Abstract: A penalty approach is presented for coupling adjacent surface patches in isogeometric Kirchhoff–Love shell analysis. This methodology imposes displacement and rotational continuity at many types of patch interfaces, addressing the issue of non-conforming geometries that exists in many complex engineering models. The penalty formulations require selection of a single, dimensionless penalty coefficient, and produce consistently accurate results for a wide variety of benchmark problems and a realistic wind turbine blade model.

Tiffany Jones, On the stability and accuracy of a dual-scale approximation to self-focusing Helmholtz problems
Abstract: This poster presents a dual-scale compact method for solving highly oscillatory Helmholtz equations in polar coordinates. Both the computational transverse domain and the governing  equations are decomposed into micro and macro domains. The coupled equations are sub- sequently discretized utilizing a compact strategy for increased efficiency and radial accuracy.
The numerical method is shown to be asymptotically stable at high-wavenumbers, facilitating the algorithmic effectiveness, reliability, and applicability. Spectrum analysis also reveals the 2-norm of the amplification matrices to be bounded. Optical beam numerical simulations, including those conducted with a range domain scaling factors, reinforce these findings.

Lara Kassab, On Infinite Multidimensional Scaling
Abstract: Multidimensional scaling (MDS) is a popular technique for mapping a finite metric space into a low-dimensional Euclidean space in a way that best preserves pairwise distances. We study a notion of MDS on infinite metric measure spaces, along with its optimality properties and goodness of fit. Furthermore, we address questions on convergence of MDS. For instance, if a sequence of metric measure spaces converges to a fixed metric measure space X, then in what sense do the MDS embeddings of these spaces converge to the MDS embedding of X? Convergence is well-understood when each metric space has the same finite number of points, but we are also interested in convergence when the number of points varies and is possibly infinite.

Daewa Kim, A Kinetic Theory Approach to Pedestrian Motion
Abstract: We consider a kinetic theory approach of the crowd evacuation from bounded domains. The interactions of a person with other pedestrians and the environment are modeled by using tools of game theory and are transferred to the crowd dynamics. The model allows to weight between two competing behaviors: the search for less congested areas and the tendency to follow the stream unconsciously. For the numerical approximation of the solution, we apply an operator splitting scheme.

Mehtap Iafci^*, Gavin K. Sabine, Hristo V. Kojouharov, Benito M. Chen-Charpentier, Sara R. McMahan, and Jun LiaoMathematical Modeling of Post-Myocardial Infarction Left Ventricular Remodeling
Abstract: We present new mathematical models to study the left ventricular remodeling after myocardial infarction (MI). First we consider a system of ordinary differential equations that models the interactions among heart cells and the immune system post-MI and without any medical interventions. Next, we consider a system that models the regeneration process of cardiomyocytes under constant oxygen supply and different possible medical interventions. Qualitative analysis of the models and numerical simulations are also presented.

Jennifer Li, The Kawamata-Morrison-Totaro Cone Conjecture for Log Calabi-Yau Surfaes
Abstract: Morrison’s cone conjecture states that for a smooth Calabi-Yau manifold X, the automorphism group of X acts on its effective nef cone with rational polyhedral fundamental domain. Totaro generalized this conjecture to Kawamata log terminal (klt) Calabi-Yau pairs \((X, D)\). In our project, we are studying pairs \((X, D)\) where X is a smooth projective surface and \(D = D_1 + \cdots + D_n\) is a reduced normal crossing divisor on X, with the properties that \(K_X +D = 0\) and D has negative-definite self intersection matrix. Results by Gross-Hacking-Keel on mirror symmetry for cusp singularities suggest that we consider the pair \((X, D)\) with a distinguished complex structure in which the mixed Hodge structure on U = X\D is split. The goal of our project is to prove Morrison’s cone conjecture in this special case. We note that this is different from Totaro’s work, because in our project the pair \((X, D)\) is not klt, and we must consider \((X, D)\) with the special complex structure (otherwise the conjecture is false as observed by Totaro). We have shown that the cone conjecture holds when D has at most six components: in these cases the nef cone is rational polyhedral and we give explicit generators for the dual cone. The cases where D has more than six components are current work in progress. Under our conditions, there exists a contraction of \((X, D)\) to a cusp singularity \((X^\prime, p)\). Cusp singularities come in mirror dual pairs, and the embedding dimension m of the dual cusp is equal to max(n, 3) where n is the number of components of the boundary divisor D. By studying the nef cone of \((X, D)\), we hope to give a description of the deformation space of the dual cusp, which is not well understood for m greater than six.

Kate Lorenzen, Constructions of Distance Laplacian Cospectral Graphs
Abstract: Graphs are mathematical objects that can be embedded into matrices. Two graphs are cospectral if they have the same set of eigenvalues with respect to a matrix. We present two constructions of cospectral graphs for the distance Laplacian matrix. The first uses vertex twins which have predictable eigenvectors and eigenvalues in the distance Laplacian. The second develops a relaxation of twins called vertex cousins. This second construction produces the only pair of bipartite distance Laplacian cospectral graphs on eight vertices.

Danielle Middlebrooks, Quantifying Flows in Time-Irreversible Markov Chains: Application to a Gene Regulatory Network
Abstract: Transition path theory (TPT) is a theoretical framework used to study the statistical properties of reactive trajectories. Reactive trajectories are those trajectories by which a random walker transits from one subset in the state-space to another disjoint subset. We develop analytical and computational tools based on TPT in order to quantify flows in time irreversible Markov Chains. These tools are applied to a gene regulatory network modeling the dynamics of the Budding Yeast cell cycle.

Duong Nguyen, Texas Women in Mathematics Symposium 2018
Abstract:  The Texas Women In Mathematics Symposium (TWIMS) is a 2-day conference for Texas- based mathematicians. The goal of TWIMS is to strengthen the network of female mathematicians in  Texas, which will encourage collaborations and mentoring relationships. In addition, participants have the opportunity to: (1) learn about the research of other women in Texas, (2) present their work in a supportive environment, (3) network with other Texas women mathematicians, and (4) explore issues surrounding being a woman in mathematics. Conference activities include parallel talk sessions Saturday and Sunday, a breakout session, professional development seminars, and a keynote address.

Elpiniki Nikolopoulou, Tumor-immune dynamics with an immune checkpoint inhibitor
Abstract: The use of immune checkpoint inhibitors is becoming more commonplace in clinical trials across the nation. Two important factors in the tumour-immune response are the checkpoint protein programmed death-1 (PD-1) and its ligand PD-L1. We propose a mathematical tumour-immune model using a system of ordinary differential equations to study dynamics with and without the use of anti-PD-1. A sensitivity analysis is conducted, and series of simulations are performed to investigate the effects of intermittent and continuous treatments on the tumour-immune dynamics. We consider the system without the anti-PD-1 drug to conduct a mathematical analysis to determine the stability of the tumour free and tumorous equilibria. Through simulations, we found that a normally functioning immune system may control tumour. We observe treatment with anti-PD-1 alone may not be sufficient to eradicate tumour cells. Therefore, it may be beneficial to combine single agent treatments with additional therapies to obtain a better antitumour response.

Omomayowa Olawoyin, Effects of Multiple Transmission Pathways on Zika Dynamics
Abstract: Although the Zika virus is transmitted to humans primarily through the bite of infected female Aedes aegypti mosquitoes, it can also be sexually and vertically transmitted within both populations. In this study, we develop a new mathematical model of the Zika virus which incorporates sexual transmission in humans and mosquitos, vertical transmission in mosquitos, and mosquito to human transmission through bites. Analysis of this deterministic model shows that the secondary transmission routes of Zika increase the basic reproductive number (R₀) of the virus by 5%, shift the peak time of an outbreak to occur 10% sooner, increase the initial growth of an epidemic, and have important consequences for control strategies and estimates of R₀. Furthermore, sensitivity analysis show that the basic reproductive number is most sensitive to the mosquito biting rate and transmission probability parameters and reveal that the dynamics of juvenile mosquito stages greatly impact the peak time of an outbreak. These discoveries deepen our understanding of the complex transmission routes of ZIKV and the consequences that they may hold for public health officials.

Carolyn Reinhart, The normalized distance Laplacian Matrix
Abstract: The adjacency matrix \(A(G)\) of a graph G is the matrix with entries 1 if two vertices are adjacent and 0 otherwise. The normalized Laplacian matrix is \(\mathcal{L}(G) = I − D(G)^{−1/2}A(G)D(G)^{−1/2}\) where \(\mathcal{D}(G)\) is the diagonal matrix of degrees of the vertices of the graph. Similarly, the distance matrix \(\mathcal{D}(G)\) of a graph G is the matrix containing the pairwise distances between vertices. The transmission of a vertex \(v_i\) in G is the sum of the distances from \(v_i\) to all other vertices and we let \(T(G)\) be the diagonal matrix of transmissions of the vertices of the graph. The new matrix the normalized distance Laplacian, denoted \(\mathcal{D^L}(G)\), is defined such that \(\mathcal{D^L}(G) = I − T(G)^{−1/2}\mathcal{D}(G)T(G)^{−1/2}\). New results to be presented include bounds on the spectral radius of \(\mathcal{D^L}\) and connections with the normalized Laplacian matrix. Methods for determining eigenvalues of \(\mathcal{D^L}\) will also be discussed, including the use of twin vertices (vertices with the same neighborhood). Finally, results on \(\mathcal{D^L}\)-cospectral graphs will be presented. Two non-isomorphic graphs G and H are M-cospectral if M(G) and M(H) have the same multiset of eigenvalues.

Aleksandra Sobieska Snyder, Minimal Free Resolutions over Rational Normal Scrolls
Abstract: Free resolutions of modules over the polynomial ring have a storied and active history of study. However, resolutions over quotients of the polynomial ring are much more mysterious; even simple examples can be infinitely long. The poster will present a minimal free resolution of the ground field over the toric ring arising from rational normal 2-scrolls, as well as a computation of the Betti numbers of the ground field for all rational normal k-scrolls.

Sarah Yoseph, An Enumeration Process of n-Quandles
Abstract: Determining if two knots or links are equivalent is a fundamental problem in knot theory. Much of the research in the field involves the study of knot and link invariants. An invariant is a mathematical object assigned to a knot or link in such a way that equivalent knots are assigned equal objects. There are knot invariants of all types, a somewhat recent knot and link invariant is called a quandle. The quandle of a knot is an algebraic structure that extends one of the most powerful algebraic knot invariants called the knot group. For every natural number n, there is a simpler invariant called the n-quandle. For some well known knots and links, this invariant is easy to compute for n = 2. We will investigate a process of enumerating the n-quandle given by a presentation using tables similar to those in the Todd- Coxeter process.
References
[1] R. Fenn and C. Rourke, “Racks and links in codimension two”, Journal of Knot Theory and Its Ramifications, Vol. 1 No. 4 (1992), 343 – 406.
[2] D. Holt, Discrete Mathematics and Its Applications: Handbook of Computational Group Theory, Chapman & Hall/CRC Press, Boca Raton, FL. 2005.
[3] D. Joyce, “A classifying invariant of knots, the knot quandle”, Journal of Pure and Applied Algebra, 23 (1982), 37 – 65.
[4] S. Winker, “Quandles, knot invariants, and the n-fold branched cover”, Ph.D. Thesis, University of Illinois at Chicago (1984).

Joy Yu, Smoothing Spline Semiparametric Density Models,
Abstract: Density estimation plays a fundamental role in many areas of statistics and machine learning. Semiparametric density models are flexible in incorporating domain knowledge and uncertainty regarding the shape of the density function. We consider a unified framework based on reproducing kernel Hilbert space for modeling, estimation, computation and theory. Our proposed general semiparametric density models include many existing models as special cases. We develop penalized likelihood based estimation methods and computational methods under different situations. We also establish joint consistency and derive convergence rates of the proposed estimators, as well as the convergence rate of the overall density function. Lastly, we validate our estimation methods empirically through simulations and an application.

Fatma Zürnacı, Çetin Dişibüyük. Generalized Taylor Series
Abstract: Divided differences are a basic tool in interpolation and approximation by polynomials and in spline theory. They are directly involved in the definition of B-splines. Recently, in [1], Zürnacı and Dişibüyük give the explicit representation of non-polynomial B-spline functions for a wide collection of spline spaces including trigonometric splines, hyperbolic splines, and special Müntz spaces of splines by using non-polynomial divided differences applied to a proper generalization of truncated-power function. With the definition of a generalized derivative operator, it is shown that as in the polynomial case, non-polynomial divided differences can be viewed as a discrete analogue of derivatives. It is the purpose of this work to obtain generalized Taylor series by this property of non-polynomial divided difference.
References
[1] F. Zürnacı, Ç. Dişibüyük (2019), Non-polynomial divided differences and B-spline functions. Journal of Computational and Applied Mathematics 349: 579-592. doi:10.1016/j.cam.2018.09.026.
[2]  Ç. Dişibüyük, R. Goldman (2016), A unified approach to non-polynomial B-spline curves based on a novel variant of the polar form. Calcolo 53(4) , 751-781. doi:10.1007/s10092-015-0172-x.
[3]  Ç. Dişibüyük, R. Goldman (2015), A unifying structure for polar forms and for Bernstein Bézier curves. Journal of Approximation Theory 192 , 234-249. doi:10.1016/j.jat.2014.12.007.
[4] Ç. Dişibüyük (2015), A functional generalization of the interpolation problem. Applied Mathematics and Computation 256, 247-251. doi:10.1016/j.amc.2014.12.152.