Scholarly Works, Mathematics

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https://hdl.handle.net/10919/24285

Research articles, presentations, and other scholarship

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Extending Elman's Bound for GMRES
Embree, Mark P. (Elsevier, 2025-07-24)
If the numerical range of a matrix is contained in the right half of the complex plane, the GMRES algorithm for solving linear systems will reduce the norm of the residual at every iteration. In his Ph.D. dissertation, Howard Elman derived a bound that guarantees convergence. When the numerical range contains the origin, GMRES need not make progress at every step and Elman's bound does not apply, even if all the eigenvalues are located in the right half-plane. However by solving a Lyapunov equation, one can construct an inner product in which the numerical range is contained in the right half-plane. One can then bound GMRES (run in the standard Euclidean norm) by applying Elman's bound in this new inner product, at the cost of a multiplicative constant that characterizes the distortion caused by the change of inner product. Using Lyapunov inverse iteration, one can build a family of such inner products, trading o the location of the numerical range with the size of constant. This approach complements techniques that Greenbaum and colleagues have recently proposed for excising the origin from the numerical range to gain greater insight into the convergence of GMRES for nonnormal matrices.
Polynomial Approximation to the Inverse of a Large Matrix
Embree, Mark P.; Henningsen, Joel A.; Jackson, Jordan; Morgan, Ronald B. (Society for Industrial & Applied Mathematics, 2025-08-25)
The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its approximation to the inverse often seems to track the accuracy of the GMRES iteration. We investigate the quality of this approximation through theory and experiment, noting the practical need to add copies of some polynomial terms to improve stability. To mitigate storage and orthogonalization costs, other approaches have appeal, such as polynomial preconditioned GMRES and deflation of problematic eigenvalues. Applications of such polynomial approximations include solving systems of linear equations with multiple right-hand sides (where the solutions to subsequent problems come simply by multiplying the polynomial against the new right-hand sides) and variance reduction in multilevel Monte Carlo methods.
An Adaptive Mixed Precision and Dynamically Scaled Preconditioned Conjugate Gradient Algorithm
Guo, Yichen; de Sturler, Eric; Warburton, Tim (2025-05-07)
We propose an adaptive mixed precision and dynamically scaled preconditioned conjugate gradient algorithm (AMP-PCG). It dynamically adjusts the precision for storing vectors and computing, exploiting low precision when appropriate, while maintaining a convergence rate and accuracy comparable to that of double precision PCG. Our mixed precision strategy consists of three main components: (1) The residual and matrix-vector product are initially computed in double precision, and the algorithm switches these to single precision based on the chosen convergence tolerance and an estimate of the residual gap. (2) Depending on the eigenvalue distribution, the preconditioned residual and search direction are either in half precision throughout the iterations or initially in double precision and then stepwise reduced to single and half precision. (3) A dynamically scaled residual is used at every iteration to mitigate underflow in half precision. We provide theoretical support for our estimates and we demonstrate the effectiveness of AMP-PCG through numerical experiments, highlighting both its robustness and the significant performance gains (1.63× speedup) achieved compared to double precision PCG on a GPU.
Macroscopic approximation of tight-binding models near spectral degeneracies and validity for wave packet propagation
Bal, Guillaume; Cazeaux, Paul; Massatt, Daniel; Quinn, Solomon (2026-02-08)
This paper concerns the derivation and validity of macroscopic descriptions of wave packets supported in the vicinity of degenerate points (K,E) in the dispersion relation of tight-binding models accounting for macroscopic variations. We show that such wave packets are well approximated over long times by macroscopic models with varying orders of accuracy. Our main applications are in the analysis of single- and multilayer graphene tight-binding Hamiltonians modeling macroscopic variations such as those generated by shear or twist. Numerical simulations illustrate the theoretical findings.
Computing supersingular endomorphism rings with inseparable endomorphisms
Fuselier, Jenny; Iezzi, Annamaria; Kozek, Mark; Morrison, Travis; Namoijam, Changningphaabi (Academic Press - Elsevier, 2023-06-08)
We give an algorithm for computing an inseparable endomorphism of a supersingular elliptic curve E defined over Fp², which, conditional on GRH, runs in expected O(p^1/2(log⁡p)²(log⁡log⁡p)³) bit operations and requires O((log⁡p)²) storage. This matches the time and storage complexity of the best conditional algorithms for computing a nontrivial supersingular endomorphism, such as those of Eisenträger–Hallgren–Leonardi–Morrison–Park and Delfs–Galbraith. Unlike these prior algorithms, which require two paths from E to a curve defined over Fp, the algorithm we introduce only requires one; thus when combined with the algorithm of Corte-Real Santos–Costello–Shi, our algorithm will be faster in practice. Moreover, our algorithm produces endomorphisms with predictable discriminants, enabling us to prove properties about the orders they generate. With two calls to our algorithm, we can provably compute a Bass suborder of End(E). This result is then used in an algorithm for computing a basis for End(E) with the same time complexity, assuming GRH. We also argue that End(E) can be computed using O(1) calls to our algorithm along with polynomial overhead, conditional on a heuristic assumption about the distribution of the discriminants of these endomorphisms. Conditional on GRH and this additional heuristic, this yields a O(p^1/2(log⁡p)²(log⁡log⁡p)³) algorithm for computing End(E) requiring O((log⁡p)²) storage.
Congruence classes of simplex structures in finite field vector spaces
Cheek, Timothy; Cooper, Joseph; Gilman, Pico; Iosevich, Alex; Jaber, Kareem; Palsson, Eyvindur; Sharan, Vismay; Shuffelton, Jenna; Tomé, Marie-Helene (2025-10-30)
We study a generalization of the Erdős-Falconer distance problem over finite fields. For a graph G, two embeddings p, p′ : V(G) → Fdq of a graph G are congruent if for all edges (vi, vj) of G we have that ||p(vi) − p(vj)|| = ||p′(vi) − p′(vj)||. What is the infimum of s such that for any subset E ⊂ Fdq with |E| ≳ qs, E contains a positive proportion of congruence classes of G in Fdq ? Bennett et al. and McDonald used group action methods to prove results in the case of k-simplices. The work of Iosevich, Jardine and McDonald as well as that of Bright et al. have proved results in the case of trees and trees of simplices, utilizing the inductive nature of these graphs. Recently Aksoy, Iosevich, and McDonald combined these two approaches to obtain nontrivial bounds on the “bowtie” graph, two triangles joined at a vertex. Their proof relies on an application of the Hadamard three-lines theorem to pass to a different graph. We develop novel geometric techniques called branch shifting and simplex unbalancing to reduce our analysis of trees of simplices to a much smaller class of simplex structures. This allows us to establish a framework that handles a wide class of graphs exhibiting a combination of rigid and loose behavior. In F2q , this approach gives new nontrivial bounds on chains and trees of simplices. In Fdq , we improve on the results of Bright et al. in many cases and generalize their work to a wider class of simplex trees. We discuss partial progress on how this framework can be extended to more general simplex structures, such as cycles of simplices and structures of simplices glued together along an edge or a face.
nlKrylov: A Unified Framework for Nonlinear GCR-type Krylov Subspace Methods
Werner, Tom; Wan, Ning; Miedlar, Agnieszka (2025-11-18)
In this paper, we introduce a unified framework for nonlinear Krylov subspace methods (nlKrylov) to solve systems of nonlinear equations. Building on classical GCR-like/type linear Krylov solvers such as GMRESR, we generalize these approaches to nonlinear problems via nested algorithmic structures. We present rigorous convergence results for problems, relying on relaxed assumptions that avoid the need for exact line searches. The framework is further extended to matrix-valued rootfinding problems using global nonlinear Krylov approaches. Extensive numerical experiments validate the theoretical insights and demonstrate the robustness and efficiency of our proposed algorithms.
Mathematical modeling of malaria vaccination with seasonality and immune feedback
Qu, Zhuolin; Patterson, Denis; Zhao, Lihong; Ponce, Joan; Edholm, Christina J.; Prosper-Feldmen, Olivia F.; Childs, Lauren M. (Public Library of Science, 2025-05)
Malaria is one of the deadliest infectious diseases globally, claiming hundreds of thousands of lives each year. The disease presents substantial heterogeneity among the population, with approximately two-thirds of fatalities occurring in children under five years old. Immunity to malaria develops through repeated exposure and plays a crucial role in disease dynamics. Seasonal environmental fluctuations, such as changes in temperature and rainfall, lead to temporal heterogeneity and further complicate transmission dynamics and the utility of intervention strategies. We employ an age-structured partial differential equation model to characterize seasonal malaria transmission and assess vaccination strategies that vary by timing and duration. Our model integrates vector-host epidemiological dynamics across different age groups and nonlinear feedback between transmission and immunity. We calibrate the model to year-round and seasonal malaria settings and conduct extensive sensitivity analyses for both scenarios to systematically assess which assumptions lead to the most uncertainty. We use time-varying sensitivity indices to identify critical disease parameters during low and high transmission seasons. We further investigate the impact of vaccination and its implementation in the seasonal malaria settings. When implementing a three-dose primary vaccination series, seasonally targeted campaigns can prevent significantly more cases per vaccination than constant year-long programs in regions with strong seasonal variation in transmission. In such scenarios, the optimal vaccination interval aligns with the peak in infected mosquito abundance and precedes the peak in malaria transmission. In contrast, seasonal booster programs may provide limited advantages over year-long vaccination. Additionally, while increasing annual vaccination counts can reduce overall disease incidence, it yields marginal improvements in cases prevented per vaccination.
Building as common property: examining Ostrom's model in an innovative university residence hall
Baird, Timothy D.; Tural, Elif; Kniola, David J.; Pingel, Thomas J.; Abaid, Nicole (Routledge, 2025-10-18)
Buildings are not only physical infrastructures but also socially and institutionally produced environments that structure access to space, resources and community life. This study draws from human–environment geography, common property theory and scholarship on built environments to conceptualize buildings as shared indoor environments that function as common pool resources and can be governed as common property regimes. Using an ethnographic approach, we examine a large, mixed-use academic–residential building at a U.S. research university to better understand how it was produced and governed as a shared resource. Data from stakeholder interviews, institutional documents and participant observation reveal governance dynamics that align closely with Ostrom’s design principles, including clear boundaries, collective choice, monitoring and sanctions. We identify both the institutional mechanisms and spatial strategies that contribute to sustainable, cooperative use of shared indoor resources. We also propose a conceptual framework that links building governance to broader national design trends, institutional mental models, and localized scarcities and abundances. Our findings offer practical insights for designers, campus planners and institutional decision-makers seeking to foster more inclusive, adaptive and sustainable building use.
Unveiling an Immunological Mystery: Deciphering the Durability Divide Between Infection and Vaccine-Elicited Antibody Responses
Lewis, George K.; Ciupe, Stanca M.; Sajadi, Mohammad (Bentham Science Publishers, 2025-07-23)
Achieving durable antibody-mediated protection remains critical in vaccine develop-ment, particularly for viral diseases like COVID-19 and HIV. We discuss factors influencing an-tibody durability, highlighting the role of long-lived plasma cells (LLPCs) in the bone marrow, which are essential for sustained antibody production over many years. The frequencies and prop-erties of bone marrow LLPC are critical determinants of the broad spectrum of antibody durability for different vaccines. Vaccines for diseases like measles and mumps elicit long-lasting antibod-ies; those for COVID-19 and HIV do not. High epitope densities in the vaccine are known to favor antibody durability, but we discuss three underappreciated variables that also play a role in long-lived antibody responses. First, in addition to high epitope densities, we discuss the im-portance of CD21 as a critical determinant of antibody durability. CD21 is a B cell antigen recep-tor (BCR) complex component. It significantly affects BCR signaling strength in a way essential for generating LLPC in the bone marrow. Second, all antibody-secreting cells (ASC) are not cre-ated equal. There is a four-log range of antibody secretion rates, and we propose epigenetic im-printing of different rates on ASC, including LLPC, as a factor in antibody durability. Third, antibody durability afforded by bone marrow LLPC is independent of continuous antigenic stim-ulation. By contrast, tissue-resident T-bet+CD21low ASC also persists in secondary lymphoid tissues and continuously produces antibodies depending on persisting antigen and the tissue mi-croenvironment. We discuss these variables in the context of making an HIV vaccine that elicits broadly neutralizing antibodies against HIV that persist at protective levels without continuous vaccination over many years.
On holomorphic conformal structures associated with lattice polarized K3 surfaces
Malmendier, Andreas; Schultz, Michael T. (American Mathematical Society, 2025-06-25)
We discuss the connection between Picard-Fuchs equations for certain families of lattice polarized K3 surfaces and the construction of integrable holomorphic conformal structures on their period domains. We then compute an explicit example of a locally conformally flat holomorphic metric associated with generic Jacobian Kummer surfaces, which allows for a novel description of the local variation of complex structure.
Bistability between acute and chronic states in a Model of Hepatitis B Virus Dynamics
Afrin, Nazia; Ciupe, Stanca M.; Conway, Jessica M.; Gulbudak, Hayriye (Elsevier, 2025-05-31)
Understanding the mechanisms responsible for different clinical outcomes following hepatitis B infection requires a systems investigation of dynamical interactions between the virus and the immune system. To help elucidate mechanisms of protection and those responsible from transition from acute to chronic disease, we developed a deterministic mathematical model of hepatitis B infection that accounts for cytotoxic immune responses resulting in infected cell death, non-cytotoxic immune responses resulting in infected cell cure and protective immunity from reinfection, and cell proliferation. We analyzed the model and presented outcomes based on three important disease markers: the basic reproduction number R0, the infected cells death rate δ (describing the effect of cytotoxic immune responses), and the liver carrying capacity K (describing the liver susceptibility to infection). Using asymptotic and bifurcation analysis techniques, we determined regions where virus is cleared, virus persists, and where clearance-persistence is determined by the size of viral inoculum. These results can guide the development of personalized intervention.
Post-recovery viral shedding shapes wastewater-based epidemiological inferences
Phan, Tin; Brozak, Samantha; Pell, Bruce; Ciupe, Stanca M.; Ke, Ruian; Ribeiro, Ruy M.; Gitter, Anna; Mena, Kristina D.; Perelson, Alan S.; Kuang, Yang; Wu, Fuqing (Springer Nature, 2025-05-22)
Background: The prolonged viral shedding from the gastrointestinal tract is well documented for numerous pathogens, including SARS-CoV-2. However, the impact of prolonged viral shedding on epidemiological inferences using wastewater data is not yet fully understood. Methods: To gain a better understanding of this phenomenon at the population level, we extended a wastewater-based modeling framework that integrates viral shedding dynamics, viral load data in wastewater, case report data, and an epidemic model. Results: Our results indicate that as an outbreak progresses, the viral load from recovered individuals gradually becomes predominant, surpassing that from the infectious population. This phenomenon leads to a dynamic relationship between model-inferred and reported daily incidence over the course of an outbreak. Sensitivity analyses on the duration and rate of viral shedding for recovered individuals reveal that accounting for this phenomenon can considerably advance prediction of transmission peak timing. Furthermore, extensive viral shedding from the recovered population toward the conclusion of an epidemic wave may overshadow viral signals from newly infected cases carrying emerging variants, which can delay the rapid recognition of emerging variants based on viral load. Conclusions: These findings highlight the necessity of integrating post-recovery viral shedding to enhance the accuracy and utility of wastewater-based epidemiological analysis.
Investigation of a Two-Patch Within-Host Model of Hepatitis B Viral Infection
Castellano, Keoni; Saucedo, Omar; Ciupe, Stanca M. (Springer, 2025-12)
Chronic infection with hepatitis B virus (HBV) can lead to formation of abnormal nodular structures within the liver. To address how changes in liver anatomy affect overall virus-host dynamics, we developed within-host ordinary differential equation models of two-patch hepatitis B infection, one that assumes irreversible and one that assumes reversible movement between nodular structures. We investigated the models analytically and numerically, and determined the contribution of patch susceptibility, immune responses, and virus movement on within-patch and whole-liver virus dynamics. We explored the structural and practical identifiability of the models by implementing a differential algebra approach and the Monte Carlo approach for a specific HBV data set. We determined conditions for viral clearance, viral localization, and systemic viral infection. Our study suggests that cell susceptibility to infection within modular structures, the movement rate between patches, and the immune-mediated infected cell killing have the most influence on HBV dynamics. Our results can help inform intervention strategies.
A combinatorial approach to avoiding weak keys in the BIKE cryptosystem
Matthews, Gretchen L.; McMillon, Emily (Springer, 2025-08-01)
Bit Flipping Key Encapsulation (BIKE) is a code-based cryptosystem that was considered in Round 4 of the NIST Post-Quantum Cryptography Standardization process. It is based on quasi-cyclic moderate-density parity-check (QC-MDPC) codes paired with an iterative decoder. While (low-density) parity-check codes have been shown to perform well in practice, their capabilities are governed by the code’s graphical representation and the choice of decoder rather than the traditional code parameters, making it difficult to determine the decoder failure rate (DFR). Moreover, decoding failures have been demonstrated to lead to attacks that recover the BIKE private key. In this paper, we demonstrate a strong correlation between weak keys and 4-cycles in their associated Tanner graphs. We give concrete ways to enumerate the number of 4-cycles in a BIKE key and use these results to present a filtering algorithm that will filter BIKE keys with large numbers of 4-cycles. These results also apply to more general parity check codes.
Reservoir Computation with Networks of Differentiating Neuron Ring Oscillators
Yeung, Alexander; DelMastro, Peter; Karuvally, Arjun; Siegelmann, Hava; Rietman, Edward; Hazan, Hananel (MDPI, 2025-10-20)
Reservoir computing is an approach to machine learning that leverages the dynamics of a complex system alongside a simple, often linear, machine learning model for a designated task. While many efforts have previously focused their attention on integrating neurons, which produce an output in response to large, sustained inputs, we focus on using differentiating neurons, which produce an output in response to large changes in input. Here, we introduce a small-world graph built from rings of differentiating neurons as a Reservoir Computing substrate. We find the coupling strength and network topology that enable these small-world networks to function as an effective reservoir. The dynamics of differentiating neurons naturally give rise to oscillatory dynamics when arranged in rings, where we study their computational use in the Reservoir Computing setting. We demonstrate the efficacy of these networks in the MNIST digit recognition task, achieving comparable performance of 90.65% to existing Reservoir Computing approaches. Beyond accuracy, we conduct systematic analysis of our reservoir’s internal dynamics using three complementary complexity measures that quantify neuronal activity balance, input dependence, and effective dimensionality. Our analysis reveals that optimal performance emerges when the reservoir operates with intermediate levels of neural entropy and input sensitivity, consistent with the edge-of-chaos hypothesis, where the system balances stability and responsiveness. The findings suggest that differentiating neurons can be a potential alternative to integrating neurons and can provide a sustainable future alternative for power-hungry AI applications.
The log Grothendieck ring of varieties
Gross, Andreas; Herr, Leo; Holmes, David; Spelier, Pim; Vogel, Jesse (Wiley, 2025-10-06)
We define a Grothendieck ring of varieties for log schemes. It is generated by one additional class “𝑃” over the usual Grothendieck ring. We show the naïve definition of log Hodge numbers does not make sense for all log schemes. We offer an alternative that does.
Matrix Multiplication: Sharing Computation without Sharing Data
Brock, Charlotte; Durisheti, Medha; Hinds, Claire; Matthews, Gretchen L.; Santos, Welington (2019-12)
Absorbing sets of LDPC codes under a Gallager B decoding variant
Dhingra, Renaya; Matthews, Gretchen L.; McDevitt, Collin; McMillon, Emily (Mathematical Sciences Publishers, 2026)
Absorbing sets are graphical structures that cause iterative decoders to fail. The sets depend on the particular iterative decoding algorithm. In this paper, we consider a variant of the wellstudied Gallager B decoding algorithm for binary low-density parity-check codes. Here, a bit node updates its value according to either the number of or fraction of unsatisfied neighboring check nodes. The traditional Gallager B algorithm is the special case in which half (meaning a fraction of ½) of the check nodes being unsatisfied is required for a bit to change its value. We study absorbing sets in these settings along with the received words which give decoder failure. We determine the how changing the update rules pick up new absorbing sets. This allows for fine-tuning the standard algorithm according to the particular channel or error probability. We also connect these notions to Boolean functions.
Fast growth rate is associated with musculoskeletal biomechanical imbalance and dorsal cranial myopathy in broiler chickens
Lourenço-Silva, Marconi Italo; Norton, Anderson H. III; Jacobs, Leonie (Public Library of Science, 2025-09-01)
Dorsal cranial myopathy is a degenerative lesion that affects the anterior Latissimus dorsi muscle in broiler chickens, with an etiology that remains unknown. The objective was to investigate the influence of musculoskeletal biomechanical balance and gait on the prevalence of dorsal cranial myopathy in three broiler chicken strains with differing growth potential. Three-hundred and ninety-six broiler chickens from three genetic strains with differing growth potential (fast, intermediate, and slow, 132 birds/ strain) were housed in 18 pens with 22 birds/pen. Five birds/pen (n = 30 birds/genetic strain) were randomly wing- or leg-banded to assess gait and musculoskeletal biomechanical balance (by calculating body angulation) at 1, 2, 3, and 3.7 kg weight sampling points. Dorsal cranial myopathy was assessed one day after birds reached final body weight. Gait and musculoskeletal balance were both negatively impacted by body weight in fast- and slow-growing strains but not in the intermediate-growing strain. Dorsal cranial myopathy was more prevalent in fast-growing broilers compared to other strains, with no case observed in the slow-growing strain. Impaired gait negatively affected musculoskeletal biomechanical balance and increased the prevalence of dorsal cranial myopathy. Our results suggest that genetic strain, musculoskeletal biomechanical imbalance, poor gait, and high body weight are all associated with the prevalence of dorsal cranial myopathy in broiler chickens. We successfully simplified a non-invasive body posture methodology to quantify the musculoskeletal biomechanical balance in broiler chickens.

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