Abstract. In this paper, we finally prove the well-posedness of the linearized R13 moment model, which describes, e.g., rarefied gas flows. As an extension of the classical fluid equations, moment models are robust and have been frequently used, yet they are challenging to analyze due to their additional equations. By effectively grouping variables, we identify a 2-by-2 block structure, allowing us to analyze well-posedness within the abstract LBB framework for saddle point problems. Due to the unique tensorial structure of the equations, in addition to an interesting combination of tools from Stokes’ and linear elasticity theory, we also need new coercivity estimates for tensor fields. These Korn-type inequalities are established by analyzing the symbol map of the symmetric and trace-free part of tensor derivative fields. Together with the corresponding right inverse of the tensorial divergence, we obtain the existence and uniqueness of weak solutions. This result also serves as the basis for future numerical analysis of corresponding discretization schemes.
Abstract. In the early stages of a newly emerged or reemerged disease, there is a rapid increase in new infections, which can potentially lead to a healthcare crisis due to constraints of available medical resources. It is important to note that the recovery period of these emerging diseases typically follows a Gamma distribution rather than being narrowly centered around the mean. In this study, we propose a susceptible-infectious-recovered (SIR) network model that incorporates a general recovery rate and a saturation treatment function. We establish the well-posedness and global stability of the disease-free equilibrium in the model by employing semigroup theory and the standard comparison principle, respectively. From an epidemiological perspective, when the delayed treatment effect exceeds a significant threshold, a phenomenon known as backward bifurcation emerges near the disease-free equilibrium. This assertion is supported by an updated version of the Lyapunov–Schmidt approach. Additionally, we conduct numerical simulations to investigate how network topology and non-Markovian processes affect the patterns of disease transmission.
A Bayesian perspective in the data-driven modeling of complex processes offers the flexibility to construct hybrid models that consider both domain knowledge and statistical assumptions. With this approach, physical or physiological constraints can be naturally enforced across estimated values and uncertainty. In this study, we develop a Bayesian hierarchical model (BHM) that models insulin secretion rate (ISR) as a mixed-effects, log-Gaussian process, ensuring positive estimates across the inferred posterior distribution. Our approach improves upon a previously developed BHM that allowed for nonphysiological negative ISR estimates. By applying a logarithmic transformation, we integrate positivity directly into the model and leverage a quadratic mean trend to capture ISR's natural rise and fall. We implement an inversion method based on Newton--Raphson, allowing for computationally efficient inference and uncertainty quantification. Applying our model to oral glucose tolerance test (OGTT) data from youth with and without cystic fibrosis (CF), we find that ISR estimates obtained using the log-Gaussian approach are generally consistent with estimates using the Gaussian approach. However, the log-Gaussian approach eliminates negative predictions for both the ISR and its credible envelopes and generally reduces uncertainty in ISR estimates. By incorporating physiological constraints directly into the model structure, this novel framework enhances ISR inference for metabolic research and offers a robust tool for assessing beta-cell function in health and disease.
In various intracellular processes, particles move in a unidirectional manner, attaching or detaching from specific sites depending on the states of neighboring sites. Inspired by the critical role of neighboring site-dependent transitions and their significant impact on driven stochastic transport systems, we investigate how these transitions influence the overall dynamics of the system. In this paper, we introduce a deterministic model called the ribosome flow model with neighbor-dependent Langmuir kinetics (RFMNDLK), in which ribosome detachment and attachment are influenced by the occupancy of adjacent sites. Our findings reveal that the RFMNDLK system, where the forward rate dominates, exhibits a globally asymptotically stable equilibrium point, as demonstrated using contraction theories of dynamical systems. We further prove that the steadystate responds monotonically to the Langmuir parameters, implying that the enhanced premature ribosome drop-off reduces translational efficiency. Finally, we simulate the model using Monte Carlo simulations, and the numerical solutions of the governing equations closely match the results obtained from the simulations. The overall study suggests that slow elongation regions can induce ribosome traffic jams that limit protein production under high initiation rates, while collision-induced ribosome drop-off alleviates these jams and sustains translation efficiency.
This paper investigates the three-dimensional interaction between deformable elastic solids and electromagnetic fields. A novel boundary integral formulation is introduced for this coupled system, through which we reveal a previously unreported subwavelength resonance phenomenon. This resonance arises in two specific high-contrast conditions: a high contrast between the solid's Lame'\ parameters (the compression and shear moduli) and the reference electric field energy density, and a high contrast between the solid's mass density and the effective mass density associated with the reference electric field energy. By analyzing an equivalent boundary integral reformulation of the coupled system, we derive rigorous asymptotic expansions characterizing these resonances. Our results have significant implications for inverse scattering problems and the design of advanced metamaterials.
We study the population dynamics of a single species modeled by a Fisher-KPP equation with an integral weighted free boundary condition in a two dimensional region. The spatial domain of the model is a rectangle with one side having a moving boundary and the remaining three sides fixed. We prove the global existence and uniqueness of solution, and show the vanishingbalancing-spreading trichotomy dynamics of the model. By numerical simulation, the steady states exhibit a finger-shaped front pattern, a long-lasting topic of interest in spatial ecology. For certain sets of parameters, the finger-shaped free boundary could vary over time.
This paper presents the global dynamics of a three-dimensional predator--prey system with asymmetric competition, where two predators compete for a single prey and only one predator employs the cooperative hunting strategy. We provide the complete topological classifications of the equilibria and prove the existence of a periodic orbit arising from the positive equilibrium via Hopf bifurcation, showing the coexistence of the three species. Then, we study the dynamics at infinity on the Poincare'\ sphere, showing the boundedness of the species dynamics and conditions for competitive exclusion. Finally, we prove that this system only has Darboux polynomials x, y, and z and possesses no algebraic first integral, which indicates that this system is complicated from the perspective of integrability.
We study the formation of chiral structures in chiral and achiral liquid crystal systems in the simplified setting of one dimension spatial dependence. For cholesteric (chiral) liquid crystals, we characterize minimizers when the intrinsic pitch conflicts with the twist imposed by the boundary conditions and it is within a certain range of values. For nematic liquid crystals composed of achiral bent-core molecules, known to exhibit periodically modulated structures, we analyze the twist-bend nematic phase, where the molecules arrange in a heliconical structure with a nanoscale pitch. We show that the heliconical formation is the ground state within a certain parameter range of the material and identify the parameter regimes that ensure the stability of strong local minimizers of the nematic phase, and guarantee the global optimality of the cholesteric phase, for appropriate anchoring conditions. Numerical simulations based on constrained minimization explore the effects of a constant applied magnetic field.
In infectious disease containment, the rational formulation and selection of control measures and isolation measures, as two classic approaches, are of utmost importance. This paper uses susceptible-infected-susceptible (SIS) epidemic reaction-diffusion models to conduct a comparative analysis of four disease containment measures: core control zone, peripheral control zone, core isolation zone, and peripheral isolation zone, in spatially heterogeneous environments. Key findings reveal that control zones universally reduce the basic reproduction number of the disease, whereas isolation zones exhibit context-dependent efficacy. Specifically, setting up isolation zones in low-risk areas can reduce the basic reproduction number of the disease, but isolation zones in high-risk areas may exacerbate the outbreak of the epidemic. Allowing infected individuals to move freely throughout the region and centralizing disease control in the core (resp., peripheral) zone is more effective than isolating them in the peripheral (resp., core) zone. For effective containment, prioritize the establishment of control zones in high-risk areas and establish isolation zones in low-risk areas where medical resources are adequate. Furthermore, in the case of extremely high disease infectivity, control zones preserve susceptible individuals within them, while isolation zones lead to collapse of the entire region. Finally, numerical simulations validate the theoretical results.
We investigate the Helmholtz equation in a two-dimensional open waveguide with a thin and high contrast core layer. We develop an asymptotic analysis of the Green function of the problem, and through it we identify and characterize the appearance of resonant frequencies. For waves originating outside of the core, the waveguide response at these resonant frequencies is vastly different from the response at nonresonant frequencies. Using this phenomenon and multifrequency measurements containing the first resonance, we propose, theoretically analyze, and numerically validate a reconstruction algorithm to identify the location, thickness, and index of refraction of the core layer.
Abstract. This work presents a mathematical model for the optimal control of thin-film flows over a flexible substrate influenced by an external force. The objective is to find the optimal distributed force acting on the topography that minimizes the differences between actual and desired thin-film profiles. A nonlinear lubrication equation governing the fluid dynamics and appropriate functional settings for this model are presented. It is also shown that this system satisfies a global energy-dissipation law for a suitable energy functional. Optimality conditions are derived for the solution of the minimization problem of a specified cost function across a time horizon. These conditions are formulated at a continuous level as a system of coupled, forward-backward PDEs, which are subsequently discretized for numerical investigation. To ensure computational efficiency and stability, first-order implicit-explicit time-stepping schemes are employed to handle the nonlinearities in the model, and a reduced gradient descent algorithm is applied to obtain a numerical approximation of the optimal control signal. Numerical results illustrate that controlling the thin film, even during rupture, achieves a precise film profile. This control strategy accelerates convergence towards a steady state, reduces instabilities, stabilizes dewetting processes, and meets the desired profile specifications.
Abstract. In this paper, we aim to construct a general research framework for the generalized planar impulsive semidynamic system (ISDS) and evaluate the effectiveness of control strategies by exploring its dynamics. To do this, we propose a Kolmogorov-type predator-prey system with nonlinear state-dependent feedback control, which means that when the combination of the number of prey species and its growth rates is below the action threshold, no control measures are implemented; once the action threshold is reached, control measures are immediately implemented. We define two one-dimensional discrete Poincaré maps, which are determined by the difference equation iterated by the pulse point sequence. Based on those, the existence and stability of order-1 periodic solutions, the existence of order-[Formula: see text] periodic solutions, and the sufficient conditions for the ISDS to undergo transcritical bifurcation with respect to the key parameters are identified. Finally, to illustrate the applications of the main results, an example in pest control and its numerical analyses are provided.
Abstract. We present a geometric reformulation of the Hückel molecular orbital/tight-binding (HMO/TB) method that integrates tools from algebraic graph theory and statistical mechanics. By extending classical notions of charge density and bond order to incorporate bond orbital energies, we construct a probabilistic framework that embeds molecular graphs into Euclidean (pseudometric) space. This embedding introduces new geometric descriptors—such as internodal distances, node angles, and electronic propagation efficiency—that correlate with chemical properties including bond lengths, reactivity, and molecular stability. Analytical results for linear and cyclic polyenes as well as polycyclic aromatic hydrocarbons illustrate the utility of these descriptors and suggest a pathway toward closer integration of HMO/TB with ab initio methods and machine learning approaches in molecular modeling.
The nematic liquid crystal droplet problem is of significant interest due to the complex interplay between variable droplet shapes and orientation fields, with numerous applications in physics and material science. Here we introduce an improved diffuse-interface Landau--de Gennes model for nematic liquid crystal droplets, which eliminates an artificial penalty term from [Wu et al., SIAM J. Math. Anal., 57 (2025), pp. 4358--4395] that is inconsistent with the classical droplet problem. We prove the existence of minimizers and \Gamma-convergence to a physically interpretable sharp-interface energy functional. We also present asymptotic solutions across the interface between the interior nematic droplet and external isotropic phase. Numerical simulations reveal optimal droplet configurations---radial, ring, and tactoid---highlighting the mutual influence between topological defects and droplet morphology. Furthermore, we construct a phase diagram of (meta)stable configurations as a function of temperature and domain size, delineating distinct stability regimes for biaxial and uniaxial phases.
Many real objects are modeled as discrete sets of points, such as corners or other salient features. For our main applications in chemistry, points represent atomic centers in a molecule or a solid material. We study the problem of classifying discrete (finite and periodic) sets of unordered points under isometry, which is any transformation preserving distances in a metric space. Experimental noise motivates the new practical requirement to make such invariants Lipschitz continuous so that perturbing every point in its \varepsilon -neighborhood changes the invariant up to a constant multiple of \varepsilon in a suitable distance satisfying all metric axioms. Since the given points are unordered, the key challenge is to compute all invariants and metrics in a near-linear time of the input size. We define the Pointwise Distance Distribution (PDD) for any discrete set and prove, in addition to the properties above, the completeness of PDD for all periodic sets in general position. The PDD can compare nearly 2 million crystals from the world's five largest databases within 2 hours on a modest desktop computer. The impact is upholding data integrity in crystallography because the PDD will not allow anyone to claim a ``new"" material as a noisy disguise of a known crystal.
Optimal transport (OT) theory and models have been widely applied in the fields regularization is most commonly used in practice due to its simple computational scheme. The other two formulations are rarely regularized since adding regularization terms brings additional difficulties to their theoretical analysis and numerical algorithms. In this paper, we propose a regularized OT model based on the dynamic formulation. We incorporate a W1,2 regularization term on the momentum field to promote smoothness, and a total variation regularization term on the density field to enhance robustness to noise. We establish the existence and uniqueness of solutions to the regularized dynamic OT model and provide a theoretical analysis based on \Gamma -convergence. Moreover, we develop a primal-dual algorithm to efficiently solve the resulting optimization problem. Numerical experiments on spatial-temporal image generation tasks demonstrate the effectiveness of the proposed model and the stability and efficiency of the proposed numerical algorithm. The proposed regularized dynamical OT model provides a flexible and theoretically grounded framework for image generation tasks in imaging science.
We introduce a new family of artificial backgrounds corresponding to averaged impedance boundary conditions formulated in an abstract framework. These backgrounds are used to define a finite number of averaged Steklov eigenvalues, which are associated with inverse scattering problems from inhomogeneous media. We prove that these special eigenvalues can be determined from full-aperture, fixed-frequency far fields using the inside-outside duality method. We then numerically demonstrate how this method can be used to reconstruct averaged values of the refractive index.
We study the shunt boundary condition for second order elliptic PDEs. A specific case of this is the PDE describing the electrostatic potential for a conductive body into which current is injected through electrodes that touch the boundary. We obtain the optimal description of the gradient of the electrostatic potential upon approach to the edge of the electrodes. This generalizes the classical Zaremba problem. The asymptotic description we obtain is then implemented to produce a substantial speed-up of numerical solvers for this boundary-value problem.
By utilizing the linear conjugacy relationship, the stability of a well-characterized system can be transferred to an unknown one. A key application lies in extending the stability of complex balanced (CB) systems to general systems linearly conjugate to them. However, introducing time delays to certain CB systems and their linearly conjugate counterparts may disrupt this conjugacy relationship, thereby hindering the transfer of stability properties. In this paper, we first establish several sufficient conditions to ensure the Lyapunov stability of the delayed version of systems linearly conjugated to CB systems. To further address the degenerated equilibrium in the stoichiometric compatibility class in such systems, we redefine the invariant sets of trajectories and extend the Lyapunov stability to achieve the local asymptotic stability with respect to the newly defined invariant sets. Illustrative examples, such as the PAK-1 network, are provided to validate the theoretical findings.
We investigate the structure of the endemic equilibrium (EE) set in a multipatch susceptible-infectious-susceptible (SIS) epidemic model with mass-action transmission mechanism. For the corresponding single-patch model, the basic reproduction number R-0 completely determines the disease dynamics: a unique stable EE exists if and only if R-0 > 1. In contrast, the multipatch setting exhibits far more complex behavior. First, we show that as the dispersal rate of susceptible individuals d(S) varies, the EE set consists of a finite union of disjoint curves. Under mild conditions, when R-0 < 1, the closure of each curve forms a loop in the d(S) x || I || (1)-plane, where || I ||(1) denotes the total infected population at equilibrium. When R-0 > 1, the EE set contains two distinct types of curves: bounded and unbounded. Moreover, the structure of the EE set as d(S) tends to zero provides explicit spatial patterns of the EEs, offering insights into how restricting susceptible movement influences disease dynamics. Second, using R-0 as the bifurcation parameter, we establish that the EE set forms a simple unbounded curve in the R-0 x || I || (1)-plane, which may exhibit an Sshape and undergo either a backward or forward transcritical bifurcation at R-0 = 1. These results refine and extend previous findings by revealing novel and intricate structures within the EE set. They underscore the interplay between population movement, total population size, and spatial heterogeneity, providing new insights into the long-term dynamics of infectious diseases.
