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In this paper, we propose a new approach to deformable image registration that captures sliding motions. The large deformation diffeomorphic metric mapping (LDDMM) registration method faces challenges in representing sliding motion since it per construction generates smooth warps. To address this issue, we extend LDDMM by incorporating both zeroth- and first-order momenta with a nondifferentiable kernel. This allows us to represent both discontinuous deformation at switching boundaries and diffeomorphic deformation in homogeneous regions. We provide a mathematical analysis of the proposed deformation model from the viewpoint of discontinuous systems. To evaluate our approach, we conduct experiments on both artificial images and the publicly available DIR-Lab 4DCT dataset. Results show the effectiveness of our approach in capturing plausible sliding motion. [ABSTRACT FROM AUTHOR]

Phages are ubiquitous in nature, but many essential factors of host-phage biology have not yet been integrated into mathematical models. In this paper, we investigate a spatial phage-bacteria model to describe the propagation of phages and bacteria in different types of nutrient media. Unlike existing models, we construct a more realistic reaction-diffusion model that incorporates inoculum and bacterial growth and movement, then rigorous mathematical analysis is challenging. We study traveling wave solutions and obtain complete information about the existence and nonexistence of nontrivial traveling wave solutions. The threshold conditions for the existence and nonexistence of traveling wave solutions are obtained by using Schauder's fixed point theorem, limiting argument, and one-sided Laplace transform. Considering different propagation media, we extend the existence of traveling wave solutions from liquid nutrition model to agar model. Moreover, in the absence of bacterial mortality, we obtain the existence of a new traveling wave solution describing phage invasion. We attempt to explain the occurrence of co-transport by the existence and nonexistence of traveling waves, and screen out the key parameters affecting the co-transport of phages and bacteria according to the definition of critical wave speed. Finally, we provide numerical simulations to verify the theoretical results and reveal the effects of key parameters on the propagation of phages and bacteria. [ABSTRACT FROM AUTHOR]

This article is concerned with a class of prey-predator systems with indirect preytaxis, which means the stimulus released by the prey causes the directional movement of the predator. Under homogeneous Neumann boundary conditions, the dynamics of such systems have been studied extensively. However, the corresponding Dirichlet problem remains largely unexplored. The purpose of this paper is to discuss the existence and nonexistence of positive steady-state solutions and make a detailed description for the global bifurcation structure of the set of positive steady-state solutions. Compared with the original systems without indirect prey-taxis, our mathematical analysis shows that the introduction of indirect prey-taxis not only makes mathematical analysis more difficult, but also affects the bifurcation structure of the set of positive steady-state solutions. We hope this work will inspire further research on prey-predator systems with indirect prey-taxis. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the energy minimization model arising in the ensemble Kohn--Sham density functional theory for metallic systems, in which a pseudo-eigenvalue matrix and a general smearing approach are involved. We study the invariance of the energy functional and the existence of the minimizer of the ensemble Kohn--Sham model. We propose an adaptive twoparameter step size strategy and the corresponding preconditioned conjugate gradient methods to solve the energy minimization model. Under some mild but reasonable assumptions, we prove the global convergence for the gradients of the energy functional produced by our algorithms. Numerical experiments show that our algorithms are efficient, especially for large scale metallic systems. In particular, our algorithms produce convergent numerical approximations for some metallic systems, for which the traditional self-consistent field iterations fail to converge. [ABSTRACT FROM AUTHOR]

Magnetic resonance electrical impedance tomography (MREIT) aims to recover the electrical conductivity distribution of an object using partial information of magnetic flux densities inside the tissue which can be measured using a magnetic resonance imaging scanner, with the advantage that a higher spatial resolution of conductivity image can be provided than existing EIT techniques involving surface measurements. Traditional MREIT reconstruction algorithms use two data sets obtained with two linearly independent injected currents. However, injection of two currents is often not possible in applications such as transcranial electrical stimulation. Recently, we proposed an iterative conductivity reconstruction algorithm called the single-current harmonic Bz algorithm that demonstrated satisfactory performance in numerical and phantom tests. In this paper, we provide a rigorous mathematical analysis of the convergence of the iterative sequence for realizing this algorithm. We prove that, applying some mild conditions on the exact conductivity, the iterative sequence converges to the true solution within an explicit error bound. Such theoretical results substantiate the reasonability and efficiency of the proposed algorithm. We also provide more numerical evidence to validate these theoretical results. [ABSTRACT FROM AUTHOR]

We propose a mathematical theory of acoustic wave scattering in one-dimensional finite high-contrast media. The system considered is constituted of a finite alternance of high-contrast segments of arbitrary lengths and interdistances, called the "resonators," and a background medium. We prove the existence of subwavelength resonances, which are the counterparts of the well-known Minnaert resonances in three-dimensional systems. One of the main contribution of the paper is to show that the resonant frequencies as well as the transmission and reflection properties of the system can be accurately predicted by a "capacitance" eigenvalue problem, analogously to the threedimensional setting. Moreover, we discover new properties which are peculiar to the one-dimensional setting, notably the tridiagonal structure of the capacitance matrix as well as the fact that the first resonant frequency is always exactly zero, implying a low-pass filtering property of a one-dimensional chain of resonators. Numerical results considering different situations with N = 1 to N = 6 resonators are provided to support our mathematical analysis and to illustrate the various possibilities offered by high-contrast resonators to manipulate waves at subwavelength scales. [ABSTRACT FROM AUTHOR]

The goal of this paper is to relate the capacitance matrix formalism to the tightbinding approximation. By doing so, we open the way to the use of mathematical techniques and tools from condensed matter theory in the mathematical and numerical analysis of metamaterials, in particular for the understanding of their topological properties. We first study how the capacitance matrix formalism, both when the material parameters are static and modulated, can be posed in a Hamiltonian form. Then, we use this result to compare this formalism to the tight-binding approximation. We prove that the correspondence between the capacitance formulation and the tight-binding approximation holds only in the case of dilute resonators. On the other hand, the tight-binding model is often coupled with a nearest-neighbor approximation, whereby long-range interactions are neglected. Even in the dilute case, we show that long-range interactions between subwavelength resonators are relatively strong and nearest-neighbor approximations are not generally appropriate. [ABSTRACT FROM AUTHOR]

Image deblurring under Rician noise has attracted considerable attention in imaging science. Frequently appearing in medical imaging, Rician noise leads to an interesting nonconvex optimization problem, termed as the MAP-Rician model, which is based on the Maximum a Posteriori (MAP) estimation approach. As the MAP-Rician model is deeply rooted in Bayesian analysis, we want to understand its mathematical analysis carefully. Moreover, one needs to properly select a suitable algorithm for tackling this nonconvex problem to get the best performance. This paper investigates both issues. Indeed, we first present a theoretical result about the existence of a minimizer for the MAP-Rician model under mild conditions. Next, we aim to adopt an efficient boosted difference of convex functions algorithm (BDCA) to handle this challenging problem. Basically, BDCA combines the classical difference of convex functions algorithm (DCA) with a backtracking line search, which utilizes the point generated by DCA to define a search direction. In particular, we apply a smoothing scheme to handle the nonsmooth total variation (TV) regularization term in the discrete MAP-Rician model. Theoretically, using the Kurdyka--Lojasiewicz (KL) property, the convergence of the numerical algorithm can be guaranteed. We also prove that the sequence generated by the proposed algorithm converges to a stationary point with the objective function values decreasing monotonically. Numerical simulations are then reported to clearly illustrate that our BDCA approach outperforms some state-of-the-art methods for both medical and natural images in terms of image recovery capability and CPU-time cost. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a mean curvature of high magnitude can be reconstructed more easily and stably. In proving this, we develop a novel mathematical scheme by analyzing the generalized polarization tensors (GPTs) and the scattering coefficients (SCs) coming from the associated scattered fields, which in turn boils down to the analysis of the layer potential operators that sit inside the GPTs and SCs via microlocal analysis. In a delicate and subtle manner, we decompose the reconstruction process into several steps, where all but one step depends on the global geometry, and one particular step depends on the mean curvature at the given boundary point. Then by a sensitivity analysis with respect to local perturbations of the curvature of the boundary surface, we establish the local resolution effects. Our study opens up a new field of mathematical analysis on wave superresolution imaging. [ABSTRACT FROM AUTHOR]

This paper is mainly concerned with a two-species competition model in open advective environments, where individuals cannot pass through the upstream boundary and do not return to the habitat after leaving the downstream boundary. By the theory of principal eigenvalue, we first obtain two critical curves (Γ1 and Γ2) in the plane of bifurcation parameters that sharply determine the local stability of the two semitrivial steady states. Then under various conditions on given parameters, we discuss the global dynamics via different techniques, including the comparison principle for eigenvalues and perturbation and compactness arguments, and show that both competitive exclusion and coexistence are possible. For general values of parameters, we take both analytic and numerical approaches to further understand the potential behaviors of Γ1 and Γ2, and we numerically observe that in addition to the competitive exclusion and coexistence, the bistable phenomenon is also possible, which is different from the known results of competitive ODE and reaction-diffusion systems (where bistability is impossible). The implication of our numerical results on future work is also discussed. [ABSTRACT FROM AUTHOR]

In this paper we characterize the regularity structure, as well as show the global-in-time existence and uniqueness, of (energy) conservative solutions to the Hunter--Saxton equation by using the method of characteristics. The major difference between the current work and previous results is that we are able to characterize the singularities of energy measure and their nature in a very precise manner. In particular, we show that singularities, whose temporal and spatial locations are also explicitly given in this work, may only appear at at most countably many times, and are completely determined by the absolutely continuous part of initial energy measure. Our mathematical analysis is based on using the method of characteristics in a generalized framework that consists of the evolutions of solutions to the Hunter--Saxton equation and the energy measure. This method also provides a clear description of the semigroup property for the solution and energy measure for all times. [ABSTRACT FROM AUTHOR]

The blade element momentum (BEM) theory introduced by Lock et al. and formulated in its modern form by Glauert provides a framework to model the aerodynamic interaction between a turbine and a fluid flow. This theory is used either to estimate turbine efficiency or as a design aid. However, a lack of mathematical interpretation limits the understanding of some of its issues. The aim of this paper is to propose an analysis of BEM equations. Our approach is based on a reformulation of Glauert's model which enables us to identify criteria to guarantee the existence of solutions, analyze the convergence of usual and new (and more efficient) solution algorithms, and study turbine design procedures. The mathematical analysis is completed by numerical experiments. [ABSTRACT FROM AUTHOR]

This work concerns the asymptotic analysis of high-frequency wave propagation in randomly layered media with fast variations and long-range correlations. The analysis takes place in the three-dimensional physical space and weak-coupling regime. The role played by the slow decay of the correlations on a propagating pulse is twofold. First we observe a random travel time characterized by a fractional Brownian motion that appears to have a standard deviation larger than the pulse width, which is in contrast with the standard O'Doherty-Anstey theory for random propagation media with mixing properties. Second, a deterministic pulse deformation is described as the solution of a paraxial wave equation involving a pseudodifferential operator. This operator is characterized by the autocorrelation function of the medium fluctuations. In case of fluctuations with long-range correlations this operator is close to a fractional Weyl derivative whose order, between 2 and 3, depends on the power decay of the autocorrelation function. In the frequency domain, the pseudodifferential operator exhibits a frequency-dependent power-law attenuation with exponent corresponding to the order of the fractional derivative, and a frequency-dependent phase modulation, both ensuring the causality of the limiting paraxial wave equation as well as the Kramers-Kronig relations. The mathematical analysis is based on an approximation-diffusion theorem for random ordinary differential equations with long-range correlations. [ABSTRACT FROM AUTHOR]

Systems of differential equations with polynomial right-hand sides are very common in applications. On the other hand, their mathematical analysis is very challenging in general, due to the possibility of complex dynamics: multiple basins of attraction, oscillations, and even chaotic dynamics. Even if we restrict our attention to mass-action systems, all of these complex dynamical behaviors are still possible. On the other hand, if a polynomial dynamical system has a weakly reversible deficiency zero (WR0) realization, then its dynamics is known to be remarkably simple: oscillations and chaotic dynamics are ruled out, and, up to linear conservation laws, there exists a single positive steady state, which is asymptotically stable. Here we describe an algorithm for finding WR0 realizations of polynomial dynamical systems, whenever such realizations exist. [ABSTRACT FROM AUTHOR]

The finite-time stabilization problem is considered for a class of multi-input multioutput nonlinear systems composed of several different subsystems that are coupled with unknown nonlinearities, possibly superlinear and nonvanishing at the origin. A novel decentralized, continuous finite-time output-feedback control algorithm is presented by dynamically compensating the unknown nonlinear couplings and applying a saturation technique. A crucial strategy is to estimate the uncertain nonlinearities and the unmeasured state by means of a finite-time extended state observer. The effectiveness of the proposed decentralized finite-time output-feedback design is validated by rigorous mathematical analysis and numerical simulations. [ABSTRACT FROM AUTHOR]

A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation. We present a comprehensive proof of Bertrand's Theorem that is accessible to undergraduate students who are familiar with basic notions from advanced calculus and differential equations. [ABSTRACT FROM AUTHOR]

Mean-field equations have been developed recently to approximate the dynamics of the Deffuant model of opinion formation. These equations can describe both fully mixed populations and the case where individuals interact only along edges of a network. In each case, interactions only occur between individuals whose opinions differ by less than a given parameter, called the confidence bound. The size of the confidence bound parameter is known to strongly affect both the dynamics and the number and location of opinion clusters. In this work, we carry out a mathematical analysis of the mean-field equations to investigate the role of the confidence bound and boundaries on these important observables of the model. We consider the limit in which the confidence bound interval is small, and we identify the key mechanisms driving opinion evolution. We show that linear stability analysis can predict the number and location of opinion clusters. Comparison with numerical simulations of the model illustrates that the early-time dynamics and the final cluster locations can be accurately approximated for networks composed of two degree classes, as well as for the case of a fully mixed population. [ABSTRACT FROM AUTHOR]

This article considers the minimization of the total number of infected individuals over the course of an epidemic in which the rate of infectious contacts can be reduced by timedependent nonpharmaceutical interventions. The societal and economic costs of interventions are taken into account using a linear budget constraint which imposes a trade-off between short-term heavy interventions and long-term light interventions. We search for an optimal intervention strategy in an infinite-dimensional space of controls containing multiple consecutive lockdowns, gradually imposed and lifted restrictions, and various heuristic controls based, for example, on tracking the effective reproduction number. Mathematical analysis shows that among all such strategies, the global optimum is achieved by a single constant-level lockdown of maximum possible magnitude. Numerical simulations highlight the need for careful timing of such interventions and illustrate their benefits and disadvantages compared to strategies designed for minimizing peak prevalence. Rather counterintuitively, adding restrictions prior to the start of a well-planned intervention strategy may even increase the total incidence. [ABSTRACT FROM AUTHOR]

The importance of myocardial perfusion at the outset of cardiac disease remains largely understudied. To address this topic we present a mathematical model that considers the systemic circulation, the coronary vessels, the myocardium, and the interactions among these components. The core of the whole model is the description of the myocardium as a multicompartment poromechanics system. A novel decomposition of the poroelastic Helmholtz potential involved in the poromechanics model allows for a quasi-incompressible model that adequately describes the physical interaction among all components in the porous medium. We further provide a rigorous mathematical analysis that gives guidelines for the choice of the Helmholtz potential. To reduce the computational cost of our integrated model we propose decoupling the deformation of the tissue and systemic circulation from the porous flow in the myocardium and coronary vessels, which allows us to apply the model also in combination with precomputed cardiac displacements, obtained form other models or medical imaging data. We test the methodology through the simulation of a heartbeat in healthy conditions that replicates the systolic impediment phenomenon, which is particularly challenging to capture as it arises from the interaction of several parts of the model. [ABSTRACT FROM AUTHOR]

We consider how transformation optics and invisibility cloaking can be used to construct models in subsets ℝ³ with a varying metric, where the time-harmonic waves for a given angular wavenumber k are equivalent to the waves in some closed orientable manifold. Our model can be used in two ways. First, the model makes it possible to do mathematical analysis on a compact 3-manifold M by smoothly embedding the set M \ L, where L is a finite union of closed curves, to an open subset of ℝ³. Second, the model could in principle be physically implemented using a device built from metamaterials. In particular the measurements in the metamaterial device given by the Helmholtz source-to-solution operator are equivalent to Helmholtz source-to-solution measurements in a universe given by (ℝ+ x M, -dt² + g), where (M, g) is a closed, orientable, C∞-smooth, 3- dimensional Riemannian manifold. Thus the obtained device could be used to simulate cosmological models using metamaterial devices. [ABSTRACT FROM AUTHOR]

Path integration is a navigational strategy through which an animal remains oriented even in the absence of stable landmarks. Bump attractor networks have played a key role in modeling and analyzing the neural processes involved in path integration. In such networks, self-motion inputs and asymmetries in the network translate an activity bump over the attractor manifold, updating the animal's perceived position in space. The self-motion inputs can be incorporated through a multiplicative model, in which neuronal inputs are combined multiplicatively, or an additive model, in which neuronal inputs are combined through summation. Multiplicative models have a simple traveling wave solution and are often used to analyze properties of these networks. Relatively little mathematical analysis has been done on additive models, which are generally considered to be more biologically realistic. We analyze a general additive bump attractor model of path integration, deriving explicit mathematical expressions describing the temporal and spatial dynamics of the activity bump. We find that intrinsic spatial deformations of the activity bump lead to small, positive path integration errors that are approximately proportional to the animal's velocity. This error is minimized through a balance of acceleration and velocity input. We also derive simple expressions for the path integration gain, showing explicitly how it depends on the model parameters. Our analytical results are supported by numerical simulations of four specific, representative models. Our results have biological implications and provide a mathematical foundation for understanding a large class of path integration models. [ABSTRACT FROM AUTHOR]

We study an integral formulation for the radiative transfer equation in anisotropic media with truncated approximation to the scattering phase function. The integral formulation consists of a coupled system of integral equations for the angular moments of the transport solution. We analyze the approximate separability of the kernel functions in these integral formulations, deriving asymptotic lower and upper bounds on the number of terms needed in a separable approximation of the kernel functions as the moment grows. Our analysis provides a mathematical understanding of when low-rank approximations to the discretized integral kernels can be used to develop fast numerical algorithms for the corresponding system of integral equations. [ABSTRACT FROM AUTHOR]