Two-dimensional Setting Research Articles

A Many-Body Index for Quantum Charge Transport

We propose an index for pairs of a unitary map and a clustering state on many-body quantum systems. We require the map to conserve an integer-valued charge and to leave the state, e.g. a gapped ground state, invariant. This index is integer-valued and stable under perturbations. In general, the index measures the charge transport across a fiducial line. We show that it reduces to (i) an index of projections in the case of non-interacting fermions, (ii) the charge density for translational invariant systems, and (iii) the quantum Hall conductance in the two-dimensional setting without any additional symmetry. Example (ii) recovers the Lieb-Schultz-Mattis theorem, and (iii) provides a new and short proof of quantization of Hall conductance in interacting many-body systems.

A Priori Estimates for a Nonlinear System with Some Essential Symmetrical Structures

In this paper, we are concerned with a nonlinear system containing some essential symmetrical structures (e.g., cross-diffusion) in the two-dimensional setting, which is proposed to model the biological transport networks. We first provide an a priori blow-up criterion of strong solution of the corresponding Cauchy problem. Based on this, we also establish a priori upper bounds to strong solution for all positive times.

Biomathematical model for simulating abnormal orifice patterns in colonic crypts

Colonic polyps, which are abnormal growths in the colon, are a major concern in colon cancer diagnosis and prevention. Medical studies evidence that there is a correlation between histopathology and the shapes of the orifices in colonic crypts. We propose a biomathematical model for simulating the appearance of anomalous shapes for the orifices of colonic crypts, associated to an abnormal cell proliferation. It couples a mechanical model that is a mixed elastic/viscoelastic quasi-static model describing the deformation of the crypt orifice, with a convection-diffusion model that simulates the crypt cell dynamics in space and time. The coupling resides in the variation of pressure generated by abnormal proliferative cells that induce a mechanical force and originate the change in shape of the crypt orifice. Furthermore the model is formulated in a two-dimensional setting, for emulating the top view of the colonic mucosa, observed in vivo in colonoscopy images. The primary focus of this study is on the modeling of this complex biological phenomenon, by defining an appropriate reduced biomathematical model. Additionally, a numerical procedure to determine its solution is also addressed. The overall numerical simulations indicate that an excess of cell proliferation, in different crypt locations, creates some of the anomalous patterns of the colonic crypt orifices, observed in vivo in medical images.

Regulation of an osteon-like concentric microgrooved surface on osteogenesis and osteoclastogenesis

Topographical cues provided by micropatterns on material surfaces have been demonstrated to control multiple cell functions. However, the majority of currently studied micropatterns fail to recapitulate the key characteristics of an osteon, which is the structural unit of natural cortical bone. Thus, in the present study, a micropatterned polycaprolactone (PCL) surface comprising a series of concentric circular microgrooves was fabricated by combining photolithography with the melt-casting method to mimic the concentric structure of an osteon in a two-dimensional setting. By culturing mouse mesenchymal stem cells (mMSCs) and osteoclast progenitor cells (RAW264.7 cells) on the osteon-like concentric microgrooved surface, the effects of this micropatterned surface on the osteogenesis of mMSCs and the osteoclastogenesis of RAW264.7 cells were systematically investigated. Osteoclastogenic differentiation was significantly inhibited in RAW264.7 cells on the fabricated osteon-like concentric microgrooved surface compared to that on the parallel linear microgrooved and flat surfaces, as indicated by the downregulated expression of key osteoclast-specific function genes (TRAP, CATK and MMP9), the lower activity of TRAP and less formation of TRAP-positive multinucleated giant osteoclasts. Further investigation indicated that RANK-NFκB signaling may have been involved in mediating the inhibited osteoclastogenesis of RAW264.7 cells by the osteon-like concentric microgrooved surface. In addition, the osteon-like concentric microgrooved surface greatly modulated the osteoclastogenic-related paracrine secretion of mMSCs (RANKL, M-CSF and OPG), despite its small effect on the osteogenesis of mMSCs. This secretory profile was found to be able to effectively inhibit osteoclastogenesis in RAW264.7 cells, confirming the enhanced osteoclastogenesis inhibitory functions of mMSCs on the osteon-like concentric microgrooved surface. Our findings demonstrate the importance of the microgroove orientation and arrangement in affecting cellular behaviors and highlight the potential benefits of incorporating osteon-like concentric microgrooved patterns on the surface of scaffolds for bone repair.

Influence of a wet wood particle form on the characteristics of its ignition in the high-temperature medium

The results of the experimental and theoretical studies of the processes of the single particles ignition of the wood biomass of various forms (a cube, a cylinder and a sphere) have been presented. According to the results of the experimental studies, it has been established that the particles form can have a fairly significant effect on the characteristics and conditions of ignition. It has been shown that the wood particles in the form of a sphere are ignited much faster than cylindrical and cubic ones (ignition delay times are less by 56%). Such a tendency is maintained for particles sizes from 1 · 10−3 m to 3 · 10−3 m. According to the results of the experiments, the mathematical model for ignition of a particle of wet wood biomass has been formulated, taking into account the joint occurrence of the main processes of the thermal preparation (radiation-convective particle heating, evaporation of water, thermal decomposition of the organic part of the fuel, output, ignition and combustion of gaseous pyrolysis products, ignition of the char residue).The formulated system of the nonlinear differential equations in the partial derivatives with the corresponding boundary conditions describes the ignition process in a two-dimensional setting, taking into account the substantial nonuniformity of the temperature field of a wood particle in the induction period. We have taken into account that the evaporation process takes place in the front. Up to the present time, the mathematical models of such a degree of detailing of the physical processes occurring together when igniting wet wood have not been formulated. A comparative analysis of the ignition delay times (tign) obtained theoretically and experimentally has showed their satisfactory correspondence. As a result of the numerical simulation, the temperature fields in the system “a fuel particle — high-temperature medium” have been obtained, illustrating the high gradients of the temperatures at the moment of ignition. It has been established that it is possible to ignite the water-saturated wood particles before they dry completely. It has been established that the ignition mode of wood particles is possible before they dry completely.

Rigorous continuum limit for the discrete network formation problem

Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on reformulating the discrete energy functional as a sequence of integral functionals and proving their Γ-convergence towards a continuum energy functional.

Enclosing a moving target with an optimally rotated and scaled multiagent pattern

We propose a novel control method to enclose a moving target in a two-dimensional setting with a team of agents forming a prescribed geometric pattern. The approach optimises a measure of the overall agent motion costs, via the minimisation of a suitably defined cost function encapsulating the pattern rotation and scaling. We propose two control laws which use global information and make the agents exponentially converge to the prescribed formation with an optimal scale that remains constant, while the team's centroid tracks the target. One control law results in a multiagent pattern that keeps a constant orientation in the workspace; for the other, the pattern rotates with constant speed. These behaviours, whose optimality and steadiness are very relevant for the task addressed, occur independently from the target's velocity. Moreover, the methodology does not require distance measurements, common coordinate references, or communications. We also present formal guarantees of collision avoidance for the proposed approach. Illustrative simulation examples are provided.

Ignition of the wood biomass particles under conditions of near-surface fragmentation of the fuel layer

The results of the experimental and theoretical studies of the ignition processes of the woody biomass single particles of a cubic form under conditions of preliminary fragmentation of the particle surface have been presented. It is established experimentally that the increase in the result of the external mechanical influences, in the heat exchange area (due to the formation of grooves and cracks) leads to an acceleration of the ignition process. According to the results of the experiments, a new mathematical model of the ignition process of the wood biomass particle under conditions of near-surface fragmentation, which differs from previously known ones, has been formulated. It describes the thermophysical processes and thermochemical processes and intense phase (evaporation of water) transformations, in conditions of change in the surface relief of the particle, and, accordingly, the density of the heat flux along the particle contour and the concentration of combustible gases in the boundary layer of the particle. The formulated system of the nonlinear partial differential equations in partial derivatives with the corresponding boundary conditions describes the process of ignition in a two-dimensional setting, taking into account the substantial nonuniformity of the temperature field of a wood particle in the induction period. Up to the present time, the mathematical models of such a degree of detailing of the physical processes occurring together during the ignition of wood under the conditions of fragmentation of its surface have not been formulated. The comparative analysis of the ignition delay times (tign), obtained theoretically and experimentally, has showed their satisfactory correspondence. As a result of the numerical simulation, the temperature fields in the “fuel particle-high-temperature medium” system have been obtained, illustrating the features of the course of the thermochemical reactions of the wood particles ignition under conditions of fragmentation of their surface.

How Tumor Cells Can Make Use of Interstitial Fluid Flow in a Strategy for Metastasis.

The phenomenon of lymph node metastasis has been known for a long time. However, the underlying mechanism by which malignant tumor cells are able to break loose from the primary tumor site remains unclear. In particular, two competing fluid sensitive migration mechanisms have been reported in the experimental literature: (i) autologous chemotaxis (Shields et al. in Cancer Cell 11:526-538, 2007) which gives rise to downstream migration; (ii) an integrin-mediated and strain-induced upstream mechanism (Polacheck et al. in PNAS 108:11115-11120, 2011). How can these two competing mechanisms be used as a means for metastatic behavior in a realistic tumor setting? Excessive fluid flow is typically produced from leaky intratumoral blood vessels and collected by lymphatics in the peritumoral region giving rise to a heterogeneous fluid velocity field and a corresponding heterogeneous cell migration behavior, quite different from the experimental setup. In order to shed light on this issue there is a need for tools which allow one to extrapolate the observed single cell behavior in a homogeneous microfluidic environment to a more realistic, higher-dimensional tumor setting. Here we explore this issue by using a computational multiphase model. The model has been trained with data from the experimental results mentioned above which essentially reflect one-dimensional behavior. We extend the model to an envisioned idealized two-dimensional tumor setting. A main observation from the simulation is that the autologous chemotaxis migration mechanism, which triggers tumor cells to go with the flow in the direction of lymphatics, becomes much more aggressive and effective as a means for metastasis in the presence of realistic IF flow. This is because the outwardly directed IF flow generates upstream cell migration that possibly empowers small clusters of tumor cells to break loose from the primary tumor periphery. Without this upstream stress-mediated migration, autologous chemotaxis is inclined to move cells at the rim of the tumor in a homogeneous and collective, but space-demanding style. In contrast, inclusion of realistic IF flow generates upstream migration that allows two different aspects to be synthesized: maintain the coherency and solidity of the the primary tumor and at the same time cleave the outgoing waves of tumor cells into small clusters at the front that can move collectively in a more specific direction.

Stability analysis for some numerical schemes of partial differential equation with extra measurements

This paper is devoted to study the stability analysis of some finite difference schemes for an inverse problem with unknowns time-dependent coefficients subject to extra measurements. We prove that the popular forward time centered space scheme is a conditional method. But the backward time centered space and Crank Nicolson methods are suitable schemes because they are unconditional methods. We justify this advantage of the stability analysis versus the some numerical methods with an example. All the results and a numerical example are in two-dimensional setting.

Computational geometry and the U.S. Supreme Court

We use the United States Supreme Court as an illuminative context in which to discuss three different spatial voting preference models: an instance of the widely used single-peaked preferences, and two models that are more novel in which vote outcomes have a strength in addition to a location. We introduce each model from a formal axiomatic perspective, briefly discuss practical motivation for each in terms of judicial behavior, prove mathematical relationships among the voting coalitions compatible with each model, and then study the two-dimensional setting by presenting computational tools for working with the models and by exploring these with judicial voting data from the Supreme Court.

Transport of marine microplastic particles: why is it so difficult to predict?

Marine microplastic particles (MPs, <5 mm) exhibit wide ranges of densities, sizes, and shapes, so that the entire MPs “ensemble” at every time instant can be characterized by continuous distributions of these parameters. Accordingly, this community of particles demonstrates distributions of dynamical properties, such as sinking or rising velocity, critical shear stress, and the re-suspension threshold. Moreover, all the MPs’ properties vary significantly with the time spent in marine environment and with particular conditions experienced by the particle on its journey. A brief review of the present-day numerical efforts towards prediction of MPs transport shows the prevalence of the Lagrangian particle tracking approach, especially for floating litter. In a broader context, the present practice of MPs transport modelling follows the “selective” strategy (e.g., only a certain sub-class of MPs, or specific processes, are considered, sometimes in only one- or two-dimensional setting). The heterogeneous nature of MPs, their enormous longevity and movability in marine environment, and the wide spectrum of the involved environmental processes suggest further integration (or coupling) of different models in future, as well as application of other types of models (ensemble modeling, chaos theory approaches, machine learning, etc.) to the problems of MPs transport and fate in the marine environment.

Two-Dimensional Capillary Origami with Inextensibility and Free Triple-Contact Points

Motivated by recent elasto-capillary experiments done in the context of capillary origami, we model the static elastic deformations of a thin sheet due to the surface tension of a liquid drop. The equilibrium equations are derived variationally in a two-dimensional setting under the assumptions that the sheet is inextensible and that the drop is not pinned to the boundary of the sheet. In this situation, the behavior of the system, for varying cross-sectional area $\mathcal{A}$ of the drop, can be fully characterized in terms of two dimensionless parameters, $\lambda$ and $\tau$, respectively, measuring the relative importance of bending versus surface tension and the balance of the interfacial tensions. Specifically, for fixed $\lambda$ and $\tau$, bifurcation diagrams are constructed, plotting the distance across the opening of the plate versus $\mathcal{A}$. Then these bifurcation diagrams are studied for $\lambda >0$ and $|\tau| < 1$, leading us to split the physical description of the system's behavior into three different parameter regimes.

A simplified multiscale model of degenerate graphite clusters in grey cast iron

To take into account the weakening effect of defects clusters in real microstructures, we propose a multiscale model in a two-dimensional setting. At a small scale, single defects are described as elliptical voids randomly distributed and randomly oriented in an isotropic matrix. Using an effective field method proposed by Tandon and Weng [5,6], the effective properties of a porous equivalent material can be estimated in a simple closed form. At a larger scale, defects clusters are modelled as single large elliptical inclusions characterized by the weakened effective properties calculated in the first step and embedded in an infinite, elastic, isotropic matrix under remote loading. The method allows to examine the dependence of defects interaction on some basic microstructure parameters (porosity density, voids aspect ratio, inclusion aspect ratio).

A Fully Fourth Order Accurate Energy Stable Finite Difference Method for Maxwell's Equations in Metamaterials

We present a novel fully fourth order in time and space finite difference method (FDM) for the time domain Maxwell's equations in metamaterials. We consider a Drude metamaterial model for the material response to incident electromagnetic fields. We consider the second-order formulation of the system of partial differential equations that govern the evolution in time of electric and magnetic fields along with the evolution of the polarization and magnetization current densities. Our discretization employs fourth-order staggering in space of different field components and the modified equation approach to obtain fourth-order accuracy in time. Using the energy method, we derive energy relations for the continuous models and design numerical schemes that preserve a discrete analogue of the energy relation. Numerical simulations are provided in one- and two-dimensional settings to illustrate fourth-order convergence as well as to compare with second-order schemes.

Hierarchical B-spline complexes of discrete differential forms

AbstractIn this paper we introduce the hierarchical B-spline complex of discrete differential forms for arbitrary spatial dimension. This complex may be applied to the adaptive isogeometric solution of problems arising in electromagnetics and fluid mechanics. We derive a sufficient and necessary condition guaranteeing exactness of the hierarchical B-spline complex for arbitrary spatial dimension, and we derive a set of local, easy-to-compute and sufficient exactness conditions for the two-dimensional setting. We examine the stability properties of the hierarchical B-spline complex, and we find that it yields stable approximations of both the Maxwell eigenproblem and Stokes problem provided that the local exactness conditions are satisfied. We conclude by providing numerical results showing the promise of the hierarchical B-spline complex in an adaptive isogeometric solution framework.

Analysis of the Effect of Physico-Mechanical Characteristics of Cumulative Liner Material on Parameters of a High-Speed Element

In this paper, we present the results of numerical simulation of explosive formation of a high-speed compact element from copper, steel, and aluminum cumulative liners combining the shape of a hemispherical segment smoothly converting into a cylindrical surface (“hemisphere–cylinder” liners). The problem is solved in a two-dimensional axisymmetric setting considering the limiting parameters of the dynamic stress-strain state causing plastic flow and destruction of the cumulative liner material. The original model of the functioning of cumulative shaped charge, which determines the effect of individual elements of the cumulative liners, including the difference in numerical characteristics of their physicomechanical properties and critical destruction conditions, on the final parameters of the high-speed compact element, was used. The plastic properties of the material and the critical conditions for its destruction were found not to affect the final velocity of the formed high-speed compact element, but they affect its shape, dimensions, and mass.

Global boundedness of solutions to a chemotaxis–haptotaxis model with tissue remodeling

We consider a cancer invasion model comprising a strongly coupled PDE–ODE system in two and three space dimensions. The system consists of a parabolic equation describing cancer cell migration arising from a combination of chemotaxis and haptotaxis, a parabolic/elliptic equation describing the dynamics of matrix degrading enzymes (MDEs), and an ODE describing the evolution and re-modeling of the extracellular matrix (ECM). We point out that this strongly coupled PDE–ODE setup presents new mathematical difficulties, which are overcome by developing new integral estimate techniques. We prove that the system admits a unique global classical solution which is uniformly bounded in time in the two-dimensional spatial setting at all cancer cell proliferation rates. We also prove that, in the case of three-dimensional convex spatial domain, when cancer cell proliferation is suitably small, the system also possesses a unique classical solution for appropriately small initial data. These results improve previously known ones.

An accurate spectral method for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media

In this paper, we propose an accurate numerical means built upon a spectral-Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for the transverse magnetic mode of Maxwell equations in Cole-Cole dispersive media in two-dimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields, and the induced electric polarization (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then, we solve the IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show that such an enriched collocation method is convergent and accurate.

Fractional Tikhonov Method for an Inverse Time-Fractional Diffusion Problem in 2-Dimensional Space

In this paper, we present a new fractional Tikhonov regularization method for solving an inverse problem for a time-fractional diffusion equation which is highly ill-posed in the two-dimensional setting. Fractional Tikhonov regularization method not only retains the advantage of classical Tikhonov method, but also overcomes the effect of over-smoothing of classical Tikhonov method. We give the selection of regularization parameters of the new method and the corresponding error estimation. Furthermore, numerical results show that the fractional Tikhonov method outperforms the classical one.