Semi-infinite System Research Articles

Primal characterizations of stability of error bounds for semi-infinite convex constraint systems in Banach spaces

This article focuses on the stability of error bounds, both local and global, for semi-infinite convex constraint systems in Banach spaces. We present primal characterizations of the stability of these error bounds under small perturbations. These characterizations are expressed through the directional derivatives of the functions that define the systems. It is shown that ensuring stability of the error bounds is closely tied to verifying that the optimal values of several minimax problems, formulated using the directional derivatives, remain outside a certain neighbourhood of zero. Furthermore, this stability condition only requires that all component functions of the system share the same linear perturbation. When applied to the sensitivity analysis of Hoffman's constants for semi-infinite linear systems, these stability results yield primal criteria that guarantee the uniform boundedness of Hoffman's constants under perturbations in the problem data.

Photothermal measurement of material properties for translucent thermal barrier coatings

In this study, a photothermal nondestructive method was proposed to measure the material parameters of semi-transparent or translucent thermal barrier coatings (TBCs). We derived a theoretical model for the photothermal signal from a two-layer semi-infinite material system with a translucent first layer after a pulse laser excitation. Its solution was verified by numerical solution. A data regression algorithm based on a least-squares fitting was used for the determination of the material parameters in the translucent first layer material. To verify this new method, an experimental system was set up with a pulse laser for thermal excitation and an infrared camera for image acquisition of the thermal emission transient from several translucent EBPVD TBC samples. The predicted coating thickness is consistent with the measured value by an optical microscope. The predicted thermal conductivity and optical attenuation coefficients in the absorption and emission band are found to be in good agreement with reference values.

Semi-infinite Simple Exclusion Process: From Current Fluctuations to Target Survival.

The symmetric simple exclusion process (SEP), where diffusive particles cannot overtake each other, is a paradigmatic model of transport in the single-file geometry. In this model, the study of currents has attracted a lot of attention, but so far most results are restricted to two geometries: (i)a finite system between two reservoirs, which does not conserve the number of particles but reaches a nonequilibrium steady state, and (ii)an infinite system which conserves the number of particles but never reaches a steady state. Here, we obtain an expression for the full cumulant generating function of the integrated current in the important intermediate situation of a semi-infinite system connected to a reservoir, which does not conserve the number of particles and never reaches a steady state. This results from the determination of the full spatial structure of the correlations, which we infer to obey the very same closed equation recently obtained in the infinite geometry and argue to be exact. Besides their intrinsic interest, these results allow us to solve two open problems: the survival probability of a fixed target in the SEP, and the statistics of the number of particles injected by a localized source.

Extraordinary phase transition revealed in a van der Waals antiferromagnet

While the surface-bulk correspondence has been ubiquitously shown in topological phases, the relationship between surface and bulk in Landau-like phases is much less explored. Theoretical investigations since 1970s for semi-infinite systems have predicted the possibility of the surface order emerging at a higher temperature than the bulk, clearly illustrating a counterintuitive situation and greatly enriching phase transitions. But experimental realizations of this prediction remain missing. Here, we demonstrate the higher-temperature surface and lower-temperature bulk phase transitions in CrSBr, a van der Waals (vdW) layered antiferromagnet. We leverage the surface sensitivity of electric dipole second harmonic generation (SHG) to resolve surface magnetism, the bulk nature of electric quadrupole SHG to probe bulk spin correlations, and their interference to capture the two magnetic domain states. Our density functional theory calculations show the suppression of ferromagnetic-antiferromagnetic competition at the surface is responsible for this enhanced surface magnetism. Our results not only show counterintuitive, richer phase transitions in vdW magnets, but also provide viable ways to enhance magnetism in their 2D form.

An Analytical Approach to Contaminant Transport with Spatially and Temporally Dependent Dispersion in a Heterogeneous Porous Medium

This study presents an analytical solution to the one-dimensional advection-dispersion equation (ADE) for a semi-infinite heterogeneous aquifer system with space and time-dependent groundwater velocity and dispersion coefficient. The dispersion coefficient is assumed to be proportional to the groundwater flow velocity. In addition, retardation factor, first-order decay and zero-order production terms are also considered. Contaminants and porous media are assumed to be chemically inert. Initially, it is assumed that some uniformly distributed solutes are already present in the aquifer domain. The input point source is considered uniformly continuous and increasing nature in a semi-infinite porous medium. The solutions are obtained analytically using the Laplace Integral Transform Technique (LITT). The nature of the concentration profile of the resulting solution for different parameters in different time domains is illustrated graphically.

A modal decomposition approach to topological wave propagation

The design of valley-Hall topological insulators is typically informed by analyzing the primitive unit cell, using the Berry curvature of the Bloch-eigenmodes to predict energy localization on the domain boundary, or the dispersion diagram of the semi-infinite lattice to predict the group velocity of topological waves. However, practical systems are finite in size and the propagating topological wave is composed of a finite bandwidth as guaranteed by the Fourier uncertainty principle. Hence, the propagation predicted by the ideally infinite lattice for a given frequency deviate from practical topological systems driven by finite bandwidth wave packets. In this work, we demonstrate that the propagating topological waves in valley Hall lattices can be interpreted using the linear degenerate modal basis. We show that a small subset of closely spaced modes comprise the topological waves, which enables the construction of analytical reduced-order models that accurately capture the topological wave. This result is related to the sparse density of the modal spectrum inside the topological band gap, which allows for the energy to be carried nearly completely by a narrow band of spectrally isolated eigenmodes that are spatially localized along the domain boundary. This result is then employed to refine group velocity predictions of propagating topological waves with respect to input signal bandwidth and frequency by matching the modal spectrum to the supercell dispersion diagram describing the semi-infinite system. Moreover, we utilize the damped modal spectrum to characterize variations in wave group velocity in damped topological lattices and predict edge-to-bulk transitions. Hence, this work establishes a framework for accurately characterizing undamped and damped topological wave propagation in quantum valley Hall systems using the machinery of classical dynamics.

Phase diagrams of semi-infinite systems by renormalization group theory and Monte Carlo simulation

In this work, we used the Migdal–Kadanoff renormalization group method to investigate two three-dimensional semi-infinite Ising systems. In the first one, the surface and bulk sites are occupied by spin-3/2 and spin-1/2 respectively, whereas those of the second one are occupied by spin-1/2 on the surface and spin-3/2 in the bulk. Based on the ratio of bulk and surface exchange interactions, we explored different topologies of phase diagrams showing various second order phase transitions, namely ordinary, extraordinary, surface and special transitions. We also found that both systems exhibited first order phase transitions, multicritical points and critical end-points. The existence of a first order phase transition at low temperatures was confirmed by plotting the derivative of the free energy. The Monte Carlo simulation was also used to verify and compare the results obtained by the renormalization group for the two semi-infinite systems.

Hydrogen diffusion into caprock: A semi-analytical solution and a hydrogen loss criterion

Depleted gas reservoirs can provide a gigaton capacity and the necessary infrastructure for large scale hydrogen storage. Therefore, underground storage of hydrogen in these geological formations is considered a valuable option. However, the high molecular diffusion of hydrogen from these storage sites is a major concern. In this paper, the idea of hydrogen diffusion through the caprock during geological storage was investigated using a series of one- and three-dimensional numerical models. In these models, the caprock and overlying formations were considered as a semi-infinite system, while the interaction between the reservoir and the caprock was defined by the pressure and gas composition at the boundary. Hydrogen diffusion was based on chemical potential differences and the influence of pre-diffused hydrocarbon gas in the caprock was considered. Furthermore, based on the analogies between the related partial differential equations of diffusion and spontaneous imbibition processes, a simple yet accurate semi-analytical solution was proposed. The presented semi-analytical solution can be used to predict the cumulative hydrogen loss over time. This solution states that the hydrogen loss increases proportionally with the square root of time. Using the semi-analytical solution, a criterion for loss by molecular hydrogen diffusion was presented that indicates the fraction of hydrogen lost by diffusion at a given time. The results obtained based on the diffusion fluxes showed that the hydrogen loss can be underestimated if the existence of hydrocarbon gas in the caprock is ignored. The analysis also indicated that hydrogen loss is directly proportional to the interface between rock and reservoir exposed to hydrogen, the caprock porosity, the gas saturation of the caprock, the square root of the diffusion coefficient, and the square root of time. While the presence of hydrocarbon gas facilitates diffusion, the thermodynamic effects at high pressures lead to a comparatively lower molar density in the caprock than in the matrix. Thus, a lower final loss of the injected hydrogen can be expected at higher pressures.

Duality for Sets of Strong Slater Points

The strong Slater condition plays a significant role in the stability analysis of linear semi-infinite inequality systems. This piece of work studies the set of strong Slater points, whose non-emptiness guarantees the fullfilment of the strong Slater condition. Given a linear inequality system, we firstly establish some basic properties of the set of strong Slater points. Then, we derive dual characterizations for this set in terms of the data of the system, following similar characterizations provided also for the set of Slater points and the solution set of the given system, which are based on the polarity operators for evenly convex and closed convex sets. Finally, we present two geometric interpretations and apply our results to analyze the strict inequality systems defined by lower semicontinuous convex functions.

Effect of electromechanical coupling on locally resonant quasiperiodic metamaterials

Electromechanical metamaterials have been the focus of many recent studies for use in simultaneous energy harvesting and vibration control. Metamaterials with quasiperiodic patterns possess many useful topological properties that make them a good candidate for study. However, it is currently unknown what effect electromechanical coupling may have on the topological bandgaps and localized edge modes of a quasiperiodic metamaterial. In this paper, we study a quasiperiodic metamaterial with electromechanical resonators to investigate the effect on its bandgaps and localized vibration modes. We derive here the analytical dispersion surfaces of the proposed metamaterial. A semi-infinite system is also simulated numerically to validate the analytical results and show the band structure for different quasiperiodic patterns, load resistors, and electromechanical coupling coefficients. The topological nature of the bandgaps is detailed through an estimation of the integrated density of states. Furthermore, the presence of topological edge modes is determined through numerical simulation of the energy harvested from the system. The results indicate that quasiperiodic metamaterials with electromechanical resonators can be used for effective energy harvesting without changes in the bandgap topology for weak electromechanical coupling.

Mathematical theory for topological photonic materials in one dimension

This work presents a rigorous theory for topological photonic materials in one dimension. The main focus is on the existence of interface modes that are induced by topological properties of the bulk structure. For a general 1D photonic structure with time-reversal symmetry, we investigate the existence of an interface mode that is induced by a Dirac point upon perturbation. Specifically, we establish conditions on the perturbation which guarantee the opening of a band gap around the Dirac point and the existence of an interface mode. For a periodic photonic structure with both time-reversal and inversion symmetry, the Zak phase is quantized, taking only two values . We show that the Zak phase is determined by the parity (even or odd) of the Bloch modes at the band edges. For a photonic structure consisting of two semi-infinite systems on the two sides of an interface with distinct topological indices, we show the existence of an interface mode inside the common gap. The stability of the mode under perturbations is also investigated. Finally, we study resonances for finite topological structures. Our results are based on the transfer matrix method and the oscillation theory for Sturm–Liouville operators. The methods and results can be extended to general topological Sturm–Liouville systems in one dimension.

From Calmness to Hoffman Constants for Linear Semi-infinite Inequality Systems

In this paper we focus on different---global, semilocal, and local---versions of Hoffman-type inequalities expressed in a variational form. In a first stage our analysis is developed for generic multifunctions between metric spaces, and we finally deal with the feasible set mapping associated with linear semi-infinite inequality systems (finitely many variables and possibly infinitely many constraints) parameterized by their right-hand sides. The Hoffman modulus is shown to coincide with the Lipschitz upper semicontinuity modulus and the supremum of calmness moduli when confined to multifunctions with a convex graph and closed images in a reflexive Banach space, which is the case for our feasible set mapping. Moreover, for this particular multifunction a formula---involving only the system's left-hand side---of the global Hoffman constant is derived, providing a generalization to our semi-infinite context of finite counterparts developed in the literature. In the particular case of locally polyhedral systems, the paper also provides a point-based formula for the (semilocal) Hoffman modulus in terms of the calmness moduli at certain feasible points (extreme points when the nominal feasible set contains no lines), yielding a practically tractable expression for finite systems.

General analytical solutions of multispecies advective-dispersive solute transport equations coupled with a complex reaction network

Stepwise reductive dechlorination of tetrachloroethene (PCE) to trichloroethene (TCE), three dichloroethylene (DCE) isomers (1,1-DCE, cis −1,2- DCE, and trans −1,2- DCE), vinyl chloride (VC), and ethane (ETH) may proceed under anaerobic conditions. However, most multispecies transport models for describing the plume migration of a chemical mixture comprising the original chlorinated solvent and its dechlorinated byproducts in the literature are often simplified to a sequential first-order reaction network which cannot account for the divergent reactions from TCE to the three DCE isomers and convergent reactions from the three DCE isomers to VC. In this study, general analytical solutions to multispecies transport equations with a complex reaction network were derived for a combination of semi-infinite and finite systems. The developed analytical solutions were robustly verified against a semi-analytical solution that can consider the same complex reaction network. The verification results indicated that the derived analytical solutions were accurate and robust. The general solutions derived in the present study were also used to investigate the effects of the outlet boundary conditions on solute transport involving a complex reaction network. The results showed that the analytical solution derived for infinite outlet BCs predicted lower concentrations of contaminants near the outlet boundary than those for finite outlet BCs. Moreover, a straight decay chain model may over- or under- estimate the DCE and VC concentrations compared to the multiple branching isomer reaction model. The developed analytical solutions with a complex reaction network provide more realistic and efficient tools for assessing the movement of chlorinated solvents and their degradation-related byproducts in soil–water systems.

Probing the bulk plasmon continuum of layered materials through electron energy loss spectroscopy in a reflection geometry

A periodic arrangement of 2D conducting planes is known to host a (bulk) plasmon dispersion that interpolates between the typical, gapped behavior of 3D metals and a gapless, acoustic regime as a function of the out-of-plane wavevector. The semi-infinite system -- the configuration relevant to Electron Energy Loss Spectroscopy (EELS) in a reflection geometry, as in High Resolution EELS (HREELS) -- is known to host a surface plasmon that ceases to propagate below a cutoff wavevector. As the f-sum rule requires a finite response whether or not there exist sharp excitations, we demonstrate that what remains in the surface loss function -- the material response probed by HREELS -- is the contribution from the (bulk) plasmon of the infinite system. In particular, we provide a one-to-one mapping between the plasmon continuum and the spectral weight in the surface loss function. In light of this result, we suggest that HREELS be considered a long wavelength probe of the plasmon continuum in layered materials.

Bi-infinite Solutions for KdV- and Toda-Type Discrete Integrable Systems Based on Path Encodings

We define bi-infinite versions of four well-studied discrete integrable models, namely the ultra-discrete KdV equation, the discrete KdV equation, the ultra-discrete Toda equation, and the discrete Toda equation. For each equation, we show that there exists a unique solution to the initial value problem when the given data lies within a certain class, which includes the support of many shift ergodic measures. Our unified approach, which is also applicable to other integrable systems defined locally via lattice maps, involves the introduction of a path encoding (that is, a certain antiderivative) of the model configuration, for which we are able to describe the dynamics more generally than in previous work on finite size systems, periodic systems and semi-infinite systems. In particular, in each case we show that the behaviour of the system is characterized by a generalization of the classical ‘Pitman’s transformation’ of reflection in the past maximum, which is well-known to probabilists. The picture presented here also provides a means to identify a natural ‘carrier process’ for configurations within the given class, and is convenient for checking that the systems we discuss are all-time reversible. Finally, we investigate links between the different systems, such as showing that bi-infinite all-time solutions for the ultra-discrete KdV (resp. Toda) equation may appear as ultra-discretizations of corresponding solutions for the discrete KdV (resp. Toda) equation.

Impact contact behaviors of elastic coated medium with imperfect interfaces

Imperfect interface of coating-substrate system, where displacement and stress are transmitted discontinuously, affects mechanical response of coated medium subjected to impact loading a lot. This paper proposes an impact contact model between an elastic moving ball and imperfectly bonded semi-infinite coating-substrate system with combined dislocation-like and force-like interfaces. Impact contact process obeys the second-order Newmark theory and contact theory for coated medium. Impact velocity, acceleration and reaction force of the ball are cost-effectively determined by a step by step iterative strategy, binary search method and conjugate gradient method (CGM). Analytical explicit expressions of displacement and stress fields of imperfectly bonded coating-substrate system are derived with the assistance of the Papkovich–Neuber potentials, which are further transformed to frequency domain aiming at improving calculating efficiency through employing a fast Fourier transform (FFT) algorithm. Six jumping coefficients for interfacial displacement and stress are introduced to describe different interface conditions, and their effects on impact reaction force, contact radius, contact pressure and subsurface stress of imperfectly bonded coating-substrate system under different impact velocities are studied.

Stability and Sensitivity of Uncertain Linear Programs

The present paper deals with uncertain linear optimization problems where the objective function coefficient vector belongs to a compact convex uncertainty set and the feasible set is described by a linear semi-infinite inequality system (finitely many variables and possibly infinitely many constrainsts), whose coefficients are also uncertain. Perturbations of both, the objective coefficient vector set and the constraint coefficient set, are measured by the Hausdorff metric. The paper is mainly concerned with analyzing the Lipschitz continuity of the optimal value function, as well as the lower and upper semicontinuity in the sense of Berge of the optimal set mapping. Inspired by Sion’s minimax theorem, a new concept of weak optimal solution set is introduced and analyzed.

Mott versus Hybridization Gap in the Low-Temperature Phase of 1T-TaS_{2}.

We address the long-standing problem of the ground state of 1T-TaS_{2} by computing the correlated electronic structure of stacked bilayers using the GW+EDMFT method. Depending on the surface termination, the semi-infinite uncorrelated system is either band insulating or exhibits a metallic surface state. For realistic values of the on-site and inter-site interactions, a Mott gap opens in the surface state, but it is smaller than the gap originating from the bilayer structure. Our results are consistent with recent scanning tunneling spectroscopy measurements for different terminating layers, and with our own photoemission measurements, which indicate the coexistence of spatial regions with different gaps in the electronic spectrum. By comparison to exact diagonalization data, we clarify the interplay between Mott insulating and band insulating behavior in this archetypal layered system.

A road modulus test method based on rigid spherical indentation

Currently, the modulus test method such as falling weight deflectometer has an inevitable problem of stress concentration, which may lead to a negative impact on the results. This paper attempted to apply the spherical indentation test equipment into road detection to solve this problem. Firstly, by following the displacement boundary conditions, a general expression of modulus, force, and displacement under the semi-infinite system was established. Compared with the Sneddon’s formula (a classic formula in materials industry), this expression was consistent with it under the small deformation situations, which verified the rationality of this method. Moreover, this expression can also be utilized in the larger deformation situations, which was more in line with the pavement environment. Then, by further improving the method, this paper established the method of calculating the structure modulus of new pavement layers. Finally, a specific test method for modulus detection based on the spherical indentation was proposed.

OPERATOR METHOD IN DIFFRACTION PROBLEM OF WAVES OF CIRCULAR WAVEGUIDE BY THE ANNULAR DISCONTINUITIES

Purpose. Diff raction problem of the H11 and E11 waves of circular waveguide by the fi nite and semi-infi nite system of similar discontinuities is considered. Four types of discontinuities are considered: iris, disk, ring, annular slot. Th e distance between irregularities is the same. Th e waveguide is fi lled by the dielectric with losses. Design/methodology/approach. To solve the problem we chose the operator method. While so-called key problem, the diff raction problem by a single discontinuity, is solved by the method of moments. The field in the domain of the obstacle is represented as a series in terms of eigenwaves of infi nite waveguide which cross section coincides with the cross section of discontinuity, with unknown amplitudes. To fi nd the amplitudes we obtain the infi nite system of equations, which is solved by the reduction. As a result, the transmission and refl ection operators of a key-problem are obtained. The properties of fi nite sequence are determined from the operator equations relatively amplitudes of the scattered field. To write these equations the iterative procedure is used. The properties of the structure, which consists of N discontinuities are obtained under assumption that the properties of the structure, which consists of N – 1 discontinuity are known. The reflection operator of the semi-infi nite system is obtained from known non-linear operator equation of the second kind. Findings. The systems of equations for every single discontinuity are obtained. The operator equations relatively amplitudes of the scattered fields for fi nite system are presented. With the help of the created model the dependences of the transmission, refl ection coeffi cients of the H11 wave as well as transformation coeffi cients of the H11 to E11 wave of the circular waveguide on the wavenumber, geometrical and material parameters for fi nite and semi-infi nite structures are studied. The comparison of the results obtained by presented method with the results obtained in HFSS is made. Conclusions. With the use of the operator method the scattering model of the H1n and E1n eigenwaves by the system of annular discontinuities of zero thickness in a circular waveguide is built. Th e study of the scattering characteristics on the parameters is made. The coincidence of the behavior of the curves obtained by the presented method with those obtained in HFSS allows us to draw a conclusion about the correctness of the results. Th e results can be used during creation of a series of the microwaves and optic devices.