Two-dimensional Extension Research Articles

Localized wave structures: Solitons and beyond

The review is concerned with solitary waves and other localized structures in the systems described by a variety of generalizations of the Korteweg–de Vries (KdV) equation. Among the topics we focus upon are “radiating solitons,” the generic structures made of soliton-like pulses, and oscillating tails. We also review the properties of solitary waves in the generalized KdV equations with the modular and “sublinear” nonlinearities. Such equations have an interesting class of solutions, called compactons, solitary waves defined on a finite spatial interval. Both the properties of single solitons and the interactions between them are discussed. We show that even minor non-elastic effects in the soliton–soliton collisions can accumulate and result in a qualitatively different asymptotic behavior. A statistical description of soliton ensembles (“soliton gas”), which emerges as a major theme, has been discussed for several models. We briefly outline the recent progress in studies of ring solitons and lumps within the framework of the cylindrical KdV equation and its two-dimensional extension. Ring solitons and lumps (2D solitons) are of particular interest since they have many features in common with classical solitons and yet are qualitatively different. Particular attention is paid to interactions between the objects of different geometries, such as the interaction of ring solitons and shear flows, ring solitons and lumps, and lumps and line solitons. We conclude our review with views of the future developments of the selected lines of studies of localized wave structures in the theory of weakly nonlinear, weakly dispersive waves.

Ultra-high order mode-assisted optical differentiator for edge detection with high tunability

An optical spatial differentiator based on the photonic spin Hall effect (PSHE) with high tunability is presented. By utilizing the characteristics of ultra-high order modes in the symmetrical metal cladding waveguide, the Fresnel reflection coefficient spectrum exhibits a narrow peak width and low trough at the resonant incident angles, resulting in high sensitivity to changes in the incident angle-induced spatial shift caused by the PSHE (the highest ∂(|r s /r p |)/∂θ value can reach 107). After polarization transformation and extinction, the output field demonstrates differential operation with respect to the input field. When applied to edge detection, our differentiator can achieve tunable resolution edge images by adjusting the incident angle. Our proposed edge detection scheme has potential applications for cellular and molecular imaging through two-dimensional extension via the target rotation.

Risk Assessment for Linear Regression Models in Metrology

The conformity assessment of products or a measured value with the given standards is carried out based on the global risk of producers and consumers’ calculations. A product may conform to specifications but be falsely rejected as non-conforming. This is about the producer’s risk. If a product does not meet the requirements but is falsely accepted as conforming, that poses a risk to the consumer. The conventional approach to risk assessment, which yields only a single numerical value for the global risk of producers and consumers, is naturally extended and utilized for assessing risk in measurement models with linear regression. The outcomes of the two-dimensional extension, along a moderate scale, are the parabolas with upwards openings. Risk surfaces were obtained through three-dimensional extension over the area limited by the moderate scale and guard band axes. Four models with different ranges of tolerance intervals were used to test this innovative method of risk assessment in linear regression. The corresponding standard measurement uncertainties were determined by applying a simplified measurement model with the use of comprehensive data on the measurement performance and by determining measurement uncertainty derived from consideration of the functional relationship obtained by linear regression analysis. Models that utilize information from linear regression analysis to determine measurement uncertainty are biased towards risks at the edges of the moderate scale. Testing the model’s performances with metrics related to the confusion matrix, such as the F1 score, further substantiated this assertion. The diagnostic odds ratio has been proven to be extremely effective in identifying the curve along the guard band axis, along which the global risks of producers and consumers are at their lowest.

Representation and interpretation about underwater sound speed gradient field in the GNSS-A observation

SUMMARY The Global Navigation Satellite System—Acoustic ranging combination technique (GNSS-A) is a seafloor geodetic observation technique that achieves an accuracy of centimetres by combining high-rate GNSS data with acoustic ranging. The technique determines the seafloor position by acoustic ranging between the sea surface and multiple seafloor stations, using GNSS data from the sea surface station. Here, the gradient state of the underwater sound speed structure (SSS) is a significant source of error. The open-source software GARPOS can reduce the effect from underwater gradient structures by retrieving the underwater disturbance as a parameter projected onto the sea surface and seafloor. To evaluate the effects of underwater disturbances, a quantitative comparison of the model parameters is necessary. In this study, we developed a representation method to evaluate features of the ocean field. Here, the expression method was described in the order of a formulation and an interpretation in the case of a 2-D cross-section and extension to the case of an actual 3-D field. This method makes it possible to evaluate SSS states in the GNSS-A observations. As an example, we showed the correlation between the anomaly of the expressed ocean state and the anomaly of the seafloor station position, showing that this expression method is an effective index for correcting bias errors. Additionally, we used the data from sites located in the Kuroshio area, a strong current near Japan, to show that the ocean state can be quantitatively interpreted using this expression method.

Collapse dynamics for two-dimensional space-time nonlocal nonlinear Schrödinger equations

The question of collapse (blow-up) in finite time is investigated for the two-dimensional (non-integrable) space-time nonlocal nonlinear Schrödinger equations. Starting from the two-dimensional extension of the well known AKNS q,r system, three different cases are considered: (i) partial and full parity-time (PT) symmetric, (ii) reverse-time (RT) symmetric, and (iii) general q,r system. Through extensive numerical experiments, it is shown that collapse of Gaussian initial conditions depends on the value of its quasi-power. The collapse dynamics (or lack thereof) strongly depends on whether the nonlocality is in space or time. A so-called quasi-variance identity is derived and its relationship to blow-up is discussed. Numerical simulations reveal that this quantity reaching zero in finite time does not (in general) guarantee collapse. An alternative approach to the study of wave collapse is presented via the study of transverse instability of line soliton solutions. In particular, the linear stability problem for perturbed solitons is formulated for the nonlocal RT and PT symmetric nonlinear Schrödinger (NLS) equations. Through a combination of numerical and analytical approaches, the stability spectrum for some stationary one soliton solutions is found. Direct numerical simulations agree with the linear stability analysis which predicts filamentation and subsequent blow-up.

A New Method for 2D-Adapted Wavelet Construction: An Application in Mass-Type Anomalies Localization in Mammographic Images

This contribution presents a wavelet-based algorithm to detect patterns in images. A two-dimensional extension of the DST-II is introduced to construct adapted wavelets using the equation of the tensor product corresponding to the diagonal coefficients in the 2D discrete wavelet transform. A 1D filter was then estimated that meets finite energy conditions, vanished moments, orthogonality, and four new detection conditions. These allow, when performing the 2D transform, for the filter to detect the pattern by taking the diagonal coefficients with values of the normalized similarity measure, defined by Guido, as greater than 0.7, and α=0.1. The positions of these coefficients are used to estimate the position of the pattern in the original image. This strategy has been used successfully to detect artificial patterns and localize mass-like abnormalities in digital mammography images. In the case of the latter, high sensitivity and positive predictive value in detection were achieved but not high specificity or negative predictive value, contrary to what occurred in the 1D strategy. This means that the proposed detection algorithm presents a high number of false negatives, which can be explained by the complexity of detection in these types of images.

An extended 2D Gao beam model

This work derives and simulates a two-dimensional extension of the nonlinear Gao beam, by adding the cross-sectional shear variable, similarly to the extension of the usual Bernoulli–Euler beam into the Timoshenko beam. The model allows for oscillatory motion about a buckled state, as well as adds vertical shear of the cross sections, thus reflecting better nonlinear thick beams. The static model is derived from the principle of virtual elastic energy, and is in the form of a highly nonlinear coupled system for the beams transverse vibrations and the motion of the cross sections. The model allows for general distributive transversal and longitudinal loads and a compressive horizontal load acting on its edges. The model is simulated numerically, using the dynamic version for better insight into the steady solutions. The terms that distinguish our numerical solutions from the solutions of the Gao beam, described in the literature, are highlighted. The numerical scheme and its characteristic finite element matrices allow us to obtain simulation results that demonstrate the type of vibrations of our extended and modified beam, and also the differences between these solutions and those of the Gao beam model.

Integration of nonsmooth 2-forms: From Young to Itô and Stratonovich

We show that geometric integrals of the type ∫Ωfdg1∧dg2 can be defined over a two-dimensional domain Ω when the functions f, g1, g2:R2→R are just Hölder continuous with sufficiently large Hölder exponents and the boundary of Ω has sufficiently small dimension, by summing over a refining sequence of partitions the discrete Stratonovich or Itô type terms. This leads to a two-dimensional extension of the classical Young integral that coincides with the integral introduced recently by R. Züst. We further show that the Stratonovich-type summation allows to weaken the requirements on Hölder exponents of the map g=(g1,g2) when f(x)=F(x,g(x)) with F sufficiently regular. The technique relies upon an extension of the sewing lemma from Rough paths theory to alternating functions of two-dimensional oriented simplices, also proven in the paper.

Symmetry Analysis for the 2D Aw-Rascle Traffic-Flow Model of Multi-Lane Motorways in the Euler and Lagrange Variables

A detailed symmetry analysis is performed for a microscopic model used to describe traffic flow in two-lane motorways. The traffic flow theory employed in this model is a two-dimensional extension of the Aw-Rascle theory. The flow parameters, including vehicle density, and vertical and horizontal velocities, are described by a system of first-order partial differential equations belonging to the family of hydrodynamic systems. This fluid-dynamics model is expressed in terms of the Euler and Lagrange variables. The admitted Lie point symmetries and the one-dimensional optimal system are determined for both sets of variables. It is found that the admitted symmetries for the two sets of variables form different Lie algebras, leading to distinct one-dimensional optimal systems. Finally, the Lie symmetries are utilized to derive new similarity closed-form solutions.

Two-dimensional model of ion transport in composite membranes active layers with TEM-scanned morphology

Ion rejection by composite membranes for RO/NF is performed by an ultrathin polyamide active layer (AL) with non-uniform thickness. Current models explain the solute and water fluxes by concentration and potential in AL of uniform properties, there are studies showing the effects of its topology on permeability. We developed a two-dimensional extension of the solution-friction model, coupling ion and water transport in irregularly-shaped charged AL, with geometry extracted from TEM images. Simulations indicated that AL with pronounced roughness lead to highly non-uniform distribution of ionic fluxes and lead to NaCl permeability greater than obtained by uniform layers. The transmembrane pressure and membrane charge can shift the relative dominance of ion transport mechanisms. In weakly charged membranes diffusion dominates, even at high water fluxes. Electromigration can counteract the convective flux in highly charged membranes. The model revealed the possibility of circular ionic currents within/around the AL. The response of the 2D model to variations in flux and salinity can be identical to the response of a 1D model, providing that the 1D model uses an appropriate equivalent membrane thickness. We provided a method to convert the detailed 2D geometry into a single number that can be computed from images' active layers.

Two-dimensional ferromagnetic extension of a topological insulator

Two-dimensional ferromagnetic extension of a topological insulator

Machine learning and polymer self-consistent field theory in two spatial dimensions.

A computational framework that leverages data from self-consistent field theory simulations with deep learning to accelerate the exploration of parameter space for block copolymers is presented. This is a substantial two-dimensional extension of the framework introduced in the work of Xuan et al. [J. Comput. Phys. 443, 110519 (2021)]. Several innovations and improvements are proposed. (1) A Sobolev space-trained, convolutional neural network is employed to handle the exponential dimension increase of the discretized, local average monomer density fields and to strongly enforce both spatial translation and rotation invariance of the predicted, field-theoretic intensive Hamiltonian. (2) A generative adversarial network (GAN) is introduced to efficiently and accurately predict saddle point, local average monomer density fields without resorting to gradient descent methods that employ the training set. This GAN approach yields important savings of both memory and computational cost. (3) The proposed machine learning framework is successfully applied to 2D cell size optimization as a clear illustration of its broad potential to accelerate the exploration of parameter space for discovering polymer nanostructures. Extensions to three-dimensional phase discovery appear to be feasible.

Existence result for a two-dimensional system of conservation laws with unilateral constraints

The aim of this paper is to study a pressureless model with unilateral constraint in two dimensions. The corresponding one-dimensional case is a traffic flow model and was studied in Berthelin et al. (2008). The two-dimensional extension is linked to a pedestrian flow. Several challenges, geometric and analytical, appear with respect to the one-dimensional case. We get the existence of weak solutions for general initial data and the constraints taken in the strong sense.

Hydrogen Bond Networks Stabilized High-Capacity Organic Cathode for Lithium-Ion Batteries.

High-capacity small organic materials are plagued by their high solubility. Here we proposed constructing hydrogen bond networks (HBN) via intermolecular hydrogen bonds to suppress the solubility of active material. The illustrated 2, 7- diamino-4, 5, 9, 10-tetraone (PTO-NH2) molecule with intermolecular hydrogen (H) bond between O in -C=O and H in -NH2, which make PTO-NH2 presents transverse two-dimensional extension and longitudinal π-π stacking structure. In-situ Fourier transform infrared spectroscopy (FTIR) has tracked the reversible evolution of H-bonds, further confirming the existence of HBN structure can stabilize the intermediate 2-electron reaction state. Therefore, PTO-NH2 with HBN structure has higher active site utilization (95%), better cycle stability and rate performance. This study uncovers the H-bond effect and evolution during the electrochemical process and provides a strategy for materials design.

On two-dimensional extensions of Bougerol’s identity in law

Let B={Bt}t≥0 be a one-dimensional standard Brownian motion and denote by At,t≥0, the quadratic variation of semimartingale eBt,t≥0. The celebrated Bougerol’s identity in law (1983) asserts that, if β={βt}t≥0 is another Brownian motion independent of B, then βAt has the same law as sinhBt for every fixed t>0. Bertoin, Dufresne and Yor (2013) obtained a two-dimensional extension of the identity involving, as the second coordinates, the local times of B and β at level zero. In this paper, we present a generalization of their extension in a situation that the levels of those local times are not restricted to zero. Our argument provides a short elementary proof of the original extension and sheds new light on that subtle identity.

Lax integrability and soliton solutions of the (2 + 1)- dimensional Kadomtsev– Petviashvili– Sawada–Kotera– Ramani equation

In this paper, a new (2 + 1)-dimensional nonlinear evolution equation is investigated. This equation is called the Kadomtsev–Petviashvili–Sawada–Kotera–Ramani equation, which can be seen as the two-dimensional extension of the Korteweg–de Vries–Sawada–Kotera–Ramani equation. By means of Hirota’s bilinear operator and the binary Bell polynomials, the bilinear form and the bilinear Bäcklund transformation are obtained. Furthermore, by application of the Hopf-Cole transformation, the Lax pair is also derived. By introducing the new potential function, infinitely many conservation laws are constructed. Therefore, the Lax integrability of the equation is revealed for the first time. Finally, as the analytical solutions, the N-soliton solutions are presented.

On the two-dimensional extension of one-dimensional algebraically growing waves at neutral stability

This work considers two linear operators which yield wave modes that are classified as neutrally stable, yet have responses that grow or decay in time. Previously, King et al. (2016) and Huber et al. (2020) examined the one-dimensional (1D) wave propagation governed by these operators. Here, we extend the linear operators to two spatial dimensions (2D) and examine the resulting solutions. We find that the increase of dimension leads to long-time behaviour where the magnitude is reduced by a factor of t−12 from the 1D solutions. Thus, regions of the solution which grew algebraically as t12 in 1D now are algebraically neutral in 2D, whereas regions which decay (algebraically or exponentially) in 1D now decay more quickly in 2D. Additionally, we find that these two linear operators admit long-time solutions that are functions of the same similarity variable that contracts space and time.

Pattern formation of elliptic particles by two-body interactions: A model for dynamics of endothelial cells in angiogenesis

A two-dimensional mathematical model for dynamics of endothelial cells in angiogenesis is investigated. Angiogenesis is a morphogenic process in which new blood vessels emerge from an existing vascular network. Recently a one-dimensional discrete dynamical model has been proposed to reproduce elongation, bifurcation, and cell motility such as cell-mixing during angiogenesis on the assumption of a simple two-body interaction between endothelial cells. The present model is its two-dimensional extension, where endothelial cells are represented as the ellipses with the two-body interactions: repulsive interaction due to excluded volume effect, attractive interaction through pseudopodia and rotation by contact. We show that the oblateness of ellipses and the magnitude of contact rotation significantly affect the shape of created vascular patterns and elongation of branches.

Editorial

<p class="Text">Wearable technology, also known as wearables, smartwear, skin electronics of fashion technology, is a category of hands-free electronic devices that can be worn as accessories or embedded into clothes or skin surface. The usage involves all kinds of industries such as medicine, sports, apparel, health monitoring and management, and artificial intelligence.</p><p class="Text">As a new journal focused on wearable technology, we hope to provide a good communication platform for scholars and experts from various industries. It is our honor to invite Prof. Jun Qiu from Tsinghua University and Prof. Pibo Ma from Jiangnan University to write articles involving different fields for the first issue. Prof. Jun Qiu lab reviewed the strategic choices for high-quality development of smart wearable sporting goods industry. In this article, a strategic policy of people-oriented is established, focusing on research and development, and scientific management. Prof. Pibo Ma and his team pointed out that the type, structure and weaving method of the conductive yarn were important factors which affect the performance and wearing comfort of knitted sensors, and the electrical characteristics of two-dimensional extension and three-dimensional deformation in the strain stretching process of knitted sensors determined the effective strain sensing range.</p><p class="Text">Furthermore, we collected several other excellent articles on application of wearable technology and mechanisms or models about this technology, hoping to show an overview of this rapidly developed subject.</p><p class="Text"> </p><p class="Text" align="right">Editor-in-chief</p>Dr. Zhen Cao

Research progress of knitted sensors in the field of sports and fitness apparel

<p>Knitted sensor has the advantages of lightness, conformity, good strain tensile recovery and formability, which provides a possibility for flexible and non-inductive motion signal monitoring and smart wearable sports health clothing preparation. This paper reviews the preparation methods of knitted sensors, analyzes the influence of yarn types, fabric microstructure and tensile sensing direction on its sensing performance, and compares the advantages and disadvantages of knitted sensors in the fields of life and health, human movement and other fields. It is pointed out that the type, structure and weaving method of the conductive yarn are important factors affecting the performance and wearing comfort of knitted sensors, and the electrical characteristics of the two-dimensional extension and three-dimensional deformation in the strain stretching process of knitted sensors determine the effective strain sensing range. This paper outlines the development opportunities and challenges faced by knitted sensors in the field of sports and health clothing.</p>