Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime

<abstract><p>We study the existence of solutions to the cubic Schrödinger system</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -\Delta u_i = \sum\limits_{j = 1}^m \beta_{ij} u_j^2u_i + \lambda_i u_i\ \hbox{in}\ \Omega,\ u_i = 0\ \hbox{on}\ \partial\Omega,\ i = 1,\dots,m, $\end{document} </tex-math></disp-formula></p> <p>when $ \Omega $ is a bounded domain in $ \mathbb R^4, $ $ \lambda_i $ are positive small numbers, $ \beta_{ij} $ are real numbers so that $ \beta_{ii} &gt; 0 $ and $ \beta_{ij} = \beta_{ji} $, $ i\neq j $. We assemble the components $ u_i $ in groups so that all the interaction forces $ \beta_{ij} $ among components of the same group are attractive, i.e., $ \beta_{ij} &gt; 0 $, while forces among components of different groups are repulsive or weakly attractive, i.e., $ \beta_{ij} &lt; \overline\beta $ for some $ \overline\beta $ small. We find solutions such that each component within a given group blows-up around the same point and the different groups blow-up around different points, as all the parameters $ \lambda_i $'s approach zero.</p></abstract>

Frugal manufacturing in smart factories for widespread sustainable development

Manufacturing is a crucial activity of product development that feeds into and is also influenced by the design process. Any material conservation gained during manufacturing directly affects the green credentials of a product. Manufacturing waste can be contrived to approach zero through a recently developed frugal design approach that quantifies resource conservation at all stages of development of a product engineered for frugality. Accordingly, this effort presents frugal manufacturing (FM), integral to the frugal design approach , for utmost reduction of waste while aiming for good surface integrity, better properties, minimal number of processes and low cost. Other than saving on energy and hence emissions, the new concept of FM also goes beyond current near net shape technologies, which advocate mainly for zero wastage and suitable properties while using a narrow range of manufacturing processes. Case studies involving high-speed machining , superplastic forming and additive manufacturing of aerospace alloys have been presented that bring out the features and benefits of FM. As such the multipronged objectives of FM should be dovetailed with those of smart factories for creating novel technologies that abet widespread sustainable development. Such enhancement of the smart factories concept has been argued to support unusual applications such as the fight against pandemics including the current one involving COVID-19.

A novel surface-integral-equation formulation for efficient and accurate electromagnetic analyses of near-zero-index structures

We consider accurate and iteratively efficient solutions of electromagnetic problems involving homogenized near-zero-index (NZI) bodies using surface-integral-equation formulations in frequency domain. NZI structures can be practically useful in a plethora of optical applications, as they possess near-zero permittivity and/or permeability values that cannot be found in nature. Hence, numerical simulations are of utmost importance for rigorous design and analyses of NZI structures. Unfortunately, small values of electromagnetic parameters bring computational challenges in numerical solutions of homogeneous models. Conventional formulations available in the literature encounter stability issues that make them inaccurate and/or inefficient as permittivity and/or permeability approach zero. We propose a novel formulation that involves a well-balanced combination of operators and that can provide both accurate and efficient solutions of all NZI cases. Numerical results are presented to demonstrate the superior properties of the developed formulation in comparison to the conventional ones.

Design of Adaptive Sliding Mode Controllers for Perturbed Nonlinear Systems with Partial Unmeasurable States and State Constraints

A sliding mode control (SMC) strategy is proposed in this paper for a class of perturbed nonlinear systems with unmeasurable states and state constraints to deal with the state tracking problems. First of all, a partial states observer is designed for solving the problems due to unmeasurable states. The estimation errors will approach zero in a finite time. Secondly, based on a designed barrier Lyapunov function, one designs the sliding surface function and an adaptive sliding mode tracking controller to ensure that the states have the ability to track the desired signals. Moreover, the tracking error is capable of converging to zero in a finite time without violating the given state's constraints. Perturbation estimator and adaptive mechanisms are also utilized so that there is no need to know the upper bounds of perturbations and perturbation estimation errors. Finally, a numerical example is provided to demonstrate the feasibility of the proposed control strategy.

The Behavioral Renaissance

This chapter looks at what will be required to rebalance the radiative balance of climate at a societally acceptable level, around 1.5°C to at most 2°C warming according to the Paris Climate Agreement. The chapter outlines the complex portfolio of measures needed to achieve this: emissions reduction, new emissions avoidance, greenhouse gas removal, and potential solar radiation management. It also shows how the relative proportions of these four different classes of measures will need to be flexible through time, in response to different needs, such as a high need for emissions reduction today that may decline with time as emissions approach zero. Flexibility will also be needed in response to the emergence of new breakthroughs, challenges, cost limits, and economic and societal constraints. The chapter considers key parameters with respect to societal change and the roles of government, corporations, and consumers, and discusses routes for channeling discontent and litigation.

Numerical Simulation and Modelling Convention of Unsteady Fluidelastic Forces of Tube Arrays

This paper presents a numerical investigation on the unsteady fluidelastic forces of tube arrays. The key focus is on the consistency between the unsteady fluidelastic force model and the quasi-steady model for tube arrays at large reduced flow velocities, as well as comparing two well-known conventions for the unsteady model. Two-dimensional unsteady Reynolds-averaged Navier-Stokes (URANS) simulations are used to prove that the viscous damping coefficients of Tanaka's convention (Tanaka and Takahara, 1981) approach their quasi-steady values as the reduced flow velocity approaches infinity, whereas the hysteretic damping coefficients of Chen's modified convention (Chen et al., 1983) always approach zero and hence result in low-resolution data plots as the reduced flow velocity becomes large. The non-constant viscous damping coefficients of Tanaka's experimental data at high reduced flow velocities (which motivated the introduction of Chen's modified convention) might be induced by a systematic identification error in the phase of the fluidelastic force. A row of three flexible cylinders is used as a numerical example to analyse the effect of systematic phase error on the predicted stability boundary of the fluidelastic instability. Although identical fluidelastic forces are simulated by using the two conventions, Tanaka's convention is recommended due to its compatibility with the quasi-steady theory and optimal resolutions of data plots over any range of reduced flow velocities.

Energetics of collapsible channel flow with a nonlinear fluid-beam model

We consider flow along a finite-length collapsible channel driven by a fixed upstream flux, where a section of one wall of a planar rigid channel is replaced by a plane-strain elastic beam subject to uniform external pressure. A modified constitutive law is used to ensure that the elastic beam is energetically conservative. We apply the finite element method to solve the fully nonlinear steady and unsteady systems. In line with previous studies, we show that the system always has at least one static solution and that there is a narrow region of the parameter space where the system simultaneously exhibits two stable static configurations: an (inflated) upper branch and a (collapsed) lower branch, connected by a pair of limit point bifurcations to an unstable intermediate branch. Both upper and lower static configurations can each become unstable to self-excited oscillations, initiating either side of the region with multiple static states. As the Reynolds number increases along the upper branch the oscillatory limit cycle persists into the region with multiple steady states, where interaction with the intermediate static branch suggests a nearby homoclinic orbit. These oscillations approach zero amplitude at the upper branch limit point, resulting in a stable tongue between the upper and lower branch oscillations. Furthermore, this new formulation allows us to calculate a detailed energy budget over a period of oscillation, where we show that both upper and lower branch instabilities require an increase in the work done by the upstream pressure to overcome the increased dissipation.

On Linear Instability of Atmospheric Quasi-hydrostatic Equations in Response to Small Shortwave Perturbations

A set of 3-dimensional atmospheric-dynamics equations with quasi-hydrostatic approximation is proposed and justified with the practical goal to optimize atmospheric modelling at scales ranging from meso meteorology to global climate. Sound waves are filtered by applying the quasi-hydrostatic approximation. In the closed system of hydro/thermodynamic equations, the inertial forces are negligibly small compared to gravity forces, and the asymptotically exact equation for vertical velocity is obtained. Investigation of the stability of solutions to this system in response to small shortwave perturbations has shown that solutions have the property of shortwave instability. There are situations when the increment of the perturbation amplitude tends to infinity, corresponding to absolute instability. It means that the Cauchy problem for such equations may be ill-posed. Its formulation can become conditionally correct if solutions are sought in a limited class of sufficiently smooth functions whose Fourier harmonics tend to zero reasonably quickly when the wavelengths of the perturbations approach zero. Thus, the numerical scheme for the quasi-hydrostatic equations using the finite-difference method requires an adequately selected pseudo-viscosity to eliminate the instability caused by perturbations with wavelengths of the order of the grid size. The result is useful for choosing appropriate vertical and horizontal grid sizes for modelling to avoid shortwave instability associated with the property of the system of equations. Implementation of pseudo-viscosities helps to smoothen or suppress the perturbations that occur during modelling.

Oxidation Mechanism of Al-Sn Bearing Alloys

Oxidation of Al-Sn bearing alloy occurs during production, processing and use, which reduces both alloy performance and performance of coatings applied to the alloy surface. Therefore, the oxidation mechanism of Al-Sn bearing alloy is studied at 25, 180, 300, and 500 °C. The oxidation morphologies of the alloy were observed by scanning electron microscopy (SEM), and the oxidation products were determined by X-ray diffraction (XRD). The oxidation weight gain curves were obtained by thermogravimetric analysis. The experimental results show that: Al-Sn bearing alloy is oxidized quickly to form Al2O3. As the oxidation temperature increases, Sn phase start to precipitate along the grain boundary and form networked spheroids of Sn on the alloy surface. The amount of precipitation increases with further increase of the oxidation temperature. Cracks and holes are left in the alloy. The oxide layer is mainly composed of Sn, SnO2, and Al2O3. At 25 °C, oxidation rate of Al-Sn alloy approach zero. At 180, 300, and 500 °C, the oxidation rate increases quickly conforming to a power function, and eventually remains stable at about 3 × 10−6 mg·mm−2·s−1.

Modeling the optical properties of twisted bilayer photonic crystals

We demonstrate a photonic analog of twisted bilayer graphene that has ultra-flat photonic bands and exhibits extreme slow-light behavior. Our twisted bilayer photonic device, which has an operating wavelength in the C-band of the telecom window, uses two crystalline silicon photonic crystal slabs separated by a methyl methacrylate tunneling layer. We numerically determine the magic angle using a finite-element method and the corresponding photonic band structure, which exhibits a flat band over the entire Brillouin zone. This flat band causes the group velocity to approach zero and introduces light localization, which enhances the electromagnetic field at the expense of bandwidth. Using our original plane-wave continuum model, we find that the photonic system has a larger band asymmetry. The band structure can easily be engineered by adjusting the device geometry, giving significant freedom in the design of devices. Our work provides a fundamental understanding of the photonic properties of twisted bilayer photonic crystals and opens the door to the nanoscale-based enhancement of nonlinear effects.