Continuous Solution Research Articles

Well‐posedness and blow‐up of the fractional Keller–Segel model on domains

AbstractThis work deals with well‐posedness and blow‐up in the setting of Lebesgue and Besov spaces to the time‐fractional Keller–Segel model for chemotaxis under homogeneous Neumann boundary conditions in a smooth domain of . The KS model consists in a coupled system of partial differential equations. In particular, we also treat the unique continuation of the solution and the persistence of continuous dependence on the initial data for the continued solution.

Axi-symmetric solutions for active vector models generalizing 3D Euler and electron–MHD equations

We study systems interpolating between the 3D incompressible Euler and electron–MHD equations, given by ∂tB+V⋅∇B=B⋅∇V,V=−∇×(−Δ)−aB,∇⋅B=0,where B is a time-dependent vector field in R3. Under the assumption that the initial data is axi-symmetric without swirl, we prove local well-posedness of Lipschitz continuous solutions and existence of traveling waves in the range 1/2<a<1. These generalize the corresponding results for the 3D axisymmetric Euler equations and should be useful in the study of stability and instability for axisymmetric solutions.

Effect of Heat Treatment on Microstructure and Properties Evolution of Stainless Steel Cladding Plate.

Heat treatments are necessary sometimes in order to improve comprehensive properties of stainless steel cladding plate (SSCP). However, carbon atoms in carbon steel diffuse into stainless cladding during the heat treatment process, thus decreasing its corrosion resistance. In this paper, optical microscopy, scanning electron microscopy, and microhardness and shear testing were employed to characterize the microstructure and mechanical properties of the bonding interface in SSCP. Then, the corrosion resistance of the stainless steel cladding surface was evaluated by electrochemical tests. The results showed that the diffusion of carbon atoms played an important role in enhancing the bonding strength of SSCP, but might lead to intergranular sensitization of the cladding surface because of chromium carbide precipitation. Notably, this precipitation could be induced by quenching and tempering treatment, and hindered by solution treatment. Hence, the cladding surface on SSCP after single solution treatment possessed the superior corrosion resistance, and SSCP with continuous solution and tempering treatment exhibited the highest bonding strength.

COMPUTABLE VS DESCRIPTIVE COMBINATORICS OF LOCAL PROBLEMS ON TREES

Abstract We study the position of the computable setting in the “common theory of locality” developed in [4, 5] for local problems on $\Delta $ -regular trees, $\Delta \in \omega $ . We show that such a problem admits a computable solution on every highly computable $\Delta $ -regular forest if and only if it admits a Baire measurable solution on every Borel $\Delta $ -regular forest. We also show that if such a problem admits a computable solution on every computable maximum degree $\Delta $ forest then it admits a continuous solution on every maximum degree $\Delta $ Borel graph with appropriate topological hypotheses, though the converse does not hold.

Existence and Global Asymptotic Behavior of Positive Solutions for Superlinear Singular Fractional Boundary Value Problems

In this paper, we provide sufficient conditions for the existence, uniqueness and global behavior of a positive continuous solution to some nonlinear Riemann-Liouville fractional boundary value problems. The nonlinearity is allowed to be singular at the boundary. The proofs are based on perturbation techniques after reducing the considered problem to the equivalent Fredholm integral equation of the second kind. Some examples are given to illustrate our main results.

Enhanced Autonomous Vehicle Positioning Using a Loosely Coupled INS/GNSS-Based Invariant-EKF Integration.

High-precision navigation solutions are a main requirement for autonomous vehicle (AV) applications. Global navigation satellite systems (GNSSs) are the prime source of navigation information for such applications. However, some places such as tunnels, underpasses, inside parking garages, and urban high-rise buildings suffer from GNSS signal degradation or unavailability. Therefore, another system is required to provide a continuous navigation solution, such as the inertial navigation system (INS). The vehicle's onboard inertial measuring unit (IMU) is the main INS input measurement source. However, the INS solution drifts over time due to IMU-associated errors and the mechanization process itself. Therefore, INS/GNSS integration is the proper solution for both systems' drawbacks. Traditionally, a linearized Kalman filter (LKF) such as the extended Kalman filter (EKF) is utilized as a navigation filter. The EKF deals only with the linearized errors and suppresses the higher orders using the Taylor expansion up to the first order. This paper introduces a loosely coupled INS/GNSS integration scheme using the invariant extended Kalman filter (IEKF). The IEKF state estimate is independent of the Jacobians that are derived in the EKF; instead, it uses the matrix Lie group. The proposed INS/GNSS integration using IEKF is applied to a real road trajectory for performance validation. The results show a significant enhancement when using the proposed system compared to the traditional INS/GNSS integrated system that uses EKF in both GNSS signal presence and blockage cases. The overall trajectory 2D-position RMS error reduced from 19.4 m to 3.3 m with 82.98% improvement and the 2D-position max error reduced from 73.9 m to 14.2 m with 80.78% improvement.

Phase Assemblage of the Li+,Na+,K+||F–,Cl–,Br– Five-Component Reciprocal System and Its LiF–KCl–KBr–NaBr–NaCl Stable Pentatope

The phase assemblage of the Li+,Na+,K+||F–,Cl–,Br– five-component reciprocal system was studied for the first time. The phase tree obtained by partition of the phase assemblage into stable elements is linear. It consists of the following stable elements: the LiF–NaF–KF–KBr–KCl pentatope, LiF–NaBr–NaCl–KCl–KBr–NaF hexatope, and NaCl–KCl–KBr–LiBr–LiCl–LiF–NaBr heptatope, linked by the LiF–NaF–KCl–KBr stable tetrahedron and the LiF–KBr–NaBr–NaCl–KCl square pyramid (pentatope). Phase equilibria in the LiF–KCl–KBr–NaBr–NaCl stable pentatope were studied by differential thermal analysis (DTA). Monovariant phase equilibrium L ⇄ LiF + NaClxBr1 – x + KClyBr1 – y occurs in the pentatope, where NaClxBr1 – x and KClyBr1 – y are continuous solid solutions (css) between NaCl and NaBr, KCl and KBr salt pairs, respectively. The composition of the mixture at point Min◻ 591 and the lowest monovariant equilibrium temperature were determined. A 3D computer model was designed as a projection of the phase assemblage on the LiF–KCl–KBr–NaBr–NaCl pentatope in KOMPAS-3D software. The volumes of crystallizing equilibrium phases were outlined.

Territory and sustainability: graphic expressions against a dystopia

The disruptive events that have occurred in recent years encourage us to determine that science does not make free assessments of the climate situation. The territory, its resources, are in a phase of depredation, the environment, attacked. The result is a process of environmental degradation without a solution of continuity. We take great care in environmental words and discourses, we ignore the value of the image, that of "an image is worth a thousand words", which refers us to the value of graphic expression. Through its analysis in this article, we claim it as a proactive weapon in the defense of the environment, in the fight for a new paradigm alien to growing systems, to an educational environment to reverse the situation and determine environmental stability. Actions are difficult, but they are possible, and in chromatic semiotics we find a fundamental ally. The emotion, the feeling, is the first asset to remove to start the path of reversal.

A monotone numerical flux for quasilinear convection diffusion equation

We propose a new numerical 2-point flux for a quasilinear convection–diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later approach generalizes an idea first proposed by Scharfetter and Gummel [IEEE Trans. Electron Devices 16 (1969), pp. 64–77] for linear drift-diffusion equations. We establish first that the new flux satisfies sufficient properties ensuring the convergence of the associate finite volume scheme, while respecting the maximum principle. Then, we pay attention to the long time behavior of the scheme: we show relative entropy decay properties satisfied by the new numerical flux as well as by the generalized Scharfetter-Gummel flux. The proof of these properties uses a generalization of some discrete (and continuous) log-Sobolev inequalities. The corresponding decay of the relative entropy of the continuous solution is proved in the appendix. Some 1D numerical experiments confirm the theoretical results.

DMIS: Dynamic Mesh-Based Importance Sampling for Training Physics-Informed Neural Networks

Modeling dynamics in the form of partial differential equations (PDEs) is an effectual way to understand real-world physics processes. For complex physics systems, analytical solutions are not available and numerical solutions are widely-used. However, traditional numerical algorithms are computationally expensive and challenging in handling multiphysics systems. Recently, using neural networks to solve PDEs has made significant progress, called physics-informed neural networks (PINNs). PINNs encode physical laws into neural networks and learn the continuous solutions of PDEs. For the training of PINNs, existing methods suffer from the problems of inefficiency and unstable convergence, since the PDE residuals require calculating automatic differentiation. In this paper, we propose Dynamic Mesh-based Importance Sampling (DMIS) to tackle these problems. DMIS is a novel sampling scheme based on importance sampling, which constructs a dynamic triangular mesh to estimate sample weights efficiently. DMIS has broad applicability and can be easily integrated into existing methods. The evaluation of DMIS on three widely-used benchmarks shows that DMIS improves the convergence speed and accuracy in the meantime. Especially in solving the highly nonlinear Schrödinger Equation, compared with state-of-the-art methods, DMIS shows up to 46% smaller root mean square error and five times faster convergence speed. Code is available at https://github.com/MatrixBrain/DMIS.

Moser-Trudinger inequalities and complex Monge-Ampère equation

Our aim is to give a version of the Moser-Trudinger inequality in the setting of complex geometry. As a very particular case, our result already gives a new Moser-Trudinger inequality for functions in the Sobolev space $W^{1,2}$ of a domain in $R^2$. We also deduce a new necessary condition for the complex Monge-Ampere equation for a given measure on a compact Kahler manifold to admit a Holder continuous solution.

Existence for a nonlocal Penrose-Fife type phase field system with inertial term

This article presents a nonlocal Penrose-Fife type phase field system with inertial term. We do not know whether we can prove the existence of solutions to the problem as in Colli-Grasselli-Ito [3] or not. In this article we introduce a time discretization scheme, then pass to the limit as the time step h approaches 0, and obtain an error estimate for the difference between the continuous solution and the discrete solution.

Platinum Black/Gold Nanoparticles/Polyaniline Modified Electrochemical Microneedle Sensors for Continuous In Vivo Monitoring of pH Value.

Continuous in vivo monitoring (CIVM) of pH value is essential for personalized medicine, as many diseases are closely related to acid-base imbalances. However, conventional pH meters are limited in their ability to perform CIVM due to excessive blood consumption, large device volume, frequent calibration, and inadequate real-time monitoring. There is thus an urgent need for a portable method for CIVM of pH value. To address this need, we propose a minimally invasive, continuous monitoring solution in the form of an implantable pH microneedle sensor (MNS) in this study. The MNS is based on the integration of an acupuncture needle (AN) and a Ag/AgCl reference electrode. We fabricate the sensor by electrochemically depositing platinum black and gold nanoparticles onto the AN and further modifying it with polyaniline to increase its sensitivity to hydrogen ions. The pH value is obtained by calculating the open circuit voltage between the modified AN and the reference electrode. The resulting MNS demonstrates excellent selectivity and a high nernstian response to pH (-57.4 mV per pH) over a broad range (pH = 4.0 to pH = 9.0). Both in vitro and in vivo experiments have verified the performance of the sensor, showcasing its potential for biomedical research and clinical practice. The MNS provides an alternative to conventional pH meters, offering a less invasive and more convenient way to perform CIVM of pH value. Moreover, this electrochemical implantable sensor based on AN and silver wires provides a simple and sensitive method for continuous in vivo detection of other biomarkers.

A Fully Discrete Approximation of a New Two-Temperature Thermoelastic Model

In this work, we study a new thermoelastic model with two temperatures from the numerical point of view. The problem is written as a coupled linear system whose unknowns are the displacement field and the conductivity and thermodynamic temperatures. An existence and uniqueness result recently proved is recalled. Then, a fully discrete approximation is introduced by using the finite element method and the classical implicit Euler scheme. A main a priori error estimates result is proved and, under some appropriate regularity conditions on the continuous solution, we obtain the linear convergence. Finally, some numerical simulations are presented to demonstrate the numerical convergence and the discrete energy decay.

A continuous Class‐EF power amplifier based on the general continuous mode design theory

SummaryContinuous mode provides better bandwidth performance and constant efficiency by offering greater design flexibility in fundamental and harmonics. However, this mode is mainly applicable to limited classes of power amplifiers (PAs) (such as Class‐F, Class‐J, and their derived types). In view of this problem, this paper proposes a general design theory for high‐efficiency broadband continuous PAs, which can easily extend any waveform of PAs to obtain the continuous impedance space. In theoretical analysis, first, the high‐efficiency equations are analyzed from the perspective of frequency domain. Furthermore, combining time domain constraints, a general continuous mode analysis and solution theory is presented. The theory is applied to the bandwidth expansion of Class‐EF PAs, resulting in a new continuous Class‐EF (C‐EF) mode that maintains constant efficiency while offering greater broadband design freedom. The proposed theory is verified by fabricating a C‐EF PA using GaN HEMT CGH40010F, which demonstrated output power (Pout) ranging from 39.5 to 41.2 dBm and drain efficiency (DE) of 69.5% to 79.2% in the frequency range of 2.5–3.6 GHz.

A PDE-constrained optimization method for 3D-1D coupled problems with discontinuous solutions

AbstractA numerical method for coupled 3D-1D problems with discontinuous solutions at the interfaces is derived and discussed. This extends a previous work on the subject where only continuous solutions were considered. Thanks to properly defined function spaces a well posed 3D-1D problem is obtained from the original fully 3D problem and the solution is then found by a PDE-constrained optimization reformulation. This is a domain decomposition strategy in which unknown interface variables are introduced and a suitably defined cost functional, expressing the error in fulfilling interface conditions, is minimized constrained by the constitutive equations on the subdomains. The resulting discrete problem is robust with respect to geometrical complexity thanks to the use of independent discretizations on the various subdomains. Meshes of different sizes can be used without affecting the conditioning of the discrete linear system, and this is a peculiar aspect of the considered formulation. An efficient solving strategy is further proposed, based on the use of a gradient based solver and yielding a method ready for parallel implementation. A numerical experiment on a problem with known analytical solution shows the accuracy of the method, and two examples on more complex configurations are proposed to address the applicability of the approach to practical problems.

Vacuum diffusion bonding between equiatomic CoCrNi-based concentrated solid solution alloys system and 316 stainless steel by spark plasma sintering

Spark plasma sintering (SPS) is a new method for diffusion bonding of dissimilar metals. In this study, equiatomic CoCrNi-based concentrated solid solution alloys (CSSAs) system (CoCrFeNiMn, CoCrFeNi, and CoCrNi) were successfully joined to 316 stainless steels by SPS diffusion bonding, and the effect of bonding temperature and concentration on diffusion behavior and tensile property of bonded joints were investigated. The tensile strength of the CSSAs/316SS joints were gradually increased with increasing bonding temperature. In addition, the tensile strength of the joints bonded at 0.8Tm were all exceeded 80% of that in 316SS. Meanwhile, the bonding at 0.8Tm seemed to have sufficient diffusion and grain growth at the interface, resulted in the disappearance of voids and the increase in strength of joint. In CoCrNi/316SS joint, large element concentration variation would amplitude in the diffusion zone, could result in rapid diffusion of atoms and large diffusion distance. The diffusion distance of diffusion zone in the CoCrNi/SS joint was increased by nearly two times compared to traditional diffusion bonding. Furthermore, it suggests that the continuous FCC solid solution structure was generated in the diffusion zone, which was beneficial to the increase of tensile property in joints. The tensile strength and elongation of diffusion bonded CSSAs/316SS joints were increased compared to laser beam welded joints.

Existence, construction and extension of continuous solutions of an iterative equation with multiplication

Existence, construction and extension of continuous solutions of an iterative equation with multiplication

Lead Release from Simulated Lead-Containing Jarosite Using Freeze–Thaw Cycling with EDTA

Lead is the primary toxic element found in jarosite residue; it is necessary to synthesize simulated lead-containing jarosite residue (SLJS) to investigate its lead release behavior and predict the slag’s stability and potential for secondary environmental pollution. This study explores the ion release behavior, leaching toxicity, and stability of SLJS during freeze–thaw cycles with EDTA (E-FTC). Experimental results demonstrate that the release of lead, iron, and sulfate from SLJS under E-FTC is contingent upon multiple factors, including solution pH, EDTA concentration, freeze–thaw cycles, freezing temperature, and freeze–thaw mode. Specifically, employing an EDTA concentration of 200 mM, a pH of 6, a freezing temperature of −20 °C, and 12 freeze–thaw cycles, the lead release reaches 15.1 mM, accounting for 94.9% of the total lead content, while iron is negligibly released, thus enabling effective separation of lead from iron. Subsequent to E-FTC, the exchangeable lead content exhibits a substantial reduction, accompanied by a marked increase in residual lead, resulting in a remarkable 98% reduction in leaching toxicity. Moreover, the equilibrium concentration of lead in the continuous stable leaching solution is 0.13 mg/L, significantly below the lead toxicity threshold (5 mg/L). Therefore, environmental stability can be greatly enhanced. This study presents a novel approach for the safe disposal of jarosite residue under mild conditions and at low temperatures, contributing to the broader field of environmentally sustainable waste management.

Efficient Discretization of Optimal Transport.

Obtaining solutions to optimal transportation (OT) problems is typically intractable when marginal spaces are continuous. Recent research has focused on approximating continuous solutions with discretization methods based on i.i.d. sampling, and this has shown convergence as the sample size increases. However, obtaining OT solutions with large sample sizes requires intensive computation effort, which can be prohibitive in practice. In this paper, we propose an algorithm for calculating discretizations with a given number of weighted points for marginal distributions by minimizing the (entropy-regularized) Wasserstein distance and providing bounds on the performance. The results suggest that our plans are comparable to those obtained with much larger numbers of i.i.d. samples and are more efficient than existing alternatives. Moreover, we propose a local, parallelizable version of such discretizations for applications, which we demonstrate by approximating adorable images.