Gradient Of Solution Research Articles

Buoyancy-Driven Micro/-Nanomotors: From Fundamentals to Applications.

The progression of self-powered micro/-nanomotors (MNMs) has rapidly evolved over the past few decades, showing applications in various fields such as nanotechnology, biomedical engineering, microfluidics, environmental science, and energy harvesting. Miniaturized MNMs transduce chemical/biochemical energies into mechanical motion for navigating through complex fluidic environments with directional control via external forces fields such as magnetic, photonic, and electric stimuli. Among various propulsion mechanisms, buoyancy-driven MNMs have received noteworthy recognition due to their simplicity, efficiency, and versatility. Buoyancy force-driven motors harness the principles of density variation-mediated force to overcome fluidic resistance to navigate through complex environments. Restricting the propulsion in one direction helps to control directional movement, making it more efficient in isotropic solutions. The changes in pH, ionic strength, chemical concentration, solute gradients, or the presence of specific molecules can influence the motion of buoyancy-driven MNMs as evidenced by earlier reports. This review aims to provide a fundamental and detailed analysis of the current state-of-the-art in buoyancy-driven MNMs, aiming to inspire further research and innovation in this promising field.

Ion current oscillation of polyelectrolyte modified micropipettes.

Here, we report for the first time that ion current oscillation (ICO) with periodic amplitude and frequency can autonomously occur at polyimidazole brush (PvimB) modified pipettes in an asymmetric solution with a pH gradient (e.g. pH 6.0/pH 8.0). Experimental results demonstrated that under a strong bias voltage, the proton responsive PvimB-modified pipettes exhibited significant current switching behavior under negative bias voltages, which contributed to the periodic oscillating ion current under constant biases. Based on this dynamic, the frequency and amplitude of the ICO phenomenon were regulated by adjusting the pH gradient in the asymmetric solution. ICOs under different bias voltages were further explored to show the voltage-dependent nature of this phenomenon. This observation of ICO phenomena offers a new strategy for designing iontronic devices with dynamic conductivity changes induced by surface chemical interactions within spatial confinements.

On divergence-free (form-bounded type) drifts

We develop regularity theory, including a posteriori Harnack inequality and higher integrability of the gradient of solutions, for divergence-form elliptic equations with singular divergence-free drift satisfying a broad form-boundedness-type condition (more generally, our drifts can have singular divergence). A key step in our proofs is a new iteration procedure (needed already to prove Caccioppoli's inequality), which is used in addition to the classical De Giorgi's iterations and Moser's method.

Modified nonmonotonic projection Barzilai-Borwein gradient method for nonnegative matrix factorization

<p>In this paper, an active set recognition technique is suggested, and then a modified nonmonotonic line search rule is presented to enhance the efficiency of the nonmonotonic line search rule, in which we introduce a new parameter formula to attempt to control the nonmonotonic degree of the line search, and thus improve the chance of discovering the global minimum. By using a modified linear search and an active set recognition technique, a global convergence gradient solution for nonnegative matrix factorization (NMF) based on an alternating nonnegative least squares framework is proposed. We used a Barzilai-Borwein step size and greater step-size tactics to speed up the convergence. Finally, a large number of numerical experiments were carried out on synthetic and image datasets, and the results showed that our presented method was effective in calculating the speed and solution quality.</p>

Gradient estimates for singular $p$-Laplace type equations with measure data

We are concerned with interior and global gradient estimates for solutions to a class of singular quasilinear elliptic equations with measure data, whose prototype is given by the p -Laplace equation -\Delta_{p} u=\mu with p\in (1,2) . The cases when p\in (2-\frac{1}{n},2) and p\in (\frac{3n-2}{2n-1},2-\frac{1}{n}] were studied in Duzaar and Mingione [J. Funct. Anal. 259 (2010), 379–418] and Nguyen and Phuc [J. Funct. Anal. 278 (2020), art. 108391] respectively. In this paper, we improve the results of Nguyen and Phuc and address the open case when p\in(1,\frac{3n-2}{2n-1}] . Interior and global modulus of continuity estimates of the gradients of solutions are also established.

A nonlinear-manifold reduced-order model and operator learning for partial differential equations with sharp solution gradients

Traditional linear subspace-based reduced order models (LS-ROMs) can be used to significantly accelerate simulations in which the solution space of the discretized system has a small dimension (with a fast decaying Kolmogorov n-width). However, LS-ROMs struggle to achieve speed-ups in problems whose solution space has a large dimension, such as highly nonlinear problems whose solutions have large gradients. Such an issue can be alleviated by combining nonlinear model reduction with operator learning. Over the past decade, many nonlinear manifold-based reduced order models (NM-ROM) have been proposed. In particular, NM-ROMs based on deep neural networks (DNN) have received increasing interest. This work takes inspiration from adaptive basis methods and specifically focuses on developing an NM-ROM based on Convolutional Neural Network-based autoencoders (CNNAE) with iteration-dependent trainable kernels. Additionally, we investigate DNN-based and quadratic operator inference strategies between latent spaces. A strategy to perform vectorized implicit time integration is also proposed. We demonstrate that the proposed CNN-based NM-ROM, combined with DNN-based operator inference, generally performs better than commonly employed strategies (in terms of prediction accuracy) on a benchmark advection-dominated problem. The method also presents substantial gain in terms of training speed per epoch, with a training time about one order of magnitude smaller than the one associated with a state-of-the-art technique performing with the same level of accuracy.

Adaptive deep neural networks for solving corner singular problems

Deep neural networks (DNNs) for numerical solutions to partial differential equations (PDEs) have exhibited their remarkable merits of meshless methods, dimensionless features, and nonlinear approximation powers. The majority of conventional DNNs are mainly concentrated on the low- or high-dimensional PDE with smooth solutions. In singular problems the DNN does not behave so efficiently as in the smooth problem because of low regularities of the solution. In this paper, we propose a novel adaptive DNN for the PDE with corner singular solutions. In the corner singular problem gradients of solution vary rapidly around some particular vertices or lines. This poses a difficulty in development of DNN solvers. We design adaptive DNN techniques for the corner singularity, which incorporates (a) an adaptive loss function that assigns radial weights to local losses, (b) an adaptive activation function that varies the choice of activation functions from neuron to neuron, and (c) an adaptive sampling strategy that selectively focuses on sampling points near the singular locations. The adaptive DNN with these adaptive schemes greatly enhances approximation accuracy and computation efficiency. This is verified by a comprehensive series of numerical experiments of 2D and 3D corner singular problems. The comparison analyzes with the existing DNN solvers are also made to demonstrate effectiveness of the new method. The L2 relative errors of the proposed adaptive DNN are reduced by an order of magnitude with respect to several conventional DNN methods.

Analyzing the presence of ten synthetic colors in selected food and drink samples using ultra high-performance liquid chromatography diode array detection analysis

AbstractColoring agents in foods and drinks have been popular for centuries. This study aims to analyze the presence of ten synthetic colors (namely, (allura red (E129), amaranth (E123), sunset yellow (E110), tetrazine (E102), fast green (E143), ponceau 4R (New Coccine) (E124), erythrosin B (E127), brilliant blue FCF (E133), brilliant black (E151) and carmoisine (E122))) in food and drink samples using ultra-high-performance liquid chromatography diode array detection (UHPLC-DAD). The present analytical method was carried out using Agilent Poroshell 120 HPH-C18 column, 3 × 100 mm, 2.7 µm, and a mobile phase consisting of 10 mM Na2HPO4, pH 7, mixed with methanol as a time-increment gradient solution until the time was 20 min, then decreased with time until the time was 26 min. The pH was set by orthophosphoric acid at 7 and 5 μL injection volume, 0.50 mL flow rate, and the elution systems were monitored at 428 nm for E102, 518 nm for E124, E110, E129, E122, 530 nm for E151, E127, 622 nm for E143, and E133, respectively. The limit of detection and quantification for all colors ranged from 0.017 to 0.025 and 0.057 and 0.082 mg L−1, respectively. The correlation coefficient values ranged between 0.9991 and 1.0. The selectivity of the assay revealed no interference from other components in the analyzed samples. The percent recovery and precision (intra- and inter-day) of the spiked samples were within the acceptable limits of the ICH guidelines. Five analytical parameters were employed, and the results showed a new, novel, and robust method according to ICH guidelines for analyzing these colors. While most of the investigated food and drinks fell within the accepted range, some fell outside. The current sample preparation and analytical methods are comprehensive and universal for extracting and measuring synthetic colors in various food and drink samples.

A study on the linkage between strategic human resource management and organizational effectiveness based on gradient descent algorithm

Abstract Using a gradient descent algorithm, a study on the linkage between strategic human resource management and organizational effectiveness. A logarithmic function with a base of 10 is used to represent the useful value of the fused elements of the association and simplify the expression formula for the usefulness of management. The usefulness of the management approach in achieving organizational effectiveness is measured using the information gain approach. The square of the difference between the predicted and true values is used as the loss function value, and the gradient descent algorithm is combined with the gradient descent algorithm to divide the effective joint data and calculate the solution gradient for all samples. Evaluation indexes such as accuracy and recall test the joint degree effective prediction model. The results show that the usefulness of the joint degree effective prediction model is classified into three degree levels: average [0,0.2), useful [0.2,0.4), and 40% or more as very useful. The team’s organizational commitment results in 50% of employees working hard, and only 5% are unmotivated to work.

Understanding natural convection in low-concentration electrochemical system: Validation of the convection modes

Natural convection could arise under density gradients of solutions caused by the electrochemical reaction occurring on the electrode surface. This density-driven convection decays rapidly with the decreasing redox concentration, and becomes negligible in dilute solutions (1–2 mM). However, steady-state voltammetry is still observed in these dilute solutions due to the presence of a convection–diffusion layer. Here we studied natural convection effects in low-concentration (5–30 mM) redox solutions and clarified the convection modes in these transient-concentration solutions using cyclic voltammetry and chronoamperometry. The combined theoretical–experimental work demonstrated the coexistence of convection–diffusion layer and density-driven convection. With the use of a 150-μm-radius Pt microelectrode, the convection–diffusion layer convection dominated natural convection in dilute solution (1–5 mM), reinforced mass transport with density-driven convection in low redox concentrations (5–11 mM), and suppressed the density-driven convection at high redox concentrations (>15 mM). The zone diagrams delineating the transition from diffusion to convection were then established to reveal the effects of electrode radius, thickness of the convection–diffusion layer, time scale, and redox concentration on natural convection. Consequently, the use of microelectrodes (∼25 μm radius) could greatly inhibit natural convection effects in 30 mM redox solution.

Energy estimate up to the boundary for stable solutions to semilinear elliptic problems

We obtain a universal energy estimate up to the boundary for stable solutions of semilinear equations with variable coefficients. Namely, we consider solutions to −Lu=f(u), where L is a linear uniformly elliptic operator and f is C1, such that the linearized equation −L−f′(u) has nonnegative principal eigenvalue. Our main result is an estimate for the L2+γ norm of the gradient of stable solutions vanishing on the flat part of a half-ball, for any nonnegative and nondecreasing f. This bound only requires the elliptic coefficients to be Lipschitz. As a consequence, our estimate continues to hold in general C1,1 domains if we further assume the nonlinearity f to be convex. This result is new even for the Laplacian, for which a C3 regularity assumption on the domain was needed.

Fractional differentiability for elliptic measure data problems with variable exponents

We investigate quasilinear elliptic equations with variable growth exponents having a Borel measure on the right-hand side. We prove fractional differentiability results for a certain nonlinear expression of the gradient of solutions.

Gradient bounds for non-uniformly quasilinear elliptic two-sided obstacle problems with variable exponents

We are concerned with a class of quasilinear elliptic variational inequalities, the so-called two-obstacle problems, that are driven by the double-phase operators with variable exponents. In this paper, by using the terminology of fractional maximal distribution functions, we prove decay estimates for level sets of the gradient of weak solutions in turn implying regularity estimates in some generalized weighted function spaces.

Bioavailability of Cd in Agricultural Soils Evaluated by DGT Measurements and the DIFS Model in Relation to Uptake by Rice and Tea Plants

The elevated accumulation of cadmium (Cd) in rice (Oryza sativa L.) and tea (Camellia sinensis L.) grown in agricultural soils may lead to a variety of adverse health effects. This study collected and analyzed crop samples along with paired rhizosphere soil samples from 61 sites in Cd-contaminated regions in Anhui Province, China. The findings revealed that both the diffusive gradients in thin-films (DGT) and soil solution were capable of effectively predicting Cd contents in crops. Conventional chemical extraction methods were inappropriate to evaluate the bioavailability of Cd. However, the effective concentrations (CE) corrected by the DGT-induced fluxes in soils (DIFS) model exhibited the strongest correlation with crop Cd contents. Except for CE, various measurement methods yielded better results for predicting Cd bioavailability in tea compared to rice. Pearson’s correlation analysis and the random forest (RF) model identified the key influencing factors controlling Cd uptake by rice and tea, including pH, soil texture, and contents of zinc (Zn) and selenium (Se) in soils, which antagonize Cd. To reduce the potential health risk from rice and tea, the application of soil liming and/or Se-oxidizing bacteria was expected to be an effective management strategy.

Higher regularity for weak solutions to degenerate parabolic problems

In this paper, we study two related features of the regularity of the weak solutions to the following strongly degenerate parabolic equation ut-divDu-1+p-1DuDu=finΩT=Ω×(0,T),\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} u_t-\ extrm{div}\\left( \\left( \\left| Du\\right| -1\\right) _+^{p-1}\\frac{Du}{\\left| Du\\right| }\\right) =f\\qquad \ ext{ in } \\Omega _T =\\Omega \ imes (0,T), \\end{aligned}$$\\end{document}where Omega is a bounded domain in mathbb {R}^{n} for nge 2, pge text {and}, T>0. We prove the higher differentiability of a nonlinear function of the spatial gradient of the weak solutions, assuming only that fin L^{2}_{textrm{loc}}left( Omega _Tright) . This allows us to establish the higher integrability of the spatial gradient under the same minimal requirement on the datum f.

Rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. Part I Partial differential equation

The aim of this paper is to provide the analysis result for the partial differential equations arising from the rigorous derivation of the degenerate parabolic-elliptic Keller-Segel system from a moderately interacting stochastic particle system. The rigorous derivation is divided into two articles. In this paper, we establish the solution theory of the degenerate parabolic-elliptic Keller-Segel problem and its non-local version, which will be used in the second paper for the discussion of the mean-field limit. A parabolic regularized system is introduced to bridge the stochastic particle model and the degenerate Keller-Segel system. We derive the existence of the solution to this regularized system by constructing approximate solutions, giving uniform estimates and taking the limits, where a crucial step is to obtain the L∞ Bernstein type estimate for the gradient of the approximate solution. Based on this, we obtain the well-posedness of the corresponding non-local equation through perturbation method. Finally, the weak solution of the degenerate Keller-Segel system is obtained by using a nonlinear version of Aubin-Lions lemma.

Spatial–temporal analysis of magnetohydrodynamics flow and energy flux of power-law nanofluid in a confined domain

Energy flux analysis of power-law fluid is a novel contribution to recent developments in computational fluid dynamics. The study of the unsteady two-dimensional flow with double diffusive effect inside a complex enclosure exhibits great potential in optimizing the heat transfer rate due to the wavy nature of the side walls. The enclosure is confined with the flow circulation due to the thermal and solutal gradients acting along the left and right wavy walls. The computational time and accuracy in results are estimated and compared by implementing the finite volume method and element-free Galerkin technique. The results are obtained in terms of streamlines, isotherms, isoconcentrations, average Nusselt number, Sherwood number, and total entropy generation due to the effect of conventional parameters, namely, power-law index, Rayleigh number, buoyancy ratio parameter, thermophoresis parameter, Brownian motion parameter, and Lewis number with a fixed Prandtl number throughout the computation. The optimized double-diffusive natural convection analysis is based on entropy generation and a calculated Bejan number. The novelty of this paper lies in the implementation of a mesh-free approach, which may be useful for the further analysis of elliptical/semi-elliptical structures.

Discrete Gagliardo–Nirenberg inequality and application to the finite volume approximation of a convection–diffusion equation with a Joule effect term

Abstract A discrete order-two Gagliardo–Nirenberg inequality is established for piecewise constant functions defined on a two-dimensional structured mesh composed of rectangular cells. As in the continuous framework, this discrete Gagliardo–Nirenberg inequality allows to control in particular the $L^4$ norm of the discrete gradient of the numerical solution by the $L^2$ norm of its discrete Hessian times its $L^\infty $ norm. This result is crucial for the convergence analysis of a finite volume method for the approximation of a convection–diffusion equation involving a Joule effect term on a uniform mesh in each direction. The convergence proof relies on compactness arguments and on a priori estimates under a smallness assumption on the data, which is essential also in the continuous framework.

Role of density gradients in the growth dynamics of 2-dimensional MoS2 using liquid phase molybdenum precursor in chemical vapor deposition

The key objective of this work was the comparison of MoS2 2-dimensional monolayer flakes grown by Chemical Vapour Deposition with different density gradients in the liquid precursors solution. Structures grown using solutions with iodixanol or glycerol were characterized by Optical Microscopy, Atomic Force Microscopy, Raman spectroscopy and Photoluminescence. Furthermore, electrical properties of structures were studied by characterizing backgated Field-Effect Transistors fabricated on individual flakes.The fundamental outcome of this study is that there are relevant differences between the structures grown with the two different liquid solutions: MoS2 structures grown with glycerol present a large variability in the lateral size of the sharp-corner triangular structure with a good crystalline quality while the employment of iodixanol (the most used substance) causes a poorer crystalline quality, demonstrated by the presence of nanometric pores, with a more homogeneous sizing of the triangular structures.The poor crystalline quality of the iodixanol samples results in reduced light emission efficiency and lower mobility.

A massively parallel implicit 3D unstructured grid solver for computing turbulent flows on latest distributed memory computational architectures

An implicit unstructured grid density-based solver–PRAVAHA based on a parallel variant of the lower upper symmetric Gauss-Seidel (LUSGS) method is developed to compute large-scale engineering problems. A four-layered parallel algorithm is designed to efficiently compute three-dimensional turbulent flows on massively parallel modern multiple instruction multiple data-stream (MIMD) computational hardware. This data parallel approach achieves multiple layers of parallelism including continuity of flow solution, transfer of solution gradients, and calculation of drag/lift/solution residuals, right up to the innermost implicit LUSGS solver sub-routine, which is relatively less explored in the literature. Domain decomposition is performed using the METIS software based on multi-level graph partitioning algorithms. Non-blocking message-passing interface library functions are used to manage inter-processor communication through explicit message passing, efficiently. Super-linear scalability of the parallel solver is established on the current state-of-the-art supercomputing facility, the 838 teraflops PARAM seva on up to 6144 cores. Linear or even super-linear speedup on problems of significant size is observed even on ad-hoc parallel computing platforms like workstations and multi-node clusters, for turbulent flow simulations.