Long-time Solution Research Articles

Prediction of Successful Amorphous Solid Dispersion Pairs through Liquid State Nuclear Magnetic Resonance.

Amorphous solid dispersions (ASDs) function in part via a "parachute effect", i.e., polymer-enabled prolonged drug supersaturation, presumably through drug-polymer interactions in the liquid state. We aim to expand the utility of liquid state nuclear magnetic resonance (1HNMR) to streamline polymer selection for ASDs. Our hypothesis is that strong molecular interactions between polymer and drug in 1HNMR anticipate reduced precipitation kinetics in supersaturation studies. For three drug-polymer pairs (i.e., etravirine with each HPMC, HPMCAS-M, and PVP-VA), 1HNMR findings were compared to more common supersaturation studies. Drug-polymer interactions were assessed by saturation transfer difference NMR (STD-NMR) and T1 relaxation time. 2D-1H NOESY experiments were also performed. Supersaturation studies involved precipitation inhibition using the solvent-shift methodology. The results from STD-NMR and T1 relaxation time indicate etravirine bound preferably to HPMCAS-M > HPMC ≫ PVP-VA. STD-NMR and T1 relaxation time yielded insight into which fragments of etravirine structure bind with HPMCAS-M and HPMC. The strong interactions from STD-NMR and T1 relaxation time changes indicated that HPMCAS-M and HPMC, but not PVP-VA, are suitable polymers to maintain etravirine supersaturation and inhibit drug precipitation. 2D-1H NOESY results corroborate the findings of STD-NMR and T1 relaxation time, showing that etravirine interacts preferably to HPMCAS-M than to PVP-VA. Supersaturation studies using solvent-shift technique corroborated our hypothesis as predissolved HPMCAS-M and HPMC, but to a less extent PVP-VA, markedly promoted etravirine supersaturation and inhibited drug precipitation. Supersaturation studies agreed with STD-NMR and T1 relaxation time predictions, as HPMC and HPMCAS-M maintained etravirine in solution for longer time than PVP-VA. The results show promise of 1HNMR to streamline polymer selection in a nondestructive and resource sparing fashion for subsequent ASD development.

Analysis of continuous data assimilation with large (or even infinite) nudging parameters

This paper considers continuous data assimilation (CDA) in partial differential equation (PDE) discretizations where nudging parameters can be taken arbitrarily large. We prove that solutions are long-time optimally accurate for such parameters for the heat and Navier–Stokes equations (using implicit time stepping methods), with error bounds that do not grow as the nudging parameter gets large. Existing theoretical results either prove optimal accuracy but with the error scaled by the nudging parameter, or suboptimal accuracy that is independent of it. The key idea to the improved analysis is to decompose the error based on a weighted inner product that incorporates the (symmetric by construction) nudging term, and prove that the projection error from this weighted inner product is optimal and independent of the nudging parameter. We apply the idea to BDF2-finite element discretizations of the heat equation and Navier–Stokes equations to show that with CDA, they will admit optimal long-time accurate solutions independent of the nudging parameter, for nudging parameters large enough. Several numerical tests are given for the heat equation, fluid transport equation, Navier–Stokes, and Cahn–Hilliard that illustrate the theory.

Optimized combination of room-temperature formability and high-temperature creep property in a dilute-alloyed Mg sheet via tailoring Nd solute segregation

The synergy of room-temperature formability and high-temperature creep property has been a long-term problem in the manufacturing of Mg alloy sheets. Aiming at saving rare-earth resource and providing a widely applicable approach to realize the synergy, this study focuses on the effects of preparation conditions on these two properties in Mg alloyed by dilute Nd (0.8 % in weight percent). Through comparing the mechanical behaviors of the samples prepared by different treatments, including low-temperature and long-time solution (Sample #1), high-temperature and short-time solution (Sample #2), and hot deformation for grain refinement (Sample #3), the best combination of stretch formability and creep property was successfully achieved in the Sample #1. The key reason was the conspicuous Nd solute segregation along grain boundaries after solution treated under the low-temperature and long-time condition. Such segregation behavior allowed dislocations moving freely in grain interiors but inhibited dislocations moving across grain boundaries. Finally, the stretch formability related to free dislocation slip in grain interiors and the creep property related to strong grain boundaries were synchronously enhanced. Thus, the low-temperature and long-time solution treatment is recommended for developing dilute Mg–Nd based alloy sheet used in the environment of alternating temperature changes.

Harmonic Flow of Quaternion-Kähler Structures

We formulate the gradient Dirichlet flow of Sp(2)Sp(1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Sp}(2)\ extrm{Sp}(1)$$\\end{document}-structures on 8-manifolds, as the first systematic study of a geometric quaternion-Kähler (QK) flow. Its critical condition of harmonicity is especially relevant in the QK setting, since torsion-free structures are often topologically obstructed. We show that the conformally parallel property implies harmonicity, extending a result of Grigorian in the G2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{G}_2$$\\end{document} case. We also draw several comparisons with Spin(7)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{Spin}(7)$$\\end{document}-structures. Analysing the QK harmonic flow, we prove an almost-monotonicity formula, which implies to long-time existence under small initial energy, via ϵ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\epsilon $$\\end{document}-regularity.We set up a theory of harmonic QK solitons, constructing a non-trivial steady example. We produce explicit long-time solutions: one, converging to a torsion-free limit on the hyperbolic plane; and another, converging to a limit which is harmonic but not torsion-free, on the manifold SU(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SU}(3)$$\\end{document}. We also study compactness and the formation of singularities.

Absence of anomalous dissipation of enstrophy for 3D incompressible Navier-Stokes equations

A valid method of studying the absence of anomalous dissipation of enstrophy for long-time averaged solutions of the forced and damped fluid equations in two dimensions was introduced in Constantin and Ramos (2007) [8]. Under the blueprint of their work, we will prove the absence of anomalous dissipation of enstrophy for the axisymmetric flow of the 3D damped and incompressible Navier-Stokes equations in this paper.

Streamwise dispersion of soluble matter in solvent flowing through a tube

For the dispersion of soluble matter in solvent flowing through a tube as investigated originally by G.I. Taylor, a streamwise dispersion theory is developed from a Lagrangian perspective for the whole process with multi-scale effects. By means of a convected coordinate system to decouple convection from diffusion, a diffusion-type governing equation is presented to reflect superposable diffusion processes with a multi-scale time-dependent anisotropic diffusivity tensor. A short-time benchmark, complementing the existing Taylor–Aris solution, is obtained to reveal novel statistical and physical features of mean concentration for an initial phase with isotropic molecular diffusion. For long times, effective streamwise diffusion prevails asymptotically corresponding to the overall enhanced diffusion in Taylor's classical theory. By inverse integral expansions of local concentration moments, a general streamwise dispersion model is devised to match the short- and long-time asymptotic solutions. Analytical solutions are provided for most typical cases of point and area sources in a Poiseuille tube flow, predicting persistent long tails and skewed platforms. The theoretical findings are substantiated through Monte Carlo simulations, from the initial release to the Taylor dispersion regime. Asymmetries of concentration distribution in a circular tube are certified as originated from (a) initial non-uniformity, (b) unidirectional flow convection, and (c) non-penetration boundary effect. Peculiar peaks in the concentration cloud, enhanced streamwise dispersivity and asymmetric collective phenomena of concentration distributions are illustrated heuristically and characterised to depict the non-equilibrium dispersion. The streamwise perspective could advance our understanding of macro-transport processes of both passive solutes and active suspensions.

A pressure-free long-time stable reduced-order model for two-dimensional Rayleigh-Bénard convection.

The present work presents a stable proper orthogonal decomposition (POD)-Galerkin based reduced-order model (ROM) for two-dimensional Rayleigh-Bénard convection in a square geometry for three Rayleigh numbers: 104 (steady state), 3×105 (periodic), and 6×106 (chaotic). Stability is obtained through a particular (staggered-grid) full-order model (FOM) discretization that leads to a ROM that is pressure-free and has skew-symmetric (energy-conserving) convective terms. This yields long-time stable solutions without requiring stabilizing mechanisms, even outside the training data range. The ROM's stability is validated for the different test cases by investigating the Nusselt and Reynolds number time series and the mean and variance of the vertical temperature profile. In general, these quantities converge to the FOM when increasing the number of modes, and turn out to be a good measure of accuracy. However, for the chaotic case, convergence with increasing numbers of modes is relatively difficult and a high number of modes is required to resolve the low-energy structures that are important for the global dynamics.

Synchronization in a system of Kuramoto oscillators with distributed Gaussian noise.

We consider a system of globally coupled phase-only oscillators with distributed intrinsic frequencies and evolving in the presence of distributed Gaussian white noise, namely, a Gaussian white noise whose strength for every oscillator is a specified function of its intrinsic frequency. In the absence of noise, the model reduces to the celebrated Kuramoto model of spontaneous synchronization. For two specific forms of the mentioned functional dependence and for a symmetric and unimodal distribution of the intrinsic frequencies, we unveil the rich long-time behavior that the system exhibits, which stands in stark contrast to the case in which the noise strength is the same for all the oscillators, namely, in the studied dynamics, the system may exist in either a synchronized, or an incoherent, or a time-periodic state; interestingly, all these states also appear as long-time solutions of the Kuramoto dynamics for the case of bimodal frequency distributions, but in the absence of any noise in the dynamics.

The long-time asymptotics of the derivative nonlinear Schr[formula omitted]dinger equation with step-like initial value

Consideration in this present paper is the long-time asymptotics of solutions to the derivative nonlinear Schrödinger (DNLS) equation with the step-like initial value q(x,0)=q0(x)=A1eiϕe2iBx,x<0,A2e−2iBx,x>0,by Deift–Zhou method. The step-like initial problem is described by a matrix Riemann–Hilbert problem. A crucial ingredient used in this paper is to introduce the g-function mechanism for solving the problem of the entries of the jump matrix growing exponentially as t→∞. It is shown that the leading order term of the long-time asymptotics solution of the DNLS equation is expressed by the Theta function Θ about the Riemann-surface of genus 3 and the subleading order term expressed by parabolic cylinder and Airy functions.

The effects of heat treatment on the extrusion-induced inhomogeneity of spray-formed aluminum alloy 7055

The local inhomogeneity along the thickness direction of the extruded plate of spray-formed aluminum alloy 7055 were investigated. The thermal extrusion process was simulated by using DEFORM-3D software to understand its formation mechanism. In order to reduce the extrusion-induced inhomogeneity and improve the mechanical properties, the evolution of the secondary phase during different heat treatments and its influence on the peak aged hardness and tensile strength of the alloy were emphatically analyzed. The results show that the strain rate and temperature during the thermal extrusion process tend to decrease from the surface to the center, which leads to the local inhomogeneity of the microstructure and properties in the thickness direction. Such inhomogeneity can be reduced by heat treatments combining long-time or slow-heating solution treatments (F2 and F3) with peak ageing at 120 °C, due to similar content of S (Al2MgCu) phase after solution treatment and η′ phase (peak-ageing) in different regions of the alloy. The alloy solution treated with a slow heating rate (F3) and then peak aged shows highest hardness and tensile strength, because the slow heating process during the solution treatment promotes the precipitation of Al3Zr particles, and there are less S (Al2MgCu) phases after solution treatment which is conducive to the η′ phase precipitation during the subsequent aging process.

Geodynamic Studies in the Pieniny Klippen Belt in 2004–2020

ABSTRACT The Pieniny Geodynamic Test Field is situated in the middle of the region between the Inner and Outer Carpathians. Geodynamic research conducted in the past in the Pieniny Klippen Belt (PKB) region were suggestive of neotectonic activity. The goal of the investigation was to determine whether the nearby structures, the Podhale Flysh (FP) and the Magura Nappe (MN), are affected by neotectonic activity in the PKB. The goal of the study was to ascertain the velocity and direction of motion of stations situated close to the Pieniny Geodynamic Test Field’s 3 main structures. Twelve GNSS stations, including 6 in the PKB, 3 in the MN, and 3 in the FP, make up the Pieniny Geodynamic Test Field. Three GNSS sites in the Tatra Mountains (TM) complete the entire geodynamic test field. The satellite observations made between 2004 and 2020 (excluding the year 2005 due to lack of observation) were investigated to identify the horizontal movements. Using the IGb14 reference system, the station’s positions and velocities were calculated. First, daily sessions were used to process the horizontal coordinates of the points for an average observation epoch in a given year. Sixteen measurement epochs were included in the long-time solution. Based on the horizontal velocity residues in the north–south and east–west directions, the station’s movement was calculated. The collected results were compared to information from the EUREF Permanent GNSS Network (EUREF) and to the findings of prior research on the tectonic activity of the PKB. The results of horizontal displacements calculated using GNSS measurements in the area of the PKB and nearby structures—the MN and the FP are presented and analyzed in this article.

Moment analysis for predicting effective transport properties in hierarchical retentive porous media

We report on a new homogenization approach to solve, with drastically improved speed and accuracy, the general advection-diffusion equation in hierarchical porous media with localized diffusion and adsorption/desorption processes, thus opening the way to a much deeper understanding of the band broadening process in chromatographic systems. The proposed robust and efficient moment-based approach allows us to compute the exact local and integral concentration moments and hence provides exact solutions for the effective velocity and dispersion coefficients of migrating solute particles. Innovative to the proposed method is also that it not only produces the exact effective transport parameters of the long-time asymptotic solution, but also their entire transient. The analysis of the transient behaviour can be used, for example, to properly identify the time and length scales needed to achieve the macro-transport conditions. If the hierarchical porous media can be represented as the periodic repetition of a unit lattice cell, the method only requires the solution of the time-dependent advection-diffusion equations for the zeroth order and first-order exact local moments, exclusively on the unit cell. This implies an enormous reduction of the computational efforts and a significant improvement of the accuracy of the results when compared to the direct numerical simulation (DNS) approaches which require flow domains that are long enough to achieve steady-state conditions, and hence often cover tens to hundreds of unit cells. The reliability of the proposed method is verified by comparing its predictions with DNS results, in one, two and three dimensions, in both transient and asymptotic conditions. The influence of top and bottom no-slip walls on the separation performance of chromatographic columns with micromachined porous and nonporous pillars is discussed in detail.

On long-time asymptotics to the nonlocal short pulse equation with the Schwartz-type initial data: Without solitons

In this work, the initial value problem of the nonlocal short pulse (NSP) equation is studied with the Schwartz-type initial data. Our aim is to adequately study the long-time asymptotic behavior of the solution of the NSP equation in view of the initial value condition. Starting with the Lax pair of the NSP equation, we give some discussion about the NSP equation, including infinite number of conservation laws and finite-dimensional Hamiltonian function. The relevant spectral analysis is discussed, among which it is important to note that we discuss it separately due to the singularity of the spectrum. According to these results, a suitable Riemann–Hilbert (RH) problem which is used to express the solution of the NSP equation is established. By using the nonlinear steepest descent method, the original RH problem is transformed into a model RH problem, which is given by a parabolic cylinder function. The long-time asymptotic solution of the NSP equation is therefore given, especially the error term is quite different from the local situation.

Infrastructure adaptation and emergence of loops in network routing with time-dependent loads.

Network routing approaches are widely used to study the evolution in time of self-adapting systems. However, few advances have been made for problems where adaptation is governed by time-dependent inputs. In this work we study a dynamical systems where the edge conductivities of a network are regulated by time-varying mass loads injected on nodes. Motivated by empirical observations, we assume that conductivities adapt slowly with respect to the characteristic time of the loads. Furthermore, assuming the loads to be periodic, we derive a dynamics where the evolution of the system is controlled by a matrix obtained with the Fourier coefficients of the input loads. Remarkably, we find a sufficient condition on these coefficients that determines when the resulting network topologies are trees. We show an example of this on the Bordeaux bus network where we tune the input loads to interpolate between loopy and tree topologies. We validate our model on several synthetic networks and provide an expression for long-time solutions of the original conductivities.

Second-order estimates for collapsed limits of Ricci-flat Kähler metrics

AbstractWe show that the singularities of the twisted Kähler–Einstein metric arising as the longtime solution of the Kähler–Ricci flow or in the collapsed limit of Ricci-flat Kähler metrics are intimately related to the holomorphic sectional curvature of reference conical geometry. This provides an alternative proof of the second-order estimate obtained by Gross, Tosatti, and Zhang (2020, Preprint, arXiv:1911.07315) with explicit constants appearing in the divisorial pole.

A compact finite difference scheme with absorbing boundary condition for forced KdV equation

Studying the long-time solution behavior of the Korteweg-de Vries (KdV) type equation with a periodic force acting at one end of the long channel is important for simulating the blood flow in artery driven by heart pulses. It is of great interest to develop an accurate numerical method for solving the forced KdV problem. In this article, we present the following methods to obtain an accurate approximation to the solution of KdV problem.•An accurate compact finite difference scheme is proposed for solving the above forced KdV problem with fourth-order accuracy.•An absorbing boundary condition at the right end of the interval is used to avoid the wave reflection.•The stability of scheme is proved by the von Neumann method and then tested by three examples. Results show that the method provides an accurate solution, and the wave propagates without reflection.

On the pluriclosed flow on Oeljeklaus–Toma manifolds

AbstractWe investigate the pluriclosed flow on Oeljeklaus–Toma manifolds. We parameterize left-invariant pluriclosed metrics on Oeljeklaus–Toma manifolds, and we classify the ones which lift to an algebraic soliton of the pluriclosed flow on the universal covering. We further show that the pluriclosed flow starting from a left-invariant pluriclosed metric has a long-time solution $\omega _t$ which once normalized collapses to a torus in the Gromov–Hausdorff sense. Moreover, the lift of $\tfrac {1}{1+t}\omega _t$ to the universal covering of the manifold converges in the Cheeger–Gromov sense to $(\mathbb H^s\times \mathbb C^s, \tilde {\omega }_{\infty })$ , where $\tilde {\omega }_{\infty }$ is an algebraic soliton.

Dynamic Heat Transfer Calculation for Ground-Coupled Floor in Emergency Temporary Housing

Generally, ground-coupled floor heat transfer is supposed as annual periodic, which is reasonable for conventional buildings. However, for emergency housing with a short life cycle, the influence of initial soil temperature needs to be considered. In a previous study, the Wiener–Hopf technique was introduced to solve the two-dimensional transient heat transfer equation with mixed Dirichlet and Robin boundary conditions. Based on that, an analytical solution of the dynamic heat transfer equation with initial soil temperature conditions was obtained. Since the solution was in the form of a double integral, its numerical evaluation method was also analyzed to improve computational efficiency. The accuracy and efficiency of the solution were validated by the finite volume method. Then, the effects of initial soil temperatures in different seasons, soil heat conductivities, and floor insulation on ground-coupled heat transfer were discussed. Results showed significant temperature differences between the current solution and the annual periodic solutions (long-time solutions), especially in hot and cold climates. Moreover, the larger the thermal capacity of the soil, the bigger temperature differences occurred. Therefore, this study is expected to provide a theoretical foundation for the indoor environment prediction and optimization design of emergency temporary housing.

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.

Diameter estimates for long-time solutions of the Kähler–Ricci flow

Diameter estimates for long-time solutions of the Kähler–Ricci flow