Leading-order Research Articles

Theory and Transport of Nearly Incompressible Magnetohydrodynamic Turbulence: High Plasma Beta Regime

Nearly incompressible magnetohydrodynamic (NI MHD) theory for β ∼ 1 (or β ≪ 1) plasma has been developed and applied to the study of solar wind turbulence. The leading-order term in β ∼ 1 or β ≪ 1 plasma describes the majority of 2D turbulence, while the higher-order term describes the minority of slab turbulence. Here, we develop new NI MHD turbulence transport model equations in the high plasma beta regime. The leading-order term in a β ≫ 1 plasma is fully incompressible and admits both structures (flux ropes or magnetic islands) and slab (Alfvén waves) fluctuations. This paper couples the NI MHD turbulence transport equations with three fluid (proton, electron, and pickup ion) equations, and solves the 1D steady-state equations from 1–75 au. The model is tested against 27 yr of Voyager 2 data, and Ulysses and NH SWAP data. The results agree remarkably well, with some scatter, about the theoretical predictions.

Reappraisal and health: How habitual reappraisal and reappraisal ability interact to protect against life stress in young adults.

Inadequate emotion regulation may underlie the development of psychopathology as well as worsened physical health, particularly in the context of life stress. Cognitive reappraisal is typically considered an adaptive strategy to manage negative emotions. However, the extent to which reappraisal is beneficial may hinge upon contextual and individual differences. Specifically, it is unclear whether and how the ability to reappraise effectively (i.e., reappraisal ability) and exposure to stressful life events moderate the association between habitual reappraisal and health. Using a series of questionnaires and an experimental task designed to measure the ability to effectively down-regulate sad emotions using reappraisal, the present study examined the interactive effects of habitual reappraisal, reappraisal ability, and exposure to stressful life events on depressive and anxiety symptoms as well as self-reported physical health in 400 young adults (62.5% women; mean age = 19.8 ± 2.5). Results indicated that habitual reappraisal may protect against elevated depressive symptoms and worsened self-reported physical health for people exposed to more stressful life events. Moreover, reappraising often appeared to be particularly beneficial for those who were less effective in their attempts. Results for anxiety symptoms were not significant, although habitual reappraisal remained significantly associated with anxiety symptoms as a lower-order term. These findings provide novel contributions to the field of emotion regulation and health by clarifying that exposure to stressful life events is a key moderator in the association between reappraisal and some areas of health and by elucidating the important roles of both habitual reappraisal and reappraisal ability. (PsycInfo Database Record (c) 2022 APA, all rights reserved).

Spherical bispectrum: a novel visualization scheme for facilitating comparisons

Recent developments of Perturbation Theory (PT), specifically the Effective Field Theory of Large Scale Structure (EFTofLSS) and its equivalents, have proven powerful in analyzing galaxy clustering statistics such as the galaxy power spectrum and bispectrum. To further this pursuit, we have devised a novel spherical-bispectrum visualization scheme that collapses configuration dependencies to highlight the scale dependence of the bispectrum. The resulting one-dimensional curves facilitate the comparison between different bispectra, for example, from simulation and PT calculation. Using the new scheme, we present a quantitative analysis of the accuracy of PT modeling by comparing PT's analytical prediction to the result from a suite of Quijote simulations. Specifically, we determine k NL, the wavenunmber below which the analytical prediction matches well with the N-body result by inspectingboth leading order (LO) and next-to-leading order (NLO) power spectrum and bispectrum at redshifts z = 0, 0.5, 1, 2, 3. We also quantify the binning effect in Fourier space and show that an appropriate correction must be applied to the analytic predictions in order to compare them with the discrete Fourier transform results obtained from N-body-simulation or real data.

Convergence Rates for Regularized Optimal Transport via Quantization

We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or Lp regularization, general transport costs, and multimarginal problems are obtained. A novel methodology using quantization and martingale couplings is suitable for noncompact marginals and achieves, in particular, the sharp leading-order term of entropically regularized 2-Wasserstein distance for marginals with a finite [Formula: see text]-moment. Funding: This work was supported by the Alfred P. Sloan Foundation [Grant FG-2016-6282] and the Division of Mathematical Sciences [Grants DMS-1812661 and DMS-2106056].

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.

On the asymptotic behavior of solutions to a structural acoustics model

This article is concerned with the long term behavior of solutions to a structural acoustics model consisting of a semilinear wave equation defined on a bounded domain Ω⊂R3 which is coupled with a Berger plate equation acting on a flat portion of the boundary of Ω. The system is influenced by several competing forces, in particular a source term acting on the wave equation which is allowed to have a supercritical exponent.Our results build upon those obtained by Becklin and Rammaha [8]. With some restrictions on the parameters in the system and with careful analysis involving the Nehari manifold we obtain global existence of potential well solutions and establish either exponential or algebraic decay rates of energy, dependent upon the behavior of the damping terms. The main novelty in this work lies in our stabilization estimate, which notably does not generate lower-order terms. Consequently, the proof of the main result is shorter and more concise.

Long-time behaviour of solutions of superlinear systems of differential equations

ABSTRACT This paper establishes the precise asymptotic behaviour, as time t tends to infinity, for nontrivial, decaying solutions of genuinely nonlinear systems of ordinary differential equations. The lowest order term in these systems, when the spatial variables are small, is not linear, but rather positively homogeneous of a degree greater than one. We prove that the solution behaves like , as , for a nonzero vector ξ and an explicit number p>0.

Impact of electrode orientation, myocardial wall thickness, and myofiber direction on intracardiac electrograms: numerical modeling and analytical solutions.

Intracardiac electrograms (iEGMs) are time traces of the electrical potential recorded close to the heart muscle. We calculate unipolar and bipolar iEGMs analytically for a myocardial slab with parallel myofibers and validate them against numerical bidomain simulations. The analytical solution obtained via the method of mirrors is an infinite series of arctangents. It goes beyond the solid angle theory and is in good agreement with the simulations, even though bath loading effects were not accounted for in the analytical calculation. At a large distance from the myocardium, iEGMs decay as 1/R (unipolar), 1/R 2 (bipolar and parallel), and 1/R 3 (bipolar and perpendicular to the endocardium). At the endocardial surface, there is a mathematical branch cut. Here, we show how a thicker myocardium generates iEGMs with larger amplitudes and how anisotropy affects the iEGM width and amplitude. If only the leading-order term of our expansion is retained, it can be determined how the conductivities of the bath, torso, myocardium, and myofiber direction together determine the iEGM amplitude. Our results will be useful in the quantitative interpretation of iEGMs, the selection of thresholds to characterize viable tissues, and for future inferences of tissue parameters.

A two-stage Polynomial Chaos Expansion application for bound estimation of uncertain FRFs

Polynomial Chaos Expansion (PCE) is a method for analysing uncertain vibratory structures with lower computational effort. It may simply be described as a curve fitting method with orthogonal basis terms, where the polynomial type, dimension and order are predefined for the uncertain responses. However, the polynomial order in PCE must be very high to accurately estimate statistical moments of the frequency response function in resonance regions of lightly damped and uncertain structures. To solve this issue different transformation techniques are reported in the literature, where implementations of PCE produce higher accuracy with a lower order polynomial. However, these transformations lose the attraction for using PCE, since they require some additional mathematical operations and, mostly, they present high accuracy if the higher orders of polynomials are again of interest. In this study, an efficient approach is presented for the upper bound estimation of the uncertain frequency response functions (FRFs) via PCE with lower order terms without performing any transformation. Rather than one-stage application of PCE for the desired response of an uncertain problem, the approach comprises a two-stage application of the classical PCE, i.e. first for the natural frequencies and then for the FRF calculations. As an example application of the approach, a thin beam for two different uncertainty cases is considered, namely local and global uncertainty. The local and global input uncertainties are generated by variability of lumped masses added at the boundary and Young's modulus, respectively. The FRF bounds are compared with extensive experimental and numerical Monte Carlo simulations, showing that low order polynomials are sufficient to calculate the bounds accurately with the technique described.

Nonlinear degenerate p(x)-Laplacian equation with singular gradient and lower order term

Nonlinear degenerate p(x)-Laplacian equation with singular gradient and lower order term

Unexpected biases between congruence classes for parts in k-indivisible partitions

For integers k,t≥2, and 1≤r≤t let Dk×(r,t;n) be the number of parts among all k-indivisible partitions of n (i.e., partitions where all parts are not divisible by k) of n that are congruent to r modulo t. Using Wright's circle method, we derive an asymptotic for Dk×(r,t;n) as n→∞ when k,t are coprime. The main term of this asymptotic does not depend on r, and so, in a weak asymptotic sense, the parts are equidistributed among congruence classes. However, inspection of the lower order terms indicates a bias towards different congruence classes modulo t. This induces an ordering on the congruence classes modulo t, which we call the k-indivisible ordering. We prove that for k≥6(t2−1)π2 the k-indivisible ordering matches the natural ordering. We also explore the properties of these orderings when k<6(t2−1)π2.

Long-time asymptotics of solution to the coupled Hirota system with 4 × 4 Lax pair

We study the Cauchy problem for the integrable coupled Hirota system with a 4×4 Lax pair on the line with decaying initial data. By deriving a Riemann–Hilbert representation for the solution, we compute the precise leading-order terms for long-time asymptotics based on the nonlinear steepest descent arguments.

A general leading-order asymptotic theory of thin microstructural plates and uncertainty quantification of the elastic performance of composite laminates

We propose an asymptotic framework that may be used as a platform to model general thin microstructural plates, and our initiative is twofold. First, it provides an explanation to why so many tailored plate theories have been introduced for modelling plates bearing microstructures. In hypotheses-based plate theories, the microstructural details within each ply are usually smeared out. But it is shown that with such in-ply details properly taken into account, one manages to address, even only with leading-order terms, several key issues, such as transverse shear strain effects, shear locking, etc., whose rationalisation is believed to necessitate higher-order terms. To a certain degree, the proposed formulation can be used as a target for tailoring kinematic shape functions in both Reissner-Mindlin types of plate theories and zig-zag theories. Second, being slender in nature, a microstructural plate should accommodate a stress state that is close to the state of plane stress. This agrees with our findings that the (leading-order) mechanical units of a microstructural plate should be two-dimensional slices parameterised with the thickness coordinate, distinguishing the proposed framework from other plate theories based on asymptotic analysis. To demonstrate what the present framework can predict beyond existing theories, the effects on the behaviour of fibrous laminates due to manufacturing uncertainties are investigated.

Positive solutions of the Gross–Pitaevskii equation for energy critical and supercritical nonlinearities

We consider positive and spatially decaying solutions to the following Gross–Pitaevskii equation with a harmonic potential: −Δu+|x|2u=ωu+|u|p−2uin Rd, where , p > 2 and ω > 0. For (energy-critical case) there exists a ground state u ω if and only if , where for d = 3 and for . We give a precise description on asymptotic behaviours of u ω as up to the leading order term for different values of . When (energy-supercritical case) there exists a singular solution for some . We compute the Morse index of in the class of radial functions and show that the Morse index of is infinite in the oscillatory case, is equal to 1 or 2 in the monotone case for p not large enough and is equal to 1 in the monotone case for p sufficiently large.

Spurious Dianeutral Advection and Methods for Its Minimization

Abstract An existing approximately neutral surface, the ω surface, minimizes the neutrality error and hence also exhibits very small fictitious dianeutral diffusivity Df that arises when lateral diffusion is applied along the surface, in nonneutral directions. However, there is also a spurious dianeutral advection that arises when lateral advection is applied nonneutrally along the surface; equivalently, lateral advection applied along the neutral tangent planes creates a vertical velocity esp through the ω surface. Mathematically, esp = u ⋅ s, where u is the lateral velocity and s is the slope error of the surface. We find that esp produces a leading-order term in the evolution equations of temperature and salinity, being similar in magnitude to the influence of cabbeling and thermobaricity. We introduce a new method to form an approximately neutral surface, called an ωu·s surface, that minimizes esp by adjusting its depth so that the slope error is nearly perpendicular to the lateral velocity. The esp on a surface cannot be reduced to zero when closed streamlines contain nonzero neutral helicity. While esp on the ωu·s surface is over 100 times smaller than that on the ω surface, the fictitious dianeutral diffusivity on the ωu·s surface is larger, nearly equal to the canonical 10−5 m2 s−1 background diffusivity. Thus, we also develop a method to minimize a combination of esp and Df, yielding the surface, which is recommended for inverse models since it has low Df and it significantly decreases esp through the surface, which otherwise would be a leading term that cannot be ignored in the conservation equations.

Wilson loops in the abelian lattice Higgs model

We consider the lattice Higgs model on $\mathbb{Z}^4$, with structure group given by $ \mathbb{Z}_n $ for $ n \geq 2 $. We compute the expected value of the Wilson loop observable to leading order when the gauge coupling constant and hopping parameter are both sufficiently large. The leading order term is expressed in terms of a quantity arising from the related but much simpler $ \mathbb{Z}_n $ model, which reduces to the Ising model when $n=2$. As part of the proof, we construct a coupling between the lattice Higgs model and the $ \mathbb{Z}_n $ model.

On the rate of convergence of an exponential scheme for the non-linear stochastic Schrödinger equation with finite-dimensional state space

We address the numerical solution of the finite-dimensional non-linear stochastic Schrödinger equation, which is a locally Lipschitz stochastic differential equation modeling, for instance, quantum measurement processes. We study the rate of weak convergence of an exponential scheme that reproduces the norm of the desired solution by projecting onto the unit sphere. This justifies the use of the Talay-Tubaro extrapolation procedure in the numerical simulation of open quantum systems. In particular, we prove that an Euler-Exponential scheme converges with weak-order one, and we obtain the leading order term of its weak error expansion with respect to the step-size. Then, applying the Talay-Tubaro extrapolation procedure to the Euler-Exponential scheme under consideration we get a second-order method for computing the mean values of smooth functions of the solution of the non-linear stochastic Schrödinger equation. We also prove that the exponential scheme under study has order of strong convergence 1/2, which gives theoretical support to the use of the multilevel Monte Carlo method in simulating open quantum systems. We present a numerical experiment with a quantized electromagnetic field in interaction with a reservoir that illustrates the good performance of the weak second-order method, and the multilevel Monte Carlo method.

Bias in the Representative Volume Element method: Periodize the Ensemble Instead of Its Realizations

We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{\ extrm{hom}}$$\\end{document} of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document} and the corresponding linear elliptic operator -∇·a∇\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$-\ abla \\cdot a\ abla $$\\end{document}. In line with the theory of homogenization, the method proceeds by computing d=3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d=3$$\\end{document} correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document}. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a_{\ extrm{hom}}$$\\end{document} and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L-1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(L^{-1})$$\\end{document}. In case of a suitable periodization of ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document}, we rigorously show that it is O(L-d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$O(L^{-d})$$\\end{document}. In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨·⟩\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\langle \\cdot \\rangle $$\\end{document} of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.

Physics-based adaptivity of a spectral method for the Vlasov–Poisson equations based on the asymmetrically-weighted Hermite expansion in velocity space

We propose a spectral method for the 1D-1V Vlasov–Poisson system where the discretization in velocity space is based on asymmetrically-weighted Hermite functions, dynamically adapted via a scaling α and shifting u of the velocity variable. Specifically, at each time instant an adaptivity criterion selects new values of α and u based on the numerical solution of the discrete Vlasov–Poisson system obtained at that time step. Once the new values of the Hermite parameters α and u are fixed, the Hermite expansion is updated and the discrete system is further evolved for the next time step. The procedure is applied iteratively over the desired temporal interval. The key aspects of the adaptive algorithm are: the map between approximation spaces associated with different values of the Hermite parameters that preserves total mass, momentum and energy; and the adaptivity criterion to update α and u based on physics considerations relating the Hermite parameters to the average velocity and temperature of each plasma species. For the discretization of the spatial coordinate, we rely on Fourier functions and use the implicit midpoint rule for time stepping. The resulting numerical method possesses intrinsically the property of fluid-kinetic coupling, where the low-order terms of the expansion are akin to the fluid moments of a macroscopic description of the plasma, while kinetic physics is retained by adding more spectral terms. Moreover, the scheme features conservation of total mass, momentum and energy associated in the discrete, for periodic boundary conditions. A set of numerical experiments confirms that the adaptive method outperforms the non-adaptive one in terms of accuracy and stability of the numerical solution.

Well‐posedness and Mittag–Leffler stability for a nonlinear fractional telegraph problem

AbstractThe ordinary diffusion models in continuous media are derived from basic principles: Conservation of energy and momentum. The classical telegraph equation is one of the earliest and most commonly used equation to describe the transmission of electromagnetic waves. In fractal and disordered media, to capture the complex diffusion feature, we consider rather fractional models. Indeed, the mean square displacement in anomalous media is no longer proportional to the time but rather proportional to a power of time whose exponent may be between zero and one or between one and two. This is the case for the present fractional telegraph equation in the presence of non‐negligible voltage wave. Here, the well‐posedness and stability of a telegraph problem with two time‐fractional derivatives is discussed. In addition to being a noninteger problem, the equation in the model is subject to a nonlinear source as well as a nonlinear lower order fractional derivative term. First, the well‐posedness in an appropriate underlying space is established by the use of resolvent operators. Then, it is proved that, despite the presence of these nonlinearities, solutions are Mittag–Leffler stable. This confirms (and extends) the fact that the lower order term plays the role of a damper in the fractional case. To this end, we combine the energy method with some properties of the Caputo fractional derivative.