Leading-order Research Articles

Extremal results on feedback arc sets in digraphs

AbstractFor an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.

Time-Asymptotic Expansion with Pointwise Remainder Estimates for 1D Viscous Compressible Flow

We construct a time-asymptotic expansion with pointwise remainder estimates for solutions to 1D compressible Navier–Stokes equations. The leading-order term is the well-known diffusion wave and the higher-order terms are a newly introduced family of waves which we call higher-order diffusion waves. In particular, these provide an accurate description of the power-law asymptotics of the solution around the origin x=0, where the diffusion wave decays exponentially. The expansion is valid locally and also globally in the L^p({mathbb {R}})-norm for all 1le ple infty . The proof is based on pointwise estimates of Green’s function.

Regularity and stability of wave equations with variable coefficients and Wentzell type boundary conditions

We investigate regularity and stability of wave equations with variable coefficients and the frictional damping, where the damping effect is only on Wentzell boundary and there is no interior damping at all. Based on the spectral approach and the Reimannian geometry method, we present a new way of studying the challenging Wentzell-type problem concerning “the lack of interior damping” and the variable coefficients adiv(A(x)∇u). First, we establish a high-order regularity estimate for the solution of variable coefficient Wentzell problem and a spectral theorem for the operator derived from this problem. And then, after a series of specialized analysis and investigation on the special structure and basic properties of some auxiliary systems, the lower-order term and the resolvent of the operator associated to the variable coefficient Wentzell system by virtue of the Riemannian geometry method and other ideas, we prove eventually that the energies of the system decay with the rate 1/t. As can be seen, even for the constant coefficient case, the decay rate 1/t obtained in this paper is essentially better than the decay rate t−4/5 given in the previous work.

Noncoercive parabolic obstacle problems

Abstract We prove an existence result for obstacle problems related to convection-diffusion parabolic equations with singular coefficients in the convective term. Our operator is not coercive, the obstacle function is time-dependent irregular, and the coefficients in the lower-order term belong to a borderline mixed Lebesgue-Marcinkiewicz space.

Characterising small objects in the regime between the eddy current model and wave propagation

AbstractBeing able to characterise objects at low frequencies, but in situations where the modelling error in the eddy current approximation of the Maxwell system becomes large, is important for improving current metal detection technologies. Importantly, the modelling error becomes large as the frequency increases, but the accuracy of the eddy current model also depends on the object topology and on its materials, with the error being much larger for certain geometries compared to others of the same size and materials. Additionally, the eddy current model breaks down at much smaller frequencies for highly magnetic conducting materials compared to non-permeable objects (with similar conductivities, sizes and shapes) and, hence, characterising small magnetic objects made of permeable materials using the eddy current at typical frequencies of operation for a metal detector is not always possible. To address this, we derive a new asymptotic expansion for permeable highly conducting objects that is valid for small objects and holds not only for frequencies where the eddy current model is valid but also for situations where the eddy current modelling error becomes large and applying the eddy approximation would be invalid. The leading-order term we derive leads to new forms of object characterisations in terms of polarizability tensor object descriptions where the coefficients can be obtained from solving vectorial transmission problems. We expect these new characterisations to be important when considering objects at greater stand-off distance from the coils, which is important for safety critical applications, such as the identification of landmines, unexploded ordnance and concealed weapons. We also expect our results to be important when characterising artefacts of archaeological and forensic significance at greater depths than the eddy current model allows and to have further applications parking sensors and improving the detection of hidden, out-of-sight, metallic objects.

Kernel quantile estimators for nested simulation with application to portfolio value-at-risk measurement

Nested simulation has been widely used in portfolio risk measurement in recent years. We focus on one risk measure, value at risk (VaR), and study a kernel quantile estimator (KQE) for nested simulation to estimate this risk. We analyze the bias, variance, and mean squared error (MSE), based on which we show that the variance is reduced in the lower-order terms, while in some cases bias could be reduced in the dominant term. For practical implementation, we propose an efficient bootstrap-based algorithm to guide kernel bandwidth selection and budget allocation in nested simulation. We also conduct numerical experiments to show that KQE works quite well at different significance levels compared with the widely used sample quantile.

Singular elliptic problem involving a Hardy potential and lower order term

We consider the following non-linear singular elliptic problem (1) − div ( M ( x ) | ∇ u | p − 2 ∇ u ) + b | u | r − 2 u = a u p − 1 | x | p + f u γ in Ω u > 0 in Ω u = 0 on ∂ Ω , where 1 < p < N; Ω ⊂ R N is a bounded regular domain containing the origin and 0 < γ < 1, a ⩾ 0 , b > 0 , 0 ⩽ f ∈ L m ( Ω ) and 1 < m < N p . The main goal of this work is to study the existence and regularizing effect of some lower order terms in Dirichlet problems despite the presence of Hardy the potentials and the singular term in the right hand side.

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.