Order Correction Terms Research Articles

Wrinkling of differentially growing bilayers with similar film and substrate moduli

Growth-induced surface wrinkling in constrained bilayers comprising a thin film attached to a thick substrate is a canonical model for understanding pattern formation in many biological systems. While the bilayer model has received much prior attention, the nonlinear behaviour for arrangements with similar film and substrate properties, or substrate growth that outpaces film growth, remains poorly understood. This paper therefore focuses on these cases in which the substrate’s elasticity dominates surface wrinkling. By combining analytical modelling and finite element simulations incorporating advanced path-following techniques, we characterise critical wrinkling states and explore the initial and advanced post-buckling behaviour for a wide range of film-to-substrate modulus and growth rate ratios. It is shown that the classical leading-order asymptotic expression for the critical strain is not of sufficient accuracy for film-to-substrate modulus ratios below 50, and higher order correction terms are presented. Leveraging a weakly nonlinear analytical model, we introduce a phase diagram categorising stable and unstable post-buckling regimes. Advanced path-following simulations are employed to unveil the evolution of wrinkling patterns, from initial sinusoidal instabilities to complex formations involving period doubling, quadrupling, and eventual surface creasing. These comprehensive phase diagrams, parameterised by film-to-substrate modulus and growth rate ratios, provide new insights into the rich dynamics of surface wrinkling. Finally, we demonstrate the existence of multi-stability in the advanced post-buckling regimes for bilayers experiencing substrate-dominated growth. This work contributes to the understanding of the mechanics underlying pattern formation in growing bilayers over the entire parameter regime, with potential implications for explaining biological morphogenesis and informing the development of novel diagnostic tools and artificial flexible electronic skins.

Electronic structure and optical properties of nitrogen-doped antimonene under biaxial strain: first-principles study.

In this thesis, the role of N atom doping and biaxial strain in modulating the electronic structure and optical properties of antimonene has been deeply investigated using a first-principles approach based on density-functional theory. The results show that N doping significantly reduces the band gap of antimonene and introduces new electronic states, thus affecting its electronic structure. In terms of optical properties, N doping reduces the static permittivity of antimonene and alters its absorption, reflection, and energy loss properties. In addition, biaxial strain further enhanced the modulation effect of these properties. This study not only provides theoretical support for the application of antimonene in the field of high-performance two-dimensional electronic and optoelectronic devices, but also reveals strain and doping as an effective means to modulate the physical properties of two-dimensional materials. For the calculations, we used the DFT-based CASTEP software package for the simulation of the electronic structure. In order to more accurately characterize the weak interactions between two-dimensional materials, we specifically introduced the Van der Waals dispersion correction. We have chosen the Perdew-Burke-Ernzerhof (PBE) exchange-correlation generalization under the generalized gradient approximation (GGA) and combined it with the Van der Waals correction term in order to fully consider the electronic structure of antimonene. For the calculation parameter settings, we set the truncation energy to 400 eV to ensure the accuracy of the calculation. Meanwhile, we adopt a 6 × 6 × 1 k-point grid for Brillouin zone sampling to obtain more accurate energy band structure and density of states information. For the convergence settings, the convergence criteria for both the system energy and the interaction force between atoms were set to 1 × 10-5 eV and 0.01 eV/Å, respectively. We selected a 3 × 3 × 1 supercell model with 18 Sb atoms. A vacuum thickness of 18 Å was established in the Z direction, which is sufficient to avoid interactions between the two atomic layers above and below the periodic structure.

Riemann-Hilbert hierarchies for hard edge planar orthogonal polynomials

abstract: We obtain a full asymptotic expansion for orthogonal polynomials with respect to weighted area measure on a Jordan domain $\mathscr{D}$ with real-analytic boundary. The weight is fixed and assumed to be real-analytically smooth and strictly positive, and for any given precision $\varkappa$, the expansion holds with an $\mathrm{O}(N^{-\varkappa-1})$ error in $N$-dependent neighborhoods of the exterior region as the degree $N$ tends to infinity. The main ingredient is the derivation and analysis of Riemann-Hilbert hierarchies---sequences of scalar Riemann-Hilbert problems---which allows us to express all higher order correction terms in closed form. Indeed, the expansion may be understood as a Neumann series involving an explicit operator. The expansion theorem leads to a semiclassical asymptotic expansion of the corresponding hard edge probability wave function in terms of distributions supported on $\partial\mathscr{D}$.

The Potential Roles of Transacylation in Intracellular Lipolysis and Related Qssa Approximations

Fatty acids (FAs) are crucial energy metabolites, signalling molecules, and membrane building blocks for a wide range of organisms. Adipose triglyceride lipase (ATGL) is the first and presumingly most crucial regulator of FA release from triacylglycerols (TGs) stored within cytosolic lipid droplets. However, besides the function of releasing FAs by hydrolysing TGs into diacylglycerols (DGs), ATGL also promotes the transacylation reaction of two DG molecules into one TG and one monoacylglycerol molecule. To date, it is unknown whether DG transacylation is a coincidental byproduct of ATGL-mediated lipolysis or whether it is physiologically relevant. Experimental evidence is scarce since both, hydrolysis and transacylation, rely on the same active site of ATGL and always occur in parallel in an ensemble of molecules. This paper illustrates the potential roles of transacylation. It shows that, depending on the kinetic parameters but also on the state of the hydrolytic machinery, transacylation can increase or decrease downstream products up to 80% respectively 30%. We provide an extensive asymptotic analysis including quasi-steady-state approximations (QSSA) with higher order correction terms and provide numerical simulation. We also argue that when assessing the validity of QSSAs one should include parameter sensitivity derivatives. Our results suggest that the transacylation function of ATGL is of biological relevance by providing feedback options and altogether stability to the lipolytic machinery in adipocytes.

A Special Relativistic Exploitation of the Second Law of Thermodynamics and Its Non-Relativistic Limit.

A thermodynamic process is a solution of the balance equations fulfilling the second law of thermodynamics. This implies restrictions on the constitutive relations. The most general way to exploit these restrictions is the method introduced by Liu. This method is applied here, in contrast to most of the literature on relativistic thermodynamic constitutive theory, which goes back to a relativistic extension of the Thermodynamics of Irreversible Processes. In the present work, the balance equations and the entropy inequality are formulated in the special relativistic four-dimensional form for an observer with four-velocity parallel to the particle current. The restrictions on constitutive functions are exploited in the relativistic formulation. The domain of the constitutive functions, the state space, is chosen to include the particle number density, the internal energy density, the space derivatives of these quantities, and the space derivative of the material velocity for a chosen observer. The resulting restrictions on constitutive functions, as well as the resulting entropy production are investigated in the non-relativistic limit, and relativistic correction terms of the lowest order are derived. The restrictions on constitutive functions and the entropy production in the low energy limit are compared to the results of an exploitation of the non-relativistic balance equations and entropy inequality. In the next order of approximation our results are compared to the Thermodynamics of Irreversible Processes.

Accurate computation of geoid-quasigeoid separation in mountainous region – A case study in Colorado with full extension to the experimental geoid region

Abstract The geoid-quasigeoid separation (GQS) traditionally uses the Bouguer anomalies to approximate the difference between the mean gravity and normal gravity along the plumb line. This approximation is adequate in flat and low elevation areas, but not in high and rugged mountains. To increase the accuracy, higher order terms of the corrections (potential and gravity gradient) to the approximation were computed in Colorado where the 1 cm geoid computation experiment was conducted. Over an area of 730 km by 560 km where the elevation ranges between 932 and 4,385 m, the potential correction (Pot. Corr.) reaches −0.190 m and its root mean square (RMS) is 0.019 m. The gravity gradient correction is small but has high variation: the RMS of the correction is merely 0.003 m but varies from −0.025 to 0.020 m. In addition, the difference between the Bouguer gravity anomaly and gravity disturbance causes about a 0.01 m bias and a maximum correction of 0.02 m. The total corrections range from −0.135 to 0.180 m, with an RMS value of 0.019 m for the region. The magnitude of the corrections is large enough and is not negligible considering today’s cm-geoid requirement. After the test in Colorado, the complete GQS term is computed in 1′ × 1′ grids for the experimental geoid 2020 (xGEOID20), which covers a region bordered by latitude 0–85° north, longitude 180–350° east. Over the land areas, the RMS of the GQS is 0.119 m and the maximum reaches 1.3 m. The RMS of the GQS increases with respect to the height until 4,000 m, then decreases unexpectedly. At the highest peaks (5,500–6,000 m) of Denali and Mount Logan, the RMS of the GQS ranges between 0.08 and 0.189 m. The small GQS at these high peaks are caused by steep slopes around the peaks that produce large Pot. Corr. caused by the topography. In addition, the higher order correction terms reach half of a meter in those peaks.

Reply to the Comment by Yubo Wang et al.

Abstract In the Reply, we suppose that although the higher order correction terms of the combined field with a tightly focused Gaussian laser pulse and an external uniform magnetic field have certain influence on the dynamics and radiation spectrum of the electron, it does not affect the issues and concerns studied in our paper [EPL, 139 (2022) 14001]. Therefore, it is feasible for us to adopt the lowest order approximation for the tightly focused Gaussian laser pulse and it does not affect the main ideas and conclusions of our paper.

Response to “Comment on ‘Even-order harmonic generation from nonlinear Thomson backscatter in a tightly focused Gaussian laser pulse’” [Phys. Plasmas 29, 114703 (2022)

In a recent Comment, Chang et al. [Phys. Plasmas 29, 114703 (2022)] have studied the electron dynamics for an electron accelerating in a tightly focused Gaussian laser pulse with the higher order correction terms, and it is found that the initial position of the electron is the reason why the electron is pushed to the −z axis at the end. In this Reply, the electron dynamics and its nonlinear Thomson backscattering in a tightly focused Gaussian laser pulse with the higher order correction terms are presented, and it is found that the result is consistent with the lowest order approximation case in our paper [Hong et al., Phys. Plasmas 29, 043102 (2022)]. Meanwhile, it is also found that when the longitudinal deceleration effect of the ponderomotive force on the electron introduced by the falling part of the tightly focused laser pulse is greater than the longitudinal acceleration effect of the rising part, the electron that is initially stationary will be slightly pushed to the −z axis at the end, which can well explain the “return” phenomenon of the electron in the longitudinal direction both in our paper and in the Comment.

Second order corrections to the sputter yield of a curved surface

We study the sputter yield Y of a curved surface that is struck by a normally incident ion for radii of curvature that are large compared to the size of the collision cascade. The leading order correction to Y is proportional to the mean curvature H at the point of impact. We demonstrate analytically that there are two second order corrections to Y. One of these is proportional to H2 and the other is proportional to the Gaussian curvature at the point of impact. The predictions of the theory are compared to the results of Monte Carlo simulations of the sputtering of a variety of silicon surface morphologies for three different noble gas ion species and three ion energies. We find that including the second order correction terms considerably extends the range of radii of curvature for which the approximate formula for Y is applicable. Finally, we highlight our theory’s implications for nanoscale pattern formation on an initially flat solid surface that is bombarded with a broad ion beam.

Spectroscopic investigation of light strange S = −1 Λ and Σ and S = −2 Ξ baryons

The present study is dedicated to light-strange with strangeness S = −1 and isospin I = 0, with S = −1 and I = 1, and baryon with S = −2 and . In this study, the hypercentral constituent quark model with linear confining potential has been employed along with a first order correction term to obtain the resonance masses up to approximately 4 GeV. The calculated states include 1S-5S, 1P-4P, 1D-3D, 1F-2F, and 1G (in a few cases) along with all possible spin-parity assignments. Regge trajectories have been explored for the linearity of the calculated masses for and . Magnetic moments have been intensively studied for ground state spin and , in addition to the configuration mixing of the first negative parity state for . Lastly, the transition magnetic moments and radiative decay widths have been presented.

Stability and asymptotic analysis of the Föllmer–Schweizer decomposition on a finite probability space

First, we consider the problem of hedging in complete binomial models. Using the discrete-time F\"ollmer-Schweizer decomposition, we demonstrate the equivalence of the backward induction and sequential regression approaches. Second, in incomplete trinomial models, we examine the extension of the sequential regression approach for approximation of contingent claims. Then, on a finite probability space, we investigate stability of the discrete-time F\"ollmer-Schweizer decomposition with respect to perturbations of the stock price dynamics and, finally, perform its asymptotic analysis under simultaneous perturbations of the drift and volatility of the underlying discounted stock price process, where we prove stability and obtain explicit formulas for the leading order correction terms.

Quantitative stability for the Heisenberg–Pauli–Weyl inequality

We prove a quantitative stability result for the Heisenberg–Pauli–Weyl inequality. This leads to a next, and next-to-next order correction terms in the inequality.

Rough volatility and CGMY jumps with a finite history and the Rough Heston model – small-time asymptotics in the regime

A small-time Edgeworth expansion is established for near-the-money European options under a general rough stochastic volatility (RSV) model driven by a Riemann-Liouville (RL) process plus an additional generalised tempered stable Lévy process with when (this relaxes the more complicated and restrictive condition which appeared in an earlier version of the article.), in the regime where log-moneyness as for z fixed, conditioned on a finite volatility history. This can be viewed as a more practical variant of Theorem 3.1 in Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] (Fukasawa [Short-time at-the-money skew and rough fractional volatility. Quant. Finance, 2017, 17(2), 189–198] does not allow for jumps or a finite history and uses the somewhat opaque Muravlev representation for fractional Brownian motion), or if we turn off the rough stochastic volatility, the expansion is a variant of the main result in Mijatovic and Tankov [A new look at short-term implied volatility in asset price models with jumps. Math. Finance, 2016, 26(1), 149–183] and Theorem 3.2 in Figueroa-López et al. [Third-order short-time expansions for close-to-the-money option prices under the CGMY model. J. Appl. Math. Finance, 2017, 24, 547–574]. The regime is directly applicable to FX options where options are typically quoted in terms of delta (.10,.25 and .50) not absolute strikes, and we also compute a new prediction formula for the Riemann-Liouville process, which allows us to express the history term for the Edgeworth expansion in a more useable form in terms of the volatility process itself. We later relax the assumption of bounded volatility, and we also compute a formal small-time expansion for implied volatility in the Rough Heston model in the same regime (without jumps) which includes an additional at-the-money, convexity and fourth order correction term, and we outline how one can go to even higher order in the three separate cases , and . (We thank Masaaki Fukasawa and Hongzhong Zhang for fruitful discussions, Adam Hesse for help with Matlab computations and Vian Dinh for help with IT issues.)

A Higher Order Perturbation Approach for Electromagnetic Scattering Problems on Random Domains

We consider time-harmonic electromagnetic scattering problems on perfectly conducting scatterers with uncertain shape. Thus, the scattered field will also be uncertain. Based on the knowledge of the two-point correlation of the domain boundary variations around a reference domain, we derive a perturbation analysis for the mean of the scattered field. Therefore, we compute the second shape derivative of the scattering problem for a single perturbation. Taking the mean, this leads to an at least third order accurate approximation with respect to the perturbation amplitude of the domain variations. To compute the required second order correction term, a tensor product equation on the domain boundary has to be solved. We discuss its discretization and efficient solution using boundary integral equations. Numerical experiments in three dimensions are presented.

A new conservative/dissipative time integration scheme for nonlinear mechanical systems

We present a conservative/dissipative time integration scheme for nonlinear mechanical systems. Starting from a weak form, we derive algorithmic forces and velocities that guarantee the desired conservation/dissipation properties. Our approach relies on a collection of linearly constrained quadratic programs defining high order correction terms that modify, in the minimum possible way, the classical midpoint rule so as to guarantee the strict energy conservation/dissipation properties. The solution of these programs provides explicit formulas for the algorithmic forces and velocities which can be easily incorporated into existing implementations. Similarities and differences between our approach and well-established methods are discussed as well. The approach, suitable for reduced-order models, finite element models, or multibody systems, is tested and its capabilities are illustrated by means of several examples.

Self-bound Bose–Fermi liquids in lower dimensions

We study weakly interacting mixtures of ultracold atoms composed of bosonic and fermionic species in 2D and 1D. When interactions between particles are appropriately tuned, self-bound quantum liquids can be formed. We show that while formation of these droplets in 2D is due to the higher order correction terms contributing to the total energy and originating in quantum fluctuations, in 1D geometry the quantum fluctuations have a negligible role on formation of the self-bound systems. The leading mean-field interactions are then sufficient for droplet formation in 1D. We analyze energetic stability for 2D and 1D systems and predict values of equilibrium densities of droplets.

Evaluation of second order optomechanical coupling strength in photonic crystal cavities including the case of degenerated modes

This paper presents the potentialities of the perturbation method to provide physical interpretations of optomechanical coupling in a simultaneous photonic and phononic crystal cavity. The classical perturbation method as applied to photonic modes is recalled. The method is then extended to the degenerated modes including second order perturbation. The method ends with semi-analytical expressions for the first and second order correction terms. These semi-analytical expressions enable, on the basis of symmetry criteria, to analyze and/or predict the modulation behaviors and ultimately lead to design rules. Special attention is given to the case of degenerated optical modes and the event of degeneracy lifting is also analyzed. In order to illustrate this method, we use the simplest example of an L1 cavity: a missing hole in a square lattice. This structure displays a miscellaneous situation exhibiting interesting behaviors that may be exploited to clearly highlight the versatility of the perturbation method. We select two cases: one where the first order is dominant and practically hides the second order behavior whatever the modes are degenerated or not; the other where the first order correction vanishes in the case of non-degenerate modes, resulting in a second order behavior, but in the case of degenerate modes, both first and second orders express themselves.

Pair and many-body interactions between ligated Au nanoparticles.

We report the results of molecular dynamics simulations of the properties of a pseudo-atom (united atom) model of dodecane thiol ligated 5-nm diameter gold nanoparticles (AuNPs) in a vacuum as a function of ligand coverage and particle separation in three states of aggregation, namely, the isolated AuNPs, the isolated pair of AuNPs, and a square lattice of four AuNPs. Our calculations show that the ligand density along a radius emanating from the core of an isolated AuNP has the same gross features for all values of the coverage; it oscillates around a constant value up to a distance along the chain corresponding to the position of the fourth pseudo-atom and then smoothly decays to zero, reflecting both the restricted conformations of the chain near the core surface and the larger numbers of conformations available further from the core. Interaction between two AuNPs generates changes in the ligand distributions of each. We examine the structure and general shape of the ligand envelope as a function of the coverage and demonstrate that the equilibrium structure of the envelope and the deformation of that envelope generated by interaction between the NPs are coverage-dependent so that the shape, depth, and position of the minimum of the potential of mean force display a systematic dependence on the ligand coverage. We propose an accurate analytical description of the calculated potential of mean force as a function of a set of parameters that scale linearly with the ligand coverage. Noting that the conformational freedom of the ligands implies that multiparticle induced deviations from additivity of the pair potential of mean force are likely important; we define and calculate a "bond stretching" effective pair potential of mean force for a square lattice of particles that contains, implicitly, both the three- and four-NP contributions. We find that the bond stretching effective pair potential of mean force in this cluster has a different minimum and a different well depth from the isolated pair potential of mean force. Previous work has found that the three-particle contribution to deviation from pair additivity is monotonically repulsive, whereas we find that the combined three- and four-particle contributions have an attractive well, implying that the three- and four-particle contributions are of comparable magnitude but opposite sign, thereby suggesting that even higher order correction terms likely play a significant role in the behavior of dense assemblies of many nanoparticles.

Thermodynamics of black holes with higher order corrected entropy

For analyzing the thermodynamical behavior of two well-known black holes, such as Reissner–Nordström – anti-de Sitter (RN-AdS) black hole with global monopole and f(R) black hole, we consider the higher order logarithmic corrected entropy. We develop various thermodynamical properties, such as entropy, specific heat, pressure, and Gibbs and Helmhotz free energies for both black holes in the presence of corrected entropy. A versatile study on the stability of black holes is made by using various frameworks, such as the ratio of heat capacities (γ), grand canonical and canonical ensembles, and phase transition in view of higher order logarithmic corrected entropy. It is observed that both black holes exhibit more stability (locally as well as globally) for growing values of cosmological constant and higher order correction terms.

How Accurate is Your Gaussian/Gamma Approximation?

Gaussian and Gamma approximations appear very frequently in communications theory applications. However, in most of the applications, numerical methods are used to illustrate the accuracy of such approximations as there are no simple general purpose analytical tools available to provide a more rigorous assessment of the error. We present a simple and accurate tool to measure the accuracy of Gaussian and Gamma approximations. Specifically, we consider the first order correction terms of the approximations and derive their upper bounds. Our tool provides an accurate approximation of the error term based on the third and the fourth moments of the random variable. To demonstrate the usefulness of our analysis we consider two topical applications based on mmWave channel modeling and distributed massive multiple-input-multiple-output systems. In both cases, the proposed technique provides valuable insights into the accuracy of the commonly used approximations.