Second-order Terms Research Articles

Vanishing viscosity limit for a one-dimensional viscous conservation law in the presence of two noninteracting shocks

Abstract In this article, we study the inviscid limit of the solution to the Cauchy problem of a one-dimensional viscous conservation law, where the second-order term is nonlinear. Under the assumption that the inviscid equation admits a piecewise smooth solution with two noninteracting entropy shocks, we prove that the solution of the viscous equation converges uniformly to the piecewise smooth inviscid solution away from the shocks, even the strength of shocks is not small.

ANALYSIS OF OPTIMAL PORTFOLIO ON FINITE AND SMALL-TIME HORIZONS FOR A STOCHASTIC VOLATILITY MODEL WITH MULTIPLE CORRELATED ASSETS

In this paper, we consider the portfolio optimization problem in a financial market where the underlying stochastic volatility model is driven by [Formula: see text]-dimensional Brownian motions. At first, we derive a Hamilton–Jacobi–Bellman equation including the correlations among the standard Brownian motions. We use an approximation method for the optimization of portfolios. With such approximation, the value function is analyzed using the first-order terms of expansion of the utility function in the powers of time to the horizon. The error of this approximation is controlled using the second-order terms of expansion of the utility function. It is also shown that the one-dimensional version of this analysis corresponds to a known result in the literature. We also generate a close-to-optimal portfolio near the time to horizon using the first-order approximation of the utility function. It is shown that the error is controlled by the square of the time to the horizon. Finally, we provide an approximation scheme to the value function for all times and generate a close-to-optimal portfolio.

Solutions of a Neutron Transport Equation with a Partly Elastic Collision Operators

In this paper, we derive sufficient conditions that guarantee an description of long-time asymptotic behavior of the solution to the Cauchy problem governed by a linear neutron transport equation with a partially elastic collision operator under periodic boundary conditions. Our strategy involves showing that the strongly continuous semigroups et(T+Ke)t≥0 and et(T+Kc+Ke)t≥0, generated by the operators T+Ke and T+Kc+Ke, respectively, have the same essential type. According to Proposition 1, it is sufficient to show that remainder term in the Dyson–Philips expansion is compact. Our analysis focuses on the compactness properties of the second-order remainder term in the Dyson–Phillips expansion related to the problem. We first show that R2(t) is compact on L2(Ω×V,dxdv), and, using an interpolation argument (see Proposition 2), we establish the compactness of R2(t) on Lp(Ω×V,dxdv)-spaces for 1<p<+∞. To the best of our knowledge, outside the one-dimensional case, this result is known only for vaccum boundary conditions in the multidimensional setting. However, our result, Theorem 1, is new for periodic boundary conditions.

Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations*

Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations.

Investigation of Coupling Effect for Adjacent Orthopedic Implants on MRI Radio-Frequency Heating

This paper investigates the mechanism of radio-frequency (RF) heating that occurs when two adjacent orthopedic implants are present together under magnetic resonance imaging (MRI) at 1.5 Tesla and 3.0 Tesla. When a patient has multiple implants close to each other, interactions between the implants may increase RF heating. Typical generic interlocking plate and antibiotic nail implants are adopted as examples. To analyze the effect of adjacent implants, the amplitude and direction of incident and scattering vector electric fields at the hot spot position are calculated and extracted using numerical simulation based on Huygens principle. It is shown that a strong coupling effect occurs due to the existence of both the incident field and a strong scattering field. Huygens principle can be used to obtain the first and second order scattering fields generated between implants. If the first- and second-order electric field terms are summed within a certain region, the RF-induced heating of this dual-implant system increases.

Real-time phase-resolved ocean wave prediction in directional wave fields: Second-order Lagrangian wave models

The Improved Choppy Wave Model (ICWM) was established to achieve a second-order Lagrangian expansion of surface waves. The second-order Lagrangian nonlinear interaction terms were found to be negligible and were therefore discarded. However, these interactions in directional wave fields remained unexplored. ICWM was successfully used, especially for floating offshore wind turbines, but it was noted that its accuracy in predicting directional sea states needed improvement. Hence, we formulated the Complementary ICWM (CICWM) with the nonlinear terms for free surface elevation in directional waves. Detailed formulations for data assimilation and wave propagation were provided, enabling real-time wave forecasting for directional seas using a simplified assimilation method. Comparing the model performances against tank-scale experiments showed that CICWM enhances the surface elevation description in directional seas. Ultimately, it was confirmed that the second-order Lagrangian model can reduce the prediction error of the linear wave model by 90% in directional seas for the experimental setups and sea states investigated in this study.

Investigating the effects of congruence between within-person associations: A comparison of two extensions of response surface analysis.

Response surface analysis (RSA) allows researchers to study whether the degree of congruence between two predictor variables is related to a potential psychological outcome. Here, we adapt RSA to the case in which the two predictor variables whose congruence is of interest refer to individual differences in within-person associations (WPAs) between variables that fluctuate over time. For example, a WPA-congruence hypothesis in research on romantic relationships could posit that partners are happier when they have similar social reactivities-that is, when they have similarly strong WPAs between the quantity of their social interactions and their momentary well-being. One method for testing a WPA-congruence hypothesis is a two-step approach in which the individuals' WPAs are first estimated as random slopes in respective multilevel models, and then these estimates are used as predictors in a regular RSA. As an alternative, we suggest combining RSA with multilevel structural equation modeling (MSEM) by specifying the WPAs as random slopes in the structural equation and using their latent second-order terms to predict the outcome on Level 2. We introduce both approaches and provide and explain their corresponding computer code templates. We also compared the two approaches with a simulation study and found that the MSEM model-despite its complexities (e.g., nonlinear functions of latent slopes)-has advantages over the two-step approach. We conclude that the MSEM approach should be used in practice. We demonstrate its application using data from a daily diary study and offer guidance for important decisions (e.g., about standardization). (PsycInfo Database Record (c) 2024 APA, all rights reserved).

Traditional output dynamics: A structural perspective

We provide theoretical underpinnings to the traditional IS curve by using the concept of basic and non-basic goods and services in the consumption basket. An extended version of this IS curve incorporates the effects of functional income distribution by appealing to rule-of-thumb behavior. An empirical analysis for both emerging and advanced economies shows that this IS curve does not suffer from the "IS puzzle," a weakness that affects its new Keynesian counterpart, when a measure of the nominal interest rate is used. The interest rate negatively and significantly affects output. Secondly, the second-order term typically specified in the output dynamics does not arise from structural consumption behavior. Thirdly, although we find significant rule-of-thumb behavior in that the labor share is statistically significant, optimal behavior dominates economies. Lastly, basics make up an overwhelming component of the consumption basket, which explains the sluggish response of output to shocks. The implication is that there is a need to extend the basic two-equation model into a 3-equation system by adding the dynamics of the labor share.

Bonding and antibonding electromagnetic coupling in two interacting toroidal metamolecules

The existence of a toroidal-like eigenmode and its electromagnetic coupling in a system of dielectric particles are studied. A constituent structure (metamolecule) is made of a ring consisting of the radial arrangement of several vertically standing dielectric disks (meta-atoms). In the eigenstate of the given metamolecule, the second-order term related to the exact electric dipole in the multipole decomposition is much greater than the first-order term. This eigenstate is defined as a toroidal-like mode. Then, the characteristics of a system (metamacromolecule) composed of two identical rings are studied in both eigen- and excited states to reveal the peculiarities of the toroidal-like mode coupling. Similar to the well-known electric dipole-dipole and magnetic dipole-dipole interactions, the interaction of toroidal-like modes also appears in symmetric (bonding) and antisymmetric (antibonding) forms. Their excitation in the metamacromolecule depends on the propagation direction and polarization of the irradiating wave. The manifestation of toroidal-like mode coupling is confirmed by checking the extinction cross section and near-field distributions obtained from the full-wave numerical simulation and microwave experiment. A clear understanding of the nature of toroidicity is important from the fundamental physics perspective and practical implementation of metamaterials operated in such exotic states. Published by the American Physical Society 2024

Tensor hypercontraction for fully self-consistent imaginary-time GF2 and GWSOX methods: Theory, implementation, and role of the Green's function second-order exchange for intermolecular interactions.

We present an efficient MPI-parallel algorithm and its implementation for evaluating the self-consistent correlated second-order exchange term (SOX), which is employed as a correction to the fully self-consistent GW scheme called scGWSOX (GW plus the SOX term iterated to achieve full Green's function self-consistency). Due to the application of the tensor hypercontraction (THC) in our computational procedure, the scaling of the evaluation of scGWSOX is reduced from O(nτnAO5) to O(nτN2nAO2). This fully MPI-parallel and THC-adapted approach enabled us to conduct the largest fully self-consistent scGWSOX calculations with over 1100 atomic orbitals with only negligible errors attributed to THC fitting. Utilizing our THC implementation for scGW, scGF2, and scGWSOX, we evaluated energies of intermolecular interactions. This approach allowed us to circumvent issues related to reference dependence and ambiguity in energy evaluation, which are common challenges in non-self-consistent calculations. We demonstrate that scGW exhibits a slight overbinding tendency for large systems, contrary to the underbinding observed with non-self-consistent RPA. Conversely, scGWSOX exhibits a slight underbinding tendency for such systems. This behavior is both physical and systematic and is caused by exclusion-principle violating diagrams or corresponding corrections. Our analysis elucidates the role played by these different diagrams, which is crucial for the construction of rigorous, accurate, and systematic methods. Finally, we explicitly show that all perturbative fully self-consistent Green's function methods are size-extensive and size-consistent.

Painlevé integrability and multiple soliton solutions for the extensions of the (modified) Korteweg-de Vries-type equations with second-order time-derivative

This work introduces two (3+1)-dimensional expansions of the Korteweg–de Vries (KdV) and modified KdV (mKdV) equations. These extensions incorporate a second-order time-derivative term, similar to the Boussinesq equation. The Painlevé test is utilized to verify the integrability of each extended model. The bilinear form is employed to investigate the existence of multiple-soliton (MS) solutions for each system under consideration. Furthermore, we provide solutions in the form of lumps for the extended KdV equation. The multidimensional KdV-type equations surpass the standard KdV equation. However, they offer enhanced accuracy by representing a broader spectrum of nonlinear phenomena in plasma physics, fluid mechanics, tsunami phenomena, and other science disciplines. Furthermore, the aforementioned equations can be employed to analyze the characteristics of various acoustic waves (AW) in different plasma models, such as their amplitude, width, frequency, and dispersion, as well as the phase shifts after collisions.

Active contour model with improved second-order differential driven term

Driven terms in active contour models (ACMs) play a significant role in edge identification and image segmentation. However, in many existing ACMs, the driven terms are iteratively updated, resulting in slower segmentation speed because the image segmentation time increases with the iteration number. To address this problem, an ACM based on an improved second-order differential driven term (ISDDT) is presented, which can extract the edge information of images. The improved second-order differential driven term is computed only once before the iterations. Therefore, the computational complexity of our presented ACM is reduced, leading to a faster image segmentation speed. In addition, an improved regularization method with mean filtering is presented to improve the robustness of our ISDDT model. As an application, a target contour tracking method is developed based on our ISDDT model. Experimental results show that our ISDDT model segments images with inhomogeneous intensities well. The image segmentation speed of our proposed model has obvious advantages.

Parameter estimation for Gibbs distributions

A central problem in computational statistics is to convert a procedure for sampling combinatorial objects into a procedure for counting those objects, and vice versa. We consider sampling problems coming from Gibbs distributions , which are families of probability distributions over a discrete space \(\Omega\) with probability mass function of the form \(\mu^{\Omega}_{\beta}(\omega)\propto e^{\beta H(\omega)}\) for \(\beta\) in an interval \([\beta_{\min},\beta_{\max}]\) and \(H(\omega)\in\{0\}\cup[1,n]\) . Two important parameters are the partition function , which is the normalization factor \(Z(\beta)=\sum_{\omega\in\Omega}e^{\beta H(\omega)}\) , and the vector of preimage counts \(c_{x}=|H^{-1}(x)|\) . We develop black-box sampling algorithms to estimate the counts using roughly \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions, where \(q=\log\frac{Z(\beta_{\max})}{Z(\beta_{\min})}\) (ignoring some second-order terms and parameters). We show this is optimal up to logarithmic factors. We illustrate with improved algorithms for counting connected subgraphs, independent sets, and perfect matchings. As a key subroutine, we estimate all values of the partition function using \(\tilde{O}(\frac{n^{2}}{\varepsilon^{2}})\) samples for integer-valued distributions and \(\tilde{O}(\frac{q}{\varepsilon^{2}})\) samples for general distributions. This improves over a prior algorithm of Huber (2015) which computes a single point estimate \(Z(\beta_{\max})\) and which uses a slightly larger amount of samples. We show matching lower bounds, demonstrating this complexity is optimal as a function of \(n\) and \(q\) up to logarithmic terms.

Flow birefringence of cellulose nanocrystal suspensions in three-dimensional flow fields: revisiting the stress-optic law

This study systematically investigates the flow birefringence of cellulose nanocrystal (CNC) suspensions. The aim is to clarify the importance of the stress component along the camera’s optical axis in the stress-optic law (SOL), which describes the relationship between birefringence, the retardation of transmitted polarized light, and the stress field. More than 100 datasets pertaining to the retardation of CNC suspensions (concentrations of 0.1, 0.3, 0.5, and 1.0 wt%) in a laminar flow field within a rectangular channel (aspect ratios of 0.1, 1, and 3) are systematically obtained. The measured retardation data are compared with the predictions given by the conventional SOL excluding the stress component along the camera’s optical axis and by the SOL including these components as second-order terms (2nd-order SOL). The results show that the 2nd-order SOL gives a significantly better agreement with the measurements. Based on the 2nd-order SOL, the retardation at the center of the channel, where the effect of the stress component along the camera’s optical axis is most pronounced, is predicted to be proportional to the square of the flow rate, which agrees with the experimental data. The results confirm the importance of considering the stress component along the camera’s optical axis in the flow birefringence of CNC suspensions at high flow rates, even for quasi-two-dimensional channel flow.

Models for predicting corrosion inhibition efficiency of common drugs on steel surfaces: A rationalized comparison among methodologies

Several anticorrosive treatments have been proposed over time to create protective layers to hinder the corrosion phenomenon. In recent years, organic molecules from plant extracts and expired drugs have been tested due to their potential corrosion inhibition properties. However, direct corrosion inhibition efficiency (IE%) evaluation requires costly reactants and a specific experimental setup. Quantitative-structure activity relationship (QSAR) proposes modeling IE% in terms of variables measured in previous experiments or determined by theoretical approaches. Computed descriptors, such as ionization energy (I), electronic affinity (A), or global hardness, were added to a database of physicochemical properties. This work compares several methodologies to obtain precise yet portable mathematical models for predicting corrosion inhibition efficiency. As an original approach from this research group, nonlinear autoregressive moving average with exogenous inputs (NARMAX), using forward regression with orthogonal least squares (FROLS), models were implemented as a robust method to get nonlinear portable models and to determine the most important variables impacting IE%. Contrastingly, ordinary least squares (OLS) methodology was employed with the novelty of applying power series expansions from the promoted FROLS variables for linear and polynomial regression with only one independent variable, which resulted in clearer graph visualization of trends and the ease of proposing thumb rules based on raw information. Finally, IBM Watson was also compared as a robust yet non-portable and highly parametrized alternative to conventional mathematical approaches, based on extra trees regressor (ETR). The models were compared using mean absolute percentage error (MAPE), mean-squared error (MSE), and root-mean-squared error (RMSE). Overall, models with fewer variables and up to second-order terms show improved performance. The main tendencies of IE%, drawn by inferences for 630 substances by second-order NARX, are analyzed. Also, the determinant role of the highest occupied molecular orbital energy was reported. Experimentalists can take advantage of a “cost-free” general approach that can obtain estimations for IE% values with errors of about 6 %, in particular the second-order NARX model.

Ultrafast Excited-State Energy Transfer in Phenylene Ethynylene Dendrimer: Quantum Dynamics with the Tensor Network Method.

Photoinduced excited-state energy transfer (EET) processes play an important role in solar energy conversions. Owing to their excellent photoharvesting and exciton-transport properties, phenylene ethynylene (PE) dendrimers display great potential for improving the efficiency of solar cells. In this work, we investigated the intramolecular EET dynamics in a dendrimer composed of two linear PE units (2-ring and 3-ring) using a fully quantum description based on the tensor network method. We first constructed a diabatic model Hamiltonian based on the electronic structure calculations. Using this diabatic vibronic coupling model, we tried to obtain the main features of the EET dynamics in terms of the several diabatic models with different numbers of vibrational modes (from 4 modes to 129 modes) and to explore the corresponding vibronic coupling interactions. The results show that the EET in this PE dendrimer is ultrafast. Four modes of A' symmetry play dominant roles in the dynamics; the remaining 86 modes of A' symmetry can dampen the electronic coherence; and the modes of A″ symmetry do not exhibit significant influence on the EET process. Overall, the first-order intrastate vibronic coupling terms show the dominant role in the EET dynamics, while the second-order intrastate vibronic coupling terms cause damping of the electronic coherence and slow down the overall EET process. This work provides a microscopic understanding of the EET dynamics in PE dendrimers.

Going beyond an old shockwave conjecture for improving upon Navier-Stokes.

Nonequilibrium molecular dynamics (NEMD) computer simulations of steady shockwaves in dense fluids and rarefied gases produce detailed shockwave profiles of mechanical and thermal properties. The Boltzmann equation, under the assumption of local thermodynamic equilibrium (LTE), leads to the first-order (linear) continuum theory of hydrodynamic flow: Navier-Stokes-Fourier (NSF). (Expansion of the LTE Boltzmann equationin higher powers of gradients yields so-called Burnett second-order terms, etc.) NEMD simulations of strong shockwaves with high gradients are not well modeled by NSF theory. Many years ago, a conjecture for going "beyond Navier-Stokes" was proposed, applying the empirical observation of anisotropic thermal enhancement in the shock front to the temperature dependence of the NSF transport coefficients, whose dissipation determines the slope at the center of the shock profile: for weak shocks, the actual coefficients in NEMD simulations appear to be smaller than in NSF predictions, leading to steeper gradients being observed, while for strong shocks, the NEMD coefficients appear to be larger, leading to less steep shock rises than predicted by NSF calculations. In this paper, we show that adding significant Burnett nonlinearity into an LTE continuum theory reproduces the early shock rise and slope of NEMD profiles, for both weak and strong shocks in dense fluids, as well as strong shockwaves in the ideal gas. Moreover, we show that "Holian's conjecture" incorporates significant Burnett nonlinearity, but like all the other LTE continuum theories, it fails to describe the slow NEMD return to equilibrium beyond the shock front. We show that Maxwell relaxation has to be applied to the hydrodynamic variables themselves (rather than attempting indirect relaxation of their gradients) in order to more accurately model the entire shockwave profile. Non-LTE Maxwell relaxation is the only way to bring the entire profile into agreement with NEMD, most noticeably for strong shockwaves.

The Suppression Effect of an Imaging System on the Geometric Tilt-to-Length Coupling in a Test Mass Interferometer

Tilt-to-length (TTL) coupling noise arises from angular misalignments of interfering beams in optical path length (OPL) measurements and significantly impacts the accuracy of interferometry measurement systems. This paper focuses on geometric TTL coupling in a test mass (TM) interferometer and examines how an imaging system influences TTL noise suppression. First, the analytical expression of the geometric TTL coupling in a TM interferometer with alignment errors is derived and confirmed through numerical simulation. Subsequently, an imaging system is incorporated into the geometric model and the corresponding analytical expressions are obtained under two common conjugate relationships. Nevertheless, the TTL coupling remains beyond the requirement of TM interferometer, as the residual TTL coupled with alignment errors persists even with the imaging system. Therefore, an optimal position of the imaging system capable of eliminating the second-order term of the TTL coupling is determined. Meanwhile, the first-order term can be mitigated through in-orbit calibrations. These findings offer valuable guidance for the design and adjustment of imaging systems in space-borne gravitational wave detection missions, which require high-precision laser interferometry.

Research on laser communication ranging Integration and VLC nonlinear compensation technology

Abstract Visible light communication (VLC) is a new spectrum communication technology that provides a new direction to solve spectrum resource problems. The nonlinearity problem in VLC will lead to a significant reduction in the performance of the communication system. To solve this problem, this paper studies the integration of laser communication ranging and VLC nonlinear compensation technology based on analyzing the principles of ranging methods and factors affecting ranging performance. Aiming at the problem that the complexity of the Random Fourier Features Kernel adaptive filter (RFF-KAF) method will increase significantly as the dimension of the mapping space increases, we introduce some second-order Volterra series square terms into the input vector of RFF-KAF and propose an improved prediction method. Experimental simulation results show that the method proposed in this article can greatly improve the performance of the RFF-KAF method while maintaining the same mapping space dimension. Compared with obtaining similar performance by increasing the mapping space dimension, the cost of this method is greatly reduced.

The anion photoelectron spectrum and diabatization of tetrazolyl.

The potential energy surface of tetrazolyl [cyclic (N4CH)] has a conical intersection seam between the two lowest-energy electronic states near the ground state minimum geometry. This work treats that molecule. The potential energy surfaces used in this study are based on a least-squares fitting procedure that includes abinitio energies, energy gradients, and derivative couplings described using polynomials up to fourth-order and abinitio data obtained from multireference configuration interaction wave functions. A five-electronic-state description was generated with a root mean square absolute energy error of 9.6cm-1, compared to 326.8cm-1 when only second-order terms were used. The time-independent multimode vibronic coupling in the KDC approximation was used to simulate and analyze the anion ultraviolet photoelectron spectrum of tetrazolide.