Second Derivative Research Articles

Stability of the isotropic conductivity reconstruction using magnetic resonance electrical impedance tomography (MREIT)

Magnetic resonance electrical impedance tomography (MREIT) is a high-resolution imaging modality that aims to reconstruct the objects’ conductivity distributions at low frequency using the measurable z-component of the magnetic flux density obtained from an MRI scanner. Traditional reconstruction algorithms in MREIT use two datum subject to two linearly independent current densities. However, the temporal resolution of such a MREIT image is relatively low. Recently, a single current harmonic Bz algorithm has been proposed to improve the temporal resolution. Even though a series of reconstruction algorithms have been proposed in the last two decades, the theoretical studies of MREIT are still quite limited. This paper presents the stability theorems for two datum and a single data-based isotropic conductivity reconstruction using MREIT. Using the regularity theory of elliptic partial differential equations, we prove that the only instability in the inverse problem of MREIT comes from taking the second-order derivative of the measured data Bz , the z-component of the magnetic flux density. To get a stable reconstruction from the noisy Bz data, we note that the edge structure of ∇Bz reveals the edge features in the unknown conductivity and provides an edge-preserving denoising approach for the ∇Bz data. We use a modified Shepp–Logan phantom model to validate the proposed theory and the denoising approach.

Weighted estimates of commutators of singular operators in generalized Morrey spaces beyond Muckenhoupt range and applications

For a certain class of radial weights, we prove weighted norm estimates for commutators with BMO coefficients of singular operators in local generalized Morrey spaces. As a consequence of these estimates, we obtain norm inequalities for such commutators in the generalized Stummel-Morrey spaces. We also discuss a.e. well-posedness of singular operators and their commutators on weighted generalized Morrey spaces. The obtained estimates are applied to prove interior regularity for solutions of elliptic PDEs in the frameworks of the corresponding weighted Sobolev spaces based on the local generalized Morrey spaces or Stummel-Morrey spaces. To this end also conditions for the applicability of the representation formula, for the second-order derivatives of solutions to elliptic PDEs, are found for the case of such weighted spaces. In both results, for commutators and applications, we admit weights beyond the Muckenhoupt range.

Deterministic and Stochastic Nonlinear Model for Transmission Dynamics of COVID-19 with Vaccinations Following Bayesian-Type Procedure

We develop a mathematical model for the SARAS-CoV-2 double variant transmission characteristics with variant 1 vaccination to address this novel aspect of the disease. The model is theoretically examined, and adequate requirements are derived for the stability of its equilibrium points. The model includes the single variant 1 and variant 2 endemic equilibria in addition to the endemic and disease-free equilibria. Various approaches are used for the global and local stability of the model. For both strains, we determine the basic reproductive numbers R1 and R2. To investigate the occurrence of the layers (waves), we expand the model to include some analysis based on the second-order derivative. The model is then expanded to its stochastic form, and numerical outcomes are computed. For numerical purposes, we use the nonstandard finite difference method. Some error analysis is also recorded.

Augmenting the Stability of Automatic Voltage Regulators through Sophisticated Fractional-Order Controllers

The transition from traditional to renewable energy sources is a critical issue in current energy-generation systems, which aims to address climate change and the increased demand for energy. This shift, however, imposes additional burdens on control systems to maintain power system stability and quality within predefined limits. Addressing these challenges, this paper proposes an innovative Modified Hybrid Fractional-Order (MHFO) automatic voltage regulator (AVR) equipped with a fractional-order tilt integral and proportional derivative with a filter plus a second-order derivative with a filter FOTI-PDND2N2 controller. This advanced controller combines the benefits of a (FOTI) controller, known for enhancing dynamic performance and steady-state response, with a (PDND2N2) controller to improve system robustness and adaptability. The proposed MHFO controller stands out with its nine tunable parameters, providing more extensive control options than the conventional three-parameter PID controller and the five-parameter FOPID controller. Furthermore, a recent optimization approach using a growth optimizer (GO) has been formulated and applied to optimally adjust the MHFO controller’s parameters simultaneously. The performance of the proposed AVR based on the MHFO-GO controller is scrutinized by contrasting it with various established and developed optimization algorithms. The comparative study shows that the AVR based on the MHFO-GO controller surpasses other AVR controllers from the stability, robustness, and dynamic response speed points of view.

Homogenization of nonlinear Cosserat plate including growth theory when the thickness of the plate and the size of the in-plane heterogeneities are of the same order of magnitude

In this work, we present a new two-scale finite-strain plate theory for highly heterogeneous plates described by a repetitive periodic microstructure. Two scales exist, the macroscopic scale is linked to the entire plate and the microscopic one is linked to the size of the heterogeneity. This work aims to propose such a theory for thick plates in a nonlinear setting when the thickness and the size of heterogeneities are of the same order of magnitude. The homogenization theory for large deformation with growth is suitable for the modelization of nearly incompressible plant tissue. This model is suitable for wavy leaves. For thick plates, the transverse normal stress and transverse shearing are modelized at both microscopic and macroscopic levels. At the macroscopic level, we consider a nonlinear Cosserat plate model. At the microscopic level, we impose that the average of contribution of the microscopic displacement to rotation angles is equal to zero. We also deal with the problem of boundary layer problem near the lateral boundary. The model recently proposed by Pruchnicki is valid for thin heterogeneous plates; we present an extension for thick plates that takes into account both transverse normal stress and shearing. This model is equivalent to the first model presented but it involves a second-order derivative of the macroscopic displacement field.

Modeling risk assessment of soil heavy metal pollution using partial least squares and fuzzy logic: A case study of a gully type coal-based solid waste dumpsite

Continuous release and migration of heavy metals from coal-based solid waste (CSW) dumpsites often results in significant encroachment on ecological lands and pollution of natural environments. As a result, there is an urgent need for long-term and rapid monitoring, analysis, and assessment to control environmental risks associated with large CSW dumpsites. We constructed a new composite model (PLS-FL) that uses partial least squares regression (PLSR) and fuzzy logic inference (FLI) to accurately predict heavy metal concentrations in soils and assess pollution risk levels. The potential application of the PLS-FL was tested through a gully type CSW case study. We compared 20 modeling strategies using the PLS-FL: five types heavy metals (Cd, Zn, Pb, Cr and As) * four spectral transformation methods (first derivative (FD), second derivative (SD), reverse logarithm (RL), and continuum removal (CR)) * one variable selection method (competitive adaptive reweighted sampling (CARS)). The results showed that the combination of derivative transformation and CARS was recommended for estimation, with R2C > 0.80 and R2P > 0.50. When comparing the PLSR model with four traditional machine learning methods (Support Vector Machines (SVM), Random Forests (RF), Extreme Learning Machines (ELM), and KNN), the PLSR model demonstrated the highest average prediction accuracy. Additionally, the FLI process no longer relies on human perception and expert opinion, enhancing the model's objectivity and reliability. The evaluation results revealed that the heavy metal contamination areas of the CSW dumpsite are concentrated at the bottom of the gully, with more severe contamination in the north. Furthermore, a high-risk zone exists in the interim storage area for CSW to the east of the dump. These findings align with the initial detections at the sampling sites and highlight the need for targeted monitoring and control in these areas. The application of the model will empower regulators to quickly assess the overall situation of large-scale heavy metal pollution and provide scientific program and data support for continuous large-scale pollution risk monitoring and sustainable risk management.

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

On a stabilization of the Ingard-Myers impedance boundary condition and its time domain implementation

It has been well-known that the Ingard-Myers impedance condition, while simple to apply, is subject to the hydrodynamic Kelvin-Helmholtz-type instability due to its use of a vortex sheet in modeling the flow at the liner boundary. Recently, in the development of a time domain boundary element method for acoustic scattering by treated surfaces, it was found that by neglecting a certain second-order spatial derivative term in the Ingard-Myers formulation, the hydrodynamic instability can be avoided. The present paper aims to provide further analysis of this modified condition, hereby referred to as the Truncated Ingard-Myers Impedance Boundary Condition (TIMIBC). It will be shown, based on the dispersion relations of linear waves, that the instability intrinsic to the Ingard-Myers condition is eliminated in the proposed new formulation. Quantitative assessments on the accuracy of the TIMIBC for scattering of acoustic waves by lined surfaces are carried out, and its effectiveness is demonstrated by a numerical example. It is found that the TIMIBC provides a good approximation to the original Ingard-Myers condition for flows of low to mid subsonic Mach numbers. As such, the proposed TIMIBC can offer a practical solution for overcoming the intrinsic instability associated with the Ingard-Myers condition. Moreover, time domain implementation of the TIMIBC is also discussed and illustrated with a numerical example using a finite difference scheme. In particular, a minimization procedure for finding the poles and coefficients of a broadband multipole expansion for the impedance function is formulated by which, unlike the commonly used vector-fitting method, passivity of the model is ensured.

A G-Modified Helmholtz Equation with New Expansions for the Earth’s Disturbing Gravitational Potential, Its Functionals and the Study of Isogravitational Surfaces

The G-modified Helmholtz equation is a partial differential equation that enables us to express gravity intensity g as a series of spherical harmonics having radial distance r in irrational powers. The Laplace equation in three-dimensional space (in Cartesian coordinates, is the sum of the second-order partial derivatives of the unknown quantity equal to zero) is used to express the Earth’s gravity potential (disturbing and normal potential) in order to represent other useful quantities—which are also known as functionals of the disturbing potential—such as gravity disturbance, gravity anomaly, and geoid undulation as a series of spherical harmonics. We demonstrate that by using the G-modified Helmholtz equation, not only gravity intensity but also disturbing potential and its functionals can be expressed as a series of spherical harmonics. Having gravity intensity represented as a series of spherical harmonics allows us to create new Global Gravity Models. Furthermore, a more detailed examination of the Earth’s isogravitational surfaces is conducted. Finally, we tabulate our results, which makes it clear that new Global Gravity Models for gravity intensity g will be very useful for many geophysical and geodetic applications.

Various duality models associated with some constrained robust nonlinear optimal control problems

This paper deals with various duality results for a new class of constrained robust nonlinear control problems. Concretely, we introduce a minimisation problem associated with a vector of functionals of multiple integral type. In addition, the considered multi-objective controlled optimisation problem is governed by several mixed constraints generated by partial derivatives of second order and uncertain data. For this new class of control problems, we establish and characterise Wolfe, Mond-Weir and mixed type robust dual models. To this aim, by using the notion of convex multiple integral functional, generated by a controlled second-order Lagrangian with uncertain data, and the concept of robust weak efficient solution to the problem under study, we build a new mathematical context to state and prove the duality theorems.

On Second-Order Derivatives of Boolean Functions and Cubic APN Permutations in Even Dimension

The big APN problem is one of the most important challenges in the theory of Boolean functions, i.e. finding a new APN permutation in even dimension. Among this class of functions, those with the lowest possible degree are cubic. Yet, none has been found so far. In this paper, we introduce new parameters for Boolean functions and for vectorial Boolean functions, mostly derived from the behavior of their second-order derivatives. These parameters are invariant under extended affine equivalence, and they are particularly relevant for small-degree functions. They allow studying bent, semi-bent and APN functions of degrees two and three. In particular, they allow tackling the big APN problem for cubic permutations. Notably, we focus on the case of dimension 8, providing some computational results.

DCES-PA: Deformation-controllable elastic shape model for 3D bone proliferation analysis using hand HR-pQCT images

Bone proliferation is an important pathological feature of inflammatory rheumatic diseases. Although recent advance in high-resolution peripheral quantitative computed tomography (HR-pQCT) enables physicians to study microarchitectures, physicians’ annotation of proliferation suffers from slice inconsistency and subjective variations. Also, there are only few effective automatic or semi-automatic tools for proliferation detection. In this study, by integrating pathological knowledge of proliferation formation with the advancement of statistical shape analysis theory, we present an unsupervised method, named Deformation-Controllable Elastic Shape model, for 3D bone Proliferation Analysis (DCES-PA). Unlike previous shape analysis methods that directly regularize the smoothness of the displacement field, DCES-PA regularizes the first and second-order derivative of the displacement field and decomposes these vector fields according to different deformations. For the first-order elastic metric, DCES-PA orthogonally decomposes the first-order derivative of the displacement field by shearing, scaling and bending deformation, and then penalize deformations triggering proliferation formation. For the second-order elastic metric, DCES-PA encodes both intrinsic and extrinsic surface curvatures into the second-order derivative of the displacement field to control the generation of high-curvature regions. By integrating the elastic shape metric with the varifold distances, DCES-PA achieves correspondence-free shape analysis. Extensive experiments on both simulated and real clinical datasets demonstrate that DCES-PA not only shows an improved accuracy than other state-of-the-art shape-based methods applied to proliferation analysis but also produces highly sensitive proliferation annotations to assist physicians in proliferation analysis.

A revisit to the pressureless Euler–Navier–Stokes system in the whole space and its optimal temporal decay

In this paper, we present a refined framework for the global-in-time well-posedness theory for the pressureless Euler–Navier–Stokes system and the optimal temporal decay rates of certain norms of solutions. Here the coupling of two systems, pressureless Euler system and incompressible Navier–Stoke system, is through the drag force. We construct the global-in-time existence and uniqueness of regular solutions for the pressureless Euler–Navier–Stokes system without using a priori large time behavior estimates. Moreover, we seek for the optimal Sobolev regularity for the solutions. Concerning the temporal decay for solutions, we show that the fluid velocities exhibit the same decay rate as that of the heat equations. In particular, our result provides that the temporal decay rate of difference between two velocities, which is faster than the fluid velocities themselves, is at least the same as the second-order derivatives of fluid velocities.

A Method for State of Charge and State of Health Estimation of LithiumBatteries Based on an Adaptive Weighting Unscented Kalman Filter

This paper considers the estimation of SOC and SOH for lithium batteries using multi-innovation Levenberg–Marquardt and adaptive weighting unscented Kalman filter algorithms. For parameter identification, the second-order derivative of the objective function to optimize the traditional gradient descent algorithm is used. For SOC estimation, an adaptive weighting unscented Kalman filter algorithm is proposed to deal with the nonlinear update problem of the mean and covariance, which can substantially improve the estimation accuracy of the internal state of the lithium battery. Compared with fixed weights in the traditional unscented Kalman filtering algorithm, this algorithm adaptively adjusts the weights according to the state and measured values to improve the state estimation update accuracy. Finally, according to simulations, the errors of this algorithm are all lower than 1.63 %, which confirms the effectiveness of this algorithm.

Hamilton-Jacobi Treatment of Singular Systems Using Fractional Calculus

In this paper, the theory of fractional singular systems is investigated with second-order derivatives. The fractional Hamilton–Jacobi treatment of these systems is examined. The fractional Hamilton–Jacobi partial differential equations (FHJPDEs) are constructed. The (FHJPDEs) can be solved to obtain the fractional Hamilton–Jacobi function. By building the fractional Hamilton–Jacobi function, the equations of motion can be obtained.

χ(2) and χ(3) enhancement by arylamine substituents in aromatic conjugated polyenyne polymers

In this work, we have determined the χ(1), χ(2), and χ(3) NLO properties of NH2substituted aromatic polyenynes in ortho, meta, and para positions. It is worth noting that Mednis-Genkin’s approximation takes into account the intraband contribution with a derivate of the dipole valence-conduction band moment transition, unlike, for example, Ward’s summation-over-sates (SOS) methodology, which considers the difference of these states. Indeed, we have selected these polymers because of the high asymmetry in their molecular wavefunction. In this way, the calculated χ(3) values of the para-NH2 substituted aromatic polyenyne with a higher degree of polymerization are near 1 X 10−7 esu. For its part, the ortho polymer has higher values, of the order of 1 X 10−6 esu, due to its high transition dipole moment. We have also found that the energy band distribution in the para position of these polymers produces van Hove points, which have a negligible proportion of the NLO properties, compared with singularity regions 1 and 2, which have a contribution of 100 %. These regions are due to the wavefunction coefficients’ first and second-order derivatives. Besides, without considering border effects nor correlation effects, and only the variation of excitation energy, we have calculated that the quaternary monomeric oligomers of these polymers have almost similar values of χ(2) and χ(3). Then, it implies building a methodology to synthesize the polymers to avoid canceling the NLO-χ(2) and -χ(3) values.

Port berth allocation and microgrid cluster joint optimization scheduling based on master-slave game

The port microgrid cluster, which integrates berth allocation and energy scheduling for joint optimization, is a highly interconnected system of logistics and energy closely coupled microgrids. It can reduce port load fluctuations and ensure the stable and reliable operation of the port energy system. A port berth allocation and microgrid cluster joint optimization scheduling method based on master-slave game is proposed to address issues such as large fluctuations in current port loads, vulnerability of port power supply system to the impact of vessel loading and unloading operations, and conflicts of interest among various participants in the power market. To ensure power quality while achieving peak shaving and valley filling, a two-stage optimization model for microgrid cluster was constructed, enabling joint optimization of berth allocation and energy scheduling. In the first stage, considering the impact of random events during vessel navigation, berth allocation is conducted with the objectives of maximizing peak shaving and valley filling rates and minimizing overall costs, thereby achieving preliminary load leveling while ensuring power supply system quality. In the second stage, dispatching is performed for power generation equipment and adjustable loads, aiming to further achieve peak shaving and valley filling while maintaining the economic viability of the microgrids. Addressing conflicting interests among stakeholders in the port's electricity market, a master-slave game strategy has been established. This strategy aims to balance the interests between the electricity trading agent and the microgrid aggregator composed of multiple microgrid models. By solving the second-order partial derivatives of each optimization variable with respect to the objective function, we demonstrate the existence and uniqueness of the equilibrium solution in the master-slave game. In response to the challenge of difficulties in solving the multi-objective optimization model due to conflicting objective functions, a Dung Beetle Optimization (DBO) algorithm incorporating non-dominated sorting strategy and crowding distance was introduced. This resulted in a novel multi-objective optimization algorithm termed Non-Dominated Sorting Dung Beetle Optimizer (NSDBO). The NSDBO algorithm was demonstrated to possess favorable characteristics in terms of distribution, convergence, and uniformity. Using a northern Chinese port as an example, berth allocation and energy scheduling for three port areas were jointly optimized, and various scenarios were compared. The results indicated that, compared to Scenario 1, which did not consider random events, the proposed joint optimization method increased the peak shaving and valley filling rate by 8.46 %, reduced network losses by 42 kW, and increased the revenue of the microgrid aggregation merchant by 2.11 %. Compared to Scenario 2, which only considered energy scheduling, the joint optimization increased the peak shaving and valley filling rate by 36.89 %, reduced network losses by 18 kW, and increased the revenue of the microgrid aggregation merchant by 2.29 %. Compared to Scenario 3, which was individually optimized, the joint optimization increased the peak shaving and valley filling rate by 48.42 %, reduced network losses by 24 kW, and increased the revenue of the microgrid aggregation merchant by 2.73 %. This validates the superiority of the proposed joint optimization method in mitigating the impact of random events, reducing network losses, stabilizing port load fluctuations, and enhancing the revenue of the microgrid cluster.

Prediction of soil organic carbon in black soil based on a synergistic scheme from hyperspectral data: Combining fractional-order derivatives and three-dimensional spectral indices

Monitoring soil organic carbon (SOC) content is crucial for climate change mitigation and sustaining ecological balance. Despite the unparalleled advantages of hyperspectral data in capturing nuanced variations in soil properties through its high spectral resolution, effectively extracting useful features from numerous bands via spectral processing techniques remains a formidable challenge. This study proposes an integrated approach combining fractional-order derivative (FOD) technique and optimal band combination algorithm using ZY1-02D satellite hyperspectral data to estimate SOC in Northeast China’s Black soil region. Three modeling strategies were compared: (1) FOD-transformed reflectance (FOD spectra), (2) FOD spectra with traditional 2D spectral indices (FOD + 2D SI), and (3) FOD spectra with new 3D spectral indices (FOD + 3D SI). These strategies were implemented using the random forest model with the aim of the optimal SOC prediction. Results showed that the application of FOD technique for spectral transformation effectively addressed the challenges posed by overlapping peaks and baseline drift inherent in the original spectral reflectance. Additionally, FOD transformation enhanced subtle soil spectral features and yielded more pronounced spectral variations with increasing fractional order, as compared to the original spectral data and conventional integer-order derivatives (i.e., first and second-order derivatives). However, as the FOD order continued to increase beyond 1.4, the spectral curve exhibited amplified noise and distortion, thereby adversely impacting subsequent model development. The 3D spectral indices correlate more robustly with SOC than 2D indices. The model that combines 0.6-order FOD and 3D spectral indices achieved the best accuracy (R2 = 0.66, RMSE = 2.99 g/kg and MAE = 2.42 g/kg), significantly outperforming the models built by 0.6-order FOD spectra (R2 = 0.48, RMSE = 3.65 g kg−1, and MAE = 2.93 g kg−1) and 0.8-order FOD + 2D SI modeling strategy (R2 = 0.55, RMSE = 3.54 g kg−1, and MAE = 2.85 g kg−1). These findings indicated that FOD and 3D spectral indices exhibit superior synergistic performance in SOC prediction, demonstrating their feasibility and providing valuable insights for large-scale soil property prediction and mapping using satellite hyperspectral data.

A molecular mechanics implementation of the cyclic cluster model

Abstract The implementation of the cyclic cluster model (CCM) for molecular mechanics is presented in the framework of the computational chemistry program deMon2k. Because the CCM is particularly well-suited for the description of periodic systems with defects, it can be used for periodic QM/MM approaches where the non-periodic QM part is treated as a defect in a periodic MM surrounding. To this end, we present here the explicit formulae for the evaluation of the Ewald sum and its first- and second-order derivatives as implemented in deMon2k. The outlined implementation was tested in molecular dynamics (MD) simulations and periodic structure optimization calculations. MD simulations of an argon system were carried out using the Nosé-Hoover chain (NHC) thermostat and the Martyna-Tobias-Klein (MTK) barostat to control the temperature and pressure of the system, respectively. For the validation of CCM structure optimization a set of molecular crystals were optimized using the Ewald method for the evaluation of the electrostatic interactions. Two optimization procedures for the determination of the atomic positions and CCM cell parameters were tested. Our results show that the simultaneous optimization of the atomic positions and cell parameters is most efficient.

Design and validation of novel maize grain cleaning loss detection system based on classification models of particle time-domain signals

Quantitative evaluation of grain cleaning loss is a central concern in maize intelligent direct harvesting. This study integrates particle time-domain signals with machine learning models to develop a novel grain cleaning loss detection system and create modular embedded companion software. Specifically, collisional time-domain signals of grains and impurities with varying mass gradients are captured, and abnormal data are eliminated utilizing the Mahalanobis Distance (MD) algorithm. Signal classification models incorporating diverse preprocessing methods are trained using Naive Bayes (NB) and Support Vector Machine (SVM) algorithms. Model performance is evaluated through internal leave-one-out-cross-validation (LOOCV) and external validation sets. The best classification model is integrated into the grain loss detection software, and bench tests verify real-time cleaning loss rates for detection systems. Results demonstrate that the SVM model with Savitzky-Golay (S-G) smoothing + Second-Order Derivative (SOD) preprocessing achieves the highest predictive accuracy, with 97.44 % and 96.15 % in training and validation sets, respectively. When the feeding rate of maize mixtures is 5 kg/s, absolute grain loss rates for three measurements are 1.62 %, 1.72 % and 1.86 %, respectively. Compared to traditional loss detection systems with multifunctional circuits, the novel loss detection system enhances the average accuracy of grain loss rate by 6.54, 6.82 and 8.93 percentage points. Hence, the proposed method and apparatus can be applied to detect grain cleaning loss quantitatively.