Biquadratic Research Articles

Rotations and Integrability

We discuss some families of integrable and superintegrable systems in -dimensional Euclidean space which are invariant under rotations. The invariant Hamiltonian is integrable with integrals of motion and an additional integral ofmotion , which are first- and fourth-order polynomials in momenta, respectively.

Optimal control of the gradient systems

In this paper we consider the optimal control problems governed by the gradient systems for the Ginzburg-Landau free energy where denotes a potential function and ε is the diffusivity. One example of gradient systems are the Schlögl equation arising in chemical waves with a quartic potential function F(y). Gradient systems are characterized by energy decreasing property . Numerical integrators that preserve the energy decreasing property in the discrete setting are called energy or gradient stable. It is known that the implicit Euler method is first order unconditionally energy stable method. The only second order unconditionally energy stable method is the average vector field (AVF) method. We discretize the gradients systems by discontinuous Galerkin method in space and by AVF integrator in time. We solve optimal control problems for the Schlögl equation with traveling and spiraling waves using sparse and regularized controls .

Modified Wave-Front Propagation and Dynamics Coming from Higher-Order Double-Well Potentials in the Allen–Cahn Equations

In this paper, we conduct a numerical investigation into the influence of polynomial order on wave-front propagation in the Allen–Cahn (AC) equations with high-order polynomial potentials. The conventional double-well potential in these equations is typically a fourth-order polynomial. However, higher-order double-well potentials, such as sixth, eighth, or any even order greater than four, can model more complex dynamics in phase transition problems. Our study aims to explore how the order of these polynomial potentials affects the speed and behavior of front propagation in the AC framework. By systematically varying the polynomial order, we observe significant changes in front dynamics. Higher-order polynomials tend to influence the sharpness and speed of moving fronts, leading to modifications in the overall pattern formation process. These results have implications for understanding the role of polynomial potentials in phase transition phenomena and offer insights into the broader application of AC equations for modeling complex systems. This work demonstrates the importance of considering higher-order polynomial potentials when analyzing front propagation and phase transitions, as the choice of polynomial order can dramatically alter system behavior.

Numerical study of the time-fractional partial differential equations by using quartic B-spline method

This paper utilizes the quartic B-spline method for the numerical resolution of time-fractional partial differential equations. The fractional Caputo derivative is employed to depict anomalous diffusion processes influenced by memory effects. The proposed numerical method utilizes quartic B-spline functions for spatial discretization and employs a finite difference method to address the time-fractional derivative. Based on the Fourier method, its stability has been evaluated to demonstrate its efficacy in addressing fractional-order models. Numerical experiments, encompassing both linear and nonlinear scenarios, are performed to illustrate the method’s effectiveness and accuracy. The results obtained are compared with exact solutions and alternative numerical methods, demonstrating improved performance in convergence and computational efficiency.

Pure-positivity-preserving methods with an optimal sufficient CFL number for fifth-order MR-WENO schemes on structured meshes

In this paper, one-dimensional and two-dimensional pure-positivity-preserving (PPP) methods are proposed for fifth-order finite volume multi-resolution WENO (MR-WENO) schemes to solve some extreme problems on structured meshes. The MR-WENO spatial reconstruction procedures only require one five-cell, one three-cell, and one one-cell stencils for achieving uniform fifth-order accuracy in smooth regions and keeping essentially non-oscillatory property in non-smooth regions in one dimension. One redefines five new cell averages vectors after performing such spatial reconstructions and design one quartic polynomials vector and three quadratic polynomials vectors based on them. After that, a new detective process is used to examine the positivity of density and pressure of three quadratic polynomials vectors inside the whole target cell. If the negativity happens, a new compression limiter is carried out to enable the positivity of density and pressure of three quadratic polynomials vectors over the whole target cell and the positivity of density and pressure of one quartic polynomials vector at the midpoint of the target cell. It is a new way to design the positivity-preserving methods to keep fifth-order accuracy and the positivity over the target cell instead of only at some discrete Gauss-Lobatto quadrature points, since the precise minimum values of the density and pressure are now available. Then a theoretically proof is given to increase the optimal sufficient CFL number from 1/12 to 1/6 for the fifth-order WENO schemes. This methodology can be expanded to multi-dimensions easily. Unlike some classical positivity-preserving methods, the PPP methods could apply a special four-point Gauss-Lobatto quadrature formula or any other quadrature formulas on condition that their numerical precision is no smaller than four. Since the optimal CFL number of 1/6 is a sufficient but not necessary condition, the novelty PPP methods for fifth-order finite volume MR-WENO schemes with a larger practical CFL number of 0.6 are also available and robust enough when simulating some extreme problems without timely halving its value.

Experimental investigation on the thermophysical properties and solidification characteristics of n-octadecane in a spherical capsule

A thorough understanding of the thermophysical properties of phase change materials and the natural convection effect during the phase transition process is crucial for the accurate modeling and designing latent heat storage systems. However, research findings on the thermophysical properties of n-octadecane and the natural convection effect during the solidification process remain insufficient. In this study, the thermophysical properties of n-octadecane, including latent heat, thermal conductivity, density, thermal expansion coefficient and dynamic viscosity, were systematically measured under varying temperature conditions. Subsequently, a comprehensive solidification experiment of n-octadecane in a spherical capsule was conducted to assess the temporal and spatial temperature distribution characteristics and the evolution patterns of the solidification front. Finally, numerical techniques were employed to quantify the impact of natural convection. The results indicate that, the relationship between solidification mass fraction and time follows a quartic polynomial pattern. Based on the temperature curve at the central point, the solidification process of n-octadecane can be divided into four distinct stages. The first two stages account for 98.2 % of the total heat release, with natural convection primarily concentrated in the sensible heat release stage of liquid phase, contributing only 2.3 % to the entire solidification process, making its effect essentially negligible.

Shaped pattern synthesis for hybrid analog–digital arrays via manifold optimization-enabled block coordinate descent

Hybrid analog–digital (HAD) architecture is a promising means to realize large-scale arrays owing to the judicious trade-off between system performance and hardware complexity. This paper investigates the beampattern shaping of HAD arrays through minimizing the beampattern matching error between desired and actual patterns. Due to the nonconvex constraints on analog and digital weights and the nonconvex nonsmooth objective function, the resultant codesign is NP-hard. To address this issue, we create an efficient algorithm by leveraging block coordinate descent (BCD) and Riemannian manifold optimization. We first equivalently represent the objective function as a smooth bi-quadratic function by introducing a uni-modulus auxiliary variable. Subsequently, we alternatingly optimize the scaling factor, digital weights, analog weights and auxiliary variable under the BCD framework, where the simultaneous update of analog weights and auxiliary variable is recast as uni-modular constrained quadratic programming which is efficiently solved by Riemannian Newton method, and the digital weights have a global optimal solution via Lagrangian multiplier method. We also derive an explicit convergence condition of this algorithm. Numerical results demonstrate that the proposed algorithm has superior performance and faster convergence speed than alternative algorithms, and produces nearly the same mainlobe gain as fully digital arrays with significantly fewer radio-frequency chains.

Application of Machine Learning in the Development of Fourth Degree Quantitative Structure-Activity Relationship Model for Triclosan Analogs Tested against Plasmodium falciparum 3D7.

Despite new therapies against malaria, this disease remains one of the main causes of death affecting humanity. The phenomenon of resistance has caused concern as the drugs no longer have the same efficacy, forcing scientific research to develop new methods that prospect new molecules. With the advancement of artificial intelligence, it becomes possible to apply machine learning (ML) techniques in the discovery and evaluation of new molecules, by employing the quantitative structure-activity relationship (QSAR), a classic method that uses regressions to create a model that allows identifying and evaluating new drug candidates. This work combined QSAR with ML and developed a supervised model that modeled a fourth degree polynomial equation capable of identifying new drug candidates derived from the triclosan compound-a classic inhibitor of Plasmodium falciparum growth, the cause of severe malaria. The model produces an R 2 greater than 80% for training and concurrent testing, as well as a correlation index greater than 80% between the calculated and experimental pEC50 (negative logarithm of half maximal effective concentration) data. In addition, a web software (PlasmoQSAR) was created that allows researchers to calculate the EC50 (half maximal effective concentration) of new molecules using the developed analytical method.

Optimization of novel sustainable Dictyota mertensii alginate films with beeswax using a simplex centroid mixture design

The environmental risks of synthetic polymers drive the need for innovative and sustainable film and packaging solutions. This study optimized the formulation of films composed of alginate derived from the brown seaweed Dictyota mertensii (ADM), glycerol, beeswax, and tween 80, aiming to improve their properties. A simplex-centroid blend design was employed to develop and optimize the composite films. The films were characterized based on their morphological properties, moisture content (MC), contact angle (CA), solubility (S), water vapor permeability (WVP), color parameters (L∗, a∗, and b∗), opacity, tensile strength (TS), elongation at break (EB), and elastic modulus (E). Contour surfaces were plotted from the studied variables, fourth-order polynomial models were predicted, the influence of the blend components was analyzed, and the response variables were subjected to analysis of variance (ANOVA) and the F test with 95% confidence. The predicted conditions were experimentally validated. MC, WVP, solubility, opacity, a∗, and b∗ were minimized, whereas CA, L∗, TS, EB, and E were maximized, resulting in MC = 6.85%, WVP = 16.37 g mm/h.kPa.m2, S = 28.87%, CA = 83.86°, Opacity = 5.60 AU.nm.mm−1, L∗ = 70.31, a∗ = 4.79, b∗ = 21.00, TS = 11.76 MPa, EB = 12.70%, and E = 96.47 MPa. The overall desirability index was 0.70, resulting in an optimal biopolymer matrix of 75.00% ADM, 5.00% glycerol, 7.50% beeswax, and 12.50% Tween 80. The results highlight the potential of alginate films made from the brown seaweed Dictyota mertensii and the sustainable use of natural marine resources.

Modeling nonlinear stress-strain model for sulfate dry-wet cycle erosion of concrete: considerations for the initial compaction stage

Sulfate dry-wet cycle erosion significantly affects the mechanical properties of concrete. Investigating the uniaxial compressive stress-strain relationship under these conditions is essential for developing accurate constitutive models. This study analyzes the uniaxial stress-strain curves of concrete subjected to dry-wet cycles in 5% and 15% sulfate solutions. The results show that the initial compaction phase in the stress-strain relationship is particularly pronounced under increasing sulfate concentrations and cycle counts. The concrete experiences an extended compaction phase, which accounts for up to 35.71% of the total strain process. This finding challenge traditional constitutive models, which struggle to accurately describe this phase. To address this issue, the study develops a nonlinear stress-strain model for concrete, incorporating the initial damage caused by sulfate dry-wet cycle erosion, based on Weibull statistical damage mechanics principles. The research indicates that the effects of sulfate concentration and cycle count are predominantly reflected in the pronounced nonlinearity of the skeleton strain function’s opening size (a) and shape characteristics (b), modeled using a fourth-degree polynomial. The model demonstrates an excellent fit to experimental data with an R2 value of 0.99989, showing that the proposed nonlinear stress-strain relationship effectively captures the uniaxial mechanical behavior of concrete under sulfate dry-wet cycle erosion and provides a robust framework for developing constitutive models in such environments.

Design considerations for a continuous mussel farm in New England Offshore waters. Part I: Development of environmental conditions for engineering design

Sustainable aquaculture in nearshore waters faces challenges such as stakeholder conflicts, environmental pollution, and spatial constraints. Offshore aquaculture offers a promising solution but requires robust engineering design to withstand extreme weather conditions. This study develops the environmental conditions essential for engineering the design of a continuous mussel dropper system in New England offshore waters. A potential farm location was identified using criteria including water depth, federal boundaries, seafloor suitability, farm size, and proximity to ports, based on bathymetric and sedimentary maps. Historical data from five wave monitoring stations and two current velocity stations were analyzed to model extreme environmental conditions, as waves and currents pose primary threats. The extreme wave and current conditions are modeled using the Weibull distribution, on the annual maximum hourly significant wave height data and the largest 0.3 % of current speeds. A newly proposed method combining Spalding’s wall function with a fourth-order polynomial is used to enhance the current profile analysis. Additionally, a joint probability density function was developed for wave height and current velocity at a specific depth, providing insights into wave height, period, and wavelength for various return periods such as 10, 25, 50, and 100 years. The results suggest a 10-yr wave of 8 m significant wave height and a current speed of 1.68 m/s, while a 50-yr values are 9.4 m and 1.96 m/s respectively. These findings offer critical data on extreme wave and current conditions in New England’s offshore waters, providing practical guidance for the engineering design of offshore mussel farms. This research advances offshore mussel farming and benefits the development of all types of offshore aquaculture systems.

Stability and Optimality Criteria for an SVIR Epidemic Model with Numerical Simulation

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

MONOGENIC QUARTIC POLYNOMIALS AND THEIR GALOIS GROUPS

Abstract A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree N is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta ,\theta ^2,\ldots ,\theta ^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta )$ , where $f(\theta )=0$ . We use the classification of the Galois groups of quartic polynomials, due to Kappe and Warren [‘An elementary test for the Galois group of a quartic polynomial’, Amer. Math. Monthly96(2) (1989), 133–137], to investigate the existence of infinite collections of monogenic quartic polynomials having a prescribed Galois group, such that each member of the collection generates a distinct quartic field. With the exception of the cyclic case, we provide such an infinite single-parameter collection for each possible Galois group. We believe these examples are new and we provide evidence to support this belief by showing that they are distinct from other infinite collections in the literature. Finally, we devote a separate section to the cyclic case.

Retrospective analysis of dual-energy x-ray absorptiometry data demonstrates body composition changes with age in dogs and cats.

Use 18 years of dual-energy x-ray absorptiometry (DEXA) scan data to characterize how body composition changes with age in dogs and cats. This was a retrospective observational study using data obtained from DEXA scans performed between 2006 and 2023. A total of 6,973 observations from 1,273 colony-housed dogs ≤ 1 to 16.1 years old and 6,593 observations from 1,096 colony-housed cats ≤ 1 to 16.9 years old were obtained. Animal ages were rounded to the nearest 1/10-year intervals. Means for each interval were calculated and quadratic, cubic, and quartic polynomial models were fit to assess trends over age. Age had an effect on all DEXA measurements. In dogs, lean mass increased early in life before slowing to a peak at age 6.3 and then declined gradually. Fat mass also increased until slowing to a peak at age 9.3 and then decreased. In cats, lean mass increased before slowing to a peak at age 4.5, decreased gradually until age 12.5, and then sharply declined. Fat mass increased until slowing to a peak at age 7.5 and then decreased gradually. This retrospective study provides a baseline for how body composition changes with age. Results suggest that lean mass loss may begin earlier than previously reported in dogs and cats. Sarcopenia and obesity are common conditions in aging pets. Results can be used to improve body composition assessment of patients and investigate the efficacy of nutritional interventions.

A Damping Modification Factor Prediction Model for Horizontal Pseudoacceleration Spectrum for Subduction Slab Earthquakes Using Offshore Ground Motions in the Japan Trench Area

ABSTRACT This study develops a damping modification factor (DMF) model for offshore ground motion from subduction slab earthquakes that is a function of the damping ratio, spectral period, moment magnitude, focal depth, and rupture distance. The DMF values for 10 damping ratios between 1% and 30% and 34 spectral periods ranging from 0.03 to 5.0 s are calculated using 2803 offshore ground-motion records obtained from the S-net in Japan. An examination of the residuals from a median model, which considers deployment methods, indicates that DMF is influenced by parameters such as magnitude, depth, and distance. The DMF model includes the following components: a median model considering deployment method, a magnitude linear term, a magnitude square term, a focal depth term, an ln(R) scaling, and a constant term. The influence of damping ratio on model coefficients can be simulated by a quadratic function of the damping ratio. In addition, we established a DMF standard deviation model, for which the influence of damping ratio and spectral period on the standard deviation can be simulated by a quadratic function of the damping ratio and a quartic polynomial of the spectral period, respectively. The uncertainty related to the path and site effects is the main source of the deviation of the DMF model. We have conducted a comparison between the offshore DMF model and onshore model proposed by Rezaeian, Al Atik, et al. (2021), revealing that for lower damping ratios, the offshore DMF is generally lower at periods less than 0.2 s and larger at periods greater than 2.0 s compared to its onshore counterparts. The offshore DMF model adjusts the acceleration spectrum proposed by Tan and Hu (2024) to generate smooth pseudoacceleration spectra for most spectral periods.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

Generation of nonlinear gravity waves based on the harmonic polynomial cell method incorporated with a mass source

A nonlinear numerical wave tank is established using the Harmonic Polynomial Cell (HPC) method, which is incorporated with a mass source. The numerical wave tank consists of the mass source wave generation region and the working region, with sponge layers at both ends for wave absorption. In the mass source region, a generalized HPC method is applied to solve the inhomogeneous elliptic boundary value problems. The Poisson equation's special solution is represented by a bi-quadratic function. In the remaining domains, the HPC method is employed with harmonic polynomials to solve the problems governed by the Laplace equation in each grid cell. The free surface is tackled by the immersed boundary method (IB-HPC), and the kinematic and dynamic conditions of the free surface are described using a semi-Lagrangian approach. A variety of waves propagation are simulated, including Second-order Stokes wave, fifth-order Stokes wave, solitary wave, fifth-order Fenton stream function wave, random wave and the interaction with a straight vertical wall. The numerical solutions are compared with theoretical solutions. The numerical simulation results demonstrate that the present method can generate arbitrary two-dimensional wave fields by specifying an appropriate source function. Additionally, the reflected waves can propagate through the wave generation region, ensuring that the process of wave generation is not affected by the reflections.

Special research on well-being and earning of rice farmers in Laos: Survey of Official Development Assistant (ODA) projects and economic analysis

Farmers in rural areas can only earn revenue from rice farming which rice farming supports their better lives. Agricultural knowledge is a potential incentive for promoting well-being. This paper was survey research, there were two sections: 1) the author utilized an econometric approach to analyze the earnings of farmers from 10 provinces from 2017–2021; there were 1120 farmers involved in the survey; 2) the author used an economic evaluation approach to analyze effectiveness and efficiency of ODA by using data from 1110 farmers in 2018; 1205 farmers in 2019; and 1330 farmers in 2020. ODA is Instrumental Variables (IV) in the mincer model, ODA promotes agricultural knowledge for farmers, and of course, agricultural knowledge promotes revenue for farmers. The 2SLS and maximum likelihood are essential for the quartic function of mincer. Evaluation approaches from OECD/DAC and PCAP/JICA were utilized to compare the results of well-being indicators called basic human needs (BHN). According to the results of this study, the earnings of farmers who joined ODA projects were higher than the earnings of farmers who did not join ODA projects. The result of the economic evaluation approach found ‘ineffectiveness’; on the other hand, the author could find the efficiency of ODA projects. Finally, the recommendation is that the government policy on BHN should be consistently linked to the ODA program made by both the government and donors. Certainly, BHN policy for farmers who are minorities is necessary such as providing rice fields to be utilized for free, this will lead to chances for farmers to enhance their well-being from rice production.

Research on Trajectory Planning and Tracking Algorithm of Crawler Paver

The implementation of unmanned intelligent construction on the concrete surfaces of an airport effectively improves construction accuracy and reduces personnel investment. On the basis of three known common tracked vehicle dynamics models, reference trajectory planning and trajectory tracking controller algorithms need to be designed. In this paper, based on the driving characteristics of the tracked vehicle and the requirements of the stepping trajectory, a quartic polynomial trajectory planning algorithm was selected with the stability of the curve as a whole and the end point as the optimization goal, combining the tracked vehicle dynamics model, collision constraints, start–stop constraints and other boundary conditions. The objective function of trajectory planning was designed to effectively plan the reference trajectory of the tracked vehicle’s step-by-step travel. In order to realize accurate trajectory tracking control, a nonlinear model predictive controller with transverse-longitudinal integrated control was designed. To improve the real-time performance of the controller, a linear model predictive controller with horizontal and longitudinal decoupling was designed. MATLAB 2023A and CoppeliaSim V4.5.1 were used to co-simulate the two controller models. The experimental results show that the advantages and disadvantages of the tracked vehicle dynamics model and controller design are verified.

Explicit solutions of the generalized Kudryashov’s equation with truncated M-fractional derivative

The main purpose of this article is to study the generalized Kudryashov’s equation with truncated M-fractional derivative, which is commonly used to describe the propagation of wide pulses in nonlinear optical fibers. By employing the complete discriminant system of fourth-order polynomials, various types of explicit solutions are systematically classified, which include periodic solutions, the trigonometric functions, the double-period solutions, and the elliptic function solutions. Additionally, a series of 2D, 3D, and contour plots are generated to visually depict the spatial distribution and evolution of various solutions. This not only advances the development of nonlinear equations in theory but also provides valuable guidance in practical applications.