Biquadratic Research Articles

On estimations of Dawn spacecraft’s capture probability into 1:1 ground-track resonance around Vesta

Abstract This paper develops semi-analytical and analytical methodologies to estimate the probability of the Dawn spacecraft being captured into a 1:1 ground-track resonance around Vesta. The spacecraft, using low-thrust propulsion, approached the asteroid Vesta, and one significant challenge during this phase is crossing ground-track resonances with the asteroid. As the capture phenomenon is dependent on the initial condition of the trajectory, it is necessary to accurately estimate the probability of such a capture. Firstly, the system dynamics are described by a model incorporating Hamiltonian perturbations from the irregular gravitational field up to the second degree and order, and continuous low-thrust that is constant in magnitude and directed in the opposite direction of the spacecraft’s velocity. The resonance region is enclosed by separatrices which are approximated with a fourth-order polynomial for quantitative analysis. The Hamiltonian, serving as a proxy for the system’s energy, changes when the spacecraft crosses the separatrices, and these changes are quantified using a global adaptive quadrature method. Finally, the probability of capture into ground-track resonance is estimated based on the energy change across the separatrices, and the accuracy and efficiency of the developed semi-analytical and analytical methodologies are investigated by comparing them to numerical simulations based on the perturbed Hamilton’s equations of motion. This research makes a significant contribution to the field of astrodynamics by providing a systematic and efficient approach to estimating the probability of resonance capture.

Prescribed-time stabilisation of stochastic high-order nonlinear systems with time-varying powers

A control strategy for stochastic nonlinear systems with time-varying high-order dynamics is presented in this paper, which is specifically designed for prescribed-time state-feedback stabilisation. The approach, based on the backstepping design, constructs a time-varying controller, utilising scaled quartic Lyapunov functions and the given scaling function is not used for coordinate transformation. The fundamental characteristic is that the system has a unique and strong solution, almost surely, is implemented for arbitrary initial conditions. The designed scheme ensures mean-square convergence of both state and input within a prescribed time. The validity of the theoretical framework is vividly verified through a numerical example, showcasing the practical application and efficiency of the control method.

Mathematical modeling and optimum design of zinc oxide nanostructures modified carbon fiber reinforced polymethylmethacrylate-based bone cement composite using machine learning methods

A novel design-modeling strategy is proposed to enhance the mechanical properties of a medical polymethylmethacrylate (PMMA) bone cement (BC) under compressive and flexural loading conditions by considering the ratio of carbon fiber (CF), zinc (Zn)-coated carbon fiber (CF/Zn), and nanostructured zinc-oxide (ZnO) coated CF as design variables. A detailed study was carried out on multiple nonlinear neuro-regression analyses and revealed the advantages of the proposed method compared to Artificial Neural Networks (ANN). Polynomial, trigonometric and hybrid mathematical models are used to define the experimental process accurately. The boundedness of the candidate models is checked after the calculation of R2training and R2testing values to reveal whether the model is realistic or not. The results show that hybrid models are the most successful when the strength and strain are considered objective functions; the fourth-order polynomial model should be preferred when the module is considered objective. Also, it is seen that the most significant advantage of the Neuro-regression approach compared to artificial neural networks is that the mathematical models can be used directly without needing any transformation. This is not possible in neural networks and, therefore, significantly limits and complicates the use of models obtained using ANN.

CALCULATING POWER INTEGRAL BASES IN SOME QUARTIC FIELDS CORRESPONDING TO MONOGENIC FAMILIES OF POLYNOMIALS

Harrington and Jones characterized monogenity of four new parametric families of quartic polynomials with various Galois groups. A short time later Voutier added a cyclic family. In this note, we intend to describe all generators of power integral bases in the number fields generated by a root of the monogenic polynomials.

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.

Analysis of the granulometric composition of a three-fraction charge for vibropressed products

Using developed equipment and a methodology for experimental studies on the workability of the grain composition of three-fraction charges under load in a cylindrical container during vibrational impact after a series of blows from a compactor, the optimal fraction content was determined to meet the conditions for achieving a high degree of compaction. The analysis of experimental results identified charges with high bulk density and compaction coefficients, highlighting the following optimal fraction composition for further research (wt.%): 20–50% of 5–10 mm granite gravel, 25–40% of 2–5 mm granite gravel, and 25–40% of 0–0.3 mm sand. All experimental results were processed using the simplex-lattice method of mathematical modeling, applying the Scheffe methodology to derive regression equations in the form of fourth-degree polynomials. The analysis of graphical interpretations of the regression equations revealed patterns in bulk density changes under load and the compaction coefficient of three-fraction charges depending on their composition.

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 spatial interpolation for a stochastic particle Fokker–Planck model using a polynomial reconstruction

The stochastic particle Fokker–Planck (FP) model describes the behavior of rarefied gases while reducing the computational cost compared to the direct simulation Monte Carlo (DSMC) method, particularly for gas flows in the continuum regime. Many studies using FP models rely on cell-averaged macroscopic properties to update particle velocities, limiting spatial resolution in regions with large macroscopic gradients. To overcome this limitation, this paper introduces a spatial interpolation method based on the polynomial reconstruction. This method provides more accurate estimations of macroscopic properties using cell-averaged values and allows for extension to higher-order spatial accuracy. The spatial interpolation method is evaluated through three numerical simulations: Couette flow, lid-driven cavity flow, and hypersonic flow over a flat plate. The results demonstrate that the polynomial reconstruction method significantly improves accuracy. The second-order polynomial reconstruction method consistently outperforms the first-order polynomial reconstruction method, while the fourth-order polynomial reconstruction method does not consistently surpass the second-order polynomial reconstruction method due to challenges in boundary treatment. The study also examines accuracy improvements by interpolating a combined property of the viscous stress and density in the hypersonic flow over a flat plate, where large viscous stress gradients are present. The result demonstrates that interpolating the combined property enhances the overall accuracy of flow predictions by capturing large gradients.

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.