Search Results (57)

The unital associative algebra structure $\mathbb{A}$ on ${\mathbb{R}}^n$ allows for defining elementary functions and functions defined by convergent power series. For these, the usual derivative has a simple form even for higher-order derivatives, which allows us to have the $\mathbb{A}$-calculus. Thus, we introduce $\mathbb{A}$-differentiability. Rules for $\mathbb{A}$-differentiation are obtained: a product rule, left and right quotients, and a chain rule. Convergent power series are $\mathbb{A}$-differentiable, and their $\mathbb{A}$-derivatives are the power series defined by their $\mathbb{A}$-derivatives. Therefore, we use associative algebra structures to calculate the usual derivatives. These calculations are carried out without using partial derivatives, but only by performing operations in the corresponding algebras. For $f\left(x\right)=x^2$, we obtain $d{f}_x\left(v\right)=vx+xv$, and for $f\left(x\right)=x^{-1}$, $d{f}_x\left(v\right)=-x^{-1}v{x}^{-1}$. Taylor approximations of order k and expansion by the Taylor series are performed. The pre-twisted differentiability for the case of non-commutative algebras is introduced and used to solve families of quadratic ordinary differential equations. Full article

Let S be the $\mathbb{N}\times \mathbb{N}$-matrix of the shift operator and let k denote the field of real or complex numbers. We consider two different deformations of the commutative algebra $k\left[S\right]$, together with the evolution equations of the deformations of the powers $\{S^i,i⩾1\}$. They are called the $k\left[S\right]$-hierarchy and the strict $k\left[S\right]$-hierarchy. For suitable Banach spaces B, we explain how LU factorizations in $GL\left(B\right)$ can be used to produce dressing matrices of both hierarchies. These dressing matrices correspond to bounded operators on B, a class far more general than the one used at a prior construction. This wider class of solutions of both hierarchies makes it possible to treat reductions of both systems. The matrix coefficients of these matrices are shown to be quotients of analytic functions. Moreover, we present a subgroup $G_{cpt}\left(B\right)$ of $GL\left(B\right)$ such that the procedure with LU factorizations works for each $g\in G_{cpt}\left(B\right)$. Full article

This paper presents a fast Crank–Nicolson L1 finite difference scheme for the two-dimensional time fractional mobile/immobile diffusion equation with weakly singular solution at the initial moment. First, the time fractional derivative is discretized using the Crank–Nicolson formula on uniform meshes, and a local truncation error estimate is provided. The spatial derivative is discretized using the central difference quotient on uniform meshes. Then, energy analysis methods are utilized to provide an optimal error estimates. On the other hand, the numerical scheme is optimized based on the sum-of-exponentials approximation, effectively reducing computation and memory requirements. Finally, numerical examples are simulated to verify the effectiveness of the algorithm. Full article

This study examines the relationships between critical thinking, metacognition, psychological well-being, and empathy using structural equation modeling. The study sample consists of 155 university students from a higher education institution in Spain, who completed the PENCRISAL, the metacognitive abilities inventory, the Ryff psychological well-being scale, and the empathy quotient, which assess these psychological constructs. The results indicate that critical thinking has a direct positive effect on metacognition, which, in turn, is significantly associated with higher levels of psychological well-being and empathy. These findings reinforce the essential role of critical thinking in fostering cognitive self-regulation and socioemotional competencies. Furthermore, this study provides empirical evidence supporting the integration of critical thinking into educational programs, emphasizing its potential to enhance reflective thinking, emotional awareness, and interpersonal understanding. Full article

In this paper, we investigate the exterior problem of parabolic Hessian quotient equations. By utilizing Perron’s method, we establish the existence of viscosity solutions that exhibit generalized asymptotic behavior at infinity. The main approach we adopt involves constructing sub- and supersolutions to handle the non-constant term on the right-hand side of the equation. Full article

Total leaf area per shoot (A_T) can reflect the photosynthetic capacity of a shoot. A prior study hypothesized that A_T is proportional to the product of the sum of the individual leaf widths per shoot (L_KS) and the maximum individual leaf length per shoot (W_KS), referred to as the Montgomery–Koyama–Smith equation (MKSE). However, empirical evidence does not support such a proportional relationship hypothesis, as A_T was found to allometrically scale with L_KSW_KS, i.e., $A_{\mathrm{T}}\propto {\left(L_{\mathrm{K}\mathrm{S}}{W}_{\mathrm{K}\mathrm{S}}\right)}^{\mathsf{\alpha }}$, where $\mathsf{\alpha }\ne 1$, referred to as the power law equation (PLE). Given that there is variation in the total number of leaves per shoot (n), little is known about whether the leaf area distribution has an explicit mathematical link with the sorted leaf area sequence per shoot, and it is unknown whether the mathematical link can affect the prediction accuracy of the MKSE and PLE. In the present study, the leaves of 500 shoots of a dwarf bamboo (Sasaella kongosanensis ‘Aureostriatus’) were scanned, and the leaf area, length, and width values were obtained by digitizing the leaf images. We selected the shoots with n ranging from 3 to 10, which accounted for 76.6% of the totally sampled shoots (388 out of 500 shoots). We used the formula for the sum of the first j terms (j ranging from 1 to n) of a geometric series (GS), with the mean of the quotients of any adjacent two terms (denoted as ${\overline{q}}_A$) per shoot as the common ratio of the GS, to fit the cumulative leaf area observations. Mean absolute percentage error (MAPE) was used to measure the goodness of fit of the GS. We found that there were 367 out of 388 shoots (94.6%) where 1 < ${\overline{q}}_A$ < 1.618 and MAPE < 15%, and these 367 shoots were defined as valid samples. The GS hypothesis for leaf area distribution was supported by the result that the MAPE values for most valid samples (349 out of 367, i.e., 95.1%) were smaller than 5%. Here, we provide a theoretical basis using the GS hypothesis to demonstrate the validity of the MKSE and PLE. The MAPE values for the two equations to predict A_T were smaller than 5%. This work demonstrates that the leaf area sequence per shoot follows a GS and provides a useful tool for the calculation of total leaf area per shoot, which is helpful to assess the photosynthetic capacity of plants. Full article

The concept of tree or stem form has been central to forest research for over a century, playing a vital role in accurately assessing tree growth, volume, and biomass. The form factor is an essential component for expressing the shape of a tree, enabling more accurate volume estimation, which is vital for sustainable forest management and planning. Despite its simplicity, flexibility, and advantages in volume estimation, the form factor has received less attention compared to other measures like taper equations and form quotient. This review summarizes the concept, theories, and measures of stem form, and describes the factors influencing its variation. It focuses on the form factor, exploring its types, parameterization, and models in the context of various tropical species and geographic conditions. The review also discusses the use of the form factor in volume estimation and the issues with using default or generic values. The reviewed studies show that tree stem form and form factor variations are influenced by multiple site, tree, and stand characteristics, including site quality, soil type, climate conditions, tree species, age, crown metrics, genetic factors, stand density, and silviculture. The breast height form factor is the most adopted among the three common types of form factors due to its comparative benefits. Of the five most tested form factor functions for predicting tree form factors, Pollanschütz’s function is generally considered the best. However, its performance is often not significantly different from other models. This review identifies the “Hohenadl” method and mixed-effects modelling as overlooked yet potentially valuable approaches for form factor modelling. Using the form factor, especially by diameter or age classes, can enhance tree volume estimation, surpassing volume equations. However, relying on default or generic form factors can lead to volume and biomass estimation errors of up to 17–35%, underscoring the need to limit variation sources in form factor modelling and application. Further recommendations are provided for improving the statistical techniques involved in developing form factor functions. Full article

According to the Donnell–Mushtari shell theory, this work presents a closed-form solution procedure for free vibration of open laminated circular cylindrical shells with arbitrary homogeneous boundary conditions (BCs). The governing differential equations of free vibration are derived from the Rayleigh quotient and solved by the iterative separation-of-variable (iSOV) method. In addition, considering axial aerodynamic pressure, simulated by the linear piston theory, the exact eigensolutions for the flutter of open laminated cylindrical shells with simply supported circumferential edges and closed laminated cylindrical shells are also achieved. The governing differential equations of cylindrical shell flutter are derived from the Hamilton variational principle and solved by the separation-of-variable (SOV) method. The influence of circumferential dimension on flutter speed is investigated for open cylindrical shells, which reveals that the number of circumferential waves in critical flutter mode increases with circumferential length, and there exists an infimum for flutter speed that is an invariant independent of circumferential length. The present results agree well with those obtained by the Galerkin method, the finite element method, and other analytical methods. Full article

In this paper, the numerical solution for two-dimensional nonlinear parabolic equations is studied using an alternating-direction implicit (ADI) Crank–Nicolson (CN) difference scheme. Firstly, we use the CN format in the time direction, and then use the CN format in the space direction before discretizing the second-order center difference quotient. In addition, we strictly prove that the proposed ADI difference scheme has unique solvability and is unconditionally stable and convergent. The extrapolation method is further applied to improve the numerical solution accuracy. Finally, two numerical examples are given to verify our theoretical results. Full article

To monitor temperature as a function of varying inductance and resistance, we propose a data-driven digital twin approach for the rapid and efficient real-time estimation of the rotor temperature in an induction motor. By integrating differential equations with online signal processing, the proposed data-driven digital twin approach is structured into three key stages: (1) transforming the nonlinear differential equations into discrete algebraic equations by substituting the differential operator with the difference quotient based on the sampled voltage and current; (2) deriving approximate analytical solutions for rotor resistance and stator inductance, which can be utilized to estimate the rotor temperature; and (3) developing a general procedure for obtaining approximate analytical solutions to nonlinear differential equations. The feasibility and validity of the proposed method were demonstrated by comparing the test results with a 1.5 kW AC motor. The experimental results indicate that our method achieves a minimum estimation error that falls within the standards set by IEC 60034-2-1. This work provides a valuable reference for the overheating protection of induction motors where direct temperature measurement is challenging. Full article

The fungicide pyraclostrobin is the main measure used to control apple alternaria blotch in production. To evaluate the potential dietary risks for consumers, the dissipation and terminal residues of pyraclostrobin were investigated using ultra-high performance liquid chromatography–mass spectrometry (UHPLC–MS/MS). Pyraclostrobin in apples was extracted by acetonitrile with 2% ammonia and then purified using primary secondary amine (PSA) and graphitized carbon black (GCB). The method showed good linearity within the concentration range of 0.005–0.1 mg L⁻¹, with a coefficient of determination (R²) ≥ 0.9958. The recoveries ranged from 96.0% to 103.8%, with relative standard deviations (RSDs) between 0.8% and 2.3%. The limit of quantification (LOQ) was 0.01 mg kg⁻¹. Pyraclostrobin dispersible oil suspension was applied in 12 apple fields across China according to good agricultural practices (GAPs). In Beijing and Shandong, the dissipation of pyraclostrobin followed first-order kinetic equations, with a half-life of 11 days. The terminal residues ranged from <0.01 to 0.09 mg kg⁻¹. The national estimated daily intake (NEDI) of pyraclostrobin was compared with the acceptable daily intake (ADI), resulting in risk quotient (RQ_c) of 80.8%. These results suggest that pyraclostrobin poses a low health risk to consumers under GAP conditions and according to recommended dosages. Full article

A destructive degradation model was applied on films made from different concentrations of starch, glycerol and nanoclay using various elongation levels as a stress variable at different stress times and stretch cycles. The log tensile quotient (logarithm of the tensile strength to the corresponding break cycle) was recorded as the response variable. The log tensile quotient increased, and the log exact break time decreased, as the elongation level increased. The treatment containing the highest starch and nanoclay and lowest glycerol content proved to be the most resistant to stress conditions and the most versatile in relation to the varying log tensile quotients, while the treatments containing the lowest nanoclay and highest glycerol contents, regardless of the starch concentration, manifested the lowest log tensile quotient at higher levels of log exact break time. According to multiple regression findings, the break cycle governed mostly the stress conditions in the degradation model, followed by the sample ID and the log exact break time. The term log tensile quotient, attempted for the first time on data concerning biodegradable films enhanced with nanoclay, seems very promising for deeper research due to its ability to retrieve predictive information from survival equations and to discriminate the difference between film structures. Full article

K-essence theories are usually studied in the framework of a single scalar field $\varphi$. Namely, the Lagrangian of K-essence is the function of the single scalar field $\varphi$ and its covariant derivative. However, in this paper, we explore a double-field pure K-essence, i.e., the corresponding Lagrangian is the function of covariant derivatives of double scalar fields without a dependency on scalar fields themselves. This is why we call it double-field pure K-essence. The novelty of this K-essence is that its Lagrangian contains the quotient term of the kinetic energies from the two scalar fields. This results in the presence of many interesting features; for example, the equation of state can be arbitrarily small and arbitrarily large. In comparison, the range of the equation of state for quintessence is $-1$ to $+1$. Interestingly, this novel K-essence can play the role of an inflation field, dark matter, or dark energy by appropriately selecting the expressions of Lagrangian. Full article

Sea level rise due to glacier melting caused by climate warming is a major global challenge, but the mechanism of the effect of salinity on soil carbon (C) mineralization in different land types is not clear. The pathways by which salinity indirectly affects soil carbon mineralization rates need to be investigated. Whether or not the response mode is consistent among different land-use types, as well as the intrinsic links and interactions between soil microbial resource limitation, environmental stress, microbial extracellular enzyme activity, and soil carbon mineralization, remain to be demonstrated. In this paper, three typical land-use types (wetland, forest, and agroforestry) were selected, and different salinity levels (0‰, 3‰, 6‰, and 32‰) were designed to conduct a 125-day laboratory incubation experiment to determine the soil CO₂ release rate, soil physicochemical properties, and soil enzyme activities, and to correlate C mineralization with biotic and abiotic factors. A correlation analysis of soil physical and chemical properties, extracellular enzyme activities, and carbon mineralization rates was conducted to investigate their intrinsic linkages, and a multiple linear regression of C mineralization at different sites was performed to explore the variability of mineralization among different site types. Structural equation models were established in the pre- and post-incubation stages to study the pathways of soil C mineralization at different incubation times, and the mechanism of mineralization was further verified by enzyme stoichiometry. The results showed that, at the end of 125 days of incubation, the 32‰ salinity addition reduced the cumulative mineralization of forest and agroforestry types by 28.41% and 34.35%, respectively, compared to the 0‰ salinity addition. Soil C mineralization in the three different land-use types was highly correlated with the active C fractions of readily oxidizable C (ROC), dissolved organic C, and microbial biomass C (MBC) in the soil, with the standardized coefficients of multivariate linear regression reaching 0.67 for MBC in the wetland and −0.843 for ROC in the forest. Under long-term salinity additions, increased salinity would reduce the microbial respiratory quotient value by inhibiting β-glucosidase activity, thus indirectly affecting the rate of CO₂ release. With added salinity, the mineralization of non-saline soil was more susceptible to the inhibitory effect of salinity, whereas the mineralization of salinized soil was more controlled by soil C pools. Full article