Equation Formula Research Articles

Exploration of Changes in Coal Pore Characteristics and Gas Adsorption Characteristics Based on Influence of Stress

As the mining depth increases, the effect of stress on the gas adsorption of coal gradually becomes significant. There are significant differences in the pore volume, specific surface area, and adsorption characteristics of coal before and after stress. In this study, the porosity variation characteristics of coal were studied using axial and confining pressure loading processes, and volumetric stress was introduced to characterize the pore variation law of coal under triaxial stress. By calculating the stress values at different burial depths, gas isothermal adsorption experiments were conducted on coal under different stress effects. The Langmuir equation, D-A equation, and Freundlich empirical formula were used to fit the adsorption experimental results. Combining experiments and models to predict the adsorbed and free gas content under stress, we described the gas adsorption law of coal under different stress effects.

Generalized Itô’s lemma and the stochastic thermodynamics of diffusion with resetting

Abstract Methods from the theory of stochastic processes are increasingly being used to extend classical thermodynamics to mesoscopic non-equilibrium systems. One characteristic feature of these systems is that averaging the stochastic entropy with respect to an ensemble of stochastic trajectories leads to a second law of thermodynamics that quantifies the degree of departure from thermodynamic equilibrium. A well known mechanism for maintaining a diffusing particle out of thermodynamic equilibrium is stochastic resetting. In its simplest form, the position of the particle instantaneously resets to a fixed position x 0 at a sequence of times generated from a Poisson process of constant rate r. Within the context of stochastic thermodynamics, instantaneous resetting to a single point is a unidirectional process that has no time-reversed equivalent. Hence, the average rate of entropy production calculated using the Gibbs–Shannon entropy cannot be related to the degree of time-reversal symmetry breaking. The problem of unidirectionality can be avoided by considering resetting to a random position or diffusion in an intermittent confining potential. In this paper we show how stochastic entropy production along sample paths of diffusion processes with resetting can be analyzed in terms of extensions of Itô’s formula for stochastic differential equations (SDEs) that include both continuous and discrete processes. First, we use the stochastic calculus of jump-diffusion processes to calculate the rate of stochastic entropy production for instantaneous resetting, and show how previous results are recovered upon averaging over sample trajectories. Second, we formulate single-particle diffusion in a switching potential as a hybrid SDE and develop a hybrid extension of Itô’s stochastic calculus to derive a general expression for the rate of stochastic entropy production. We illustrate the theory by considering overdamped Brownian motion in an intermittent harmonic potential. Finally, we calculate the average rate of entropy production for a population of non-interacting Brownian particles moving in a common switching potential. In particular, we show that the latter induces statistical correlations between the particles, which means that the total entropy is not given by the sum of the 1-particle entropies.

Research on temperature performance prediction of vortex tubes based on artificial neural networks

Abstract This study constructs a hybrid neural network model by integrating the physical constraints of the Bernoulli equation and Nikolaev's formula. The model is designed to explore and predict the variation pattern of the cold end temperature in a vortex tube. The input parameters include inlet pressure, inlet temperature, and cold mass fraction, with the cold end temperature as the output parameter. The network employs a multilayer feedforward model and the Levenberg-Marquardt learning algorithm, using a hyperbolic tangent function as the activation function. To evaluate the statistical validity of the developed model, the coefficient of determination (R²) and root mean square error (RMSE) are utilized, along with an analysis of the model's uncertainty and robustness. The hybrid model achieves an R² of 0.9936 and an RMSE of 0.3392, demonstrating strong performance in terms of uncertainty and robustness. These results indicate that the model accurately predicts the cold end temperature variation in the vortex tube. Furthermore, the findings reveal an optimal pressure range (0.49 MPa to 0.76 MPa) and cold mass fraction range (0.1 to 0.2) that minimize the cold end temperature.

Equivalent conversion of different multiaxial stress rupture criteria for creep materials based on skeletal point stresses

Multiaxial creep life prediction of components has been challenging. To cope with the challenge, efforts have been made to develop suitable multiaxial stress rupture criteria (MSRC) based on representative stress concept. Of all MSRCs, the ones due to Sdobyrev and Hayhurst-Leckie (denoted as SHL MSRC), and to Cane (denoted as Cane MSRC) are commonly-used. In this work, the equivalent conversion between the SHL MSRC and the Cane MSRC is investigated based on skeletal point stresses. Equivalent conversion formulae are proposed based on the skeletal point stresses of circumferentially-notched tension (CNT) specimen, and then validated by comparison with the experimental data available in the literature. The equivalent conversion formula between the two MSRCs is found to be dependent on the signs of principal stresses. Therefore, different conversion formulae should be chosen depending on the signs of principal stresses. Afterwards, the applicability of different equivalent conversion formulae is demonstrated for the CNT specimen when skeletal point stresses are available, and for creep crack growth specimen when skeletal point stresses are not available. Finally, the equivalence of the two MSRCs is discussed.

Novel efficient iterative schemes for linear finite element approximations of elliptic optimal control problems with integral constraint on the state

Centering on a distributed optimal control problem governed by elliptic equations with a unilateral integral constraint on the state, we recall the first-order optimality conditions to explore variational formulations. The classical linear finite element method is employed to approximate the equivalent formulae. Taking into account the distinctive structural characteristics of discretized schemes, we divide the discretized optimal conditions into two equivalent matrix schemes. Subsequently, we introduce an efficient iterative algorithm designed to solve the approximation schemes. We demonstrate the convergence of our iterative algorithm and investigate its robustness and uniform optimality under a specified constraint, particularly focusing on small regularization parameters. Finally, numerical tests are conducted using various h and α to illustrate the efficiency of our proposed algorithm. These tests also validate a tremendous reduction in the number of required iterations.

The Carbon Stocks Estimation on Tangga Community Forests of Lombok Utara

Forests are one of the largest carbon dioxide absorbers, the forest's function as a carbon dioxide absorber causes an increase in greenhouse gases in the atmosphere, and carbon dioxide is stored in forest biomass. Forest development to increase carbon dioxide absorption can be carried out in state forest areas or private forests, which also include community forests. This research aims to estimate carbon stocks in the Tangga community forest (HKm) , in the Tangga HKm, Lombok utara, NTB. This research uses quantitative methods. The data collected is standing vegetation data. Standing vegetation data includes saplings, poles and trees by measuring diameter and height using purposive sampling collection techniques. Carbon stocks are estimated using the allometric equation formula. The total carbon reserves in the HKm Tangga obtained from each stand were 9.972,11 ton/ha. The highest carbon stock value is at the tree level with a value of 8.785,08 tons/ha, followed by the sapling level at 731,34 ton/ha, and the pole level at 455,69 tons/ha with the highest amount at the candlenut tree level (Aleurites moluccana) at 2.200,63 ton/ha and the lowest was the breadfruit (Artocarpus altilis) with a value of 1,96 ton/ha.

Recurrence formula for some higher order evolution equations

Riccati’s differential equation is formulated as abstract equation in finite or infinite dimensional Banach spaces. Since the Riccati’s differential equation with the Cole–Hopf transform shows a relation between the first order evolution equations and the second order evolution equations, its generalization suggests the existence of recurrence formula leading to a sequence of differential equations with different order. In conclusion, by means of the logarithmic representation of operators, a transform between the first order evolution equations and the higher order evolution equation is presented. Several classes of evolution equations with different orders are given, and some of them are shown as examples.

Understanding the Role of Economic Education on Generation Z's Lifestyle and Interest in Saving: Why Do Financial Literacy and Circle of Friends Matter?

The lifestyle of the people of a country can also be an illustration of how that country will experience growth in the economic sector. An individual's lifestyle and interest in saving must, of course, be accompanied by knowledge and literacy that can control this. This research aims to determine the influence of economic education on the generation's lifestyle and interest in saving, which is mediated by financial literacy and a circle of friends. The population in this study is generation Z, who live in Petasia District. The sampling technique used the Limeshow equation formula with a determined sample size of 700 respondents. The data collection method uses a questionnaire. The data analysis technique is partial least squares structural equation modeling (PLS-SEM) analysis. The results of this research show that economic education (X) has a positive and significant influence on Generation Z's lifestyle and interest in saving. Then, financial literacy and a circle of friends are able to mediate the influence between economic education and Generation Z's interest in saving. These findings encourage generation Z to optimize economic education, both formal and informal, as well as increase financial literacy, in order to maintain their lifestyle. It is also necessary to optimize learning access and provide great opportunities for Generation Z to increase their financial literacy.

MANAJEMEN STOK PANGAN RUMAH TANGGA PETANI PADI ANGGOTA LUMBUNG PANGAN DI KECAMATAN AMBARAWA KABUPATEN PRINGSEWU

This study aimed firstly to analyze the management of household food stocks and secondly to analyze the various factors affecting household food stocks for rice farmers. All rice farmers were members of the food barn in Ambarawa Sub-district, Pringsewu District. The samples were selected by simple random sampling consisting of 7 granaries with a total of 70 members. The data were collected from January to February 2023. To obtain the first objective, the data were analyzed by using the calculation of food stocks referring to the Chafid equation formula (2017), i.e. the amount of food inventory was subtracted by its sale quantity and its utilization. To obtain the second objective, the data were analyzed by using the multiple linear regression. The results showed that the food stocks of rice farmers who were members of the food barn were 573 kg of MDG in the first season and 450 kg of MDG in the second season, which were not an ideal amount for food stock. This was because the ideal amounts of needs were 586.93 kg of MDG for the first season and 591.51 kg of MDG for the second season. Some of the factors affecting the food stock of the rice farming households incorporated in food barns were land area and household income. Key words: regression analysis, sale delay analysis, stocks, storage.

Feynman-Kac formula for general diffusion equations driven by TFBM with Hurst index H ∈ (0,1)

We consider the general diffusion equation driven by tempered fractional Brownian motion (TFBM)∂u(t,x)∂t=Lu(t,x)+u(t,x)dBH,λ(t)dt,(t,x)∈R+×Rd, where L is the infinitesimal generator of some time-homogeneous Markov process {Xt}t≥0 and {BH,λ(t)}t∈R is a tempered fractional Brownian motion with Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0 which is independent of {Xt}t≥0. Based on approximating TFBM with a family of Gaussian processes possessing absolutely continuous sample paths, a unified framework of the Feynman-Kac formulau(t,x)=Ex[f(Xt)]eBH,λ(t) is established for the general stochastic diffusion equation driven by TFBM with the initial value f, Hurst index H∈(0,12)∪(12,1) and tempering parameter λ>0. The difficulty is that the Hurst parameter H can be allowed to be less than 1/2. The idea is to explore the above simplest form, to utilize the techniques of Malliavin calculus. By using the properties of TFBM and especially that of the modified Bessel function of the second kind, we prove that the process defined by the Feynman-Kac formula is the mild and weak solutions of the general diffusion equation driven by TFBM. From the Feynman-Kac formula, we exhibit the smoothness of the density of the solution and the sharp Hölder regularity. We further show that limt→∞⁡ln⁡‖u(t,⋅)‖pg(t)=0 in the almost surely sense where p∈R and the function g:R+→R+ grows at least as fast as a linear function. The Feynman-Kac formula for the stochastic general diffusion equation in the Skorohod sense is also achieved by exploring the relationship between the Stratonovich and Skorohod integrals.

One-dimensional analysis of pressure variations induced by trains passing each other in a tunnel

In this study, the asymptotic solutions of the pressure variations induced by two trains passing each other in a tunnel are theoretically investigated. The one-dimensional inviscid compressible airflow is analysed, and two methods to obtain numerically exact solutions and $M_{H}$ expansion formulas for approximate equations are presented, where $M_{H}$ is the Mach number of the high-speed train. The pressure coefficient, corresponding to the maximum value of the magnitude of the pressure, is expressed as $|c_{p}|_{max}=|c_{p,min}|=[({R}/({1-R}))$ $(1+\alpha )^{2}+({R(1-R)}/{(1-2R)^{2}})(1-\alpha )^{2}]+O[M_{H}]$ , where $c_{p,min}<0$ , $\alpha =U_{L}/U_{H}$ and $U_{L}$ and $U_{H}$ denote the speeds of the low- and high-speed trains, respectively, and $R$ is the cross-sectional area ratio of the train to the tunnel. The theoretical results indicate the dependence of the speeds of the two trains on the pressure distribution and that the maximum magnitude of the asymptotic pressure for a fixed value of $M_{H}$ is obtained for $\alpha =1$ and $\alpha =0$ when $R< R_{c}$ and $R>R_{c}$ , respectively, where $R_{c}$ denotes the critical blockage ratio. Because the airflow along the side of the low-speed train, induced by the low-speed train, is along the running direction of the high-speed train and reduces the relative velocity of the high-speed train as the two trains pass each other, $|c_{p}|_{max}$ for $\alpha =0$ is larger than $|c_{p}|_{max}$ for $\alpha =1$ when $R>R_{c}$ . It is theoretically demonstrated that, as conventional high-speed railway systems satisfy $R< R_{c}$ , a conservative pressure estimation can be established assuming $\alpha =1$ .

Feynman–Kac Formulas for Difference-differential Equations of Retarded Type

Feynman–Kac Formulas for Difference-differential Equations of Retarded Type

On the Itô–Alekseev–Gröbner formula for stochastic differential equations

On the Itô–Alekseev–Gröbner formula for stochastic differential equations

Study of the dynamic interactions between helix-stiffened cement mixing piles and soft soil under oblique incidence of SV waves in arbitrary space

Helix stiffened cement mixing (HSCM) piles are extensively utilized in soft soil sites due to their excellent seismic and pullout resistance. Initially, the equivalent nodal force formulas for each face under the oblique incidence of SV waves in any spatial direction are derived based on the viscous-spring artificial boundary. The method's applicability in studying pile–soil interactions is verified based on existing pile–soil tests. Subsequently, the HSCM pile–soil model is constructed using finite element software through a Python program. Various analyses of accelerations, relative displacements, soil reactions, and pile–soil dynamic p-y curves are conducted using two different directions of incidence angles (horizontal and vertical) and various types of seismic waves. Subsequently, the backbone line of the dynamic p-y curve is plotted, revealing the pile–soil interaction mechanism under spatial oblique incidence. Finally, Matlock's formula is modified to ensure its applicability to pile–soil interactions under spatially oblique incidence. The results demonstrate that loading the Kobe wave causes the peak acceleration in the x direction, the peak pile–soil relative displacement, and the peak pile-side soil reaction force to increase by 22.64 %, −18.90 % and 18.42 %, respectively, compared to those of the El Centro wave.

A new stable method to compute mean value coordinates

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments.

A Comparative Analysis of Sediment Concentration Using Artificial Intelligence and Empirical Equations

Morphological changes in canals are greatly influenced by sediment load dynamics, whose estimation is a challenging task because of the non-linear behavior of the sediment concentration variables. This study aims to compare different techniques including Artificial Intelligence Models (AIM) and empirical equations for estimating sediment load in Upper Chenab Canal based on 10 years of sediment data from 2012 to 2022. The methodology involves utilization of a newly developed empirical equation, the Ackers and White formula and AIM including 20 neural networks with 10 training functions for both Double and Triple Layers, two Artificial Neuro-Fuzzy Inference System (ANFIS), Particle Swarm Optimization, and Ensemble Learning Random Forest models. Sensitivity analysis of sediment concentration variables has also been performed using various scenarios of input combinations in AIM. A state-of-the-art optimization technique has been used to identify the parameters of the empirical equation, and its performance is tested against AIM and the Ackers and White equation. To compare the performance of various models, four types of errors—correlation coefficient (R), T-Test, Analysis of Variance (ANOVA), and Taylor’s Diagram—have been used. The results of the study show successful application of Artificial Intelligence (AI) and empirical equations to capture the non-linear behavior of sediment concentration variables and indicate that, among all models, the ANFIS outperformed in simulating the total sediment load with a high R-value of 0.958. The performance of various models in simulating sediment concentration was assessed, with notable accuracy achieved by models AIM11 and AIM21. Moreover, the newly developed equation performed better (R = 0.92) compared to the Ackers and White formula (R = 0.88). In conclusion, the study provides valuable insights into sediment concentration dynamics in canals, highlighting the effectiveness of AI models and optimization techniques. It is suggested to incorporate other AI techniques and use multiple canals data in modeling for the future.

Equivalent resistance of a periodic and asymmetric 2 × n resistor network

This article proposes a class of periodic and asymmetric 2 × n resistor network model, and adopts the RT-V theory to conduct in-depth research on this problem, achieving new theoretical breakthroughs. This study derived three original equivalent resistance formulae for this complex circuit network, and also discusses the analytical expressions for equivalent resistance in different special cases, verifying that the conclusions in special cases are completely consistent with the actual circuit or previous existing conclusions, indirectly verifying the correctness of the theoretical results obtained in this article. The resistor network model in this article has multifunctional characteristics, for example, when taking r1→∞, the model degenerates into a cobweb network model. The research methods and results of the article will provide a new theoretical basis for related scientific and engineering research.

Perturbation algorithm for refueling reactivity coefficient in the pebble-bed high temperature reactor

The fuel management mode of the pebble-bed high temperature reactor (PB-HTR) is on-line continuous refueling, that is, the reactor is refueling at the same time of operation, and the reactivity is continuously supplemented by continuous refueling. In this paper, the perturbation algorithm for refueling reactivity coefficient is proposed, and the internal mechanism of reactivity introduced by refueling in the PB-HTR is revealed, that is, refueling reactivity depends on the dot product of the pebble flow velocity and the differential of atomic density in space. Based on the neutron diffusion equation, the difference and differential formulas of refueling reactivity coefficient are derived. The application of differential formula, which can be directly calculated according to the current parameters of the reactor core before the refueling, is simpler than the difference formula. Then, the equilibrium core and running-in phase of HTR-PM are used to verify the effect of the perturbation algorithm and the calculation results agree well with the reference value from the VSOP code. The distribution of refueling reactivity in different spatial regions can also be analyzed based on the difference and differential formulas.

Ground Infiltration in Various Land Uses in Kawatuna Sub- Watershed

The research was conducted in the Kawatuna Sub-watershed with the aim of studying the physical properties of the soil and knowing the infiltration rate of several land uses in the Kawatuna Sub-watershed. This study used the Horton equation formula and took soil samples for analysis at the Laboratory of Soil Science, Faculty of Agriculture, Tadulako University, Palu. The results showed that the physical properties of the soil greatly influenced the infiltration process, the highest organic matter was secondary soil 6.10%, dry land agriculture 4.50% and the lowest was shrubs 3.04%. Secondary forest soil textures are loam and dusty loam, dry land agriculture has sandy loam textures, and shrubs have loamy, sandy loam and dusty loam textures. In terms of physical properties of water content, dry land agriculture had the highest water content 21.48%, secondary forest 19.19% and shrubs the lowest 15.54%. The highest infiltration rate was secondary forest land (12.74 cm/hour) with fast criteria, dry land agriculture (7.80 cm/hour) medium fast category and shrub land (6.19 cm/hour) medium category.

Approximate theory of armour perforation by ductile hole expansion

The present study suggests a complete analytical model for armour perforation by ductile hole enlargement process. The new approximate model is applicable for entry, tunnelling and exit phases of a rigid nose-pointed projectile with an arbitrary nose-shape. The theory is based on dividing the target into infinitesimal thin layers, in which a cylindrical hole expansion is assumed. The elastoplastic work that is consumed by the target for infinitesimal projectile advancement is based on the specific cavitation energy and leads to reduction of the projectile kinetic energy. Several equivalent formulae for ballistic limit velocity, and a formula for the ballistic limit thickness are found to be independent of projectile nose-shape and nose length. The approximate model is compared with numerical simulations for AA6061-T651 targets that are impacted by two different ogive nose-shape projectiles at several striking velocities. A small nose-shape effect on the ballistic limit velocity is observed in the numerical simulations and is attributed mainly to the target axial effect and the bulge formation at the target rear surface, which are not considered in the approximate model.