Mathematical Equations Research Articles

Effect of Axial Modification on the Meshing Performance of Involute Beveloid Gear Pair

In order to enhance the load-bearing capacity of the involute beveloid gear pair, this study proposes research on improving the contact stress through axial modification. Under the condition of segmented parabolic axial modification, the mathematical equation for the tooth flank of the beveloid gear is derived. A simulation meshing model for the beveloid gear pair is constructed to investigate the effects of different amounts and lengths of axial modification on the meshing performance, changes in flank and root stress, and transmission error before and after modification. The results indicate that as the modification amount increases, the maximum equivalent stress initially decreases and then exhibits a tendency towards stability. Moreover, an increase in modification length concentrates the contact zone towards the middle of the tooth flank while expanding the range of tooth root stress distribution, and it also leads to a decrease in maximum equivalent stress. The implementation of the modification has been observed to result in a reduction in cumulative transmission error and an effective reduction in edge load on the beveloid gear pair, which has been demonstrated to enhance the bearing capacity and transmission stability extension while also impacting the dynamic meshing performance.

Marginal and average considerations in LCA and their role for defining emission factors and characterization factors

Abstract Introduction In the literature on LCA, one often finds the terms “marginal” and “average,” often in combination with words like “data,” “process,” “emission,” or “characterisation factor.” However, the meaning of these terms appears to differ between sources. This paper aims to clarify the situation. Critical analysis We review the various definitions and interpretations of the terms “marginal” and “average” in economics, LCI and LCIA. We also study the role of various related terms, such as “linear” and “incremental.” It turns out that the term “marginal” is used for characterizing processes in some sources and for characterizing the data that describes processes in other sources. These two interpretations are shown to differ substantially in a hypothetical example. We also note that the situation in the LCIA literature differs markedly from that in the LCI literature. Conclusion and discussion We propose to distinguish three concepts, marginal, average, and average marginal, and offer verbal definitions, mathematical equations, and a numerical example with a graphical interpretation. We also draw an agenda to research the implications for the attributional-consequential debate, the development of databases and software, and several other topics. This may also help to bring more insights in the continuing controversy on consequential versus attributional LCA.

An advanced machine learning-based mathematical equation for predicting compression index from readily-available soil physical properties

ABSTRACT The Compression index, known as Cc, plays a crucial role in geotechnical design. However, the conventional approach to ascertain Cc is not only expensive but also time-intensive. This paper proposes an innovative modeling approach to develop a Deep Neural Network (DNN)-based formula for the prediction of Cc incorporating readily accessible physical properties of soil, including specific gravity (G), saturation degree (Sr), liquid limit (LL), silt content (SC), and clay content (CC). To assess the predictive performance of the proposed formula, various metrics were employed, such as determination coefficient (R²), root mean square error (RMSE), mean absolute error (MAE), and Nash–Sutcliffe Efficiency (NSE). From the collected dataset, 608 samples were used as the training set for the DNN model, while the remaining 152 samples were designated as the testing set. The R², RMSE, MAE, and NSE values for the training and testing sets were 0.9821 and 0.9935; 0.126 and 0.098; 0.083 and 0.074; 0.9812 and 0.9922, respectively. These results indicate that the developed equation demonstrates outstanding predictive accuracy for estimating Cc. Furthermore, the comparative analysis underscores that the proposed DNN-based formula significantly outperforms empirical equations and multilinear regression (MLR) model in predicting Cc with higher accuracy.

Enhancing CO oxidation performance by controlling the interconnected pore structure in porous three-way catalyst particles.

Highly ordered porous structured particles comprising three-way catalyst (TWC) nanoparticles have attracted attention because of their remarkable catalytic performance. However, the conditions for controlling their pore arrangement to form interconnected pore structures remain unclear. In particular, the correlation between framework thickness (distance between pores) or macroporosity and the diffusion of gaseous reactants to achieve a high catalytic performance has not been extensively discussed. Here, the interconnected pore structure was successfully controlled by adjusting the precursor components (i.e., template particle concentration) via a template-assisted spray process. A cross-sectional image analysis was conducted to comprehensively examine the internal structure and porous properties (framework thickness and macroporosity) of the porous TWC particles. In addition, we propose mathematical equations to predict the framework thickness and macroporosity, as well as determine the critical conditions that caused the formation of interconnected pores and broken structures in the porous TWC particles. The evaluation of CO oxidation performance revealed that porous TWC particles with an interconnected pore structure, thin framework, and high macroporosity exhibited a high catalytic performance owing to the effective diffusion and utilization of their internal parts. The study findings provide valuable insights into the design of porous TWC particles with interconnected pore structures to enhance exhaust gas emission control in real-world applications.

Organizational Performance Evaluation Using European Foundation for Quality Management (EFQM) Model: Case Study of the Central Bank of Iraq

This study seeks to assess the organizational performance of the Central Bank of Iraq using the European Foundation for Quality Management (EFQM 2025) model. By identifying strengths, weaknesses, and performance gaps, the research provides practical insights aimed at achieving organizational excellence and sustainability in emerging economies.The study used EFQM-aligned checklists to collect data, which were then analyzed using weighted averages and mathematical equations. Statistical tools, including SPSS and the least significant difference (LSD) test, were used to rank EFQM criteria according to their impact on improving performance quality.The Central Bank of Iraq achieved a 66.4% compliance rate with EFQM standards and excelled in areas such as organizational culture and leadership, and stakeholder engagement (88%). However, significant deficiencies were detected in strategic and operational performance (31%), highlighting critical areas for targeted improvement initiatives.While the study is limited to the Central Bank of Iraq, the EFQM framework provides a versatile tool for improving organizations in diverse contexts. Future research could explore its application across different sectors and geographies to validate the findings.This research represents a pioneering effort to apply the EFQM 2025 criteria to assess organizational performance in an emerging economy. Its findings provide practical recommendations for policy makers and practitioners, contributing to the broader discourse on quality management in dynamic environments.

Photothermal Material-Based Solar-Driven Cogeneration of Water and Electricity: An Efficient and Promising Technology.

With the increasing demand for fresh-water and electricity in modern society, various technologies are being explored to obtain fresh-water and electricity. Due to advances in materials science, solar-driven interfacial evaporation (SDIE) systems have attracted widespread attention because they require only solar energy, and possess a high evaporation rate and little pollution. The researchers combined energy harvesting measures into the system to output electricity, further improving energy utilization. However, more in-depth research and review remain on using SDIE systems for efficient water-electricity cogeneration. Therefore, the mechanisms of different photothermal materials that utilize solar energy to produce thermal energy are first summarized in this paper. Subsequently, the mechanism and application of thermal, mechanical, chemical, and evaporation energy to produce electrical power in SDIE water-electricity cogeneration systems are discussed. Concurrently, vital mathematical equations and widely used mathematical simulation methods for performance evaluation and practical applications are presented. The design and operation of water-electricity cogeneration systems based on photothermal materials are analyzed and summarized. Based on a review and in-depth understanding of these aspects, the future development direction of cogeneration is proposed to address the problems faced in basic research and practical applications.

A fuzzy based computational model to analyze the influence of mitochondria, buffer, and ER fluxes on cytosolic calcium distribution in neuron cells.

A free calcium ion in the cytosol is essential for many physiological and physical functions. Also, it is known as a second messenger as the quantity of free calcium ions is an essential part of brain signaling. In this work, we have attempted to study calcium signaling in the presence of mitochondria, buffer, and endoplasmic reticulum fluxes. Small organelles called mitochondria are found in the nervous system and are involved in several cellular functions, including energy production, response to stress, calcium homeostasis regulation, and pathways leading to cell death. It has been discovered that buffer, endoplasmic reticulum, and mitochondria significantly affect calcium signaling. To investigate how various circumstances impact the quantity of calcium in the cytosol, a mathematical model ofa second-order linear partial differential equation with fuzzy boundary conditionshas been developed. Systems having ambiguous or imprecise boundaryvalues can be effectively modeled and simulated with the help of fuzzy boundary conditions. Models can provide more dependable and instructive outcomes and become adaptable to real-world circumstances by implementing fuzzy logic into boundary conditions. In this paper, we observed the Fuzzy Laplace Transform to solve variable coefficient fuzzy differential equations using triangular fuzzy numbers. It isnoted that maintaining the delicate calcium ion balance, which controls essential cellular functions, depends on the buffer affinity. Also, neurodegenerative illnesses like Alzheimer's, Parkinson's, etc. are linked to disruptions in the control of components such as buffers, mitochondria, and the endoplasmic reticulum.

Based on AVL-CRUISE Hybrid and Electric Vehicle Simulation and FTP-72 Circle Power Consumption Factor Analysis

With the rapid advancement of technology and the growing scarcity of energy resources, finding effective ways to reduce power consumption has become increasingly crucial. Hybrid and electric vehicles play a vital role in this endeavor, warranting deeper exploration and focus. In this study, I utilized AVL-Cruise software, a powerful tool for simulation and data modeling, to conduct comprehensive simulations of hybrid and electric vehicle performance, with a specific focus on analyzing the FTP72 cycle.The simulation results are highly encouraging. The vehicle achieved a maximum speed of 164 km/h, an acceleration rate of 3 m/s, a 0 to 100 km/h acceleration time of 9 seconds, and a maximum climbing slope of approximately 74%. These figures not only demonstrate that hybrid and electric vehicles can meet performance expectations but also highlight their capability to deliver on key metrics. Furthermore, the total energy consumption was recorded at 4200 kJ, underscoring the efficiency of these vehicles.In addition to the simulation outcomes, I delved into various factors that influence vehicle performance, including vehicle mass, drag coefficient, and motor power.Using MATLAB, I conducted a detailed analysis of the relationships between these factors and energy consumption, incorporating residual analysis and deriving a mathematical equation to describe these interactions.Through this extensive data analysis and model fitting, we gain a deeper understanding of hybrid and electric vehicle dynamics. This research not only contributes to the field but also has the potential to drive further development, broadening the focus and scope of hybrid and electric vehicle technology

Characterization, analysis and prediction of damage onset in adhesively bonded joints using fracture mechanics and acoustic emission monitoring technique

This study presents a new method for predicting the fatigue life of aluminum adhesive-bonded joints. The approach involves experimental tests to establish fatigue failure criteria using acoustic emission monitoring to detect damage onset, along with a finite element (FE) model to analyse changes in energy release rate at the embedded crack tip. The proposed method comprises three key steps. In the initial stage, a 3D failure surface criterion is experimentally generated, connecting the maximum total energy release rate (GT) and the mixed mode ratio (GII/GT) to the number of cycles (N) required for initiating the crack propagation. In the second step, the total energy release rate (GT) and the corresponding mixed mode ratio (GII/GT) at the crack tip of a single lap joint under different external loads are numerically determined utilizing the virtual crack closure technique. Mathematical equations linking the applied load (P) to the associated values of GTmax and GII/GT are established. Ultimately, once the energy release rate and mixed mode ratio for a given load are determined, the number of cycles required for initiation of crack growth can be extracted from the experimentally derived failure surface in the initial step. The predictive model shows excellent correlation with experimental data, depending solely on the adhesive system rather than joint design.

Moment of momentum integral analysis of turbulent boundary layers with pressure gradient

Turbulent boundary layers on immersed objects can be significantly altered by the pressure gradients imposed by the flow outside the boundary layer. The interaction of turbulence and pressure gradients can lead to complex phenomena such as relaminarization, history effects and flow separation. The angular momentum integral (AMI) equation (Elnahhas & Johnson, J. Fluid Mech., vol. 940, 2022, A36) is extended and applied to high-fidelity simulation datasets of non-zero pressure gradient turbulent boundary layers. The AMI equation provides an exact mathematical equation for quantifying how turbulence, free-stream pressure gradients and other effects alter the skin friction coefficient relative to a baseline laminar boundary layer solution. The datasets explored include flat-plate boundary layers with nearly constant adverse pressure gradients, a boundary layer over the suction surface of a two-dimensional NACA 4412 airfoil and flow over a two-dimensional Gaussian bump. Application of the AMI equation to these datasets maps out the similarities and differences in how boundary layers interact with favourable and adverse pressure gradients in various scenarios. Further, the fractional contribution of the pressure gradient to skin friction attenuation in adverse-pressure-gradient boundary layers appears in the AMI equation as a new Clauser-like parameter with some advantages for understanding similarities and differences related to upstream history effects. The results highlight the applicability of the integral-based analysis to provide quantitative, interpretable assessments of complex boundary layer physics.

Type-2 fuzzy logic controller-based maximum power point tracking for photovoltaic system

Introduction. Photovoltaic (PV) systems play a crucial role in converting solar energy into electricity, but their efficiency is highly influenced by environmental factors such as irradiance and temperature. To optimize power output, Maximum Power Point Tracking (MPPT) techniques are used. This paper introduces a novel approach utilizing a Type-2 Fuzzy Logic Controller (T2FLC) for MPPT in PV systems. The novelty of the proposed work lies in the development of a T2FLC that offers enhanced adaptability by managing a higher degree of uncertainty, we introduce an original method that calculates the error between the output voltage and a dynamically derived reference voltage, which is obtained using a mathematical equation. This reference voltage adjusts in real-time based on changes in environmental conditions, allowing for more precise and stable MPPT performance. The purpose of this paper is to design and validate the effectiveness of a T2FLC-based MPPT technique for PV systems. This approach seeks to enhance power extraction efficiency in response to dynamic environmental factors such as changing irradiance and temperature. The methods used in this study involve the implementation of T2FLC to adjust the duty cycle of a DC-DC converter for continuous and precise MPPT. The system was simulated under various environmental conditions, comparing the performance of T2FLC against T1FLC. The results show that the T2FLC MPPT system significantly outperforms traditional methods in terms of tracking speed, stability, and power efficiency. T2FLC demonstrated faster convergence to the MPP, reduced oscillations, and higher accuracy in rapidly changing environmental conditions. The findings of this study confirm the practical value of T2FLC logic in improving the efficiency and stability of PV systems, making it a promising solution for enhancing renewable energy technologies. References 33, tables 4, figures 10.

Microcrack feature extraction method for chaotic optical surface of silicon nitride ceramic bearing roller based on multi-scale wavelet transform enhancement and optimized PSO-FCM coupling

To address the issue of chaotic optical gradient feature and non-target interference factors affecting the feature extraction effect of chaotic optical surface microcracks in silicon nitride ceramic bearing rollers, an algorithm for extracting microcrack features from chaotic optical surfaces of silicon nitride ceramic bearing rollers based on multi-scale wavelet transform enhancement and optimized PSO-FCM coupling is proposed. Through the db4 basis function of multi-scale decomposition of microcrack features, the soft threshold function is constructed to deeply denoise the image. The normalized fusion features of microcracks after multi-scale vector decomposition are enhanced, and the gradient information is enhanced while retaining the microcrack features on the chaotic optical surface. The particle swarm optimization mathematical equation with a decay function is built to optimize the FCM chaotic clustering model. The nonlinear decayed particle velocity equation is constructed to update the particle position and iteratively refine the optimal clustering center positions to realize the feature extraction of microcracks. The experiment showed that the reinforcement index RESE and PSNR in the chaotic optical surface microcracks of silicon nitride ceramic bearing rollers reach 15.76 and 24.19, respectively, effectively overcoming the problems of chaotic features in the surface microcracks while retaining the defect features. The segmentation indices Miou, F1 score, accuracy, and recall of the optimal clustering center [231 161] reached 0.912, 0.972 08, 0.998 32, and 0.985 60, respectively, overcoming the influence of non-target interference factors and chaotic optical gradient features to achieve complete feature extraction of surface microcracks.

Flow of SWCNT and MWCNT based hybrid nanofluids in a semi-circular enclosure with corrugated wall

Heat transfer mechanisms participate to fulfill the necessities of modern technologies. The phenomenon of heat transport has implementations in multiple fields including metal working, heating systems, thermal management in spacecraft, solar energy, and automobile engines. In the current novel work, heat transfer mechanism in a semi-circular enclosure with corrugated circular wall is studied numerically. A hybrid nanofluid consisting of water as the base fluid and solid particles of single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs) is considered. Prevailing mathematical equations are explained by finite element method with the help of COMSOL Multiphysics. The study examines various volume fractions of solid particles over a broad range. The SWCNT volume fraction is adjusted from 0.02% to 0.1%, while the MWCNT volume fraction ranged from 0.01% to 0.04%. Velocity, temperature, and pressure contours are imagined. Local and average Nusselt numbers are studied for each of the cases. The results indicate that heat transfer in the semi-circular enclosure improves with higher volume fractions of solid particles. As the volume fraction of solid particles increases, the average Nusselt number over the heated surface decreases.

Technological transformation in the delivery of public services: A study of public governance in the emerging market

The State of Haryana offers more than 650 services/schemes through its unique one-stop SARAL portal. While exploring the SARAL performance, determined using a mathematical equation SARAL portal, it has been observed that the districts’ performance is asymmetrical in the technological impact of governance in the state. To study this research problem, the objective was to trace the factors responsible for this asymmetrical technological transformation in the state. To meet the research objective, the authors selected the Karnal, Jind, and Nuh districts, considering the top, middle, and bottom performers in SARAL score indicators by using a purposive sampling technique. This quantitative research has been analytical and empirical based on the secondary data. The study has found that not only technical but there are other significant factors viz., social, political, administrative, awareness, and economic factors influence the technological transformation in public service delivery. The findings may help the government policymakers in reengineering the technological development policy in the state so as to move towards fulfilling the spirit of Article 38 of the Indian Constitution and Sustainable Development Goal (SDG) 16 prescribed by the United Nations (UN). Further research can supplement this study by collecting primary data from the service providers and the service recipients.

Dissipative disorder analysis of Homann flow of Walters B fluid with the applications of solar thermal energy absorption aspects

Encountering of entropy generation is meaningful while investigating the energy loss during the operational mechanical system. In particular, the flow of fluid experiencing friction drag and due to which a significant amount of heat transfer occurred. Thus, the thermodynamic system energy conversion is one of the measures of the lost available work and is known as irreversibility. Avoiding of such energy loss can be minimized by introducing the concept of hybridization during the liquid dynamics. This work is initiated to formally characterize and address the significance of irreversible process during the typical Homann flow of viscoelastic liquid. The flow with heat and mass balance aspects are further characterize with the inclusion of thermophoretic and Brownian motion factors. The flow configuration is interpreted in terms of gravitationally affected vertical cylindrical disk, for a better understanding of the impact of irreversible processes, more physical effects in terms of heating source/sink, chemical reaction and solar thermal radiation. New physical impacts are described numerically in terms of flow speed temperatures, nanoparticle volume fraction, displacement thicknesses and entropy generation. Perturbation method is utilized for the reduction of the fourth-order mathematical equation for reducing the problem in to well-posed from ill-posed status. The numerical analysis is carried out by applying one of the built-in commands while using MATLAB software. The buoyancy force factor enhanced the liquid speed, and the concentration of the liquid was determined with uplifted conduct for higher values of chemical reaction parameters. The overall entropy rate is reduced as the Brinkman number and magnetic parameter are increased. The heat transfer flow is increased by internal heat generation. Higher Prandtl and Schmidt numbers significantly affected the isotherms and volume fraction contours.

Enhancing synthesized engine oil thermal performance through the use of tri-hybrid nanofluids over a bi-directed sheet with Coriolis force

Synthetic engine oil offers excellent protection against wear, withstands high temperatures, and maintains its viscosity under various operating conditions. It also enhances oxidative stability and reduces sludge buildup, leading to longer engine life and improved performance. To optimize these benefits, incorporating specific nanoparticles into the synthetic oil may be advantageous. This study investigates the emulsification of three types of nanoparticles-aluminum oxide [Formula: see text], gold [Formula: see text], and graphene oxide [Formula: see text]-suspended in synthetic oil. The physical insights gained are translated into mathematical equations that describe fluid motion and thermal sensitivity. This dynamical system is simplified into a set of single-variable differential equations, making the analysis more manageable. The dynamical equations are solved numerically using an efficient Spectral Chebyshev Collocation Method (SCCM). The findings indicate that the presence of nanoparticles significantly increases the rate of heat transfer. The shape of the nanoparticles influences thermal performance; platelet-shaped nanoparticles demonstrate superior thermal properties compared to spherical and cylindrical shapes. For a specific Eckman number [Formula: see text], skin friction increases by 141.6% from mono to hybrid nanofluids, and by 83.9% from hybrid to tri-hybrid nanofluids. Within a thermal Biot number range of [Formula: see text], the Nusselt number is enhanced by 97.6%, 97.0%, and 96.2%, respectively. Furthermore, the Darcy effect shows opposing behavior at primary and secondary velocities, while thermal convection is improved near the wall as the Biot number increases.

An Analysis of High School Students' Mathematical Representation Skills on Relations and Functions

This research was carried out with the aim of analyzing the abilities of high school level students in solving relationship and function problems. This research method is a qualitative method with descriptive research type. This research was carried out at one of Pekanbaru High School with subjects in classes X.1 and X.2 at YLPI Pekanbaru High School, with a total of 6 students. The data collection technique used in this research is a written test technique in the form of a description consisting of 2 questions. After the students have completed the questions, the results of the answers will be analyzed by the researcher as a percentage of the students' results and the researchers will conduct interviews with the students regarding the answers they made. Based on the research results, the percentage of the "Visual Representation" indicator was 79.16%, which is included in the high category. Then the indicator "Mathematical Equations or Expressions" was 66.66%, which was included in the high category. Finally, the "Words or Written Text" indicator is 50%, which is in the medium category. Furthermore, the percentage of questions for mathematical representation ability is 80% for question number 1, in the high category and 88.33% for question number 2, in the very high category. So a conclusion can be drawn from the score obtained by the students, namely 84.16%, which is included in the very high category. It is hoped that the results of this research will provide maximum guidance for teachers to develop students' abilities in mathematics, especially relation and function material. Keywords: Analysis of Student Abilities, Mathematical Representations, Relations and Functions

A computational study on the sensitivity of slip constraints in the blade coating process of an electrically conducting oldroyd 4-constant fluid

The blade coating process holds significant importance due to its widespread application in manufacturing products such as newspapers, photographic film, fibers, catalogs, and magnetic storage media. Its economic impact has driven extensive research aimed at deepening the understanding of the underlying physical mechanisms, ultimately leading to improvements in process efficiency and optimization. Therefore, this article investigates the blade coating process of an electrically conducting Oldroyd 4-constant liquid with velocity slippage on the blade surface. The impression of viscous dissipation is also inspected through the energy equation. The mathematical equations are modeled with the use of lubrication approximation theory and the normalized equations of the Oldroyd 4-constant fluid and are numerically solved by the shooting method. To offer valuable insights, the pressure, pressure gradient, velocity, temperature, and load metrics are calculated and displayed in graphs and tables. It is found that liquid velocity and pressure decrease as the Hartmann number increases. As the Brinkman number increases, the temperature distribution increases, with the peak temperature appearing in the narrowest region of the flow.

ANALISA DAN PERANCANGAN SISTEM INFORMASI ENKRIPSI DATA DENGAN ALGORITMA VIGENERE CHIPER BERBASIS WEB

Cryptography is a field of knowledge that uses mathematical equations to perform encryption and decryption processes. This technique is used to convert data into certain codes, with the aim that the stored information cannot be read by anyone except those who have the right. In this study, a cryptography learning application was designed that implements encryption and decryption methods using the Vigenere cipher method. The results of the study are a cryptography learning application using the Vigenere cipher method that can encode secret messages and also help increase the understanding of users who want to learn about cryptography, especially the Vigenere cipher method.

Dynamic stochastic model of allometric equations and cumulative distribution for biomass-carbon in Pinus hartwegii Lindl., facing climate change

Objective: to construct a dynamic stochastic model with validated biospheric-interaction, estimating allometric equations for the total volumetric increase and cumulative distribution of biomass for Pinus hartwegii Lindl in the states of Mexico and Puebla, considering climate change. Design/ Methodology/ Approach: the methodology included the use of SiBiFor numerical databases, NASA Power data, Ordinary Least Squares mathematical models, the Random-Forest software, Ridge model with regression, R algorithms and Newton volumetric estimation equations. Results: estimated allometric equations were obtained for the total volume of trees in 2023, highlighting the importance of linear regression models and the validity of the variables used. Newton's mathematical equations and theoretical models for excurrent tree-form types were found to have the best accuracy in estimating the total volume of the barked tree. Limitations of the study/ Implications: this study has limitations in terms of generalization to other forest types and the availability of data in Mexico. However, it highlights the importance of understanding forest responses to environmental changes and the need for validated dynamic stochastic models to estimate allometric equations and assess carbon sequestration. Findings/ Conclusions: this study highlights the importance of understanding and assessing the carbon storage capacity of forests, especially in the context of climate change. In addition, it underlines the usefulness of linear regression models and variable validation to estimate carbon sequestration in Pinus hartwegii forests.