Mathematical Expressions Research Articles

Graphical User Interface for Handwritten Mathematical Expression Employing RNN-based Encoder-decoder Model

Aim: Scientific, technical, and educational research domains all heavily rely on handwritten mathematical expressions. The extensive use of online handwritten mathematical expression recognition is a consequence of the availability of strong computational touchscreen appliances, such as the recent development of deep neural networks as superior sequence recognition models. Background: Further investigation and enhancement of these technologies are vital to tackle the contemporary obstacles presented by the widespread adoption of remote learning and work arrangements as a result of the global health crisis. Objective: Handwritten document processing has gained more attention in the last ten years due to notable developments in deep neural network-based computer vision models and sequence recognition, as well as the widespread proliferation of touch and pen-enabled smartphones and tablets. It comes naturally to people to write by hand in daily interactions. Method: In this patent article, authors implemented Hand written expressions using RNNbased encoder for the CROHME dataset. Later, the proposed model was validated using CNNbased encoder and End-to-end encoder decoder techniques. The proposed model is also validated on other datasets. Results: The RNN-based encoder model yields 82.78%, while the CNN-based encoder model and end-to-end encoder-decoder technique yield 81.38% and 80.73%, respectively. Conclusion: 1.6% accuracy improvement was attained over CNN-based encoder while 2.4% accuracy improvement over end-to-end encoder-decoder. CROHME dataset 2019 version results in better accuracy than other datasets.

Vision-language pre-training for graph-based handwritten mathematical expression recognition

Vision-language pre-training for graph-based handwritten mathematical expression recognition

Dipole-induced transitions from Schottky to Ohmic contact at Janus MoSiGeN4/metal interfaces.

Janus MoSiGeN4 monolayers exhibit exceptional mechanical stability and high electron mobility, which make them a promising channel candidate for field-effect transistors (FETs). However, the high Schottky barrier at the contact interface would limit the carrier injection efficiency and degrade device performance. Herein, using density functional theory calculations and machine learning methods, we investigated the interfacial properties of the Janus MoSiGeN4 monolayer and metal electrode contacts. The results demonstrated that the n-type/p-type Schottky and n-type Ohmic contacts can be realized in metal/MoSiGeN4 by changing the built-in electric dipole orientation of MoSiGeN4. Specifically, the contact type of Cu/MoSiGeN4 (Au/MoSiGeN4) transfers from an n-type Schottky (p-type Schottky) contact to an n-type Ohmic (n-type Schottky) contact when the contact side of MoSiGeN4 switches from Si-N to Ge-N. In addition, the Fermi level pinning (FLP) effect of metal/MoSiGeN4 with the Si-N side is weaker than that of metal/MoSiGeN4 with the Ge-N side due to the effect of intrinsic dipole and interface dipole. Notably, a simplified mathematical expression ΔV/WM is developed to describe the Schottky barrier height at metal/MoSiGeN4 interfaces using the machine learning method. These findings offer valuable guidance for the design and development of high-performance Janus MoSiGeN4-based electronic devices.

Dissipation of Current Carriers in Lead Telluride Films

The mechanisms of current carrier scattering in polycrystalline lead telluride (PbTe) films deposited on glass substrates in the temperature range 77-300 K have been studied. The main attention is paid to scattering at the grain-to-grain limits, which is the dominant mechanism affecting the mobility of current carriers. In this work, a mathematical expression is obtained for calculating the mobility of current carriers in thin films, considering scattering on the surface and intergrain limits, which makes it possible to evaluate the influence of these mechanisms on the mobility of current carriers. The dependence of current carrier mobility in PbTe films on thickness and temperature has been studied. It has been established that the dominant scattering mechanism is scattering at grain boundaries. The activation energy of electrical conductivity was estimated, which is approximately 0.04-0.07 eV.

ANALYSIS OF OUTAGE PROBABILITY AND INTERCEPT PROBABILITY TRADE-OFF FOR SECURE TWO-WAY RELAYING SCHEMES BETWEEN TWO CLUSTERS OF NODES USING FOUNTAIN CODES

This paper presents two secure two-way relaying schemes between two clusters of nodes: i) 3-phase two-way relaying using Digital Network Coding; ii) conventional 4-phase two-way relaying. In the proposed schemes, one node selected from each cluster will use Fountain codes to encode the original data, and then send encoded packets to nodes in the remaining cluster, with the help of a common relay node. Explicit mathematical expressions are derived to evaluate the outage probability (system outage probability) and intercept probability (system intercept probability) of the proposed schemes over Rayleigh fading channel. Our results show a reliability-security trade-off as well as the impact of system parameters on the network performances.

Extraction of parameters of a stochastic integrate-and-fire model with adaptation from voltage recordings

Integrate-and-fire models are an important class of phenomenological neuronal models that are frequently used in computational studies of single neural activity, population activity, and recurrent neural networks. If these models are used to understand and interpret electrophysiological data, it is important to reliably estimate the values of the model’s parameters. However, there are no standard methods for the parameter estimation of Integrate-and-fire models. Here, we identify the model parameters of an adaptive integrate-and-fire neuron with temporally correlated noise by analyzing membrane potential and spike trains in response to a current step. Explicit formulas for the parameters are analytically derived by stationary and time-dependent ensemble averaging of the model dynamics. Specifically, we give mathematical expressions for the adaptation time constant, the adaptation strength, the membrane time constant, and the mean constant input current. These theoretical predictions are validated by numerical simulations for a broad range of system parameters. Importantly, we demonstrate that parameters can be extracted by using only a modest number of trials. This is particularly encouraging, as the number of trials in experimental settings is often limited. Hence, our formulas may be useful for the extraction of effective parameters from neurophysiological data obtained from standard current-step experiments.

Steady-State Analysis of Asymmetrical Pulse-Width-Modulated Series Resonant Converter for Light Load Condition

This paper proposes a steady-state analysis of the asymmetrical pulse-width-modulated series resonant converter, commonly employed under light load conditions for effective voltage regulation. The proposed method achieves precise and explicit converter waveforms by applying the Laplace-based theorem to the converter’s ordinary differential equation with periodic and discontinuous inputs, without relying on approximations. By numerically determining the intermediate variables that define converter waveforms, the analysis provides accurate steady-state results based on system parameters. Furthermore, it derives an analytical solution to identify the load conditions suitable for asymmetrical pulse-width modulation operation, as well as the boundary load conditions for discontinuous conduction mode within this mode. The mathematical expressions derived for the converter waveforms and operational modes are validated through simulations using a switching model in PSIM software, as well as through experimental results.

Identifying determinants of γ’ phase coarsening behavior in Co/CoNi-based superalloys with explainable artificial intelligence (XAI)

The coarsening of γ’ phases in superalloys is influenced by multiple factors and significantly impacts mechanical properties, such as strength and creep resistance. While classical kinetic theory delineates the coarsening mechanisms of γ’ phases, it has limitations when excessive coarsening occurs. The challenge of identifying the pivotal factors that markedly affect the kinetic behavior of γ’ phase coarsening remains largely unresolved. This study utilized eXtreme Gradient Boosting (XGBoost) models to identify the CP and CPAE features that characterize the coarsening behavior of γ’ phases in Co/CoNi-based superalloys. Through meticulous analysis of CP features [aging temperature (T ), the atomic fraction of the Ti element (c Ti), the atomic fraction of the V element (c V), and the atomic fraction of the Ta element (c Ta)] and CPAE features [T , Young’s modulus (E ), electronegativity (EN ), atomic radius (r ), c Ti, the differences of atomic radius (σr ), and the differences of electronegativity (σEN )], we derived explicit mathematical expressions using symbolic classification approach, which offers improved interpretability compared to traditional black-box models. Our findings indicate that T , c Ti, c Ta, E and σr are dominant factors influencing γ’ coarsening behavior across different Co/CoNi-based superalloys. Notably, T , c Ti, c Ta and σr exhibit an inverse correlation with the γ’ coarsening behavior, whereas E shows a positive correlation. These insights provide valuable guidance for determining whether coarsening occurs in newly designed alloys. Furthermore, they facilitate the design of innovative anti-coarsening Co/CoNi-based superalloys. This knowledge is instrumental in enhancing alloy formulations, ensuring their performance and longevity in demanding environments where thermal stability is paramount.

Analysis of Students' Mathematical Communication Skills Through Culturally Responsive Teaching and Differentiated Learning Based on Local Wisdom

This study aims to analyze students' mathematical communication skills in solving problems related to measures of central tendency in real-life contexts, focusing on the aspects of drawing, mathematical expression, and written text, as well as types of Newman’s errors. The analysis involved 25 students, with three selected as representatives based on their mathematical communication skill levels. The findings indicate that, on average, students exhibited a low level of errors across the three aspects, with average scores of 7.5 for drawing, 8.0 for mathematical expression, and 8.5 for written text. In the drawing aspect, students were able to present solutions visually well, although minor inaccuracies were noted. In the mathematical expression aspect, students understood the problems but tended to omit explicit solution steps. For the written text aspect, students often provided numerical results without formulating conclusions in sentence form. The main factors causing errors were a lack of habit in writing solution steps and conclusions, as well as a need for improved accuracy. A differentiated learning approach based on local wisdom positively contributed to reducing student error rates. However, strengthening practice in writing solution steps and composing conclusions is necessary to improve students' overall mathematical communication skills.

Testing Bench for Reciprocating Hydraulic Cylinders with Energy Recovery: Structure, Simulation, and Calculation

Introduction. Energy conservation is an urgent topic of research in the field of mechanical engineering all over the world. A particular direction in these studies is the search for energy-efficient methods of testing technical systems, which allow the most accurate prediction of the reliability of the designed equipment. The duration of resource tests established by technical conditions and GOSTs causes an irreversible loss of energy, amounting to more than 1.5 resource of the tested machine, and the energy lost in the form of “harmful heat” is released into the environment. Therefore, the problem of energy saving during testing is given special attention. One of the ways of energy saving under testing of hydraulic machines is energy recovery. However, in papers devoted to energy recovery during testing of hydraulic machines, the problems of energy recovery during testing of rotary hydraulic machines were solved, and for reciprocating hydraulic machines, the problem of energy recovery during testing of plunger hydraulic cylinders was solved. The results of these studies cannot be directly used for testing reciprocating hydraulic cylinders. In this regard, the objective of this work has been formulated — to develop the structure and basic scheme of a test bench for reciprocating hydraulic cylinders, providing the recovery of part of the energy spent on testing, due to which the energy efficiency of the testing process is significantly increased.Materials and Methods. The paper used methods for modeling the process of stand functioning based on the application of the theory of volumetric rigidity. To carry out preliminary calculations of the stand operation process, a computer program based on the SimInTech software package was developed.Results. A structural and schematic diagram of a test bench for reciprocating hydraulic cylinders has been developed. A mathematical expression has been obtained that provides a preliminary assessment of the test efficiency coefficient. A computer program based on the SimInTech software package has been created, which allows the design parameters of the stand affect its operational characteristics, including the coefficient of energy efficiency of the test process.Discussion and Conclusion. The preliminary calculations of the operating characteristics of the stand showed that the efficiency coefficient of the proposed stand was 1.7. It can be increased by conducting additional studies aimed at obtaining rational design parameters of the stand. The proposed stand provides testing of reciprocating hydraulic cylinders with recovery of part of the expended energy, and its mathematical model allows using the numerical methods in calculations. This significantly simplifies the calculation process and increases the accuracy of the calculations. At the same time, it is possible to obtain rational parameters of the stand already at the design stage, without resorting to expensive and labor-intensive field studies.

A springback minimization study using the experimental design methods for 15-5 PH stainless steel material in the hydroforming process

Purpose This study aims to examine the springback behavior of 15-5 PH stainless steel, which is widely used in the aviation industry. Design/methodology/approach The shaping sheet metal with a diaphragm is a widely used shaping method in aviation. In this process, the sheet metal is attached to the mold, placed in the form machine and pressure is applied. The diaphragm, which swells under the influence of gas, transmits pressure to the sheet metal surface and allows the mold to take its shape. One of the behaviors that occurs in the part after processing is springback. The mold design is shaped according to this springing amount. Some prominent parameters in the amount of springback are bending radius, bending angle, material thickness and material type. Material thickness is taken as 0.813, 1.27, 1.6 and 2 mm for experimental studies. This study compares the general full factorial design with the 2k factorial design. Then, the authors get analysis of variance (ANOVA) tables for the experiments and finite element analysis results, respectively. Findings At the end of the study, a mathematical expression was obtained that gives the springback rate with an accuracy of over 90%, depending on the process parameters, in 15-5 PH material. In addition, the main effects of the working parameters on the springback behavior were revealed by ANOVA method. Originality/value To the best of the authors’ knowledge, this is a new application that provides robust results to the springback minimization for the sheet metal bending process.

A Navigation Algorithm Based on the Reinforcement Learning Reward System and Optimised with Genetic Algorithm

Regarding autonomous vehicle navigation, reinforcement learning is a technique that has demonstrated significant results. Nevertheless, it is a technique with a high number of parameters that need to be optimised without prior information, and correctly performing this is a complicated task. In this research study, a system based on the principles of reinforcement learning, specifically on the concept of rewards, is presented. A mathematical expression was proposed to control the vehicle’s direction based on its position, the obstacles in the environment and the destination. In this equation proposal, there was only one unknown parameter that regulated the degree of the action to be taken, and this was optimised through the genetic algorithm. In this way, a less computationally expensive navigation algorithm was presented, as it avoided the use of neural networks. The controller’s time to obtain the navigation instructions was around 6.201·10−4 s. This algorithm is an efficient and accurate system which manages not to collide with obstacles and to reach the destination from any position. Moreover, in most cases, it has been found that the proposed navigations are also optimal.

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

Underwater Laguerre-Gaussian beam propagation and phase compensation method

Laguerre-Gaussian (LG) beams experience phase twist after long-distance transmission, making the orbital angular momentum (OAM) indistinguishable. This phenomenon becomes more severe in water due to the higher refractive index. Based on the physical principles of LG beams, this paper derives the mathematical expression for LG beam transmission in water to address this issue. It organizes and analyses the physical significance of each term. The exponential term responsible for phase twist is separated and phase compensation is applied to the initial LG beam. Simulation results show that after transmission through water, traditional LG beams exhibit a clockwise twisted distribution of isophase lines. By applying the phase compensation method proposed in this paper, the phase of the initial LG beam is modulated and when the LG beam reaches the observation plane, the isophase lines become straight, verifying the effectiveness of the method. This compensation method holds significant value for LG beams in advanced physics research.

Deep Symbolic Regression-Based MPPT Control for a Standalone DC Microgrid System

Enhancing the performance of maximum power point tracking (MPPT) methods is essential for optimizing the operation of solar systems under any weather condition. This improvement is crucial because MPPT methods serve as the controllers for power converters, directly influencing the ability to maintain a steady and efficient energy output from solar installations. Therefore, we propose a deep symbolic regression (DSR)-based MPPT method in standalone direct current microgrids, which is particularly useful under drastically shifted meteorological conditions. This study aimed to improve the significant performance parameters of the MPPT approach, such as the rate of convergence during transient periods, tracking accuracy, steady-state stability, and overshoot/undershoot reduction, creating more reliable and efficient solar microgrid systems. To evaluate the proposed method, performance comparisons were made with those of existing advanced algorithms, namely particle swarm optimization-, artificial neural network-, and adaptive neuro-fuzzy inference system-based MPPT methods. The comparative results demonstrate a significant improvement with the proposed method. Moreover, the simple mathematical expression obtained by the DSR process minimizes the computational complexity of the control.

Experimental study on the motion characteristics and critical hydraulic parameters of microplastics in a freshwater environment.

The migration behavior of microplastics in water is affected by many factors; in particular, the migration mechanism of microplastics in the terrestrial freshwater environment is more complicated than that in the marine environment. In order to understand the migration behavior of microplastics in the freshwater environment, the hydraulic parameter thresholds of different types of microplastics in water were identified based on hydraulic experiments and force analysis methods. The results show that the motion state of microplastics is affected by their own internal factors and external environmental factors, and the flow rate is the key external factor affecting the change of their motion state. In the vertical direction, the higher the density, the rougher the environment, and the closer the shape to the flake, the greater the critical starting flow velocity and the critical resuspension flow velocity. The settling velocities, critical initiation velocities, and critical resuspension velocities of microplastics range from 0.05 to 0.17 m s-1, 0.03 to 0.44 m s-1, and 0.251 to 0.83 m s-1, respectively. Horizontally, the bottom rolling velocities of microplastics vary significantly. These velocities are positively correlated with water flow velocity but are inversely proportional to the density of the microplastics and the roughness of the substrate. By combining experimental data, mathematical expressions for the critical hydraulic parameters of microplastics were derived, showing improved accuracy compared to traditional methods. This paper explores the trajectory of different types of microplastics after entering the water body and analyzes their migration mechanism in the river. The research results have certain theoretical guiding significance for revealing the migration law of microplastics in the freshwater environment.

Explanation of the principles of the Booth algorithm, Verilog implementation and simulation

In an era marked by heightened constraints on computer processing speed and storage capacity, optimizing computational efficiency and space utilization assumes paramount importance. Consequently, the deployment of proficient algorithmic tools holds significant significance within the realm of computing. The principal categories of multipliers encompass conventional multipliers, shift-and-add multipliers, Look-Up Table (LUT) multipliers, and Booth algorithm multipliers. Among these, the Booth algorithm substantially influences multiplier performance enhancement. This paper investigation commences with a comprehensive elucidation of the mathematical underpinnings of the Booth algorithm. Subsequently, it involves the development of code corresponding to the mathematical expressions, which is executed within the VIVADO environment. Given the inherent simplicity of the experiment, virtual simulation is directly conducted in VIVADO. The analysis is grounded in the examination of data waveforms. Ultimately, the output waveform is correlated with the BOOTH mathematical formula, culminating in the successful implementation of the BOOTH algorithm. This paper aims to effectively curtail the presence of '1's in computer data, thereby reducing the computational steps and conserving operational space. This, in turn, leads to a notable acceleration in the computational speed, accomplishing the objective of enhancing computational efficiency.

Arithmetic optimization algorithm with three-dimensional chaotic mapping in spherical coordinate system for combined economic emission dispatch problem

The objective of the Combined Economic Emissions Dispatch (CEED) problem is to reduce pollutant emissions by lowering the total cost of generating electricity while complying with all other constraints. The multiple objective functions of the CEED problem may be converted into a single objective by using a price penalty factor. Then it was solved with the Arithmetic Optimization Algorithm (AOA) with three-dimensional chaotic mapping in spherical coordinate system. Firstly, five three-dimensional chaotic mappings in Cartesian coordinate system are embedded in the Mathematical Optimization Accelerator (MOA) and the Mathematical Optimization Probability (MOP) of the original algorithm. Secondly, 15 three-dimensional chaotic mappings in spherical coordinate system are constructed from the values of 5 three-dimensional chaotic mappings in Cartesian coordinate system through the mathematical expressions of mode length, polar angle and azimuth angle in spherical coordinate system. Then through a large number of simulation experiments, five best three-dimensional chaotic mappings in the spherical coordinate system are selected to be embedded into the two parameters (MOA and MOP) of the original algorithm to better balance the algorithm's global and local searching ability. The superiority of the improved algorithm is verified by employing 12 benchmark test functions in CEC2022. Eventually, the improved optimal algorithm (IAOA-r) to solve the power system CEED problem is tested on six generating units with four different loads (150 MW, 175 MW, 200 MW and 225 MW). The results of the improved algorithm are compared and analyzed with the results of the Reptile Search Algorithm (RSA), Prairie Dog Optimization (PDO), Bat Algorithm (BAT), Whale Optimization Algorithm (WOA), Harris Hawk Optimization (HHO), Rat Swarm Optimization (RSO). The results demonstrate that the improved algorithm is capable of obtaining optimal fuel costs and smaller pollution emissions in every test case.

Marine propulsion shafting-oriented status detection algorithm for uncertainty time-varying

The marine propulsion shafting is the lifeline of navigation. The marine propulsion shafting vibration signals tend to contain rich connotations that suggest the shafting status. In this paper, aiming at the difficulty in the capture of time dependence in uncertainty time-varying signals of marine propulsion shafting, the concept of Degree of Membership (DM) in fuzzy mathematics is introduced into Recurrent Neural Network (RNN), called Degree of Membership Recurrent Neural Network (DM-RNN). A marine propulsion shafting status detection algorithm based on DM-RNN is proposed. Specifically, the mathematical expressions for forward propagation and back propagation of DM-RNN are theoretically derived. The self-propulsion test scheme based on a full-scale boat is designed. Actual vibration signals under multiple working conditions are collected. The progressiveness of the present algorithm under shafting vibration signals of the full-scale boat is verified by the experiment. The results of the two comparison experiments indicate that the present algorithm significantly improved the detection accuracy compared with the classical ANNs, up to 93.25% ± 0.71%. Furthermore, a DM function with a simple form and fewer parameters was more conducive to the high detection accuracy of DM-RNN.

Spiritual Function: Mathematical Expression and Application of Spiritual Activity’s Process

Spiritual Function: Mathematical Expression and Application of Spiritual Activity’s Process