The aim of this research paper is to provide a comprehensive comparison between Riemann integral and Lebesgue integral. Integration is described as the inverse process of differentiation, which is used to determine the original function. Riemann integration is a specific type of definite integral applied to find the exact area under a function graph between two limits in a closed interval. Lebesgue integration, on the other hand, provides a more generalized framework for integration theory. Integrals are essential in mathematical modelling and analysis tools. Studying and comparing the similarities and differences between these two integration methods can help us better understand the essence and properties of integrals, so as to more accurately apply integration methods to solve practical problems. This paper provides a systematic analysis of the basics, definitions, concepts, and properties of Riemann integral and Lebesgue integral. Reasoning, proofs, and examples are consolidated to explain the properties and characteristics of these two integral methods. Finally, this paper explores the strengths and limitations of each integration methods, summarising their advantages and applicability to practical problems and providing insights into their respective computational methods and applicability in different contexts.

Every morning rush hour is the most crowded time for elevators in office buildings, and many office workers will be late because they can't wait for the elevator. How to reasonably call the existing elevator resources has become a problem we need to pay attention to. In this paper, we will discuss the modeling of elevator scheduling optimization problem for office buildings. In this paper, the background and significance of the research is firstly explained, and then the elevator group linkage control system is briefly understood and analyzed, and the evaluation indexes are determined according to its multi-objective and diversified characteristics, and then the group control method focusing on making the elevator run time the shortest is adopted, and the principle of the ratio of elevator running period/total running time equal to the number of passengers transported/total passengers transported by the elevator in one cycle is followed to compare several common elevator running methods, and the results are summarized in the following table. Common elevator operation methods are compared, the most effective way to arrive at, and finally use dynamic programming to solve the partition point, to establish the optimal scheme of elevator scheduling for the office building division. Since it is not possible to expand the elevator in the office building, reducing the number of stops of each elevator to shorten the time required for the elevator to carry passengers becomes the most feasible method. After comparing the common elevator operation methods, it can be found that zoning can shorten the final average arrival time of passengers more than no zoning. That is, the elevators in an office building are divided into different zones, and each zone has a designated elevator serving a designated floor segment. In order to solve the floor segment allocation problem, the shortest path algorithm is used in this paper. The paper concludes with a summary and future outlook of the resulting optimization model.

This study investigates the synchronization phenomenon inherent to mechanical metronomes. Synchronization properties were probed under diverse conditions, with modifications to parameters such as the number of metronomes, the tabletop material, the additional weight on the platform, and the friction coefficient between the roller and platform. Experimental outcomes demonstrate that the synchronization is significantly influenced by the platform's movement, which is subsequently determined by the properties of both the platform and roller. An analytical modeling approach provided insights into these effects, particularly when analyzing two metronomes placed on a flat platform equipped with rollers. Central to the findings is the revelation that both the coefficient of friction and the mass of supplementary weights play crucial roles in shaping synchronization behavior, a conclusion in line with experimental observations. The numerical segment of the study leveraged the Kuramoto model. Observations centered on the phase variation of resonators and a time-responsive order parameter, shedding light on the role of coupling strength in synchronization and its absence. Overall, this study furnishes a thorough theoretical and experimental exploration of mechanical metronome synchronization, targeting an enhanced understanding and application of the observed phenomenon.

Fractal geometry plays a crucial role in the study of fractals, with wide-ranging applications and it is closely related to architectural design creation, notably in architecture. Fractals are becoming more prevalent in today’s society and can be used to describe a variety of extremely irregular items in the actual world. This article aims to provide an overview and analysis of the significant implications of fractal geometry in architecture, exploring its relationship with elements in nature, such as trees. By searching, organizing data, and elucidating examples, the article delves into the origin and basic background of fractal geometry. The use of fractal geometry in architecture demonstrates the perfect symbiosis between fractal and nature through analysis of TOD, the ancient gardens in Suzhou, and other practical applications. One notable fractal algorithm is the L-system, a modeling approach expressing fractal concepts, enabling parameterization and architectural design. Fractal geometry successfully expands possibilities for the harmonious coexistence of architecture and nature, embodying numerous natural features and human-centric ideas.

Let φ denote Euler’s Totient function. There are some properties about φ(n), when n is a prime or n=p_1^(r_1 )⋯p_k^(r^k ). The Euler’s function equation, kφ(n)=n-1(1), where k is a positive integer, and n is a composite number, is called Lehmer’s conjecture. Lehmer mentioned a series of properties of n that satisfy the equation in his own thesis and provided some proof. Afterwards, Ke Zhao and Sun Qi conducted further research. In previous studies, this conjecture was considered correct, but it is difficult to prove it. The case k=2 has been discussed and proved that when k=2 and n=p_1 p_2,... p_i are different prime numbers. Also, some properties of the composite numbers that satisfy the equation have also been proven. Some conclusions can be proven, by using elementary number theory methods. Using these conclusions, we can conclue that when k=2, the solution of (1) is at least the product of 12 odd prime numbers.

A major impetus in the early development of group theory in the 19th century was the study of geometrical symmetries. Inspired by the theory regarding orbifold notation developed by John Conway, we understand and analyze the theory of the classification of symmetrical patterns in three different spaces. One of the triumphs was the full understanding of 2-dimensional planar symmetries, precisely, the classification theorem of wallpaper groups. Then we use the same way to classify the symmetrical patterns in spherical and hyperbolic spaces. There were also scattered theories on the general notion called crystallographic groups. In this paper, we reproduce the classical result that there are 17 types of wallpaper groups using a topological method; we also conclude that there are 14 family cases in the spherical space and infinite conditions in the hyperbolic spaces. During this process, we first consider the three different spaces, then, analyze the symmetrical patterns case by case. The characteristic we consider to classify these patterns is orbifold; finally, we use the formulas to calculate the result.

This paper explores the necessary conditions for extremum in both constrained and unconstrained problems by extracting fundamental principles of constraint conditions. We provide a precise geometric understanding of the Lagrange Multiplier, prioritizing analytical insight. Beginning with a geometric interpretation of the gradient, we leverage the expansion of functions and their images to comprehend extremum and detail the Lagrangian derivation process.We expand the base vectors of the constraint surface into those of the full space and use a transition matrix to assess the function's extremum. This demonstrates how the second derivative matrix is transformed into its full-space representation to discern extrema in optimization problems. Additionally, we introduce incremental variables to optimize the second-order derivative matrix in full space, providing a novel perspective to solve extremal necessary conditions.

Utilizing the powerful feature extraction capabilities of deep learning, a text classification algorithm with multi-dimensional and high-domain adaptability is designed in this study. This method enhances the model’s understanding of topics and content by incorporating the Polarized Self-Attention (PSA) module, which strengthens the spatial structure and semantic features of textual information. The loss function is redesigned to assign smaller losses to misclassifications of neighboring categories, allowing the model to optimize classification accuracy while learning hierarchical structural information between categories. Finally, experimental verification is conducted on a publicly available news dataset, demonstrating improved results in text classification achieved by the proposed algorithm.

Microchannel heat sink is a potential procedure for scatter heat as a result of its little size and high effectiveness. Thermal resistance and pumping power are two significant parameters to assess the behavior of microchannel heat sink. In this review, heat transfer and fluid behavior have been concentrated mathematically in a microchannel heat sink with the counter stream and different shape of the groove. Four unique cases are discussed in this paper, microchannel heat sink with rectangular groove (R-MCHS), microchannel heat sink with circular groove (C-MCHS), microchannel heat sink with triangular groove (T-MCHS), and straight microchannel heat sink (S-MCHS). Three-dimensional numerical simulation is conducted with the Reynolds number, Re, shifting from 100 to 500 with expanding 100 for each step. As the coolant, Single-stage fluid water streams in the heat sink. The thermal and hydraulic performance of these cases are evaluated by comparing the average temperature of the solid domain, Nusselt number, Thermal Resistance, and Pumping Power. Contrasted with the S-MCHS, the thermal performance of MCHSs with grooves is increased, and the Nusselt number increased at the furrow area. Likewise, the pumping power is bigger for all cases with grooves contrasted and S-MCHS. In all grooved cases, R-MCHS has the lowest average temperature on the bottom surface and the highest average temperature in the solid domain. The C-MCHS has the least thermal resistance for all Re between 100 to 500. In light of the component, the plan standards to move toward the presentation of inclination are proposed.

Health concerns have become a significant focus in people’s daily lives. Currently, with the increasing demand for monitoring human health, some social organizations need to strengthen health detection technology, and various related technologies are emerging. The aim of this study is to develop a social organization disease prevention and assistance system that integrates the advantages and resources of social organizations, social service institutions, social workers, caregivers, service recipients, and their families. The main system is designed based on a front-end and back-end separation architecture, using core technology: the Lorenz-RR scatterplot classification algorithm to achieve the selection of classification algorithms, the development of AlexNet algorithm, and the optimization of dataset expansion algorithm, Thus, the improvement of the Lorenz-RR scatterplot classification algorithm of the AlexNet model is achieved, and the development of a social organization disease prevention and assistance system is completed. It has high accuracy and sensitivity in the analysis of relevant indicators of service objects, and its application value is significant, with widespread promotion significance.

Engines, the mechanical workhorses powering modern societies, have a rich historical evolution, the Industrial Revolution marks a turning point, James Watt’s improvements, the rise of internal combustion engines, first with Nikolaus Otto’s four-stroke cycle, later Rudolf Diesel’s compression-ignition engine. These engines fueled the automotive and aviation revolutions. In contrast, the Stirling engine, patented by Robert Stirling in 1816, offered a unique closed-cycle operation. Engines, from steam to internal combustion, continue to underpin technological advancements, shaping economies, industries, and daily life. This paper comprehensively analyzes the development and significance of four major engine types: the steam engine, the internal combustion engine, the Stirling engine, and the turbojet engine. The analysis encompasses various aspects, including their principles of operation, historical contexts, and practical applications. The paper concludes that these engines have played pivotal roles in shaping human history and technological progress. From the steam engine’s impact on industrialization to the internal combustion engine’s revolution of transportation, the Stirling engine’s potential for sustainable power generation, and the turbojet engine’s transformation of aviation and military capabilities, each engine type has made a unique and vital contribution to our world’s advancement. This narrative of engine evolution reflects human ingenuity and our ceaseless pursuit of technological innovation.

The COVID-19 pandemic has severely affected numerous individuals’ lives and inevitably lowered their happiness levels. In the process of recovering from the effects of the pandemic, a series of measures must be taken to enhance people’s happiness as soon as possible. The research used multiple linear regression and random forest to analyze the data from World Happiness Report 2023 in order to identify some effective predictive factors of happiness, and made recommendations to policymakers. The result of this study shows social support and GDP are the main indexes policymakers should put effort into improving. Governments can enhance the level of social support not only from the classical perspective but also from improving health care. Reformations of the health care system are necessary if current contingency measures are not sufficient for such severe situations as COVID-19. Policies on economics are continuously beneficial to a population’s overall well-being and thus are important for policymakers to work on.

Since the construction of the first thermal power plant in 1875, mankind has achieved great success in the field of energy. Today, the pursuit of clean energy is also a major breakthrough. Through examples and comparisons, this paper classifies human energy by means of power generation, and expounds the evolution, history and development experience of energy. Mankind is now entering the era of nuclear energy on a large scale, which is an important node in the history of power generation. This article summarises the past, reviews the era of nuclear energy, and discusses the future development of energy. In the history of energy evolution, from steam power to clean energy, mankind has experienced constant innovation and progress. The advent of the nuclear energy era has provided mankind with more efficient and sustainable energy choices, and also led the development direction of future energy.

There are lots of differences between traditional engines and alternatives. For example, most of petrol car and electric vehicles have lots of differences. The author will talk about the difference between spark-ignition engine and electric motor (advantages and disadvantages) briefly, and this essay will explain them in a few parts carefully. People will consider of the efficiency of cars, fuel source and other different properties that spark-ignition engine and electric motor have. In efficiency, the author will talk about how to calculate the efficiency of Spark-ignition engine and how to calculate the efficiency of electric motor. in fuel source and emission, the author also will analysis the benefits that electric can do and the Spark-ignition engine cannot do, the environmental problems health problems and resource limited are all important. This essay will explain the difference between Spark-ignition engine and electric motor deeper in the following page.

Horse racing is very popular in East Asia, especially in Hong Kong and Japan. But whether there is a specific way to win in this kind of gambling game is something that many people are curious about. There are many statistical analyzes in the field of mathematics that deal specifically with various forms of gambling. Many researchers also hope to use mathematics to calculate the winning method of horse racing. The aim of this article is to obtain, through mathematical analysis, the correlation between the effect of the weight of the horse on the outcome of a race in a horse race, under the influence of different distances. In the article, purely linear relationships and Bernoulli analysis were used to determine the results, and then Bernoulli analysis was used to obtain the relationship between the weight of the horse and the final result. In the plotting process, I used python’s PyMC3 library to help construct the mathematical model, and jupyter notebook to make the data more intuitive to show.

At present, robots show great application value in various industries, and the progress of the industry constantly puts forward higher requirements for the performance of robots, and robot control is an important part of robot application. It is difficult for the traditional PID controller to implement online tuning when it is faced with actual objects. Consequently, the neural network algorithm has been included into robot control methods in recent years; the PID controller based on BP neural network is the subject of this research. This study presents the current state of research, the basic concept behind BP neural networks, how they are used in controller design, and how to optimize the PID parameters of BP neural networks using a particle swarm optimization method. To increase system stability, the neural network is coupled with a PID controller. The adaptive learning capability of the neural network is utilized to modify PID control parameters online in real time. Targeting the trouble that it slips into a local minimum easily during the BP-PID self-learning process and the refined PSO algorithm is used to improve it. It makes sure that the BP-PID system converges to the global optimal solution. The proposed method can effectively improve the system control accuracy and control stability.

Tidal phenomena, a ubiquitous spectacle along coastlines, have captivated human curiosity for centuries. It can be involved in some small activities like fishing on the sea. Also, some big activities like the prediction of the position of the Earth should take the consideration of tides. This paper is aimed to have a summary of the formation and the principles of tides. Then, according to the essential theory, some applications are discussed based on the fields of astrology, geography, and clean energy. The principles of these applications are summarized and concluded into some useful information. The information that people derive from tides is a significant amount which can help humans make more efficient methods for the prediction of some phenomena like earthquakes and the movement of moons. As a result, The value of tides is still a potential topic that can help human progress in human beings.

The price changes of bitcoin and gold are frequent, as observed in previous studies, and market participants can base their daily investment plans on historical data. Market traders can maximise their investment returns by buying low and selling high on volatile assets based on the day's market conditions, as markets fluctuate every day. Gold and bitcoin are two of the most common volatile investments. In this project, we are trying create an algorithm of high degree of integration which can provide advice to traders before they made their decisions. We used ARIMA algorithm based on the Genetic Algorithm to predict the data and let the algorithm to make based on the predicted data. The complete code could be seen in appendix. The codes were mostly based on MATLAB language, for which it integrated the features of “simple and easy to get started”; “dispatchable sensitivity”; “adaptable and reliable”. Which allows the traders to use simple operations to get reliable and visualizable results. Which can let the traders to have the opportunity to get suggestions on when they should trade.

Network intrusion detection refers to monitoring and analysing network traffic, system logs and other information to identify abnormal behaviours and attacks in the network, and take timely and appropriate countermeasures to protect the security and stability of the network. This paper investigates the application of seven machine learning methods in network intrusion detection, and evaluates each model by indicators such as precision, accuracy, recall and F1 score. The results show that XGBoost, Random Forest and Decision Tree models have the best prediction results, while Support Vector Machines and Plain Bayes models have poor prediction results. XGBoost, Random Forest and Decision Tree models all belong to the category of integrated learning, which have strong generalisation ability and robustness, can handle high-dimensional and complex datasets and are not prone to overfitting. In addition, they are able to handle non-linear relationships and are suitable for complex classification problems. Catboost and logistic regression models have better prediction results, but their prediction results are also affected by feature engineering. They may under- or over-fit when dealing with high-dimensional, complex datasets. Support Vector Machines and Plain Bayesian Models have poorer prediction results, which is related to their limitations. Support vector machines may experience computational difficulties when dealing with high-dimensional, complex datasets and are weak in dealing with non-linear relationships. The plain Bayesian model assumes that the features are independent of each other, which may not hold true in practical applications, thus affecting the prediction results. The conclusions of this paper are instructive for research and application in the field of network security, and can provide reference and inspiration for research in related fields.

This abstract provides a brief overview of the importance of force analysis in motion analysis within the context of intelligent vehicle systems. It highlights the significance of force analysis in understanding the dynamics of motion in these systems and its applications for optimizing vehicle performance and safety. Force analysis plays a crucial role in motion analysis within the field of intelligent vehicle systems. By analyzing the forces acting on a vehicle, researchers and engineers can gain insights into its acceleration, trajectory, stability, and overall performance. This analysis aids in optimizing vehicle control strategies, such as steering, braking, and acceleration, to ensure smooth and safe operation. Furthermore, force analysis helps identify critical factors that influence vehicle dynamics, such as tire-road interactions, aerodynamics, and vehicle weight distribution. By understanding these forces, engineers can design intelligent vehicle systems that enhance stability, maneuverability, and fuel efficiency. In conclusion, force analysis is instrumental in motion analysis within intelligent vehicle systems. Its applications range from optimizing vehicle control strategies to improving stability, safety, and fuel efficiency. By leveraging force analysis, researchers and engineers can contribute to the development of advanced intelligent vehicles that meet the demands of a rapidly evolving transportation landscape.