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.