This document explains and explores the basics of the jet engine, from basic physical knowledge to the inner workings of the engine, as well as the maintenance aspect. It touches on the mechanical properties that allow the jet engine to function, the design considerations, the mechanisms within the engine, and finally the maintenance of the engine.

In this paper, we first introduced the significance of complex variables as a field in the first section. Then, for the main body section, we explored the history and development of complex numbers. In addition, we demonstrated the key concepts and theorems that are crucial in the fundamentals of this field. We dive in depth to the such fundamental knowledge, which is the preliminaries of complex numbers. We analyzed the preliminaries via three smaller sections: complex numbers and the complex plane, functions on the complex plane, and integration along curves. Finally, we applied these knowledges into deriving a proof for the product of different sine expressions, which results in a generalized formula. Ultimately, the last section will be the conclusion of the paper.

Contemporarily, quantum hall effect (QHE), as the quantum version of classic Hall effect, represented a turning point in the understanding of material physics, metrology, high energy physics, astrophysics. In the category of QHE, there are plenty of sub-types including fractional QHE (FQHE), anomalous QHE (QAHE), spin QHE (SPHE) etc. This paper will mainly focus on the latest progress and the state-of-art applications regarding to QHE. To be specific, a brief introduction of the principle of Hall effect and the differences among different types will be introduced. Afterwards, the applications will be discussed and demonstrated. Overall, these results would provide the audience a general view on QHE, and some recent applications regards it, which shed light on guiding further exploration for QHE.

Translation is a challenge for humans since it needs a good command of two or more languages. When it comes to computer programs, it is even more complex as it is difficult for computers to imitate human translators. With the emergence of deep learning algorithms, especially neural network architectures, neural machine translation (NMT) models gradually outperformed previous machine translation models and became the new mainstream in practical machine translation (MT) systems. Nowadays, NMT has been developing for several years and has been applied in many fields. This paper is focused on studies on four different application categories of NMT models: 1) Text NMT; 2) Automatic program repair (based on NMT); 3) Simultaneous translation. Our work provides a summary of the latest research on different applications of NMT and makes comments on their development in the future. This paper also mentioned the shortcomings of existing studies in this essay and pointed out some possible research directions.

Intelligent dialogue systems, as a subfield of artificial intelligence, have very important research significance and application value. Today’s AI dialogue systems are still in a relatively early stage, but they are developing very rapidly. In recent years, intelligent dialogue systems have been applied in many fields, such as intelligent customer service in online transactions, intelligent voice assistants in smartphones, and virtual chatbots. This paper introduces the background of intelligent dialogue systems and the current research status of key technologies and discusses some challenges in this field and some recent research to improve the system. Most of the current intelligent dialogue systems can perform effective human-computer interaction and respond accordingly. But for the next generation of intelligent dialogue system, more human characteristics are needed so that it can better understand and express human language, have its own personality, and maintain the consistency and logic of dialogue.

Encryption is an important factor during online communication. It is useful to protect users’ privacy and prevent eavesdroppers listening. RSA encryption and quantum encryption are two mainstream encryption methods applied nowadays. This paper focuses on the evaluation and comparison between these two encryptions. It adopts the basic theory of RSA encryption and quantum encryption and provides an analysis of the benefits and shortcomings of these encryptions. It can be concluded that RSA (a type of mathematical encryption) is more popular than quantum encryption (a kind of physical encryption), but is less secure.

In the field of astrophysics and cosmology, astronomical telescopes are the main and key tool for observing and capturing information of celestial bodies. Generally, there are two types of telescopes, i.e., terrestrial telescopes and space telescopes. This study will be mainly focused on the differences and similarities between the two types mentioned above. According to the analysis, the ground telescope is larger and collect data with processing instead of real picture. The space telescope, on the other hand, is smaller but hard to repair, which is able to send back the real pictures of the space. The research aims of this investigation are helping the further researchers to have better understanding to the two different types of telescopes and offering a guideline for them to choose the more appropriate resources to use for the future research. These results shed light on guiding further exploration of different types of telescopes and pave a path for the future inventions of the telescopes.

The black hole has been a mystery and a source of detection motivation for the human race ever since our quest of the discovery and research of other space. In general, the black hole can be defined as a special time-space structure where the gravity is strong enough to trap the photons, i.e., the light will be unable to penetrate or escape from it. In this paper, the current theorem and state-of-art detection paradigms are discussed in detail. Primarily, the categories of the black holes are demonstrated. The four different types of black holes all present their unique features. the quest for black hole exploration has never ceased. It has achieved stunning breakthrough with the first ever photo taken of the black hole by NASA in 2019. The future analysis of black hole excavation and research has been brought a bright future, as technology and outer space exploration plans continue to evolve. These results shed light on guiding further exploration of cosmology as well as the astrophysics.

In order for investors to maximize their benefit by having better forecasts of the complex dynamics of the stock market, there are many factors that affect the stock market, from a company's financial ratios to investor sentiment and reactions to financial news. This project aims to collect UK business news from the Guardian and uses NLP techniques to transform unstructured text data into usable structured sentiment data to predict the movement of the FTSE100 index. The program uses two different libraries TEXTBLOB and VADER to extract sentiments from both the headlines and main bodies of the business news articles. Four machine learning algorithms including Logistic Regression, Naive Bayes, K-Nearest Neighbours and Support Vector Machines and a voting classifier were used to predict FTSE100 index movement given the business news sentiments of the previous day.

The most fundamental way to comprehend mathematics is through its history. It is becoming more widely acknowledged that studying and teaching mathematics history has important epistemological implications. However, learning about the history of mathematics is also a good way to get a liberal education. This paper, through a method of literature review, introduces the functions of mathematics history education and provides several suggestions for mathematics history education.