This study examines the dark matter density in the Galaxy by fitting recent data on stellar rotational velocities. We employ a model including the disk, the bulge, and the dark matter halo. The disk’s density is parameterized using a modified Bessel function, and the bulge is modeled using both the De Vaucouleurs and Exponential Sphere functions to better fit large and small radii respectively. The dark halo is modeled using the generalized Navarro-Frenk-White profile, with slope parameters for standard NFW and Moore profiles, and an Isothermal profile for comparison. The fit of the model parameters is established using local Dark Matter density and total Dark Matter mass as boundary conditions, with circular velocities derived via a Newtonian approach. The least chi-square (χ 2 ) method is used for rotation curve fitting. Results demonstrate successful fitting of rotational velocity data and significant influence of the DH at large radial distances from the Galactic center.

The drive to attain ever higher speeds, to be able to travel ever faster fuels the research and development for a commercial supersonic aircraft. This has previously led to the Concorde which travelled at more than twice the speed of sound. Now, in addition to business considerations about economic viability, supersonic aircraft must be quieter and emit less emissions. Considering the 20 years that have elapsed since Concorde’s retirement, this study aims to reevaluate the current challenges and limitations to achieving commercial supersonic flight again, in the context of noise. Identifying sonic booms and jet exhaust noise as two main challenges, it reviews current shape optimization methods, plasma as a sonic boom mitigator, sonic boom circumvention, chevron nozzles, variable cycle engines, engine positioning, and their corresponding limitations. Some of the methods have been refined for use and application in final stage design and manufacture of certain supersonic aircraft which indicates a certain feasibility.

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .

Nanotechnology is the future research area that could benefit human beings in many different directions. In order to build that technology for the human beings, it is necessary to do some research on the simulators to find out some important mechanical properties of these robots in the environment they are in, in other word, low Reynolds environments. To check the availability of the cell model, it is necessary to construct it by using MATLAB and SOLIDWORKS. After construction, the testing process would show that behavior difference of the model in relative high Reynolds environments and low Reynolds environments. The cell simulator would provide a faster swirling speed but a slower velocity in high viscosity solutions than in low viscosity solutions. That would be contradictory to the theory of the Reynolds number which potentially indicate a failure in the experiment. Thus, the macroscopic simulation model might not provide an accurate result for future nanorobotics.

Between the years 2020 and 2022, the COVID-19 pandemic is anticipated to emerge as the most severe global epidemic. The objective of this study is to examine the utilization of biostatistics in the domains of medication development, analysis of epidemic trends, and survival model analysis within the context of the COVID-19 pandemic. Through the utilization of a literature review method, this research delved into the examination of prospective therapeutic interventions employed in the realm of drug development studies. Specifically, the paper explored the efficacy and potential of camostat mesylate and remdesivir, alongside the exploration of immunotherapeutic strategies. Furthermore, the study examines the use of mathematical modeling in forecasting the trajectory of epidemic dissemination, and the significance of survival model analysis in comprehending patient longevity. The study revealed that medication development and immunotherapy play a crucial role in effectively combating novel coronavirus pneumonia. Furthermore, the utilization of mathematical modeling can provide valuable insights into forecasting the propagation of the epidemic. Additionally, survival model analysis can offer guidance in the allocation of medical resources and aid in decision-making processes. The findings of this research will contribute to a deeper comprehension and more effective mitigation of worldwide public health issues, such as the ongoing COVID-19 pandemic.

Given the probabilities (which remain unchanged at each step) of travelling a certain distance in a certain direction, a random walk is a method used in probability theory to determine the likely position of a point depending on random movements. Markov processes, in which future behavior is independent of previous behavior, include random walks as an illustration. This study will mainly pay attention to three different applications of random walks in aspects of multilayer networks, predation and stock market. A real and intricate network could be solved by a mathematical equation depending on a random walk model. There are totally four different types of random walks that properly represent the motion of an animal searching for preys. Among these four types, CCRW is the best explanation for the route of a predator. Nevertheless, by analyzing the panel unit root tests, random walks fail to simulation the variability of share prices. This study promotes scholars to come up with an idea of the principle and application of random walks, and therefore realize the significance of such simulations.

Nowadays, the aircraft has become a very important part of human life. The invention of the aircraft is a shining point in human civilization. During the development of aircraft, there have been significant changes in aircraft’s appearance, power systems, design theories, and applications. In order to provide a clearer understanding of the development process of aircraft and lay the groundwork for future development of the aircraft, this article reviews the three important stages in the development of aircraft and elaborates on the development of aircraft from the perspectives of structural changes, theoretical updates, and applications. While introducing the development process, this article emphasizes the difficulties at each stage and the measures to solve them. In a hundred years of development, human beings have evolved from relying solely on the experience and imitation of birds in the early stages to developing and applying aerodynamics and then combining theory and practice to design.

A rigid exoskeleton has been developed for decades, and its feasibility has been proven in many areas, such as rehabilitation. Unlike the rigid exoskeleton, the soft exosuit provides a new insight for wearable robotics development and has drawn much attention as the external muscles instead of exoskeletons, especially for supporting users’ activities of daily living (ADL) and human body augmentation. This paper reviews the upper-limb soft exosuit studies in the last three years, including the core technologies and the current challenges that need to be addressed. Then, the actuator designs were described, including motor-tendon unit, pneumatic artificial muscle, hydraulic artificial muscle, and textile-based actuation. Their advantages and disadvantages were given and the applications were listed. Also, as the other part of core technologies described in this paper, the controller design which contains low-level and high-level control was discussed. Finally, the challenges were listed, which could be the further directions of research.

Unlike black hole binary merger, the merger between a neutron star and a black hole will produce an abundant number of gravitational waves and electromagnetic waves. Using this information, scientists can easily find many properties of the universe and test the general relativity and some other gravitational theories. The detection of the gravitational wave from source is essential to develop the current knowledge of the gravitational force. From last century, scientists were trying to detect the gravitational waves, and as the time passes, the method of detection has already developed from on land detector to space detector in order to take more precise readings. This paper provides some basic information of the neutron star and the black hole, together with the formation of the binary neutron star-black hole system. The relationship between the neutron star and black hole is explained in this paper. The knowledge of the current methods of detecting gravitational waves is also provided and the paper specifically elaborated the space laser interferometry.

Einstein's general relativity theory includes the gravitational wave as a key prediction. One of the most crucial areas of contemporary physics is gravitational wave detection. A fantastic addition to conventional electromagnetic radiation astronomy, gravitational wave astronomy is a brand-new field of study based on the discovery of gravitational waves. In this paper, the prediction and characteristics of gravitational waves are discussed, and the detection methods of gravitational waves are given, such as the limitations of the resonant rod of gravitational waves, the working principle and basic structure of the gravitational wave probe, the laser interferometer. The gravitational wave signal of the binary black hole merger detected by the LIGO laser interferometer gravitational wave detector in the United States for the first time on September 14, 2015, which opens a new "gravitational wave window" for human astronomy research. It is foreseeable that in the near future gravitational wave research will explore the unknown information of the universe from various gravitational wave frequency bands.

The investigation of topological insulator materials plays a crucial role in the exploration of the quantum anomalous Hall effect. A topological insulator is a distinct type of insulator characterized by its band structure with non-trivial topological features. Topological insulators are characterized by the occurrence of topological phase transitions in the electron energy bands at the Fermi level, which can be attributed to the combined effects of spin-orbit coupling and an external magnetic field. These transitions give rise to the emergence of Hall conductive boundary states, facilitating the manifestation of quantum Hall conductance even in the absence of magnetic fields. The quantum anomalous Hall effect exhibits promising prospects for various applications. For instance, it can serve as a viable means of current transmission in low-power electronic devices, or alternatively, as a medium for constructing qubits in topological quantum computing systems. Furthermore, the utilization of the quantum anomalous Hall effect extends to the development of magnetic sensors with superior performance characteristics and the creation of energy-efficient spintronic devices. This work endeavors to conduct a comprehensive examination and evaluation of the theory and practical implementation of the quantum Anomaly Hall effect by the analysis and review of relevant literature. In addition, it intends to provide potential avenues for future applications in this field.

Nowadays, due to the advancement of 5G technology and the emergence of smart cities, people's working and production environments are gradually transitioning from outdoors to indoors. Consequently, an increasing number of corporate and governmental organizations are demanding high-precision indoor positioning solutions. The utilization of ultra-wideband (UWB) signals in indoor positioning systems possesses multiple benefits, such as high transmission rates, strong anti-jamming capabilities, simple structures, and superior security measures. As a result, UWB stands out among other typical indoor positioning methods and has vast market potential. This paper introduces the basic concepts of ultra-wideband positioning techniques and thoroughly discusses the current state of domestic and international research as well as the main positioning models. The trilateral measurement method and TOA positioning algorithms are the primary focus of this paper. In the future, by exploring the combination of intelligent algorithms and wireless positioning technology by combining ultra-wideband and a variety of positioning methods and integrating these technologies together, it is possible to significantly improve positioning accuracy.

The relation between local and global solution of an equation can be discussed with the method of class field theory and algebraic number theory. In this piece of writing, the author will introduce the behavior of both local and global m-th power in some specific number field. Of course, the result in this paper can be extended into the function field, but it will not be involved in this paper. This paper will prove that if k(ω_(2^t ))/k is cyclic then the Local m-th powers everywhere is equivalent to the global m-th power. In the Non-cyclic case this decomposition becomes P(m,S)=k^m∪δk^m. This paper will also prove some useful propositions in topological group theory, which will be used in the proof of Grunwald-Wang theorem. Grunwald-Wang theorem states that we can find a cyclic extension with given local behaviors. To describe the extension, this paper combines character theory with a topological group, one can depict the cyclic extension. This theorem can be used in the further exploration of central simple algebra.

In mathematics, the logarithm, log_a⁡〖b,〗 where a∈(0,1)∪(1,∞) and b>0, is always defined as the real number x, such that a^x=b. Moreover, in the field of number theory, a similar concept called the discrete logarithm can be defined as follows: For a given positive integer m(m≥2), let a∈N^(+ ) with (a,m)=1, and r is the primitive root of m, x=〖ind〗_r a if r^x≡a (mod m). Here, x is the discrete logarithm. The Discrete Logarithm Problem, which is a famous problem in number theory, is formulized as: For a positive integer b and a prime number p, and a is the primitive root of p, the goal is to find the exact value of i, such that a^i≡b (mod p), in other words, it is targeted at finding the exact value of 〖ind〗_a b. The goal of this research is to give several solutions to the Discrete Logarithm Problem, so firstly, some background concept like order and primitive root will be introduced with the proof of some foundational theories of these two concepts, then this essay will give two methods that can solve the Discrete Logarithm Problem called Shanks' Babystep-Giantstep Algorithm and Pohlig-Hellman Discrete Logarithm Algorithm.

Over thousands of years, many great physicists, such as Carnot, Boltzmann, Plank, Clausius, to name only a few, have put great endeavor into unwinding the mysteries of thermodynamic studies. In recent decades, many innovations have been made by groundbreaking modern technologies, such as refrigerators and air conditioning, and both recent findings and formerly discovered concepts of thermodynamics have been explained deeply by modern scientists. The paper briefly introduces the definition of the second law of thermodynamics and the relative concepts and their application. This paper delves into the history and definition of the Carnot Theorem, the Irreversible Carnot engine and refrigerator principle, as well as the Coefficient of Performance (COP). After that, this paper will introduce a few relative concepts, including entropy, exergy, and the Clausius-Duhem inequality. In summary, this paper covers a basic overall explanation of the aforementioned concepts, as well as drawing conclusions about the experimental achievements of the early explorers of thermodynamics.

In recent years, more and more scientists have used ratios to explore Milky Way. This paper delves into the importance of 21cm radio astronomy for investigating the structure and evolution of the Milky Way. Through an exhaustive review of the literature, the paper emphasizes the benefits of radio astronomy relative to optical and infrared astronomy and elaborates on the distinct attributes of the 21cm radio wavelength. Fundamental concepts of 21cm radio astronomy, encompassing the Doppler effect, neutral hydrogen examination, and the era of reionization, are explored in detail. The paper underscores the significance of comprehending the Milky Way's structure, dynamics, and matter distribution. Forthcoming developments in the domain, such as cutting-edge radio telescopes, data handling techniques, and interdisciplinary collaborations, are examined, presenting the potential for additional advancements in understanding our home galaxy and the cosmos at large.

Aerospace materials play a crucial role in modern aerospace technology. This paper is mainly a comprehensive overview of aerospace materials, including their classification, properties and applications. Aerospace materials are used in a wide range of applications, including aircraft and spacecraft, propulsion systems, and electronics. The paper also summarizes the importance and diversity of aerospace materials in various applications. The classification of aerospace materials includes metal materials, composite materials, ceramic materials and nanomaterials. Each material has unique properties that make it suitable for different applications. Aerospace materials are used in a wide range of applications, including aircraft and spacecraft, propulsion systems, and electronic devices. Materials for aircraft and spacecraft are particularly demanding because they are subjected to extreme temperatures, pressures and vibrations. Materials for propulsion systems and electronics require properties such as high strength, high electrical conductivity and high corrosion resistance. This paper also summarizes the importance and diversity of aerospace materials in various applications. Through the reading of this article, readers will understand the key role of aerospace materials in modern aerospace technology.

Micro-turbojet engine refers to a closed cycle gas turbine engine with thrust below 500kg and power below 100kW. The performance indicators of domestic products are relatively low, and the main problems of domestic products are low thrust-to-weight ratio, poor reliability, and low heat resistance. The low working efficiency of the components and the short life limit the performance of the aircraft. In this paper, a micro-turbojet engine with a higher thrust-to-weight ratio and higher heat tolerance has been designed. The hollow centrifugal impeller with a built-in skeleton structure is successfully developed. The integrated design of the evaporation tube guide and shaft sleeve diffuser solves the problems of poor stability and high production cost, reduces the heat loss, and improves the overall efficiency. Ceramic matrix composite integral turbine disk makes the engine achieve breakthroughs in thrust-to-weight ratio, reliability, and heat tolerance. In addition, the engine has been tested to prove its feasibility. This paper may offer a reference for the design of micro-turbojet engine.

A Vertical Take-Off and Landing (VTOL) aircraft is an aircraft with the ability to take off and land vertically. The ability of removing the need for a runway allows VTOL aircraft to be used for applications that standard aircraft cannot be used for. These applications include military, firefighting, and transportation applications. For the purposes of this paper, helicopters will be considered to be VTOL aircraft, as they fit the general criteria and are among the first to be widely used. VTOL aircraft can be split into several categories, including hybrid VTOL aircraft, VTOL Unmanned Aerial Vehicles, and electric VTOL aircraft. These categories of VTOL aircraft are compared. Famous VTOL aircrafts like Osprey and Harrier are introduced. The technology behind VTOL aircraft is explained, and the ways in which VTOL aircraft can be made useful are reviewed. In addition, the prospect of VTOL aircraft has been made. This paper may offer a reference for the future works about VTOL.

This essay delves deep into one of the most intriguing mathematical puzzles of all time: the Continuum Hypothesis. Beginning with a robust foundational exploration, it sheds light on the key concepts of cardinality and power sets, which are pivotal to the realm of set theory. These foundational ideas set the stage for a deeper investigation into the relationship that the Continuum Hypothesis shares with real numbers and natural numbers. Historically, the Continuum Hypothesis has tantalized mathematicians. This paper takes a journey through time, highlighting the various endeavors to either prove or refute this hypothesis. Some of the most brilliant minds have grappled with its complexities, leaving behind a rich tapestry of mathematical thought. Furthermore, a significant portion of our discussion is centered on situating the Continuum Hypothesis within the context of Zermelo-Fraenkel Set Theory (ZFC). The intricate interplay between the hypothesis and ZFC offers profound insights and raises thought-provoking questions about the very nature of mathematical truth.