Table of contents

The 2021 5th International Conference on Mechanics, Mathematics and Applied Physics (ICMMAP 2021) was successfully held on July 23-25, 2021 in Guilin, China. ICMMAP 2021 is to bring together innovative academics and industrial experts in the field of mechanics, mathematics and applied physics to a common forum. The primary goal of the conference is to promote research and developmental activities in mechanics, mathematics and applied physics and another goal is to promote scientific information interchange between researchers, developers, engineers, students, and practitioners working all around the world. The conference will be held every year to make it an ideal platform for people to share views and experiences in mechanics, mathematics and applied physics and related areas.

The conference model was divided into two sessions, including oral presentations and keynote speeches. In the first part, some scholars, whose submissions were selected as the excellent papers, were given 15 minutes to perform their oral presentations one by one. Then in the second part, keynote speakers were each allocated 30-45 minutes to hold their speeches. In this conference, we were greatly honored to have three professors to serve as our Conference Chairman. There were over 120 experts and scholars in the area of Mechanics, Mathematics and Applied Physics representing different famous universities and institutes around the globe to attend our conference.

During the conference, we were pleased to invite six distinguished experts to present their insightful speeches. The first keynote speaker, Prof. Boming Yu, from Huazhong University of Science and Technology. His main research interests include "fractal geometry theory and transport physics of porous media". Prof. Guoping Zhao, from Sichuan Normal University. His research interests: Spintronics, theory in Skyrmions, spintronic information communication technology, magnetic recording, micromagnetics calculations and modeling in exchange spring, exchange bias and nanoclusters. And then we had Prof. Kuahai Yu, from School of Civil Engineering, Henan University of Science and Technology. Prof. Weiwei Zhang, from Mechanics Department of Taiyuan University of Science & Technology. In recent years, his research interests focus on the theoretical and experimental methods of micro defect detection in structures and materials, the history of mechanics and methodology. Prof. C W LIM, from City University of Hong Kong. Our finale keynote speaker, Prof. Zhi Zong, from Dalian University of Technology. Their insightful speeches had triggered heated discussion. Every participant praised this conference for disseminating useful and insightful knowledge.

The participants of the conference were from most part of the world, with background of either academia or industry, even well-known enterprise. The success and prosperity of the conference is reflected high level of the papers received. The proceedings are a compilation of the accepted papers and represent an interesting outcome of the conference. This book covers 3 chapters: 1. Mechanics; 2. Mathematical Physics; 3. Physics.

We would like to acknowledge all of those who supported ICMMAP 2021. Each individual and institutional help were very important for the success of this conference. Especially we would like to thank the organizing committee for their valuable advices in the organization and helpful peer review of the papers. We sincerely hope that ICMMAP 2021 will be a forum for excellent discussions that will put forward new ideas and promote collaborative researches. We are sure that the proceedings will serve as an important research source of references and the knowledge, which will lead to not only scientific and engineering progress but also other new products and processes.

The Committee of ICMMAP 2021

List of Committee members are available in this pdf.

All papers published in this volume of Journal of Physics: Conference Series have been peer reviewed through processes administered by the Editors. Reviews were conducted by expert referees to the professional and scientific standards expected of a proceedings journal published by IOP Publishing.

• Type of peer review: Double-blind

• Conference submission management system: AI Scholar System

• Number of submissions received: 269

• Number of submissions sent for review: 257

• Number of submissions accepted: 151

• Acceptance Rate (Number of Submissions Accepted / Number of Submissions Received X 100): 56.1%

• Average number of reviews per paper: 2

• Total number of reviewers involved: 80

• Any additional info on review process:

• Step 1. Each of selected paper will be reviewed by two/three professional experts in the related subject area.

• Step 2. Review Reports received from the experts will be judged by one of the editors either Review Reports are logical or not?

• Step 3. If not logical, then editor can assign new reviewer or can also judge at his/her own.

• Step 4. If logical, then Review Reports will be sent to authors to modify the manuscript accordingly.

• Step 5. Authors will be required to revise their papers according to the points raised.

• Step 6. Revised version will then be evaluated by the editor for the incorporation of the points raised by the reviewers.

• Step 7. Then the editor will send the revised manuscript to the reviewers again for re-evaluation.

• Step 8. If the reviewers approve the revise version of the manuscript, then will be accepted for publication.

• Contact person for queries:

Xuexia Ye

publication@keoaeic.org

AEIC Academic Exchange Information Centre

The dynamic behaviour of a self-propelled rod in a three-dimensional system with cut-and-shifted Lennard-Jones interaction is studied by stochastic Eulerian Lagrangian method which coupled the coarse-grained microstructure degrees of freedom to continuum stochastic field, and the relaxation and thermal fluctuation of the fluid dynamics mode are taken into account. The diffusion of the self-propelled rod is found to have four regimes. The distributions of the horizontal displacements tend to bimodal non-Gaussian at long time when the self-propelled forces are introduced. Furthermore, we study the distributions of the rod velocities in parallel and perpendicular to the rod axis in the body frame. They are all Gaussian, and their standard deviations increase when the self-propelled forces increase.

The development of fluid dynamics is essential to many related topics in mechanics. With the emerging high technology, people have begun to discover more functions suitable for fluid mechanics. This article mainly summarizes the following two aspects. One is to review the historical development of fluid mechanics on the time axis, and to summarize the important achievements of fluid mechanics in recent years. The second is to summarize the practical applications of fluid mechanics in aviation, engineering application, shipbuilding and daily life. It points out in detail the shortcomings and deficiencies in the development of computational fluid mechanics so far, and points out the direction for the future development of computational fluid mechanics.

In this paper, a one - dimensional high order model based on a set of basic deformation modes of cross section is proposed to analyse one-dimensional dynamic model of cold-formed channel beam Considering the Deformable Cross-section. The model considers the displacement field through the linearly superimposing of 36 basis deformation modes, which stem from the discretization of the cross-section into 9 points and 8 segments. The corresponding deformation function is approximated with Hermite Interpolation. The governing equation is deduced from the principle of Hamilton, then use quadratic Lagrange interpolation for finite element realization. Numerical examples have also been presented and the comparison with ANSYS shell model showed its accuracy, efficiency, and applicability in reproducing three-dimensional behaviors of cold-formed channel beam.

Gong is widely employed in Chinese folk bands, but its sound mechanism and mathematical model is still not fully resolved. This work employs MEMS gyroscope to collect the vibration angular rate of two types of Hand Gong including median pitch and high pitch Gongs. This paper further employs Sparse Nonlinear Dynamics Identification (SINDy) [1] to identify a reduced order model of Gong. The model is using angular velocity as state variable and candidate library of nonlinearity is chosen up to fifth-order polynomial. The results indicate that the nonlinearity is prominent up to the third order (cubic) and the cubic nonlinearity is not appearing for the rotation around the axis perpendicular to Gong surface. These results confirm previous observations that the oscillator with a cubic nonlinearity is able to well characterize the nonlinear vibration behavior of Gong [2]. One potential application of this work is on musical psychology.

Since the discovery of cryogenic superconductivity, liquid helium-based cryogenic superconductivity has been successfully applied in many disciplines. And continuous flow cry-ostats have great importance for the efficient and stable operation of the entire cryogenic system, which can provide a cryogenic working environment for large scale cryogenic applications. In this paper, a thermodynamic model of the liquid helium thermostat is developed based on the principle of continuous flow helium cryostat and the thermodynamic processes between liquid helium and gas helium, using mass conservation, energy conservation and pressure-volume-temperature relationships. The research work can provide theoretical guidance for the stability analysis of continuous flow cryostats, and also has practical value in large-scale cryogenic applications.

Gasoline-air explosion in confined space is a process involving complex fluid mechanics and explosion mechanics. The flame behavior of gasoline-air explosion in a tube with an open end is mainly discussed in this paper. Firstly, on the basis of experimental observation and theoretical analysis, the axial position of the flame front, the flame velocity and the duration of the spherical flame are deduced and calculated. Then, the validity of the theoretical analysis is verified by comparing the theoretical calculation with the experimental data. The results show that the flame velocity is about 2.29m/s at the initial stage of gasoline-air explosion in the tube. The duration of the spherical flame is about 11.7ms. The conclusions of this paper can not only provide valuable data reference for combustible gas explosion safety engineering, fire protection design and evaluation, but also lay a foundation for further research on the flame behavior of gasoline-air explosion in the tubes with an open end.

In allusion to the aerodynamic lift difference between tank test model and full scale plane, a simulation method of lift to tank test model is put forward, and conduct model test on rough water at the model status with boundary layer control technology (BLC) and without BLC, and then we study the influence of aerodynamic lift on trim and heaving characteristic of the test model systematically, the results turn out to be that: the influence of aerodynamic on trim is affected by test velocity, at relatively low speed, this difference can be ignored, but at higher speed, the maximum trim value difference lies the range of 2%∼7%, and the wave length corresponding to the maximum value decreases. The influence of aerodynamic on heaving is affected by wave length, at relative shorter wave length, the heaving value decreases a little between the model with BLC and without BLC, this difference lies about 10%, but this difference increases to 15%∼40% at higher wave length.

Nozzle is a kind of fluid equipment with a wide range of applications. Research on the internal flow of the nozzle is very important for improving the performance of the nozzle or improving the design of the nozzle. In this study, the flow conditions in the convergent-divergent nozzle, as well as the convergent or divergent nozzles are solved and explored by numerical simulations based on Euler equations. It is found that the fluid Mach number is affected by many factors. With the increase of the density, temperature, and pressure, the fluid Mach number will decrease. In the study of the nozzle shape, a discovery for the nozzle flow rate is that the change of the physical parameters of the fluid will fluctuate due to the increase of the cross-sectional area, and specifically it will slow down this change.

Taking a prestressed concrete box girder as the research object, aiming at the problem that the concrete bridge durability is reduced due to the appearance of temperature cracks in the concrete box girder, this paper adopted different types of heat insulation materials, and theoretically analyzed the structural temperature field of concrete box girder under high temperature difference and low temperature environment from the aspects of whether the heat source was set inside the concrete box girder, which provides a reference for the technical application and promotion of thermal insulation materials in concrete bridges.

Open caisson foundation sinks by overcoming the buoyancy and soil resistance by means of its own gravity and sinking auxiliary measures. Bottom resistance is a dominant variable in the sinking stage of super-sized caisson foundation. The magnitude of bottom resistance is of great importance to the safe sinking of the caisson foundation. Based on the super-sized caisson foundation of the main tower of a long-span bridge, a method is proposed for calculating the bottom resistance on the basis of the field monitoring data of earth pressure on the food blade, the bottom resistance of different zones within the bottom of the caisson foundation is calculated by the proposed method, and the distribution characteristics and variation of the bottom resistance are analyzed. The calculating results of the engineering practice show that in the process of soil sampling from the internal area to the external area within the bottom, the bottom resistance of inner bulkhead and inner partition walls decreases, while the bottom resistance of outer bulkhead and outer partition walls increases. The distribution characteristics and variation of the bottom resistance can actually reflect the global mechanical behavior of caisson foundation, which helps guide the safe sinking of the caisson.

When using the conventional segmental superposition method to make the bending moment diagram, it is not only complicated to calculate, but also easy to make mistakes. In this paper, it is discussed how to flexibly segment and simplify the calculation with examples. At the same time, according to the differential relationship, integral relationship and static equilibrium conditions between load and shear force, the fast and accurate pithy formula for shear force diagram making is concluded. The deep application of integral relationship between load and shear force is also discussed.

Tension (compression) bending combined deformation is a commonly used combined deformation form in engineering. The current material mechanics textbooks use the superposition method to calculate the stress on the cross section of the combined deformed member, whose characteristics are simple and easy to grasp. However, the premise of superposition principle is that the internal force, stress, strain and displacement have a linear relationship with external force. If the above linear relationship cannot be guaranteed, the superposition principle cannot be applied. In this case, the stress-strain relationship must be used for calculation. Based on the assumption of plane and unidirectional force, in this paper, rods with arbitrary cross-sectional shape under arbitrary load were taken as the research object, and the general calculation methods of the stress on the cross-section during the combined deformation of tension and bending were studied. This paper aimed at expanding students' thinking mode and improving students' learning ability in the process of material mechanics teaching.

The ship industry has been developed considerably in the engineering applications of deep-sea energy extraction, and the hydrodynamic performance and noise level of ships have become the focus of research for scholars and industries. In order to study the flow characteristics and acoustic characteristics of the sea suction valve, this paper conducts numerical simulation of the internal flow field and acoustic characteristics of the sea suction valve and the noise elimination trunk structure based on the numerical method of computational fluid dynamics. Through the study and analysis of different sea suction valve structures and different depths of noise elimination trunk on the flow characteristics and acoustic characteristics, it is helpful to provide guidance for engineering optimization design.

Due to the fast development in biology and chemistry, scientists find that it is important to get a suitable sorter since it can enhance the accuracy and speed of the cell sorting application. For instance, researchers in the biological field need precise sorters to help them separate different kinds of components. The need for micro-fluidic based sorting, a useful technique in medicine or biology, has promoted the development of microscale sorters. This paper reviews two types of sorters, including active and passive sorters. The principles, mechanisms, merits and demerits, avenues for improvement, and application in the future of those two sorters are summarized. In conclusion, this paper could be a useful reference for those who would like to know the microfluidic sorter's overall knowledge.

Einstein-Smoluchowski's theory and Langevin's theory are two main theories to describe Brownian motion. The Einstein-Smoluchowski theory regards Brownian motion as random walking of Brownian particles, while the Langevin theory establishes a random differential equation describing the motion of Brownian particles. Both theories involve random numbers, i.e., only the statistical results can be discussed. Based on the above theories, this paper presents the corresponding simulations, which verifies the statistical conclusions as well as compares their similarities, differences, and applicable conditions. These results offer a guideline for further studies on Brownian motion.

The vibration of the crane bridge structure and the swing of the load will cause fatigue damage to the bridge structure and affect the precise positioning of the load. The moving mass constituted by the load, the crane trolley and the flexible main girder is passed through the multi-body rigid-flexible coupling dynamics system of the bridge, and the virtual prototype of the physical model is established. The influence of the swing Angle of the load on the vibration of the main girder, the influence of the running speed of the crane trolley on the vibration of the main girder and the vertical displacement of the running trolley is simulated and analyzed. The simulation results show that the running speed of the crane car affects the vibration frequency of the bridge structure, and has little effect on the vertical displacement of the crane car. The load swing affects the vibration amplitude of the bridge structure and the vertical displacement of the crane trolley. The vibration of the main beam structure has little effect on load swing.

Prestress has a great bearing on the performance of flextensional transducer. In order to enhance the adjustability of prestress, a design method for the flextensional transducer with adjustable prestress was proposed in this study. Through the conversion design of rotational and rectilinear motions of the transition block, the clamping length of major axis of class IV flextensional transducer shell was changed, enabling the adjustment of prestress. Next, the related experiment was further carried out. The results show that the return displacement of adjustable transition block is slightly greater than the travel displacement, and both displacements show high overlap ratios in the two tests. Because of the prestress adjustment, the frequency response of flextensional transducer is reduced to some extent, within one frequency band (40-63 Hz), but its sound pressure level (SPL) substantially increases nearby the resonant frequency (about 75 Hz).

This paper presents an optical measurement method of welding deformation. Through the use of target detection, camera calibration, 3D metric reconstruction, we obtain the precision 3D location of targets, which are deformed in welding processing. With the data, the deformation of the workpiece in welding processing can be performed, and the manufacture and design level of welding can be improved.

This paper presented the dynamic behavior of tire fragments and their impact on aluminum alloy plates by the methods of experiment and numerical simulation. In the experiment, the impacts of tire fragments on aluminum alloy plates under quadrilateral constraint, bilateral constraint and four-corner constraint were studied. According to the experimental results and based on Ogden model and Johnson-Cook model, a simulation calculation model for fragment impact was proposed. Analysis of plastic deformation area, failure mode, etc. verified that the proposed model could stimulate the impact of tire fragments on aluminum alloy plates. By comparing different constraint conditions, it is found that within the impact range, the plate under quadrilateral constraint can resist the strongest impact.

With the progress of engineering technology, steel structure has become the mainstream structural form of construction engineering, and beam column composite structure is the most commonly used connection form in steel structure buildings. In order to improve the design and construction performance of beam-column joint, a new type of beam-column assembled rigid joint is proposed and its parameter optimization is analysed based on equal strength design method and finite element software. The results show that the new beam-column assembled rigid joint has excellent performance for ductility and energy-dissipating capacity and the joints and splice plates can both meet strength requirements. The number of bolts and the length of cantilever beam have great influence on the ultimate bearing capacity and ductility of the joint. The new type of beam-column assembled joint is easy to construct and has good mechanical properties, and its ductility and elastoplasticity are better than those of ordinary joints without splicing.

The disturbance of shield construction in coastal soft soil area poses a threat to the use and even safety of adjacent municipal transportation facilities. In order to study the influence of under-crossing shield construction on municipal transportation facilities in soft soil area, taking a practical project as an example, the author adopted numerical analysis method to conduct a comparative study of the influence of foundation reinforcement on under-crossing shield open-cut tunnels and bridges. The results show that for open-cut tunnels and bridges, foundation reinforcement can greatly reduce the deformation and internal force of the structure after under-crossing shield. At the same time, different reinforcement depths and methods will also lead to different results. The full reinforcement of 2 meters is obviously deeper than that of 3 meters, and strip-shaped reinforcement with 6-meter spacing has worse performance. Determining a reasonable reinforcement depth is of great significance for saving costs.

Al-Li alloy stiffened panel is the main structural form of aircraft fuselage. Laser welding will produce welding deformation and residual stress, on the other hand, low velocity impact will also affect the residual bearing capacity of stiffened panel structure. In this study, the nonlinear finite element (FE) method is used to simulate the laser welding process of stiffened panel, and the welding residual stress is directly introduced into the subsequent impact after compression (CAI) process of stiffened panel. The effects of different impact positions and different impact energy on the residual CAI strength of stiffened panel are studied. The results show that the impact damage has the greatest effect on the residual compressive strength of stiffened panel when the punch impacts the stiffener, and the greater the impact energy is, the greater the influence is.

The nozzle is a kind of widely used fluid equipment. It is very important to study its internal flow for improving nozzle performance or improving nozzle design. In this paper, the numerical simulation method based on the Euler equation is used to solve the flow in the nozzle. Euler equations are solved by an explicit finite difference method, including discretization, pre-estimation and correction steps. Boundary treatment for the subsonic inlet and supersonic outlet are discussed. The flow characteristics inside a Laval nozzle are analysed based on the numerical results. The influence of Laval nozzle throat width on the nozzle flow is investigated by comparing three nozzles. It is observed that the dimensionless pressure, temperature, and density of the three nozzles have similar trends. While the maximum Mach number at the outlet decreases with the increase of the throat width. In short, the throat width should be decreased if a larger outlet speed of the flow is needed.

In the process of deflection measurement for lifting device dynamic and static load test (DSLT), the traditional measurement method has some obvious problems, such as complex test process, slow measurement speed, poor accuracy, low efficiency, and difficult dynamic measurement. Therefore, a real-time binocular vision method for measuring deflection in the lifting device DSLT is proposed. The results of verification show that the method can realize the real-time and fast deflection measurement in the lifting device DSLT, and has the advantages of non-contact, high precision and dynamic measurement. Compared with the traditional methods, the method proposed in this paper is simple, universal and easy to implement.

All the small disturbance equation (SDE) methods, which are a useful technique to enhance the stability and security of the aircraft, play a significant role in the development of the aviation area. This review paper briefly introduces the development of the SDE and summarizes the two calculation methods combining with the SDE and application of the SDE method in the aviation area. The calculation methods mainly focus on the approximate factorization (AF) algorithm with the SDE and Newton-Orthomin method with the SDE. AF algorithm with the SDE is robust enough, and it can choose the calculation methods according to accuracy rather than stability that significantly saves the calculation cost. The newton-Orthomin method with the SDE can improve computing efficiency by 2.1 to 4.5 times compared with the monotone AF method. The direction of application focus on the flutter analysis, gust analysis and resizing of some component in the aircraft and the disadvantages are explored for each type. In conclusion, the combination of the SDE with the other algorithms can increase calculation efficiency, and the SDE has a wide range of applications in the aviation area. This paper can provide relevant references for the subsequent combination of the SDE and other algorithms development or practical application of SDE.

Aiming at the strength problem of amphibious landing on water, the pressure distribution of ship bottom is studied. RANS multi-phase flow method is used to solve the coupling problem of gas, liquid and solid in the process of amphibious landing on the water surface. Overlapping grid technology is used to simulate the landing under different initial horizontal velocity and vertical velocity. The influence of initial conditions on the motion response of amphibious is analyzed, and the variation law of pressure on each section of ship bottom with time is obtained, This paper analyzes the variation of pressure with time and position, and provides technical reference for the design and application of water loads in the development of amphibious.

To investigate the problem of the influence of the depth of the isolated D-type roughness element on the hypersonic flow around the blunt wedge boundary, the high-order accuracy finite difference method is used to carry out the direct numerical simulation, the influence of the depth of the isolated D-type roughness element on the interaction between the free flow and the wall is analysed, and the influence of the D-type roughness element on the wall pressure, the wall friction resistance and the wall heat flow is discussed. The results show that the strength of compression wave and expansion wave increases with the increase of the depth of D-type roughness element, but the position of compression wave and expansion wave does not change; With the increase of the depth of the D-type element, the change speed of the flow parameters will be accelerated, the peak value will increase and the trough value will decrease, and the position of the peak and trough will gradually move away from the center of the element with the increase of the depth of the element. When the depth of the D-type element is greater than or equal to 0.08, a vortex will form in the D-type element, and the length of the vortex increases with the deepening of the element. The D-type element will not change the flow state of the shear layer near it.

The building No.1 project located in the district of Jinjian Yingxiyuan at the city of Taiyuan is the first high-rise steel structure building in Shanxi Province by using hidden frame-brace system with the special-shaped wide concrete-filled steel tubular (WCFT) columns. In this paper, elastic analysis and pushover analysis of the new building are carried out by using the finite element software MIDAS/Gen, which is based on the theory of equivalent sectional flexural stiffness. And the results show that the various parameters under the action of frequent earthquake and rare earthquake meet requirements of the code, the failure mechanism of the structure is reasonable, and the seismic performance is good. Finally, the hidden danger and the key problems about the new structure system are put forward.

The prefabricated pier structure has the advantages of low energy consumption, high mechanized level, which meets the green development trend of the construction industry, with broad application prospects. This paper analyzes the process and construction principle of the prefabricated pier structure and the pier and the top platform of the prefabricated bearing pier, which describes the principle of the prefabricated bearer structure, and the structural operation of the prefabricated bearer structure, and proposed a new type of prefabricated pier structure.

How to select an appropriate instability evaluation criterion is a key issue in the application of the finite element strength reduction method. Commonly used evaluation criteria for slope instability mainly include non-convergence criterion of finite element numerical calculation, displacement saltation criterion at characteristic point, and penetration criterion of plastic zone. The researches on these three types of instability criteria are still continuing, but there is no uniform conclusion yet. Based on the different characteristics of the three types of instability evaluation criteria, this paper compares and analyzes the evaluation results of three criteria on the slope stability by trial calculations, and then discusses the characteristics and effectiveness of three types of instability evaluation criteria. In actual numerical calculation, it is necessary to minimize the influence of human factors and comprehensively consider the three types of instability evaluation criteria to improve the consistency of the evaluation results. when the evaluation results are inconsistent, priority should be given to the feature point displacement saltation criterion at characteristic point or penetration criterion of plastic zone. In addition, a reasonable dilatancy angle is needed to accurately consider the influence of dilatancy on slope stability.

Based on the engineering background of the roller of steel rolling, researching the contact problem of two long parallel cylinders and considering the infection of friction. Gained the stress expressions when the vertical load and the friction act on the rolls separately, then according with superposition theorem of elastic mechanics, gained the stress expressions with the contact pressure and the friction force combined action.

An ultrasonic fatigue testing system was used to carry out very high cycle fatigue (VHCF) tests on 2024 aluminum alloy in three different sizes of specimens. The influence of specimen size and optimization on the experimental results was discussed and the VHCF properties of 2024 aluminum alloy were obtained at the same time. The results show that: The S-N curves of all specimens show a continuous decreasing tendency and the curve of the dog-bone-shaped specimen (the half-length of the testing volume L₀ > 0) is lower than that of the hourglass-shaped specimen (L₀ = 0) at low-cycle and high-cycle fatigue stages, which is opposite at the VHCF stage; The optimized specimen solves the problem of fracture position deviation and effectively improves the accuracy of experimental data; 2024 aluminum alloy will still fracture when the fatigue life exceed 10⁷, and there exist no fatigue limit; All cracks initiate at surface and the fracture morphologies show characteristics of the crack initiation and propagation.

In this paper, the ultimate load of a typical hat type composite stiffened panel under axial compression is simulated by finite element method. Based on the ultimate load value and failure mode obtained by numerical simulation, the thickness of the lower flange of the stiffener in the engineering method is modified by using the stiffness equivalent method. The modified engineering method is in good agreement with the finite element results, which has a certain reference significance for the rapid calculation and analysis of stiffened plate in the early design stage.

The interfacial bonding method and mechanical properties of new repairable epoxy and its composites were studied. Ethylene glycol was used to connect the resin interface. The effects of temperature, time and resin content on the strength of interface bonding were systematically studied, and the optimal bonding scheme was determined. The results show that this connection method has significant advantages, and the tensile strength reaches 73% of the intact resin. Furthermore, the reparable resin matrix fiber-reinforced composite was prepared, and this method was successfully extended to the composite connecting. The interfacial shear strength of the fiber composites reached 21.88 MPa by single lap test, which has a good application prospect.

Concrete pavement is one of the typical highway structures at home and abroad. Most of the existing research on the performance of concrete pavement is based on the macroscopic angle of view. The complex meso-structure inside is ignored. Due to the incomplete research on concrete pavement structure, it is difficult to reveal the physical mechanism of concrete pavement deformation and damage. In order to understand the failure mechanism of concrete pavement more clearly, this paper started from the microscopic perspective. Firstly, based on the experimental phenomena, a multi-grade aggregate model was randomly placed in the concrete slab, and the interface between the aggregate and mortar was generated at the same time to establish a three-dimensional three-phase mesoscopic bone and the finite element calculation model of the material. Secondly, the damage effect of concrete specimens in homogeneous modeling, meso-modeling and experimental phenomena were compared to verify the accuracy of the selected concrete material parameters and the feasibility of meso-modeling method. Finally, the concrete failure form under three different aggregate contents and the influence of internal aggregate content of the concrete pavement on the mechanical properties of the concrete pavement were analyzed. The calculation results showed that compared with the macroscopic simulation, the mesoscopic model was more consistent with the experimental phenomena of the failure effect of the concrete specimens. And the numerical simulation results with the aggregate content of 46.6% were the most consistent with the experimental results. Comparing the failure modes of concrete specimens with different aggregate contents, it was found that the concrete specimens with an aggregate content of 60.6% have a lighter degree of damage. When the aggregate content was 30.6%, the impact tooth penetrated half of thickness of the concrete specimen, and the aggregate content is 60.6% of the impact teeth penetrated one third of the concrete specimen. Therefore, the higher the aggregate content is, the stronger the ability of the concrete specimens has to resist damage.

Cooling is a common phenomenon in daily life. Among different means of cooling, evaporative cooling through the evaporation of liquids is a prominent method, for it removes large quantities of heat. Our prompt is aimed at evaluating the impacts of different parameters on the speed of evaporative cooling. We evaluate the degree of impact by the means of experiment and simulation of experiments with finite element simulation software COMSOL, using containers of different shapes, different wind speeds, and controlling the existence of evaporation. Research shows that the speed of evaporative cooling is mainly determined by the difference in the amount of evaporation, followed by the geometric shape category of the upper surface of the container. Simply changing the proportion of the upper surface of the container does not pose a large impact on the speed of evaporative cooling due to the impact from turbulence around the container and the conductive heat transfer of the container itself. In the process of the research, the experiment and simulation results achieve a high degree of agreement, confirming the accuracy of the simulation.

We presented a silicon microring resonator based on thin film silicon nitride waveguide for biosensor applications. Design and fabrication were conducted. The device was fabricated on a silicon nitride wafer with 250 nm top film on 3 μm buried oxide by using electron-beam lithography and inductively coupled plasma etching. The radius of the microring is as smaller as 50 μm. The quality factor of the resonator is 8610. The spectra and resonance shift for bulk sensing were measured by flowing sodium chloride solutions with different concentrations on the surface of the sensor. The sensitivity and the LOD of the sensor are 384.58 nm/RIU and 4.68×10⁻⁴ RIU, respectively.

When a tire bursts, the flying fragments may break through the wing, damage the precision instruments inside it, and cause huge economic losses. In this paper, the impact of tire fragments on the fuselage structure is simulated, and the dynamic response of the structure under different postures of impacting different structures and fragments is analyzed. The damage caused by different impact modes is analyzed and compared, the harshest impact conditions are selected, and the corresponding improvement measures are given according to the impact results. It is found that the strain of the frame truss structure is the smallest when the fragments impacts on the front side, but it may lead to the fracture of the frame truss on the back side, resulting in secondary fragments splashing. When impacting in different postures, the edge impact is the most dangerous. Although it penetrates the skin, it is very close to the failure value.

The incident wave pressure acting on the wall of the spherical charge in the rock and soil is an important factor affecting the size of the explosion cavity, and the pressure of the incident wave is related to the impedance of both the explosive and the rock and soil. In this paper, based on the quasi-static model of explosive spherical charge in rock and soil, the impact impedance was introduced to calculate the size of explosion cavity in rock and soil medium. Through the field underground explosion experiment, it is found that the calculation result of the impact impedance model is 6.15% higher than the original model, and the influence rule of several geotechnical medium parameters on the cavity size was also analyzed.

When an airplane is forced to ditching on water, slight state changes, such as weight, center of gravity, initial attitude, etc., will have great influence on the ditching performance and floating ability of the airplane. In order to study the floating ability of the aircraft under the above parameters, this paper takes a fixed-wing aircraft as the basis model, and based on hydrostatics, divides the water inflow into several tiny water inflow segments. By studying the floating attitude and floating time of different aircraft states, the influence law of the center of gravity position parameters on the floating time is mastered. The research shows that the floating time and water inflow volume of the aircraft will be significantly increased when the higher center of gravity position and the non-most unfavorable longitudinal range of the center of gravity are adopted, and forced landing on water in this state can provide emergency evacuation and escape for the aircraft personnel.

The median fins of tuna can morph in position and shape to enhance their hydrodynamic performance. Based on the C-turn kinematic equations and the self-propulsion swimming numerical model, the hydrodynamics of the median fins in different morphing forms (erected and depressed forms) are analyzed under the condition of three-degree of freedom (3DoF) self-propulsion, and then the assisted-turning mechanism of the median fins is clarified. The results show that the erected median fins have higher hydrodynamic forces, which can not only increase the yaw speed, but also reduce the turning radius to a certain extent.

The self weight of steel box girder is light, and the stability effect is smaller than that of concrete bridge. In this paper, the anti overturning research of the main line bridge and ramp bridge is carried out. It is found that for the variable cross-section continuous beam bridge with wind barrier along the whole line, when the pressure reserve of the side pier is small under the action of self weight, the compression of the bearing under the action of basic combination can not ensure that the bearing will not be empty under the action of standard value combination; Centrifugal force, wind load and temperature load caused by vehicle load have great influence on bridge overturning, which can not be ignored; Temperature load and foundation displacement may lead to stability or instability according to the stiffness distribution of the main beam. In this paper, it is proposed that both the basic combination and the standard value combination should be considered when checking the bearing void. It is recommended to adopt the calculation method of "the stability effect is the standard value effect that makes the superstructure stable, and the instability effect is the standard value effect that makes the superstructure unstable" as the stability coefficient.

In this manuscript, utilizing the controlled variable method, the three different profiles of thin paper with fixed ends are investigated. The results indicate that the specific profile of the paper depends on the length of the paper and the fixed distance between the sheets. Meanwhile, the mass of the paper has little effect on its profiles and so it can be ignored. Also, the symmetry-broken phenomenon about the thin paper is observed. Finally, according to the Principle of Least Action, it proposes a simple model to simulate the thin paper profiles

In this paper, the electric field intensity of the electrode structure of the thermally emitted electron gun of UTEM is simulated, and the relation between the electric field intensity distribution and the electrode plate is analyzed. Using the electronic optics software, the optical system of the electron gun is simulated, the performance parameters of the electron gun are obtained, and the advantages and disadvantages of the whole optical system are analyzed. The research results of this paper can provide a reference for the design and simulation of similar electron gun.

Acoustics releaser plays an important role in the recovery of marine surveying system. This paper analyses and introduces present situation of the acoustics releaser in our country. Finally, main technology development direction and strategy of localization are put forward in the paper. The paper mainly includes research status quo outside and in China as well as key technical problems in design of underwater acoustic releasers.

The parabolic reflector provides one of the common sound wave convergence methods in acoustic engineering. To reduce the difficulty of fabricating large-size reflectors and improve the convergence effect, this paper proposes a multi-segment linear reflector design method. Through the normal vibration radiation analysis of the flextensional transducer shell, the reflector interface reflection was optimized, the reflection effect was improved and the reflector size was reduced. The transmission theory was used to analyze the impact of different material parameters and sound wave frequency on the reflector transmission coefficient, and the gain effect of the multi-segment linear reflector was further tested. The research results show that the reflector has a significant gain effect in the test frequency band of 20∼100Hz, and the gain at 70Hz I reaches the maximum of 11.2dB.

In this paper, the parabolic singularly perturbed convection-diffusion problem is discretized by using the backward Euler method in time and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh in space. This method is shown to be first-order convergent in time and space. Finally numerical experiments confirm the theoretical results.

The nozzle is a kind of equipment with a wide range of usage in the aerospace industry, of which its performance is vital for rocket engines by changing the geometry of the inner wall of the pipe section to accelerate the airflow. There are two types of nozzles commonly used: one is a tapered nozzle, and the other is a Laval nozzle. To analyze the flow phenomenon in nozzle is vital before transferring prototype design into industrial production, whether by laboratory experiment or computational simulation. In this paper, two different numerical methods are adopted to simulate gas behavior inside a simplified nozzle with upper and bottom symmetrical bumps. The first is to solve one dimensional Euler equations, and the other is to solve a scalar variable named velocity potential with small disturbance equations (SDE). The solutions under various inlet Mach numbers are compared by analysing the velocity fields and Mach number contours obtained by these two approaches. Similarities and differences between the Euler method and the potential SDE method for subsonic flow and supersonic flow are the key emphases in this work. For either subsonic or supersonic flow, the Mach number distribution along the nozzle's center line shows a consistent trend for both methods. In contrast, the values of maximum or minimum Mach number have corresponding differences. Moreover, by using potential SDE simulation, several types of shock waves are successfully captured. All results show that the incoming airflow decelerates at the leading edge of the nozzle then accelerates when passing through the bumps, and finally decelerates back to the speed of inlet flow. The difference is that flows with varied Mach numbers has distinct velocity distributions in the nozzle.

Complex variables have aroused strong interest in the community of mathematics in the past few decades. Our aim is to obtain the integral results of a particular equation $\begin{eqnarray}&&\begin{array}{ccc}{\rm{f}}({\rm{Z}})=\frac{{{\rm{z}}}^{3}+(1+2{{\rm{i}}){\rm{z}}}^{2}-7{\rm{z}}+1-2{\rm{i}}}{{{\rm{z}}}^{4}-1},\end{array}\end{eqnarray}$ in three defined contours. To determine the value of integral, the information we need to find is the residue values ABCD on four specific points and find out what residue values each contour includes. After finding the residue values and analysing four contours, we should use Cauchy integral formula to find the integral value. Firstly, by using the partial fraction to separate the equation into four independent parts, we obtain four values on each of the four fractions are the value of integral. Thus we can find exactly the four residue values, which are -1 on (1,0), 3 on (0,i), -2 on (-1.0) and 1 on (-I,0) as shown in the following expression.

$\begin{eqnarray}&&\begin{array}{ccc}{\rm{f(Z)=}}\frac{{{\rm{z}}}^{3}+(1+2{{\rm{i)z}}}^{2}-7{\rm{z+1}}-2{\rm{i}}}{{{\rm{z}}}^{4}-1}=\frac{-1}{{\rm{Z}}-1}+\frac{3}{{\rm{Z}}-{\rm{i}}}+\frac{-2}{Z+1}+\frac{1}{Z+i},\end{array}\end{eqnarray}$ and then we must analyse each contour and find out the residue values they include. By drawing a diagram of each of the three contours. We find that contour 1 contains A, contour 2 contains A and D, and contour 3 contains ABCD. The final step to obtaining the value of residue is to use Cauchy integral formula $\begin{eqnarray}&&\displaystyle \int {}_{{\rm{y}}}{\rm{f}}({\rm{z}}){\rm{d}}{\rm{z}}=2\pi {\rm{i}}\displaystyle {\sum }_{{\rm{k}}=1}^{{\rm{n}}}{\rm{Res}}\,{\rm{f}}({\rm{z}})\begin{array}{ccc}.\end{array}\end{eqnarray}$ According to Cauchy integral formula, the value of integral of the equation on a specific contour is obtained by multiplying the sum of the residue values included in that contour by 2πi, the value of integral of the equation can be found as we now have all the necessary information. For the first contour $\begin{eqnarray}&&\begin{array}{ccc}{\rm{y}}(t)=1+{e}^{it}\,\,\,\,,\,{\rm{where}}\,0\le t\le 2\pi,\end{array}\end{eqnarray}$ the value of integral is calculated by using 2πi multiply by -1. We can get the result -2πi. For the second contour $\begin{eqnarray}&&\begin{array}{ccc}y(t)=\frac{1-i}{2}+\sqrt{2}{e}^{it}\,\,,\,{\rm{where}}\,0\le t\le 2\pi,\end{array}\end{eqnarray}$ the value of integral is calculated by using 2πi multiply by the sum of -1 and 1. As we can see, the sum is 0. We can get the result 0. For the third contour $\begin{eqnarray}&&\begin{array}{ccc}y(t)=2{e}^{it},\,\,{\rm{where}}\,0\le t\le 2\pi,\end{array}\end{eqnarray}$ the value of integral is calculated by using 2πi multiply by the sum of -1,3, -2, and 1.

As we can see, the sum of these residue values 1. We can get the result 2πi. Hence all of the values of integral of the equation $\begin{eqnarray}&&\begin{array}{ccc}{\rm{f}}({\rm{z}})=\frac{{{\rm{z}}}^{3}+(1+2{{\rm{i)z}}}^{2}-7{\rm{z}}+1-2{\rm{i}}}{{{\rm{z}}}^{4}-1},\end{array}\end{eqnarray}$ is found.

The problem of string vibration prompted mathematicians to find a suitable equation to express this phenomenon, which led to the birth of the Fourier series and integrals. The physical law of string vibration can be expressed by wave equation and solved by Fourier series. The study on the wave equation is fraught with mathematical and physical controversy. D.Bernoulli based on results from physical experiments to make a bold conjecture that the shape of a vibrating string can be described as a combination of trigonmetric series, which was inspired and validated by work from Fourier on study on heat equation. This study reviews the wave equation using Fourier as a tool. This paper not only gives the derivation of the wave equation but also explores its solution and corresponding properties.

Taking a large-caliber gun as the research object, a projectile and gun coupling finite element numerical simulation model during the extrusion process is established according to the finite element simulation technology. The dynamic simulation of the extrusion process is carried out by using ABAQUS/EXPLAICT solver, and the stress variation law of the projectile belt during the groove and the barrel during the extrusion process is obtained. In the post-processing module of the finite element software, the data of displacement, velocity and acceleration at the center of mass of the projectile in the extrusion process are extracted, and the motion law of the projectile in the extrusion process is analyzed, which provides some reference value for the study of the extrusion process of large caliber artillery projectile.

Advanced algebra is a required course of undergraduate mathematics, which plays a fundamental role in completing the study of other professional courses for students. Matrix theory is an important branch of mathematics, it is not only a basic subject, but also themost practical value. Widely used in mathematical theory, matrix is an important basic concept in matrix theory, is a major study of algebra, Positive definite matrix is a kind of important matrix, occupies an important position in "higher algebra", its theory has rich content, positive definite quadratic form and Euclidean space, physics, probability and optimization control theory are related to it.

The graph-parameters mainly studies vertex degree and distance between unordered vertices. There are many graph-parameters, and this paper mainly focuses on the Gutman index. For the sake of discussion, this paper puts forward the concept of "contribution" based on the relationship between Gutman index of graph and vertex-Gutman index. On this basis, we discussed the Gutman index of unicyclic graph with pendent edges in odd and even parts, and gave the corresponding extremal graphs, which lays a foundation for the study of the Gutman index of other cycle graphs.

To accelerate the convergence rate and obtain robust optimal results with clear profiles of structural topologies, this paper proposes a hybrid multi-population genetic algorithm (MPGA) and bi-directional evolutionary structural optimization (BESO) method for structural topology optimization. Each element in the design domain is treated as an individual and the elemental sensitivity is taken as the fitness function of one individual. Based on these treatments, MPGA operators, including crossover, mutation, migration and selection, are modified to adapt to compliance minimization problems. Additionally, some key parameters are controlled to guarantee a convergent solution and to solve the structural unconnectivity problem. A case is used to verify the effectiveness and efficiency of the proposed method. The numerical results show that the proposed method is efficient, and compared with the BESO method and combined simple genetic algorithm and BESO method, the proposed method provides a powerful ability in searching for better robust solutions and improving convergence speed.

This work presents comprehensive numerical research on the impact load of the trans-medium aircraft (TMA) using the coupled Eulerian-Lagrangian (CEL) method during a water-entry event. The water-entry velocity, angle, and attitude play a significant role in the impact load characteristics of the water-entry trajectory. In this paper, a numerical model of a typical TMA structure is established to study the water-entry load with the velocity 0, 2m/s, 4m/s, 6m/s, 8m/s, the angle 90°, 80°, 70°, 60°, 50°, the attitude 90°, 80°, 70°, 60°, 50°. Subsequently, the variation laws of the impact load with different water-entry velocity, angle, and attitude are analyzed. The results obtained from this investigation can supply a good reference to structural design of the TMA.

Fourier transform is a linear transform that applicants in the field of engineering and physics. In this research study, we summarize the fundamental properties of the Fourier transform and its application in solving the wave equation. First, we review the definition of Fourier transform and its inverse form. Then, we show Schwartz space is the invariant subspace of the Fourier transform. We also prove the Plancherel theorem. At last, we apply the Fourier transform to solve the wave equation.

Aiming at the numerical solution problem of given interval dichotomy, the concept of one-sided monotonic function is proposed, and the unique existence of the solution is proved when the two ends of the function take different signs. Therefore, this significantly relaxes the discrimination conditions and effectively expands the numerical solution range of the dichotomy method. On this basis, it is proved that the one-sided monotonic function is a monotonic function in a relatively small interval containing the zero point. The introduction of the concept of one-sided monotonic function provides a theoretical basis for further expanding the application range of the dichotomy numerical solution, and has certain theoretical and practical significance for dealing with scientific and engineering problems as well as related numerical calculation problems.

In this work, we summarized several fundamental theorems of the entire functions and explored how their zeros determine those functions. The background of this work is based on Elias M. Stein and Rami Shakarachi, and their lectures about functions that are holomorphic in the whole complex plane that once were taught at Princeton University. First, we introduced Jensen's formula and gave detailed proof. This relates the values of a meromorphic function inside a disk with its boundary values on the circumference and with its zeros and poles. Next, we studied the proof of Weierstrass infinite products and the definition of canonical factors. Last, Hadamard's factorization theorem and a few main lemmas were introduced. We showed the proof of the Hadamard factorization theorem and solved its several applications through using examples. Hadmard's theory is well demonstrated in this article, and we showed the rigor and scientific validity of the theory through examples.

Usually, people first learn about Euler characteristic in three-dimensional space, related to the famous Euler formula. This invariant is explained in a top-down manner. In particular, there is an invariant first defined on general smooth manifold. Then, looking at different properties of this invariant leads to the discover of coincidence with Euler formula in lower dimension case. This paper extends the definition of Euler characteristic to more general geometries by glueing, and seeks for some possible applications. Result shows that Euler Charisteristic is an useful invariant that ties many subject in topology together. By studying Euler Characteristic in various cases, one may found interesting relationship between objects that are not seemingly related.

This mathematical exploration is around the topic of traffic. Traffic is a constant problem for many small and large cities with a significant population density and costs people efficiency. Traffic congestion adversely impacts metropolitan cities' living standards, reflecting on travelers' time cost, the number of resulting accidents, and trivially fuel consumption. Therefore good transportation is called for contributions everywhere today in the modern world. In this essay, mathematical theory is used to integrate the most influential cause - the culprit of traffic congestions.

This paper aims to introduce some prevalent techniques which have been used to solve linear systems. Firstly, we introduce the simplest method: how to eliminate variables in which the original systems would not be changed. We then introduce Gaussian eliminations, which work on the augmented matrix derived from a linear system. To the end, we present Cramer's rule, which computes the solution to a linear system based on matrix determinants. These methods have their application scenarios. For instance, eliminating variables has no limited condition for its operations. However, Cramer's rule needs to be under the condition of square matrices. At the same time, Gaussian elimination requires three elementary row operations, and Gaussian elimination paves the way for computing the rank of matrices. The choice between these methods depends on coefficient matrices in terms of dimensions and matrix properties such as singularity.

In this work, numerical prediction of the plastic deformation behaviors of perforated sheet metals, specifically AISI 1018 low carbon steel, C260 Cu-Zn brass, under uniaxial tension was carried out. An image-based mesh generation technique was applied to better represent and discretize the perforated geometry of sheet metal specimens. Digital image correlation measured displacement boundary conditions were also implemented into the finite element modeling. The predicted results of perforated sheet metal under tension as nominal stress-strain curves and localized plastic strain distribution agree well with the experimental observations. Possible error sources and uncertainty of this image-based finite element modeling were also discussed.

Since the outbreak of the novel coronavirus, the epidemic has received extensive attention all over the world. In this article, an evaluation system was established to analyze the epidemic prevention and control situation of some countries. And the ARIMA model was built to predict the epidemic situation in a short period of time. Then taking the United States as an example, the predicted values of the number of newly diagnosed cases, death rate, and cure rate in 10 days were obtained, which were then compared with the actual data. It is shown from the results that the ARIMA model can be used to predict the epidemic and provide a decision-making basis for the current world's epidemic prevention and control.

As the central object in the theory of complex analysis, Holomorphic functions have many elegant mathematical properties. Holomorphic functions are extremely valuable because, on the one hand, they are unexpectedly common, and on the other hand, they may be used to establish extremely powerful theorems. For example, To establish theorems like the prime number theorem, analytic number theorists commonly create holomorphic or meromorphic functions that hold number-theoretic information, such as the Riemann zeta function. Given knowledge about a holomorphic function in a relatively small part of its domain, one may extract information about the function's behavior in other a priori unrelated sections of its domain, according to Cauchy's integral formula and the identity theorem (and this is what allows things like contour integration to work). Due to difficulties in obtaining primitives of some real function, the fundamental theorem of Calculus does not work in most cases. In this article, we review the Cauchy theorem and use it as a tool to compute several real integrals.

The nonlinear Schrödinger equation (NLS) is widely used in different branches of physics. The GP equation is a kind of NLS equation with potential function term. In this paper, the soliton solutions of a generalized GP equation are explored. Firstly, we construct the Lax pair and Darboux transform(DT) of the equation. Then, we solve the single and double soliton solutions of the equation. Lastly, we draw the images of the single and double soliton solutions, and investigate the properties of the solitons. It is found that the amplitude of a single solitary wave does not change, and the shape and amplitude of the double solitary wave remain unchanged after the collision.

In recent years, the design and development of space-borne large-scale deployable antennas represented by ring truss deployable mesh antennas has attracted the attention of scholars in the aerospace industry from all over the world. Force density is the most widely used form finding algorithm for mesh antennas. The force density method assumes that the truss nodes of the antenna are rigidly connected, In this paper, an integral form finding algorithm for cable truss antenna is proposed, and the difference between the two algorithms is compared.

Suppose that $F\rlap{/}{\equiv }0$ is an entire function with hyper-order ${\sigma }_{2}(F)\lt 1$ and $m,n(n\gt m\ge 1)$ are two integers. If $F,\,{\Delta }_{\eta }^{m}F$ and ${\Delta }_{\eta }^{m}F$ share two values IM, then either $F\equiv {\Delta }_{\eta }^{m}F$ or ${\Delta }_{\eta }^{m}F\equiv {\Delta }_{\eta }^{m}F$. Moreover, if n = km for some integer k ≥ 2, then ${\Delta }_{\eta }^{m}F\equiv {\Delta }_{\eta }^{m}F$.

By non-dimensionalizing the radiation hydrodynamics equations in the form of Boltzmann-Euler coupling, the two moments approximate model written with non-dimensional variables is proposed and the non-dimensionalization leads to two parameters. An operator splitting method is developed to solve the dimensionless radiation hydrodynamics equations. The advantage of this method is that instead of solving nonlinear equations, it solves a set of ordinary differential equations, and these ordinary differential equations are decoupled. Lastly, the paper describes the Su-Olson problem which has a semi-analytic solution. The analysis shows that the solutions of numerical simulation agree well with the semi-analytic solutions in equilibrium diffusion limit.

The calculation of improper integral is of great importance. In this paper, we consider a kind of real improper integral using the method of complex analysis and prove the equation $\displaystyle {\int }_{0}^{\infty }\frac{{x}^{\alpha }}{(1+{x}^{\beta })x}dx=\frac{\pi }{\beta \,\sin \frac{\alpha \pi }{\beta }}$. Firstly we give the condition for the convergence of the improper integral $\displaystyle {\int }_{0}^{\infty }\frac{{x}^{\alpha }}{(1+{x}^{\beta })x}dx$ when $0\lt \alpha \lt \beta$, so we can see the integrand as a complex function. This function is defined on a simply connected set not containing zero to ensure that the function is holomorphic. We use three methods to prove the equation. In the first method, we choose a certain closed path as the boundary of the simply connected set, compute the integral along the path by the Residue Formula according to Cauchy's Theorem, and obtain the value of the real improper integral. In the second method, we use Mellin transform, while the idea is similar to that of the Residue Formula. In the third method, we find that the path we choose becomes simpler by a variable substitution. The function that is integrated along the new path does not have the problem of multivaluedness, so we do not have to define it on a simply connect set. Moreover, using the method of complex analysis, we prove that the equation holds when α, β are complex numbers, and the condition is about the real parts of α and β, i.e., $0\lt \Re \alpha \lt \Re \beta$. The real improper integral is just a particular case where α, β are real numbers. We compute the value of such kind of complex improper integral by some calculation and simplification, which is exactly $\frac{\pi }{\beta \,\sin \frac{\alpha \pi }{\beta }}$. We find the relationship between this integral and gamma function. The equation can prove a property of gamma function. We prove that the equation holds when α, β are complex numbers, and the condition becomes $0\lt \Re \alpha \lt \Re \beta$. We hope that this can be used in the research of more properties of the gamma function. The proof of the equation reminds us of a way of calculating such kinds of real improper integrals.

With the rapid development of social science and technology, various industries have higher and higher requirements for the quality of electronic products. SMT surface mount technology is the most popular process in the electronics assembly industry, and reflow soldering is one of the SMT mounting processes. Reflow soldering is mainly used to solder the circuit boards with components already attached. The solder paste is melted by heating to fuse the chip components and the circuit board pads together, and then the solder paste is cooled by the cooling of the reflow soldering, and the components are soldered. The plates are solidified together, and the furnace temperature curve records the temperature changes of the circuit board during this process. This article focuses on the printed circuit board welding production problem, through optimizing the furnace temperature curve for high-efficiency and quality-guaranteed production, and researches on the changes of the furnace temperature curve at different temperatures.

Complex numbers play a fundamental role in every branch of mathematics. First, we review the definition of the complex numbers, discuss their arithmetic operation properties and give proof of the Lagrange identity. At last, we take the investigation of topological notations of the complex plane

The generalized finite difference formula plays an important role in the meshless method for solving differential equations. The main purpose of this paper is to find the numerical solution of Helmholtz equation, an elliptic partial differential equation describing electromagnetic waves in physics, by using the generalized finite difference formula. Firstly, this paper introduces a simple and practical nodal distribution, which not only guarantees the uniqueness of multivariate polynomial interpolation, but also makes the matrix triangular, so that the constructed basic polynomials can be transformed into Lagrange basis polynomials. Secondly, the generalized finite difference formula is created by polynomial interpolation. Finally, a numerical example of Helmholtz equation under general boundary conditions is given.

The Kompaneets and the Ross-McCray equations can be used to describe the Comptonization process. However, the classical Kompaneets equation fails to describe down-Comptonization while Ross-McCray equation can be applied to describe down-Comptonization but inappropriate for a blackbody equilibrium. Fortunately, the Kompaneets equation extended by frequency and the Kompaneets equation extended by momentum can solve both problems. The different physical connotations behind the four equations bring about formal differences that allow the spectral evolution under different conditions to show different characteristics. In order to compare the differences between the four equations, the evolution of our common radiation spectra in astrophysics is numerically calculated in this paper. Besides, the differences between the four diffusion equations are discussed from their physical significance to the evolutionary phenomena. According to the results, the four equations evolve at different rates during down-Comptonization, from fast to slow in order of the Kompaneets equation extended by frequency, the Kompaneets equation extended by momentum, the classical Kompaneets equation and the Ross-McCray equation. The regression of the four equations is consistent during up-Comptonization. The Ross-McCray equation eventually converges to a different end state at near equilibrium. These results shed light on analytical development of Comptonization process.

In the construction and maintenance of bridge, it is very important to measure the strain of bridge accurately. However, the bridge strain measured in practical engineering is often mixed with a lot of signal noise, how to effectively remove the signal noise is a problem. Variational mode decomposition (VMD) is a denoising algorithm proposed in recent years, which has been applied in some engineering fields. Based on the vehicle bridge coupling theory, the strain of the bridge is obtained by numerical simulation, and the Gauss white noise is added to the strain to simulate the actual engineering situation. Based on VMD theory, a new bridge strain denoising method is proposed, and some key parameters are analyzed. The results of numerical simulation show that the denoising algorithm is effective. For example, when the signal-to-noise ratio of bridge strain data is 10dB, it can be increased to 21.67dB after using VMD denoising algorithm.

Multi-objective optimization is a major branch of the optimization category. On the basis of (V, η)-I symmetric invariant convex functions, the concept of generalized uniform (V, η)-I symmetric invariant convex function is proposed. Which is generalized uniform (V, η)-I type. Under the new generalized convexity hypothesis, several optimality sufficient conditions for multi-objective programming are proved.

Explosion yield estimation based on seismic data (ground motion in the near field) is one common approach to estimate near-surface explosion yield. The near-surface yield estimation models of different site media are usually established through chemical explosion experiments. However, the site media of current chemical explosion experiments are mainly soil and hard rock, lacking the experiment data and yield estimation model of soil-rock-mixture site. To solve the above problem, the seismic (ground motion) data of chemical explosion experiments at different depths in soil-rock-mixture site were gathered to establish the near-surface yield estimation model for soil-rock-mixture site. Besides, analysis was made regarding yield estimation accuracy and effects of rock-soil types on accuracy. The results show that the rock-soil type has great influence on the yield estimation accuracy of near-surface explosions, and the near-surface yield estimation model of soil-rock-mixture site has high accuracy in estimating explosion yield.

In this paper, we prove a monotonicity formula on cigar soliton. Using the monotonicity, we obtain a three-ball type theorem. There might be some people who have proved the three-ball theorem on Cigar Soliton before, but we use a new method. We add a weight function in the norm of u. By doing so, we make the calculation process much easier because when using integral by part, the term that integrates at the surface of ball would disappear as the weight function we add would be zero.

In this paper, the Euler-based numerical simulation method is employed to investigate the flow characteristics inside the Laval nozzle. In detail, the central difference method is used to discretize the flow field of the nozzle. And two kinds of flow conditions are simulated numerically: the isentropic flow from subsonic to supersonic, and the isentropic flow at full subsonic speed. The results show that the velocity increases and then decreases slowly, reaching the peak point near the nozzle throat. The pressure decreases non-linearly, and the density increases at first and then drops, following by a sharp rise later. The shock wave oscillation also appears in the flow from subsonic to supersonic, and this is because the artificial viscous term is not enough in discrete solution. Then we analyze the law of change of physical quantity by changing the shape of the nozzle. The results demonstrate that the larger the nozzle's curvature, the faster the change rate of a physical quantity is. And the velocity at the nozzle inlet approaches zero when the area at the inlet and outlet approaches infinity.

The aim of the present paper is to report on our recent results for GPU accelerated simulations of the gaseous detonation structure. Reactive Euler equations with a one-step Arrhenius chemistry model have been used for numerical simulation. And the NND space discretization scheme combined with Steger-Warming split method has been used. For time discretization we have applied the explicit third order Runge-Kutta method. We have obtained a speedup of 8 times ( in comparison to 30 threads openmp program) for the gaseous detonation simulation on a structure grid of 320 million points.

Fourier transform is a linear transform that applicants in the field of engineering and physics. The main goal of this research study is to give a self-contained study of the Fourier transform. As a modern analysis tool, the Fourier Transform has multiple applications in many fields, from the study of partial differential equations to dynamical systems. Fourier transformation has played a key role in many research fields. In this paper, we summarize the basic computational properties of the Fourier transformation and give proof of the Plancherel formula.

Regional innovation ability evaluation is a powerful engine to promote high-quality economic development, and also an important part of the strategy ofstrengthening the province through innovation. In this paper, the regional innovation capacity evaluation index system is constructed from three dimensions of regional innovation input, regional innovation environment and regional innovation output. By using the grey correlation analysis method, the regional innovation capacity of 13 cities and states in Hubei Province in 2019 is evaluated and ranked. The results show that the regional innovation capacity of Hubei Province is roughly distributed in ladder shape. In this paper, 13 cities and prefectures in Hubei province are divided into three levels, and some suggestions are put forward to improve the overall ability of scientific and technological innovation and to balance the development of regional innovation in Hubei province.

Numerical simulation refers to a computer research method which runs calculation based on a specific mathematical model to simulate actual physical processes. It is a powerful tool to analyse complex engineering problems. In this paper, a numerical method for solving the steady one-dimensional Euler equations using a two-step second-order difference scheme is developed. The method is implemented by a Python code. The method is applied to numerical simulation of flows in a Laval nozzle and used to investigate the influence of shapes of Laval nozzle. It is found that the physical quantities of the nozzle flow show a positive correlation trend with different throat positions. In terms of the temperature, density, and pressure, they increase during the initial evolutions, then reach the maximum point and produce significant fluctuations, indicating the flow flows into the transonic stage. Subsequently, those physical properties gradually tend to a stable value during the supersonic stage. Moreover, it is observed that the closer the throat is to the exit, the lower the Mach number at which it eventually stabilizes. Finally, a suggestion about the best shape of the nozzle which can realize the max efficiency is concluded. Despite using an inviscid flow model, the steady pressures are quite satisfactorily predicted over the range of frequencies studied.

Bubble flow is a phenomenon in which gas flows with liquid in the form of bubbles, which widely exists in production and life. As an important part of fluid mechanics, bubble flow also has a certain impact on the actual production process. This work briefly summarizes the development of research on bubble flow in recent years. Firstly, a brief introduction to the development process and current situation of bubble flow in chronological order is presented, and the main difficulties related to the research and development of bubble flow are also pointed out, including the complicated gas-liquid distribution pattern and the exchange of materials and energy between gas and liquid phases, which slow down scholars' research progress. Then, several theoretical models for bubble flow research are introduced. These models take bubble parameters into account, including the ultimate velocity of bubble rising, the transfer of energy and matter in the coalescence, and rupture of the bubble, the phase distribution of the bubble in the tube, and the interaction between the bubble and the liquid (the shear stress produced by bubbles affecting the turbulence field). This work briefly explains the ideas of each model and the results obtained, compares and analyzes their advantages, disadvantages and limitations, and predicts the theoretical and technical means required for subsequent theoretical model construction. Finally, part of the experimental research is summarized, along with the process and technical means of each experiment as well as the relationship between the experimental results and the theoretical research. It is hoped that this work could help to have a preliminary understanding of the research process and research methods of bubble flow, and provide some perspectives for incorporating bubble flow hydrodynamic theory into industrial applications and real life.

The technology of microfluidics is widely adopted in various fields such as biomedicine, microanalysis, and microelectronics. For example, pharmaceutical scientists often use microfluidics as a tool for drug delivery or cell separation. The LBM (Lattice Boltzmann method) is a commonly used numerical simulation in microfluidic researches. LBM is used extensively for simulations containing complicated boundary conditions and multiphase interfaces as it needs relatively low computing power compared to other numerical simulation methods in complex situations. The brilliant capability in parallelism also allows it to have a high multitasking performance, increasing overall efficiency. In this paper, we reviewed several typical applications of LBM in the following three fields: (1) particle regulation; (2) flow control; (3) drug delivery. We concluded defects in current studies and proposed potential improvements to be investigated in the future.

In recent years, many experiments, for example, like AMS-02, CALET, DAMPE, HESS and Fermi-LAT, have enhanced their precision in detecting the flux of cosmic rays, especially for high energy particles. We aim to interpret the electrons and positrons data above 1 TeV by using a model considering supernova remnants (SNRs) and the single pulsar as the major sources of electrons and positrons. We explain the rationality of applying the continuous scenario model for the single pulsar. We use the data above and choose seven parameters (d, t, γ_e, E_tot, Q_0,SNR, γ_SNR, E_c,SNR) to perform the best fit for electrons and positrons flux at Earth. We perform the fit for the data above 1 Tev and 10 GeV to analyze how the single pulsar and SNRs contribute to the total flux. We also find that the fit ranges we choose for the parameters have a significant influence on the result. By considering this effect, a further analysis on the best fit by constraining γ_e and E_c,SNR shows that the pulsar PSR B0656+14 can provide most of the flux to the data above 1 TeV.

The "Crossing the Desert" mini-game needs to consider a variety of mutually restrictive factors. Players must coordinate funding and path issues to find the optimal strategy to reach the end within the specified time. Based on the study of crossing the desert when the weather conditions are known, this paper establishes a topology model, and uses priority search and game theory to solve the player's optimal strategy for passing through the barriers. First, the entire map is digitized and represented by a general topological map; then, given the map and weather distribution, the highest profit of each path is calculated. Specific calculation details require further analysis to arrive at a more practical and efficient route planning plan. The model proposed in this paper only provides a more effective solution for this problem from the perspective of mathematical modeling.

The de Laval nozzle has been widely studied and used in many industries ranging from the aerospace to the dairy industry. This study investigated two numerical methods used to simulate nozzle flow, namely one-dimensional Euler equations and the small disturbance equations (SDE) based simulation. The simulation results reveal that the SDE method accurately captured the shock bubbles in the nozzle flow. However, the one-dimensional Euler method is not sufficient to handle two-dimensional nozzle flow simulation because at any particular nozzle location, a mix of subsonic and supersonic flow can occur. This research allows an informed choice between Euler and SDE method.

In this paper, the exponential stability in mean square of the exponential Euler method for semi-linear stochastic differential equation with piecewise continuous arguments is obtained. Firstly, the exponential Euler scheme of the equation is given. Secondly, under Lipschitz condition, sufficient conditions of exponential stability to the exact solution are achieved in mean square. Furthermore, it is proved that exponential Euler method preserves exponential stability of the exact solution without any restriction on the step-size. Finally, an example is provided to illustrate our theories.

A nozzle is a versatile fluid device that is utilized to control the characteristics of a fluid flow. In order to enhance the performance or better the design of the nozzle, it is critical to carry out extensive research into its inner flow. Thus, the investigation of the nozzle flow remains an important challenge. This article used a numerical simulation method based on Euler equations to solve the internal flow of the divergent, convergent-divergent, and convergent nozzles. Specifically, we simplify all three equations of the one-dimensional Euler equation (the continuity equation, the momentum equation, and the energy balance equation) by assuming a constant nozzle shape, then discretizing them using a central differential scheme. After conducting systematic research on the influence on the internal nozzle flow due to the change of nozzle shape, we found that the mass flow rate varied as it fluctuated and peaked at the middle point of the Laval nozzle. However, with the nozzle becomes flatter, the mass flow rate appeared to restore conserved. Moreover, it appeared that the flatter the Laval nozzle is, the smaller acceleration and smaller decrease of the pressure of the fluid flow gets. In short, the results suggested that it is dependable to apply this numerical method when analysing a relatively flat Laval nozzle.

The Laval nozzle is a very widely used device that has made great contributions due to its invention in aerospace, mechanical engineering, applied chemistry etc. The development of many modern technologies relies on the establishment of nozzles. In this paper, the current research advances on the numerical simulation methods of the nozzle flow and nozzle-related investigation or applications in different fields are reviewed. For example, the analytical and numerical simulation of nozzles in subsonic and supersonic modes are mentioned in this article. In summary, although the applications for various fields of nozzles are now very mature, the solution of the Navier-Stokes equation for nozzles is still a future direction where science and mathematics need to be strongly developed.

The furnace temperature curve of the reflow furnace records the temperature changes during the soldering process of the printed circuit board components, which is the main factor influencing the quality of component soldering. Given the temperature of each temperature zone and the furnace passing speed, use the conveyor belt model, Fourier's law and Newton's law of cooling to establish a mathematical model to find the midpoint temperature of each temperature zone and obtain the furnace temperature curve. A search algorithm is further set up to obtain the maximum conveyor belt passing speed.

Complex numbers play a fundamental role in multiple engineering fields. In this paper, we first review what the algebraic form of a complex number is and define several operations on complex numbers. At last, we introduce Blaschke factors and discuss inequalities of complex numbers.

A Fourier series is one of the methods to represent a function as a summation of orthogonal functions. In this paper, we give a detailed proof of a property regarding the behavior of a function at its continuity points. Finally, we show that an integrable function vanishes at its continuity points if all of its Fourier coefficients vanish.

The free boundary value problem of reaction-diffusion equation is studied. In the first chapter, the Logistic diffusion equation with free boundary and the main work of this paper are introduced. The second chapter is the preparation work, explaining the existence and uniqueness of the solution of the free boundary Logistic diffusion equation. In the third chapter, we prove that messages propagate or disappear in finite time by constructing the asymptotic properties of upper and lower solutions. In chapter 4, the criterion of diffusion disappearance is given, and the threshold of information propagation or disappearance in finite time is given according to the initial value. The results show that when the given value is larger than the threshold value, the information will propagate in the whole region. Otherwise the message will stop propagating for a limited time.

A produce based on a guaranteed posterior error estimates is proposed to estimate the accuracy of solution of the time-dependent heat equation. The time-dependent heat equation is discretized by the backward Euler scheme in time and conforming finite elements in space. The error between the exact solution and the numeric solution in space is bounded by a guaranteed posterior error estimates which is calculated at every time step. Numerical results demonstrate that the produce can be used to evaluate the accuracy of numerical solutions.

Due to the wide application of the Delaware nozzle in many fields, the flow characteristics of the subsonic-supersonic isentropic flow in the De-Laval nozzle are analyzed numerically. The flow in the nozzle can be simplified to a quasi-one-dimensional flow problem. First, the MacCormack format is employed to discretize the control equations in conservative form. Then, the results with and without artificial viscosity are compared. Grid independence is also discussed. The results show that the numerical solution and the theoretical solution agree very well, indicating that the numerical simulation results are very reliable. In addition, a higher pressure will reduce the peak and valley values of Mach number, pressure, density, temperature and velocity in the nozzle, and these extreme values of subsonic supersonic isentropic nozzle appear earlier. Additionally, the shock wave is accurately captured, and the shock wave is behind the throat. This research is of great significance to understand the flow characteristics in the nozzle.

The sweeping development of mathematics is largely due to the introduction of complex numbers. Although complex numbers are just absurd notations for most people, complex numbers play essential roles in engineering fields. When mathematicians began to take an investigation of complex numbers, human beings enter a marvelous world. Complex numbers convince scientists that our world is magical, full of wonderful insights and even miraculous. In this paper, I first review several basic properties of complex numbers. The set of complex numbers is a group not only under the operation of the multiplication but under the operation of addition. Then, I show a visualisation of complex numbers through building a bijective connection between complex numbers and points on the complex plane. I also give several alternative expression forms of complex numbers, namely the trigonometric form and the general form. By invoking the arithmetic properties of complex numbers, I prove two trigonometric identities.

The fracture toughness and load displacement relationship of the three point bending (3pb) asphalt mixture specimens with different notch position were experimentally and numerical investigated. Firstly, the extended finite element was used to simulate the three-point bending and fracture process of asphalt mixture beam, and the load and displacement curves were obtained, which were compared with the test results to verify the accuracy of the XFEM model. Secondly, J contour integral method in ABAQUS software was used to solve the stress intensity factor under the three-point bending of asphalt mixture, and the fracture parameters of asphalt mixture were provided by the numerical calculation results. The results showed that the offset distance increases, the influence of the type II crack in the fracture process increases, but the type I crack still was dominant.

Because subsonic-supersonic Laval nozzles are widely used in many fields, the flow characteristics of the subsonic-supersonic Laval nozzle are studied by numerical simulation. First, the MacCormack scheme is used to discrete the non-conservation and conservation forms of the governing equations of the flow in the subsonic-supersonic Laval nozzle. The flow states inside the nozzle in both formats are then compared. The results show that the numerical results of the two schemes converge well. The variation of the fluid velocity and outlet pressure is more pronounced in the non-conservative form. For the conservation form, the error of the mass flow rates at different positions is constant and acceptable, which is about 1%. For the non-conservation form, the error at different locations within the nozzle is different, reaching a minimum error at the throat. In addition, studies on grid length have demonstrated that the finer the grid, the more accurate and stable the results Moreover, for the subsonic, supersonic isentropic flow in the quasi-one-dimensional nozzle, the Mach number and velocity become larger along with the nozzle. In contrast, other variables such as pressure and temperature become smaller. There is a subsonic region in front of the throat and a supersonic region behind the throat. The Mach number of the nozzle exit can be very high. Finally, the study reveals the internal flow state of the subsonic, supersonic nozzle, which can provide theoretical reference for the application of the nozzle.

Numerical simulations were conducted for calorically perfect gas and high temperature thermochemical nonequilibrium gas by CFD methods. Hypersonic Flow with High Temperature Thermochemical Nonequilibrium Effect was studied. The results showed that high temperature effects change the flow field and reduce heat transfer observably. For high temperature flow the kinetic energy associated with hypervelocity flight is converted into increasing the temperature of the air and endothermic reactions including dissociation and ionization of the air past shock. Thus, temperature in the shock layer dramatically decreases. All these works build foundations for further hypersonic aerodynamical computations and engineering applications.

The nozzle is a widely used device in daily life, such as water fountains to rocket engines. It is important to find out the influence of the position of the nozzle throat for the application or the design of the nozzle. To that end, the finite difference method was employed to solve the 1D Euler equations to obtain the flow inside the nozzle. To implement the method, an in-house python code was developed. The relationship among the velocity, pressure and density in the convergent-divergent nozzle flow was found. It is observed that: the velocity rose quickly along with the nozzle and reached the top before a rapid decrease; pressure remained constant initially, which eventually began to drop; density dropped steadily and had a turning point. Moreover, the influence of the nozzle throat position is investigated thoroughly. It is observed that the position of the nozzle throat influences the velocities at the nozzle exit. The faster the flow reaches the throat, the higher the velocity or Mach number at the exit boundary.

The traditional inertial impact switch based on experience design often appears in the problem of early closure within the threshold. To solve this problem, a new method of inertial switch design is presented in this paper. Firstly, the force analysis of inertia switch is carried out and a mechanical model is established. Through the model, it is concluded that the closing threshold of inertia switch is related to the spring resistance, the mass and center of mass of contact pole, the spring radius of contact pole and the horizontal projection of the minimum distance between the spring and the reference point. After that, the mechanical model is verified by simulation, and the correctness of the mechanical model is proved. The experimental results show that the proposed model can effectively solve the problem of premature closing of the inertial switch within the threshold and improve the pass rate of the product.

The quality of urban drinking water has a great influence on the life and work of the residents. Only when the water quality reaches the National Standard, the safety of the residents and the development of society and economy can be guaranteed. In order to study the water quality of tap water in Guangzhou, 42 indicators from 8 water plants in Guangzhou were selected as sample data, and the grey correlation method was used to analyze and evaluate the indicators. The results showed that the water quality of each waterworks met the national sanitary standard for drinking water, among which the water quality of D, G and H waterworks was better than that of E and F waterworks. Based on the comparative analysis of the test indexes, some suggestions for improving the water quality of the factory are put forward. The grey correlation method can be used to evaluate the water quality of waterworks, which can not only make full use of the original data, but also be easy to calculate and the evaluation result is objective, it's very practical.

Lithium-ion battery, one of the most commonly used portable power supply devices, still has many drawbacks, such as low rate of charging and discharging, poor safety and low energy density. Through using the appropriate two-dimensional (2D) anode materials, the performance of lithium-ion batteries (LIBs) can be greatly improved. In this study, through first principles calculation, we compare two 2D materials as anode materials, VCl₂ and VBr₂, and we find that: (1) VCl₂ cannot stably adsorb lithium, because lithium breaks down the original crystal structure of VCl₂; (2) VBr₂ not only adsorbs lithium stably, but also has good properties of voltage, conductivity and diffusion barrier. Given these advantages, VBr₂ is predicted to be promising for being applied as an anode material of LIBs. It is hopeful that this study has a guiding significance for the research and development of two-dimensional materials as battery anode materials.

The equations of time-delay fractional order financial dynamic system are studied. Fractional calculus, as the expansion of the corresponding integer parts, has more complex definition and update laws. It is found that modeling by fractional-order differential equation in some nature phenomenon and engineering field can be consistent with essentiality of the phenomenon and matters, such as in mathematical modeling of viscoelastic materials, electromagnetic wave and other fields. Moreover, there exist more complex dynamical behaviors in fractional-order dynamical system than in integral-order one, and it has more preferable history memory ability. Therefore, it is of theoretical value and practical significance researching the fractional-order nonlinear dynamical systems on the base of the researches on the traditional integral-order nonlinear dynamical system.

Gödel's Incompleteness Theorem is considered a significant finding in logic and mathematics and has many indications in physics. This paper analyzes and concludes that Gödel's Incompleteness theorem potentially parallels problems of reference frame in classical physics and other issues considering self-inclusion with regard to consciousness, which not necessarily relevant to quantum mechanics. It also demonstrates further investigations on Gödel's Incompleteness theorem, which can add to our understanding of problems in classical physics and philosophy of mind that still require in-depth research.

Quantum physics is changed in an unexpected way. Scientists believe that the study of quantum physics will play an important role in future technology. Therefore, it is meaningful to study quantum physics. The aim of this paper is to provide an initial explanation of the quantum world through the classical single particle double-slit interpretation experiment, and to answer a few questions about the experiment using mainstream quantum mechanical interpretations, such as the Copenhagen interpretation and the Many-Worlds interpretation. The superposition state, which is the core of the Copenhagen interpretation, is then introduced. The Einstein-Podolsky-Rosen paradox and locality are also introduced, and the relationship between the latter and special relativity is analyzed. The author also discusses whether non-locality leads to retrocausality. Finally, the author discusses the relationship between multi-world interpretation and Schrodinger's cat, but this may raise some philosophical questions. However, realist and hidden variable interpretation are also consistent as long as locality is abandoned.

The spin of the black hole is one of the deterministic qualities for itself, providing us with the knowledge of its birth and evolution. In this review, I described two kinds of the spin measurements, respectively based upon electromagnetic waves and gravitational waves, and discuss the meaning for the black hole spin evolution indicated by the spin results measured by these methods. We find that for X-ray binaries, the spin results are much bigger than those results for natal spin, and for binary black holes the spin result are much smaller than those in X-ray binaries, which may be resulted from their different evolutions. For binary black holes detected through gravitational waves, formation channels of the binary systems are essential for interpreting spin results in these binary systems.

The Alpha Magnetic Spectrometer (AMS-02) experiment on board of the International Space Station is revolutionizing our understanding of cosmic rays. AMS-02 Collaboration has released the most precise data for several cosmic-ray species ever. Electrons are among the rarest cosmic particles and have been measured between 0.5 and 1000 GeV by AMS-02 with very high precision. This paper explores the possibility that the origin of cosmic-ray electrons is associated with the acceleration of particles from supernova remnants. In this paper we calculate the flux of cosmic electrons from two nearby and radio bright sources in our Galaxies: Vela and Cygnus supernova remnants. We concluded that they contribute at the few percent level at 1 TeV. Finally, we calculate the flux of other bright sources taken from radio catalog of supernova remnants, finding that their cumulative contribution is about 10% at 1 TeV. Further exploration of the AMS-02 data could lead to the conclusion that supernova remnants are among the important accelerators of cosmic-ray electrons in the galaxy.

Quantum entanglement and spacetime are strongly associated fields in modern physics. This paper investigated the connection between quantum entanglement and spacetime, particularly focusing on how they mutually form one another. The quantum mechanics and relativity theory cannot peacefully coexist in some cases, so it is difficult to combine them together. This article analyzes studies which suggest quantum entanglement can fabricate the continuous spacetime, and studies that imply spacetime geometry can explain quantum entanglement. These findings may bring us a better vision of the relation between quantum mechanics and relativity, as well as pave a path to the unification of quantum mechanics and relativity.

The evolution of galaxies is a very important topic for understanding of the evolution of Universe, and this evolution results in that galaxies are not evenly distributed in space. The probability of distribution of galaxies in space can be computed according to probabilistic models. Although the evolution of galaxies has been studied from various angles, the probability of distribution of galaxies in space has yet to be studied with respect to their positions over time. Of galaxies, the positions of galaxies in Local Group have been studied more thoroughly. In this study, we computed the distribution probabilities of galaxies in Local Group, i.e. the distribution probabilities of galaxies at present and previous positions, and the probability for an even distribution of galaxies in space. The results show that galaxies in Local Group have an extremely small chance to distribute evenly, and the distribution probability of galaxies in Local Group increases from the past to the present. The implication is that the accelerating expansion of the universe is increasing the distribution probability of galaxies, i.e. the accelerating expansion of the universe is a probabilistically probable event.

A stereo PIV technique is utilized for the analysis of the flow field over a 70° delta wing. Instantaneous velocity vectors are measured at a series of stations in the chordwise direction. The angle of attack from 0° to 45° are investigated with the free stream velocities of 20, 30, 40, 50 and 60 m/s, respectively. The velocity measurements conducted in a small-sized subsonic wind tunnel (1.8m×1.4m) at China Areodynamics Research and Development Center (CARDC) are compared with the previously obtained PSP results. The comparison shows a good agreement with each other, providing the validity of the PIV technique as well as indicating its effectiveness to investigate complex flow fields in a fairly large-scale wind tunnel facility.

Quantum entanglement, an unknown "spooky action" that confused scientists like Einstein, has become one of the research hotspots in quantum mechanics. Based on the technique, applications distributed among diverse fields have appeared, e.g., cryptography, computer science, distant communication, etc, which has attracted interest from researchers in varied fields. This article reviews the basics of manifestations of quantum entanglement, namely quantum key distribution, quantum computation, and quantum teleportation. An overview of milestones is presented, e.g., Bennett-Brassard 1984 protocol, entanglement-based quantum computers, Shor's and Grover's algorithm as well as the process of transporting quantum states at an unprecedented speed. Moreover, the implementations of quantum entanglement are demonstrated, evolving these technologies to higher levels. Finally, a brief conclusion and expectations for future developments on entanglement are given.

Standard solar models built from spectroscopy and from the helioseismology predicted inconsistent solar metallicity. Solar neutrino fluxes can be used to test two models. Hypothesis test was performed using ⁷Be, ⁸B CNO neutrino fluxes. Current experimental results already disfavored the low metallicity standard solar model at a significance of 2.1 σ. Considering new experiments that can improve the precision of the measured CNO neutrino flux are being planned, it is important to study their potentiality to discriminating two standard solar models. We evaluated the expected significance to reject the low metallicity solar model where the assumed central value and uncertainty of the measured CNO neutrino flux vary within the range of 3.5 × 10⁸ -- 8 × 10⁸ s⁻¹cm⁻² and 1%--20%, respectively. It was found that the potentiality of future experiments to reject the low metallicity standard solar model strongly depends on the central value of the CNO neutrino flux measurement. When the central value is the same as the value measured by Borexino in 2020, the required precision to reach 3 σ and 5 σ are 20% and 8%, respectively, which are both achievable.

In order to improve the radiation performance of low-frequency sound device, a design method for a long-stroke low-frequency sound device is proposed, which mainly involves the design of the drive structure and radiation structure and the drive structure is designed with an electric long stoke. The flat plate and conical structures of piston plate of radiation structure are compared, and a long-stroke radiation structure with a low-damping guide support motion pair uniformly distributed in the circumferential direction is proposed. The numerical simulation results show that, with six straight-line bearings circumferentially and uniformly distributed, the displacement and stress distribution of piston plate are better than those of piston plate with three straight-line bearings installed in the same manner. The sound device is further tested in the experiment, and it is found that the vibration displacement amplitude at the center of piston plate is 3.9 mm (20 Hz), and the maximum sound pressure level of 40 Hz is found at the location 10 m from the center of sound device along the axial direction, reaching 105.3 dB.

Contemporarily, the gravitational wave has played an important role in the field of astronomy. A systematic review of the gravitational wave is presented in this paper. Primarily, the principle of gravitational waves is introduced in both Newton and Einstein's concept of gravity. Besides, we deduced the solution of Einstein's field equation. Moreover, we described the detection of gravitational waves with the mechanisms of the two gravitational wave detectors LIGO and Virgo Finally, the applications of gravitational waves are discussed in detecting black hole mergers, neutron star mergers, gamma-ray burst, and core-Collapse Supernova explosion. The gravitational wave could contribute to more scientific findings in the future and solve more mysteries of the Universe.

The theory of dark matter (DM) has been perfected developments throughout the past ten decades. Although DM are still not being detected nowadays, plenty of models were proposed in detail. This paper focus on supersymmetry, axions, and neutrinos, which are three typical models of DM. Specifically, the basic descriptions of these kinds of DM are introduced, including definition, principals, detections, etc. Besides, the importance and latest studies/proposals are discussed. These results will offer a guideline for further research in DM.

Chaos Theory is not only one of the most active branches of the dynamical system, but also one of the most important topics of the nonlinear scientific field. The main purpose of this thesis is to introduce and illustrate the basic concepts, characteristics, and various applications of Chaos Theory. Chaos Theory is widely applied in physics, biology, and economics fields etc., so the thesis will also exhibit interdisciplinary research on Chaos Theory. The conclusion is that the most important refinement of Chaos Theory is its sensitivity to initial values, and subtle changes will affect the final shape and state of the entire structure.

A high mass X-ray binary is a binary system consisting of an accreting neutron star or black hole and an early-type star with more than 10 solar masses. A compact star produces X-ray radiation during the accretion of material from the stellar wind or Roche lobe. High-mass X-ray binaries are ideal places to study dense stars, stellar winds, material exchange between binary stars, and the evolution of massive binary stars. In this review, the observational properties, classification, and evolution of high mass X binaries are described. According to the mass and types of X-ray resources, this review classifies the X-ray binaries and list out their features.

The study of particle system is inevitable to study its statistical law, and understanding the distribution of particles is fundamental and important. In this paper, the author points out the relation between the number of states of free particles in the relativistic and non-relativistic cases, and gives formulae for each case. Furthermore, the author enumerates some popular and basic particle systems, as well as classifies them in terms of relativistic and non-relativistic cases, also their basic formulae are deduced and enumerated in this paper. These particle systems are applicable to different formulae for the density of states for relativistic or non-relativistic cases. It is concluded that the different densities of states in different cases are related to each other through the de Broglie relation and the energy formula.

While the research of the celestial body is essential to understand the universe, it requires considerable costs in time and equipment. However, the computation methods can be used to analyze the astrophysics process in a simple approach. In this paper, a model of computing the motion of solar system based on Euler's method is established. The model divided the process into infinite small time steps, and iteration is used to represent integration and update the position of planets over the process. The NumPy and matplotlib are used to visualize the motion. By the analysis of momentum and energy through the process and the application in the 2-body situation, the model is proved to be valid and accurate.

Primordial black holes (PBHs) are probable candidates of dark matter (DM). In this review, we gathered data, evidence, and opinions from previous literatures to support the theory that PBHs are a form of DM. Besides, the formation of PBHs is introduced as well to give a more comprehensive description of PBHs. According to the results, the PBHs could assemble DM, or be DM itself, or generates DM, or influence DM. On the other hand, DM could have the similar influence on PBHs. Different relationships are discussed between PBHs and DM, including entropy, microlensing, distribution, growth rate, mass, and different detections (e.g., gammy ray). These results offer a guideline for future studies of PBHs.

Spin is an important property of black holes. The formation of jet and black hole is related to spin. At present, there are two commonly used spin measurement methods, the X-ray reflection method, and the continuum-fitting method. By analyzing the distribution of black hole spin in black hole binary system measured by these two methods, we find that in high-mass X-ray binaries, black holes have high spin, this is because the companion stars have a large mass and short lifetime, the black holes don't have enough time to accelerate the spin by accretion so must be bored with high spin. However, in low-mass X-ray binaries, the spin will be widely distributed because they have enough time to increase spin by accreting gas. And according to the existing data, we can also get the relationship between spin and jet. However, both methods are affected by the complex physics of the accretion disk, in the future, gravitational waves will become another effective tool to measure the spin of black holes.

This paper concerns the properties of submanifolds of doubly warped product space. For Riemannian manifold B and interval I, we investigate some surfaces properties of doubly warped product manifolds _hI×_fB, and obtain the scalar curvature for the doubly warped product space.

It is suggested that some comets originated from solar activity and were ejected by the sun in late period. In order to verify this point of view, the new star ejected from the sun' s surface is regarded as a particle, and the orbital evolution problem is simplified to the problem of a particle moving around a sphere, whose motion is disturbed by the asteroid belt and planets. By analyzing the influence of gravity, Coriolis force and viscous force on the trajectory of the new star, the trajectory of the new star is obtained. The results show that the orbits of small objects ejected from the surface of the sun are very similar to those of comets, which also explain the rotation of comets along the direction of the sun' s rotation, etc., indicating the physical rationality of the comet hypothesis.

Chongqing Bus falling into the river shocked the whole country. In order to find out whether the Coriolis illusion is one of the real causes of Chongqing Bus falling into the river, the mechanism of Coriolis acceleration is studied, and the corresponding analysis of the phenomenon caused by the Coriolis acceleration is carried out. The driver's head in Chongqing Bus falling into the river is modelled and calculated. It is concluded that the effect of Coriolis acceleration can be ignored when the object's motion speed is low, that is, the main cause of Chongqing Bus falling into the river is not Coriolis illusion.

Hubble constant (H0) is one of the most important parameters in cosmology. There are mainly two ways to determine the value of Hubble constant, which measure the properties of early universe and the late universe, namely cosmic microwave background radiation (CMB) and Type Ia supernovae (SNe Ia). Those who had used these two methods to measure the Hubble constant won the Nobel Prize in Physics respectively, in 1978 and 2011. This article introduces the principle of accelerating universe and the methods to measure the Hubble constant. We analyze each method and discuss the uncertainties of them. In addition, we investigate possible reasons for Hubble constant discrepancy based on previous studies. We discuss about the conclusion and prospects of Hubble constant measurement.

In severe polluted areas, objects tend to be redder because of the refraction of light by particulate pollutants through the atmosphere. In this study, a particle scattering model is established to simulate the scattering of pollutants based on assumptions. Then the actual data of lunar extinction is obtained by using an astronomical telescope to measure the RGB color differentiation at different altitude angles of the lunar surface in Shanghai caused by pollutants scattering. The results are combined with the particle scattering model calculation to obtain the estimated pollutant thickness in the atmosphere.

The need for efficient cell sorting, which is a useful technique in medical and biological research, has boosted the development of microfluidic cell sorter. This review paper concludes the basic principle of microfluidic sorter which covers active and passive sorting techniques. The work process, sorting efficiency, advantages, disadvantages, and avenue for improvement are explored for each type by introducing several typical sorters. The industrial implementation and potential development are also conveyed. In conclusion, developing efficient microfluidic cell sorter offers a greater control over cell distribution and is fundamental in realizing efficacious cell sorting systems.

The relationship between time and space has been studied by people for a long time. Although it is still in a development stage, it is necessary to be researched. This paper aims to introduce the space-time relationship from classical physics to modern physics. Subsequently, the author analyses Galilean transformation and its limitations to lead naturally to Lorentz transformation. After that, the special relativity is shown by giving some examples. Special relativity is based on two postulates, including the Principle of Relativity and the Principle of Invariant Light Speed. Mickelson-Morley experiment and mass-energy equivalence can explain the Principle of Invariant Light Speed well. The paper then introduces intuitively Minkowski spacetime, which can help readers understand four-dimensional spacetime as well. In addition, the distinctions of different dimensional spaces will be mentioned. The essay also discusses the general relativity that time is also affected by gravity and the principle of equivalence, but it's really only a quick cheat-sheet summary. Finally, this paper points out the possibility of reaching the speed of light, and if it can be achieved, spacetime will be changed so that lots of unbelievable consequences will come one after another. The article draws a conclusion that time and space are like a whole, one change makes all change.

Contemporarily, convincing evidence is lacking for the existence of micro black holes due to their small masses. In this paper, a specific situation is discussed in which a Schwarzschild micro black hole, with a mass of up to 10¹³ kg and hence the lifetime longer than the universe's age flies towards the Earth with the typical speed of an asteroid, ${\mathscr{O}}$(10 km/s). The analysis of the two-body problem shows that Earth's surface will be tidally affected only if the initial configuration space, namely space of the impact parameter and initial velocity, satisfies a very strict relationship. Even if the micro black hole directly goes through the Earth core, its small mass restricts its accretion rate so much that its accreted mass is negligible. We also consider the possibility of detecting the black hole optically by its optical distortion. Analysis of the size of its Einstein ring indicates that one would need accurate instruments to detect such an effect. Finally, we analyze the emission spectrum of the micro black hole by assuming the validity of the Hawking radiation and conclude that its unique energy spectrum might be detected and used as evidence of a bypassing micro black hole.

Liquid drops exist everywhere in daily life. However, such a common phenomenon involves complex mechanisms, and therefore has never been investigated thoroughly—the achievements of the studies are limited and unsatisfactory. In this article, the phenomenon of a liquid drop falling in another immiscible fluid (mostly its settling speed) was investigated. A series of experiments, in which liquid drops emerging from a syringe falls inside a measuring cylinder filled another fluid, was performed. Fluids used in the experiment include ethanol solutions and rice bran oil; for rice bran oil, its temperature-dependent viscosity was approximated by a function, which was proved effective on other similar oils. The dependence of the settling speed of the liquid drop on the density difference between the liquid drop and the surrounding fluid, as well as on the temperature of the surrounding fluid, was measured. Finite-Element Modeling (FEM) simulations were then carried out to model these situations, and lead to results that agreed with the experiments. It also visualized the flow field and revealed more details of the two-phase flow that were not detectable by our devices. Furthermore, inspired by some previous formulae, quantitative equations were derived semi-analytically by modifying an approximate drag formula originally developed for a rigid object at moderate Reynolds numbers. The form of the modification was never introduced in any other previous works. The equations were validated by simulations under ideal conditions.

This paper is based on the new theoretical system of physics in the 21st century published by the British Physical Society IOP Journal of physics in 2020. It is a challenge to some popular views of physics today! It negates the big bang theory, general relativity, epistemology of gravitational wave existence and epistemology of antimatter existence. It denies the research method of relying on the electron Collider to find the smallest particle in the material world! It is put forward that "the state of matter light" is the epistemology of the fifth physical state; stars are not composed of hydrogen and helium, nor are they nuclear reactions; the formation reason of Galactic System and the epistemology of cosmic unit. This paper is a summary and deduction of the new theoretical system of physics in the 21st century.

In this paper, the finite square well and its application are investigated, namely the quantum state discrimination. The finite square well is treated in all standard textbooks on introductory quantum mechanics. It is used as a simple 'model of departure' in many areas of physics. In atomic and molecular physics, it may be used as a model of an electron moving in the mean field of a linear molecule It also arises as the partial wave radial equation for a spherically symmetric, finite square-well potential. The Schrodinger equation for finite square well is solved. The domain is divided into three regions by the existing potential V₀, so for convenience, those three regions are named Region I, Region II, and Region III, respectively. Specifically, the potential of Region I and III are V₀, and that of Region II is 0. Also, a constant is needed to make this wave function normalized. Then since it is a wave function, it is necessary to make sure the function is continuous and differentiable everywhere within the domain. Besides, the wave function needs to be either odd or even just like infinite square well. After that, the wave functions for the three intervals can be obtained, and the exact quantum state is distinguished from a group of different quantum states. In the end, two wave functions are obtained; one for even form and one for odd form. Then all the energy level of the corresponding function with different wavelength needs to be found and listed. Local discrimination of orthogonal quantum states has attracted much attention during the last twenty years. The results are applied to the quantum-information task of state discrimination, by using the obtained six states in finite square well. It is assumed that all the quantum states are locally distinguishable, and the six states are distinguished using the hypothesis of quantum measurements.

The study of exoplanetary systems can help us understand the formation and evolution of the solar system itself and search for terrestrial planets in the habitable and extrasolar lives in exoplanetary systems. Exoplanets have become an important area of astrophysics in the last two decades. This paper reviews five different methods to detect exoplanets, including direct imaging, astrometry, radial velocity, transit event observation, and microlensing. These approaches could expand the sample size of exoplanets and further our understanding of the types, formation and evolution of exoplanets.

Contemporarily, the dramatic progress that proceeded in astrophysical simulation stems from the computational upgrade. Simultaneously, the relative algorithms are from naive to mature. Some imaginary algorithms spring up like bamboo. The typical methods are Monte-Carlo, Particle-in-Cell, and Magnetohydrodynamic considering the relativistic effect. This paper reviews the typical progress of the above three methods as well as their hybrid method in astrophysics. These results offer a guideline for further research in astrophysics simulation.