Why Is an Integral an Accurate Value?

The derivative and integral in calculus are both exact values. To explain this reason, the integration interval can be infinitely subdivided. The difference in area between curved trapezoids and rectangles can be explained by the theory of higher-order infinitesimal, leading to the conclusion that the difference between the two is an infinitesimal value. From this, it can be inferred that the result obtained by integration is indeed an accurate value.

The Theoretical Significance and Reality of Imaginary Number

It is a fact that imaginary numbers do not have practical significance. But the role of imaginary numbers is very broad and enormous, due to the existence of Euler’s formula. Due to Euler’s formula, imaginary numbers have been applied in many theoretical theories. One of the biggest functions of imaginary numbers is to represent changes in phase, which is indispensable in signal analysis theory. The imaginary numbers in quantum mechanics pose a greater mystery: do the imaginary numbers really exist? This question still needs further scientific development to be answered.

Why Are There as Many Elements in the Cantor Set as There Are Real Numbers?

There are many important concepts in linear algebra, such as linear correlation and linear independence, eigenvalues and eigenvectors, and so on. The article provides a graphical explanation of how to distinguish between the concepts of linear correlation and linear independence. The conclusion points out that linear independence means that there are no two (base) vectors with the same direction in a vector graph;otherwise, it is a linear correlation.

The Meaning of Divergence

The concept of the divergence is fundamental in electromagnetic field theory, yet they are especially difficult mathematical concepts. Understanding this concept requires strong spatial and abstract thinking. When teaching, through the graphical and quantitative methods presented herein, the significance of the divergence is displayed by a quantitative method for the first time. Through these methods, the concepts of the divergence can be grasped more easily. These explanations will be helpful for students to strengthen the understanding of this concept and has a certain reference significance.

Analytical solution for longitudinal deformation of shield tunnel induced by overcrossing tunnelling considering circumferential joints

This paper presents a new analytical solution for assessing the longitudinal deformation of shield tunnel associated with overcrossing tunnelling in consideration of circumferential joints.A simplified longitudinal beam-spring model(SLBSM)is established to model the longitudinal behaviours of shield tunnel,which can consider the opening and dislocation between segmental rings simultaneously.Then,the existing tunnel is treated as the SLBSM resting on the elastic foundation.The state equations including tunnel displacements and internal forces are constructed to solve the discontinuous deformation of circumferential joint-segmental ring.The feasibility of the proposed solution is verified through three well-documented cases.The predictions from the proposed method are also compared with other analytical methods.It is found that the proposed method can well capture the deformation of tunnel segmental rings and joints,where the rigid displacement mainly occurs in the segmental rings while the rotation and dislocation occur in the circumferential joints.Some dominant parameters are also analysed to explore the effects on existing tunnel deformation,including the rotation stiffness and shearing stiffness of joints,the skew angle and the clearance between new and old tunnels.

Numerical Analysis of Meat Quality Traits of Ningdu Yellow Rooster

In order to study the meat quality characteristics of Ningdu yellow rooster, the 7 indexes of 120 healthy Ningdu yellow roosters after slaughter at the age of 16 weeks were determined, and correlation analysis, regression analysis and principal component analysis were carried out among the indexes. The results showed that the variation coefficient of PH value was smaller than that of water drop rate and meat colorimetric, and the lactic acid produced by anaerobic glycolysis of glycogen in chicken meat was slower. Through correlation analysis, the 24-hour drip rate was strongly correlated with the 48-hour drip rate (coefficient ≥ 0.7, p ≤ 0.05), and the other indicators were independent of each other. By means of regression analysis, the regression equation (fitting degree R2 = 0.984) of 24- and 48-hour drip rate was established. Principal component analysis synthesized 7 indicators into 3 principal components, with a cumulative contribution rate of 70.13%, indicating that the meat quality of Ningdu Yellow chicken rooster is mainly selected by water drop rate, PH value and meat color test.

The Application of Three Order Cumulants Coupling Features in Fault Diagnosis

Although mechanical vibration is extremely complicated, each fault signal produced by it has its own inherent features, The distinction may be most prominent between the certain components hidden in those features and the same components of normal signals. Three-order cumulant can reduce the Gaussian background noise automatically and its complex formal includes different coupling information of its signal. In the experiment, through these different coupling modes, the same coupling components are fetched from specific fault signal and normal signal, then these components are used to diagnose that certain fault. The results show that the method can fetch the most prominent distinction between normal signal and the specific fault signal, so the specific fault diagnosis by this method is satisfactory.

Depth Analysis of Divergence in Teaching

The concept of divergence is the basis of electromagnetic field theory, and has been a special difficulty in mathematical theory. In this paper, the connotation of divergence itself is truly displayed by quantitative method for the first time. Through these methods, the concepts of divergence can be grasped more easily. The explanation of divergence is helpful to strengthen the understanding of the concept and has a certain reference significance.

Depth Analysis of Gradient in Teaching

Gradient and divergence are the basis of electromagnetic field theory, and have been a special difficulty in mathematical theory. Understanding these concepts requires strong spatial and abstract thinking. In this paper, through the graphical and quantitative methods, the significance of gradient representation is clearly displayed in the form of graphics.

An Asynchronous Clustering and Mobile Data Gathering Schema Based on Timer Mechanism in Wireless Sensor Networks

Recently,Wireless sensor networks(WSNs)have become very popular research topics which are applied to many applications.They provide pervasive computing services and techniques in various potential applications for the Internet of Things(IoT).An Asynchronous Clustering and Mobile Data Gathering based on Timer Mechanism(ACMDGTM)algorithm is proposed which would mitigate the problem of“hot spots”among sensors to enhance the lifetime of networks.The clustering process takes sensors’location and residual energy into consideration to elect suitable cluster heads.Furthermore,one mobile sink node is employed to access cluster heads in accordance with the data overflow time and moving time from cluster heads to itself.Related experimental results display that the presented method can avoid long distance communicate between sensor nodes.Furthermore,this algorithm reduces energy consumption effectively and improves package delivery rate.

Application of Three Order Cumulants in Fault Diagnosis

The three-order cumulants’ complex forms of different definitions include different coupling information of signals, and the information can be used to diagnose fault. In the experiment of pressure reducing valve’s fault diagnosis, through these different coupling information, the features of fault signals and normal signals were extracted by wavelet in different directions, then these features were inputted to diagnose the fault. The experiment shows that this method can achieve a satisfactory result.

Fault Diagnosis of Overflow Valve Based on Trispectrum

The high-order spectrum can effectively remove Gaussian noise. The three-spectrum and its slices represent random signals from a higher probability structure. It can not only qualitatively describe the linearity and nonlinearity of vibration signals closely related to mechanical failures, Gaussian and non-Gaussian Performance, and can greatly improve the accuracy of mechanical fault diagnosis. The two-dimensional slices of trispectrum in normal and fault states show different peak characteristics. 2-D wavelet multi-level decomposition can effectively compress 2-D array information. Least squares support vector machine can obtain the global optimum under limited samples, thus avoiding the local optimum problem, and has the advantage of reducing computational complexity. In this paper, 2-D wavelet multi-level decomposition is used to extract features of trispectrum 2-D slices, and input LSSVM to diagnose the fault of the pressure reducing valve, which has achieved good results.

The Application of Coupled Three Order Cumulants’ Differential Feature in Fault Diagnosis

Complex three-order cumulant has different definition forms. Different forms conclude different information. For studying the effection of frequency in the coupled signals to fault diagnosis, the differential method to the three order cumulants of coupled signals is adopted. By using the differential of complex three order cumulants before and after respectively, then their ?dimensional spectrum is calculated, and the results are used to fault diagnosis. The experimental results show that, the increase frequency item in three order cumulants after differentiated impacts on the results of fault diagnosis and the degree of effection is relative to the differential times. And the correct rate of fault diagnosis can be raised by changing the differential times of three order cumulants.

Analysis of the Precision of Bispectral Estimation Methods

In order to evaluate the accuracy of bispectral estimation method, signals of cosine function were adopted. Because the cosine signals’ three order moment spectrum and three order cumulant spectrum are zero, Non zero part of the bispectrum estimated by no matter which method is the estimation error. Through the comparison of three kinds of estimation methods: the direct method, indirect method and AR parameter method, errors of various estimation methods were obtained, then changing the values of different parameters in all methods, and observing the bispectral error values changes with the parameters, so as to provide a basis for the various bispectrum estimation method and the selection.

The Application of Cumulants Slices in Fault Diagnosis

Complex third-order cumulant has different definition forms. Different forms have different coupling properties, and the generated complex cumulants slices contain different coupling information of signals. In experiments, using the different definitions, the same coupling method is applied to both specific fault signals and normal signals. Furthermore, complex third-order cumulant slices spectrum is defined, and it is used to analyse the coupling features of normal signals and fault signals. Experiments indicate that the detection accuracy rate on the same fault is not the same with the different coupling method, thus, it provides an alternative method to diagnose the specific fault.

Fictitious soil pile model for dynamic analysis of pipe piles under high-strain conditions

A fictitious soil pile(FSP)model is developed to simulate the behavior of pipe piles with soil plugs undergoing high-strain dynamic impact loading.The developed model simulates the base soil with a fictitious hollow pile fully filled with a soil plug extending at a cone angle from the pile toe to the bedrock.The friction on the outside and inside of the pile walls is distinguished using different shaft models,and the propagation of stress waves in the base soil and soil plug is considered.The motions of the pile—soil system are solved by discretizing them into spring-mass model based on the finite difference method.Comparisons of the predictions of the proposed model and conventional numerical models,as well as measurements for pipe piles in field tests subjected to impact loading,validate the accuracy of the proposed model.A parametric analysis is conducted to illustrate the influence of the model parameters on the pile dynamic response.Finally,the effective length of the FSP is proposed to approximate the affected soil zone below the pipe pile toe,and some guidance is provided for the selection of the model parameters.

Classification of hyperspectral images using fusion of CNN and MiniGCN with SVM

Convolutional neural networks(CNNs)have gained popularity for categorizing hyperspectral(HS)images due to their ability to capture representations of spatial-spectral features.However,their ability to model relationships between data is limited.Graph convolutional networks(GCNs)have been introduced as an alternative,as they are effective in representing and analyzing irregular data beyond grid samplingconstraints.WhileGCNs have traditionally.been computationally intensive,minibatch GCNs(miniGCNs)enable minibatch training of large-scale GCNs.We have improved the classification performance by using miniGCNs to infer out-of-sample data without retraining the network.In addition,fuzing the capabilities of CNNs and GCNs,through concatenative fusion has been shown to improve performance compared to using CNNs or GCNs individually.Finally,support vector machine(SvM)is employed instead of softmax in the classification stage.These techniques were tested on two HS datasets and achieved an average accuracy of 92.80 using Indian Pines dataset,demonstrating the effectiveness of miniGCNs and fusion strategies.

Experimental investigation on the deformation characteristics of locking-steel-pipe(LSP)pile retaining structure during excavation in sand

Locking-steel-pipe(LSP)piles connect with adjacent joints to form a pile row enclosure structure.Due to the advantages of quick construction,efficiency in installation,and recycle utilization,the connected LSP piles are frequently used as retaining structure in deep excavation.However,systematic studies of the deformation mechanism of the LSP pile retaining structure are rarely reported,and it still lack of experimental evidence to optimize the design.In this study,a braced supported excavation experimental model test in sand was designed and conducted to investigate the deformation characteristics of LSP pile retaining structure.Three dimensional(3D)printing technique was creatively applied to manufacture LSP model piles.The experimental results show that,a“S”shaped distribution of bending moments is observed along pile shaft when excavation is executed;the deflection of pile shaft develops deep-seated movements toward the excavation side as excavation went deeper,resulting in a“bowl”ground settlement.With the deflection of LSP piles,a rotating trend was occurred between pairs of locking joint,and the severe open deformation of locking joint arose on excavation side.There was a gradual reduction in earth pressure behind the LSP pile retaining wall with excavation depth.The earth pressure between two struts level had no obvious changing,owing to the supported effect of inner struts.