主成分分析法数学推导方法的研究

主成分分析法虽然已被广泛的应用,但其数学原理及其数学推导并不被广泛的了解.为了研究者们更全面和熟练地应用主成分分析法,本文详细介绍了主成分分析的数学推导.

Fractal Analysis of Spatial Fault Distribution in the Southern South China Sea

Fractal geometry can effectively quantify naturally nonlinear or much complicated phenomena, and fractal dimension is the quantitative parameter. In this paper, we applied fractal geometry to analyze spatial fault distribution of the southern South China Sea. Calculation results indicate that the spatial fault distribution of this area have remarkable fractal features in a non-scale interval of 25 - 250 km, with fractal dimensions of 1.660 1, 1.387 5, and 1.269 3 for all faults, NE-trending faults and NW-trending faults, respectively. These dimensions （capacity dimensions）, obtained by box-counting method, represent well the characteristics of spatial fault distribution. Displayed by isoline-maps of fractal dimensions, faults in the study area arrange symmetrically along the spreading axis of the Southwest Sub-basin, and density of the NE-trending faults supplements to that of the NW-trending faults, revealing a property of mutual restriction between these two groups of faults. At last, by correlating the fractal features of spatial fault distribution with the evolution of the South China Sea and the distribution of oil or gas basins, we tried to reveal their relationships.

FLOW FIELD ANALYSES OF PLANE JET AT LOW REYNOLDS NUMBERS USING LATTICE BOLTZMANN METHOD

A two-dimensional（2-D） incompressible plane jet is investigated using the lattice Boltzmann method（LBM） for low Reynolds numbers of 42 and 65 based on the jet-exit-width and the maximum jet-exit-velocity. The results show that the mean centerline velocity decays as x-1/3 and the jet spreads as x2/3 in the self-similar region, which are consistent with the theoretical predictions and the experimental data. The time histories and PSD analyses of the instantaneous centerline velocities indicate the periodic behavior and the interaction between periodic components of velocities should not be neglected in the far field region, although it is invisible in the near field region.

Multiobjective particle swarm inversion algorithm for two-dimensional magnetic data

Regularization inversion uses constraints and a regularization factor to solve ill- posed inversion problems in geophysics. The choice of the regularization factor and of the initial model is critical in regularization inversion. To deal with these problems, we propose a multiobjective particle swarm inversion （MOPSOI） algorithm to simultaneously minimize the data misfit and model constraints, and obtain a multiobjective inversion solution set without the gradient information of the objective function and the regularization factor. We then choose the optimum solution from the solution set based on the trade-off between data misfit and constraints that substitute for the regularization factor. The inversion of synthetic two-dimensional magnetic data suggests that the MOPSOI algorithm can obtain as many feasible solutions as possible; thus, deeper insights of the inversion process can be gained and more reasonable solutions can be obtained by balancing the data misfit and constraints. The proposed MOPSOI algorithm can deal with the problems of choosing the right regularization factor and the initial model.

Simulation study on non-linear effects of initial melt temperatures on microstructures during solidification process of liquid Mg_7Zn_3 alloy

The non-linear effects of different initial melt temperatures on the microstructure evolution during the solidification process of liquid Mg7Zn3 alloys were investigated by molecular dynamics simulation, The microstructure transformation mechanisms were analyzed by several methods. The system was found to be solidified into amorphous structures from different initial melt temperatures at the same cooling rate of 1×10^12 K/s, and the 1551 bond-type and the icosahedron basic cluster （12 0 12 0 ） played a key role in the microstructure transition. Different initial melt temperatures had significant effects on the final microstructures. These effects only can be clearly observed below the glass transition temperature Tg; and these effects are non-linearly related to the initial melt temperatures, and fluctuated in a certain range. However, the changes of the average atomic energy of the systems are still linearly related with the initial melt temperatures, namely, the higher the initial melt temperature is, the more stable the amorphous structure is and the stronger the glass forming ability will be.

Calculation of ion diffusion coefficient related to concrete deterioration based on Debye-Hückel-Onsager theory

This paper applies the Debye-Hückel-Onsager electrolyte solution theory to investigate the diffusivity of ions in concrete pore solutions.First,a model of the diffusion coefficient associated with the ionic species,solution concentration and ambient temperature is proposed in the saturated concrete.Secondly,as an example,the effects of sodium chloride solution concentration,which are associated with influencing factors such as the ionic cloud radius,electrophoresis and relaxation,on the chloride diffusion coefficient are analyzed.It is found that the diffusion coefficient decreases with the increase in solution concentration,and the electrophoresis and ionic cloud radius are two important factors influencing the ionic diffusivity.Finally,the experiments,in which the chloride diffusion coefficients in specimens under different water-cement ratios are measured by the rapid chloride migration（RCM）method,are carried out to validate the effectiveness of the proposed model,and the results indicate that there is a generally reasonable agreement between the experimental and the proposed model results.

Effect of ultrasonic on structure and electrochemical performance of α-Ni(OH)_2 electrodes

Al/Co co-doped α-Ni（OH）2 samples were prepared by either ultrasonic co-precipitation method （Sample B） or co-precipitation method （Sample A）. The crystal structure and particle size distribution of the prepared samples were examined by X-ray diffraction （XRD） and laser particle size analyzer, respectively. The results show that Sample B has more crystalline defects and smaller average diameter than Sample A. The cyclic voltammetry and electrochemical impedance spectroscopy measurements indicate that Sample B has better electrochemical performance than Sample A, such as better reaction reversibility, lower charge-transfer resistance and better cyclic stability. Proton diffusion coefficient of Sample B is 1.96×10-10cm2/s, which is two times as large as that （9.78×10-11cm2/s） of Sample A. The charge-discharge tests show that the discharge capacity （308 mA·h/g） of Sample B is 25 mA·h/g higher than that of Sample A （283 mA·h/g）.

Selecting the optimum location of the corner using gravity gradient method

The conventional gravity gradient method to plot the geologic body location is fuzzy. When the depth is large and the geologic body is small, the Vzz and Vzx derivative errors are also large. We describe that using the status distinguishing factor to optimally determine the comer location is more accurate than the conventional higher-order derivative method. Thus, a better small geologic body and fault resolution is obtained by using the gravity gradient method and trial theoretical model calculation. The actual data is better processed, providing a better basis for prospecting and determination of subsurface geologic structure.

Recovering implied risk-neutral probability density function using SVR

Using support vector regression （SVR）, a novel non-parametric method for recovering implied risk-neutral probability density function （IRNPDF） is investigated by solving linear operator equations. First, the SVR principle for function approximation is introduced, and an SVR method for solving linear operator equations with knowing some values of the right-hand function and without knowing its form is depicted. Then, the principle for solving the IRNPDF based on SVR and the method for constructing cross-kernel functions are proposed. Finally, an empirical example is given to verify the validity of the method. The results show that the proposed method can overcome the shortcomings of the traditional parametric methods, which have strict restrictions on the option exercise price; meanwhile, it requires less data than other non-parametric methods, and it is a promising method for the recover of IRNPDF.

Investigation on Non-covalent Complexes of Cyclodextrins with Li＋ in Gas Phase by Mass Spectrometry

To investigate the non-covalent interaction between cyclodextrins （CD） and lithium ion, a stoichiometry of α-CD, β-CD, heptakis（2,6-di-O-methyl）-β-CD （DM-β-CD）, or heptakis（2,3,6-tri-O-methyl）-β-CD （TM-β-CD） was mixed with lithium salt, respectively, and then incubated at room temperature for 10 min to reach the equilibrium. In posi- tive mode, the electrospray ionization mass spectrometry （ESI-MS） results demonstrated that lithium ion can conjugate to α-, β-, DM-β- or TM-β-CD and form 1：1 stoichiometric non-covalent complexes. The binding of the complexes was further confirmed by collision- induced dissociation. The dissociation constants Kdl of four complexes （Li＋α-CD, Li＋β- CD, Li＋DM-β-CD, and Li＋TM-β-CD） were determined by mass spectrometric titration. The results showed Kdl were 18.7, 26.7, 33.6, 30.5 μmol/L for the complexes of Li＋ with α-CD, β-CD, DM-β-CD, and TM-β-CD, respectively. Kdl for the Li＋ complexes of/3-CD is smaller than that of DM-β-CD due to its steric effect of the partial substituted -CH3. The Kdl for the Li＋ complexes of DM-β-CD is nearly in agreement with that of TM-β-CD, indicating Li＋ is more likely to locate in the small rim of DM-β-CD＇s hydrophobic cavity. The DFT results showed through electrostatic interaction, one Li＋ can strongly conjugate to four neighboring oxygen atoms. For the （α-CD＋Li）＋ complex, one Li＋ may also situate the small rim of α-CD＇s hydrophobic cavity to form a non-specific host-guest complex.

An Estimate of the Enumeration Function of Multiplicative Partitions

Let f(n) denote the number of multiplicative partitions of a positive intger. We prove that if n is an odd number,then f(n)≤n/15+7/5.

Statistical analysis of fracture properties based on particle swarm optimization and Pearson correlation coefficient method

Prediction of reservoir fracture is the key to explore fracture-type reservoir. When a shear-wave propagates in anisotropic media containing fracture,it splits into two polarized shear waves: fast shear wave and slow shear wave. The polarization and time delay of the fast and slow shear wave can be used to predict the azimuth and density of fracture. The current identification method of fracture azimuth and fracture density is cross-correlation method. It is assumed that fast and slow shear waves were symmetrical wavelets after completely separating,and use the most similar characteristics of wavelets to identify fracture azimuth and density,but in the experiment the identification is poor in accuracy. Pearson correlation coefficient method is one of the methods for separating the fast wave and slow wave. This method is faster in calculating speed and better in noise immunity and resolution compared with the traditional cross-correlation method. Pearson correlation coefficient method is a non-linear problem,particle swarm optimization( PSO) is a good nonlinear global optimization method which converges fast and is easy to implement. In this study,PSO is combined with the Pearson correlation coefficient method to achieve identifying fracture property and improve the computational efficiency.

Hybrid particle swarm optimization with chaotic search for solving integer and mixed integer programming problems

A novel chaotic search method is proposed,and a hybrid algorithm combining particle swarm optimization(PSO) with this new method,called CLSPSO,is put forward to solve 14 integer and mixed integer programming problems.The performances of CLSPSO are compared with those of other five hybrid algorithms combining PSO with chaotic search methods.Experimental results indicate that in terms of robustness and final convergence speed,CLSPSO is better than other five algorithms in solving many of these problems.Furthermore,CLSPSO exhibits good performance in solving two high-dimensional problems,and it finds better solutions than the known ones.A performance index(PI) is introduced to fairly compare the above six algorithms,and the obtained values of(PI) in three cases demonstrate that CLSPSO is superior to all the other five algorithms under the same conditions.

Nested Bethe Ansatz for Spin Ladder Model with Open Boundary Conditions

The nested Bethe ansatz (BA) method is applied to find the eigenvalues and the eigenvectors of the transfer matrix for spin-ladder model with open boundary conditions. Based on the reflection equation, we find the general diagonal solution, which determines the generalboundary interaction in the Hamiltonian. We introduce the spin-ladder model with open boundary conditions. By finding the solution K± of the reflection equation which determines the nontrivial boundary terms in the Hamiltonian, we diagonalize the transfer matrix of the spin-ladder model with open boundary conditions in the framework of nested BA.

A rigidity theorem for submanifolds in S^(n+p) with constant scalar curvature

Let Mn be a closed submanifold isometrically immersed in a unit sphere Sn . Denote by R, H and S, the normalized +p scalar curvature, the mean curvature, and the square of the length of the second fundamental form of Mn, respectively. Suppose R is constant and ≥1. We study the pinching problem on S and prove a rigidity theorem for Mn immersed in Sn +pwith parallel nor- malized mean curvature vector field. When n≥8 or, n=7 and p≤2, the pinching constant is best.

Application of Lyapunov exponent algorithm in balise signal chaotic oscillator detection

To improve the detection accuracy of the balise uplink signal transmitted in a strong noise environment,we use chaotic oscillator to detect the balise uplink signal based on the characteristics of the chaotic system that is sensitive to initial conditions and immune to noise.Combining with the principle of Duffing oscillator system used in weak signal detection and uplink signal feature,the methods and steps of using Duffing oscillator to detect the balise signal are presented.Furthermore,the Lyapunov exponent algorithm is used to calculate the critical threshold of the Duffing oscillator detection system.Thus,the output states of the system can be quantitatively judged to achieve demodulation of the balise signal.The simulation results show that the chaotic oscillator detection method for balise signal based on Lyapunov exponent algorithm not only improves the accuracy and efficiency of threshold setting,but also ensures the reliability of balise signal detection.

K2_SPH Method and its Application for 2-D Water Wave Simulation

Smoothed Particle Hydrodynamics (SPH) is a Lagrangian meshless particle method. However, its low accuracy of kernel approximation when particles are distributed disorderly or located near the boundary is an obstacle standing in the way of its wide application. Adopting the Taylor series expansion method and solving the integral equation matrix, the second order kernel approximation method can be obtained, namely K2_SPH, which is discussed in this paper. This method is similar to the Finite Particle Method. With the improvement of kernel approximation, some numerical techniques should be adopted for different types of boundaries, such as a free surface boundary and solid boundary, which are two key numerical techniques of K2_SPH for water wave simulation. This paper gives some numerical results of two dimensional water wave simulations involving standing wave and sloshing tank problems by using K2_SPH. From the comparison of simulation results, the K2_SPH method is more reliable than standard SPH.

Decomposable Algorithm for Large Dynamic Programming

An approach about large dynamic programming based on discrete linear system with a quadratic index function is proposed by importing two Lagrange multipliers.

Investigating the factors affecting traffic violations based on electronic enforcement data:A case study in Shangyu,China

To study the influencing factors of traffic violations,this study investigated the effects of vehicle attribution,day of week,time of day,location of traffic violations,and weather on traffic violations based on the electronic enforcement data and historical weather data obtained in Shangyu,China.Ten categories of traffic violations were determined from the raw data.Then,chi-square tests were used to analyze the relationship between traffic violations and the potential risk factors.Multinomial logistic regression analyses were conducted to further estimate the effects of different risk factors on the likelihood of the occurrence of traffic violations.By analyzing the results of chi-square tests via SPSS,the five factors above were all determined as significant factors associated with traffic violations.The results of the multinomial logistic regression revealed the significant effects of the five factors on the likelihood of the occurrence of corresponding traffic violations.The conclusions are of great significance for the development of effective traffic intervention measures to reduce traffic violations and the improvement of road traffic safety.

Accelerated Matrix Recovery via Random Projection Based on Inexact Augmented Lagrange Multiplier Method

In this paper, a unified matrix recovery model was proposed for diverse corrupted matrices. Resulting from the separable structure of the proposed model, the convex optimization problem can be solved efficiently by adopting an inexact augmented Lagrange multiplier (IALM) method. Additionally, a random projection accelerated technique (IALM+RP) was adopted to improve the success rate. From the preliminary numerical comparisons, it was indicated that for the standard robust principal component analysis (PCA) problem, IALM+RP was at least two to six times faster than IALM with an insignificant reduction in accuracy; and for the outlier pursuit (OP) problem, IALM+RP was at least 6.9 times faster, even up to 8.3 times faster when the size of matrix was 2 000×2 000.