The Convergence Rate of Fréchet Distribution under the Second-Order Regular Variation Condition

In this article we consider the asymptotic behavior of extreme distribution with the extreme value index γ>0 . The rates of uniform convergence for Fréchet distribution are constructed under the second-order regular variation condition.

Theoretical Calculations on the Second-order Nonlinear Optical Property of Chiral Bis-corroles

The second-order nonlinear optical （NLO） properties of 5,10,15-triphenylcorrole （TPC）, 5,10,15,20-tetraphenylporphyrin （TPP） and L-amino acid bridged bis-corroles 1,2,3 and 4 have been calculated by using TDHF/PM3 method based on the RHF/6-31G （TPC and TPP） or semiempirical PM3 （1, 2, 3, 4） optimized geometries. Calculation results showed TPC and TPP have C1 and D2h symmetry, respectively when N-H protons are localized on the nitrogen atoms. TPC is the second-order NLO active chromophore due to the cancellation of centrosymmetrical structure and its first hyperpolarizability β increases to 11.524×10^-30 esu. Under electrical dipole approximation, βvalues of bis-corroles 1, 2, 3 and 4 vary from 9.831×10^-30 to 14.221×10^-30 esu, and no much improvement in the first hyperpolarizability was observed as compared to TPC monomer.However, β values of bis-corroles 1, 2, 3 and 4 are improved by about 4 times as compared to their bis-porphyrin counterparts. The analysis of β components indicates that β of this kind of bis-corroles is mainly contributed from its radial component βr. With the variation of amino acid side chains, βHRS, β,βxyz, βr and βa of bis-corroles change remarkably. Chiral L-amino acid bridged bis-corroles 2, 3 and 4 have a right-handed helix structure, and their chiral component βxyz matches βxyz ∝ r^2 ζ/L^4 （helix parameters）, showing the second-order chiral NLO response of these bis-corroles could be described by one-electron helical model theory. It was found that the radial component βr of chiral helix bis-corroles also matches βr ∝ r^2 ζ/L^4.

A HIGH ACCURACY DIFFERENCE SCHEME FOR THE SINGULAR PERTURBATION PROBLEM OF THE SECOND-ORDER LINEAR ORDINARY DIFFERENTIAL EQUATION IN CONSERVATION FORM

In this paper, combining the idea of difference method and finite element method, we construct a difference scheme for a self-adjoint problem in conservation form. Its solution uniformly converges to that of the original differential equation problem with order h3.

AN INFEASIBLE-INTERIOR-POINT PREDICTOR-CORRECTOR ALGORITHM FOR THE SECOND-ORDER CONE PROGRAM

A globally convergent infeasible-interior-point predictor-corrector algorithm is presented for the second-order cone programming （SOCP） by using the Alizadeh- Haeberly-Overton （AHO） search direction. This algorithm does not require the feasibility of the initial points and iteration points. Under suitable assumptions, it is shown that the algorithm can find an -approximate solution of an SOCP in at most O（√n ln（ε0/ε）） iterations. The iteration-complexity bound of our algorithm is almost the same as the best known bound of feasible interior point algorithms for the SOCP.

CONSENSUS OF THE SECOND-ORDER MULTI-AGENT SYSTEMS WITH AN ACTIVE LEADER AND COUPLING TIME DELAY

This article investigates the consensus problem of the second-order multi-agent systems with an active leader and coupling time delay in direct graph. One decentralized state control rule is constructed for each agent to track the active leader and it is proved that the proposed control scheme enables the consensus to be obtained when the adjacency topology is fixed/switched. Simulation results show effectiveness of the proposed theoretical analysis.

Horizontal dynamic response of a large-diameter pipe pile considering the second-order effect of axial force

The second-order effect of axial force on horizontal vibrating characteristics of a large-diameter pipe pile is theoretically investigated.Governing equations of the pile-soil system are established based on elastodynamics.Threedimensional wave equations of soil are decoupled through differential transformation and variable separation.Consequently,expressions of soil displacements and horizontal resistances can be obtained.An analytical solution of the pile is derived based on continuity conditions between the pile and soil,subsequently from which expressions of the complex impedances are deduced.Analyses are carried out to examine the second-order effect of axial force on the horizontal vibrating behavior of the pipe pile.Some conclusions can be summarized as follows： stiffness and damping factors are decreased with the application of axial force on the pile head; distributions of the pile horizontal displacement and rotation angle are regenerated due to the second-order effect of the applied axial force; and redistributions of the bending moment and shearing force occur due to the second-order effect of the applied axial force.

Static response analysis of structures with interval parameters using the second-order Taylor series expansion and the DCA for QB

In this paper, based on the second-order Taylor series expansion and the difference of convex functions algo- rithm for quadratic problems with box constraints （the DCA for QB）, a new method is proposed to solve the static response problem of structures with fairly large uncertainties in interval parameters. Although current methods are effective for solving the static response problem of structures with interval parameters with small uncertainties, these methods may fail to estimate the region of the static response of uncertain structures if the uncertainties in the parameters are fairly large. To resolve this problem, first, the general expression of the static response of structures in terms of structural parameters is derived based on the second-order Taylor series expansion. Then the problem of determining the bounds of the static response of uncertain structures is transformed into a series of quadratic problems with box constraints. These quadratic problems with box constraints can be solved using the DCA approach effectively. The numerical examples are given to illustrate the accuracy and the efficiency of the proposed method when comparing with other existing methods.

An analytical study on the second-order resonance system of tide in an ideal partially-enclosed bay

In the numerical studies of a real tide M4 resonance system, the Xiangshan Port which is a partially-closed bay, Dong et al. [1999. Acta Oceanologica Sinica, 21 （3）： 1-6] found the interesting phenomenon that the advection plays an important role in inhibiting the growth of the amplitude of the tidal second-order resonance response （M4）. This result is contrary to the general traditional ideas for a non-resonance system. How this phenomenon is interpreted and what internal mechanism is behind the phenomenon are the main focuses of this study. The followings are examined： （1） the dynamic features of a second-order resonance system of tide; （2） the dominating factors on the second-order resonance responses; （3） the effects of both the friction and the advection on the second-order resonance responses; and （4） their roles in dominating the second-order resonance response and internal mechanisms by using the analytical methods. The respective results show that： （1） Both the bottom friction and the advection play significant roles in dominating the magnitude of the amplitude of the second-order resonance responses; （2） the effect of the friction on the second-order resonance response depends on the distribution ratio of the work-done of the system to friction force exhausted into between the damping of the first-order system and the inner excitation of the second-order system; （3） the advection plays a positive role in increasing the amplitude of the second-order non-resonance response in the second order non-resonance of tide; （4） in a second-order resonance system of tide, the effect of the advection may be either to increase or to decrease the amplitudes of the second-order resonance responses of tide, which depends on the distribution ratio mentioned above.

Synthesis, Structure, and the Second-order Nonlinear Optical Properties of a Borate-carboxylic Acid Compound: KB[L-（-）-C4H4O5]2·H2O

A new organic-inorganic hybrid noncentrosymmetric potassium bis(malic acid)-borates KB(L-(-)-C4H4O5)2·H2O was synthesized by solution method. It was characterized by elemental analysis, FT-IR, TGA and single-crystal X-ray crystallography. It crystallizes in the monoclinic space group P21/c with a = 5.546(6), b = 11.985(13), c = 9.952(11) ?, β = 97.522(17)°, V = 655.8(13) ?3, Z = 2, Dc = 1.682 g/cm3, μ(MoKa) = 0.46 mm–1 and F(000) = 340. 5009 reflections were measured and 2893 independent reflections(Rint = 0.051) were used for further refinement. Single-crystal X-ray diffraction reveals that the complex exhibits a threedimensional pseudo tunnel structure consisting of fundamental building block [B(L-(-)-C4H4O5)2]– anions. The small cavities are occupied by the H2O molecules, which stabilize the whole structure by O–H×××O hydrogen bonds. The complex exhibits nonlinear optical effect as high as 1.5 times that of KDP standard.

Establishment of Numerical Wave Flume Based on the Second-Order Wave-Maker Theory

With growing computational power, the first-order wave-maker theory has become well established and is widely used for numerical wave flumes. However, existing numerical models based on the first-order wave-maker theory lose accuracy as nonlinear effects become prominent. Because spurious harmonic waves and primary waves have different propagation velocities, waves simulated by using the first-order wave-maker theory have an unstable wave profile. In this paper, a numerical wave flume with a piston-type wave-maker based on the second-order wave-maker theory has been established. Dynamic mesh technique was developed. The boundary treatment for irregular wave simulation was specially dealt with. Comparisons of the free-surface elevations using the first-order and second-order wave-maker theory prove that second-order wave-maker theory can generate stable wave profiles in both the spatial and time domains. Harmonic analysis and spectral analysis were used to prove the superiority of the second-order wave-maker theory from other two aspects. To simulate irregular waves, the numerical flume was improved to solve the problem of the water depth variation due to low-frequency motion of the wave board. In summary, the new numerical flume using the second-order wave-maker theory can guarantee the accuracy of waves by adding an extra motion of the wave board. The boundary treatment method can provide a reference for the improvement of nonlinear numerical flume.

INTRUSION DETECTION BASED ON THE SECOND-ORDER STOCHASTIC MODEL

This paper presents a new method based on a second-order stochastic model for computer intrusion detection.The results show that the performance of the second-order stochastic model is better than that of a first-order stochastic model.In this study,different window sizes are also used to test the performance of the model.The detection results show that the second-order stochastic model is not so sensitive to the window size,comparing with the first-order stochastic model and other previous researches.The detection result of window sizes 6 and 10 is the same.

An Approach to Fault Diagnosis of Rotating Machinery Using the Second-Order Statistical Features of Thermal Images and Simplified Fuzzy ARTMAP

Thermal image, or thermogram, becomes a new type of signal for machine condition monitoring and fault diagnosis due to the capability to display real-time temperature distribution and possibility to indicate the machine’s operating condition through its temperature. In this paper, an investigation of using the second-order statistical features of thermogram in association with minimum redundancy maximum relevance (mRMR) feature selection and simplified fuzzy ARTMAP (SFAM) classification is conducted for rotating machinery fault diagnosis. The thermograms of different machine conditions are firstly preprocessed for improving the image contrast, removing noise, and cropping to obtain the regions of interest (ROIs). Then, an enhanced algorithm based on bi-dimensional empirical mode decomposition is implemented to further increase the quality of ROIs before the second-order statistical features are extracted from their gray-level co-occurrence matrix (GLCM). The highly relevant features to the machine condition are selected from the total feature set by mRMR and are fed into SFAM to accomplish the fault diagnosis. In order to verify this investigation, the thermograms acquired from different conditions of a fault simulator including normal, misalignment, faulty bearing, and mass unbalance are used. This investigation also provides a comparative study of SFAM and other traditional methods such as back-propagation and probabilistic neural networks. The results show that the second-order statistical features used in this framework can provide a plausible accuracy in fault diagnosis of rotating machinery.

The Second-Order Differential Equation System with the Feedback Controls for Solving Convex Programming

In this paper, we establish the second-order differential equation system with the feedback controls for solving the problem of convex programming. Using Lagrange function and projection operator, the equivalent operator equations for the convex programming problems under the certain conditions are obtained. Then a second-order differential equation system with the feedback controls is constructed on the basis of operator equation. We prove that any accumulation point of the trajectory of the second-order differential equation system with the feedback controls is a solution to the convex programming problem. In the end, two examples using this differential equation system are solved. The numerical results are reported to verify the effectiveness of the second-order differential equation system with the feedback controls for solving the convex programming problem.

A multi-scale second-order autoregressive recursive filter approach for the sea ice concentration analysis

To effectively extract multi-scale information from observation data and improve computational efficiency,a multi-scale second-order autoregressive recursive filter(MSRF)method is designed.The second-order autoregressive filter used in this study has been attempted to replace the traditional first-order recursive filter used in spatial multi-scale recursive filter(SMRF)method.The experimental results indicate that the MSRF scheme successfully extracts various scale information resolved by observations.Moreover,compared with the SMRF scheme,the MSRF scheme improves computational accuracy and efficiency to some extent.The MSRF scheme can not only propagate to a longer distance without the attenuation of innovation,but also reduce the mean absolute deviation between the reconstructed sea ice concentration results and observations reduced by about 3.2%compared to the SMRF scheme.On the other hand,compared with traditional first-order recursive filters using in the SMRF scheme that multiple filters are executed,the MSRF scheme only needs to perform two filter processes in one iteration,greatly improving filtering efficiency.In the two-dimensional experiment of sea ice concentration,the calculation time of the MSRF scheme is only 1/7 of that of SMRF scheme.This means that the MSRF scheme can achieve better performance with less computational cost,which is of great significance for further application in real-time ocean or sea ice data assimilation systems in the future.

Correction of the second-order degree of coherence measurement

The measurement of the second-order degree of coherence [g（2）（t）] is one of the important methods used to study the dynamical evolution of photon-matter interaction systems. Here, we use a nitrogen-vacancy center in a diamond to compare the measurement of g（2） （t） with two methods. One is the prototype measurement process with a tunable delay. The other is a start-stop process based on the time-to-amplitude conversion （TAC） and multichannel analyzer （MCA） system, which is usually applied to achieve efficient measurements. The divergence in the measurement results is observed when the delay time is comparable with the mean interval time between two neighboring detected photons. Moreover, a correction function is presented to correct the results from the TAC-MCA system to the genuine g（2）（t）. Such a correction method will provide a way to study the dynamics in photonic systems for quantum information techniques.

SCATTERING OF A GRAZING LONGITUDINAL WAVE PULSE BY A RIBBON-TYPE CRACK—THE SECOND-ORDER APPROXIMATE SOLUTION

Scattering of an incident grazing longitudinal ultrasonic wave pulse by a traction - free ribbon - type crack in solid is theoretically analyzed. An exact series solution of the scattered field is obtained by iteration. Each term of the series corresponds roughly to one of the rescattering processes by the respective crack tips. The first two terms, which combine to form the second order approximate solution, are discussed in detail. This 2nd - order solution includes scattered cylindrical longitudinal waves, cylindrical shear waves , head waves and surface waves on the crack surfaces, and it gives good description of the early - time wave field.

Second-Order Raman Scattering from n- and p-Type 4H-SiC

The results of second-order Raman-scattering experiments on n- and p-type 4H-SiC are presented,covering the acoustic and the optical overtone spectral regions.Some of the observed structures in the spectra are assigned to particular phonon branches and the points in the Brillouin zone from which the scattering originates.There exists a doublet at 626/636cm -1 with energy difference about 10cm -1 in both n- and p-type 4H-SiC,which is similar to the doublet structure with the same energy difference founded in hexagonal GaN,ZnO, and AlN.The cutoff frequency at 1926cm -1 of the second-order Raman is not the overtone of the A 1(LO) peak of the n-type doping 4H-SiC,but that of the undoping one.The second-order Raman spectrum of 4H-SiC can hardly be affected by doping species or doping density.

Anti-periodic solutions to a class of second-order evolution equations

In this paper we discuss the anti-periodic problem for a class of abstractnonlinear second-order evolution equations associated with maximal monotone operators in Hilbertspaces and give some new assumptions on operators. We establish the existence and uniqueness ofanti-periodic solutions, which improve andgeneralize the results that have been obtained. Finally weillustrate the abstract theory by discussing a simple example of an anti-periodic problem fornonlinear partial differential equations.

Anti-interference beam pattern design based on second-order cone programming optimization

When signal-to-interference ratio is low, the energy of strong interference leaked from the side lobe of beam pattern will infect the detection of weak target. Therefore, the beam pattern needs to be optimized. The existing Dolph-Chebyshev weighting method can get the lowest side lobe level under given main lobe width, but for the other non-uniform circular array and nonlinear array, the low side lobe pattern needs to be designed specially. The second order cone programming optimization （SOCP） algorithm proposed in the paper transforms the optimization of the beam pattern into a standard convex optimization problem. Thus there is a paradigm to follow for any array formation, which not only achieves the purpose of Dolph-Chebyshev weighting, but also solves the problem of the increased side lobe when the signal is at end fire direction The simulation proves that the SOCP algorithm can detect the weak target better than the conventional beam forming.

Stability analysis for a second-order continuous finite-time control system subject to a disturbance

In this paper, using finite-time control method, we consider the disturbance analysis of a second-order system with unknown but bounded disturbance. We show that the states of the second-order system will be stabilized to a region containing the origin. The radius of this region is determined by the control parameters and can be rendered as small as desired. The rigorous stability analysis is also given. Compared with the conventional PD control law, the finite-time control law yields a better disturbance rejection performance. Numerical simulation results show the effectiveness of the method.