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.

Bulk-boundary-transport correspondence of the second-order topological insulators

The bulk-boundary correspondence of the second-order topological insulator(SOTI)has been well established,but a universal transport signature for open systems is still absent.For a variety of SOTIs induced by applying in-plane magnetic fields in Z2-invariant first-order TIs,rotating this magnetic field features the spin pump mechanism while maintaining the SOTI phase.We demonstrate that,this spin pump can generate quantized pure spin current when tuning the magnetic field strength,which corresponds to the formation of topological corner states characterizing SOTI in two-dimensional(2D)systems.Quantized spin pump is discovered in various 2D and 3D SOTI models evolved from Z2-invariant TIs,which is robust against disorder and universally independent of system parameters including Fermi energy,system size,magnetic field strength,and pumping frequency.These findings suggest that this universal quantized spin pump can characterize the bulk-boundary-transport correspondence of SOTIs.Quantized spin pump can also be realized by combining pseudo spin such as the orbital degree of freedom with the rotating magnetic field,which could be achieved in higher-order photonic or acoustic topological systems.Such a quantized spin pump is promising as an accurate and stable single-spin source.

Closed-form steady-state solutions for forced vibration of second-order axially moving systems

Second-order axially moving systems are common models in the field of dynamics, such as axially moving strings, cables, and belts. In the traditional research work, it is difficult to obtain closed-form solutions for the forced vibration when the damping effect and the coupling effect of multiple second-order models are considered.In this paper, Green's function method based on the Laplace transform is used to obtain closed-form solutions for the forced vibration of second-order axially moving systems. By taking the axially moving damping string system and multi-string system connected by springs as examples, the detailed solution methods and the analytical Green's functions of these second-order systems are given. The mode functions and frequency equations are also obtained by the obtained Green's functions. The reliability and convenience of the results are verified by several examples. This paper provides a systematic analytical method for the dynamic analysis of second-order axially moving systems, and the obtained Green's functions are applicable to different second-order systems rather than just string systems. In addition, the work of this paper also has positive significance for the study on the forced vibration of high-order systems.

Three-dimensional mixed convection stagnation-point fow past a vertical surface with second-order slip velocity

This study is concerned with the three-dimensional(3D)stagnation-point for the mixed convection flow past a vertical surface considering the first-order and secondorder velocity slips.To the authors’knowledge,this is the first study presenting this very interesting analysis.Nonlinear partial differential equations for the flow problem are transformed into nonlinear ordinary differential equations(ODEs)by using appropriate similarity transformation.These ODEs with the corresponding boundary conditions are numerically solved by utilizing the bvp4c solver in MATLAB programming language.The effects of the governing parameters on the non-dimensional velocity profiles,temperature profiles,skin friction coefficients,and the local Nusselt number are presented in detail through a series of graphs and tables.Interestingly,it is reported that the reduced skin friction coefficient decreases for the assisting flow situation and increases for the opposing flow situation.The numerical computations of the present work are compared with those from other research available in specific situations,and an excellent consensus is observed.Another exciting feature for this work is the existence of dual solutions.An important remark is that the dual solutions exist for both assisting and opposing flows.A linear stability analysis is performed showing that one solution is stable and the other solution is not stable.We notice that the mixed convection and velocity slip parameters have strong effects on the flow characteristics.These effects are depicted in graphs and discussed in this paper.The obtained results show that the first-order and second-order slip parameters have a considerable effect on the flow,as well as on the heat transfer characteristics.

Six New Sets of the Non-Elementary Jef-Family-Functions that Are Giving Solutions to Some Second-Order Nonlinear Autonomous ODEs

In this paper, we define some new sets of non-elementary functions in a group of solutions x(t) that are sine and cosine to the upper limit of integration in a non-elementary integral that can be arbitrary. We are using Abel’s methods, described by Armitage and Eberlein. The key is to start with a non-elementary integral function, differentiating and inverting, and then define a set of three functions that belong together. Differentiating these functions twice gives second-order nonlinear ODEs that have the defined set of functions as solutions. We will study some of the second-order nonlinear ODEs, especially those that exhibit limit cycles. Using the methods described in this paper, it is possible to define many other sets of non-elementary functions that are giving solutions to some second-order nonlinear autonomous ODEs.

Second-Order MaxEnt Predictive Modelling Methodology. II: Probabilistically Incorporated Computational Model (2nd-BERRU-PMP)

This work presents a comprehensive second-order predictive modeling (PM) methodology based on the maximum entropy (MaxEnt) principle for obtaining best-estimate mean values and correlations for model responses and parameters. This methodology is designated by the acronym 2nd-BERRU-PMP, where the attribute “2nd” indicates that this methodology incorporates second- order uncertainties (means and covariances) and second (and higher) order sensitivities of computed model responses to model parameters. The acronym BERRU stands for “Best-Estimate Results with Reduced Uncertainties” and the last letter (“P”) in the acronym indicates “probabilistic,” referring to the MaxEnt probabilistic inclusion of the computational model responses. This is in contradistinction to the 2nd-BERRU-PMD methodology, which deterministically combines the computed model responses with the experimental information, as presented in the accompanying work (Part I). Although both the 2nd-BERRU-PMP and the 2nd-BERRU-PMD methodologies yield expressions that include second (and higher) order sensitivities of responses to model parameters, the respective expressions for the predicted responses, for the calibrated predicted parameters and for their predicted uncertainties (covariances), are not identical to each other. Nevertheless, the results predicted by both the 2nd-BERRU-PMP and the 2nd-BERRU-PMD methodologies encompass, as particular cases, the results produced by the extant data assimilation and data adjustment procedures, which rely on the minimization, in a least-square sense, of a user-defined functional meant to represent the discrepancies between measured and computed model responses.

Second-Order MaxEnt Predictive Modelling Methodology. I: Deterministically Incorporated Computational Model (2nd-BERRU-PMD)

This work presents a comprehensive second-order predictive modeling (PM) methodology designated by the acronym 2nd-BERRU-PMD. The attribute “2nd” indicates that this methodology incorporates second-order uncertainties (means and covariances) and second-order sensitivities of computed model responses to model parameters. The acronym BERRU stands for “Best- Estimate Results with Reduced Uncertainties” and the last letter (“D”) in the acronym indicates “deterministic,” referring to the deterministic inclusion of the computational model responses. The 2nd-BERRU-PMD methodology is fundamentally based on the maximum entropy (MaxEnt) principle. This principle is in contradistinction to the fundamental principle that underlies the extant data assimilation and/or adjustment procedures which minimize in a least-square sense a subjective user-defined functional which is meant to represent the discrepancies between measured and computed model responses. It is shown that the 2nd-BERRU-PMD methodology generalizes and extends current data assimilation and/or data adjustment procedures while overcoming the fundamental limitations of these procedures. In the accompanying work (Part II), the alternative framework for developing the “second- order MaxEnt predictive modelling methodology” is presented by incorporating probabilistically (as opposed to “deterministically”) the computed model responses.