Study of scattering from time-varying Gerstners sea surface using second-order small slope approximation

Backscattered fields from one-dimensional time-varying Gerstners sea surface are calculated utilising the secondorder small slope approximation. It is well known that spectral properties of the backscattered echoes relate to the velocity of the small elementary scatterers on sea surface profiles. Therefore, modeling Doppler spectra from the ocean requires an accurate description of the sea surface motion. The profile of nonlinear Gerstners sea surface shows verticalskewness of sea waves, it is sharper at the crest and flatter at the trough than linear waves, and its maximum slope position is closer to the crest than to the trough. Furthermore, the horizontal component of the small elementary scatterers orbit velocity on the sea surface, which yields noticeable influence on Doppler spectra, can be obtained conveniently by Gerstners sea surface model. In this study the characteristics of Doppler spectra of backscattered fields from time-varying Gerstners sea surface are investigated and the dependences of the Doppler frequency and the Doppler bandwidth on the parameters, such as the wind speed, the radar frequency, the incident angle, etc. are discussed. It is shown that the Doppler bandwidth of microwave scattered fields from Gerstners sea surface is considerably broadened. For the case of high frequency backscattered fields, the values of the higher-order spectrum peaks are larger than those obtained by linear sea surface.

Dynamics of Rabi model under second-order Born Oppenheimer approximation

We apply the second-order Born-Oppenheimer （BO） approximation to investigate the dynamics of the Rabi model, which describes the interaction between a two-level system and a single bosonic mode beyond the rotating wave approxi- mation. By comparing with the numerical results, we find that our approach works well when the frequency of the two-level system is much smaller than that of the bosonic mode.

Study of(e,2e) process on potassium at 6 e V-60 eV above threshold in a second-order Born approximation

The standard distorted wave Born approximation （DWBA） method has been extended to second-order Born amplitude in order to describe the multiple interactions between the projectile and the atomic target. Second-order DWBA calculations have been preformed to investigate the triple differential cross sections （TDCS） of coplanar doubly symmetric （e, 2e） collisions for the alkali target potassium at excess energies of 6 eV-60 eV. Compared with the previous first-order DWBA calculations, the present theoretical model improves the degree of agreement with experiments, especially for the backward scattering angle region of TDCS. This indicates that the present second-order Born term is capable of giving a reasonable correction to the DWBA model in studying coplanar symmetric （e, 2e） problems in low and intermediate energy ranges.

Approximate Controllability of Second-Order Neutral Stochastic Differential Equations with Infinite Delay and Poisson Jumps

The modelling of risky asset by stochastic processes with continuous paths, based on Brow- nian motions, suffers from several defects. First, the path continuity assumption does not seem reason- able in view of the possibility of sudden price variations （jumps） resulting of market crashes. A solution is to use stochastic processes with jumps, that will account for sudden variations of the asset prices. On the other hand, such jump models are generally based on the Poisson random measure. Many popular economic and financial models described by stochastic differential equations with Poisson jumps. This paper deals with the approximate controllability of a class of second-order neutral stochastic differential equations with infinite delay and Poisson jumps. By using the cosine family of operators, stochastic analysis techniques, a new set of sufficient conditions are derived for the approximate controllability of the above control system. An example is provided to illustrate the obtained theory.

Second-order Born calculation of coplanar symmetric(e, 2e) process on Mg

The second-order distorted wave Born aPl6roximation （DWBA） method is employed to investigate the triple differen- tial cross sections （TDCS） of coplanar doubly symmetric （e, 2e） collisions for magnesium at excess energies of 6 eV-20 eV. Comparing with the standard first-order DWBA calculations, the inclusion of the second-order Born term in the scattering amplitude improves the degree of agreement with experiments, especially for backward scattering region of TDCS. This indicates that the present second-order Born term is capable to give a reasonable correction to DWBA model in studying coplanar symmetric （e, 2e） problems of two-valence-electron target in low energy range.

Second-order two-scale method for bending behavior analysis of composite plate with 3-D periodic configuration and its approximation

This paper considers the bending behaviors of composite plate with 3-D periodic configuration.A second-order two-scale(SOTS)computational method is designed by means of construction way.First,by 3-D elastic composite plate model,the cell functions which are defined on the reference cell are constructed.Then the effective homogenization parameters of composites are calculated,and the homogenized plate problem on original domain is defined.Based on the Reissner-Mindlin deformation pattern,the homogenization solution is obtained.And then the SOTS’s approximate solution is obtained by the cell functions and the homogenization solution.Second,the approximation of the SOTS’s solution in energy norm is analyzed and the residual of SOTS’s solution for 3-D original in the pointwise sense is investigated.Finally,the procedure of SOTS’s method is given.A set of numerical results are demonstrated for predicting the effective parameters and the displacement and strains of composite plate.It shows that SOTS’s method can capture the 3-D local behaviors caused by3-D micro-structures well.

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.

Approximate Controllability of Second-order Semilinear Evolution Systems with Finite Delay

In this work,we study the approximate controllability for a class of semilinear second-order control systems with finite delay.Sufficient conditions for approximate controllability are established by constructing fundamental solutions and using the resolvent condition and techniques on cosine family of linear operators.To illustrate the applications of the obtained results,an example is provided in the end.

Formulations and Approximations of Branch Flow Model for General Power Networks

The formulations and approximations of the branch flow model for general(radial and mesh) power networks(General-BranchFlow) are given in this paper. Using different sets of the power flow equations, six formats of the exact General-BranchFlow model are listed. The six formats are mathematically equivalent with each other. Linear approximation and second-order cone programming(SOCP) are then used to derive the six formats of the convex General-BranchFlow model. The branch ampacity constraints considering the shunt conductance and capacitance of the transmission line Π-model are derived. The key foundation of deriving the ampacity constraints is the correct interpretation of the physical meaning of the transmission line Π-model. An exact linear expression of the ampacity constraints of the power loss variable is derived. The applications of the General-BranchFlow model in deriving twelve formats of the exact optimal power flow(OPF) model and twelve formats of the approximate OPF model are formulated and analyzed. Using the Julia programming language, the extensive numerical investigations of all formats of the OPF models show the accuracy and computational efficiency of the General-BranchFlow model. A penalty function based approximation gap reduction method is finally proposed and numerically validated to improve the AC-feasibility of the approximate General-BranchFlow model.

Bi-iterative least squares algorithms for blind channel identification and equalization with second-order statistics

We present an adaptive algorithm for blind identification and equalization of single-input multiple-output （SIMO） FIR channels with second-order statistics. We first reformulate the blind channel identification problem into a low-rank matrix approximation solution based on the QR decomposition of the received data matrix. Then, a fast recursive algorithm is developed based on the bi-iterative least squares （Bi-LS） subspace tracking method. The new algorithm requires only a computational complexity of O（md2） at each iteration, or even as low as O（md） if only equalization is necessary, where m is the dimension of the received data vector （or the row rank of channel matrix） and d is the dimension of the signal subspace （or the column rank of channel matrix）. To overcome the shortcoming of the back substitution, an inverse QR iteration algorithm for subspace tracking and channel equalization is also developed. The inverse QR iteration algorithm is well suited for the parallel implementation in the systolic array. Simulation results are presented to illustrate the effectiveness of the proposed algorithms for the channel identification and equalization.

Successive MISOCP Algorithm for Islanded Distribution Networks with Soft Open Points

An optimal operation scheme is of great significance in islanded distribution networks to restore critical loads and has recently attracted considerable attention.In this paper,an optimal power flow(OPF)model for islanded distribution networks equipped with soft open points(SOPs)is proposed.Unlike in the grid-connected mode,the adequacy of local power generation in distribution networks is critical for islanded systems.The proposed approach utilizes the power output of local distributed generations(DGs)and the benefits of reactive power compensation provided by SOPs to allow maximum loadability.To exploit the available resources,an optimal secondary droop control strategy is introduced for the islanded distribution networks,thereby minimizing load shedding.The formulated OPF problem is essentially a mixed-integer nonlinear programming(MINLP)model.To guarantee the computation efficiency and accuracy.A successive mixed-integer second-order cone programming(SMISOCP)algorithm is proposed for handling the nonlinear islanded power flow formulations.Two case studies,incorporating a modified IEEE 33-bus system and IEEE 123-bus system,are performed to test the effectiveness of the proposed approach.

On the local convergence of a stochastic semismooth Newton method for nonsmooth nonconvex optimization

In this work,we present probabilistic local convergence results for a stochastic semismooth Newton method for a class of stochastic composite optimization problems involving the sum of smooth nonconvex and nonsmooth convex terms in the objective function.We assume that the gradient and Hessian information of the smooth part of the objective function can only be approximated and accessed via calling stochastic firstand second-order oracles.The approach combines stochastic semismooth Newton steps,stochastic proximal gradient steps and a globalization strategy based on growth conditions.We present tail bounds and matrix concentration inequalities for the stochastic oracles that can be utilized to control the approximation errors via appropriately adjusting or increasing the sampling rates.Under standard local assumptions,we prove that the proposed algorithm locally turns into a pure stochastic semismooth Newton method and converges r-linearly or r-superlinearly with high probability.

MNP: A Class of NP Optimization Problems

A large class of NP optimization problems called MNP are studied. It is shown that Rmax(2) is in this class and some problems which are not likely in Rmax(2) are in this class. A new kind of reductions, SL-reductions, is defined to preserve approximability and nonapprokimability, so it is a more general version of L-reductions and A-reductions. Then some complete problems of this class under SL-reductions are shown and it is proved that the max-clique problem is one of them. So all complete problems in this class are as difficult to approximate as the max-clique problem.

MODIFIED NEWTON AND SECANT METHODS FOR SOLVING AN ORDER O(h^4) FINITE-DIFFERENCE PROBLEM

In this paper, we present Newton-like methods (modified Newton method and modified secant method), which explore the special structure of the finitedifference approximation of fourth order to the nonlinear two-point boundary value problems At each iteration, modified secant method only calls and computes one function vector (i.e., no additional cast in function evaluations), and it has a R- convergence rate, and modified Newton method only calls two function vectors,and it has a Q-quadratic convergence rate. At last, our numerical results show the new methods are very effective.