Generalized Wave Operators in Von Neumann Algebras

Let M⊆B(H)be a countable decomposable properly infinite von Neumann algebra with a faithful normal semifinite tracial weightτwhere B(H)is the set of all bounded linear operators on Hilbert space H.The main purpose of this article is to introduce generalized weak wave operators Wf_(±),generalized weak abelian wave operators ■ and generalized stationary wave operators U_(±) in M and then to explore the relation among W_(±),■,U_(±) and generalized wave operators W_(±).

INVARIANTS-PRESERVING DU FORT-FRANKEL SCHEMES AND THEIR ANALYSES FOR NONLINEAR SCHRÖDINGER EQUATIONS WITH WAVE OPERATOR

Du Fort-Frankel finite difference method(FDM)was firstly proposed for linear diffusion equations with periodic boundary conditions by Du Fort and Frankel in 1953.It is an explicit and unconditionally von Neumann stable scheme.However,there has been no research work on numerical solutions of nonlinear Schrödinger equations with wave operator by using Du Fort-Frankel-type finite difference methods(FDMs).In this study,a class of invariants-preserving Du Fort-Frankel-type FDMs are firstly proposed for one-dimensional(1D)and two-dimensional(2D)nonlinear Schrödinger equations with wave operator.By using the discrete energy method,it is shown that their solutions possess the discrete energy and mass conservative laws,and conditionally converge to exact solutions with an order of for ofο(T^(2)+h_(x)^(2)+(T/h_(x))^(2))1D problem and an order ofο(T^(2)+h_(x)^(2)+h_(Y)^(2)+(T/h_(X))^(2)+(T/h_(y))^(2))for 2D problem in H1-norm.Here,τdenotes time-step size,while,hx and hy represent spatial meshsizes in x-and y-directions,respectively.Then,by introducing a stabilized term,a type of stabilized invariants-preserving Du Fort-Frankel-type FDMs are devised.They not only preserve the discrete energies and masses,but also own much better stability than original schemes.Finally,numerical results demonstrate the theoretical analyses.

H^m-Modified Wave Operator for Nonlinear Hartree Equation in Space of Dimension n≥2

We consider the scattering problem for the Hartree equation with potential|x|-1in a space of dimension n≥2.We prove the existence of Hm-modified wave operator for Hartree equation on a dense set of a neighborhood of zero in Hm（Rn）,meanwhile,we obtain also the global existence for the Cauchy problem of Hartree equation in a space of dimension n≥2.

Accuracy analysis of the optimal separable approximation method of one-way wave migration

An approximation for the one-way wave operator takes the form of separated space and wave-number variables and makes it possible to use the FFT, which results in a great improvement in the computational efficiency. From the function approximation perspective, the OSA method shares the same separable approximation format to the one-way wave operator as other separable approximation methods but it is the only global function approximation among these methods. This leads to a difference in the phase error curve, impulse response, and migration result from other separable approximation methods. The difference is that the OSA method has higher accuracy, and the sensitivity to the velocity variation declines with increasing order.

Certain averaging operators on Triebel-Lizorkin spaces

In this article,we study the boundedness properties of the averaging operator S_(t)^(γ) on Triebel-Lizorkin spaces F_(p,q)^(α)(R^(n))for various p,q.As an application,we obtain the norm convergence rate for S_(t)^(γ)(f)on Triebel-Lizorkin spaces and the relation between the smoothness imposed on functions and the rate of norm convergence of S_(t)^(γ) is given.

Effects of Rotor Solidity on the Performance of Impulse Turbine for OWC Wave Energy Converter

Impulse turbine, working as a typical self-rectifying turbine, is recently utilized for the oscillating water column（OWC） wave energy converters, which can rotate in the same direction under the bi-directional air flows. A numerical model established in Fluent is validated by the corresponding experimental results. The flow fields, pressure distribution and dimensionless evaluating coefficients can be calculated and analyzed. Effects of the rotor solidity varying with the change of blade number are investigated and the suitable solidity value is recommended for different flow coefficients.

Effect of Wave Accessibility on Lower Hybrid Wave Current Drive in Experimental Advanced Superconductor Tokamak with H-Mode Operation

The effect of the wave accessibility condition on the lower hybrid cm＇rent drive in the experimental advanced superconductor Tokamak （EAST） plasma with H-mode operation is studied. Based on a simplified model, a mode conversion layer of the lower hybrid wave between the fast wave branch and the slow wave branch is proved to exist in the plasma periphery for typical EAST H-mode parameters. Under the framework of the lower hybrid wave simulation code （LSC）, the wave ray trajectory and the associated current drive are calculated numerically. The results show that the wave accessibility condition plays an important role on the lower hybrid current drive in EAST plasma. For wave rays with parallel refractive index n｜｜= 2.1 or n｜｜ = 2.5 launched from the outside midplane, the wave rays may penetrate the core plasma due to the toroida] geometry effect, while numerous reflections of the wave ray trajectories in the plasma periphery occur. However, low current drive efficiency is obtained. Meanwhile, the wave accessibility condition is improved if a higher confined magnetic field is applied. The simulation results show that for plasma parameters under present EAST H-mode operation, a significant lower hybrid wave current drive could be obtained for the wave spectrum with peak value n｜｜ = 2.1 if a toroidal magnetic field BT =2.5 T is applied.

High-order generalized screen propagator migration based on particle swarm optimization

Various migration methods have been proposed to image high-angle geological structures and media with strong lateral velocity variations; however, the problems of low precision and high computational cost remain unresolved. To describe the seismic wave propagation in media with lateral velocity variations and to image high-angle structures, we propose the generalized screen propagator based on particle swarm optimization （PSO-GSP）, for the precise fitting of the single-square-root operator. We use the 2D SEG/EAGE salt model to test the proposed PSO-GSP migration method to image the faults beneath the salt dome and compare the results to those of the conventional high-order generalized screen propagator （GSP） migration and split-step Fourier （SSF） migration. Moreover, we use 2D marine data from the South China Sea to show that the PSO-GSP migration can better image strong reflectors than conventional imaging methods.

A stabilized least-squares imaging condition with structure constraints

Conventional shot-gather migration uses a cross-correlation imaging condition proposed by Clarebout （1971）, which cannot preserve imaging amplitudes. The deconvolution imaging condition can improve the imaging amplitude and compensate for illumination. However, the deconvolution imaging condition introduces instability issues. The least-squares imaging condition first computes the sum of the cross-correlation of the forward and backward wavefields over all frequencies and sources, and then divides the result by the total energy of the forward wavefield. Therefore, the least-squares imaging condition is more stable than the classic imaging condition. However, the least-squares imaging condition cannot provide accurate results in areas where the illumination is very poor and unbalanced. To stabilize the least-squares imaging condition and balance the imaging amplitude, we propose a novel imaging condition with structure constraints that is based on the least-squares imaging condition. Our novel imaging condition uses a plane wave construction that constrains the imaging result to be smooth along geological structure boundaries in the inversion frame. The proposed imaging condition improves the stability of the imaging condition and balances the imaging amplitude. The proposed condition is applied to two examples, the horizontal layered model and the Sigsbee 2A model. These tests show that, in comparison to the damped least-squares imaging condition, the stabilized least-squares imaging condition with structure constraints improves illumination stability and balance, makes events more consecutive, adjusts the amplitude of the depth layers where the illumination is poor and unbalanced, suppresses imaging artifacts, and is conducive to amplitude preserving imaging of deep layers.

INTERACTION MORAWETZ ESTIMATE AND A SIMPLIFIED PROOF ON THE ENERGY SCATTERING FOR HARTREE EQUAITONS

In this article, we give a simple proof on the energy scattering for the Hartree equations using the interaction Morawetz estimate that was originally introduced in [5].

The Scattering Theory of Nonlinear Schroedinger Equations with Interaction Terms

In this paper, we give a simplified proof on the energy scattering for the nonlinear Schroedinger equations with interaction terems by use of the interaction Morawetz estimate, which is originally introduced in [4].

Design and operation strategies of the system for destroying time-sensitive target based on system effectiveness

To improve the effect of destroying time-sensitive target （TST）, a method of operational effectiveness evaluation is presented and some influential factors are analyzed based on the combat flow of system for destroying TST. Considering the possible operation modes of the system, a waved operation mode and a continuous operation mode are put forward at first. At the same time, some relative formulas are modified. In examples, the influential factors and operation modes are analyzed based on the system effectiveness. From simulation results, some design and operation strategies of the system for destroying time sensitive targets are concluded, which benefit to the improvement of the system effectiveness.

Development and Preliminary Commissioning Results of a Long Pulse 140 GHz ECRH System on EAST Tokamak(Invited)

A long pulse electron cyclotron resonance heating（ECRH）system has been developed to meet the requirements of steady-state operation for the EAST superconducting tokamak,and the first EC wave was successfully injected into plasma during the 2015 spring campaign.The system is mainly composed of four 140 GHz gyrotron systems,4 ITER-Like transmission lines,4 independent channel launchers and corresponding power supplies,a water cooling,control ＆inter-lock system etc.Each gyrotron is expected to deliver a maximum power of 1 MW and be operated at 100-1000 s pulse lengths.The No.1 and No.2 gyrotron systems have been installed.In the initial commissioning,a series of parameters of 1 MW 1 s,900 k W 10 s,800 k W 95 s and650 k W 753 s have been demonstrated successfully on the No.1 gyrotron system based on calorimetric dummy load measurements.Significant plasma heating and MHD instability suppression effects were observed in EAST experiments.In addition,high confinement（H-mode）discharges triggered by ECRH were obtained.

Optic Axes and Elliptic Cone Equation in Coordinate-Invariant Treatment

We derive for crystal optics in coordinate-invariant way the cone approximation of refraction vectors in the neighborhood of optic axes and determine its invariants and eigenvectors. It proved to describe an elliptic cone. The second invariant of the operator of the wave equation with respect to similarity transformations determines the special cases of degeneration including the optic axes where the polarization of the waves due to self-intersection of the dispersion surface is not uniquely determined. This second invariant is included in all investigations and it is taken into account in the illustrations. It is biquadratic in the refraction vectors and the corresponding forth-order surface in three-dimensional space splits in two separate shells and a non-rational product decomposition describing this is found. We give also a more general classification of all possible solutions of an equation with an arbitrary three-dimensional operator.

STABILITY FOR IMPOSING ABSORBING BOUNDARY CONDITIONS IN THE FINITE ELEMENT SIMULATION OF ACOUSTIC WAVE PROPAGATION＊

It is well-known that artificial boundary conditions are crucial for the efficient and accurate computations of wavefields on unbounded domains. In this paper, we investigate stability analysis for the wave equation coupled with the first and the second order absorbing boundary conditions. The computational scheme is also developed. The approach allows the absorbing boundary conditions to be naturally imposed, which makes it easier for us to construct high order schemes for the absorbing boundary conditions. A thirdorder Lagrange finite element method with mass lumping is applied to obtain the spatial discretization of the wave equation. The resulting scheme is stable and is very efficient since no matrix inversion is needed at each time step. Moreover, we have shown both abstract and explicit conditional stability results for the fully-discrete schemes. The results are helpful for designing computational parameters in computations. Numerical computations are illustrated to show the efficiency and accuracy of our method. In particular, essentially no boundary reflection is seen at the artificial boundaries.

UNIQUE CONTINUATION ON A HYPERPLANE FOR WAVE EQUATION

One kind of unique continuation property for a wave equation is discussed. The authors show that, if one classical solution of the wave equation vanishes in an open set on a hyperplane, then it must vanish in a larger set on this hyperplane. The result can be viewed as a localized version of Robbiano’s result[9]. The approach involves the localized Fourier-Gauss transformation and unique continuation on a line in the Laplace equation.

Effects of large-amplitude internal solitary waves on ROV operation--A numerical study

Remotely operated vehicles(ROVs) are unique tools for underwater industrial exploration and scientific research of offshore areas and the deep ocean. With broadening application of ROVs, the study of factors that affect their safe operation is important. Besides the technical skills to control ROV movement, the dynamical ocean environment may also play a vital role in the safe operation of ROVs. In this paper, we investigate the influence of large-amplitude internal solitary waves(ISWs), focusing on the forces exerted on ROVs by ISWs. We present a methodology for modeling ISW-induced currents based on Kd V model and calculate the ISW forces using the Morrison equation. Our results show that an extremely considerable load is exerted by the ISW on the ROV, resulting in a strong disturbance of the vehicle's stability, affecting the ROV control. The numerical results of this work emphasize the importance of considering dynamical conditions when operating underwater vessels, such as ROV. Further laboratory and field investigation are suggested to gain more understanding of this subject.

Derivative Estima tes of It era ted Spherical Average and Ext ension

In the paper,we study derivative estimates of the iterated spherical averages(At)^(N)(f).We obtain the optimal range of exponents(α,N,p)to ensure the L^(p)boundedness of P(■/■x)(A1)^(N)(f)for 1≤p≤∞,where P is a homogeneous polynomial of degree a.The main theorem extends some known results.As an application,we obtain the smallest N such that(A1)^(N):L^(p)(R^(n))→L_(α)^(p)(R^(n)).

Evolutionary algorithm based on discrete ITO process for travelling salesman problems

Since dozens years ago, various metaheuristic methods, such as genetic algorithm, antcolony algorithms, have been successfully applied to combinational optimization problem. However, as one of the members, ITO algorithm has only been employed in continuous optimization, it needs further design for combinational optimization problem.In this paper, a discrete ITO algorithm inspired by ITO stochastic process is proposedfor travelling salesman problems (TSPs). Some key operators, such as move operator,wave operator, are redesigned to adapt to combinational optimization. Moreover, theperformance of ITO algorithm in different parameter selections and the maintenance ofpopulation diversity information are also studied. By combining local search methods(such as 2-opt and LK-opt) with ITO algorithm, our computational results of the TSPproblems show that ITO algorithm is currently one of the best-performing algorithmsfor these problems.

Scattering Map for the Vlasov-Poisson System

We construct(modified)scattering operators for the Vlasov-Poisson system in three dimensions,mapping small asymptotic dynamics as t→−∞to asymptotic dynamics as t→+∞.The main novelty is the construction of modified wave operators,but we also obtain a new simple proof of modified scattering.Our analysis is guided by the Hamiltonian structure of the Vlasov-Poisson system.Via a pseudo-conformal inversion,we recast the question of asymptotic behavior in terms of local in time dynamics of a new equation with singular coefficients which is approximately integrated using a generating function.