ALMOST SURE GLOBAL WELL-POSEDNESS FOR THE FOURTH-ORDER NONLINEAR SCHR?DINGER EQUATION WITH LARGE INITIAL DATA

We consider the fourth-order nonlinear Schr?dinger equation(4NLS)(i?t+εΔ+Δ2)u=c1um+c2(?u)um-1+c3(?u)2um-2,and establish the conditional almost sure global well-posedness for random initial data in Hs(Rd)for s∈(sc-1/2,sc],when d≥3 and m≥5,where sc:=d/2-2/(m-1)is the scaling critical regularity of 4NLS with the second order derivative nonlinearities.Our proof relies on the nonlinear estimates in a new M-norm and the stability theory in the probabilistic setting.Similar supercritical global well-posedness results also hold for d=2,m≥4 and d≥3,3≤m<5.

Computer Geometries for Finding All Real Zeros of Polynomial Equations Simultaneously

In this research article,we construct a family of derivative free simultaneous numerical schemes to approximate all real zero of non-linear polynomial equation.We make a comparative analysis of the newly constructed numerical schemes with a well-known existing simultaneous method for determining all the distinct real zeros of polynomial equations using computer algebra system Mat Lab.Lower bound of convergence of simultaneous schemes is calculated using Mathematica.Global convergence property of the numerical schemes is presented by taking random starting initial approximation and their convergence history are graphically presented.Some real life engineering applications along with some higher degree polynomials are considered as numerical test problems to show performance and efficiency of the derivative free family of numerical methods with comparison of an existing method of same order in literature.Local computational order of convergence,CPU time,graph of computational order of convergence and residual error graphs elaborate efficiency,robustness and authentication of the suggested family of numerical methods in its domain.

Curative effect of acupuncture and moxibustion on insomnia: a randomized clinical trial

OBJECTIVE: To observe the effect of acupuncture and moxibustion on insomnia and explore its mechanism. METHODS: One hundred and twenty patients were randomly divided into an experiment group and a control group. Sixty patients in the experiment group were treated once a day with acupuncture at Baihui (GV 20), Sishencong (EX-HN 1), Shenmai (BL 62), and Zhaohai (KI 6) and with moxibustion at Baihui (GV 20) and Sishencong (EX-HN 1). Sixty patients in the control group were acupunctured once a day at Shenmen (HT 7), Neiguan (PC 6), and Sanyinjiao (SP 6).The Pittsburgh Sleep Quality Index (PSQI) was used to compare sleep improvement between the two groups. RESULTS: The total effective rate was 87.7% in the experiment group and 76.3% in the control group. The PSQI scores and the total score were lower after treatment than before treatment in bothgroups. However, the reduction in the experiment group was greater than that in the control group in sleeping quality, time to fall asleep, sleeping disorder, and daytime function (P<0.05). CONCLUSION: Acupuncture and moxibustion at Baihui (GV 20), Sishencong (EX-HN 1), Shenmai (BL 62), and Zhaohai (KI 6) significantly improved insomnia symptoms in the experiment group compared with the control group.

A MODEL OF INTEGRATED RANDOM INITIAL VALUES AND RANDOM FORCING AND ITS PRELIMINARY EXPERIMENT

A simple quasi-geostrophic barotropic vorticity equation model is used as the dynamic frame of the model in this paper.Considering that there are many random errors in model's initial values of meteorolo- gical data,and that it is not perfectly complete about model's physical processes (for example,take no ac- count of the interaction between atmosphere and underlying surface,radiation,etc.),we add the random for- ced term to the model and use the Monte-Carlo method with random initial values.A statistical-dynamic integrated model is thus built up,and a numerical forecasting experiment of 500hPa monthly mean height field of January 1983 has been carried out.The experiment result proves that the forecasting result of the model, considering random forcing and random initial values at the same time,is better than that by the pure dynamic model,the random initial value model and the random forced model.

Double resonance in star networks of bistable units with disordered signals

It has been found that a hub node is better able to amplify weak external signals than leaf nodes in star networks.But hub-enhanced amplification is only attained by a single hub node and is limited to weak coupling strength.We show here that random initial phases in external weak signals do not affect the hub-enhanced amplification at weak coupling strength.Instead,they can improve the responses of all the leaf nodes to external signals at large coupling strength,resulting in a double resonance-like signal amplification.We use a reduced model to analyze the influence of the star structure and random initial phases on the emergence of double resonance.

Gibbs Measure for the Higher Order Modified Camassa-Holm Equation

This paper is devoted to constructing a globally rough solution for the higher order modified Camassa-Holm equation with randomization on initial data and periodic boundary condition.Motivated by the works of Thomann and Tzvetkov(Nonlinearity,23(2010),2771–2791),Tzvetkov(Probab.Theory Relat.Fields,146(2010),4679–4714),Burq,Thomann and Tzvetkov(Ann.Fac.Sci.Toulouse Math.,27(2018),527–597),the authors first construct the Borel measure of Gibbs type in the Sobolev spaces with lower regularity,and then establish the existence of global solution to the equation with the helps of Prokhorov compactness theorem,Skorokhod convergence theorem and Gibbs measure.

Gradient Descent for Symmetric Tensor Decomposition

Symmetric tensor decomposition is of great importance in applications.Several studies have employed a greedy approach,where the main idea is to first find a best rank-one approximation of a given tensor,and then repeat the process to the residual tensor by subtracting the rank-one component.In this paper,we focus on finding a best rank-one approximation of a given orthogonally order-3 symmetric tensor.We give a geometric landscape analysis of a nonconvex optimization for the best rank-one approximation of orthogonally symmetric tensors.We show that any local minimizer must be a factor in this orthogonally symmetric tensor decomposition,and any other critical points are linear combinations of the factors.Then,we propose a gradient descent algorithm with a carefully designed initialization to solve this nonconvex optimization problem,and we prove that the algorithm converges to the global minimum with high probability for orthogonal decomposable tensors.This result,combined with the landscape analysis,reveals that the greedy algorithm will get the tensor CP low-rank decomposition.Numerical results are provided to verify our theoretical results.