Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stocha...Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stochastic proximal gradient method performs well. However, research on its accelerated version remains unclear. This paper proposes a proximal stochastic accelerated gradient (PSAG) method to address problems involving a combination of smooth and non-smooth components, where the smooth part corresponds to the average of multiple block sums. Simultaneously, most of convergence analyses hold in expectation. To this end, under some mind conditions, we present an almost sure convergence of unbiased gradient estimation in the non-smooth setting. Moreover, we establish that the minimum of the squared gradient mapping norm arbitrarily converges to zero with probability one.展开更多
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-...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.展开更多
With the income increase of elderly people in Beijing and the improvement of leisure facilities since 2000, the quality of elderly people leisure life in Beijing has obviously increased com- pared to ten years ago. Ac...With the income increase of elderly people in Beijing and the improvement of leisure facilities since 2000, the quality of elderly people leisure life in Beijing has obviously increased com- pared to ten years ago. According to the investigation and com- parative analysis on more than 900 elderly people, who usually exercise in public parks in Beijing City, we have found out several new features of elderly people＇s outdoor recreation in Belting： the increase of the leisure time, the expansion of leisure space, the in- dividuation of interest and the diversification of consumption. The leisure life quality evaluation index system of urban elderly people can be built based on these analyses. In order to accommodate the development tendency of elderly people＇s demand for leisure diversification, we should strengthen the organization and manage- ment of elderly people＇s leisure activity in urban construction, and pay more attention to function complementation between commu- nities and parks.展开更多
The mechanism of the shift of the band-gap in phononic crystal （PC） with different initial confining pressures is studied experimentally and numerically. The experimental results and numerical analysis simultaneousl...The mechanism of the shift of the band-gap in phononic crystal （PC） with different initial confining pressures is studied experimentally and numerically. The experimental results and numerical analysis simultaneously indicate that the confining pressure can efficiently tune the location in and the width of the band-gap. The present work provides a basis for tuning the band-gap of phononic crystal in engineering applications.展开更多
文摘Proximal gradient descent and its accelerated version are resultful methods for solving the sum of smooth and non-smooth problems. When the smooth function can be represented as a sum of multiple functions, the stochastic proximal gradient method performs well. However, research on its accelerated version remains unclear. This paper proposes a proximal stochastic accelerated gradient (PSAG) method to address problems involving a combination of smooth and non-smooth components, where the smooth part corresponds to the average of multiple block sums. Simultaneously, most of convergence analyses hold in expectation. To this end, under some mind conditions, we present an almost sure convergence of unbiased gradient estimation in the non-smooth setting. Moreover, we establish that the minimum of the squared gradient mapping norm arbitrarily converges to zero with probability one.
基金supported by the NationalNatural Science Foundation of China(12001236)the Natural Science Foundation of Guangdong Province(2020A1515110494)。
文摘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.
基金support of Key Project of Chinese Ministry of Education (Grant No. 170170)
文摘With the income increase of elderly people in Beijing and the improvement of leisure facilities since 2000, the quality of elderly people leisure life in Beijing has obviously increased com- pared to ten years ago. According to the investigation and com- parative analysis on more than 900 elderly people, who usually exercise in public parks in Beijing City, we have found out several new features of elderly people＇s outdoor recreation in Belting： the increase of the leisure time, the expansion of leisure space, the in- dividuation of interest and the diversification of consumption. The leisure life quality evaluation index system of urban elderly people can be built based on these analyses. In order to accommodate the development tendency of elderly people＇s demand for leisure diversification, we should strengthen the organization and manage- ment of elderly people＇s leisure activity in urban construction, and pay more attention to function complementation between commu- nities and parks.
基金Project supported by the National Natural Science Foundation of China(Grant No.10732010,10972010,and 11028206)
文摘The mechanism of the shift of the band-gap in phononic crystal （PC） with different initial confining pressures is studied experimentally and numerically. The experimental results and numerical analysis simultaneously indicate that the confining pressure can efficiently tune the location in and the width of the band-gap. The present work provides a basis for tuning the band-gap of phononic crystal in engineering applications.
