In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o...In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.展开更多
Cognitive enhancement is essential for maintaining the quality of life in healthy individuals and improving the ability of those with mental impairments.In recent years,noninvasive neuromodulation techniques(such as t...Cognitive enhancement is essential for maintaining the quality of life in healthy individuals and improving the ability of those with mental impairments.In recent years,noninvasive neuromodulation techniques(such as transcranial magnetic stimulation,transcranial direct-current stimulation,and transcranial ultrasound stimulation)have shown significant potential in enhancing cognitive functions[1,2].Existing technologies are limited mainly to superficial cortical regions,with limited efficacy in targeting deep brain areas and inadequate methods for evaluating their modulatory effects.Selecting stimulation parameters(such as locus,depth,and intensity)and assessing the impact of neuromodulation remains incompletely understood.展开更多
The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variabl...The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.展开更多
IN function of one complex variable, the inner function on the unit disc plays an important role in the theory of H^p functions. In the mid-1960s, W. Rudin and A. Vitushkin independently raised the question of whether...IN function of one complex variable, the inner function on the unit disc plays an important role in the theory of H^p functions. In the mid-1960s, W. Rudin and A. Vitushkin independently raised the question of whether there exist nonconstant inner functions on the unit ball B_n. Then quickly amassed considerable evidence indicated that such functions would be so pathological that they could not exist. Lately Rudin posed the conjecture on the nonexistence展开更多
This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e f...This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e form with negative semi--definite derivative (i.e. V≤0).展开更多
In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence o...In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.展开更多
In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudin...In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudinger functional. Since the prob-lems involve more complicated norm and multiple singular points, not only we can＇tuse the symmetrization to deal with a one-dimensional inequality, but also the pro-cesses of the blow-up analysis become more delicate. In particular, the new inequalityis more general than that of [1, 2].展开更多
This paper deals with the problem of the stabilization for multi-input polytopic nonlinear systems. Based on the robust control Lyapunov function, a sufficient condition for the existence of time-invariant, continuous...This paper deals with the problem of the stabilization for multi-input polytopic nonlinear systems. Based on the robust control Lyapunov function, a sufficient condition for the existence of time-invariant, continuous, asymptotically stabilizing state feedback controller is derived. It is shown that the obtained sufficient condition is also necessary if there exists a state feedback controller such that the closed-loop system has a robust Lyapunov function for all possible uncertainties. Moreover, a universal formula for constructing stabilizing controller is proposed and the existence of the corresponding Lyapunov function is proven. Particularly, a Lyapunov function is constructed for the polytopic nonlinear system in canonical form. Finally, the feasibility of the proposed control law is verified by a numerical example.展开更多
In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extrema...In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extremal functions for this singular geometric inequality in whole space.In general,W^(1,n)(R^(n))→L^(q)(R^(n))is a continuous embedding but not compact.But in our case we can prove that W^(1,n)(R^(n))→L^(n)(R^(n))is a compact embedding.Combining the compact embedding W^(1,n)(R^(n))→Lq(R^(n),|x|^(−s)dx)for all q≥n and 0<s<n in[18],we establish the theorems for any 0≤α<1.展开更多
As China's cities shift from breakneck expansion to improving the layout and functions of existing urban spaces,the questions facing policymakers are no longer how to attract migrants,but how to integrate them;not...As China's cities shift from breakneck expansion to improving the layout and functions of existing urban spaces,the questions facing policymakers are no longer how to attract migrants,but how to integrate them;not how to build new megaprojects,but how to make cities more livable and inclusive.These themes were front and center at the China Sustainable Urbanization Forum:Theory,Policy and Practice2025,held at Tsinghua University on November 13,where officials,scholars and industry experts gathered to weigh the challenges facing the next planning cycle and the reforms needed to meet them.展开更多
文摘In this paper we deal with the existence of infinitely many critical points of the even functional I(u)=integral from n=Q to (F(x,u,Du))+integral from n=(?)Q to (G(x,u)), u∈W^(1,p)(Ω),where G(x, u)=integral from n=o to u (g(x,t)dt), under the weak structure conditions on F(x, u, q) by the Mountain Pass Lemma.
基金supported by the National Natural Science Foundation of China(82172018 and 62333002).
文摘Cognitive enhancement is essential for maintaining the quality of life in healthy individuals and improving the ability of those with mental impairments.In recent years,noninvasive neuromodulation techniques(such as transcranial magnetic stimulation,transcranial direct-current stimulation,and transcranial ultrasound stimulation)have shown significant potential in enhancing cognitive functions[1,2].Existing technologies are limited mainly to superficial cortical regions,with limited efficacy in targeting deep brain areas and inadequate methods for evaluating their modulatory effects.Selecting stimulation parameters(such as locus,depth,and intensity)and assessing the impact of neuromodulation remains incompletely understood.
文摘The optimal use of intervention strategies to mitigate the spread of Nipah Virus (NiV) using optimal control technique is studied in this paper. First of all we formulate a dynamic model of NiV infections with variable size population and two control strategies where creating awareness and treatment are considered as controls. We intend to find the optimal combination of these two control strategies that will minimize the cost of the two control measures and as a result the number of infectious individuals will decrease. We establish the existence for the optimal controls and Pontryagin’s maximum principle is used to characterize the optimal controls. The numerical simulation suggests that optimal control technique is much more effective to minimize the infected individuals and the corresponding cost of the two controls. It is also monitored that in the case of high contact rate, controls have to work for longer period of time to get the desired result. Numerical simulation reveals that the spread of Nipah virus can be controlled effectively if we apply control strategy at early stage.
文摘IN function of one complex variable, the inner function on the unit disc plays an important role in the theory of H^p functions. In the mid-1960s, W. Rudin and A. Vitushkin independently raised the question of whether there exist nonconstant inner functions on the unit ball B_n. Then quickly amassed considerable evidence indicated that such functions would be so pathological that they could not exist. Lately Rudin posed the conjecture on the nonexistence
文摘This paper is concerned with the problem of absolute stability for a control system with severalexecutive elements. Necessary and sufficient conditions are obtained for the existence of Liapunovfunction of Lur'e form with negative semi--definite derivative (i.e. V≤0).
文摘In this paper, using Fourier series, we study the problem of the existence of periodic solutionsof a type of periodic neutral differential difference system. Some necessary and sufficient conditionsfor the existence of periodic solutions of a type of neutral functional equation system are obtained,and at the same time, we present a method with formula shows how to find the periodicsolutions.
文摘In this paper, we derive the singular Moser-Trudinger inequality which in-volves the first eigenvalue and several singular points, and further prove the existenceof the extremal functions for the relative Moser-Trudinger functional. Since the prob-lems involve more complicated norm and multiple singular points, not only we can＇tuse the symmetrization to deal with a one-dimensional inequality, but also the pro-cesses of the blow-up analysis become more delicate. In particular, the new inequalityis more general than that of [1, 2].
文摘This paper deals with the problem of the stabilization for multi-input polytopic nonlinear systems. Based on the robust control Lyapunov function, a sufficient condition for the existence of time-invariant, continuous, asymptotically stabilizing state feedback controller is derived. It is shown that the obtained sufficient condition is also necessary if there exists a state feedback controller such that the closed-loop system has a robust Lyapunov function for all possible uncertainties. Moreover, a universal formula for constructing stabilizing controller is proposed and the existence of the corresponding Lyapunov function is proven. Particularly, a Lyapunov function is constructed for the polytopic nonlinear system in canonical form. Finally, the feasibility of the proposed control law is verified by a numerical example.
文摘In this paper,we investigate a singular Moser-Trudinger inequality involving L^(n) norm in the entire Euclidean space.The blow-up procedures are used for the maximizing sequence.Then we obtain the existence of extremal functions for this singular geometric inequality in whole space.In general,W^(1,n)(R^(n))→L^(q)(R^(n))is a continuous embedding but not compact.But in our case we can prove that W^(1,n)(R^(n))→L^(n)(R^(n))is a compact embedding.Combining the compact embedding W^(1,n)(R^(n))→Lq(R^(n),|x|^(−s)dx)for all q≥n and 0<s<n in[18],we establish the theorems for any 0≤α<1.
文摘As China's cities shift from breakneck expansion to improving the layout and functions of existing urban spaces,the questions facing policymakers are no longer how to attract migrants,but how to integrate them;not how to build new megaprojects,but how to make cities more livable and inclusive.These themes were front and center at the China Sustainable Urbanization Forum:Theory,Policy and Practice2025,held at Tsinghua University on November 13,where officials,scholars and industry experts gathered to weigh the challenges facing the next planning cycle and the reforms needed to meet them.
