The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspectiv...The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspective of energy balance. The work rates of external forces and internal energy dissipation are calculated. An analytical solution of necessary face pressures is derived. The optimal upper-bound solution of the face pressures is obtained by optimization. The results show that the three dimensionless parameters A, T, n of nonlinear Baker failure criterion have different effects on the necessary face pressures and the pattern failure mechanisms ahead of tunnel face. A is the most important one;n takes the second place, and T is the least one. The computed necessary face pressures are nonlinearly increasing when A is reduced. Combined with the actual monitoring data of Taxia tunnel, the calculation results in this paper is verified. It is suggested that the tunnel face supports should be strengthened timely in soft rocks to prevent the occurrence of face collapse.展开更多
LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many que...LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.展开更多
In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immed...In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.展开更多
基金Projects(51674115,51804113)supported by the National Natural Science Foundation of ChinaProject(17B095)supported by the Excellent Youth Subsidy Project of Hunan Provincial Department of Education,China
文摘The nonlinear Baker failure criterion is introduced into the upper-bound limit analysis to examine the face stability of a shallow tunnel. The tunnel face under the ultimate limit state is analyzed from the perspective of energy balance. The work rates of external forces and internal energy dissipation are calculated. An analytical solution of necessary face pressures is derived. The optimal upper-bound solution of the face pressures is obtained by optimization. The results show that the three dimensionless parameters A, T, n of nonlinear Baker failure criterion have different effects on the necessary face pressures and the pattern failure mechanisms ahead of tunnel face. A is the most important one;n takes the second place, and T is the least one. The computed necessary face pressures are nonlinearly increasing when A is reduced. Combined with the actual monitoring data of Taxia tunnel, the calculation results in this paper is verified. It is suggested that the tunnel face supports should be strengthened timely in soft rocks to prevent the occurrence of face collapse.
文摘LET Z,N,Q be the sets of integers, positive integers and rational numbers, respectively. The solutions (x, y, m, n) of the exponential Diophantine equation X^2+2~m=y^n,x,y,m,n∈N,2y,n【2 (1)are connected with many questions in number theory and combinatorial theory. In the recent fifty years, there were many papers concerned with the equation written by Ljunggren, Nagell, Brown, Toyoizumi and Cohn. In 1986, ref. [1] claimed that all solutions of (1) had been determined. However, we have not seen the proof so far. Therefore, the solution of (1) has not been found yet. In this note, using Baker’s method, we prove the following result.
基金Supported by the Scientific Research Fund of Hunan Provincial Education Department (Grant No06C245)
文摘In this paper, we consider Newton＇s method for a class of entire functions with infinite order. By using theory of dynamics of functions meromorphic outside a small set, we find there are some series of virtual immediate basins in which the dynamics converges to infinity and a series of immediate basins with finite area in the Fatou sets of Newton＇s method.
