This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the correspon...This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).展开更多
This paper presents an improved BP algorithm. The approach can reduce the amount of computation by using the logarithmic objective function. The learning rate μ(k) per iteration is determined by dynamic o...This paper presents an improved BP algorithm. The approach can reduce the amount of computation by using the logarithmic objective function. The learning rate μ(k) per iteration is determined by dynamic optimization method to accelerate the convergence rate. Since the determination of the learning rate in the proposed BP algorithm only uses the obtained first order derivatives in standard BP algorithm(SBP), the scale of computational and storage burden is like that of SBP algorithm,and the convergence rate is remarkably accelerated. Computer simulations demonstrate the effectiveness of the proposed algorithm展开更多
It poses the inverse problem that consists in finding the logarithm of a function. It shows that when the function is holomorphic in a simply connected domain , the solution at the inverse problem exists and is unique...It poses the inverse problem that consists in finding the logarithm of a function. It shows that when the function is holomorphic in a simply connected domain , the solution at the inverse problem exists and is unique if a branch of the logarithm is fixed. In addition, it’s demonstrated that when the function is continuous in a domain , where is Hausdorff space and connected by paths. The solution of the problem exists and is unique if a branch of the logarithm is fixed and is stable;for what in this case, the inverse problem turns out to be well-posed.展开更多
In this paper,based on coupled network generated by chaotic logarithmic map,a novel algorithm for constructing hash functions is proposed,which can transform messages and can establish a mapping from the transformed m...In this paper,based on coupled network generated by chaotic logarithmic map,a novel algorithm for constructing hash functions is proposed,which can transform messages and can establish a mapping from the transformed messages to the coupled matrix of the network.The network model is carefully designed to ensure the network dynamics to be chaotic.Through the chaotic iterations of the network,quantization and exclusive-or (XOR) operations,the algorithm can construct hash value with arbitrary length.It is shown by simulations that the algorithm is extremely sensitive to the initial values and the coupled matrix of the network,and has excellent performance in one-way,confusion and diffusion,and collision resistance.展开更多
Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we est...Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).展开更多
Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + ...Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.展开更多
文摘This paper is devoted to studying the existence of solutions for the following logarithmic Schrödinger problem: −div(a(x)∇u)+V(x)u=ulogu2+k(x)| u |q1−2u+h(x)| u |q2−2u, x∈ℝN.(1)We first prove that the corresponding functional I belongs to C1(HV1(ℝN),ℝ). Furthermore, by using the variational method, we prove the existence of a sigh-changing solution to problem (1).
文摘This paper presents an improved BP algorithm. The approach can reduce the amount of computation by using the logarithmic objective function. The learning rate μ(k) per iteration is determined by dynamic optimization method to accelerate the convergence rate. Since the determination of the learning rate in the proposed BP algorithm only uses the obtained first order derivatives in standard BP algorithm(SBP), the scale of computational and storage burden is like that of SBP algorithm,and the convergence rate is remarkably accelerated. Computer simulations demonstrate the effectiveness of the proposed algorithm
文摘It poses the inverse problem that consists in finding the logarithm of a function. It shows that when the function is holomorphic in a simply connected domain , the solution at the inverse problem exists and is unique if a branch of the logarithm is fixed. In addition, it’s demonstrated that when the function is continuous in a domain , where is Hausdorff space and connected by paths. The solution of the problem exists and is unique if a branch of the logarithm is fixed and is stable;for what in this case, the inverse problem turns out to be well-posed.
基金supported by the Program for New Century Excellent Talents in University of China(No.NCET-06-0510)National Natural Science Founda-tion of China(No. 60874091)Six Projects Sponsoring Talent Summits of Jiangsu Province(No. SJ209006)
文摘In this paper,based on coupled network generated by chaotic logarithmic map,a novel algorithm for constructing hash functions is proposed,which can transform messages and can establish a mapping from the transformed messages to the coupled matrix of the network.The network model is carefully designed to ensure the network dynamics to be chaotic.Through the chaotic iterations of the network,quantization and exclusive-or (XOR) operations,the algorithm can construct hash value with arbitrary length.It is shown by simulations that the algorithm is extremely sensitive to the initial values and the coupled matrix of the network,and has excellent performance in one-way,confusion and diffusion,and collision resistance.
基金Research supported by National Natural Science Foundation of China.
文摘Let X be a d-dimensional random vector with unknown density function f(z) = f (z1, ..., z(d)), and let f(n) be teh nearest neighbor estimator of f proposed by Loftsgaarden and Quesenberry (1965). In this paper, we established the law of the iterated logarithm of f(n) for general case of d greater-than-or-equal-to 1, which gives the exact pointwise strong convergence rate of f(n).
文摘Let stand for the polar coordinates in R2, ?be a given constant while satisfies the Laplace equation in the wedge-shaped domain or . Here αj(j = 1,2,...,n + 1) denote certain angles such that αj αj(j = 1,2,...,n + 1). It is known that if r = a satisfies homogeneous boundary conditions on all boundary lines ?in addition to non-homogeneous ones on the circular boundary , then an explicit expression of in terms of eigen-functions can be found through the classical method of separation of variables. But when the boundary?condition given on the circular boundary r = a is homogeneous, it is not possible to define a discrete set of eigen-functions. In this paper one shows that if the homogeneous condition in question is of the Dirichlet (or Neumann) type, then the logarithmic sine transform (or logarithmic cosine transform) defined by (or ) may be effective in solving the problem. The inverses of these transformations are expressed through the same kernels on or . Some properties of these transforms are also given in four theorems. An illustrative example, connected with the heat transfer in a two-part wedge domain, shows their effectiveness in getting exact solution. In the example in question the lateral boundaries are assumed to be non-conducting, which are expressed through Neumann type boundary conditions. The application of the method gives also the necessary condition for the solvability of the problem (the already known existence condition!). This kind of problems arise in various domain of applications such as electrostatics, magneto-statics, hydrostatics, heat transfer, mass transfer, acoustics, elasticity, etc.
