In this paper, an efficient weight initialization method is proposed using Cauchy’s inequality based on sensitivity analy- sis to improve the convergence speed in single hidden layer feedforward neural networks. The ...In this paper, an efficient weight initialization method is proposed using Cauchy’s inequality based on sensitivity analy- sis to improve the convergence speed in single hidden layer feedforward neural networks. The proposed method ensures that the outputs of hidden neurons are in the active region which increases the rate of convergence. Also the weights are learned by minimizing the sum of squared errors and obtained by solving linear system of equations. The proposed method is simulated on various problems. In all the problems the number of epochs and time required for the proposed method is found to be minimum compared with other weight initialization methods.展开更多
This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z =...This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.展开更多
A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have been shown.
Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influ...Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.展开更多
How to effectively describe the rotation of a deforming body is still a difficult problem for the researchers of mechanics; and there exist a lot of essential problems unsolved yet. Early in the middle of last century...How to effectively describe the rotation of a deforming body is still a difficult problem for the researchers of mechanics; and there exist a lot of essential problems unsolved yet. Early in the middle of last century, Cauchy (1841) mentioned his idea of mean rotation which has quite simple and obvious geometric meanings. How-展开更多
It is well-known that partial stability is an active modern problem. About stationary linear systems, a lot of results on partial stability have been obtained. For time-varying linear systems, however, the general the...It is well-known that partial stability is an active modern problem. About stationary linear systems, a lot of results on partial stability have been obtained. For time-varying linear systems, however, the general theories of partial stability have not been established.展开更多
文摘In this paper, an efficient weight initialization method is proposed using Cauchy’s inequality based on sensitivity analy- sis to improve the convergence speed in single hidden layer feedforward neural networks. The proposed method ensures that the outputs of hidden neurons are in the active region which increases the rate of convergence. Also the weights are learned by minimizing the sum of squared errors and obtained by solving linear system of equations. The proposed method is simulated on various problems. In all the problems the number of epochs and time required for the proposed method is found to be minimum compared with other weight initialization methods.
文摘This article studies on Cauchy’s function f （z） and its integral, （2πi）J[ f （z）] ≡ ■C f （t）dt/（t z） taken along a closed simple contour C, in regard to their comprehensive properties over the entire z = x ＋ iy plane consisted of the simply connected open domain D ＋ bounded by C and the open domain D outside C. （1） With f （z） assumed to be C n （n ∞-times continuously differentiable） z ∈ D ＋ and in a neighborhood of C, f （z） and its derivatives f （n） （z） are proved uniformly continuous in the closed domain D ＋ = [D ＋ ＋ C]. （2） Cauchy’s integral formulas and their derivatives z ∈ D ＋ （or z ∈ D ） are proved to converge uniformly in D ＋ （or in D = [D ＋C]）, respectively, thereby rendering the integral formulas valid over the entire z-plane. （3） The same claims （as for f （z） and J[ f （z）]） are shown extended to hold for the complement function F（z）, defined to be C n z ∈ D and about C. （4） The uniform convergence theorems for f （z） and F（z） shown for arbitrary contour C are adapted to find special domains in the upper or lower half z-planes and those inside and outside the unit circle ｜z｜ = 1 such that the four general- ized Hilbert-type integral transforms are proved. （5） Further, the singularity distribution of f （z） in D is elucidated by considering the direct problem exemplified with several typ- ical singularities prescribed in D . （6） A comparative study is made between generalized integral formulas and Plemelj’s formulas on their differing basic properties. （7） Physical sig- nificances of these formulas are illustrated with applicationsto nonlinear airfoil theory. （8） Finally, an unsolved inverse problem to determine all the singularities of Cauchy function f （z） in domain D , based on the continuous numerical value of f （z） z ∈ D ＋ = [D ＋ ＋ C], is presented for resolution as a conjecture.
基金Supported by the Natural Science Foundation of Hubei Province!(992P0 30 7) the National Natural Science Foun-dation of Chi
文摘A simple method for solving Cauchy’s problem of wave equations in higher space dimensions with initial condition of separated variables, has been given by using D’Alembert’s formula and some examples have been shown.
文摘Characteristics of Mode I crack near the interface of elasticity matched but plasticity and strength mismatched materials differ from those of the crack in a homogenous body. Interface body of different strength influences the plastic or cohesive zone at the crack tip in parent body. The mathematical model for load line opening of the crack near the interface in linear elastic regime involves singular integrals. The paper presents explicit solution of these integrals with the help of Cauchy’s principal value theorem. Cases of thin and thick welds between the materials are investigated. Solutions of the integrals are well substantiated. Final results are provided in a consolidated form.
文摘How to effectively describe the rotation of a deforming body is still a difficult problem for the researchers of mechanics; and there exist a lot of essential problems unsolved yet. Early in the middle of last century, Cauchy (1841) mentioned his idea of mean rotation which has quite simple and obvious geometric meanings. How-
基金Project supported by the National Natural Science Foundation of China
文摘It is well-known that partial stability is an active modern problem. About stationary linear systems, a lot of results on partial stability have been obtained. For time-varying linear systems, however, the general theories of partial stability have not been established.
