The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a ...The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a bivariate function by given scattered data has been obtained in a simple analytical form as a special case.展开更多
Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=...Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=L(t,x,)+(t ,x,) also possesses similar exponential estimate. For α=0, a similar result is given.展开更多
Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and repr...Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.展开更多
In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>（x）f<sup>1</sup>（x）dx+ sum from f=0 to i<sub>1&...In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>（x）f<sup>1</sup>（x）dx+ sum from f=0 to i<sub>1</sub>(b<sub>1</sub>f<sup>1</sup>（x<sub>1</sub>）)]) by a simpler linear functional Lf=sum from i=1 to m（a<sub>1</sub>f（x<sub>1</sub>）） In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.展开更多
We consider a functional partially linear additive model that predicts a functional response by a scalar predictor and functional predictors. The B-spline and eigenbasis least squares estimator for both the parametric...We consider a functional partially linear additive model that predicts a functional response by a scalar predictor and functional predictors. The B-spline and eigenbasis least squares estimator for both the parametric and the nonparametric components proposed. In the final of this paper, as a result, we got the variance decomposition of the model and establish the asymptotic convergence rate for estimator.展开更多
By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f（t, u） ...By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f（t, u） is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0, 1] positive solutions as well as the C1 [0, 1] positive solutions.展开更多
This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectru...This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters.展开更多
An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） a...An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.展开更多
Identification of the Wiener system with the nonlinear block being a piecewiselinear function is considered in the paper, generalizing the results given by H. E. Chen to the case of noisy observation. Recursive algori...Identification of the Wiener system with the nonlinear block being a piecewiselinear function is considered in the paper, generalizing the results given by H. E. Chen to the case of noisy observation. Recursive algorithms are given for estimating all unknown parameters contained in the system, and their strong consistency is proved. The estimation method is similar to that used by H. E. Chen for Hammerstein systems with the same nonlinearity. However, the assumption imposed by H. E. Chen on the availability of an upper bound for the nonsmooth points of the piecewise-linear function has been removed in this paper with the help of designing an additional algorithm for estimating the upper bound.展开更多
In this work,we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space.Our objective is to study,using the Crandall-Liggett app...In this work,we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space.Our objective is to study,using the Crandall-Liggett approach,the solutions as a semigroup of non-linear operators.展开更多
In this paper, we establish several inequalities for the the generalized linear distortion function λ（a, K） by using the monotonicity and convexity of certain combinations λ（a, K）.
In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Erro...In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.展开更多
Traditionally, extra binary variables are demanded to formulate a fuzzy nonlinear programming(FNLP) problem with piecewise linear membership functions(PLMFs). However, this kind of methodology usually suffers increasi...Traditionally, extra binary variables are demanded to formulate a fuzzy nonlinear programming(FNLP) problem with piecewise linear membership functions(PLMFs). However, this kind of methodology usually suffers increasing computational burden associated with formulation and solution, particularly in the face of complex PLMFs. Motivated by these challenges, this contribution introduces a novel approach free of additional binary variables to formulate FNLP with complex PLMFs, leading to superior performance in reducing computational complexity as well as simplifying formulation. A depth discussion about the approach is conducted in this paper, along with a numerical case study to demonstrate its potential benefits.展开更多
A simple method for disturbance decoupling for matrix second-order linear systems is proposed directly in matrix second-order framework via Luenberger function observers based on complete parametric eigenstructure ass...A simple method for disturbance decoupling for matrix second-order linear systems is proposed directly in matrix second-order framework via Luenberger function observers based on complete parametric eigenstructure assignment. By introducing the H2 norm of the transfer function from disturbance to estimation error, sufficient and necessary conditions for disturbance decoupling in matrix second-order linear systems are established and are arranged into constraints on the design parameters via Luenberger function observers in terms of the closed-loop eigenvalues and the group of design parameters provided by the eigenstructure assignment approach. Therefore, the disturbance decoupling problem is converted into an eigenstructure assignment problem with extra parameter constraints. A simple example is investigated to show the effect and simplicity of the approach.展开更多
In this paper we consider the empirical Bayes (EB) estimation problem for estimable function of regression coefficient in a multiple linear regression model Y=Xβ+e. where e with given β has a multivariate standard n...In this paper we consider the empirical Bayes (EB) estimation problem for estimable function of regression coefficient in a multiple linear regression model Y=Xβ+e. where e with given β has a multivariate standard normal distribution. We get the EB estimators by using kernel estimation of multivariate density function and its first order partial derivatives. It is shown that the convergence rates of the EB estimators are under the condition where an integer k > 1 . is an arbitrary small number and m is the dimension of the vector Y.展开更多
In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satis...In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility,and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal;they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.展开更多
In this paper, the normal Luenberger function observer design for second-order descriptor linear systems is considered. It is shown that the main procedure of the design is to solve a so-called second-order generalize...In this paper, the normal Luenberger function observer design for second-order descriptor linear systems is considered. It is shown that the main procedure of the design is to solve a so-called second-order generalized Sylvester-observer matrix equation. Based on an explicit parametric solution to this equation, a parametric solution to the normal Luenberger function observer design problem is given. The design degrees of freedom presented by explicit parameters can be further utilized to achieve some additional design requirements.展开更多
In this paper, the key nature of general hybrid urthogonal functions(GHOF)is given.With it,a more concise system model fur identification is ob-tained.Usins this model,the modified recursive algorithni of parameter es...In this paper, the key nature of general hybrid urthogonal functions(GHOF)is given.With it,a more concise system model fur identification is ob-tained.Usins this model,the modified recursive algorithni of parameter estimation issiniple, rapid and coiivenient fur practical use,and the store space of computer willbe reduced considerably.展开更多
A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the...A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.展开更多
文摘The best recovery of a linear functional Lf, f=f(x,y), on the basis of given linear functionals L jf,j=1,2,...,N in a sense of Sard has been investigated, using analogy of Peano's theorem. The best recovery of a bivariate function by given scattered data has been obtained in a simple analytical form as a special case.
基金Research supported by China National Science Foundation
文摘Assume that the fundamental solution matrix U (t, s ) of x’(t)=L(t, x,) satisfies |U(t,s)|≤Ke-e(t-s) for t≥s.If|(t,φ)|≤δ|φ(0)|with δ【a/K, then the fundamental solution matrix of the perturbed equation x’(t)=L(t,x,)+(t ,x,) also possesses similar exponential estimate. For α=0, a similar result is given.
文摘Modelling and simulation of projectile flight is at the core of ballistic computer software and is essential to the study of performance of rifles and projectiles in various engagement conditions.An effective and representative numerical model of projectile flight requires a relatively good approximation of the aerodynamics.The aerodynamic coefficients of the projectile model should be described as a series of piecewise polynomial functions of the Mach number that ideally meet the following conditions:they are continuous,differentiable at least once,and have a relatively low degree.The paper provides the steps needed to generate such piecewise polynomial functions using readily available tools,and then compares Piecewise Cubic Hermite Interpolating Polynomial(PCHIP),cubic splines,and piecewise linear functions,and their variant,as potential curve fitting methods to approximate the aerodynamics of a generic small arms projectile.A key contribution of the paper is the application of PCHIP to the approximation of projectile aerodynamics,and its evaluation against a set of criteria.Finally,the paper provides a baseline assessment of the impact of the polynomial functions on flight trajectory predictions obtained with 6-degree-of-freedom simulations of a generic projectile.
文摘In numerical analysis, it is significant to approximate the linear functional Ef=sum from i=0 to m-1([integral from a to b(a<sub>1</sub>（x）f<sup>1</sup>（x）dx+ sum from f=0 to i<sub>1</sub>(b<sub>1</sub>f<sup>1</sup>（x<sub>1</sub>）)]) by a simpler linear functional Lf=sum from i=1 to m（a<sub>1</sub>f（x<sub>1</sub>）） In this paper, making use of natural Tchebysheff spline function, we give existence theorem and uniqueness theorem of L that is exact for the degree m to F; we also give three sufficient and necessary conditions in which L is the Sard best approximation to F.
文摘We consider a functional partially linear additive model that predicts a functional response by a scalar predictor and functional predictors. The B-spline and eigenbasis least squares estimator for both the parametric and the nonparametric components proposed. In the final of this paper, as a result, we got the variance decomposition of the model and establish the asymptotic convergence rate for estimator.
基金. Supported by National Natural Science Foundation of China （Grant No. 10871116）, Natural Science Foundation of Shandong Province of China （Grant No. ZR2010AM005） and the Doctoral Program Foundation of Education Ministry of China （Grant No. 200804460001）Acknowledgements The authors would like to thank the referees for their valuable comments.
文摘By using the upper and lower solution method and fixed point theory, we investigate some nonlinear singular second-order differential equations with linear functional boundary conditions. The nonlinear term f（t, u） is nonincreasing with respect to u, and only possesses some integrability. We obtain the existence and uniqueness of the C[0, 1] positive solutions as well as the C1 [0, 1] positive solutions.
基金Project partially supported by the China Postdoctoral Science Foundation (Grant No. 20060400705)Tianjin University Research Foundation (Grant No. TJU-YFF-08B06)
文摘This paper presents a new chaotic Hopfield network with a piecewise linear activation function. The dynamic of the network is studied by virtue of the bifurcation diagram, Lyapunov exponents spectrum and power spectrum. Numerical simulations show that the network displays chaotic behaviours for some well selected parameters.
文摘An alternative method of solving Lagrange＇s first-order partial differential equation of the form（a1x ＋b1y＋C1z）p＋ （a2x ＋b2y＋c2z）q =a3x ＋b3y＋c3z,where p = Эz/Эx, q = Эz/Эy and ai, bi, ci （i = 1,2,3） are all real numbers has been presented here.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 60221301. 60334040. And 60474004)
文摘Identification of the Wiener system with the nonlinear block being a piecewiselinear function is considered in the paper, generalizing the results given by H. E. Chen to the case of noisy observation. Recursive algorithms are given for estimating all unknown parameters contained in the system, and their strong consistency is proved. The estimation method is similar to that used by H. E. Chen for Hammerstein systems with the same nonlinearity. However, the assumption imposed by H. E. Chen on the availability of an upper bound for the nonsmooth points of the piecewise-linear function has been removed in this paper with the help of designing an additional algorithm for estimating the upper bound.
文摘In this work,we are concerned with a general class of abstract semilinear autonomous functional differential equations with a non-dense domain on a Banach space.Our objective is to study,using the Crandall-Liggett approach,the solutions as a semigroup of non-linear operators.
基金Supported by the National Natural Science Foundation of China(11071069, 11171307)the Natural Science Foundation of Hunan Province(09JJ6003)
文摘In this paper, we establish several inequalities for the the generalized linear distortion function λ（a, K） by using the monotonicity and convexity of certain combinations λ（a, K）.
文摘In this paper orthogonal matrix polynomials with respect to a right matrix moment functional an introduced. Basic results, important examples and applications to the approximation of matrix integrals are studied. Error bounds for the proposed matrix quadrature rules are given.
文摘Traditionally, extra binary variables are demanded to formulate a fuzzy nonlinear programming(FNLP) problem with piecewise linear membership functions(PLMFs). However, this kind of methodology usually suffers increasing computational burden associated with formulation and solution, particularly in the face of complex PLMFs. Motivated by these challenges, this contribution introduces a novel approach free of additional binary variables to formulate FNLP with complex PLMFs, leading to superior performance in reducing computational complexity as well as simplifying formulation. A depth discussion about the approach is conducted in this paper, along with a numerical case study to demonstrate its potential benefits.
文摘A simple method for disturbance decoupling for matrix second-order linear systems is proposed directly in matrix second-order framework via Luenberger function observers based on complete parametric eigenstructure assignment. By introducing the H2 norm of the transfer function from disturbance to estimation error, sufficient and necessary conditions for disturbance decoupling in matrix second-order linear systems are established and are arranged into constraints on the design parameters via Luenberger function observers in terms of the closed-loop eigenvalues and the group of design parameters provided by the eigenstructure assignment approach. Therefore, the disturbance decoupling problem is converted into an eigenstructure assignment problem with extra parameter constraints. A simple example is investigated to show the effect and simplicity of the approach.
文摘In this paper we consider the empirical Bayes (EB) estimation problem for estimable function of regression coefficient in a multiple linear regression model Y=Xβ+e. where e with given β has a multivariate standard normal distribution. We get the EB estimators by using kernel estimation of multivariate density function and its first order partial derivatives. It is shown that the convergence rates of the EB estimators are under the condition where an integer k > 1 . is an arbitrary small number and m is the dimension of the vector Y.
基金Supported by the Natural Science Foundation of China(11171221)
文摘In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption;cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility,and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal;they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.
基金This work was supported by National Natural Science Foundation of China(No.60710002)Program for Changjiang Scholars and Innovative Research Team in University(PCSIRT).
文摘In this paper, the normal Luenberger function observer design for second-order descriptor linear systems is considered. It is shown that the main procedure of the design is to solve a so-called second-order generalized Sylvester-observer matrix equation. Based on an explicit parametric solution to this equation, a parametric solution to the normal Luenberger function observer design problem is given. The design degrees of freedom presented by explicit parameters can be further utilized to achieve some additional design requirements.
文摘In this paper, the key nature of general hybrid urthogonal functions(GHOF)is given.With it,a more concise system model fur identification is ob-tained.Usins this model,the modified recursive algorithni of parameter estimation issiniple, rapid and coiivenient fur practical use,and the store space of computer willbe reduced considerably.
文摘A Cauchy problem for the semi-linear elliptic equation is investigated. We use a filtering function method to define a regularization solution for this ill-posed problem. The existence, uniqueness and stability of the regularization solution are proven;a convergence estimate of H?lder type for the regularization method is obtained under the a-priori bound assumption for the exact solution. An iterative scheme is proposed to calculate the regularization solution;some numerical results show that this method works well.
