Support vector machines (SVMs) have been extensively studied and have shown remarkable success in many applications. A new family of twice continuously differentiable piecewise smooth functions are used to smooth th...Support vector machines (SVMs) have been extensively studied and have shown remarkable success in many applications. A new family of twice continuously differentiable piecewise smooth functions are used to smooth the objective function of uncon- strained SVMs. The three-order piecewise smooth support vector machine (TPWSSVMd) is proposed. The piecewise functions can get higher and higher approximation accuracy as required with the increase of parameter d. The global convergence proof of TPWSSVMd is given with the rough set theory. TPWSSVMd can efficiently handle large scale and high dimensional problems. Nu- merical results demonstrate TPWSSVMa has better classification performance and learning efficiency than other competitive base- lines.展开更多
The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study...The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.展开更多
This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piec...This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piecewise smooth systems with 3-parameters is investigated in this paper using the theory of differential inclu-sion and tools of Poincar′e maps.展开更多
It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. ...It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for展开更多
This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often pro...This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.展开更多
The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator-prey model with predator harvesting.We consider a harvesting strategy that allows constant catches if the population size is abo...The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator-prey model with predator harvesting.We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value(to obtain predictable yield)and no catches if the population size is below the threshold(to protect the population).It is shown that boundary equilibrium bifurcation and sliding grazing bifurcation can happen as the threshold value varies.We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles,the coexistence of them with standard limit cycles.Some numerical simulations are given to demonstrate ourresults.展开更多
In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole do...In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.展开更多
In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system ...In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system may have a focus,saddle or node.Our results show that this piecewise smooth differential system will have two limit cycles and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation,Hopf bifurcation and critical crossing bifurcation CC.Some examples are given to illustrate our results.展开更多
This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems.We first establish the Poincaré map near a periodic orbit,and deduce the first order derivative of...This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems.We first establish the Poincaré map near a periodic orbit,and deduce the first order derivative of the map for general piecewise smooth systems on the plane.Then,we obtain a sufficient condition for determining the stability of limit cycles for these systems.展开更多
Spiral springs have a wide range of applications in various fields.As a result of the complexity of friction,few theoretical analyses of spring belts under static loading have been carried out.Considering the piecewis...Spiral springs have a wide range of applications in various fields.As a result of the complexity of friction,few theoretical analyses of spring belts under static loading have been carried out.Considering the piecewise smooth property of the whole contact area,a simplified static model of spiral springs under loading is established in this paper.Besides,three main stress and friction distribution areas of the spring belt are proposed,namely,internal,transitional,and external regions.Since the outermost side of the spring is not subject to any pressure,a recursive method is adopted from the outside to the inside.The model provides the parameter conditions,i.e.,the internal and external forces are independent or dependent.Therefore,the case that the whole contact region of the spring belt has one subregion,two subregions,and three subregions is obtained.The model gives a theoretical basis for the parameter optimization of spiral springs.展开更多
We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k+...We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k+1 of first category, defined in the range of the initial datum.展开更多
We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, ...We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, x)‖ which reflects the influence of the position of the x's and ω(f^(r+1),δ)j,j = 0, 1,... , s,on the error of approximation.展开更多
In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the...In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.展开更多
It is known that principal value plays vital roles in the study of singular integral.In this paper,one type of Cauchy–Fantappièintegral with multiple indexes is introduced.Since the structure of the kernel is co...It is known that principal value plays vital roles in the study of singular integral.In this paper,one type of Cauchy–Fantappièintegral with multiple indexes is introduced.Since the structure of the kernel is complexity,the Jacobian determinant included in the kernel is expanded for making clearly the expression of the kernel.Moreover,one differential operator is utilized for setting up relations between integrals with higher and usual orders.The work also concerns the convergent properties of the integral.In order to study Hadamard principal value and composite formula of this integral,finite and divergent parts will be estimated and separated.As an application,solvability of the system of integral equations with higher order singularity kernel is discussed.展开更多
Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equation...Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains,creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient.This work expands the use of the polynomial annihilation edge detector,(Archibald,Gelb and Yoon,2005),to locate discontinuities in the gradient given irregularly sampled point values of a continuous function.The idea is to preprocess the given data by calculating the derivative,and then to use the polynomial annihilation edge detector to locate the jumps in the derivative.We compare our results to other recently developed methods.展开更多
基金supported by the National Natural Science Foundation of China(6110016561100231+6 种基金5120530961472307)the Natural Science Foundation of Shaanxi Province(2012JQ80442014JM83132010JQ8004)the Foundation of Education Department of Shaanxi Province(2013JK1096)the New Star Team of Xi’an University of Posts and Telecommunications
文摘Support vector machines (SVMs) have been extensively studied and have shown remarkable success in many applications. A new family of twice continuously differentiable piecewise smooth functions are used to smooth the objective function of uncon- strained SVMs. The three-order piecewise smooth support vector machine (TPWSSVMd) is proposed. The piecewise functions can get higher and higher approximation accuracy as required with the increase of parameter d. The global convergence proof of TPWSSVMd is given with the rough set theory. TPWSSVMd can efficiently handle large scale and high dimensional problems. Nu- merical results demonstrate TPWSSVMa has better classification performance and learning efficiency than other competitive base- lines.
基金The NSF (10671082) of Chinathe postgraduate program of 985 (20080239) of Jilin University
文摘The goal of this paper is to investigate the bifurcation properties of stationary points of a class of planar piecewise smooth systems with 3 parameters using the theory of differential inclusions. We especially study the existence of the stationary points on the line of discontinuity of this kind of planar piecewise smooth system.
文摘This paper is concerned with the bifurcation properties on the line of discontinuity of planar piecewise smooth systems. The existence of equilibria and periodic solutions with sliding motion in a class of planar piecewise smooth systems with 3-parameters is investigated in this paper using the theory of differential inclu-sion and tools of Poincar′e maps.
基金This paper is supported by the National Foundations.
文摘It is important to study the propagation and interaction of progressing waves of nonlinear equations in the class of piecewise smooth function. However, there has not been many works on that in multidimensional case. In 1985, J, Rauch & M. Reed have provad the existence and uniqueness of piecewise smooth solution for
基金supported in part by NSF-DMS 1502640,NSF-DMS 1912685,AFOSR FA9550-18-1-0316Office of Naval Research MURI grant N00014-20-1-2595.
文摘This paper presents an application of the sparse Bayesian learning(SBL)algorithm to linear inverse problems with a high order total variation(HOTV)sparsity prior.For the problem of sparse signal recovery,SBL often produces more accurate estimates than maximum a posteriori estimates,including those that useℓ1 regularization.Moreover,rather than a single signal estimate,SBL yields a full posterior density estimate which can be used for uncertainty quantification.However,SBL is only immediately applicable to problems having a direct sparsity prior,or to those that can be formed via synthesis.This paper demonstrates how a problem with an HOTV sparsity prior can be formulated via synthesis,and then utilizes SBL.This expands the class of problems available to Bayesian learning to include,e.g.,inverse problems dealing with the recovery of piecewise smooth functions or signals from data.Numerical examples are provided to demonstrate how this new technique is effectively employed.
基金This work is supported by NNSFC(No.11871022)Shanghai Key Laboratory of PMMP.
文摘The aim of this paper is to study the dynamical behaviors of a piecewise smooth predator-prey model with predator harvesting.We consider a harvesting strategy that allows constant catches if the population size is above a certain threshold value(to obtain predictable yield)and no catches if the population size is below the threshold(to protect the population).It is shown that boundary equilibrium bifurcation and sliding grazing bifurcation can happen as the threshold value varies.We provide analytical analysis to prove the existence of sliding limit cycles and sliding homoclinic cycles,the coexistence of them with standard limit cycles.Some numerical simulations are given to demonstrate ourresults.
基金This work was supported in part by the National Natural Science Foundation of China(No.11431004)the Innovation Program of Shanghai Municipal Education Commission.
文摘In this paper,we study smoothing approximations for some piecewise smooth functions.We first present two approaches for one-dimensional case:a global approach is to construct smoothing approximations over the whole domain and a local approach is to construct smoothing approximations within appropriate neighborhoods of the nonsmooth points.We obtain some error estimate results for both approaches and discuss whether the smoothing approximations can inherit the convexity of the original functions.Furthermore,we extend the global approach to some multiple dimensional cases.
基金partially supported by Postgraduate research and innovation ability cultivation plan of Huaqiao Universitypartially supported by NNSF of China grant11671040+3 种基金Cultivation Program for Outstanding Young Scientific talents of Fujian Province in 2017Program for Innovative Research Team in Science and Technology in Fujian Province UniversityQuanzhou High-Level Talents Support Plan under Grant2017ZT012Promotion Program for Young and Middle-aged Teacher in Science and Technology Research of Huaqiao University(ZQN-YX401)
文摘In this paper,we consider a planar piecewise smooth differential system consisting of a linear system and a quadratic Hamiltonian system.The quadratic system has some folds on the discontinuity line.The linear system may have a focus,saddle or node.Our results show that this piecewise smooth differential system will have two limit cycles and a sliding cycle.Moreover,this piecewise smooth system will undergo pseudo-homoclinic bifurcation,Hopf bifurcation and critical crossing bifurcation CC.Some examples are given to illustrate our results.
基金supported by the National Natural Science Foundation of China(Grant No.11931016)。
文摘This paper is concerned with the problem of stability discrimination of limit cycles for piecewise smooth systems.We first establish the Poincaré map near a periodic orbit,and deduce the first order derivative of the map for general piecewise smooth systems on the plane.Then,we obtain a sufficient condition for determining the stability of limit cycles for these systems.
基金the National Natural Science Foundation of China(No.11972055)the National Defense Science and Technology Fund in the Technical Field of the Foundation Strengthening Plan(No.2020-JCJQ-JJ-009)the Civil Aerospace Pre-research Project(No.D020206)。
文摘Spiral springs have a wide range of applications in various fields.As a result of the complexity of friction,few theoretical analyses of spring belts under static loading have been carried out.Considering the piecewise smooth property of the whole contact area,a simplified static model of spiral springs under loading is established in this paper.Besides,three main stress and friction distribution areas of the spring belt are proposed,namely,internal,transitional,and external regions.Since the outermost side of the spring is not subject to any pressure,a recursive method is adopted from the outside to the inside.The model provides the parameter conditions,i.e.,the internal and external forces are independent or dependent.Therefore,the case that the whole contact region of the spring belt has one subregion,two subregions,and three subregions is obtained.The model gives a theoretical basis for the parameter optimization of spiral springs.
基金supported by National Natural Foundation of China(10671116 and 10871133)
文摘We prove that for a given strictly increasing initial datum in Ck, the solution of the initial value problem is piecewise Ck smooth except for flux functions of nonconvex conservation laws in a certain subset of C^k+1 of first category, defined in the range of the initial datum.
基金Supported by the National Nature Science Foundation.
文摘We study some approximation properties of Lagrange interpolation polynomial based on the zeros of (1-x^2)cosnarccosx. By using a decomposition for f(x) ∈ C^τC^τ+1 we obtain an estimate of ‖f(x) -Ln+2(f, x)‖ which reflects the influence of the position of the x's and ω(f^(r+1),δ)j,j = 0, 1,... , s,on the error of approximation.
基金supported by National Natural Science Foundation of China(Grant Nos.11931016 and 11771296)Hunan Provincial Education Department(Grant No.19C1898).
文摘In the study of the number of limit cycles of near-Hamiltonian systems,the first order Melnikov function plays an important role.This paper aims to generalize Horozov-Iliev’s method to estimate the upper bound of the number of zeros of the function.
基金Supported by the National Natural Science Foundation of China(Grant No.11771357)。
文摘It is known that principal value plays vital roles in the study of singular integral.In this paper,one type of Cauchy–Fantappièintegral with multiple indexes is introduced.Since the structure of the kernel is complexity,the Jacobian determinant included in the kernel is expanded for making clearly the expression of the kernel.Moreover,one differential operator is utilized for setting up relations between integrals with higher and usual orders.The work also concerns the convergent properties of the integral.In order to study Hadamard principal value and composite formula of this integral,finite and divergent parts will be estimated and separated.As an application,solvability of the system of integral equations with higher order singularity kernel is discussed.
基金This work was partially supported by NSF grants CNS 0324957DMS 0617867+2 种基金DMS 0608844(AG)DMS 0510813(AG and HM)DMS 0421846(AG and HM).
文摘Detection of edges in piecewise smooth functions is important in many applications.Higher order reconstruction algorithms in image processing and post processing of numerical solutions to partial differential equations require the identification of smooth domains,creating the need for algorithms that will accurately identify discontinuities in a given function as well as those in its gradient.This work expands the use of the polynomial annihilation edge detector,(Archibald,Gelb and Yoon,2005),to locate discontinuities in the gradient given irregularly sampled point values of a continuous function.The idea is to preprocess the given data by calculating the derivative,and then to use the polynomial annihilation edge detector to locate the jumps in the derivative.We compare our results to other recently developed methods.