Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every r...Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)<deg(f,y')or n max i=0{deg(a_(i),y)−2(n−i)}>0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.展开更多
This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments.The authors first transform the ...This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments.The authors first transform the problem into a parameterized polynomial optimization problem,and obtain a numerical ranking function using polynomial sum-of-squares relaxation via semidefinite programming(SDP).A rational vector recovery algorithm is deployed to recover a rational polynomial from the numerical ranking function,and some symbolic computation techniques are used to certify that this polynomial is an exact ranking function of the loop programs.At last,the authors demonstrate on some polynomial loop programs from the literature that our algorithm successfully yields nonlinear ranking functions with rational coefficients.展开更多
The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which...The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors ?rst determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.展开更多
Tool path generation is a fundamental problem in 5-axis CNC machining, which consists of tool orientation planning and cutter-contact(CC) point planning. The planning strategy highly depends on the type of tool cutter...Tool path generation is a fundamental problem in 5-axis CNC machining, which consists of tool orientation planning and cutter-contact(CC) point planning. The planning strategy highly depends on the type of tool cutters. For ball-end cutters, the tool orientation and CC point location can be planned separately;while for flat end cutters, the two are highly dependent on each other. This paper generates a smooth tool path of workpiece surfaces for flat end mills from two stages: Computing smooth tool orientations on the surface without gouging and collisions and then designing the CC point path. By solving the tool posture optimization problem the authors achieve both the path smoothness and the machining efficiency. Experimental results are provided to show the effectiveness of the method.展开更多
This issue contains the following seven papers as the proceeding of the Eleventh Conference of Computer Mathematics (CM2019), held at Chengdu, Sichuan Province, October 24–27,2019. In the conference, the contributed ...This issue contains the following seven papers as the proceeding of the Eleventh Conference of Computer Mathematics (CM2019), held at Chengdu, Sichuan Province, October 24–27,2019. In the conference, the contributed papers were selected by the Program Committee for presentation at the symposium and went through a standard refereeing process after the symposium. We are therefore extremely grateful to the Program Committee members and the reviewers for their work in evaluating the submissions before and after the conference.展开更多
Curve interpolation with B-spline is widely used in various areas. This problem is classic and recently raised in application scenario with new requirements such as path planning following the tangential vector field ...Curve interpolation with B-spline is widely used in various areas. This problem is classic and recently raised in application scenario with new requirements such as path planning following the tangential vector field under certified error in CNC machining. This paper proposes an algorithm framework to solve Hausdorff distance certified cubic B-spline interpolation problem with or without tangential direction constraints. The algorithm has two stages: The first stage is to find the initial cubic B-spine fitting curve which satisfies the Hausdorff distance constraint;the second stage is to set up and solve the optimization models with certain constraints. Especially, the sufficient conditions of the global Hausdorff distance control for any error bound are discussed, which can be expressed as a series of linear and quadratic constraints. A simple numerical algorithm to compute the Hausdorff distance between a polyline and its B-spline interpolation curve is proposed to reduce our computation.Experimental results are presented to show the advantages of the proposed algorithms.展开更多
基金supported by Beijing Natural Science Foundation under Grant No.Z190004the National Key Research and Development Project under Grant No.2020YFA0713703the Fundamental Research Funds for the Central Universities.
文摘Let f(t,y,y')=∑ _(i=0)^(n )a_(i)(t,y)y'^(i)=0 be an irreducible first order ordinary differential equation with polynomial coefficients.Eremenko in 1998 proved that there exists a constant C such that every rational solution of f(t,y,y')=0 is of degree not greater than C.Examples show that this degree bound C depends not only on the degrees of f in t,y,y' but also on the coefficients of f viewed as the polynomial in t,y,y'.In this paper,the authors show that if f satisfies deg(f,y)<deg(f,y')or n max i=0{deg(a_(i),y)−2(n−i)}>0,then the degree bound C only depends on the degrees of f in t,y,y',and furthermore we present an explicit expression for C in terms of the degrees of f in t,y,y'.
基金supported in part by the National Natural Science Foundation of China under Grant Nos.10901055,61021004,91118007by NKBRPC 2011CB302802,2011CB70690by the Fundamental Research Funds for the Central Universities under Grant No.78210043
文摘This paper presents a hybrid symbolic-numeric algorithm to compute ranking functions for establishing the termination of loop programs with polynomial guards and polynomial assignments.The authors first transform the problem into a parameterized polynomial optimization problem,and obtain a numerical ranking function using polynomial sum-of-squares relaxation via semidefinite programming(SDP).A rational vector recovery algorithm is deployed to recover a rational polynomial from the numerical ranking function,and some symbolic computation techniques are used to certify that this polynomial is an exact ranking function of the loop programs.At last,the authors demonstrate on some polynomial loop programs from the literature that our algorithm successfully yields nonlinear ranking functions with rational coefficients.
基金supported by Beijing Nova Program under Grant No.Z121104002512065The author PerezDíaz S is a member of the Research Group ASYNACS(Ref.CCEE2011/R34)
文摘The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, the authors focus on the rational developable surfaces. For a given algebraic surface, the authors ?rst determine whether it is developable by geometric inspection, and then give a rational proper parametrization in the affrmative case. For a rational parametric surface, the authors also determine the developability and give a proper reparametrization for the developable surface.
基金supported by the National Natural Science Foundation of China under Grant No.11688101,61872332Beijing National Natural Science Foundation under Grant No.Z190004+1 种基金National Center for Mathematics and Interdisciplinary SciencesYouth Innovation Promotion Association of the Chinese Academy of Sciences。
文摘Tool path generation is a fundamental problem in 5-axis CNC machining, which consists of tool orientation planning and cutter-contact(CC) point planning. The planning strategy highly depends on the type of tool cutters. For ball-end cutters, the tool orientation and CC point location can be planned separately;while for flat end cutters, the two are highly dependent on each other. This paper generates a smooth tool path of workpiece surfaces for flat end mills from two stages: Computing smooth tool orientations on the surface without gouging and collisions and then designing the CC point path. By solving the tool posture optimization problem the authors achieve both the path smoothness and the machining efficiency. Experimental results are provided to show the effectiveness of the method.
文摘This issue contains the following seven papers as the proceeding of the Eleventh Conference of Computer Mathematics (CM2019), held at Chengdu, Sichuan Province, October 24–27,2019. In the conference, the contributed papers were selected by the Program Committee for presentation at the symposium and went through a standard refereeing process after the symposium. We are therefore extremely grateful to the Program Committee members and the reviewers for their work in evaluating the submissions before and after the conference.
基金partially supported by the National Key Research and Development Program of China under Grant No. 2020YFA0713703the National Science Foundation of China under Grant Nos. 11688101, 12371384+1 种基金12271516the Fundamental Research Funds for the Central Universities。
文摘Curve interpolation with B-spline is widely used in various areas. This problem is classic and recently raised in application scenario with new requirements such as path planning following the tangential vector field under certified error in CNC machining. This paper proposes an algorithm framework to solve Hausdorff distance certified cubic B-spline interpolation problem with or without tangential direction constraints. The algorithm has two stages: The first stage is to find the initial cubic B-spine fitting curve which satisfies the Hausdorff distance constraint;the second stage is to set up and solve the optimization models with certain constraints. Especially, the sufficient conditions of the global Hausdorff distance control for any error bound are discussed, which can be expressed as a series of linear and quadratic constraints. A simple numerical algorithm to compute the Hausdorff distance between a polyline and its B-spline interpolation curve is proposed to reduce our computation.Experimental results are presented to show the advantages of the proposed algorithms.
