This paper presents a generalization of the authors＇ earlier work. In this paper, the two concepts, generic regular decomposition （GRD） and regular-decomposition-unstable （RDU） variety introduced in the authors＇...This paper presents a generalization of the authors＇ earlier work. In this paper, the two concepts, generic regular decomposition （GRD） and regular-decomposition-unstable （RDU） variety introduced in the authors＇ previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors＇ previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors＇ previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.展开更多
In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces repre...Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently,Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic,we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f（z） and g（z） which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f（z） and g（z） can be divided into two cases. One case is that f（z） is a constant and g（z） is a quadratic polynomial, and another case is that the degree of f（z） and g（z） are 2 and 1 respectively. For these two cases,we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover,we study some geometric properties of the bi-quintic harmonic surfaces based on the B′ezier representation. Finally, some numerical examples are demonstrated to verify our results.展开更多
We consider the problem of complex root classification,i.e.,finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots.It is well known that suc...We consider the problem of complex root classification,i.e.,finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots.It is well known that such conditions can be written as conjunctions of several polynomial equalities and one inequality in the coefficients.Those polynomials in the coefficients are called discriminants for multiplicities.It is also known that discriminants can be obtained using repeated parametric greatest common divisors.The resulting discriminants are usually nested determinants,i.e.,determinants of matrices whose entries are determinants,and so on.In this paper,we give a new type of discriminant that is not based on repeated greatest common divisors.The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees.展开更多
In this paper,a new method to analyze Boolean functions is proposed.By this method,one can analyze the balancedness,the nonlinearity,and the input-output correlation of vectorial Boolean functions.The basic idea of th...In this paper,a new method to analyze Boolean functions is proposed.By this method,one can analyze the balancedness,the nonlinearity,and the input-output correlation of vectorial Boolean functions.The basic idea of this method is to compute the refined covers of some parametric Boolean polynomial systems which are equivalent to these problems.By a refined cover,the parameter space is divided into several disjoint components,and on each component,the parametric Boolean polynomial system has a fixed number of solutions.An efficient algorithm based on the characteristic set method to compute refined covers of parametric Boolean polynomial systems is presented.The experimental results about some instances generated from cryptanalysis show that this new method is efficient and can solve some instances which can not be solved in reasonable time by other methods.展开更多
A complete solution classification of the perspective-three-point(P3P) problem is given by using the Gr?bner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be...A complete solution classification of the perspective-three-point(P3P) problem is given by using the Gr?bner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Gr?bner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.展开更多
Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted muc...Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted much attention over the past several decades, and many effcient algorithms have been proposed. Moreover, these algorithms have been applied to many different ?elds,such as parametric polynomial equations solving, geometric theorem proving and discovering, quanti?er elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.展开更多
The propagation of oscillating disturbances with various frequencies in multi stage turbine passages in a rocket is analyzed using the oscillating fluid mechanics theorem and the parametric polynomial method. The r...The propagation of oscillating disturbances with various frequencies in multi stage turbine passages in a rocket is analyzed using the oscillating fluid mechanics theorem and the parametric polynomial method. The results show that oscillating disturbances can be rapidly dissipated when the disturbance occurs at the inlet except for very high frequency oscillation such as 50 kHz. Dangerous low frequency oscillations occur at the outlet. The effects of the flow parameter variations on the oscillating disturbance propagation are also studied. The analysis will facilitate safe operation of the whole rocket system.展开更多
基金supported by by the National Natural Science Foundation of China under Grant Nos.11271034,11290141the Project SYSKF1207 from SKLCS,IOS,the Chinese Academy of Sciences
文摘This paper presents a generalization of the authors＇ earlier work. In this paper, the two concepts, generic regular decomposition （GRD） and regular-decomposition-unstable （RDU） variety introduced in the authors＇ previous work for generic zero-dimensional systems, are extended to the case where the parametric systems are not necessarily zero-dimensional. An algorithm is provided to compute GRDs and the associated RDU varieties of parametric systems simultaneously on the basis of the algorithm for generic zero-dimensional systems proposed in the authors＇ previous work. Then the solutions of any parametric system can be represented by the solutions of finitely many regular systems and the decomposition is stable at any parameter value in the complement of the associated RDU variety of the parameter space. The related definitions and the results presented in the authors＇ previous work are also generalized and a further discussion on RDU varieties is given from an experimental point of view. The new algorithm has been implemented on the basis of DISCOVERER with Maple 16 and experimented with a number of benchmarks from the literature.
基金The work was partially supported by NSFC # 69735052
文摘In this paper, we study the factorization of bi-orthogonal Laurent polynomial wavelet matrices with degree one into simple blocks. A conjecture about advanced factorization is given.
基金Supported by the National Natural Science Foundation of China(11401077,11671068,11271060)the Fundamental Research of Civil Aircraft of China(MJ-F-2012-04)the Fundamental Research Funds for the Central Universities of China(DUT16LK38)
文摘Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently,Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic,we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f（z） and g（z） which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f（z） and g（z） can be divided into two cases. One case is that f（z） is a constant and g（z） is a quadratic polynomial, and another case is that the degree of f（z） and g（z） are 2 and 1 respectively. For these two cases,we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover,we study some geometric properties of the bi-quintic harmonic surfaces based on the B′ezier representation. Finally, some numerical examples are demonstrated to verify our results.
基金supported by U.S.National Science Foundations(Grant Nos.2212461 and 1813340)supported by National Natural Science Foundation of China(Grant Nos.12261010 and 11801101)。
文摘We consider the problem of complex root classification,i.e.,finding the conditions on the coefficients of a univariate polynomial for all possible multiplicity structures on its complex roots.It is well known that such conditions can be written as conjunctions of several polynomial equalities and one inequality in the coefficients.Those polynomials in the coefficients are called discriminants for multiplicities.It is also known that discriminants can be obtained using repeated parametric greatest common divisors.The resulting discriminants are usually nested determinants,i.e.,determinants of matrices whose entries are determinants,and so on.In this paper,we give a new type of discriminant that is not based on repeated greatest common divisors.The new discriminants are simpler in the sense that they are non-nested determinants and have smaller maximum degrees.
基金the National Natural Science Foundation of China under Grant Nos.61977060 and 61877058。
文摘In this paper,a new method to analyze Boolean functions is proposed.By this method,one can analyze the balancedness,the nonlinearity,and the input-output correlation of vectorial Boolean functions.The basic idea of this method is to compute the refined covers of some parametric Boolean polynomial systems which are equivalent to these problems.By a refined cover,the parameter space is divided into several disjoint components,and on each component,the parametric Boolean polynomial system has a fixed number of solutions.An efficient algorithm based on the characteristic set method to compute refined covers of parametric Boolean polynomial systems is presented.The experimental results about some instances generated from cryptanalysis show that this new method is efficient and can solve some instances which can not be solved in reasonable time by other methods.
基金supported by the National Nature Science Foundation of China under Grant Nos.11371356 and 61121062
文摘A complete solution classification of the perspective-three-point(P3P) problem is given by using the Gr?bner basis method. The structure of the solution space of the polynomial system deduced by the P3P problem can be obtained by computing a comprehensive Gr?bner system. Combining with properties of the generalized discriminant sequences, the authors give the explicit conditions to determine the number of distinct real positive solutions of the P3P problem. Several examples are provided to illustrate the effectiveness of the proposed conditions.
基金supported in part by the CAS Project QYZDJ-SSW-SYS022the National Natural Science Foundation of China under Grant No.61877058the Strategy Cooperation Project AQ-1701
文摘Weispfenning in 1992 introduced the concepts of comprehensive Gr?bner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then,this research ?eld has attracted much attention over the past several decades, and many effcient algorithms have been proposed. Moreover, these algorithms have been applied to many different ?elds,such as parametric polynomial equations solving, geometric theorem proving and discovering, quanti?er elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.
基金Supported by the State Key Developments Plan Project of China( No.G19990 2 2 3 0 4 )
文摘The propagation of oscillating disturbances with various frequencies in multi stage turbine passages in a rocket is analyzed using the oscillating fluid mechanics theorem and the parametric polynomial method. The results show that oscillating disturbances can be rapidly dissipated when the disturbance occurs at the inlet except for very high frequency oscillation such as 50 kHz. Dangerous low frequency oscillations occur at the outlet. The effects of the flow parameter variations on the oscillating disturbance propagation are also studied. The analysis will facilitate safe operation of the whole rocket system.
