In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithm...In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.展开更多
Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (...Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).展开更多
In this paper,we use the theory of value distribution and study the uniqueness of meromorphic functions.We will prove the following result:Let f(z)and g(z)be two transcendental meromorphic functions,p(z)a polynomial o...In this paper,we use the theory of value distribution and study the uniqueness of meromorphic functions.We will prove the following result:Let f(z)and g(z)be two transcendental meromorphic functions,p(z)a polynomial of degree k,n≥max{11,k+1}a positive integer.If fn(z)f(z)and gn(z)g(z)share p(z)CM,then either f(z)=c1ec p(z)dz, g(z)=c2e ?c p(z)dz ,where c1,c2 and c are three constants satisfying(c1c2) n+1 c2=-1 or f(z)≡tg(z)for a constant t such that tn+1=1.展开更多
The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations.As an illus...The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations.As an illustration of the method,the Devil's Problem of Pommaret is solved in details.展开更多
We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal a...We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets. If the polynomial system is zero dimensional, the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.展开更多
To improve the vibration isolation performance of suspensions,various new structural forms of suspensions have been proposed.However,there is uncertainty in these new structure suspensions,so the deterministic researc...To improve the vibration isolation performance of suspensions,various new structural forms of suspensions have been proposed.However,there is uncertainty in these new structure suspensions,so the deterministic research cannot refect the performance of the suspension under actual operating conditions.In this paper,a quasi-zero stifness isolator is used in automotive suspensions to form a new suspension−quasi-zero stifness air suspension(QZSAS).Due to the strong nonlinearity and structural complexity of quasi-zero stifness suspensions,changes in structural parameters may cause dramatic changes in suspension performance,so it is of practical importance to study the efect of structural parameter uncertainty on the suspension performance.In order to solve this problem,three suspension structural parameters d_(0),L_(0) and Pc_(0) are selected as random variables,and the polynomial chaos expansion(PCE)theory is used to solve the suspension performance parameters.The sensitivity of the performance parameters to diferent structural parameters was discussed and analyzed in the frequency domain.Furthermore,a multi-objective optimization of the structural parameters d_(0),L_(0) and Pc_(0) of QZSAS was performed with the mean and variance of the root-mean-square(RMS)acceleration values as the optimization objectives.The optimization results show that there is an improvement of about 8%−1_(0)%in the mean value and about 4_(0)%−55%in the standard deviation of acceleration(RMS)values.This paper verifes the feasibility of the PCE method for solving the uncertainty problem of complex nonlinear systems,which provide a reference for the future structural design and optimization of such suspension systems.展开更多
Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+...Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.展开更多
In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈...In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.展开更多
This paper treats the problem of root distribution invariance of polynomial families. We first establish the generalized zero exclusion principle for root distribution of polynomial families, and prove the complex bou...This paper treats the problem of root distribution invariance of polynomial families. We first establish the generalized zero exclusion principle for root distribution of polynomial families, and prove the complex boundary theorem and complex edge theorem for robust stability in parameter space. Based on the generalized zero-exclusion principle, the corresponding results on root distribution in parameter and coefficient space are obtained. Moreover, for real edge theorem on robust stability in coefficient space, we show that the assumptions made on the stability region can further be weakened. For some more geometrically characterized polytopes, and some specific stability regions, the edge theorem can be improved, i.e. the number of edges to be checked is independent of the number of edges of the polytope. Finally, a Nyquist-type criterion is proposed for verification of the root distribution of an edge.展开更多
In this paper, we deal with basic properties of some pretzel links and properties of the Jones polynomials of some pretzel links. By using these properties, the zero distribution of pretzel links is studied. We discus...In this paper, we deal with basic properties of some pretzel links and properties of the Jones polynomials of some pretzel links. By using these properties, the zero distribution of pretzel links is studied. We discuss the properties of the Jones polynomial of non-tame pretzel links and give that zeros of the Jones polynomial of these pretzel links are distributed on the planar curves.展开更多
Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and unif...Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds axe discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.展开更多
IT is well known that for any real coefficient polynomial f(x) of degree n,how to determineits number of real zero points is a very important problem in theory of polynomial,as well as inapplications in mathematics,me...IT is well known that for any real coefficient polynomial f(x) of degree n,how to determineits number of real zero points is a very important problem in theory of polynomial,as well as inapplications in mathematics,mechanics,physics and other subjects.The way to solve thisproblem by using Euclidean mutual division as pointed out by Sturm is not simple and conve-nient for us to use the theoretic analyses for the above problem because the relation展开更多
In this paper we extend Enestrom -Kakeya theorem to a large class of polyno- mials with complex coefficients by putting less restrictions on the coefficients. Our results generalise and extend many known results in th...In this paper we extend Enestrom -Kakeya theorem to a large class of polyno- mials with complex coefficients by putting less restrictions on the coefficients. Our results generalise and extend many known results in this direction.展开更多
Recently, various results on the existence of deficient value or infinitely many zeros of several special classes of differential polynomials of an entire function or a transcendental meromorphic function f satisfying...Recently, various results on the existence of deficient value or infinitely many zeros of several special classes of differential polynomials of an entire function or a transcendental meromorphic function f satisfying δ(∞, f)=1 have been obtained. In this paper, we have summarized some of them and extended these results by obtaining some quantitative estimations on the zeros of several general classes of differential polynomials of an arbitrary transcendental meromorphic function. The proofs utilize the improved version of the Clunie lemma on differential polynomials and carefully count the multiplicities of the zeros of various auxiliary functions.展开更多
The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., ...The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.展开更多
文摘In this paper, a Ritt-Wu＇s characteristic set method for ordinary difference systems is proposed, which is valid for any admissible ordering. New definition for irreducible chains and new zero decomposition algorithms are also proposed.
基金Supported by the Natural Science Foundation of Beijing
文摘Many fields require the zeros of orthogonal polynomials. In this paper, the middle variable was improved to give a new asymptotic approximation, with error bounds, for the Jacobi polynomials P (α,β) n(cosθ) (0≤θ≤π/2,α,β>-1), as n→+∞. An accurate approximation with error bounds is also constructed for the zero θ n,s of P (α,β) n(cosθ)(α≥0,β>-1).
基金Supported by the Natural Science Foundation of Jiangsu Education Department（07KJD110086）
文摘In this paper,we use the theory of value distribution and study the uniqueness of meromorphic functions.We will prove the following result:Let f(z)and g(z)be two transcendental meromorphic functions,p(z)a polynomial of degree k,n≥max{11,k+1}a positive integer.If fn(z)f(z)and gn(z)g(z)share p(z)CM,then either f(z)=c1ec p(z)dz, g(z)=c2e ?c p(z)dz ,where c1,c2 and c are three constants satisfying(c1c2) n+1 c2=-1 or f(z)≡tg(z)for a constant t such that tn+1=1.
基金The present paper is in honor of late Professor R.Thom as a great mathematician, a great scientist,also a great thinker of modern times.
文摘The char-set method of polynomial equations-solving is naturally extended to the differential case which gives rise to an algorithmic method of solving arbitrary systems of algebrico-differential equations.As an illustration of the method,the Devil's Problem of Pommaret is solved in details.
文摘In this paper we establish L^q inequalities for polynomials, which in particular yields interesting generalizations of some Zygmund-type inequalities.
基金This work was partially supported by the National Key Basic Research Project of China(Grant No.2004CB31800)
文摘We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets. If the polynomial system is zero dimensional, the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.
基金Supported by National Natural Science Foundation of China(Grant No.51875256)Open Platform Fund of Hunan Institute of Technology of China(Grant No.KFA20009)Hong Kong,Macao and Taiwan Science and Technology Cooperation Project in Jiangsu Province of China(Grant No.BZ2020050)。
文摘To improve the vibration isolation performance of suspensions,various new structural forms of suspensions have been proposed.However,there is uncertainty in these new structure suspensions,so the deterministic research cannot refect the performance of the suspension under actual operating conditions.In this paper,a quasi-zero stifness isolator is used in automotive suspensions to form a new suspension−quasi-zero stifness air suspension(QZSAS).Due to the strong nonlinearity and structural complexity of quasi-zero stifness suspensions,changes in structural parameters may cause dramatic changes in suspension performance,so it is of practical importance to study the efect of structural parameter uncertainty on the suspension performance.In order to solve this problem,three suspension structural parameters d_(0),L_(0) and Pc_(0) are selected as random variables,and the polynomial chaos expansion(PCE)theory is used to solve the suspension performance parameters.The sensitivity of the performance parameters to diferent structural parameters was discussed and analyzed in the frequency domain.Furthermore,a multi-objective optimization of the structural parameters d_(0),L_(0) and Pc_(0) of QZSAS was performed with the mean and variance of the root-mean-square(RMS)acceleration values as the optimization objectives.The optimization results show that there is an improvement of about 8%−1_(0)%in the mean value and about 4_(0)%−55%in the standard deviation of acceleration(RMS)values.This paper verifes the feasibility of the PCE method for solving the uncertainty problem of complex nonlinear systems,which provide a reference for the future structural design and optimization of such suspension systems.
基金supported partially by the National Science Foundation of China(Grant#11771304)the Fundamental Research Funds for the Central Universities.
文摘Let F be a field,and let e,k be integers such that 1≤e≤|F\{0}|and k≥0.We show that for any subset{a1,……,ae}■F\{0},the curious identity∑(i1+……ie)∈Z^(e)≥0,i1+……+ie=k a_(1)^(i1)…a_(e)^(ie)=∑i=1 e a_(i)^(k+e-1)/∏i≠j=1 e(a_(i)-a_(j))holds with Z≥0 being the set of nonnegative integers.As an application,we prove that for any subset{a_(1)…,a_(e)}■F_(q)\{0}with F_(q)being the finite field of q elements and e,l being integers such that 2≤e≤q-1 and 0≤l≤e-2,∑(i_(1),…,i_(e))∈Z^(e)≥0,i_(1)+…i_(e)=q-e+l a_(1)^(i1)…a_(e)^(ie)=0 Using this identity and providing an extension of the principle of cross-classification that slightly generalizes the one obtained by Hong in 1996,we show that if r is an integer with 1≤r≤q-2,then for any subset{a_(1),…a_(r)}■F_(q)^(*)we have x^(q-1)-1/∏i=1 r(x-a_(i))-∑i=1 q-1-r(∑i_(1)+…+i_(r)=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i).This implies#{x∈Fq*|∑i=0 q-1-4(∑_(i1)+…+ir=q-1-r-i^(a_(1)^(i1)…a_(r)^(ir)))x^(i)=0}=q-1-r.
文摘In this paper we consider a class of polynomials P（z）=a0＋∑v^n=tavZv, t≥ 1, not vanishing in ｜z｜ 〈k, k≥1 and investigate the dependence of max｜z｜=1 ｜P（Rz） -P（rz）] on max｜z｜=1｜P（z）｜, where 1≤ r 〈 R. Our result generalizes and refines some known polynomial inequalities.
文摘This paper treats the problem of root distribution invariance of polynomial families. We first establish the generalized zero exclusion principle for root distribution of polynomial families, and prove the complex boundary theorem and complex edge theorem for robust stability in parameter space. Based on the generalized zero-exclusion principle, the corresponding results on root distribution in parameter and coefficient space are obtained. Moreover, for real edge theorem on robust stability in coefficient space, we show that the assumptions made on the stability region can further be weakened. For some more geometrically characterized polytopes, and some specific stability regions, the edge theorem can be improved, i.e. the number of edges to be checked is independent of the number of edges of the polytope. Finally, a Nyquist-type criterion is proposed for verification of the root distribution of an edge.
基金The NSF(11071106)of Chinathe Program(LR2011031)for Liaoning Excellent Talents in University
文摘In this paper, we deal with basic properties of some pretzel links and properties of the Jones polynomials of some pretzel links. By using these properties, the zero distribution of pretzel links is studied. We discuss the properties of the Jones polynomial of non-tame pretzel links and give that zeros of the Jones polynomial of these pretzel links are distributed on the planar curves.
基金Project supported by Scientific Research Common Program of Beijing Municipal Commission of Education of China (No.KM200310015060)
文摘Krawtchouk polynomials are frequently applied in modern physics. Based on the results which were educed by Li and Wong, the asymptotic expansions of Krawtchouk polynomials are improved by using Airy function, and uniform asymptotic expansions are got. Furthermore, the asymptotic expansions of the zeros for Krawtchouk polynomials are again deduced by using the property of the zeros of Airy function, and their corresponding error bounds axe discussed. The obtained results give the asymptotic property of Krawtchouk polynomials with their zeros, which are better than the results educed by Li and Wong.
文摘IT is well known that for any real coefficient polynomial f(x) of degree n,how to determineits number of real zero points is a very important problem in theory of polynomial,as well as inapplications in mathematics,mechanics,physics and other subjects.The way to solve thisproblem by using Euclidean mutual division as pointed out by Sturm is not simple and conve-nient for us to use the theoretic analyses for the above problem because the relation
文摘In this paper we extend Enestrom -Kakeya theorem to a large class of polyno- mials with complex coefficients by putting less restrictions on the coefficients. Our results generalise and extend many known results in this direction.
基金Project partially supported by U. P. G. C., Hong Kong and partially supported by the National Natural Science Foundation of China.
文摘Recently, various results on the existence of deficient value or infinitely many zeros of several special classes of differential polynomials of an entire function or a transcendental meromorphic function f satisfying δ(∞, f)=1 have been obtained. In this paper, we have summarized some of them and extended these results by obtaining some quantitative estimations on the zeros of several general classes of differential polynomials of an arbitrary transcendental meromorphic function. The proofs utilize the improved version of the Clunie lemma on differential polynomials and carefully count the multiplicities of the zeros of various auxiliary functions.
文摘The value distribution of differential polynomials is studied. The results in this paper improve and generalize some previous theorems given by Yang Chungchun （On deficiencies of differential polynomials, Math. Z., 116（1970）, 197- 204）, H. S. Gopalakrishna and S. S. Bhoosnurmath （On distribution of values of differential polynomials, Indian 3. Pure Appl. Math., 17（1986）, 367-372）, I. Lahiri （A note on distribution of nonhomogeneous differential polynomials, Hokkaido Math. J., 31（2002）, 453-458） and Yi Hongxun （On zeros of differential polynomials, Adv. in Math., 18（1989）, 335-351） et al. Examples show that the results in this paper are sharu.
