By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn...By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.展开更多
Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is k...Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.展开更多
In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The...In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.展开更多
Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary m...Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.展开更多
This paper deals with two concepts of polynomial dichotomy for linear difference equations which are defined in a Banach space and whose norms can increase not faster than exponentially.Some illustrating examples clar...This paper deals with two concepts of polynomial dichotomy for linear difference equations which are defined in a Banach space and whose norms can increase not faster than exponentially.Some illustrating examples clarify the relations between these concepts.Our approach is based on the extension of techniques for exponential dichotomy to the case of polynomial dichotomy.The obtained results are generalizations of well-known theorems about the exponential stability and exponential dichotomy.展开更多
This paper is devoted to the problem of stabilizing a Hopfield-type neural network with bi-directional ring architecture and two delays. The delay-independent and delay-dependent stability conditions are explicitly pr...This paper is devoted to the problem of stabilizing a Hopfield-type neural network with bi-directional ring architecture and two delays. The delay-independent and delay-dependent stability conditions are explicitly presented by the method of the characteristic roots and the skill of mathematical analysis. Moreover, if a link between the adjacent two neurons is cut, the ring neural network turns to a linear one, and the stability results are also established. Furthermore, a comparative analysis for the ring and linear network shows that the stability domain is enlarged after the breaking.展开更多
A polynomially exponential time restrained analytical hierarchy is introduced with the basic proper ties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to an...A polynomially exponential time restrained analytical hierarchy is introduced with the basic proper ties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level. And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.展开更多
Inhomogeneous Oscillator Representations of P(n)Ling Chen Abstract We study inhomogeneous oscillator representations of the strange Lie superalgebras P(n) on supersymmetric polynomial algebras and on spaces of supersy...Inhomogeneous Oscillator Representations of P(n)Ling Chen Abstract We study inhomogeneous oscillator representations of the strange Lie superalgebras P(n) on supersymmetric polynomial algebras and on spaces of supersymmetric exponentialpolynomial functions.We obtain the composition series for these representations.The obtained irreducible modules are infinite dimensional.Some of them are not of highest-weight type and even not weight modules.展开更多
The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on...The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.展开更多
In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The re...In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin＇s classical results.展开更多
基金supported partly by the National Natural Science Foundation of China(12171050,11871260)National Science Foundation of Guangdong Province(2018A030313508)。
文摘By looking at the situation when the coefficients Pj(z)(j=1,2,…,n-1)(or most of them) are exponential polynomials,we investigate the fact that all nontrivial solutions to higher order differential equations f((n))+Pn-1(z)f((n-1))+…+P0(z)f=0 are of infinite order.An exponential polynomial coefficient plays a key role in these results.
基金supported by the China Scholarship Council (CSC)supported in part by the Academy of Finland #121281
文摘Recently, C.-C. Yang and L Laine have investigated finite order entire solutions f of non- linear differential-difference equations of the form f^n ＋ L（z, f） -= h, where n ≥ 2 is an integer. In particular, it is known that the equation f（z）^2 ＋ q（z）f（z ＋ 1） = p（z）, where p（z）, q（z） are polynomials, has no transcendental entire solutions of finite order. Assuming that Q（z） is also a polynomial and c E C, equations of the form f（z）^n ＋ q（z）e^Q（Z）f（z ＋ c） = p（z） do posses finite order entire solutions. A classification of these solutions in terms of growth and zero distribution will be given. In particular, it is shown that any exponential polynomial solution must reduce to a rather specific form. This reasoning relies on an earlier paper due to N. Steinmetz.
基金Supported by the National Natural Science Foundation of China(No.60873181 and No.u0935004)
文摘In this paper we develop a novel approach to construct non-stationary subdivision schemes with a tension control parameter which can reproduce functions in a finite-dimensional subspace of exponential polynomials. The construction process is mainly implemented by solving linear systems for primal and dual subdivision schemes respectively, which are based on different parameterizations. We give the theoretical basis for the existence, uniqueness, and refinement rules of schemes proposed in this paper. The convergence and smoothness of the schemes are analyzed as well. Moreover, conics reproducing schemes are analyzed based on our theory, and a new idea that the tensor parameter ωk of the schemes can be adjusted for conics generation is proposed.
基金The authors would like to thank Prof. Y.D. Zhang for selfless helps and valuable discussions.
文摘Making use of the transformation relation among usual, normal, and antinormal ordering for the multimode boson exponential quadratic polynomial operators （BEQPO＇s）I we present the analytic expression of arbitrary matrix elements for BEQPO＇s. As a preliminary application, we obtain the exact expressions of partition function about the boson quadratic polynomial system, matrix elements in particle-number, coordinate, and momentum representation, and P representation for the BEQPO＇s.
基金Supported by the Fundamental Research Funds for the Central Universities(Grant No.2012LWB53)the Natural Science Foundation of Hubei Province(Grant No.2014CFB629)
文摘This paper deals with two concepts of polynomial dichotomy for linear difference equations which are defined in a Banach space and whose norms can increase not faster than exponentially.Some illustrating examples clarify the relations between these concepts.Our approach is based on the extension of techniques for exponential dichotomy to the case of polynomial dichotomy.The obtained results are generalizations of well-known theorems about the exponential stability and exponential dichotomy.
文摘This paper is devoted to the problem of stabilizing a Hopfield-type neural network with bi-directional ring architecture and two delays. The delay-independent and delay-dependent stability conditions are explicitly presented by the method of the characteristic roots and the skill of mathematical analysis. Moreover, if a link between the adjacent two neurons is cut, the ring neural network turns to a linear one, and the stability results are also established. Furthermore, a comparative analysis for the ring and linear network shows that the stability domain is enlarged after the breaking.
基金Research supported by the Youth NSF grant JJ890407.
文摘A polynomially exponential time restrained analytical hierarchy is introduced with the basic proper ties of the hierarchy followed.And it will be shown that there is a recursive set A such that A does not belong to any level of the p-arithmetical hierarchies.Then we shall prove that there are recursive sets A and B such that the different levels of the analytical hierarchy relative to A are different and for some n every level higher than n of the analytical hierarchy relative to B is the same as the n-th level. And whether the higher levels are collapsed into some lower level is neither provable nor disprovable in set theory and several other results.
文摘Inhomogeneous Oscillator Representations of P(n)Ling Chen Abstract We study inhomogeneous oscillator representations of the strange Lie superalgebras P(n) on supersymmetric polynomial algebras and on spaces of supersymmetric exponentialpolynomial functions.We obtain the composition series for these representations.The obtained irreducible modules are infinite dimensional.Some of them are not of highest-weight type and even not weight modules.
基金supported by the Natural Science Foundation of Shanghai(Grant No.21ZR1468500)the Fundamental Research Funds for the Central Universities(Grant No.22120240299)。
文摘The numerical computation of partial differential equations(PDEs)is highly important across numerous scientific and engineering disciplines.The accuracy and convergence of integration-based methods depend primarily on the ability to perform analytical integration over complex domains.Owing to the inherent challenges posed by the complexities of irregular integration domains and general integrands,this paper introduces an innovative analytical method for nonpolynomial integration over complex domains for the first time.This method is initially applied within the framework of the numerical manifold method(NMM)to address the inevitable trigonometric and exponential polynomial integrations encountered in the analysis of the Laplace equation problem.First,a comprehensive overview of the fundamentals of the NMM and the simplex integration(SI)method is provided in this paper.Subsequently,the NMM framework for solving the Laplace equation is elaborated upon,with a focus on deriving closed-form formulas for trigonometric and exponential polynomial integration.Finally,a series of rigorous numerical experiments is conducted,where the proposed method demonstrates improved accuracy and efficiency.In conclusion,this study innovatively enhances the NMM by introducing the SI method for nonpolynomial functions over complex domains,which is a promising approach for increasing accuracy and convergence across various integration-based methods.This groundbreaking achievement has not yet been reported in the publicly available literature.
基金Project supported by the National Natural Science Foundation of China (No.10371005)the Scientific Research Foundation of the Ministry of Education of China for Returned Overseas Chinese Scholars
文摘In this paper, the completeness and minimality properties of some random exponential system in a weighted Banach space of complex functions continuous on the real line for convex nonnegative weight are studied. The results may be viewed as a probabilistic version of Malliavin＇s classical results.
