In this paper,the growth characteristic of meromorphic solutions for the following difference equation An(z)f(z+n)+…+A1(z)f(z+1)+A0(z)f(z)=0 with no dominating coefficient is studied.By imposing certain restriction o...In this paper,the growth characteristic of meromorphic solutions for the following difference equation An(z)f(z+n)+…+A1(z)f(z+1)+A0(z)f(z)=0 with no dominating coefficient is studied.By imposing certain restriction on the entire coefficients associated with Petrenko's deviation of the above equation,we obtain some results and partially address a question posed byⅠ.Laine and C.C.Yang.Furthermore,for the entire solutions f(z)of the difference equation An(z)f(z+n)+…+A1(z)f(z+1)+A0(z)f(z)=F(z),where Aj(z)(j=0,…,n),F(z)are entire functions,we discover a close relationship between the measure of common transcendental directions associated with classical difference operators of f(z)and Petrenko's deviations of the coefficients.展开更多
In this paper, using Nevanlinna theory of the value distribution of meromorphic functions, the problem of growth order of solutions of a class of system of complex difference equations is investigated, some results ar...In this paper, using Nevanlinna theory of the value distribution of meromorphic functions, the problem of growth order of solutions of a class of system of complex difference equations is investigated, some results are improved and generalized. More precisely,some results of the growth order of solutions of system of differential equations to difference equations are extended.展开更多
By using asymptotic method,we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series meth...By using asymptotic method,we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series method.The classical problem on finding conditions on the polynomial coefficients P_(j)(z)(j=0,1,2)and F(z)to guarantee that all nontrivial solutions of complex second order difference equation P_(2)(z)f(z+2)+P_(1)(z)f(z+1)+P_(0)(z)f(z)=F(z)has slowly growing solutions with order 1/2 is detected.展开更多
In this article, for a transcendental entire function f（z） of finite order which has a finite Borel exceptional value a, we utilize properties of complex difference equations to prove the difference counterpart of B...In this article, for a transcendental entire function f（z） of finite order which has a finite Borel exceptional value a, we utilize properties of complex difference equations to prove the difference counterpart of Bruck＇s conjecture, that is, if △f（z） = f（z ＋ η） - f（z） and f（z） share one value a （≠α） CM, where η ∈ C is a constant such that f（z ＋η） ≠ f（z）,then△f（z）-a/f（z）-a=a/a-α.展开更多
In this paper, meromorphic solutions of Riccati and linear difference equations are investigated. The growth and Borel exceptional values of these solutions are discussed, and the growth, zeros and poles of difference...In this paper, meromorphic solutions of Riccati and linear difference equations are investigated. The growth and Borel exceptional values of these solutions are discussed, and the growth, zeros and poles of differences of these solutions are also investigated. Furthermore, several examples are given showing that our results are best possible in certain senses.展开更多
Let f be a transcendental meromorphic function and △f（z） = f（z ＋ 1） -- f（z） A number of results are proved concerning the existences of zeros and fixed points of Af（z） and △f（z）/f（z） when f（z） is of o...Let f be a transcendental meromorphic function and △f（z） = f（z ＋ 1） -- f（z） A number of results are proved concerning the existences of zeros and fixed points of Af（z） and △f（z）/f（z） when f（z） is of order σ（f）=1. Examples show that some of the results are sharp.展开更多
Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimate...Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.展开更多
Let f（z） be a finite order meromorphic function and let c ∈ C ／ {0} be a constant. If f（z） has a Borel exceptional value α∈ C, it is proved that max{τ（f（z） ）, τ（ △cf（z） ） } = max{τ（f（z） ）, τ...Let f（z） be a finite order meromorphic function and let c ∈ C ／ {0} be a constant. If f（z） has a Borel exceptional value α∈ C, it is proved that max{τ（f（z） ）, τ（ △cf（z） ） } = max{τ（f（z） ）, τ（f（z ＋ c））} = max{T（ τ（△cf（z） ）, τ（f（z ＋ c））} = σ（f（z） ）. If f（z） has a Borel exceptional value b ∈ （C ／ {0}） ∪ {∞}, it is proved that max{τ（f（z））,τ（△cf（z）/f（z）}=max（τ（△cf（z）/f（z））,τ（f（z＋c））}=σ（f（z））unless f（z） takes a special form. Here T（g（z）） denotes the exponent of convergence of fixed points of the meromorphic function g（z）, and a（g（z）） denotes the order of growth of g（z）.展开更多
Extractants play an important role in the separation and purification of rare earth elements(REEs),and extraction performance is the most effective tactic to evaluate whether an extractant shows complete involvement a...Extractants play an important role in the separation and purification of rare earth elements(REEs),and extraction performance is the most effective tactic to evaluate whether an extractant shows complete involvement and fast action in the separation of REEs.In this paper,the complexation differences between lanthanum/lutetium(La/Lu)and four phosphoric acid extractants(C272,P227,P507 and P204)with very similar structures but different extraction abilities are analyzed by theoretical calculations for the first time.The results show that REEs are coordinated with two oxygen atoms in each dimer extractant to form three octatomic ring complexes.La is more likely to be complexed in the form of eight water molecules in aqueous solution,while Lu takes action with the extractant in the form of pure ions.Furthermore,La mainly hybridized with coordinated oxygen atoms via 4f,5d and 6p orbitals,while Lu hybridized via 6s and 5d orbitals,and the difference between La and four extractants originates from 5d and 6p electrons while it mainly comes from the 6s orbital for Lu.It is really remarkable that the number of electrons transferred from La/Lu coordination oxygen atoms to REEs decreased in the order of P204>P507>P227>C272,indicating that the extraction ability of P204 is the strongest,which is consistent with the experimental results.Based on this result,an extractant containing NH-substituted oxygen(HEHAEP)with stronger extraction capacity than P204 was designed.Our theoretical study laid a steadfast foundation for the design of extractants,and opened up a new way for the experimental synthesis of new extractants.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11661043)and the ScienceTechnology Research Project of Jiangxi Provincial Department of Education(Grant No.GJJ2200320).
文摘In this paper,the growth characteristic of meromorphic solutions for the following difference equation An(z)f(z+n)+…+A1(z)f(z+1)+A0(z)f(z)=0 with no dominating coefficient is studied.By imposing certain restriction on the entire coefficients associated with Petrenko's deviation of the above equation,we obtain some results and partially address a question posed byⅠ.Laine and C.C.Yang.Furthermore,for the entire solutions f(z)of the difference equation An(z)f(z+n)+…+A1(z)f(z+1)+A0(z)f(z)=F(z),where Aj(z)(j=0,…,n),F(z)are entire functions,we discover a close relationship between the measure of common transcendental directions associated with classical difference operators of f(z)and Petrenko's deviations of the coefficients.
基金Project Supported by the fundamental research funds for the Central Universities project of China(No.11614801)Combining with the project of Guangdong Province production(No.2011A090200044)
文摘In this paper, using Nevanlinna theory of the value distribution of meromorphic functions, the problem of growth order of solutions of a class of system of complex difference equations is investigated, some results are improved and generalized. More precisely,some results of the growth order of solutions of system of differential equations to difference equations are extended.
文摘By using asymptotic method,we verify the existence on the slowly growing solutions to second order difference equations discussed by Ishizaki-Yanagihara’s Wiman-Valiron method and Ishizaki-Wen’s binomial series method.The classical problem on finding conditions on the polynomial coefficients P_(j)(z)(j=0,1,2)and F(z)to guarantee that all nontrivial solutions of complex second order difference equation P_(2)(z)f(z+2)+P_(1)(z)f(z+1)+P_(0)(z)f(z)=F(z)has slowly growing solutions with order 1/2 is detected.
基金supported by the National Natural Science Foundation of China(11171119)
文摘In this article, for a transcendental entire function f（z） of finite order which has a finite Borel exceptional value a, we utilize properties of complex difference equations to prove the difference counterpart of Bruck＇s conjecture, that is, if △f（z） = f（z ＋ η） - f（z） and f（z） share one value a （≠α） CM, where η ∈ C is a constant such that f（z ＋η） ≠ f（z）,then△f（z）-a/f（z）-a=a/a-α.
基金supported by National Natural Science Foundation of China(1122609011171119)Guangdong Natural Science Foundation(S2012040006865)
文摘In this paper, meromorphic solutions of Riccati and linear difference equations are investigated. The growth and Borel exceptional values of these solutions are discussed, and the growth, zeros and poles of differences of these solutions are also investigated. Furthermore, several examples are given showing that our results are best possible in certain senses.
基金supported by the NSF of Shandong Province, China (ZR2010AM030)the NNSF of China (11171013 & 11041005)
文摘Let f be a transcendental meromorphic function and △f（z） = f（z ＋ 1） -- f（z） A number of results are proved concerning the existences of zeros and fixed points of Af（z） and △f（z）/f（z） when f（z） is of order σ（f）=1. Examples show that some of the results are sharp.
基金supported by National Natural Science Foundation of China (Grant No. 10871076)Brain Pool Program of Korean Federation of Science and Technology Societies (Grant No. 072-1-3-0164)
文摘Let f be a transcendental meromorphic function and g(z)=f(z+c1)+f(z+c2)-2f(z) and g2(z)=f(z+c1)·f(z+c2)-f2(z).The exponents of convergence of zeros of differences g(z),g2(z),g(z)/f(z),and g2(z)/f2(z) are estimated accurately.
基金Supported by Training Plan Fund of Outstanding Young Teachers of Higher Learning Institutions of Guangdong Province(Grant No.Yq20145084602)National Natural Science Foundation of China(Grant Nos.11226090,11171119)Guangdong National Natural Science Foundation(Grant Nos.2014A030313422,2016A030313745)
文摘Let f（z） be a finite order meromorphic function and let c ∈ C ／ {0} be a constant. If f（z） has a Borel exceptional value α∈ C, it is proved that max{τ（f（z） ）, τ（ △cf（z） ） } = max{τ（f（z） ）, τ（f（z ＋ c））} = max{T（ τ（△cf（z） ）, τ（f（z ＋ c））} = σ（f（z） ）. If f（z） has a Borel exceptional value b ∈ （C ／ {0}） ∪ {∞}, it is proved that max{τ（f（z））,τ（△cf（z）/f（z）}=max（τ（△cf（z）/f（z））,τ（f（z＋c））}=σ（f（z））unless f（z） takes a special form. Here T（g（z）） denotes the exponent of convergence of fixed points of the meromorphic function g（z）, and a（g（z）） denotes the order of growth of g（z）.
基金supported by NSAF(no.U2130114)supported by the National Key Research and Development Project of China(2019YFC0605003)the Strategic Priority Research Program of CAS(XDA02030100)。
文摘Extractants play an important role in the separation and purification of rare earth elements(REEs),and extraction performance is the most effective tactic to evaluate whether an extractant shows complete involvement and fast action in the separation of REEs.In this paper,the complexation differences between lanthanum/lutetium(La/Lu)and four phosphoric acid extractants(C272,P227,P507 and P204)with very similar structures but different extraction abilities are analyzed by theoretical calculations for the first time.The results show that REEs are coordinated with two oxygen atoms in each dimer extractant to form three octatomic ring complexes.La is more likely to be complexed in the form of eight water molecules in aqueous solution,while Lu takes action with the extractant in the form of pure ions.Furthermore,La mainly hybridized with coordinated oxygen atoms via 4f,5d and 6p orbitals,while Lu hybridized via 6s and 5d orbitals,and the difference between La and four extractants originates from 5d and 6p electrons while it mainly comes from the 6s orbital for Lu.It is really remarkable that the number of electrons transferred from La/Lu coordination oxygen atoms to REEs decreased in the order of P204>P507>P227>C272,indicating that the extraction ability of P204 is the strongest,which is consistent with the experimental results.Based on this result,an extractant containing NH-substituted oxygen(HEHAEP)with stronger extraction capacity than P204 was designed.Our theoretical study laid a steadfast foundation for the design of extractants,and opened up a new way for the experimental synthesis of new extractants.
