The wheat above-ground biomass(AGB)is an important index that shows the life activity of vegetation,which is of great significance for wheat growth monitoring and yield prediction.Traditional biomass estimation method...The wheat above-ground biomass(AGB)is an important index that shows the life activity of vegetation,which is of great significance for wheat growth monitoring and yield prediction.Traditional biomass estimation methods specifically include sample surveys and harvesting statistics.Although these methods have high estimation accuracy,they are time-consuming,destructive,and difficult to implement to monitor the biomass at a large scale.The main objective of this study is to optimize the traditional remote sensing methods to estimate the wheat AGBbased on improved convolutional features(CFs).Low-cost unmanned aerial vehicles(UAV)were used as the main data acquisition equipment.This study acquired image data acquired by RGB camera(RGB)and multi-spectral(MS)image data of the wheat population canopy for two wheat varieties and five key growth stages.Then,field measurements were conducted to obtain the actual wheat biomass data for validation.Based on the remote sensing indices(RSIs),structural features(SFs),and CFs,this study proposed a new feature named AUR-50(multi-source combination based on convolutional feature optimization)to estimate the wheat AGB.The results show that AUR-50 could estimate the wheat AGB more accurately than RSIs and SFs,and the average R^(2) exceeded 0.77.In the overwintering period,AUR-50_(MS)(multi-source combination with convolutional feature optimization using multispectral imagery)had the highest estimation accuracy(R^(2) of 0.88).In addition,AUR-50 reduced the effect of the vegetation index saturation on the biomass estimation accuracy by adding CFs,where the highest R^(2) was 0.69 at the flowering stage.The results of this study provide an effective method to evaluate the AGB in wheat with high throughput and a research reference for the phenotypic parameters of other crops.展开更多
The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(...The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(2)=p(z)or[p1(z)f′(z)]^(2)+[p2(z)f(z+c)+p3(z)f(z)]^(2)=p(z);where c is a nonzero complex number,p1;p2 and p3 are polynomials in C satisfying p1p3■0;and p is a nonzero irreducible polynomial in C.展开更多
In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z)...In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z),g(z))withρ(f,g)<∞such thatλ(f)<ρ(f)andλ(g)<ρ(g),where P_(1()z),Q_(1)(z),P_(2)(z)and Q_(2)(z)are non-vanishing polynomials.展开更多
In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The prop...In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.展开更多
On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear ...On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].展开更多
Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=...Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=θ.In this paper,we mainly investigate the dynamical properties of Julia sets of entire solutions of the complex differential equations F(z)f^(n)(z)+P(z,f)=0,and f^(n)+A(z)P(z,f)=h(z),where P(z,f)is a differential polynomial in f and its derivatives,F(z),A(z)and h(z)are entire functions.We demonstrate the existence of close relationships Petrenko's deviations of the coefficients and the measures of limiting directions of entire solutions of the above two equations.展开更多
In this paper,the existence and growth of entire solutions of some type of nonlinear delay-differential equations are studied.Using Cartan's second main theorem and Nevanlinna theory of meromorphic functions,we ob...In this paper,the existence and growth of entire solutions of some type of nonlinear delay-differential equations are studied.Using Cartan's second main theorem and Nevanlinna theory of meromorphic functions,we obtain the exact forms of its entire solutions with hyperorder less than one.展开更多
In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where ...In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.展开更多
Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is...Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .展开更多
We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H...We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].展开更多
In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex d...In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.展开更多
If a partial contralateral C7 nerve is transferred to a recipient injured nerve, results are not satisfactory. However, if an entire contralateral C7 nerve is used to repair two nerves, both recipient nerves show goo...If a partial contralateral C7 nerve is transferred to a recipient injured nerve, results are not satisfactory. However, if an entire contralateral C7 nerve is used to repair two nerves, both recipient nerves show good recovery. These findings seem contradictory, as the above two methods use the same donor nerve, only the cutting method of the contralateral C7 nerve is different. To verify whether this can actually result in different repair effects, we divided rats with right total brachial plexus injury into three groups. In the entire root group, the entire contralateral C7 root was transected and transferred to the median nerve of the affected limb. In the posterior division group, only the posterior division of the contralateral C7 root was transected and transferred to the median nerve. In the entire root ＋ posterior division group, the entire contralateral C7 root was transected but only the posterior division was transferred to the median nerve. After neurectomy,the median nerve was repaired on the affected side in the three groups. At 8, 12, and 16 weeks postoperatively, electrophysiological examination showed that maximum amplitude, latency, muscle tetanic contraction force, and muscle fiber cross-sectional area of the flexor digitorum superficialis muscle were significantly better in the entire root and entire root ＋ posterior division groups than in the posterior division group. No significant difference was found between the entire root and entire root ＋ posterior division groups. Counts of myelinated axons in the median nerve were greater in the entire root group than in the entire root ＋ posterior division group, which were greater than the posterior division group. We conclude that for the same recipient nerve, harvesting of the entire contralateral C7 root achieved significantly better recovery than partial harvesting, even if only part of the entire root was used for transfer. This result indicates that the entire root should be used as a donor when transferring contralateral C7 nerve.展开更多
This study investigated the linkages between river water quality and land use in river catchments in Yama- guchi Prefecture, the western Japan, in order to examine the effect of land use changes of both entire catchme...This study investigated the linkages between river water quality and land use in river catchments in Yama- guchi Prefecture, the western Japan, in order to examine the effect of land use changes of both entire catchment and buffer zone on river water quality. Dissolved Oxygen (DO), pH, Biological Oxygen Demand (BOD), Suspended Solids (SS), Escherichia coli, Total Nitrogen (TN) and Total Phosphorus (TP) were considered as river water quality indica- tors. Satellite images were applied to generating the land use map. Multiple regression model was applied to linking the changes in the river water quality with the land uses in both entire catchment area and buffer zone. The results in- dicate that the integrative application of land use data from the entire catchment and the buffer zone could give rise to more robust model to predict the concentrations of Suspended Solids (r2=0.88) and Total Nitrogen (r2=0.90), rather than models which separately considered land use data in catchment and buffer zone.展开更多
In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets...In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets of solutions,their derivatives and their primitives must have a definite range of measure.展开更多
In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous fun...In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.展开更多
In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of ...In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.展开更多
In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and pr...In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.展开更多
In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f...In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f of degree d(≤n-2),p1,p2 are small meromorphic functions of e^(z),andα1,α2 are nonzero constants.Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order.As its applications,we also find some type of nonlinear difference equations having no finite order entire solutions.展开更多
This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).
基金supported by the Postgraduate Research&Practice Innovation Program of Jiangsu Province,China(SJCX23_1973)the National Natural Science Foundation of China(32172110,32071945)+7 种基金the Key Research and Development Program(Modern Agriculture)of Jiangsu Province,China(BE2022342-2,BE2020319)the Anhui Province Crop Intelligent Planting and Processing Technology Engineering Research Center Open Project,China(ZHKF04)the National Key Research and Development Program of China(2023YFD2300201,2023YFD1202200)the Special Funds for Scientific and Technological Innovation of Jiangsu Province,China(BE2022425)the Priority Academic Program Development of Jiangsu Higher Education Institutions,China(PAPD)the Central Publicinterest Scientific Institution Basal Research Fund,China(JBYW-AII-2023-08)the Science and Technology Innovation Project of the Chinese Academy of Agricultural Sciences(CAAS-CS-202201)the Special Fund for Independent Innovation of Agriculture Science and Technology in Jiangsu Province,China(CX(22)3112)。
文摘The wheat above-ground biomass(AGB)is an important index that shows the life activity of vegetation,which is of great significance for wheat growth monitoring and yield prediction.Traditional biomass estimation methods specifically include sample surveys and harvesting statistics.Although these methods have high estimation accuracy,they are time-consuming,destructive,and difficult to implement to monitor the biomass at a large scale.The main objective of this study is to optimize the traditional remote sensing methods to estimate the wheat AGBbased on improved convolutional features(CFs).Low-cost unmanned aerial vehicles(UAV)were used as the main data acquisition equipment.This study acquired image data acquired by RGB camera(RGB)and multi-spectral(MS)image data of the wheat population canopy for two wheat varieties and five key growth stages.Then,field measurements were conducted to obtain the actual wheat biomass data for validation.Based on the remote sensing indices(RSIs),structural features(SFs),and CFs,this study proposed a new feature named AUR-50(multi-source combination based on convolutional feature optimization)to estimate the wheat AGB.The results show that AUR-50 could estimate the wheat AGB more accurately than RSIs and SFs,and the average R^(2) exceeded 0.77.In the overwintering period,AUR-50_(MS)(multi-source combination with convolutional feature optimization using multispectral imagery)had the highest estimation accuracy(R^(2) of 0.88).In addition,AUR-50 reduced the effect of the vegetation index saturation on the biomass estimation accuracy by adding CFs,where the highest R^(2) was 0.69 at the flowering stage.The results of this study provide an effective method to evaluate the AGB in wheat with high throughput and a research reference for the phenotypic parameters of other crops.
基金Supported by the National Natural Science Foundation of China(11871260,11761050)the Jiangxi Natural Science Foundation(#20232ACB201005)+1 种基金the Shandong Natural Science Foundation(#ZR2024MA024)Doctoral Startup Fund of Jiangxi Science and Technology Normal University(#2021BSQD30).
文摘The main purpose of this paper is to try to find all entire solutions of the Fermat type difference-differential equation[p1(z)f(z+c)]^(2)+[p2(z)f(z)+p3(z)f′(z)]^(2)=p(z);or[p1(z)f(z)]^(2)+[p2(z)f′(z)+p3(z)f(z+c)]^(2)=p(z)or[p1(z)f′(z)]^(2)+[p2(z)f(z+c)+p3(z)f(z)]^(2)=p(z);where c is a nonzero complex number,p1;p2 and p3 are polynomials in C satisfying p1p3■0;and p is a nonzero irreducible polynomial in C.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344).
文摘In this paper,we mainly investigate the forms of entire solutions for certain Fermattype partial differential-difference equations in C^(2)by using Nevanlinna’s theory of several complex variables.
基金Supported by the National Natural Science Foundation of China(Grant No.11971344).
文摘In this paper,we will prove that the system of differential-difference equations{((f(z)f′(z))^(n)+p^(2)_(1)(z)g m(z+η)=Q1(z),(g(z)g′(z))^(n)+p^(2)_(2)(z)f m(z+η)=Q_(2)(z),has no transcendental entire solution(f(z),g(z))withρ(f,g)<∞such thatλ(f)<ρ(f)andλ(g)<ρ(g),where P_(1()z),Q_(1)(z),P_(2)(z)and Q_(2)(z)are non-vanishing polynomials.
基金supported partly by the National Natural Science Foundation of China(11926201,12171050)the National Science Foundation of Guangdong Province(2018A030313508)。
文摘In this paper,we consider entire solutions of higher order homogeneous differential equations with the entire coefficients having the same order,and prove that the entire solutions are of infinite lower order.The properties on the radial distribution,the limit direction of the Julia set and the existence of a Baker wandering domain of the entire solutions are also discussed.
基金Supported by the National Natural Science Foundation of China(12261023,11861023)the Foundation of Science and Technology project of Guizhou Province of China([2018]5769-05)。
文摘On one hand,we study the existence of transcendental entire solutions with finite order of the Fermat type difference equations.On the other hand,we also investigate the existence and growth of solutions of nonlinear differential-difference equations.These results extend and improve some previous in[5,14].
基金Supported by the National Natural Science Foundation of China(11971344)。
文摘Assume that f is a transcendental entire function.The ray arg z=θ∈[0,2π]is said to be a limiting direction of the Julia set J(f)of f if there exists an unbounded sequence{z_(n)}■J(f)such that lim rn→∞ arg z_(n)=θ.In this paper,we mainly investigate the dynamical properties of Julia sets of entire solutions of the complex differential equations F(z)f^(n)(z)+P(z,f)=0,and f^(n)+A(z)P(z,f)=h(z),where P(z,f)is a differential polynomial in f and its derivatives,F(z),A(z)and h(z)are entire functions.We demonstrate the existence of close relationships Petrenko's deviations of the coefficients and the measures of limiting directions of entire solutions of the above two equations.
基金Supported by the National Natural Science Foundation of China(Grant No.12261044)the STP of Education Department of Jiangxi Province(Grant No.GJJ210302)。
文摘In this paper,the existence and growth of entire solutions of some type of nonlinear delay-differential equations are studied.Using Cartan's second main theorem and Nevanlinna theory of meromorphic functions,we obtain the exact forms of its entire solutions with hyperorder less than one.
文摘In this paper, we investigate the growth of transcendental entire solutionsof the following algebraic differential equation a(z)f'~2 +(b_2(z)f^2 +b_1(z)f +b_0(z))f'=d_3(z)f^3+d_2(z)f^2 +d_1(z)f +d_0(z), where a(z), b_i(z) (0<- i <=2) and d_j (z) (0<=j<= 3) are allpolynomials, and this equation relates closely to the following well-known algebraic differentialequation C(z,w)w'~2 + B(z,w)w' + A(z,w) =0, where G(z,w)not ident to 0, B(z,w) and A(z,w) are threepolynomials in z and w. We give relationships between the growth of entire solutions and the degreesof the above three polynomials in detail.
文摘Let f and g be two permutable transcendental entire functions. In this paper, we first prove that J(fg)=J(f n g m) for any positive integers n and m . Then we prove that the function h(p(z))+az ∈/ B , where h(z) is any transcendental entire function with h′(z)=0 having infinitely many solutions, p(z) is a polynomial with deg p ≥2 and a(≠0) ∈ C .
基金supported by NSF of Fujian Province,China(S0750013),supported by NSF of Fujian Province,China(2008J0190)the Research Foundation of Ningde Normal University(2008J001)the Scientific Research Foundation for the Returned Overseas Chinese Scholars,State Education Ministry
文摘We deal with the problem of entire functions sharing one value weakly. Moreover, we improve and generalize some former results obtained by J.-F.Chen, et al. [6], Y.Xu and H.L.Qiu [4], M.L. Fang [5], C.C. Yang, and X.H. Hua [3].
文摘In this paper,we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations,and obtain some interesting results.It extends some results concerning complex differential（difference） equations to the systems of differential-difference equations.
基金supported by the National Natural Science Foundation of China,No.H0605/81501871
文摘If a partial contralateral C7 nerve is transferred to a recipient injured nerve, results are not satisfactory. However, if an entire contralateral C7 nerve is used to repair two nerves, both recipient nerves show good recovery. These findings seem contradictory, as the above two methods use the same donor nerve, only the cutting method of the contralateral C7 nerve is different. To verify whether this can actually result in different repair effects, we divided rats with right total brachial plexus injury into three groups. In the entire root group, the entire contralateral C7 root was transected and transferred to the median nerve of the affected limb. In the posterior division group, only the posterior division of the contralateral C7 root was transected and transferred to the median nerve. In the entire root ＋ posterior division group, the entire contralateral C7 root was transected but only the posterior division was transferred to the median nerve. After neurectomy,the median nerve was repaired on the affected side in the three groups. At 8, 12, and 16 weeks postoperatively, electrophysiological examination showed that maximum amplitude, latency, muscle tetanic contraction force, and muscle fiber cross-sectional area of the flexor digitorum superficialis muscle were significantly better in the entire root and entire root ＋ posterior division groups than in the posterior division group. No significant difference was found between the entire root and entire root ＋ posterior division groups. Counts of myelinated axons in the median nerve were greater in the entire root group than in the entire root ＋ posterior division group, which were greater than the posterior division group. We conclude that for the same recipient nerve, harvesting of the entire contralateral C7 root achieved significantly better recovery than partial harvesting, even if only part of the entire root was used for transfer. This result indicates that the entire root should be used as a donor when transferring contralateral C7 nerve.
文摘This study investigated the linkages between river water quality and land use in river catchments in Yama- guchi Prefecture, the western Japan, in order to examine the effect of land use changes of both entire catchment and buffer zone on river water quality. Dissolved Oxygen (DO), pH, Biological Oxygen Demand (BOD), Suspended Solids (SS), Escherichia coli, Total Nitrogen (TN) and Total Phosphorus (TP) were considered as river water quality indica- tors. Satellite images were applied to generating the land use map. Multiple regression model was applied to linking the changes in the river water quality with the land uses in both entire catchment area and buffer zone. The results in- dicate that the integrative application of land use data from the entire catchment and the buffer zone could give rise to more robust model to predict the concentrations of Suspended Solids (r2=0.88) and Total Nitrogen (r2=0.90), rather than models which separately considered land use data in catchment and buffer zone.
基金supported by Shanghai Center for Mathematical Sci-ences,China Scholarship Council(201206105015)National Science Foundation of China(11171119,11001057,11571049)Natural Science Foundation of Guangdong Province in China(2014A030313422)
文摘In this paper,we mainly investigate the dynamical properties of entire solutions of complex differential equations.With some conditions on coefficients,we prove that the set of common limiting directions of Julia sets of solutions,their derivatives and their primitives must have a definite range of measure.
文摘In this paper, the nonexistence of positive entire solutions for div(|Du| p-2 Du)q(x)f(u),x∈R N, is established, where p】1,Du=(D 1u,...,D Nu),q∶R N→(0,∞) and f∶(0,∞)→(0,∞) are continuous functions.
基金supported by the Natural Science Foundation of Guangdong Province in China(2014A030313422,2016A030310106,2016A030313745)
文摘In this article, we mainly devote to proving uniqueness results for entire functionssharing one small function CM with their shift and difference operator simultaneously. Letf（z） be a nonconstant entire function of finite order, c be a nonzero finite complex constant, and n be a positive integer. If f（z）, f（z＋c）, and △n cf（z） share 0 CM, then f（z＋c）≡Af（z）, where A（≠0） is a complex constant. Moreover, let a（z）, b（z）（ O） ∈ S（f） be periodic entire functions with period c and if f（z） - a（z）, f（z ＋ c） - a（z）, △cn f（z） - b（z） share 0 CM, then f（z ＋ c） ≡ f（z）.
基金supported by Shanghai Center for Mathematical Science China Scholarship Council(201206105015)the National Science Foundation of China(11171119,11001057,11571049)the Natural Science Foundation of Guangdong Province in China(2014A030313422)
文摘In this paper, we mainly investigate entire solutions of complex differential equations with coefficients involving exponential functions, and obtain the dynamical properties of the solutions, their derivatives and primitives. With some conditions on coefficients, for the solutions, their derivatives and their primitives, we consider the common limiting directions of Julia set and the existence of Baker wandering domain.
基金supported by the National Natural Science Foundation of China(11661044)
文摘In this article,the existence of finite order entire solutions of nonlinear difference equations f^(n)+P_(d)(z,f)=p1 e^(α1 z)+p2 e^(α2 z)are studied,where n≥2 is an integer,P_(d)(z,f)is a difference polynomial in f of degree d(≤n-2),p1,p2 are small meromorphic functions of e^(z),andα1,α2 are nonzero constants.Some necessary conditions are given to guarantee that the above equation has an entire solution of finite order.As its applications,we also find some type of nonlinear difference equations having no finite order entire solutions.
文摘This paper proves a result that if two entire functions f(z) and g(z) share four small functions aj(z) (j = 1,2,3,4) in the sense of Ek)(aj, f) = Ek)(aj,g), (j = 1,2,3,4) (k ≥ 11), then there exists f(z) = g(z).
