This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of...This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).展开更多
Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defin...Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.展开更多
Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P&...Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +展开更多
Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are ...Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.展开更多
The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes ...The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes directly caused the coloration and variation of 4nRR skin(red in RCC,brownish-yellow in4nRR).To further explore the molecular mechanisms underlying coloration formation and variation in 4nRR,we performed transcriptome profiling and molecular functional verification in RCC and 4nRR.Results revealed that scarb1,associated with carotenoid metabolism,underwent significant down-regulation in 4nRR.Efficient editing of this candidate pigment gene provided clear evidence of its significant role in RCC coloration.Subsequently,we identified four divergent scarb1 homeologs in 4nRR:two original scarb1 homeologs from RCC and two duplicated ones.Notably,three of these homeologs possessed two highly conserved alleles,exhibiting biased and allelespecific expression in the skin.Remarkably,after precise editing of both the original and duplicated scarb1homeologs and/or alleles,4nRR individuals,whether singly or multiply mutated,displayed a transition from brownishyellow skin to a cyan-gray phenotype.Concurrently,the proportional areas of the cyan-gray regions displayed a gene-dose correlation.These findings illustrate the subfunctionalization of duplicated scarb1,with all scarb1genes synergistically and equally contributing to the pigmentation of 4nRR.This is the first report concerning the functional differentiation of duplicated homeologs in an autopolyploidfish,substantiallyenrichingour understanding of coloration formation and change within this group of organisms.展开更多
In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several differen...This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.展开更多
By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(...By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].展开更多
This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is pres...This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.展开更多
By means of an extension of Mawhin's continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, ...By means of an extension of Mawhin's continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.展开更多
In this paper,the boundary value problems of p-Laplacian functional differential equation are studied.By using a fixed point theorem in cones,some criteria for the existence of positive solutions are given.
The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces.The criteria of the invariant and attracting sets are obtained.Particularly, the sufficient condition of asymptot...The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces.The criteria of the invariant and attracting sets are obtained.Particularly, the sufficient condition of asymptotic stability of the equilibrium point is given as the system has an equilibrium point.Several examples are also worked out to demonstrate the validity of the results.展开更多
In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel function...In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z(Bκ+1^c f(z))′= κBκ^c f(z)-(κ- 1)Bκ+1^c f(z),where b, c, p ∈ C and κ = p +(b + 1)/2 ∈ C / Z0^-(Z0^-= {0,-1,-2, … }). The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.展开更多
In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen's inequality. Some new stability theorems are obtained. Examples are given...In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen's inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.展开更多
The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equa...The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.展开更多
In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s p...In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.展开更多
In this paper, we investigate the existence of multiple positive periodic solutions for functional differential equations with infinite delay by applying the Krasnoselskii fixed point theorem for cone map and the Legg...In this paper, we investigate the existence of multiple positive periodic solutions for functional differential equations with infinite delay by applying the Krasnoselskii fixed point theorem for cone map and the Leggett-Williams fixed point theorem.展开更多
The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1...The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1 cixi(t) +g(x(t-τ) ) =p(t) =p(t+2π) is obtained.The approach is based on the abstract continuation theorem from Mawhin and the a- priori estimate of periodic solutions展开更多
The aim of this work is to study the existence of a periodic solution for some neutral partial functional differential equations. Our approach is based on the R-boundedness of linear operators Lp-multipliers and UMD-s...The aim of this work is to study the existence of a periodic solution for some neutral partial functional differential equations. Our approach is based on the R-boundedness of linear operators Lp-multipliers and UMD-spaces.展开更多
In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switch...In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.展开更多
基金The second named author was supported in part by an NSERC Postdoctoral Fellowship,Canada and a CR F Grant,University of Alberta
文摘This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).
文摘Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.
文摘Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +
文摘Continuously differentiable radial basis functions (C<sup>∞</sup>-RBFs), while being theoretically exponentially convergent are considered impractical computationally because the coefficient matrices are full and can become very ill- conditioned. Similarly, the Hilbert and Vandermonde have full matrices and become ill-conditioned. The difference between a coefficient matrix generated by C<sup>∞</sup>-RBFs for partial differential or integral equations and Hilbert and Vandermonde systems is that C<sup>∞</sup>-RBFs are very sensitive to small changes in the adjustable parameters. These parameters affect the condition number and solution accuracy. The error terrain has many local and global maxima and minima. To find stable and accurate numerical solutions for full linear equation systems, this study proposes a hybrid combination of block Gaussian elimination (BGE) combined with arbitrary precision arithmetic (APA) to minimize the accumulation of rounding errors. In the future, this algorithm can execute faster using preconditioners and implemented on massively parallel computers.
基金supported by the National Natural Science Foundation of China (32172972,U19A2040)Science and Technology Innovation Program of Hunan Province (2021RC4028)+4 种基金Earmarked Fund for China Agriculture Research System (CARS-45)Hunan Provincial Science and Technology Department (2019RS5001)Special Funds for Construction of Innovative Provinces in Hunan Province (2021NK1010)Special Science Found of Nansha-South China Agricultural University Fishery Research Institute,Guangzhou (NSYYKY202305,NSYYKY202306)Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province。
文摘The autotetraploid Carassius auratus(4nRR,4n=200,RRRR)is derived from whole-genome duplication of Carassius auratus red var.(RCC,2n=100,RR).In the current study,we demonstrated that chromatophores and pigment changes directly caused the coloration and variation of 4nRR skin(red in RCC,brownish-yellow in4nRR).To further explore the molecular mechanisms underlying coloration formation and variation in 4nRR,we performed transcriptome profiling and molecular functional verification in RCC and 4nRR.Results revealed that scarb1,associated with carotenoid metabolism,underwent significant down-regulation in 4nRR.Efficient editing of this candidate pigment gene provided clear evidence of its significant role in RCC coloration.Subsequently,we identified four divergent scarb1 homeologs in 4nRR:two original scarb1 homeologs from RCC and two duplicated ones.Notably,three of these homeologs possessed two highly conserved alleles,exhibiting biased and allelespecific expression in the skin.Remarkably,after precise editing of both the original and duplicated scarb1homeologs and/or alleles,4nRR individuals,whether singly or multiply mutated,displayed a transition from brownishyellow skin to a cyan-gray phenotype.Concurrently,the proportional areas of the cyan-gray regions displayed a gene-dose correlation.These findings illustrate the subfunctionalization of duplicated scarb1,with all scarb1genes synergistically and equally contributing to the pigmentation of 4nRR.This is the first report concerning the functional differentiation of duplicated homeologs in an autopolyploidfish,substantiallyenrichingour understanding of coloration formation and change within this group of organisms.
文摘In this paper,a new basic method of constructing smooth production functions isgiven,and many ordinary production functions are drawn. It is obvious thatsome more production functions can be obtained from the method.
基金Supported by NSFC (11001091)Chinese UniversityResearch Foundation (2010MS129)
文摘This paper establishes the Razumikhin-type theorem on stability for neutral stochastic functional differential equations with unbounded delay. To overcome difficulties from unbounded delay, we develop several different techniques to investigate stability. To show our idea clearly, we examine neutral stochastic delay differential equations with unbounded delay and linear neutral stochastic Volterra unbounded-delay-integro-differential equations.
基金National Natural Science Foundation of China( 198710 0 5 )
文摘By means of an abstract continuation theorem, the existence criteria are established for the positive periodic solutions of a neutral functional differential equation d N d t=N(t)[a(t)-β(t)N(t)-b(t)N(t-σ(t))-c(t)N′(t-τ(t))].
基金Supported by the National Natural Science Foundation of China (10571050 10871062)Hunan Provincial Innovation Foundation For Postgraduate
文摘This paper is concerned with the existence and approximation of solutions for a class of first order impulsive functional differential equations with periodic boundary value conditions. A new comparison result is presented and the previous results are extended.
基金sponsored by the National Natural Science Foundation of China(11271197)the Science and Technology Foundation in Ministry of Education of China(207047)the Science Foundation of NUIST of China(20090202 and 2012r101)
文摘By means of an extension of Mawhin's continuation theorem and some analysis methods, the existence of a set with 2kT-periodic solutions for a class of second order neutral functional differential systems is studied, and then the homoclinic solutions are obtained as the limit points of a certain subsequence of the above set.
文摘In this paper,the boundary value problems of p-Laplacian functional differential equation are studied.By using a fixed point theorem in cones,some criteria for the existence of positive solutions are given.
基金Supported by the National Natural Science Foundation of China( 1 9831 0 30 ) ,( 1 0 1 71 0 72 ) .
文摘The paper is devoted to the asymptotic properties of functional differential equations in Banach spaces.The criteria of the invariant and attracting sets are obtained.Particularly, the sufficient condition of asymptotic stability of the equilibrium point is given as the system has an equilibrium point.Several examples are also worked out to demonstrate the validity of the results.
基金partly supported by the Natural Science Foundation of China(11271045)the Higher School Doctoral Foundation of China(20100003110004)+2 种基金the Natural Science Foundation of Inner Mongolia of China(2010MS0117)athe Higher School Foundation of Inner Mongolia of China(NJZY13298)the Commission for the Scientific Research Projects of Kafkas Univertsity(2012-FEF-30)
文摘In the present paper, we derive some third-order differential subordination results for analytic functions in the open unit disk, using the operator Bcκf by means of normalized form of the generalized Bessel functions of the first kind, which is defined as z(Bκ+1^c f(z))′= κBκ^c f(z)-(κ- 1)Bκ+1^c f(z),where b, c, p ∈ C and κ = p +(b + 1)/2 ∈ C / Z0^-(Z0^-= {0,-1,-2, … }). The results are obtained by considering suitable classes of admissible functions. Various known or new special cases of our main results are also pointed out.
基金supported by the National Natural Science Foundation of China (No. 10871063)Scientific Research Fund of Hunan Provincial Education Department (No. 07A038)
文摘In this paper, we investigate the stability of a class of impulsive functional differential equations by using Lyapunov functional and Jensen's inequality. Some new stability theorems are obtained. Examples are given to demonstrate the advantage of the obtained results.
基金Sponsored by HUST Foundation(0125011017)the National NSFC under grant(70671047)
文摘The stability of stochastic functional differential equation with Markovian switching was studied by several authors,but there was almost no work on the stability of the neutral stochastic functional differential equations with Markovian switching.The aim of this article is to close this gap.The authors establish Razumikhin-type theorem of the neutral stochastic functional differential equations with Markovian switching,and those without Markovian switching.
文摘In this paper. four sufficiency theorems of existence of periodic solutions for aclass of retarded functional differential equations are given. The result of thesetheorems is better than the well-known Yoshizawa’s periodic solution theorem. Anexample of application is given at the end.
文摘In this paper, we investigate the existence of multiple positive periodic solutions for functional differential equations with infinite delay by applying the Krasnoselskii fixed point theorem for cone map and the Leggett-Williams fixed point theorem.
基金Supported by the National Natural Science Foundation of China(1 9971 0 2 6 )
文摘The sufficient condition for the existence of2 π- periodic solutions of the following third- order functional differential equations with variable coefficients a(t) x (t) +bx″2 k- 1(t) +cx′2 k- 1(t) + 2 k- 1 i=1 cixi(t) +g(x(t-τ) ) =p(t) =p(t+2π) is obtained.The approach is based on the abstract continuation theorem from Mawhin and the a- priori estimate of periodic solutions
文摘The aim of this work is to study the existence of a periodic solution for some neutral partial functional differential equations. Our approach is based on the R-boundedness of linear operators Lp-multipliers and UMD-spaces.
文摘In this paper, the Razumikhin approach is applied to the study of both p-th moment and almost sure stability on a general decay for a class of impulsive stochastic functional differential systems with Markovian switching. Based on the Lyapunov-Razumikhin methods, some sufficient conditions are derived to check the stability of impulsive stochastic functional differential systems with Markovian switching. One numerical example is provided to demonstrate the effectiveness of the results.