Using liellrami-Schaefer stress funtion in the theory of elasticity in this paper, we derive the stress functions of torsion, plane problem, axisymmetric deformation in solid of revolution and torsion on solid of revo...Using liellrami-Schaefer stress funtion in the theory of elasticity in this paper, we derive the stress functions of torsion, plane problem, axisymmetric deformation in solid of revolution and torsion on solid of revolution.展开更多
In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on margina...In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data,and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered.In addition,we also consider two other clustering methods,k-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis.Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index.The approaches are further illustrated through empirical analysis of human mortality data.展开更多
In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in un...In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.展开更多
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
The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen-...The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.展开更多
In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their...In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.展开更多
By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or we...By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.展开更多
The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen (2004) to a subclass of nonsymmetric tensor functions satisfying the commu...The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen (2004) to a subclass of nonsymmetric tensor functions satisfying the commutative condition. This subclass of tensor functions is more general than those investigated by the existing methods. In the case of three distinct eigenvalues, the commutativity makes it possible to introduce two scalar functions, which will be used to construct the general nonsymmetric tensor functions and their derivatives. In the cases of repeated eigenvalues, the results are acquired by taking limits.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
Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) fo...Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.展开更多
The molecular geometries, frontier molecular orbital properties, and absorption and emission properties of three 4-phenoxy-1,8-naphthalimide derivatives, namely 4-phenoxy-N-(2-hydroxyethyl)-1,8-naphthalimide(1),4-...The molecular geometries, frontier molecular orbital properties, and absorption and emission properties of three 4-phenoxy-1,8-naphthalimide derivatives, namely 4-phenoxy-N-(2-hydroxyethyl)-1,8-naphthalimide(1),4-(2-tert-butylphenoxy)-N-(2-hydroxyethyl)-1,8-naphthalimide(2), and 4-[2,4-di(tert-butyl)]phenoxy-N-(2-hydroxyethyl)-1,8-naphthalimide(3), are investigated by density functional theory(DFT) and time-dependent density functional theory(TD-DFT) calculations in conjunction with polarizable continuum models(PCMs). Four functionals and ten basis sets are employed for 1 to calculate the electron transition energies, which were compared with the experimental observations. Our results reveal that the B3LYP/6-311+G(d,p) method is the best choice to reproduce the experimental spectra. Moreover, the effects of substituents on the molecular geometries, electronic structures, absorption and emission spectra are also studied at the B3LYP/6-311+G(d,p) level. We find that the gap between the highest occupied molecular orbital(HOMO) and the lowest unoccupied molecular orbital(LUMO) decreases with increasing the number of tert-butyl substituents onto the phenoxy groups, suggesting red-shift of the absorption and emission bands. This is related to the increase of conjugation from 1 to 2 and 3. Our calculations are in good agreement with the experimental results.展开更多
The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function b...The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function by the partial sum of its Fourier series is inves tigated.Moreover,the order of approximation is describe d with the 2th continuous modulus.展开更多
An environmentally benign and efficient method has been developed for the synthesis of functionalized tetrahydroindole derivatives in aqueous media under catalyst-free conditions,by simply combining a phenylglyoxal mo...An environmentally benign and efficient method has been developed for the synthesis of functionalized tetrahydroindole derivatives in aqueous media under catalyst-free conditions,by simply combining a phenylglyoxal monohydrate,an enaminone,and a barbituric acid.The advantages of this method are that it is catalyst free,has an easy workup,provides good yields,and uses water as solvent,which make this procedure facile and practical.展开更多
In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant n...In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.展开更多
An excellent introduction to the topic of poset matroids is due to M.Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rankaxioms for poset matroids. In this paper, we stud...An excellent introduction to the topic of poset matroids is due to M.Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rankaxioms for poset matroids. In this paper, we study the special integral function f and obtain a newclass of poset matroids from the old ones, and then we generalize this result according to theproperties of f. Almost all of these results can be regarded as the application of global rankaxioms for poset matroids. The main results in our paper have, indeed, investigated the restrictionof the basis of the poset matroid, and we give them the corresponding geometric interpretation.展开更多
This article studies a new kind of stability judgment for dynamic systems, the definiteness Lyapunov derivative function approach based on the minimum function. The new results are described by introducing the normal ...This article studies a new kind of stability judgment for dynamic systems, the definiteness Lyapunov derivative function approach based on the minimum function. The new results are described by introducing the normal positivity. The article analyzes the relationship of a general system and a polynomial system, the judgment and measurement of positive definiteness, the polar coordinate approach, and constructing the algorithms for positive definite v·(x). Therefore, a new and systematic theory on stability judgment is clearly described.展开更多
We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear(PN) functions,including the properties of the derivative of the components of those functions.So...We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear(PN) functions,including the properties of the derivative of the components of those functions.Some sufficient and necessary conditions have been explored to judge when a function is a PN function.These conditions may be useful in constructing new PN functions.We also construct some functions with differential 4-uniformity that have rarely been studied in the literature.Some of the constructed functions with differential 4-uniformity have high nonlinearity as well.Finally,a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.展开更多
文摘Using liellrami-Schaefer stress funtion in the theory of elasticity in this paper, we derive the stress functions of torsion, plane problem, axisymmetric deformation in solid of revolution and torsion on solid of revolution.
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
基金supported by National Natural Science Foundation of China (Grant Nos.12261007)Natural Science Foundation of Guangxi Province (Grant No.2020GXNSFAA297225)。
文摘In this paper,we consider the clustering of bivariate functional data where each random surface consists of a set of curves recorded repeatedly for each subject.The k-centres surface clustering method based on marginal functional principal component analysis is proposed for the bivariate functional data,and a novel clustering criterion is presented where both the random surface and its partial derivative function in two directions are considered.In addition,we also consider two other clustering methods,k-centres surface clustering methods based on product functional principal component analysis or double functional principal component analysis.Simulation results indicate that the proposed methods have a nice performance in terms of both the correct classification rate and the adjusted rand index.The approaches are further illustrated through empirical analysis of human mortality data.
基金supported by NNSF of China(10471048)SRFDP(20050574002)
文摘In this article,the authors define the derived function of an algeboidal function in the unit disc,prove it is an algabriodal function,and study the order of algebroidal function and that of its derived function in unit circular disc.
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The object of this paper is to establish the pointwise estimations of approximation of functions in C^1 and their derivatives by Hermite interpolation polynomials. The given orders have been proved to be exact in gen- eral.
文摘In this paper,we study a special class of fractal interpolation functions,and give their Haar-wavelet expansions.On the basis of the expansions,we investigate the H(o|¨)lder smoothness of such functions and their logical derivatives of order α.
文摘By means of expansions of rapidly in infinity decreasing functions in delta functions and their derivatives, we derive generalized boundary conditions of the Sturm-Liouville equation for transitions and barriers or wells between two asymptotic potentials for which the solutions are supposed as known. We call such expansions “moment series” because the coefficients are determined by moments of the function. An infinite system of boundary conditions is obtained and it is shown how by truncation it can be reduced to approximations of a different order (explicitly made up to third order). Reflection and refraction problems are considered with such approximations and also discrete bound states possible in nonsymmetric and symmetric potential wells are dealt with. This is applicable for large wavelengths compared with characteristic lengths of potential changes. In Appendices we represent the corresponding foundations of Generalized functions and apply them to barriers and wells and to transition functions. The Sturm-Liouville equation is not only interesting because some important second-order differential equations can be reduced to it but also because it is easier to demonstrates some details of the derivations for this one-dimensional equation than for the full three-dimensional vectorial equations of electrodynamics of media. The article continues a paper that was made long ago.
基金the National Natural Science Foundation of China(No.50539030)
文摘The present paper generalizes the method for solving the derivatives of symmetric isotropic tensor-valued functions proposed by Dui and Chen (2004) to a subclass of nonsymmetric tensor functions satisfying the commutative condition. This subclass of tensor functions is more general than those investigated by the existing methods. In the case of three distinct eigenvalues, the commutativity makes it possible to introduce two scalar functions, which will be used to construct the general nonsymmetric tensor functions and their derivatives. In the cases of repeated eigenvalues, the results are acquired by taking limits.
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 +
基金Supported by International Research and Exchanges Board Supported by Hungarian National Science Foundation Grant No. 1910
文摘Let g∈C^q[-1, 1] be such that g^((k))(±1)=0 for k=0,…,q. Let P_n be an algebraic polynomial of degree at most n, such that P_n^((k))(±1)=0 for k=0,…,[_2~ (q+1)]. Then P_n and its derivatives P_n^((k)) for k≤q well approximate g and its respective derivatives, provided only that P_n well approxi- mates g itself in the weighted norm ‖g(x)-P_n(x) (1-x^2)^(1/2)~q‖ This result is easily extended to an arbitrary f∈C^q[-1, 1], by subtracting from f the polynomial of minnimal degree which interpolates f^((0))…,f^((q)) at±1. As well as providing easy criteria for judging the simultaneous approximation properties of a given Polynomial to a given function, our results further explain the similarities and differences between algebraic polynomial approximation in C^q[-1, 1] and trigonometric polynomial approximation in the space of q times differentiable 2π-periodic functions. Our proofs are elementary and basic in character, permitting the construction of actual error estimates for simultaneous approximation proedures for small values of q.
基金supported by the Scientific Research Fund of Hubei Provincial Education Department(No.Q20122909)the Open Fund of Key Laboratory of Biologic Resources Protection and Utilization of Hubei Province(No.PKLHB1314)the First-class Discipline of Forestry in Hubei Minzu University
文摘The molecular geometries, frontier molecular orbital properties, and absorption and emission properties of three 4-phenoxy-1,8-naphthalimide derivatives, namely 4-phenoxy-N-(2-hydroxyethyl)-1,8-naphthalimide(1),4-(2-tert-butylphenoxy)-N-(2-hydroxyethyl)-1,8-naphthalimide(2), and 4-[2,4-di(tert-butyl)]phenoxy-N-(2-hydroxyethyl)-1,8-naphthalimide(3), are investigated by density functional theory(DFT) and time-dependent density functional theory(TD-DFT) calculations in conjunction with polarizable continuum models(PCMs). Four functionals and ten basis sets are employed for 1 to calculate the electron transition energies, which were compared with the experimental observations. Our results reveal that the B3LYP/6-311+G(d,p) method is the best choice to reproduce the experimental spectra. Moreover, the effects of substituents on the molecular geometries, electronic structures, absorption and emission spectra are also studied at the B3LYP/6-311+G(d,p) level. We find that the gap between the highest occupied molecular orbital(HOMO) and the lowest unoccupied molecular orbital(LUMO) decreases with increasing the number of tert-butyl substituents onto the phenoxy groups, suggesting red-shift of the absorption and emission bands. This is related to the increase of conjugation from 1 to 2 and 3. Our calculations are in good agreement with the experimental results.
基金supported by the Ningbo Youth Foundation(0 2 J0 1 0 2 - 2 1 )
文摘The concept of convex type function is introduced in this paper,from which a kin d of convex decomposition approach is proposed.As one of applications of this a pproach,the approximation of the convex type function by the partial sum of its Fourier series is inves tigated.Moreover,the order of approximation is describe d with the 2th continuous modulus.
基金This work was partially supported by the Major Basic Research Project of the Natural Science Foundation of the Jiangsu Higher Education Institutions(No.10KJA150049)the Natural Science Foundation of Jiangsu Province(BK20131160)+1 种基金the Natural Science Foundation of the Jiangsu Higher Education Institutions(No.11KJB150014)A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions and the Foundation of Key Laboratory of Organic Synthesis of Jiangsu Province(No.KJS1210).
文摘An environmentally benign and efficient method has been developed for the synthesis of functionalized tetrahydroindole derivatives in aqueous media under catalyst-free conditions,by simply combining a phenylglyoxal monohydrate,an enaminone,and a barbituric acid.The advantages of this method are that it is catalyst free,has an easy workup,provides good yields,and uses water as solvent,which make this procedure facile and practical.
基金supported by the National Natural Science Foundation of China under grant No.12271108Shanghai Municipal Science and Technology Commission under grant No.22WZ2501900Innovation Program of Shanghai Municipal Education Commission
文摘In this work,we try to build a theory for random double tensor integrals(DTI).We begin with the definition of DTI and discuss how randomness structure is built upon DTI.Then,the tail bound of the unitarily invariant norm for the random DTI is established and this bound can help us to derive tail bounds of the unitarily invariant norm for various types of two tensors means,e.g.,arithmetic mean,geometric mean,harmonic mean,and general mean.By associating DTI with perturbation formula,i.e.,a formula to relate the tensor-valued function difference with respect the difference of the function input tensors,the tail bounds of the unitarily invariant norm for the Lipschitz estimate of tensor-valued function with random tensors as arguments are derived for vanilla case and quasi-commutator case,respectively.We also establish the continuity property for random DTI in the sense of convergence in the random tensor mean,and we apply this continuity property to obtain the tail bound of the unitarily invariant norm for the derivative of the tensor-valued function.
基金Supported partially by the National Natural Science Foundation of China(Grant No.10371048)
文摘An excellent introduction to the topic of poset matroids is due to M.Barnabei, G. Nicoletti and L. Pezzoli. On the basis of their work, we have obtained the global rankaxioms for poset matroids. In this paper, we study the special integral function f and obtain a newclass of poset matroids from the old ones, and then we generalize this result according to theproperties of f. Almost all of these results can be regarded as the application of global rankaxioms for poset matroids. The main results in our paper have, indeed, investigated the restrictionof the basis of the poset matroid, and we give them the corresponding geometric interpretation.
文摘This article studies a new kind of stability judgment for dynamic systems, the definiteness Lyapunov derivative function approach based on the minimum function. The new results are described by introducing the normal positivity. The article analyzes the relationship of a general system and a polynomial system, the judgment and measurement of positive definiteness, the polar coordinate approach, and constructing the algorithms for positive definite v·(x). Therefore, a new and systematic theory on stability judgment is clearly described.
基金Supported by the National Natural Science Foundation of China (60673068)the Fundamental Research Funds for the Central Universities (2009B27414)the Natural Science Foundation of Hohai University (2084/409270)
文摘We study the functions with low differential uniformity,and concentrates mainly on the properties of perfect nonlinear(PN) functions,including the properties of the derivative of the components of those functions.Some sufficient and necessary conditions have been explored to judge when a function is a PN function.These conditions may be useful in constructing new PN functions.We also construct some functions with differential 4-uniformity that have rarely been studied in the literature.Some of the constructed functions with differential 4-uniformity have high nonlinearity as well.Finally,a class of functions with differential 4-uniformity which are not extended affine equivalent to any power functions are constructed.