By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n...By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.展开更多
The Beurling-Ahlfors’ extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors’ theorem can be obtained under the M-condition.
In this paper, we prove that the control function of the dilatation function of Beurling-Ahlfors extension is convex. Using the quasi-symmetric function ρ, we get a relatively sharp estimate of the dilatation functio...In this paper, we prove that the control function of the dilatation function of Beurling-Ahlfors extension is convex. Using the quasi-symmetric function ρ, we get a relatively sharp estimate of the dilatation function: D(x,y)≤ 17/32 (ρ(x, y) + 1) (ρ(x + y/2, y/2) +ρ(x - y/2, y/2) +2) , which improves the results before. We also show that the above result is asymptotically precise.展开更多
Triangular norm is a powerful tool in the theory research and application development of fuzzy sets. In this paper, using the triang norm, we introduce some concepts such as fuzzy algebra, fuzzy a algebra and fuzzy mo...Triangular norm is a powerful tool in the theory research and application development of fuzzy sets. In this paper, using the triang norm, we introduce some concepts such as fuzzy algebra, fuzzy a algebra and fuzzy monotone class, and discuss the relations among them,obtaining the following main conclusions:Theorem 1: Let (I,S,T,C) be a norm spetem, S and T be dual norm,(Ⅰ) If is a fuzzy σ algebra, then is also a fuzzy monotooe class;(Ⅱ ) If a fuzzy algebra is a fuzzy monotone class, then is also a fuzzy σ algebra.Theorem 2: If φ(X) is a fuzzy algebra, then m (φ) =σ(φ).展开更多
Previous literature claims that the density functional theory for non-uniform non-hard sphere interaction potential fluid can be improved on by treating the tail part by the third order functional perturbation expansi...Previous literature claims that the density functional theory for non-uniform non-hard sphere interaction potential fluid can be improved on by treating the tail part by the third order functional perturbation expansion approximation (FPEA) with the symmetrical and intuitive consideration-based simple function C0(3)(r1, r2, r3) =ζ∫ dr4a(r4 - r1)a(r4 - r2)a(r4 - r3) as the uniform third order direct correlation function (DCF) for the tail part,here kernel function a(r) = (6/πσ3)Heaviside(σ/2 - r). The present contribution concludes that for the mean spherical approximation-based second order DCF, the terms higher than second order in the FPEA of the tail part of the non-uniform first order DCF are exactly zero. The reason for the partial success of the previous a kernel function-based third order FPEA for the tail part is due to the adjustable parameter ζ and the short range of the a kernel function.Improvement over the previous theories is proposed and tested.展开更多
A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of...A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.展开更多
Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both r...Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.展开更多
基金Supported by the Nature Science foundation of Henan Province(0211050200)
文摘By using small function method, the following result is obtained. If f（z） is transcendental meromorphic and that ψ（z） is non-zero meromorphic and that T（r,ψ） = S（r, f）, then（n＋1）T（r,f）≤N^-（r,1/f＇f^n-ψ）＋2N^-（r,1/f）＋N^-（r,f）＋S（r,f）.
文摘The Beurling-Ahlfors’ extension is studied under relatively general conditions and its dilatation fonction is estimated. Particularly, the classic Deurling -Ahlfors’ theorem can be obtained under the M-condition.
基金Supported by the National Natural Science Foundation of China(10271077)Supported by the Educational Department of Zhejiang Province Natural Science Project(20030768)
文摘In this paper, we prove that the control function of the dilatation function of Beurling-Ahlfors extension is convex. Using the quasi-symmetric function ρ, we get a relatively sharp estimate of the dilatation function: D(x,y)≤ 17/32 (ρ(x, y) + 1) (ρ(x + y/2, y/2) +ρ(x - y/2, y/2) +2) , which improves the results before. We also show that the above result is asymptotically precise.
文摘Triangular norm is a powerful tool in the theory research and application development of fuzzy sets. In this paper, using the triang norm, we introduce some concepts such as fuzzy algebra, fuzzy a algebra and fuzzy monotone class, and discuss the relations among them,obtaining the following main conclusions:Theorem 1: Let (I,S,T,C) be a norm spetem, S and T be dual norm,(Ⅰ) If is a fuzzy σ algebra, then is also a fuzzy monotooe class;(Ⅱ ) If a fuzzy algebra is a fuzzy monotone class, then is also a fuzzy σ algebra.Theorem 2: If φ(X) is a fuzzy algebra, then m (φ) =σ(φ).
基金国家自然科学基金，the Scientific Research Fund of Hunan Provincial Education Department
文摘Previous literature claims that the density functional theory for non-uniform non-hard sphere interaction potential fluid can be improved on by treating the tail part by the third order functional perturbation expansion approximation (FPEA) with the symmetrical and intuitive consideration-based simple function C0(3)(r1, r2, r3) =ζ∫ dr4a(r4 - r1)a(r4 - r2)a(r4 - r3) as the uniform third order direct correlation function (DCF) for the tail part,here kernel function a(r) = (6/πσ3)Heaviside(σ/2 - r). The present contribution concludes that for the mean spherical approximation-based second order DCF, the terms higher than second order in the FPEA of the tail part of the non-uniform first order DCF are exactly zero. The reason for the partial success of the previous a kernel function-based third order FPEA for the tail part is due to the adjustable parameter ζ and the short range of the a kernel function.Improvement over the previous theories is proposed and tested.
基金supported by National Natural Science Foundation of China (Grant No. 10871106)
文摘A finite group G is called exceptional if for a Galois extension F/k of number fields with the Galois group G,in the Brauer-Kuroda relation of the Dedekind zeta functions of fields between k and F,the zeta function of F does not appear.In the present paper we describe effectively all exceptional groups of orders divisible by exactly two prime numbers p and q,which have unique subgroups of orders p and q.
文摘Ideal class groups H(K) of algebraic quadratic function fields K are studied. Necessaryand sufficient condition is given for the class group H(K) to contain a cyclic subgroup of anyorder n, which holds true for both real and imaginary fields K. Then several series of functionfields K, including real, inertia imaginary, and ramified imaginary quadratic function fields, aregiven, for which the class groups H(K) are proved to contain cyclic subgroups of order n.
