In this paper, we study the approximation problem on the closed convex cone, and prove that there exists a unique solution of the approximation problem, then give the algorithm to compute the unique solution.
By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the clo...By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.展开更多
Let C＇（α,β） be the class of functions f（z） =analytic in D ={z： ｜z｜ 〈 1}, satisfying for some convex function g（z） with g（O） = g＇（O） - 1 =- 0 and for allz in D the condition zf＇（z）-1/g（z）/zf＇（z...Let C＇（α,β） be the class of functions f（z） =analytic in D ={z： ｜z｜ 〈 1}, satisfying for some convex function g（z） with g（O） = g＇（O） - 1 =- 0 and for allz in D the condition zf＇（z）-1/g（z）/zf＇（z）/g（z）＋1-2α）｜ 〈β for some α β （0 ≤ α〈1,0 〈 β 〈 1）. A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C＇（α ,β ）. The results extend the work of C. Selvaraj.展开更多
By investigating the properties of Hellinger-Toeplitz topologies, we establish a general version of Kalton's cioed graph theorem. From this general version, we deduce a number of new closed graph theorems, which a...By investigating the properties of Hellinger-Toeplitz topologies, we establish a general version of Kalton's cioed graph theorem. From this general version, we deduce a number of new closed graph theorems, which are convenient for application. Particularly we improve some results of Kalton.展开更多
An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,...An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.展开更多
Let A p(n)(p, n∈N=｛1,2,…｝) denote the class of functions of the form f(z)=z p+a p+n z p+n +… which are analytic in the unit disc E=｛z:｜z｜<1｝. By using the method of differential subordinati ons we give som...Let A p(n)(p, n∈N=｛1,2,…｝) denote the class of functions of the form f(z)=z p+a p+n z p+n +… which are analytic in the unit disc E=｛z:｜z｜<1｝. By using the method of differential subordinati ons we give some sufficient conditions for a function f(z)∈A p(n) to be a certain subclass R p(n,k) of p-valently close-to-convexity funct ions.展开更多
In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provid...In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provided to show that these conditions cannot be omitted in general.展开更多
In this paper, let ∑ R2n be a symmetric compact convex hypersurface which is （r, R）-pinched with. Then Z carries at least two elliptic symmetric closed characteristics; moreover,∑ carries at least E[n-1/2] ＋ E[n-...In this paper, let ∑ R2n be a symmetric compact convex hypersurface which is （r, R）-pinched with. Then Z carries at least two elliptic symmetric closed characteristics; moreover,∑ carries at least E[n-1/2] ＋ E[n-1/3] non-hyperbolic symmetric closed characteristics.展开更多
Let k≥2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying P^(k)=I_(2n) and ker(P^(j)-I_(2n)=0,1≤j<k.For any compact convex hypersurface ∑■R^(2n) with n≥2 which is P-cyclic symmetric,i...Let k≥2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying P^(k)=I_(2n) and ker(P^(j)-I_(2n)=0,1≤j<k.For any compact convex hypersurface ∑■R^(2n) with n≥2 which is P-cyclic symmetric,i.e.,x∈∑implies Px∈∑,we prove that if ∑ is(r,R)-pinched with R/r<√(2k+2)/k,then there exist at least n geometrically distince P-cyclic symmetric closed characteristics on ∑ for a broad class of matrices P.展开更多
In this paper, let n 2 be an integer, P = diag(-In-κ, Iκ,-In-κ, Iκ) for some integer κ∈ [0, n),and Σ∈ R2 nbe a partially symmetric compact convex hypersurface, i.e., x ∈Σ implies P x ∈Σ. We prove that if ...In this paper, let n 2 be an integer, P = diag(-In-κ, Iκ,-In-κ, Iκ) for some integer κ∈ [0, n),and Σ∈ R2 nbe a partially symmetric compact convex hypersurface, i.e., x ∈Σ implies P x ∈Σ. We prove that if Σ is(r, R)-pinched withRr<2^(1/2), then there exist at least n- κ geometrically distinct P-symmetric closed characteristics on Σ, as a consequence, Σ carry at least n geometrically distinct P-invariant closed characteristics.展开更多
In this paper, we construct first a new concrete example of asymmetric convex compact C 1,1-hypersurfaces in R 2n possessing precisely n closed characteristics. Then we prove multiplicity results on the closed charact...In this paper, we construct first a new concrete example of asymmetric convex compact C 1,1-hypersurfaces in R 2n possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R 2n pinched by not necessarily symmetric convex compact hypersurfaces.展开更多
Let ∑be a C^3 compact symmetric convex hypersurface in R^8.We prove that when ∑ carries exactly four geometrically distinct closed characteristics,then all of them must be symmetric.Due to the example of weakly non-...Let ∑be a C^3 compact symmetric convex hypersurface in R^8.We prove that when ∑ carries exactly four geometrically distinct closed characteristics,then all of them must be symmetric.Due to the example of weakly non-resonant ellipsoids,our result is sharp.展开更多
基金Research supported by National Natural Science Foundation of China(10171031).
文摘In this paper, we study the approximation problem on the closed convex cone, and prove that there exists a unique solution of the approximation problem, then give the algorithm to compute the unique solution.
文摘By introducing the notions of L-spaces and L_r-spaces, a complete generalization of Kalton's closed graph theorem is obtained. It points out the class of L_r-spaces is the maximal class of range spaces for the closed graph theorem when the class of domain spaces is the class of Mackey spaces with weakly * sequentially complete dual.Some examples are constructed showing that the class of L_r-spaces is strictly larger than the class of separable B_r-complete spaces.Some properties of L-spaces and L_r-spaces are discussed and the relations between B-complete (resp. B_r-complete) spaces and L-spaces (resp. L_r-spaces) are given.
文摘Let C＇（α,β） be the class of functions f（z） =analytic in D ={z： ｜z｜ 〈 1}, satisfying for some convex function g（z） with g（O） = g＇（O） - 1 =- 0 and for allz in D the condition zf＇（z）-1/g（z）/zf＇（z）/g（z）＋1-2α）｜ 〈β for some α β （0 ≤ α〈1,0 〈 β 〈 1）. A sharp coefficient estimate, distortion theorems and radius of convexity are determined for the class C＇（α ,β ）. The results extend the work of C. Selvaraj.
文摘By investigating the properties of Hellinger-Toeplitz topologies, we establish a general version of Kalton's cioed graph theorem. From this general version, we deduce a number of new closed graph theorems, which are convenient for application. Particularly we improve some results of Kalton.
文摘An analytic function f in the unit disk D ：= {z ∈ C ： ｜z｜ 〈 1}, standardly normalized, is called close-to-convex with respect to the Koebe function k（z） ：= z/（1-z）2, z ∈ D, if there exists δ ∈ （-π/2,π/2） such that Re {eiδ（1-z）2f′（z）} 〉 0, z ∈ D. For the class C（k） of all close-to-convex functions with respect to k, related to the class of functions convex in the positive direction of the imaginary axis, the Fekete-Szego problem is studied.
文摘Let A p(n)(p, n∈N=｛1,2,…｝) denote the class of functions of the form f(z)=z p+a p+n z p+n +… which are analytic in the unit disc E=｛z:｜z｜<1｝. By using the method of differential subordinati ons we give some sufficient conditions for a function f(z)∈A p(n) to be a certain subclass R p(n,k) of p-valently close-to-convexity funct ions.
文摘In this article, under quite weaker conditions, theorems of Helly’s type for(not necessarily finite) family of partially closed half-spaces are presented in both analytic and geometric forms. Examples are also provided to show that these conditions cannot be omitted in general.
基金Partially supported by NNSF, RFDP of MOE of China
文摘In this paper, let ∑ R2n be a symmetric compact convex hypersurface which is （r, R）-pinched with. Then Z carries at least two elliptic symmetric closed characteristics; moreover,∑ carries at least E[n-1/2] ＋ E[n-1/3] non-hyperbolic symmetric closed characteristics.
基金supported by the National Natural Science Foundation of China(Grant Nos.11771341,12022111).
文摘Let k≥2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying P^(k)=I_(2n) and ker(P^(j)-I_(2n)=0,1≤j<k.For any compact convex hypersurface ∑■R^(2n) with n≥2 which is P-cyclic symmetric,i.e.,x∈∑implies Px∈∑,we prove that if ∑ is(r,R)-pinched with R/r<√(2k+2)/k,then there exist at least n geometrically distince P-cyclic symmetric closed characteristics on ∑ for a broad class of matrices P.
基金supported by China Postdoctoral Science Foundation(Grant No.2013M540512)National Natural Science Foundation of China(Grant Nos.10801078,11171341 and 11271200)Lab of Pure Mathematics and Combinatorics of Nankai University
文摘In this paper, let n 2 be an integer, P = diag(-In-κ, Iκ,-In-κ, Iκ) for some integer κ∈ [0, n),and Σ∈ R2 nbe a partially symmetric compact convex hypersurface, i.e., x ∈Σ implies P x ∈Σ. We prove that if Σ is(r, R)-pinched withRr<2^(1/2), then there exist at least n- κ geometrically distinct P-symmetric closed characteristics on Σ, as a consequence, Σ carry at least n geometrically distinct P-invariant closed characteristics.
基金Partially supported by the 973 Program of STM,Funds of EC of Jiangsuthe Natural Science Funds of Jiangsu(BK 2002023)+1 种基金the Post-doctorate Funds of Chinathe NNSF of China(10251001)Partially supported by the 973 Program of STM,NNSF,MCME,RFDP,PMC
文摘In this paper, we construct first a new concrete example of asymmetric convex compact C 1,1-hypersurfaces in R 2n possessing precisely n closed characteristics. Then we prove multiplicity results on the closed characteristics on convex compact hypersurfaces in R 2n pinched by not necessarily symmetric convex compact hypersurfaces.
基金Hui Liu Partially supported by NSFC(No.11401555)China Postdoctoral Science Foundation No.2014T70589,CUSF(No.WK0010000037)Yiming Long Partially supported by NSFC。
文摘Let ∑be a C^3 compact symmetric convex hypersurface in R^8.We prove that when ∑ carries exactly four geometrically distinct closed characteristics,then all of them must be symmetric.Due to the example of weakly non-resonant ellipsoids,our result is sharp.
