15 June 2011, Volume 27 Issue 6

    

• Select all
  |

Articles
• Select
  New Algebraic Approaches to Classical Boundary Layer Problems

  Xiao Ping XU

  Acta Mathematica Sinica. 2011, 27(6): 1023-1070. https://doi.org/10.1007/s10114-011-9414-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Classical non-steady boundary layer equations are fundamental nonlinear partial differential equations in the boundary layer theory of fluid dynamics. In this paper, we introduce various schemes with multiple parameter functions to solve these equations and obtain many families of new explicit exact solutions with multiple parameter functions. Moreover, symmetry transformations are used to simplify our arguments. The technique of moving frame is applied in the three-dimensional case in order to capture the rotational properties of the fluid. In particular, we obtain a family of solutions singular on any moving surface, which may be used to study turbulence. Many other solutions are analytic related to trigonometric and hyperbolic functions, which reflect various wave characteristics of the fluid. Our solutions may also help engineers to develop more effective algorithms to find physical numeric solutions to practical models.  
• Select
  Hamilton-type Gradient Estimates for a Nonlinear Parabolic Equation on Riemannian Manifolds

  Bin QIAN

  Acta Mathematica Sinica. 2011, 27(6): 1071-1078. https://doi.org/10.1007/s10114-011-8474-7

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let (M,g) be a complete noncompact Riemannian manifold. In this note, we derive a local Hamilton-type gradient estimate for positive solution to a simple nonlinear parabolic equation ∂_tu = Δu + au log u + qu on M × (0,∞), where a is a constant and q is a C² function. This result can be compared with the ones of Ma (JFA, 241, 374-382 (2006)) and Yang (PAMS, 136, 4095-4102 (2008)). Also, we obtain Hamilton’s gradient estimate for the Schödinger equation. This can be compared with the result of Ruan (JGP, 58, 962-966 (2008)).  
• Select
  Existence and Uniqueness of Positive Solutions for a Class of Semilinear Elliptic Systems

  Ren Hao CUI, Jun Ping SHI, Yu Wen WANG

  Acta Mathematica Sinica. 2011, 27(6): 1079-1090. https://doi.org/10.1007/s10114-011-9299-0

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  The authors prove the uniqueness and existence of positive solutions for the semilinear elliptic system which involves nonlinearities with sublinear growth conditions.  
• Select
  Some Results on Difference Riccati Equations

  Zong Xuan CHEN, Kwang Ho SHON

  Acta Mathematica Sinica. 2011, 27(6): 1091-1100. https://doi.org/10.1007/s10114-011-9175-y

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Some properties of solutions for the difference Riccati equations are obtained. The existence and forms of rational solutions, and the Borel exceptional value, zeros, poles and fixed points of transcendental solutions are researched.  
• Select
  Some Fixed Point Theorems for Expansion onto Mappings on Cone Metric Spaces

  Chintaman T. AAGE, Jagannath N. SALUNKE

  Acta Mathematica Sinica. 2011, 27(6): 1101-1106. https://doi.org/10.1007/s10114-011-9606-9

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  This paper presents some fixed point theorems for expansion selfmaps on complete cone metric spaces.  
• Select
  Eigenvalues and Diameter

  Hui Qing LIU, Mei LU

  Acta Mathematica Sinica. 2011, 27(6): 1107-1114. https://doi.org/10.1007/s10114-011-8681-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let G be a connected graph of order n. The diameter of a graph is the maximum distance between any two vertices of G. In this paper, we will give some bounds on the diameter of G in terms of eigenvalues of adjacency matrix and Laplacian matrix, respectively.  
• Select
  A Note on the Lehmer Problem over Short Intervals

  Ya Ming LU, Yuan YI

  Acta Mathematica Sinica. 2011, 27(6): 1115-1120. https://doi.org/10.1007/s10114-011-8554-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p - 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 a + a, where a is an integer with 1 ≤ a ≤ p - 1, aa ≡ c (mod p). It is well known that r(N, c; p) = (1/2)N +O(p^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ_c=1^p-1(r(N, c; p) - (1/2)N)².  
• Select
  The Global Attractor of a Non-Local PDE Model with Delay for Population Dynamics in Rⁿ

  Xiang LI, Zhi Xiang LI

  Acta Mathematica Sinica. 2011, 27(6): 1121-1136. https://doi.org/10.1007/s10114-011-8539-7

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we consider a non-local PDE model with delay for population dynamics in Rⁿ. First, we prove the existence and uniqueness of weak solutions under some suitable decayed assumptions on non-local term at infinity. Then, we obtain the global attractor by proving ω-limit compactness property of the solution operator semigroup.  
• Select
  The Fine Structures of Three Symmetric Latin Squares with Even Orders

  Er Qiang FENG, Yan Xun CHANG

  Acta Mathematica Sinica. 2011, 27(6): 1137-1148. https://doi.org/10.1007/s10114-011-8538-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Denote by SFin(v) the set of all integer pairs (t, s) for which there exist three symmetric Latin squares of order v on the same set having fine structure (t, s). We completely determine the set SFin(2n) for any integer n ≥ 5.  
• Select
  A New Zero-density Result of L-functions Attached to Maass Forms

  Zhao XU

  Acta Mathematica Sinica. 2011, 27(6): 1149-1162. https://doi.org/10.1007/s10114-011-8310-0

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let f denote a normalized Maass cusp form for SL(2, Z), which is an eigenfunction of all the Hecke operators T(n) as well as the reflection operator T_-1: z → -z. We obtain a zero-density result of the L-function attached to f near σ = 1. This improves substantially the previous results in this direction.  
• Select
  New Improvements on Connectivity of Cages

  Hong Liang LU, Yun Jian WU, Qing Lin YU, Yu Qing LIN

  Acta Mathematica Sinica. 2011, 27(6): 1163-1172. https://doi.org/10.1007/s10114-011-8279-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  A (δ, g)-cage is a δ-regular graph with girth g and with the least possible number of vertices. In this paper, we show that all (δ, g)-cages with odd girth g ≥ 9 are r-connected, where (r - 1)² ≤ δ + √δ - 2 < r² and all (δ, g)-cages with even girth g ≥ 10 are r-connected, where r is the largest integer satisfying (r(r-1)²)/4 +1+2r(r - 1) ≤ δ. These results support a conjecture of Fu, Huang and Rodger that all (δ, g)-cages are δ-connected.  
• Select
  The Relationship between the Nonexistence of Generalized Bent Functions and Diophantine Equations

  Feng Mei LIU, Qin YUE

  Acta Mathematica Sinica. 2011, 27(6): 1173-1186. https://doi.org/10.1007/s10114-011-8198-8

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Two new results on the nonexistence of generalized bent functions are presented by using properties of the decomposition law of primes in cyclotomic fields and properties of solutions of some Diophantine equations, and examples satisfying our results are given.  
• Select
  Prime in Quadratic Progressions on Average

  Guang Shi Lü, HaiWeiSUN

  Acta Mathematica Sinica. 2011, 27(6): 1187-1194. https://doi.org/10.1007/s10114-011-8059-5

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper we study the distribution of primes in quadratic progressions on average. Our result improves a previous result of Baier and Zhao in some aspects.  
• Select
  Harnack and HWI Inequalities on Infinite-Dimensional Spaces

  Jing Hai SHAO

  Acta Mathematica Sinica. 2011, 27(6): 1195-1204. https://doi.org/10.1007/s10114-011-8021-6

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, the dimensional-free Harnack inequalities are established on infinite-dimensional spaces. More precisely, we establish Harnack inequalities for heat semigroup on based loop group and for Ornstein-Uhlenbeck semigroup on the abstract Wiener space. As an application, we establish the HWI inequality on the abstract Wiener space, which contains three important quantities in one inequality, the relative entropy “H”, Wasserstein distance “W”, and Fisher information “I”.  
• Select
  The Almost Global and Global Existence for Quasi-linear Wave Equations with Multiple-Propagation Speeds in High Dimensions

  Yi DU, Zheng An YAO

  Acta Mathematica Sinica. 2011, 27(6): 1205-1220. https://doi.org/10.1007/s10114-011-8017-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we consider the Cauchy problem for systems of quasi-linear wave equations with multiple propagation speeds in spatial dimensions n ≥ 4. The problem when the nonlinearities depend on both the unknown function and their derivatives is studied. Based on some Klainerman- Sideris type weighted estimates and space-time L² estimates, the results that the almost global existence for space dimensions n = 4 and global existence for n ≥ 5 of small amplitude solutions are presented.  
• Select
  Commutative Rings Whose Zero-divisor Graph Is a Proper Refinement of a Star Graph

  Qiong LIU, Tong Suo WU

  Acta Mathematica Sinica. 2011, 27(6): 1221-1232. https://doi.org/10.1007/s10114-011-8010-9

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  A graph is called a proper refinement of a star graph if it is a refinement of a star graph, but it is neither a star graph nor a complete graph. For a refinement of a star graph G with center c, let G_c^* be the subgraph of G induced on the vertex set V (G)\{c or end vertices adjacent to c}. In this paper, we study the isomorphic classification of some finite commutative local rings R by investigating their zero-divisor graphs G = Γ(R), which is a proper refinement of a star graph with exactly one center c. We determine all finite commutative local rings R such that G_c^* has at least two connected components. We prove that the diameter of the induced graph G_c^* is two if Z(R)² = {0}, Z(R)³ = {0} and G_c^* is connected. We determine the structure of R which has two distinct nonadjacent vertices α, β ∈ Z(R)^*\{c} such that the ideal [N(α)∩N(β)]∪{0} is generated by only one element of Z(R)^*\{c}. We also completely determine the correspondence between commutative rings and finite complete graphs Kⁿ with some end vertices adjacent to a single vertex of Kⁿ.