15 July 2019, Volume 35 Issue 7

    

• Select all
  |

Articles
• Select
  Rearrangements in Carnot Groups

  Juan J. MANFREDI, Virginia N. VERA DE SERIO

  Acta Mathematica Sinica. 2019, 35(7): 1115-1127. https://doi.org/10.1007/s10114-019-8271-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper we extend the notion of rearrangement of nonnegative functions to the setting of Carnot groups. We define rearrangement with respect to a given family of anisotropic balls B_r, or equivalently with respect to a gauge ||x||, and prove basic regularity properties of this construction. If u is a bounded nonnegative real function with compact support, we denote by u^★ its rearrangement. Then, the radial function u^★ is of bounded variation. In addition, if u is continuous then u^★ is continuous, and if u belongs to the horizontal Sobolev space W_h^1,p, then D_hu^★(x)/|D_h(||x||)|is in L^p. Moreover, we found a generalization of the inequality of Pólya and Szegö
  
  here p ≥ 1.

• Select
  Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

  Pan ZHANG

  Acta Mathematica Sinica. 2019, 35(7): 1128-1142. https://doi.org/10.1007/s10114-019-8220-0

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let V be an asymptotically cylindrical Kähler manifold with asymptotic cross-section D. Let (E_D, φ_D) be a stable Higgs bundle over D, and (E, φ) a Higgs bundle over V which is asymptotic to (E_D, φ_D). In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang-Mills metric on (E, φ).
• Select
  Global Existence and Blow-up in Finite Time for a Class of Finitely Degenerate Semilinear Pseudo-parabolic Equations

  Hua CHEN, Hui Yang XU

  Acta Mathematica Sinica. 2019, 35(7): 1143-1162. https://doi.org/10.1007/s10114-019-8037-x

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we study the initial-boundary value problem for the semilinear pseudoparabolic equations u_t-△_Xu_t-△_Xu=|u|^p-1u, where X=(X₁, X₂,..., X_m) is a system of real smooth vector fields which satisfy the Hörmander's condition, and △_X=Σ_j=1^m X_j² is a finitely degenerate elliptic operator. By using potential well method, we first prove the invariance of some sets and vacuum isolating of solutions. Then, by the Galerkin method and the concavity method we show the global existence and blow-up in finite time of solutions with low initial energy or critical initial energy. The asymptotic behavior of the global solutions and a lower bound for blow-up time of local solution are also given.
• Select
  Weighted Least Squares Method for the Accelerated Failure Time Model with Auxiliary Covariates

  Ling Hui JIN, Yan Yan LIU, Lang WU

  Acta Mathematica Sinica. 2019, 35(7): 1163-1178. https://doi.org/10.1007/s10114-019-8232-9

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  This paper deals with the analysis of accelerated failure time model when the primary covariate is subject to missing. We assume that the true covariate is measured precisely on a randomly chosen validation set, whereas auxiliary information for primary covariate is available to all study subjects. The asymptotic properties for the proposed estimator are developed and the simulation studies show that the efficiency gain is remarkable compared to the method using only the validation sample. A real example is also provided as an illustration.
• Select
  Finite p-groups All of Whose Minimal Nonabelian Subgroups are Nonmetacyclic of Order p³

  Qin Hai ZHANG

  Acta Mathematica Sinica. 2019, 35(7): 1179-1189. https://doi.org/10.1007/s10114-019-7308-x

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Assume p is an odd prime. We investigate finite p-groups all of whose minimal nonabelian subgroups are of order p³. Let P₁-groups denote the p-groups all of whose minimal nonabelian subgroups are nonmetacyclic of order p³. In this paper, the P₁-groups are classified, and as a by-product, we prove the Hughes' conjecture is true for the P₁-groups.
• Select
  Boundary Behavior of Large Solutions to the Monge-Ampère Equation in a Borderline Case

  Zhi Jun ZHANG

  Acta Mathematica Sinica. 2019, 35(7): 1190-1204. https://doi.org/10.1007/s10114-019-7524-4

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  This paper is concerned with the boundary behavior of strictly convex large solutions to the Monge-Ampère equation detD²u(x)=b(x)f(u(x)), u > 0, x ∈ Ω, where Ω is a strictly convex and bounded smooth domain in R^N with N ≥ 2, f is normalized regularly varying at infinity with the critical index N and has a lower term, and b ∈ C^∞(Ω) is positive in Ω, but may be appropriate singular on the boundary.
• Select
  On Eccentric Connectivity Index and Connectivity

  Vivian MUKUNGUNUGWA, Simon MUKWEMBI

  Acta Mathematica Sinica. 2019, 35(7): 1205-1216. https://doi.org/10.1007/s10114-019-7320-1

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let G be a finite connected graph. The eccentric connectivity index ξ^c(G) of G is defined as ξ^c(G)= _{v∈V (G)} ec(v)deg(v), where ec(v) and deg(v) denote the eccentricity and degree of a vertex v in G, respectively. In this paper, we give an asymptotically sharp upper bound on the eccentric connectivity index in terms of order and vertex-connectivity and in terms of order and edge-connectivity. We also improve the bounds for triangle-free graphs.
• Select
  Gradient Estimates and Harnack Inequalities for Positive Solutions of Lu=∂u/∂t on Self-shrinkers

  Ye Cheng ZHU, Qing CHEN

  Acta Mathematica Sinica. 2019, 35(7): 1217-1226. https://doi.org/10.1007/s10114-019-8137-7

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In this paper, we investigate the positive solutions of Lu=∂u/∂t on self-shrinkers, then get some gradient estimates and Harnack inequalities for the positive solutions.
• Select
  A Toughness Condition for Fractional (k, m)-deleted Graphs Revisited

  Wei GAO, Juan L. G. GUIRAO, Yao Jun CHEN

  Acta Mathematica Sinica. 2019, 35(7): 1227-1237. https://doi.org/10.1007/s10114-019-8169-z

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional (k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional (k, m)-deleted graphs improving the existing one. Finally, we state an open problem.

• Select
  Extremality of Graph Entropy Based on Degrees of Uniform Hypergraphs with Few Edges

  Dan HU, Xue Liang LI, Xiao Gang LIU, Sheng Gui ZHANG

  Acta Mathematica Sinica. 2019, 35(7): 1238-1250. https://doi.org/10.1007/s10114-019-8093-2

  Abstract ( ) Download PDF ( )   Knowledge map   Save

  Let H be a hypergraph with n vertices. Suppose that d₁, d₂,..., d_n are degrees of the vertices of H. The t-th graph entropy based on degrees of H is defined as
  
  where t is a real number and the logarithm is taken to the base two. In this paper we obtain upper and lower bounds of I_d^t(H) for t=1, when H is among all uniform supertrees, unicyclic uniform hypergraphs and bicyclic uniform hypergraphs, respectively.